plasma spectroscopy

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OFMONOGRAPHS ON PHYSICS

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J .B IRMANS. F. EDWARDSR. FRIENDM . R E E SD. SHERRINGTONG. VENEZIANO

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125. S. Atzeni, J. Meyer-ter-Vehn: Inertial Fusion124. C. Kiefer: Quantum Gravity123. T. Fujimoto: Plasma Spectroscopy122. K. Fujikawa, H. Suzuki: Path integrals and quantum anomalies121. T. Giamarchi: Quantum physics in one dimension120. M. Warner, E. Terentjev: Liquid crystal elastomers119. L. Jacak, P. Sitko, K. Wieczorek, A. Wojs: Quantum Hall systems118. J. Wesson: Tokamaks, Third edition117. G. Volovik: The Universe in a helium droplet116. L. Pitaevskii, S. Stringari: Bose-Einstein condensation115. G. Dissertori, I.G. Knowles, M. Schmelling: Quantum chromodynamics114. B. DeWitt: The global approach to quantum field theory113. J. Zinn-Justin: Quantum field theory and critical phenomena, Fourth edition112. R. M. Mazo: Brownian motion — fluctuations, dynamics, and applications111. H. Nishimori: Statistical physics of spin glasses and information processing - an introduction110. N. B. Kopnin: Theory of nonequilibrium superconductivity109. A. Aharoni: Introduction to the theory offerromagnetism, Second edition108. R. Dobbs: Helium three107. R. Wigmans: Calorimetry106. J. Kübler: Theory of itinerant electron magnetism105. Y. Kuramoto, Y. Kitaoka: Dynamics of heavy electrons104. D. Bardin, G. Passarino: The Standard Model in the making103. G. C. Branco, L. Lavoura, J. P. Silva: CP Violation102. T. C. Choy: Effective medium theory101. H. Araki: Mathematical theory of quantum fields100. L. M. Pismen: Vortices in nonlinear fields99. L. Mestel: Stellar magnetism98. K. H. Bennemann: Nonlinear optics in metals97. D. Salzmann: Atomic physics in hot plasmas96. M. Brambilla: Kinetic theory of plasma waves95. M. Wakatani: Stellarator and heliotron devices94. S. Chikazumi: Physics of ferromagnetism91. R. A. Bertlmann: Anomalies in quantum field theory90. P. K. Gosh: Ion traps89. E. Simánek: Inhomogeneous superconductors88. S. L. Adler: Quaternionic quantum mechanics and quantum fields87. P. S. Joshi: Global aspects in gravitation and cosmology86. E. R. Pike, S. Sarkar: The quantum theory of radiation84. V. Z. Kresin, H. Morawitz, S. A. Wolf: Mechanisms of conventional and high Tc super-conductivity83. P. G. de Gennes, J. Prost: The physics of liquid crystals82. B. H. Bransden, M. R. C. McDowell: Charge exchange and the theory of ion-atom collision81. J. Jensen, A. R. Mackintosh: Rare earth magnetism80. R. Gastmans, T. T. Wu: The ubiquitous photon79. P. Luchini, H. Motz: Undulators and free-electron lasers78. P. Weinberger: Electron scattering theory76. H. Aoki, H. Kamimura: The physics of interacting electrons in disordered systems75. J. D. Lawson: The physics of charged particle beams73. M. Doi, S. F. Edwards: The theory of polymer dynamics71. E. L. Wolf: Principles of electron tunneling spectroscopy70. H. K. Henisch: Semiconductor contacts69. S. Chandrasekhar: The mathematical theory of black holes68. G. R. Satchler: Direct nuclear reactions51. C. Møller: The theory of relativity46. H. E. Stanley: Introduction to phase transitions and critical phenomena32. A. Abragam: Principles of nuclear magnetism27. P. A. M. Dirac: Principles of quantum mechanics23. R. E. Peierls: Quantum theory of solids

THE INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS

Plasma Spectroscopy

TAKASHI FUJIMOTO

Department of Engineering Physics and MechanicsGraduate School of Engineering

Kyoto University

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PREFACE

Throughout the history of spectroscopy, plasmas have been the source of radia-tion, and they were studied for the purpose of spectrochemical analysis and alsofor the investigation of the structure of atoms (molecules) and ions constitutingthese plasmas. About a century ago, the spectroscopic investigations of theradiation emitted from plasmas contributed to establishing quantum mechanics.However, the plasma itself has been the subject of spectroscopy to a lesser extent.This less-developed state of plasma spectroscopy is attributed partly to the com-plicated relationships between the state of the plasma and the spectral char-acteristics of the radiation it emits. If we are concerned with the intensitydistribution of spectral lines over a spectrum, we have to understand the popu-lation density distribution over the excited levels of atoms and ions in the plasma.Since the latter distribution is governed by a collection of an enormous number ofatomic processes, e.g. electron impact excitation, deexcitation, ionization,recombination, and radiative transitions, and since the spatial transport of theplasma particles and the temporal development are sometimes essential as well, itis rather difficult, by starting from these elementary processes, to deduce astraightforward consequence concerning the population distribution. For certainlimiting conditions of the plasma, e.g. for the low- or high-density limit, severalconcepts like corona equilibrium and local thermodynamic equilibrium have beenproposed, but they have been accepted on a rather intuitive basis.

This book is intended first to provide a theoretical framework in which we cantreat various features of the population density distribution over the excited levels(and the ground state) of atoms and ions and give their interpretation in a unifiedand coherent way. In this new framework several concepts, some of which arealready known and some newly derived, are properly defined. For these purposes,we take hydrogen-like ions (and neutral hydrogen) as an example of an ensembleof atoms and ions immersed in a plasma. Following the first three introductorychapters, these problems are discussed in the subsequent two chapters. The fol-lowing three chapters are devoted to several facets which are useful in performinga spectroscopy experiment. This volume concludes with a chapter treating severalphenomena characteristic of dense plasmas. This chapter may be regarded as anapplication of the theoretical methods developed in the first part of the volume.

The main body of this book is based on my half-year course given at theGraduate School, Kyoto University, for more than a decade. This book isintended mainly for graduate students, but it should also be useful for researchersworking in this field. A reader who wants to obtain only the basic ideas may skipthe chapters and sections marked with an asterisk.

vi PREFACE

In writing this book I owe thanks to many colleagues and students. First of all,Professor Otsuka is especially thanked for his careful reading of the wholemanuscript and for pointing out errors, and giving me critical comments andvaluable suggestions. Professor Kato and M. Goto provided me with their valu-able unpublished spectra for Chapter 1. Various materials in Chapters 4 and 5were taken from publications by my former students, K. Sawada, T. Kawachi, andM. Goto. These and A. Iwamae, my present colleague, created many beautifulfigures for this book. I am also grateful to Dr Baronova for her comments, whichhave made this book more or less comprehensive. She also helped in some parts ofthe book. If this book is quite straightforward for beginners, that is due to mystudents, Y. Kimura and M. Matsumoto, who gave me various comments andquestions as students. I would like to express my thanks to workers who permittedme to reproduce their figures in this book. Professors Xu and Zhu even modifiedtheir original figure so as to fit better into the context of this book. Mrs. Hooper Jr.who gave me permission to use a figure on behalf of her recently deceased hus-band. The names of these workers and the copyright owners are mentioned in thereference section and the figure captions in each chapter.

CONTENTS

List of symbols and abbreviations ix

1 Introduction 11.1 Historical background and outline of the book 11.2 Various plasmas 121.3 Nomenclature and basic constants 131.4 z-scaling 141.5 Neutral hydrogen and hydrogen-like ions 151.6 Non-hydrogen-like ions 19

2 Therniodynaniic equilibrium 222.1 Velocity and population distributions 222.2 Black-body radiation 25

3 Atomic processes 303.1 Radiative transitions 313.2 Radiative recombination 423.3 Collisional excitation and deexcitation 483.4 lonization and three-body recombination 59

*3.5 Autoionization, dielectronic recombination, and satellite lines 64*3.6 Ion collisions 72

Appendix 3A. Scaling properties of ions in isoelectronic sequence 76*Appendix 3B. Three-body recombination "cross-section" 79

4 Population distribution and population kinetics 834.1 Collisional-radiative (CR) model 834.2 Ionizing plasma component 964.3 Recombining plasma component - high-temperature case 1114.4 Recombining plasma component - low-temperature case 1204.5 Summary and concluding remarks 131

*Appendix 4A. Validity of the statistical populations amongthe different angular momentum states 134

*Appendix 4B. Temporal development of excited-level populationsand validity condition of the quasi-steady-state approximation 136

5 Ionization and recombination of plasma 1505.1 Collisional-radiative ionization 1515.2 Collisional-radiative recombination - high-temperature case 1575.3 Collisional-radiative recombination - low-temperature case 163

viii CONTENTS

5.4 lonization balance 1675.5 Experimental illustration of transition from ionizing

plasma to recombining plasma 182Appendix 5A. Establishment of the collisional-radiative

rate coefficients 188Appendix 5B. Scaling law 190

*Appendix 5C. Conditions for establishing local thermodynamicequilibrium 191

*Appendix 5D. Optimum temperature, emission maximum,and flux maximum 202

6 Continuum radiation 2056.1 Recombination continuum 2056.2 Continuation to series lines 2076.3 Free-free continuum - Bremsstrahlung 211

*7 Broadening of spectral lines 2137.1 Quasi-static perturbation 2147.2 Natural broadening 2187.3 Temporal perturbation - impact broadening 2197.4 Examples 2247.5 Voigt profile 233

*8 Radiation transport 2368.1 Total absorption 2368.2 Collision-dominated plasma 2408.3 Radiation trapping 245Appendix 8A. Interpretation of Figure 1.5 252

*9 Dense plasma 2579.1 Modifications of atomic potential and level energy 2579.2 Transition probability and collision cross-section 2619.3 Multistep processes involving doubly excited states 2669.4 Density of states and Saha equilibrium 277

Index 286

LIST OF SYMBOLS AND ABBREVIATIONS

a0 first Bohr radiusau atomic unitsAa(p,nl1) autoionization probability for (p,nl ' )A(p, q) Einstein's A coefficient or transition probability for p —> qAL line absorptionAr stabilizing radiative transition probabilityB(p, q) Einstein's B coefficient for photoabsorption and for induced

emissionb(p) population normalized by the Saha-Boltzmann valueB z–1(Te) partition functionBV(T) black-body radiation distribution or Planck's distribution functionC(p, q) excitation rate coefficientE kinetic energy of an electron, energy of levelEG energy of Griem's boundary level with respect to the ground stateE(p, q) energy separation between level p and qEi(–x) exponential integralf(u), f(E) electron velocity (energy) distribution functionfqoqscillator strength for transition p —> qfp,c oscillator strength for photoionization from level pF hctric field strength of the plasma microfieldF0 normal field strengthF(q,p) deexcitation rate coefficientG scale factor for excitation or deexcitation rate coefficientg(p) statistical weight of level pg(E/R) density of states per unit energy intervalge degeneracy of electron (=2)gbb, gbt, gft Gaunt factorG(a) reduced density of statesh Planck's constant, ratio of quasi-static broadening to impact

broadeningh Planck's constant divided by 2pH scale factor for radiative decay rate/ scale factor for continuum radiationk Boltzmann's constantK scale factor for radiative decay rate\gx log^Inx logex

qqq

hh

h

hc

)ition p q

ld

)

)

x LIST OF SYMBOLS AND ABBREVIATIONS

Hn principal quantum number, population of upper level

nnHn 0 (p ) recombining plasma componentn1(p) ionizing plasma component

2 parabolic quantum numbersN perturber particle density, density of ground-state atomsp designation of a level, momentum of an electronPG Griem's boundary level/IB Byron's boundary levelPR

p(v) recombination continuum radiation powerPL

p(v) line radiation powerPR+B(v) radiation power of recombination continuum and Bremsstrahlungrd(p, nl') dielectronic capture rate coefficient for (p, nl')r 0 (P) , r 1 (p) population coefficientR Rydberg constant, Radius of cylinderRD Debey radiusR0 mean distance between perturbers, ion sphere radiusRy Rydberg unitsSCR collisional-radiative ionization rate coefficientS(p) ionization rate coefficienttres response time of populations of excited levelstr1(p) relaxation time of population n(p)ttr(p) transient time for population of level pT(x, y) Stark profile with ion and electron broadeningTe electron temperatureTeo optimum temperatureu excitation or ionization energy in threshold unitsv speed of an electronW three-body recombination flux, equivalent widthW(b) field distribution functionz ze is the nuclear charge of the next ionization-stage ionZ Ze is the charge of perturber particlesZ(p) Saha-Boltzmann coefficientZp(pq) Saha-Boltzmann coefficient with respect to the energy position of

level pa fine structure constantaCR collisional-radiative recombination rate coefficienta(p) three-body recombination rate coefficient

n1, n

)p

)

e

z

h electron mass, magnetic quantum numberion mass

effective principal quantum number

electron density

density of ions in the next ionization stagepopulation (density) of level p

n*

m

))

)

)

)

LIST OF SYMBOLS AND ABBREVIATIONS xi

ad(l, nl')bb(p)r7ADXDwD

Dw1/2

n

nv

Kv

qq(Te,ne)pPoPm

sr, si

sp,q(u), sp,q(E)sp,q(u)sp,e(v)

se,p(e)

sp,c(u)

T

Tn

TB

Ttr

Tv

C

F(s), F(s)F(p,q)

x(p)w0

Wp,q(E)

suffixF

suffixH

suffix0

suffix¥

suffix¥ +suffixB

suffixD

suffixE

dielectronic recombination rate coefficient into (1, nl')FlF0

radiative recombination rate coefficientcoupling parameterfull width at half-maximum, decay rate of energyshift of the line centerlowering of ionization potentialDoppler half-widthhalf-width at half-maximumreduced electron density, ne/z

7

emission coefficientabsorption coefficientreduced electron temperature, Te/z

2

correction factor to Saha equationimpact parameterimpact parameter of Weisskopf radiusmean distance between perturbersreal and imaginary parts of the impact broadening cross-sectionexcitation or deexcitation cross-sectionphotoabsorption cross-section for excitation p —> qphotoionization cross-sectionradiative recombination cross-sectionionization cross-sectionmean free timeperiod of one revolution of the electron in level nperiod of one revolution of the electron in the first Bohr orbittransit timeoptical thicknessatomic unit velocityautocorrelation functionintensity or radiant flux or radiant power of emission radiationfor transition p —> qionization potential of level pcentral (angular) frequency of a spectral linecollision strength

free electronindicating the quantity for neutral hydrogenquantity in the low-density limitquantity in region IIquantity in region IIIByron's boundaryDopplerrelationship valid in thermodynamic equilibrium

h

xii LIST OF SYMBOLS AND ABBREVIATIONS

suffixG Griem's boundarysuffixH HoltsmarksuffixIB ionization balancesuffixL Lorentziansuffixv Voigtsuffixw Weisskopf

CR collisional-radiativeDL dielectronic capture ladder-likeFWHM full width at half-maximuml.h.s. left-hand sideLTE local thermodynamic equilibriumQSS quasi-steady stater.h.s. right-hand side

1INTRODUCTION

1.1 Historical background and outline of the book

The history of spectroscopy began more than three hundred years ago with theexperiment by Newton in which sunlight was dispersed by a prism into light rayswhich bore the colors of a rainbow (Fig. 1.1). Later, mainly in the nineteenthcentury, when the instrument called the spectroscope was used to observe thespectra of radiation emitted from various plasmas, i.e. flames, the Sun and severalstars, and later electric arcs and sparks (see Fig. 1.2), an enormous number ofspectral lines were found as emission or absorption lines. As a result of the inventionof the photographic plate, or of the spectrograph, spectroscopy developed into ascience of very high precision in terms of wavelength of observed lines. Numerousattempts were made to find regularities manifested by these lines. In the beginningof the twentieth century the experimentally established laws governing thewavelengths, or the frequencies, of the lines characteristic of atoms and ions,together with the spectral characteristics of the black-body radiation, played anessential role in establishing quantum mechanics.

Atomic spectroscopy, which deals primarily with wavelengths of spectral lines,is still actively studied to establish the energy-level structure of complicated atoms

FIG 1.1 The sketch of "the critical experiment" drawn by Newton himself.(By permission of the Warden and Fellows, New College, Oxford.)

INTRODUCTION

FIG 1.2 The "map" of various plasmas. NFR means the future nuclear fusionplasma; "laser" means laser-produced plasmas. The oblique line shows thescaling law for neutral hydrogen and hydrogen-like ions according to thenuclear charge z. See the text for details.

and highly ionized ions. The intensities of these lines are of concern mainly fromthe viewpoint of determining the ionization stage of the ions emitting the line andthe strength of the transition, i.e. the oscillator strength and the multipolarity.(These terms are explained in a subsequent part of this book.) Other character-istics of the spectrum, e.g. broadening of the lines, have less significance to atomicspectroscopists.

2

HISTORICAL BACKGROUND AND OUTLINE OF THE BOOK 3

The reason why characteristics except for the wavelengths are unimportant inatomic spectroscopy lies in the fact that these spectral characteristics are ephem-eral rather than basic, or they are dependent on the conditions under which thisparticular plasma is produced and on the parameter values this plasma has. Thisvery fact constitutes the starting point of plasma spectroscopy. Thus, plasma spec-troscopy deals with these variable characteristics of the radiation emitted fromthe plasma in relation to the plasma itself, which is regarded as an environment ofthe atoms and ions emitting the radiation.

It may be interesting to note that the intensity (the radiant flux or the radiantpower is a more precise term; see later) has been a quantity which is difficult todetermine experimentally. In particular, its absolute value could be determinedonly in favorable situations. However, developments in techniques, i.e. photo-multipliers for the last half-century, and multichannel detectors with digital signalprocessing techniques in recent years, have enabled us to perform quantitativespectroscopy much more easily. This situation is favorable for the development ofquantitative plasma spectroscopy as treated in this book.

When we look at a plasma, laboratory or celestial as shown in Fig. 1.2, througha spectrometer, we find a spectrum of radiation emitted from this plasma. This is apattern of spectral lines (and continuum), with varying intensities, distributed overa certain wavelength range. Figure 1.3 shows an example of the spectrum from aplasma. This plasma is produced from a helium arc discharge plasma streamingalong magnetic field lines into a dilute helium gas. The distribution pattern of linesin terms of wavelength reflects, of course, the energy-level structure of atoms, orthe composition of the plasma: what atomic species constitute the plasma. Thespectrum of Fig. 1.3 is of neutral helium. Figure 1.4 is the energy-level diagram,called the Grotorian diagram, of neutral helium. It is straightforward to identifythe lines in Fig. 1.3 with transitions each connecting two levels in this diagram.The thin solid lines show these identifications.

We sometimes find that two plasmas having identical wavelength-distributionpatterns show different intensity-distribution patterns. Figure 1.5 shows an example;both the spectra are of the resonance series lines of ionized helium (hydrogen-likehelium), terminating on the ground state as shown in Fig. 1.6. The plasmasproducing these spectra are essentially the same as that for Fig. 1.3. It may be saidthat the two plasmas in Fig. 1.5 are of the same composition but show different"colors". This difference might be attributed to different temperatures of theseplasmas; a plasma with a higher temperature tends to emit intense lines havingshorter wavelengths. This suggestion may be supported for two reasons: first, thehigher the energy of electrons in the plasma the higher the energy of atomic statesexcited by them (see Fig. 1.6) and thus the shorter the wavelengths of the lightoriginating from these states; second, the higher the temperature the shorter thepeak wavelength of the black-body radiation, i.e. Wien's displacement law (seeChapter 2). We will see later (Chapters 5 and 8) whether our conclusion here isadequate or not.

FIG 1.3 An example of the spectra observed from a plasma. This is the near-ultraviolet part of the spectrum of radiationemitted from a helium plasma. Several series of lines of neutral helium and recombination continua are seen. (Plasmaproduced in the TPD-I machine, Institute for Plasma Physics, Nagoya. Quoted from Otsuka M., 1980 Japanese Journal ofOptics (in Japanese), 9, 149; with permission from The Japanese Journal of Optics.)


FIG 1.4 Energy-level diagram of neutral helium. The thin solid lines show thetransitions corresponding to the emission lines in Fig. 1.3. The dashed linesshow the transitions of Fig. 3.6. The dotted line shows the emission line whichappears in Fig. 5.17(b). The dash-dot lines appear as emission lines in Figs. 7.4and 7.6.

When we look closely at a single spectral line, we sometimes find that itsintensity is distributed over a narrow but finite wavelength region, with its peakdisplaced from the original position of the line where it is found under normalconditions. We can find several broadened lines in Fig. 1.3 in the longer-wavelength region, as contrasted with the accompanying sharp lines. We also findprominent examples in Fig. 1.7.

We may also find continuum spectra underlying the spectral lines. An exampleis seen in Fig. 1.3. Figure 1.7 shows another example. For certain plasmas likethese examples the continuum is weaker than the lines but for some others it iseven stronger than the lines.

The plasma of Fig. 1.7 is produced from a hydrogen pellet (frozen hydrogenice) injected into a high-temperature plasma. A dense plasma is produced fromevaporated hydrogen. Several broadened lines, tending to a continuum, of neutralhydrogen atoms are seen. These lines correspond in Fig. 1.6 to the transitions

INTRODUCTION

FIG 1.5 Two spectra from plasmas produced under slightly different conditions.The spectral lines are the resonance series lines (1 2S —«2P) of ionized helium.(TPD-I. By courtesy of Professor T. Kato.) The asterisk shows the real peakposition when the saturation effect of the detector is corrected for.

terminating on the n = 2 levels. Of course, the transition energies are about one-quarter for these lines, because Fig. 1.6 is for hydrogen-like helium ions, notneutral hydrogen in Fig. 1.7.

As mentioned earlier, all these variable characteristics of the spectrum ofatoms and ions are dependent on the nature of the plasma which emits theradiation. In other words, the spectrum contains information about the plasma:i.e. it is the fingerprint of the plasma. This notion constitutes the basis of plasmadiagnostics or using the observed spectrum to infer the characteristics of theplasma, e.g. its temperature, density, and particle transport property over space.

The first task of plasma spectroscopy would naturally be to find and establishthe relationships between the characteristics of the emission-line (and continuum)intensities from a plasma and the nature of this plasma. Since a spectral line

6

HISTORICAL BACKGROUND AND OUTLINE OF THE BOOK

FIG 1.6 Energy-level diagram of ionized helium, where the different / and mistates are reduced to a single level specified by n. The series of transitionsterminating on the ground state correspond to the emission lines in Fig. 1.5.The transitions terminating on the n = 1 level correspond to the emission linesin Fig. 1.7, except that this diagram is for hydrogen-like helium and Fig. 1.7 isfor neutral hydrogen.

is emitted by excited atoms or ions (see Figs. 1.4 and 1.6) its intensity is given bythe number density of these atoms or ions. Here we have assumed that theidentification of the spectral line, or the correspondence of the line to the upperand lower levels of atoms or ions, is established and that the transition probabilityis known for this transition. We have also ignored intricate factors, like polar-ization and reabsorption of radiation (see Chapter 8), both of which can affect theobserved line intensity. Thus, the problem of the intensity distribution reduces tothat of the population-density (called simply the population henceforth) distribu-tion of atoms and ions over excited levels (and in the ground state, too). Oncethese relationships are established, several conventional concepts will turn out tobe incorrect. For example, the interpretation mentioned above concerning thetemperature and the "color" (Fig. 1.5) will be found to be too naive; sometimes ahot plasma may look more "red" than some colder plasmas having the samecomposition.

7

INTRODUCTION

FIG 1.7 A spectrum of neutral hydrogen atoms from a plasma produced bypellet (a solid hydrogen ice) injection into a high-temperature plasma.(Produced at the LHD in the National Institute for Fusion Science, Toki. Bycourtesy of Dr. M. Goto.)

The primary objective of this book is to provide the reader with a sound basisfor interpreting various features manifested by a spectrum of radiation emittedfrom a plasma in terms of the characteristics of the plasma.

The first two chapters are intended so that the reader acquires the backgroundnecessary to proceed to the main part of the book developed in subsequentchapters. First, thermodynamic equilibrium relationships are discussed for thediscrete-level populations, for the ionization balance, and for the radiation field.The subsequent chapter discusses the atomic processes important in plasmas,i.e. the spontaneous radiative transition and the transitions due to electron impact.It is pointed out that, for a pair of levels, a single parameter, the absorptionoscillator strength, which gives the radiative transition probability, also determinesthe collisional excitation cross-section, although to a limited extent. Anotherimportant fact worth noting is that various features associated with high-lyingexcited states continue smoothly across the ionization limit to those associatedwith low-energy continuum states. This is a natural consequence of the continuityof the corresponding wavefunctions of the atomic electron and the ion (thecontinuum-state electron).

Chapters 4 and 5 present a theoretical framework in which the experimentallyobserved population distribution is interpreted in terms of various characteristicsof the plasma. In Chapter 4 we introduce the method known as the collisional-radiative (CR) model or the method of the quasi-steady-state (QSS) solution.

8


By this method we treat in a coherent manner the population couplings among theexcited levels (and the ground state) in the population formation by the collectionof atomic processes. Figure 1.8 shows schematically the energy-level structure; por q denotes a level and/> = 1 means the ground state. As an example of ensemblesof atoms and ions immersed in a plasma and emitting radiation we take hydrogen-like ions (and neutral hydrogen) for the purpose of illustration. A discussion ofthe validity of this method, or of the QSS solution, is given in Appendix 4B. Thenan excited-level population is expressed as a sum of the ionizing plasma compo-nent and the recombining plasma component. Figure 1.9 shows schematically thestructure of the populations. For both the components the population distributionand its kinetics are examined in detail. Figure 1.10 is the "map" of the populationsof these components, or the summary of our investigations in Chapter 4. Severalcharacteristic population distributions, e.g. the minus sixth power distribution, andthe corresponding population kinetics, i.e. the ladder-like excitation-ionization

FIG 1.8 Schematic energy-level diagram of an atom or ion with symbols used inthis book.

FIG 1.9 The structure of the excited-level populations in the collisional-radiativemodel. The population n(p) is the sum of the ionizing plasma component n^(p)which is proportional to the ground-state population «(1) and the recombiningplasma component n0(p) proportional to the "ion" density nz. Full explana-tions are given in Chapter 4.

FIG 1.10 The "map" of the excited-level populations of neutral hydrogen and hydrogen-like ions in plasma. This diagram isthe summary of our investigations to be developed in Chapter 4, so that a reader who has started to read this book doesnot need to understand the details of this diagram. The abscissa is the (reduced) electron density and the ordinate is theprincipal quantum number of excited levels, (a) The ionizing plasma component; (b) the recombining plasma component.

Griem's boundary pG, given by eq. (4.25), (4.29), or (4.59), divides the whole area into a low-density region and high-density region. Byron's boundary />B, given by eq. (4.55) or (4.56), divides the high-density region into low-lying levels andhigh-lying levels. In each area, the name of the phase I, the population distribution, and the dominant population kineticsare shown for level p with which we are concerned. For the capture-radiative-cascade (CRC) phase in (b) the near-Saha-Boltzmann population is for the high-temperature case. In practical situations of the ionizing plasma (a) Byron'sboundary lies far below p = 2, and only the saturation phase with the multistep ladder-like excitation-ionizationmechanism appears in the high-density region. (Quoted from Fujimoto, T. 1980 Journal of the Physical Society of Japan,49, 1591, with permission from The Physical Society of Japan.)


mechanism, are established. Two important boundaries, Griem's boundary andByron's boundary, are derived for electron density and temperature, or for excitedlevels. It is noted that the strong population coupling among the excited levels andits continuation to the ionic (continuum) states play an essential role in deter-mining the population distributions in both the components.

In a plasma an ensemble of atoms or ions as a whole may be in a dynamicalprocess of ionization or recombination, depending on the time history and thespatial structure of the plasma. In these processes, in addition to the direct ioniza-tion and recombination, excited levels play essential roles, too, and they affect theeffective rate of ionization and that of recombination. This subject is examined inChapter 5. An important finding is that the ionization process is associated withthe ionizing plasma component of populations and thus the magnitude of theionization flux is proportional to the excited-level population. A similar propor-tionality is also valid for the recombination flux and the recombining plasmacomponent. In both the cases, the proportionality factors naturally depend on theparameters of the plasma. Thus, an emission line intensity is a measure of theionization flux and that of the recombination flux, depending on the nature ofthe plasma.

There is a class of plasmas in which the ionization flux and the recombinationflux balance with each other, or plasmas in ionization balance. An importantconclusion is reached for this class of plasmas: the ratio of the contributions to anexcited-level population from the ionizing plasma component and from therecombining plasma component could be comparable in magnitude. This findingleads to another even more important conclusion: for many actual plasmas whichare out of, sometimes far from, ionization balance, only one of the two compo-nents dominates the actual populations while the other gives a negligibly smallcontribution. Thus we have reached an important step for a correct explanation ofthe different spectra in Fig. 1.5. Another important finding is that, among plasmasin various states of ionization-recombination, a plasma in ionization balance givesrise to the minimum of radiation intensity. Plasmas out of ionization balance emitmuch stronger radiation.

Following these two chapters of primary importance, we now turn to otherfacets which together constitute plasma spectroscopy. Chapter 6 treats the con-tinuum radiation. The spectral characteristics are examined for the recombinationcontinuum, and a smooth continuation is established for its intensity to those ofthe accompanying series lines, as is observed in Figs. 1.3 and 1.7. This con-tinuation is interpreted as the continuation of the "populations" of the discretestates to the continuum states.

The problems of broadening and shift of spectral lines follow in Chapter 7.We have already seen some examples in Figs. 1.3 and 1.7. This aspect is importantfor determining the atom (ion) temperature or the plasma density. Besides theDoppler broadening and Stark broadening in the quasi-static approximation,natural broadening and impact broadening is treated in a rather elementary way.The latter class of broadening is regarded as the relaxation of optical coherence.

12 INTRODUCTION

Chapter 8 treats the phenomena associated with radiation transport. We firstexamine how the absorption line profile develops in an absorbing medium.Then we describe how the observed intensity and profile of an emission linedevelops in a plasma when the plasma becomes optically thick to this line. We thenconsider the situation in which the excited-level population is controlled by asequence of processes of emission and reabsorption of radiation, i.e. radiationtrapping. We examine the phenomenon on the basis of two approaches whichare complementary to each other. At this point we are able to give the correctinterpretation to the spectra in Figs. 1.5 and 1.7.

When our plasma is dense, various new phenomena may appear which areabsent in "everyday" plasmas. Although part of this problem has already beendiscussed in the context of excited-level populations and line broadening, furtherdiscussions deserve a separate chapter. In Chapter 9 we investigate how the atomicstate energy and the collision cross-section are affected by the screening of theCoulomb interactions by the plasma particles surrounding the atom or the ion.We then examine additional new excitation and deexcitation processes of ionsinvolving the doubly excited states. Contributions from the resonance process tothe excitation cross-section are also found to be affected in a dense plasma. Directrecombination of ions in an excited level can be important under certain condi-tions. Finally, we investigate modification to the density of atomic states overenergy and its consequence incurred in the thermodynamic equilibrium relation-ship of the densities of atoms and ions in the plasma, a modification to the resultof Chapter 2.

1.2 Various plasmas

Figure 1.2 shows various kinds of plasmas on the plane ne—Te, Its abscissa andordinate are the most important parameters of a plasma: the number density ofelectrons, or simply the electron density, «e, in units of m~3, or in cm~3, and theelectron temperature, Te, in units of K. Occasionally, kTe expressed as eV (elec-tron volts: the energy of an elementary charge e accelerated by a potential of 1 V)is used: kTe= 1 eV corresponds to Te= 11,605 K. The abscissas and the ordinatesof Fig. 1.2 are expressed in these units. So, a plasma is located somewhere on thisplane according to its ne—Te values. The most modest plasmas are flames likecandles, and those in internal combustion engines, which are produced and heatedby chemical reactions. We have enormous numbers of plasmas produced byelectric discharge. The class of glow discharges, which are produced in a low-pressure gas, includes many laboratory plasmas as well as plasmas encountered inour everyday life; an example is the plasma in fluorescent lamp tubes. Accordingto the discharge current drawn, «e varies over several orders, but Te lies in a rathernarrow range, and kTe is one through a couple of eV. Later in Chapter 4, we willencounter an example of this class of plasma. Many kinds of processing plasmas,which are used for the purpose of manufacturing, e.g. semiconductor devices, areproduced by radio-frequency or microwave discharges in chemically active gases.

NOMENCLATURE AND BASIC CONSTANTS 13

They have similar parameter values. If an electric discharge is made continuouslyin a gas under atmospheric or even higher pressure, we have arc discharge plasmas.This kind includes many kinds of lamps for illumination, e.g. mercury dischargelamps. Interestingly, Te of these plasmas has a very narrow range aroundkTe = 1 eV. The plasma shown in Fig. 1.7 happens to be very similar to this class.When the heating of electrons by electric power input stops, the plasma decays intime or, in the case of a flowing plasma, in space. The plasmas in these decayingprocesses are called afterglows, and have low Te. In Chapter 4, we will find a fewexamples of this class of plasma. It will turn out that the plasma of Fig. 1.3 alsobelongs to this class. If an electric breakdown of a high voltage takes place in anatmospheric-pressure gas, we have a spark discharge, or even lightning. Owing tothe sudden input of energy into a thin area of the gas, a plasma with a rather highTe is produced.

In contrast to these "classical" plasmas we now have more "powerful" plasmaswhich have been developed in the last couple of decades. One of the motivations ofthese developments came from the possibility of realizing nuclear fusion reactionsfor a future energy source. One class of such plasmas is called the magneticallyconfined plasma. A high-temperature and moderate-density plasma is confinedwithin the toroidal-shaped vessel made with a magnetic field. With the enormousprogress in scientific and technological developments, plasma machines with theconfiguration called tokamak now produce plasmas close to the practical conditionfor nuclear fusion reactions. Helical configuration plasmas are also being vigor-ously investigated. In Fig. 1.2 the NFR region means the parameters of the futurenuclear fusion reactor. Another class is high-energy-density plasmas which areproduced by putting a vast amount of energy into a small volume of a gas or asolid in a very short time. A rather traditional approach is pinch plasmas, sometimescalled a vacuum spark, a plasma focus, etc. Other more modern plasmas areproduced by irradiating a solid or gas or even clusters by short-pulsed laserradiation. These plasmas occupy quite a large area of the parameters: the natureof a plasma strongly depends on the experimental conditions, and its parametersare different during laser irradiation and in the decaying period after that. Thesehigh-energy-density plasmas may be used as an x-ray light source or even as anx-ray laser source.

Plasmas in nature should not be forgotten. It is sometimes said that more than99 percent of the material in the universe is in the form of plasma. Just twoexamples are given Fig. 1.2. The Earth is surrounded by several layers of iono-sphere. It starts at about 100 km above the Earth's surface and extends up to some500 km. Another example is the solar corona surrounding the Sun; this plasmagreatly inspired the development of plasma spectroscopy.

1.3 Nomenclature and basic constants

In this book the term plasma has dual meanings; the first is in the ordinary sense toexpress a material which is composed of electrons, ions, and some neutral atoms

14 INTRODUCTION

TABLE 1.1 Basic constants.

or even molecules. Several examples are shown in Fig. 1.2. The second usage is toexpress an ensemble of atoms or ions immersed in a plasma. The latter may soundstrange, but it will turn out that this is rather natural.

As we have already seen important parameters to characterize a plasma (in thefirst sense) are «e and Te. Since electrons are usually much more active than ionsand neutrals in determining excited level populations, a plasma in the first sensesometimes means simply an electron gas having a certain ne and Te.

Constants which are used in this book without explanation are given inTable 1.1. The Rydberg constant, which is virtually equal to the ionizationpotential of neutral hydrogen, is expressed as R. Figure 1.8 shows the schematicenergy-level diagram of ions with several symbols. Suppose we are interested inthe ion or the atom denoted with (z — 1). z indicates the charge ze of the ions in thenext ionization stage, or roughly speaking, the effective core charge felt by theoptical electron of the (z — 1) ion. Here we call the electron that plays the dom-inant role in a transition and emits radiation the optical electron. If we are treatingsingly ionized helium in Figs. 1.5 and 1.6, for example, (z — 1) is 1 (i.e. singlyionized) and z is therefore 2. nz indicates the density of the ions in the nextionization stage. For hydrogen-like ions (and neutral hydrogen), with which thedominant part of this book is concerned, z is equal to the nuclear charge, p or q isused to indicate a discrete state. nz_i(p), gz-i(p) and XZ-I(P) indicate, respectively,the population (in units of m~3), the statistical weight and the ionization potential(a positive quantity), of level p. Ez_i(p,q) is the energy difference between thelower level p and the upper level q, i.e. Ez_i(p,q) = Xz-i(p) — Xz-i(<l)- The sub-script z — 1 is omitted whenever confusion is unlikely to occur.

1.4 z-scaling

For neutral hydrogen and hydrogen-like ions all the radiative transition prob-abilities and the collision cross-sections scale according to z except for the

m = 9.109x 1(T31

M= 1.6605 xl(T27

e= 1.6021 xl(T19

c = 2.998 x l O 8

h = 6. 626 x 1(T34

ft =1. 0545 x 10~34

fc= 1.3805 x 10~23

e0 = 8.854 x lO~ 1 2

a = e2/2hce0= 1/137.0flo = £0h

2/Tmie2 = 5.292 x 10~u

tf = e2/87re0a0 = 2.1799 x i(r18

= 13.605

[kg][kg][C][m/s][Js][Js][J/K][C/Vm]

[m][J][eV]

electron rest massatomic mass unitselectron chargespeed of light in free spacePlanck's constant

Boltzmann's constantdielectric constant of vacuumfine structure constantfirst Bohr radiusRydberg constant

NEUTRAL HYDROGEN AND HYDROGEN-LIKE IONS 15

excitation and ionization cross-section values near the threshold; they haveappreciable z-dependences (Chapter 3). This latter point will be shown in Fig. 3.11later. As will be discussed in Appendixes 3A and 5B, these scaling properties leadus to a certain scaling law for the plasma parameters.

In Fig. 1.2 an oblique line is drawn. The position of this line does nothave any significance, but its slope and the scale attached on it are important.The slope is 2/7 on the logarithmic scales of this figure. If we are interested in agroup of ions having a certain z, e.g. z = 26 for hydrogen-like iron, present in aplasma which have a certain Te, these ions feel and behave as if neutral hydrogen,z= 1, feels and behaves in a plasma having re/z

2. So, Te of the plasma in thefirst sense scales according to z2. Likewise, «e scales according to z7. Later in thisbook, the quantities Te/z

2 and ne/z7 are called the reduced electron temperature

and the reduced electron density, respectively. The line in the figure shows thisscaling: hydrogen-like iron ions in a plasma with ne= 1025 m~3 (1019 cm~3) andJg = 108 K (9 keV) behave like neutral hydrogen would do in a plasma with«e= 1.2 x 1015 irT3 (1.2 x 109 cm"3) and Te= 1.5 x 105 K (13 eV). On the basis ofthis scaling, we can infer the characteristics of the former ions from our knowledgeof the latter. It is noted, however, that this simple scaling law is by no meansperfect. This is partly because of the z-dependent cross-section values near thethreshold, as noted already. Another more important exception is the ionization-recombination relationship, as will be seen in Chapter 5. Even so, this scaling lawis useful to understand, or estimate, qualitatively the properties of the ions underconsideration.

In this section, we have been mainly concerned with neutral hydrogenand hydrogen-like ions. Even for ions other than hydrogen-like, the propertiesof excited states are not much different from those of hydrogen-like states,especially for highly excited states. In this sense, the above scaling law is also validfor nonhydrogen-like ions to a certain extent. For this reason many formulasderived in Chapters 4 and 5 are expressed in terms of the reduced temperature anddensity.

1.5 Neutral hydrogen and hydrogen-like ions

We start with the classical picture: an electron with charge —e moves in theCoulomb attraction field exerted by the ion with charge ze, fixed at the origin. Theorbit of the electron is elliptic and it may be specified by the major axis, ellipticity,and the direction of the axis of revolution. Instead of these parameters we chooseas parameters the total energy E, i.e. the sum of the kinetic and potential energies,which is negative; the period of one revolution over the orbit r; the angularmomentum A; and its projection onto a certain direction, say, the z-axis, /j,x. Wetake the quantity — 1Er as the ordinate in Fig. 1.1 l(a). The abscissas are 2-TrAand 2-Kfix- Then all the possible orbits are included within the volume enclosed bythree planes: two vertical planes that include the ordinate axis and the JJL\ = ±Apoints; these planes correspond to the orbits with the axis directed parallel or

16 INTRODUCTION

FIG 1.11 (Continued)

NEUTRAL HYDROGEN AND HYDROGEN-LIKE IONS 17

antiparallel to the z-axis. Another plane is the oblique plane for which 2-7rA =—1Er holds; this plane corresponds to circular orbits. All these parameters aremeasured in units of h. We divide the whole volume into elementary volumes byusing half-integer values of —2Er/h and 1-n \i\jh and integer values of 2ir\. Twoexamples of the elementary volumes are shown in this figure. We allocate aninteger n to the volumes of (n — 0.5) < (—2Er/h) <(n + 0.5) and name this integerthe principal quantum number. The classical states with —2Er/h<Q.5 are notallowed in reality. We allocate an integer /to the volumes with /< (2-K\/ti) <(/+!)and call it the angular momentum quantum number. The integer azimuthal quantumnumber mt is allocated to (w/—0.5)<(2-7r/iA//i)<(w/ + 0.5). mt is also called themagnetic quantum number. The examples of the elementary volumes in this figureare (n, I, mf) = (3,1,0) and (3,2,1). It is noted that all the elementary volumes, orthe cells, have a volume of h3. This cell may be regarded as a quantum cell; all thestates of electron motion expressed by points within a cell cannot be distinguishedin reality and therefore they are regarded as a single state. Thus we have quantizedthe classical states, and we represent all the classical states within a single quantumcell with a point at the center of this cell. Owing to the internal structure of theelectron, the spin, each point is doubly degenerate. Suppose that the whole volumeis transparent except for these points. If we look through Fig. 1.11 (a) from thedirection of the /j,x axis and rescale the ordinate according to the energy E, weobtain Fig. 1.1 l(b), the Grotorian diagram. The levels having different mt

belonging to a single / level are called the magnetic sublevels. The state (n, /, mf) hasthe principal quantum number n, angular momentum Ih, and the projection of theangular momentum onto the quantization axis mfi. In reality, the magnitude ofthe angular momentum is \/l(l + l)h.

FIG 1.11 (a) The quantization scheme of the classical electron motion of ahydrogen-like ion. The principal quantum number n is defined from the energy(with the sign reversed) multiplied by the duration of one revolution of theelectron in the orbit; the angular momentum quantum number / is defined fromthe angular momentum A; and the azimuthal quantum number (the magneticquantum number) w/ is the projection of the angular momentum onto thequantization axis, /JA- Each elementary cell has volume h3. The two examples ofthe quantum cells are shown for («,/, w/) = (3,1,0) and (3,2,1). Thequantization sheme is due to Professor R. More, (b) The energy-level diagram(the Grotorian diagram) of neutral hydrogen. States with different mt in (a) arereduced to a single / level, but different / levels are resolved. Different / levelscan be further reduced to a single level to lead to a simplified energy-leveldiagram like the one shown in Fig. 1.6. Note that, in the case of neutralhydrogen, the energy is reduced to about one-quarter of that in Fig. 1.6. In thiscase, a level is specified with the principal quantum number n, or in laterchapters, p or q. This scheme is appropriate when we can assume statisticalpopulations among the different / levels in (b), or equal populations among thedifferent mt and / states belonging to the same n in (a).

See Table 1.1 for R, In terms of the classical Bohr picture, the radius of the circularorbit (or the semi-major axis in the case of an elliptic orbit) is given by

It is noted that eq. (1.4) is valid also for elliptic orbits. The energy separationbetween the levels with principal quantum numbers differing by one is

In a plasma, electron, and ion collisions, which are random, tend to populate thedifferent / and mi states having the same n with equal probabilities. In the classicalpicture of Fig. 1.11 (a), the populations are uniformly distributed among the samen cells. We say in this situation that the population is distributed statistically. Inthis case the total population of level n is 2«2 times the population in a single cell,where 2«2 is the number of the quantum cells, i.e. n2 times the degeneracy factor 2.If the statistical populations are actually established (its validity will be discussedin Appendix 4A; see Fig. 4A.2) the principal quantum number is enough to specifya level. In such a case p or q is understood to denote the principal quantum

* One atomic unit length is a0, and one atomic unit velocity is £. One atomic unit time is a0/£, orrau = h/2R = 2A2 x 10~17s. This corresponds to the time for the n= 1 electron to traverse one radianlength over the circular orbit of radius GO- Then the relationship corresponding to eq. (1.6) becomesr™ • A_£"(«)AB=I = fi, where r™ is defined from rm similarly to eq. (1.4).

18 INTRODUCTION

The energy of levels having different / and mi but the same n is given by

The speed of the electron in the circular orbit is

For a0 and a see Table 1.1. The period of one revolution of the electron over theorbit is

with

This quantity is also understood as an energy width allocated to the level havingprincipal quantum number n. See Fig. 1.11 (a) again. It is interesting to find thatthe two quantities of eqs. (1.4) and (1.5) satisfy the relationship*

NON-HYDROGEN-LIKE IONS 19

number in place of n, so that, for example,/? = 1 means the ground state. Therefore,gz-i(p) = 2p2 and XZ-I(P) = z2R/p2. The Grotorian diagram of Fig. 1.1 l(b) reducesto a simplified energy-level diagram which is equal to Fig. 1.6 except that theordinate is reduced to one quarter. We adopt the convention that p < q meansE(p,q) = x(p)-X(([)>0-

1.6 Non-hydrogen-like ions

The energy-level structure of ions having many electrons is complicated, some-times extremely complicated. For ions with a small number of electrons, however,it is rather simple, and may be regarded as a modification from that of a hydrogen-like ion, Fig. l.ll(b). Neutral helium and helium-like ions are good examples. SeeFig. 1.4. Atoms and ions having one or two electrons outside of the closed shellsare other examples. For these atoms and ions an excited level (denoted by p) isdesignated by the principal quantum number n of the excited electron, the sum ofthe orbital angular momenta Lh* of all the electrons, and the sum of the spinangular momenta Sh* This scheme of combination of the angular momenta iscalled the L— S coupling, which describes well the energy-level structure of theseatoms and ions. A level is designated by n(2S+ l)L. The suffix (25* + 1) is called themultiplicity; S = 0 is a singlet, ^ is a doublet, 1 is a triplet, and so on. To levels withL = Q, 1, 2, 3 , . . . we assign the symbol S, P, D, F , . . . , respectively.1" Figure 1.4carries this nomenclature. The level p= n(2S+l)L has the statistical weightg(p) = (2S+ 1)(2L + 1). This level is further split into the fine-structure levels, andeach component is designated by the total angular momentum Jh* In the case ofL>S, we have J=L — S, L S+l,...,L + S, These fine-structure levels aredesignated by n(2S+r>LJ. For example, in Fig. 1.4, the 23P ("two triplet P") levelconsists of three closely lying levels 23P0, 2

3P1; and 23P2. The statistical weight ofeach of them is (2/+ 1), and their sum XX2/+ 1) is equal to (2S + 1)(2L+ 1). Inthe present example, g(2 3P) = 9. This designation is also adopted for hydrogenatoms and hydrogen-like ions. Figure l.ll(b) carries this nomenclature. In thiscase, L = / and S = .

We defined the optical electron as the electron playing the dominant role inmaking a transition and emitting radiation. We also define z; this quantity indic-ates the effective core charge ze felt by the optical electron when it is at a largedistance from the core. This happens to be equal to the roman numeral used todenote the ionization stage of a spectral line, e.g. HI, CIII, OV.

* As noted above, the actual magnitudes of these angular momenta are \/L(L + l)fi, \/S(S + l)h,and \/J(J + 1), respectively.

t These symbols stem from the early nomenclature of the series lines, "sharp", "principal","diffuse", and "fundamental". These characteristics can be recognized in Fig. 1.3: in the long-wavelength region the series of pairs of sharp and diffuse lines are seen, each originating from the S andD levels, respectively, in Fig. 1.4. Other lines in Fig. 1.3 are of the principal series, which originate fromP levels in Fig. 1.4.

Here 6 (= n — «*) is called the quantum defect and n* the effective principal quantumnumber. Again for s states 8 is large and n* is appreciably smaller than n. For d andhigher-/ states 8 is very small and n* is almost equal to n. An example is seen inFig. 1.4.

In the following even in the case of nonhydrogen-like ions p = 1 is understoodto denote the ground state.

As seen in Figs. 1.4 and 1.1 l(b) high-lying levels form a series of levels con-verging to the ionization limit. Their energies measured from the limit are given byeq. (1.1) or eq. (1.7) (or eq. (1.7a)) with large values of n. We call these levels theRydberg levels (states).

In atomic spectroscopy an emission (absorption) line and the correspondingtransition is customarily written like: Hel A 318.8 nm (23S—43P); this means thatthis spectral line (one of the lines in Fig. 1.3) is of neutral helium (called the firstspectrum) with wavelength 318.8 nm for transition with lower level 2 3S and upperlevel 4 3P. The lower level comes first. In the following we follow this convention.Figure 1.3 carries an alternative notation. Instead of nm, units of A are sometimesused; 1 A is 0.1 nm.

The first prominent line terminating on, or, in the case of absorption, startingfrom, the ground state is called the resonance line. An example is the Hell A 30.3 nm(12S—22P) line shown in Fig. 1.5 and identified in Fig. 1.6. Another example is thetransition in Fig. 1.4 of Hel (1 :S—2 :P) with the wavelength of 58.4nm.

Finally, units of energy are mentioned. 1 au (atomic units) is equal to2_R = 27.2eV. When R is used as units of energy (Rydberg units) energy isexpressed as Ry. Units cm^1 is sometimes used to express a spectral line frequency(v), and equal to vie in the cgs units. 1 cm^1 is the energy difference correspondingto a transition wavelength of A= 1 cm in vacuum; i.e. 1 Ry= 1.0974 x 105cm^1.

20 INTRODUCTION

The energy of a level p with principal quantum number n may be expressed as amodification of eq. (1.1) by

where the parameter 0 is introduced which accounts for the degree of completenessof the screening of the nuclear charge by the electrons other than the opticalelectron. If the screening is complete 0 equals zero. For the optical electron havingan orbit penetrating deep into the core electron orbits, e.g. the s electrons, (seeFig. 3.2(b) later), the screening is not complete, and 0 is a positive quantity, usuallysmaller than 1. An alternative way of expressing the energy is

REFERENCES 21

References

Several books are available for plasma spectroscopy in general and for its variousfacets which are treated in the later part of this book. A few of them are listedbelow.

Cooper, J. 1966 Rep. Prog. Phys. 22, 35.Griem, H.R. 1964 Plasma Spectroscopy (McGraw-Hill, New York).Griem, H.R. 1997 Principles of Plasma Spectroscopy (Cambridge University,

Cambridge).Huddlestone, R.H. and Leonard, S.L. (eds.) 1965 Plasma Diagnostic Techniques

(Academic Press, New York).Lochte-Holtgreven,W.(ed.)l 968 Plasma Diagnostics (North-Holland, Amsterdam).

There are excellent books on atomic structure and atomic spectra. Only a few arementioned.Bethe, H.A. and Salpeter, E.E. 1977 Quantum Mechanics of One- and Two-Electron

Atoms (Plenum, New York; reprint of 1957).Condon, E.U. and Shortley, G.H. 1967 Theory of Atomic Spectra (Cambridge

University Press, London; reprint of 1935).Hertzberg, G. 1944 Atomic Spectra and Atomic Structure (Dover, New York).Shore, B.W. and Menzel, D.H. 1968 Principles of Atomic Spectra (John Wiley and

Sons, New York).Thorne, A., Litzen, U. and Johansson, S. 1999 Spectrophysics (Springer, Berlin).White, H.E. 1934 Introduction to Atomic Spectra (McGraw-Hill, New York).

2

THERMODYNAMIC EQUILIBRIUM

with normalization, ff(E)dE= 1. The average energy is E = 3kTe/2.

Boltzmann and Saha-Boltzmann distributions

In this book, the word "ions" is used to denote both ions and neutral atoms.However, the word "atoms" is used when it is more convenient to distinguishatoms and ions in adjacent ionization stages.

It is well known from statistical mechanics that, in thermodynamic equilib-rium, the ratio of the number of ions per unit volume in two different energylevels, or the population density ratio, is given by the Boltzmann distribution

where we have assumed levels/? < q in ionization stage (z — 1). See Fig. 1.8. Actuallevel schemes are shown in Figs. 1.4, 1.6, and l.ll(b). If the ions in this ionizationstage have many levels including p and q, we may plot the populations per unitstatistical weight in a semilogarithmic plot. This plot is called the Boltzmann plot,

2.1 Velocity and population distributions

Maxwell distribution

In this book we consider a plasma, in the first sense, as consisting of electrons,ions, atoms, and even molecules. If ne is high and Te is low so that the meandistance between electrons becomes comparable to or shorter than the de Brogliewavelength of electrons with thermal energy, A = h/'^fI^nnkT~e, quantum effectsprevail, and the electron velocity distribution is given by the Fermi-Dirac dis-tribution. In the following we assume the opposite, i.e. low «e and high Te:

Then, we have the classical Maxwell-Boltzmann distribution (called simply theMaxwell distribution)

which satisfies the normalization condition, ff(v)dv = 1. The average speedis v = ^/SkTe/irm, the root mean square speed is \/(v}2 = ^/3kTe/m, and themost probable speed is t>p = ^/2kTe/m. The corresponding energy distributionfunction is

VELOCITY AND POPULATION DISTRIBUTIONS 23

and we obtain a straight line, the slope of which corresponds to Te. We will seeexamples later in Chapters 4 and 5.

Equation (2.3) applies to levels of ions in a particular ionization stage. If wetake a hydrogen-like ion (z — 1), each of these levels corresponds to a level asshown in Fig. 1.6. The above thermodynamic relationship may be extended tohigher energies across the ionization limit to the continuum states of the electron,having positive energies. We approximate here the continuum states as free-electron states. Discussions concerning this approximation will be given inChapter 9. Since the level energy is continuous we consider free states of electronshaving speed v within the range dv. This upper "level" is regarded as the collectionof states of free electrons paired to the core ion (in the ground state) in theionization stage z. Then eq. (2.3) is rewritten as

where nz(l,v)dv and gz(l,v)dv denote, respectively, the "population" and the"statistical weight" of the upper "level", and A£" is the energy difference betweenthis upper level and the lower level p, i.e. AE = mv /2 + xz-i(p)-

We now introduce the phase space, i.e. a six-dimensional space for the motionof a free electron: three dimensions for the spatial coordinate (x,y,z) and threedimensions for the momentum coordinate (px,py,pz). In the x-px plane, we definea cell Sx • Spx having area h. This is another quantum cell (remember Fig. l.ll(a))and has the significance that all the states of motion, the corresponding points(x, PX) of which fall in this cell, are regarded as a single state. Similar argumentsapply to the y—py and the z—p2 planes. Thus, the "number of states" deriving fromthe motion of the electrons is given as

where Ax, Aj, and Az are the spatial coordinate widths allocated to one of thefree electrons and t±px, Apy, and Apz are similarly the momentum coordinatewidths. The former widths make a volume A V allocated to the electron, which isequal to l/ne. Since we assume the electron motion to be isotropic we use the polarcoordinate system

Equation (2.5) gives the number of states for the electron motion. The "statisticalweight" of the "upper level" is thus given as

where ge and gz(T) are the statistical weights originating from the inner structure ofan electron and that of the ground-state ion, respectively. The former comes fromthe electron spin and is ge = 2, It is noted that eq. (2.5a) is also encountered in solidstate physics as the density of states of electrons in the free-electron model.Equation (2.4) is transformed as

Equation (2.7) or (2.7a) is called the Saha-Boltzmann distribution. Under certainconditions, thermodynamic equilibrium may be established in our plasma and thepopulation of an excited level p is actually given by eq. (2.7) or (2.7a). If this is thecase, we say that "level p is in local thermodynamic equilibrium (LTE) withrespect to ion z." This situation is called partial LTE. In the case that the LTEpopulation, eq. (2.7a), extends down to the ground state p=\, this situation isdefined as complete local thermodynamic equilibrium (complete LTE). Theseproblems will be treated later in Chapter 5.

24 THERMODYNAMIC EQUILIBRIUM

or

where we have used eq. (2.2). In many situations we are not interested in the"population ratio" as given by eq. (2.6). Rather, the quantity of interest would bethe ratio of the "ion" density nz(l) and the "atom" density nz_i(p) in level p. Theformer quantity is obtained by integration of the "populations" over the speeds ofthe free electrons, i.e. nz(l) = fnz(l,v)dv. By using the normalization condition forthe Maxwell distribution we obtain the Saha-Boltzmann distribution

or

where Z(p) is called the Saha-Boltzmann coefficient. It is interesting to note thatZ(p) is expressed in terms of the thermal de Broglie wavelength (see eq. (2.1)),

BLACK-BODY RADIATION 25

In traditional plasma spectroscopy, the term Saha equilibrium has been, andstill is, used to describe the density ratio of the "atoms" (z — 1) and the "ions" z,when complete LTE is assumed for this system. The "atom density" is the sum ofall the "atomic" level populations,

The summation in the r.h.s. (right-hand side) of this equation is called the partitionfunction, and denoted as Bz_i(T&). We define the "ion" density Nz in a similarmanner by introducing the partition function of the ions Bz(Te), Then, the densityratio of the "atoms" and the "ions" is given as

It is readily seen that, because of the nature of the statistical weight, g(p) = 2p2

for the case of hydrogen atoms and hydrogen-like ions, the partition functionsdiverge. This difficulty comes partly from eq. (2.8) itself, i.e. the notion that all thepopulations in excited levels of atoms belong to the "atom." We will see inChapters 4 and 5 that this understanding is rather unrealistic; excited atoms arestrongly coupled to ions rather than to the ground-state atoms. The difficulty ofthe divergence itself will be resolved toward the end of Chapter 9.

One important fact is noted here: in deriving eq. (2.6) we "filled" the densityof states for free electrons, eq. (2.5a), with the Boltzmann distribution, eq. (2.4),with appropriate statistical weights incorporated. Equation (2.6) itself includesthe Maxwell distribution function, eq. (2.2). This fact clearly indicates that theMaxwell distribution is nothing but the Boltzmann distribution extended overthe free-electron states. The normalization factor in eq. (2.2) or eq. (2.2a) makesthis point less obvious. This aspect will be further examined in Chapter 9. Seeeq. (9.19a).

2.2 Black-body radiation

We consider an ensemble of ions having lower level p and upper level q, and theradiation field, the wavelength of which corresponds to the transition energybetween these levels. The temporal development of the upper-level population n(q)is given by the rate equation

where /„ [Wm 2sr :s] (sr means steradian or a unit solid angle) is the spectralintensity of the radiation field at the transition frequency v = E(p, q)/h. Figure 2.1illustrates the situation of eq. (2.10). The quantity Iv is called the spectral radiance

26 THERMODYNAMIC EQUILIBRIUM

FIG 2.1 The emission-absorption processes of atoms in a radiation field.

in radiometry. It should be noted that we have assumed that, in eq. (2.10), theradiation field is isotropic and has virtually a constant intensity over the lineprofile of the transition.* The first term represents excitation of the upper-levelions by absorption of photons, and the second and third terms denote deexcitationby spontaneous transition and by induced emission, respectively. A(q,p), B(p, q)and B(q,p) are called Einstein's A and B coefficients. We further suppose that thissystem is surrounded by "mirror" walls that reflect radiation and ions completely.After a sufficiently long time a stationary state is reached and the time derivativevanishes. Then we have

On our above assumptions, we may expect that our system is in thermo-dynamic equilibrium. Then, the population ratio should be given by the Boltzmanndistribution, eq. (2.3),

* Equation (2.10) cannot describe the atomic system in a radiation field that violates theseconditions. This is the case for radiation fields encountered in many practical cases; an extremeexample is the field produced by a laser beam. This field is highly directional, monochromatic, andsometimes polarized. In such cases we have to employ an alternative approach by introducing theconcept of an absorption cross-section for the transition, as defined by eq. (3.9) later.

t Equation (2.10) is sometimes written in terms of the spectral energy density, (4w/c)Iv, in place ofthe spectral intensity Iv. Then the second relationship of eq. (2.12) takes a different form.

We note the interrelationships between the coefficients A and B*

This distribution is called the Rayleigh-Jeans law, and is of an entirely classicalnature as is understood from the absence of h. This distribution diverges with v.At the other extremum, for hv ;$> kT, we obtain

Equation (2.18) is called the Stefan-Boltzmann law.

with


Substitution of eqs. (2.12) and (2.13) into eq. (2.11) leads to the intensity of theradiation field,

As has been noted the radiation field is assumed isotropic, i.e. no angulardependence and unpolarized. We have derived eq. (2.14) for a particular frequencyregion corresponding to the transition p <-> q. We may readily extend the aboveargument to other regions by introducing other transition frequencies, and finally,to obtain eq. (2.14) for the whole spectral range. Equation (2.14) is called Planck'sdistribution or the black-body radiation. Figure 2.2(a) shows examples of theblack-body radiation for several temperatures.

Several properties of the black-body radiation are discussed. It has a maximumintensity at a certain frequency, depending on the temperature. The frequencywhich gives the maximum is readily obtained from the derivative of eq. (2.14),

See Fig. 2.2(a).In the low-frequency region oihv^_kT, eq. (2.14) reduces to

The energy density of the black-body radiation contained in a unit volume is(^/c)Bv(T)dv, in units of [Jm~3], and the total energy density is

FIG 2.2 Planck's distribution of the black-body radiation: (a) eq. (2.14);(b) eq. (2.14a). The region of visible light is indicated. Note the relationshipof eqs. (2.15) and (2.15a), respectively, as indicated with the dotted lines.

where T is measured in [K]. The relationship of eq. (2.15) or (2.15a) is calledWien's displacement law. We take as an example the surface of the Sun;its temperature is 5770 K. From eq. (2.15) we have j/max~3.36 x 1014 s^1, orA ~ 891 nm, in the near-infrared. From eq. (2.15a) we have Amax ~ 503 nm, a bluecolor. See Fig. 2.2(a) and (b), respectively.


It is sometimes convenient to rewrite eq. (2.14) in terms of wavelength A,

Figure 2.2(b) shows the distribution, eq. (2.14a), for several temperatures. Thepeak wavelength in this expression is given by

3

ATOMIC PROCESSES

In a plasma, atoms and ions undergo transitions between their quantum statesthrough radiative and collisional processes. Among these processes, the mostimportant are spontaneous radiative transitions and collisional transitionsinduced by electron impact (collisions). In the following, we review these pro-cesses. In doing so we emphasize two points:

1. The radiative transition probability and the collision cross-section are notunrelated nor independent. Rather, they share some common tendenciesthrough a parameter called the absorption oscillator strength.

2. Various properties of high-lying states continue smoothly across the ionizationlimit to those of the low-energy continuum states. This fact is reflected in theatomic processes involving these states.

Figure 3.1 shows schematically transitions which are included in our theory insubsequent chapters. These transitions are:

In the above, we have assumed that levels p and q are of the atoms or ions(sometimes called "ions" for the purpose of simplicity) in the ionization stage(z — 1) and that z is the ground state of the ions in the next ionization stage, e in theinitial state (left-hand side or l.h.s.) means the incident electron inducing thetransition and in the final state (the right-hand side or r.h.s.) is the scatteredelectron(s). hv is a photon with frequency v emitted in the transition. The symbolsshown above the arrows represent the rate constants for the transitions. A rateconstant represents quantitatively the likelihood of that reaction taking place in aplasma. It is noted that A(q,p) is a probability and has units of [s^1], C(p,q),F(q,p), (3(p), and S(p) are called the rate coefficients and have units [m3 s^1], and

spontaneous radiative transition

excitation by electron impact

deexcitation by electron impact

radiative recombination

ionization by electron impact

three-body recombination

RADIATIVE TRANSITIONS 31

FIG 3.1 The transitions included in the rate equation of populations insucceeding chapters.

a(p) is also called the rate coefficients and has units [m6 s^1]. On the right, the pairof arrows connected by the vertical thin line indicates that these two transitionsare inverse processes to each other so that their rate coefficients are relatedinternally by a thermodynamic relationship. The transitions connected by thethick lines have properties which are common in their natures, as will be discussedin the following.

3.1 Radiative transitions

The probability of a spontaneous radiative transition q^p + hv has been intro-duced already in Section 2.2 as Einstein's A coefficient, which is simply called thetransition probability. This is given in terms of the absorption oscillator strength fp>?,

The absorption oscillator strength is a measure of the ability of the atom in state pto absorb light in making the transition p + hv —> q. This is defined by

with the electric dipole matrix element, or the dipole moment

Here, ^>p and if}q are the wavefunctions and r is the position vector of the electrontaking part in this transition, the optical electron. The radiative transition of thistype is called the electric dipole transition, or the optically allowed transition.Other kinds of transitions, e.g. the electric quadrupole or magnetic dipole tran-sitions, also exist. These optically forbidden transitions are usually quite weak: for

32 ATOMIC PROCESSES

example, electric quadrupole transitions have oscillator strengths (the definitionof which is different from eq. (3.3)) smaller than those of dipole transitions byabout 10~7. We neglect these optically forbidden transitions in the followingdiscussions. We now take neutral hydrogen for the purpose of illustration. Thewavefunction is expressed as tfjnim(t") = Rni(r)Yim(0,<f>) with the radial part Rni(r)and the spherical harmonic Yim(0, <f>) for the angular part.* The angular part isfurther written as Yim(0,(f>) = (l/V2jr')Pf (cos6>)eim</>, where Pf(cosff) is the asso-ciated Legendre function. Figure 3.2(a) illustrates examples of Pf(cosO) forseveral small values of /'s and m's, and the factor em4> to be multiplied. If we takethe ground state Is as the initial state, the oscillator strength has non-zero valuesonly for the final states np. See Fig. 1.1 l(b) and Table 3.1. This selection rule stemsfrom the integration of r with the two spherical harmonics over the angularcoordinate. It is obvious from Fig. 3.2(a) that the s (1=0) and d (1=2) statewavefunctions are spatially even functions, and r is an odd function, so that thematrix element vanishes for the initial Is state and the final s or d states. For the f(/= 3) and still higher-/ final states, the reasoning of the vanishing matrix elementis more involved. The selection rule that A/ cannot be larger than 1 may also beunderstood from the fact that the photon carrying away the energy difference ofthe initial and final states has unit angular momentum h. Figure 3.2(b) shows theradial wavefunction as the form rRB/(r) (in atomic units, in which e = m = h=\)}In the integration (3.3) for p= Is and q = np, apart from the integration over theangular coordinate, which is common to all the np states, the magnitude of thematrix element, or of the oscillator strength, is approximately given by the degreeof "overlapping" of the radial wavefunctions of the initial and final states. As iseasily seen in Fig. 3.2(b) the main contribution to the integral of the radialfunctions comes from the first peak of the np wavefunction. With an increase in nthe amount of the overlap decreases, and the oscillator strength /is>Bp decreases.Table 3.1 shows several examples of the radial integrals of r (squared) and the

* In this chapter until eq. (3.6), p or q is understood to stand for nl in the example of hydrogen-likeions. It is noted that, in the r.h.s. of eqs. (3.2) and (3.2a), we ignored the presence of m, or thedegeneracy of the levels. The correct expression of eq. (3.2) is

where p stands for «'/'. Each of the above matrix elements corresponds to the transition from one ofthe cells in Fig. l.ll(a) to another. For example, for/2p3d these transitions are between the three cellsof n' = 2, /' = 1 and the five cells of n = 3 and / = 2. Here we neglect the presence of the electron spin.Therefore, within this framework, we have g(n'l') = 3. The expression ( q \ r \ p ) 2ineq. (3.2) should beunderstood to be the averaged value of (nlm \ r n'l'm') 2 over the lower and upper levels, i.e.J5)i5)EL=-/EL=-f \(nlm\r\n'l'm')\2. Note that, in reality, each cell in Fig. l.ll(a) is doublydegenerate owing to the electron spin and this degeneracy should be taken into account both in thesummations and the statistical weights. The resulting oscillator strength value is unchanged.

t The radial wavefunction for an s state, Rns(r), tends to a finite value for r^O, while otherwavefunctions, Rni(r) with /^ 0, tend to zero as can be seen from the straight line starting from zero forthe former case and the finite curvature for the latter.


FIG 3.2 Several examples of the wavefunctions of atomic hydrogen, (a) Theassociated Legendre function Pf(cosff) for several / and m. Pficosff)multiplied by eim<t> yields the spherical harmonics (apart from the normalizationfactor), Yim(0, <j)), which is the angular part of the atomic wavefunction. (b) Theradial wavefunction of ns, np, and a few nl states in the form of rRni(r), for

34 ATOMIC PROCESSES

oscillator strengths. The above feature is clearly seen. It is worth noting that theasymptotic value of both the quantities for large n is proportional to n~3. It maybe interesting to find that this factor has already appeared in eq. (1.5), the energywidth allocated to a level having principal quantum number n.

Equation (3.2a) suggests that, in the case of E(p, q) < 0, or when the final statelies below the initial state (p > q), the oscillator strength takes a negative value. Inthis case, eq. (3.2a) is called the emission oscillator strength and is defined (forE(q,p)<Q)by

For the final states, we have to include the continuum states. The transition to thecontinuum states is photoionization, which is the inverse process to radiativerecombination, and we treat these processes in the next section. Table 3.1 illus-trates the sum rule: for the initial state Is, for example, the oscillator strength sumover the discrete states is 0.565 and that over the continuum states is 0.435,satisfying the rule. For an excited initial state the sum rule is obeyed with theemission oscillator strengths included. See the state 2p, for example.

Starting from the following chapter, we consider ions (and atoms, of course) ina plasma. Electrons and ions play a major role in inducing transitions between thelevels in these ions. As we will see later, they tend to induce transitions between thelevels that have smaller energy separations. In the case of neutral hydrogen andhydrogen-like ions, the states having a common principal quantum number n arealmost completely degenerate. As a result, the populations in these individualstates with different angular momenta / tend to be distributed according to theirstatistical weights, 2(27 + 1). In other words, in Fig. 1.11 (a), the atomic electrontends to enter into the cells in the same horizontal plane with equal probability. As

n<4. The ordinate is in atomic units, (c) The radial wavefunctions of thecontinuum KS states with E=0, 0.25R, and R in the same format as in (b). Notethat some of the wavefunctions, Rn/(r) Yim(6, <f>), with n < 3 and / < 2 in (a) and (b)correspond to the classical states depicted in Fig. 1.1 l(a).

In the example of the ls^2p transition eq. (3.4) gives/2pjis = —(2/6)/1Sj2p, whichis actually seen in Table 3.1. This table also shows other examples, e.g./3Sj2p,/3pjis.

An important property of the oscillator strength is the sum rule. For a parti-cular initial state the summation of the oscillator strengths, absorption or emission,of transitions to all the final states is equal to the number of electrons participatingin these transitions. In the case of the hydrogen atom and hydrogen-like ions thisnumber is 1:

TABLE 3.1 (a) Squares of the radial integralfrom Bethe and Salpeter (1977).)

Initial

Final

«=12345678n = 9 to oo

togetherAsymptotic

Discretespectrum

Continuousspectrum

Total

Is

np

_

1.6660.2670.0930.0440.0240.0150.0100.032

4.7/T3

2.151

0.849

3.000

2s

np

_

27.009.181.660.600.290.170.100.31

44.0«~3

39.30

2.70

42.00

2p

«s

1.6727.000.880.150.0520.0250.0140.0090.025

3.7«-3

29.820

0.180

30.00

nd

_-

22.522.920.950.410.240.150.42

58. 6«~3

27.62

2.38

30.00

\(nl\r

3s

np

_

0.9162.029.9

5.11.90.90.51.4

169«"

n'l'}\ = (I RniRn>i>r3dr) for neutral hydrogen in atomic units [afc]. (Adopted

3p 3d

«s nd np

0.39.2 - 22.5

162.0 101.2 101.26.0 57.0 1.70.9 8.8 0.230.33 3.0 0.080.16 1.4 0.030.09 0.8 0.020.22 2.0 0.05

~3 28«~3 248«~3 5/T3

202.56 179.18 174.54 125.88

4.44 0.82 5.46 0.12

207.00 180.00 180.00 126.00

nf

_--

104.711.03.21.40.81.8

198/T3

122.85

3.15

126.00

4s

np

_

0.156.0

540.072.611.95.72.14.3

445«~3

642.7

5.3

648.0

4p

«s

0.091.66

29.9540.021.22.91.40.61.0

102«~3

598.7

1.3

600.0

nd

_-1.7

432.0121.9

19.37.73.25.9

655/T3

591.7

8.3

600.0

4d

np nf

_ _2.9

57.0432.0 252.0

9.3 197.81.3 26.90.5 8.60.2 3.90.3 6.9

33«~3 687«~3

503.50 496.0

0.50 8.0

504.00 504.0

nd

_-

104.7252.0

2.750.320.080.040.07

6«-3

359.95

0.05

360.0

4f

«g

_---

314.027.6

7.33.04.5

393/T3

356.4

3.6

360.0

2 2

TABLE 3.1(b) Oscillator strengths for hydrogen. (Adopted and modified from Bethe and Salpeter (1977).)

Initial

Final

n = l2345678n = 9to oo

AsymptoticDiscretespectrum

Continuousspectrum

Total

Is

«p

_

0.41620.07910.02900.01390.00780.00480.00320.0109

1.6n~3

0.5650

0.4350

1.000

2s 2p

np ns nd

-0.139-

0.4349 0.014 0.6960.1028 0.0031 0.1220.0419 0.0012 0.0440.0216 0.0006 0.0220.0127 0.0003 0.0120.0081 0.0002 0.0080.0268 0.0007 0.023

3.7n~3 O.ln~3 3.3n~3

0.6489 -0.119 0.928

0.3511 0.008 0.183

1.000 -0.111 1.111

2

n

-0.104-

0.6410.1200.0450.0220.0120.0080.024

3.5n-3

0.769

0.231

1.000

3s

np

_

-0.041-

0.4840.1210.0520.0270.0160.048

6.2n~3

0.707

0.293

1.000

3p

ns

-0.026-0.145

-0.0320.0070.0030.0020.0010.002

0.3n~3

-0.121

0.010

-0.111

nd

_

--

0.6190.1390.0560.0280.0170.045

6.1n~3

0.904

0.207

1.111

3d

np

_

-0.417-

0.0110.00220.00090.00040.00020.0007

0.07n~3

-0.402

0.002

-0.400

nf

_

--

1.0160.1560.0530.0250.0150.037

4.4n~3

1.302

0.098

1.400

3

n

-0.009-0.285

-0.8410.1500.0560.0270.0160.042

5.3n~3

0.840

0.160

1.000

4s

np

_

-0.009-0.097

-0.5450.1380.0600.0330.082

9.3n~3

0.752

0.248

1.000

4p

ns

-0.010-0.034-0.161

-0.0530.0120.0060.0030.006

0.7n~3

-0.126

0.015

-0.111

nd

_

--0.018

-0.6100.1490.0630.0330.075

9.1n~3

0.912

0.199

1.111

4d

np

_

-0.073-0.371

-0.0280.0060.0020.0010.002

0.3n~3

-0.406

0.006

-0.400

nf

_

---

0.8900.1870.0720.0370.081

8.6n~3

1.267

0.133

1.400

4f

nd

_

--0.727

-

0.0090.00160.00050.00030.0006

0.05n~3

-0.715

0.001

-0.714

ng

_

---

1.3450.1830.0580.0270.045

3.5n-3

1.658

0.056

1.714

4

n

-0.002-0.030-0.446

-1.0380.1800.0650.0320.066

6.8n~3

0.876

0.124

1.000


we will see later (Appendix 4A), this situation is actually realized under many condi-tions of practical interest. From now on, for neutral hydrogen and hydrogen-likeions, we assume this situation actually to be the case. Then, we may bundle theseindividual m and / states into one level, which is designated only by its principalquantum number. Figure 1.6 represents the reduced energy-level diagram. Theoscillator strength in this scheme is given in a similar manner to the expression inthe first footnote in p. 32:

where gbb is the Gaunt factor, or the quantum correction factor. The subscript"bb" means the bound-bound transition. Here "bound" is identical to "negativeenergy" and denotes the discrete levels. A few examples of the Gaunt factors areshown in Fig. 3.3 for the case of p= 1; their magnitudes are of the order of 1.Examples of the oscillator strengths are shown in Fig. 3.4 for p= 1,2, , 15.Two important features are seen:

1. With an increase in q of the upper level, fPA tends to be proportional to q~3. Wehave already seen this feature in Table 3.1. See also eq. (1.5). This feature willlead to important consequences later.

2. For the transition to the adjacent higher-lying level, p—>(p + 1), the oscillatorstrength is the largest in this series (remember the arguments concerningFig. 3.2(b)), and is well approximated byfp^p+1 ~ (/? + l)/5, or even by

In the following we use p or q in place of n' or n to denote the principal quantumnumber specifying the level. In this convention the oscillator strength is expressedby a simple formula which is based on the classical Kramers formula

as is seen in Fig. 3.4.

The transition probability is expressed from eqs. (3.1) and (3.6a) as

where rq is the period of one revolution of the electron of the upper level ion,eq. (1.4), in the Bohr atom picture. It is interesting to note that this expression

38 ATOMIC PROCESSES

FIG 3.3 An example of the Gaunt factors for the bound-bound transitions(Is—np) and the bound-free (Is—/cp) transitions for hydrogen. The lower levelis the ground state p = ls.

consists of the fine structure constant, the ratio of the typical atomic energy to theelectron rest-mass energy, the frequency of the classical orbit motion, and otherquantities of the order of 1 or smaller. Figure 3.5 shows several examples of thetransition probabilities. For an initial level g> 1, and for a low-lying final levelp (<^q), A(q,p) is approximately proportional to p~l. For a closely lying levelp (~q) it is approximately proportional to (,q—p)~l, as seen in eq. (3.8).

As is suggested by the term "absorption" oscillator strength, this quantityrepresents the ability of an atom in level p to absorb a photon, making thetransition p —> q. We may call this process photoexcitation. For various reasons(see Chapter 7) the absorption line is not monochromatic, rather it has a profilewith a finite width. Figure 3.6 shows an example of the photoabsorption spectra ofatoms; the initial state is the ground state of neutral helium.* The transitionsshown in this figure are indicated with the dashed lines in Fig. 1.4.

* Actually, this spectrum is the energy loss spectrum of high-energy electrons, 2.5 keV, passingthrough a dilute helium gas. The energy loss spectrum in this energy range is almost exactly equivalentto the photoabsorption spectrum of the same initial state, as can be understood in the discussion laterin Section 3.3. In the present spectrum, however, the line profile is determined by the resolution of themeasurement apparatus, 55 meV, including the energy spread of the electron beam. This is inconsistentwith our assumption of the line profile in the text. However, this inconsistency leads to no difficultyin our discussion to follow. Therefore, we regard this spectrum as the photoabsorption spectrum,averaged over this energy width.


FIG 3.4 The absorption oscillator strength f p q , eq. (3.6a), for several transitionsof hydrogen. The approximation of eq. (3.7) is shown.

We may express the characteristics of an absorption line in terms of theabsorption cross-section ap<q(v). A cross-section is a quantity that representsquantitatively the likelihood of the reaction to take place; it has dimensions ofarea. It may be imagined that if the target ion has that area then a reaction,photoexcitation in this case, takes place when the projectile, a photon in this case,hits that area. Then, we have

As can be seen in Fig. 3.6 we have a series of absorption lines, which we call theseries, corresponding to the states, w1? in this example (Fig. 1.4). For absorption,say, from level p to one of the levels q we now consider an "averaged" absorptioncross-section over the energy width allocated to this line. The integration over theline JVdz/ is replaced by (a) • Az/A?= i, where Az/A?= i is the frequency width and

FIG 3.5 The spontaneous transition probability for hydrogen, eq. (3.8). In (a) the approximation in eq. (4.13) is comparedwith the exact values. In (b) also shown is the radiative recombination rate coefficient to various levels for very lowtemperature, Te= 103 K.


Energy loss (eV)

FIG 3.6 Electron energy loss spectrum of neutral helium in the ground state. Thisis almost equivalent to the photoabsorption spectrum of Is2 1S—Iswp :P andIs2 :S—ISKP :P, where Kp means the continuum p states. In the vicinity of theionization limit the continuation from the series-line absorption to thecontinuum absorption, corresponding to the continuation from eqs. (3.9b) toeq. (3.13), is clearly seen. (Quoted from Liu et al. (2001); copyright 2001, withpermission from The American Institute of Physics.)

is given by (2z2R/hq3), See also eq. (1.5). Then from eqs. (3.9) and (3.6a) we havefor hydrogen-like ions

In this expression, note that the relationship of eq. (1.2) appears. It is also notedthat this "averaged" absorption cross-section depends on q only through hv andthat the expression in the second parentheses tends to 1 toward the series limit.This means that, even if we average the cross-sections over a larger frequencywidth, or the energy width, eq. (3.9b) is valid. This is actually seen in Fig. 3.6, whichis for neutral helium, as a small nearly constant cross-section value for large nvalues up to the ionization limit n—>oo, i.e. £"=24.59 eV. See Fig. 1.4. Equation(3.9b) suggests that this almost constant value continues to still lower levels, whichis seen to be actually the case in Fig. 3.6. This property leads to important con-clusions later.

42 ATOMIC PROCESSES

where qe represents the continuum state. Figure 3.2(c) shows examples of thecontinuum wavefunctions for the es states.* It is seen in Fig. 3.2(b) that, startingfrom the low-energy discrete states, with an increase in energy, the number ofnodes of the radial wavefunction increases. The overall "shape" of the wave-function changes smoothly from the discrete states ns (E < 0) across the ionizationlimit (£"=0) to the continuum states es (E>0). For the transitions between thediscrete levels, eqs. (3.1)-(3.3), if we take/>= Is as an example, we have seen thatthe dominant contribution to the matrix element, eq. (3.3), comes from theoverlapping of the Is wavefunction with the first peak of an «p wavefunction. SeeFig. 3.2(b). This is also true for the continuum ep wavefunctions, the "shape" ofwhich can be imagined from the ns and es wavefunctions. This continuationproperty has significant consequences, which will be discussed later.

The final states are continuously distributed in energy, and we consider pho-toionization into the final states within the energy width de centered at e. Thecross-section is sometimes expressed in terms of the differential oscillator strength

with d;/ corresponding to de = h dv. Note that this expression is quite similar toeq. (3.9).

* Two points are noted here. Since the upper level qc is in the continuum states the wavefunctionq£) is normalized over a unit energy interval, so that (p r q£)

2 has units of [m2 J~']. Equation(3.10), like eq. (3.2), ignores the presence of degeneracy. The correct expression is

where the summations over [p] and [qe] are understood to be summations over all the magneticsublevels of the lower level p and the upper level qe.

3.2 Radiative recombination

For the purpose of deriving the cross-section for radiative recombination we startwith its inverse process, i.e. photoionization.

Photoionization

Figure 3.7 illustrates the photoionization process of an ion in state/? by absorbinga photon having frequency v. The final state is one of the continuum states havingenergy e. The photoionization cross-section is given as

RADIATIVE RECOMBINATION 43

FIG 3.7 The schematic diagram for explanation of photoionization and radiativerecombination.

In the case of hydrogen-like ions the formula, eq. (3.6a), can be extended tothat for photoionization; the "principal quantum number" of the final state isimaginary, so that q is replaced by i/c, with real K, and gbb by gw, the bound-freeGaunt factor. Here "free" means the levels in the continuum states with positiveenergy. See Fig. 3.7. (i^)3 is replaced by K3.

An example of the bound-free Gaunt factor is shown in Fig. 3.3 for p=\. Thisformula represents the absorption oscillator strength for the final states lyingwithin width d/c = 1. Therefore, we have

dfp,e = fp,KdK

and

From eqs. (3.10a)-(3.12) we have

44 ATOMIC PROCESSES

In deriving the last line we have utilized the relation hv = z2R(p 2 + K 2). Figure 3.8shows examples of photoionization cross-sections for several levels of neutralhydrogen, z= 1; the cross-section has a sharp threshold and above that it is givenby eq. (3.13). It is noted here that eq. (3.13) is exactly the same as eq. (3.9b),the averaged cross-section for photoexcitation over the series lines, except forthe Gaunt factor. As Fig. 3.3 shows, with an increase in the final state energy, thisfactor continues smoothly from the discrete states, g\,\,, across the series limit orthe threshold for photoionization at (hv/z2K) = 1, to the continuum states, gbt. Animportant conclusion is thus reached: the magnitude of photoabsorption of serieslines from a particular lower level continues smoothly across the series limit or theionization limit to the photoionization continuum absorption from the same level.There is no break of absorption spectrum at the threshold of photoionization, asmight be imagined from Fig. 3.8. This feature is clearly seen in Fig. 3.6; i.e. nobreak is seen at the energy of 24.59 eV, corresponding to the ionization limit. Inthis figure actual cross-section values of neutral helium are estimated from theordinate value of about 0.07 (eV^1) to be 7 x 10~22 m2 from eqs. (3.9) and (3.10a).This value is close to the threshold photoionization cross-section of the Ishydrogen in Fig. 3.8.

FIG 3.8 Examples of the photoionization cross-sections from some of the low-lying levels of neutral hydrogen.


From eq. (3.12) we may define the integrated oscillator strength for photo-ionization,

Radiative recombination cross-section and rate coefficient

In Fig. 3.7, an ion in level p in ionization stage (z — 1) may be photoionized by aphoton with frequency i/, producing a pair of a ground state ion in ionizationstage z and an electron having energy e. In a plasma, suppose there are nz_i(p)ions present per unit volume and they are photoionized by a radiation field,which is isotropic and has intensity Iv&v at v over the frequency width dz/. Thenumber of photoionization events per unit time in unit volume (we call thisquantity the photoionization flux, which has units of m^3s^1) is given asnz-\(p)\^TfIv/hv\(Tptl,(y)d.v. The inverse process to photoionization is radiativerecombination, in which a ground state ion in ionization stage z captures anelectron having energy e to form an ion in level p in ionization stage (z — 1) byemitting a photon v. If we introduce the radiative recombination cross-sectionffe,P(s) the number of events in the plasma, or the radiative recombination flux,is given as nz(l)nef(s)<je^p(s)vds* Here the energy width de corresponds to the

* A target has area <JE:f(e) for this reaction, and if we suppose that the motion of electrons isunidirectional with speed u, then the number of collisions in one second on this target is ne<jEtp(s)v. Thenumber of targets per unit volume is nz(l). Thus, the number of collisions per unit volume per unit timeinducing this reaction is nz(l)neij£ip(e)v. Even if the velocity distribution is isotropic, this expression isvalid. This quantity is called the radiative recombination flux. Since the electron speed is distributed wereach the expression in the text. Note that this quantity is proportional to nz(l) and ne, the densities ofthe reacting agents.

where VQ = (z2R/hp2) is the threshold frequency for photoionization. This oscil-lator strength is nothing but the oscillator strength sum over the continuum statesmentioned with regard to the sum rule, eq. (3.5). From eq. (3.13) together witheq. (3.10a) we may obtain

where (gb{) is the Gaunt factor averaged over the continuum states. Table 3.1bcontains/p>c for/? with different / levels resolved. It may be interesting to note that,in eq. (3.14), the dominant contribution tofp>c comes from the differential oscil-lator strength (dfpf/dv), or the photoionization cross-section (eq. (3.10a)), in theenergy region close to the ionization threshold. See Fig. 3.8 and eq. (3.13).

46 ATOMIC PROCESSES

frequency width dz/. If we assume thermodynamic equilibrium for our plasmathese processes should balance with each other so that these fluxes should beequal:

which is called Milne's formula. Thus the cross-section for radiative recombina-tion to produce a hydrogen-like ion p is obtained from eq. (3.13):

* Actually a nucleus like a proton or a deuteron could have an internal structure stemming from theexistence of nuclear spin. However, the statistical weight due to this structure is common to an ion zand an atom (z — 1), so that it does not affect eq. (3.17).

where gz(l) = 1 has been used, since the bare ions including protons have nointernal structure.* It is noted that the radiative recombination cross-section hasno threshold energy, and it diverges toward the null energy. For low-energyelectrons of e <C z2R/p2, the cross-section is proportional to p~le~l, and for highenergies of e ;$> z2R/p2, it is proportional to p~3s~2. It may further be noted that,in these limiting cases, the cross-section is proportional to z2 and z4, respectively.These positive dependences on z are in sharp contrast to the negative dependenceof the area of the electron orbit of the bound state; it is proportional to z~2 as seenfrom eq. (1.2).

As noted already, for a beam of electrons having a speed v, the radiativerecombination flux is nz(l)ne<je>p(s)v. The magnitude of this flux divided by nz(l)ne

is called the radiative recombination rate coefficient. In the present case, the rate

where we have included on the l.h.s. the exponential factor in order to account forthe process of induced emission, just like in the case of transitions between discretelevels. See eq. (2.10) with eq. (2.13). The subscript E is understood to mean thatthis equation holds in thermodynamic equilibrium.

Since we assume thermodynamic equilibrium the ionization ratio should begiven by the Saha-Boltzmann equation, (2.7), and the radiation field is given bythe Plancks' distribution, eq. (2.14). By substituting these equations and eq. (2.2)into eq. (3.16) we obtain


coefficient is av. Likewise, for plasma electrons having energy distribution f(E)dE,the radiative recombination rate coefficient is given as

with the exponential integral

FIG 3.9 The radiative recombination rate coefficient j3(p) for several levelsof neutral hydrogen. The arrows show the temperature at which kTe = x(p)-The temperature dependence for low Te and for high Te are shown. See alsoFig. 3.5(b).

3.3 Collisional excitation and deexcitation

Excitation cross-section

We consider excitation of an ion (atom) by electron impact. For the initial andfinal states of the ion/? and q, respectively, the excitation process may be written asp + e^q + e, where e stands for the incident or scattered electron. See Fig. 3.1.For an incident electron, the likelihood of this process to take place depends on itsenergy, which must be higher than the excitation threshold, and is expressedquantitatively in terms of the excitation cross-section. Figure 3.10 shows severalexamples of measured or calculated excitation cross-sections for two transitions:(a) for 1:S + e—>2 1P + e of helium-like iron, or 24 times ionized iron ion, and(b) for the corresponding transition of neutral helium (see Figs. 1.4 and 3.6). Thesecorrespond to the resonance lines. In Fig. 3.10(a), starting from the excitationthreshold of 6.7 keV, several theoretically determined cross-sections are plottedwith small symbols. Let the incident and scattered electrons have momentum hk0

and hka, respectively. The final state of the scattered electron is expressed as aspherical wave : [/(/> —> q; 9, <f))/r]exp(ikar). If the incident wave is normalized sothat its flux is 1, the excitation cross-section is defined by

48 ATOMIC PROCESSES

In deriving eq. (3.19a) we have assumed the Maxwellian distribution, eq. (2.2a),for/(e), and gbf = 1- The exponential integral is well approximated for a small orlarge argument by

where 7 = 0.5772 is Euler's constant. Figure 3.9 shows examples of the radiativerecombination rate coefficients for neutral hydrogen calculated from eq. (3.19)without the assumption of gbf= 1. Corresponding to the above limiting cases ofthe cross-section the rate coefficient has the following dependences: for lowtemperatures of kTe <C z2R/p2, we have (3(p) <xp~lT~°-5, and for high tempera-tures of kTe > z2R/p2, we have (3(p) <xp~2-5T~1-5, where we have approximatedln(l/jc) to 0.7(l/^)°'5 for (1/x) of the order of 10.

The simplest approximation to calculate the cross-section is to express theincident electron as a plane wave for a neutral atom target, or a Coulomb wave foran ion target, and treat the interaction between the electron and the ion (atom) asa small perturbation. Only the first-order matrix element of the interation

FIG 3.10 Examples of excitation cross-sections, (a) For the transition 11S^21P of helium-like iron. Results of thetheoretical calculations with various sophistications are shown. An experimental result is shown with the closed circle. Seethe text for details. (Quoted from AMDIS.) (b) For 1 :S —> 2 :P of neutral helium. Several examples of the results of themost sophisticated calculations and recent experiments are shown. See the text for details. (Quoted from Goto 2003;copyright 2003, with permission from Elsevier.)

50 ATOMIC PROCESSES

potential is retained. This method is called the Born approximation for a neutraltarget and the Coulomb-Born approximation for an ion target. In Fig. 3.10(a) theCoulomb-Born approximation gives the largest cross-section just above theexcitation threshold. The incident electron can "kick out" one of the target electronswhile being captured by the target. This process of replacement of the two electronsis called exchange. The cross-section calculated with exchange taken into account(Coulomb-Born-Oppenheimer approximation) is the next largest result. Thismethod, however, sometimes gives incorrect results, and further improvementsare needed. The result of one such method (the Coulomb-Born-Oppenheimer-Ochkur-Rudge-Bely approximation: the reader doesn't need to be bothered bysuch nomenclature) is still smaller. Another modification is to use more accuratewavefunctions for incident and scattered electrons, as contrasted to the Coulomb-Born approximation. This is called the distorted-wave approximation. Severalresults of this method, some of which include some further sophisticated mod-ifications, are shown as a group of cross-sections having the smallest values.

For a long time, experimental confirmation of the theoretical calculation wasnot obtained for highly ionized ions, because it was virtually impossible to pro-duce a strong enough beam of such highly ionized ions to enable a crossed beamexperiment to be performed. Recently, a device called an EBIT (electron beam iontrap) has been developed: highly ionized ions are created and held in a small spaceby a magnetic-electric trap with the help of a high-current electron beam, whichexcites the ions. From the observation of the emission radiation of the xuv(extreme ultraviolet) line (A = 0.185 nm in this example), the cross-section isdetermined for this highly ionized iron, as shown by the closed circle in Fig. 3.10(a).The electron beam both ionizes and sustains the ions in the trap, and the beamenergy cannot be changed freely. So, the cross-section is obtained only for oneenergy. This data is compared with the results of the sophisticated theories. Theagreement is not perfect, but considering the difficulty of this experiment,especially in obtaining the absolute value, we may regard this agreement assurprisingly good.

Now we look at Fig. 3.10(b). This is for the transition of neutral heliumcorresponding to the ion transition we discussed above. Helium is the mostthoroughly studied atomic species, and this transition is the resonance transition.This figure contains only very sophisticated calculations and recent experiments.One method of calculating a cross-section is the close-coupling method. In thismethod, the wavefunctions during the collision process are expanded in terms ofthe atomic eigenstate wavefunctions, and the coupled equations describing thecollision process are solved numerically. The number of atomic states is as large aspractically possible, e.g. 15 or 29. The convergent close-coupling method is afurther extension, in which "all" the atomic states, including the continuum states,are virtually taken into account. The result of this calculation is shown in thisfigure with the closed circles. Another sophisticated method is the R-matrixtheory, in which the atomic wavefunction is treated differently in the core regionand the outer region, making it possible to increase the number of atomic states

COLLISIONAL EXCITATION AND DEEXCITATION 51

included. The result of a further sophistication, the R-matrix with pseudo-states, isshown with the dotted line in this figure.

There are two types of experiments for determining the cross-section of neutralatoms. The first is, by hitting a dilute helium gas in a cell or a helium beam withthe electron beam, we determine the number of excitation events by counting thenumber of photons emitted by the excited atoms. The diamonds in Fig. 3.10(b)give the result. Another technique is to count the transmitted electrons with therelevant energy loss, 21.2 eV in this example. Figure 3.6 is an example of thisenergy loss spectrum, where the incident electron energy is quite high, 2.5 keV, thehigh-energy end of Fig. 3.10(b). In the experiment determining the cross-sectionfor 1 :S —> 2 1P, the magnitude of the first peak in Fig. 3.6 is measured. The resultsof two experiments of this type are shown with the open triangles and the squares.Agreement among the data is good, except in the region immediately above theexcitation threshold.

As can be seen in Fig. 3.10(a) and (b), these cross-sections for the ion and theatom have rather similar energy dependences. But, the reader may have recognizedan important difference. For the neutral atom, with the energy approaching thethreshold, the cross-section value tends to diminish, while for the ion it tends to afinite limit. This is a quite universal tendency. This point is more explicitly shown inFig. 3.11 (a): for excitation ls^2p of neutral hydrogen and of hydrogen-like ions,the cross-sections are shown which have been scaled against the nuclear charge z.The abscissa is in threshold units: u = E/E(ls, 2p), and u=\ means the energy at theexcitation threshold. The excitation cross-section of neutral hydrogen starts from 0and reaches a maximum at around u = 3 ~ 4. For z = 2, ionized helium (Figs. 1.5and 1.6), the threshold value becomes finite, and for z = oo, which stands for ionswith z^>2, the threshold value is even larger. These differences come from thedifference of the "motion" of the electron incident on (and scattered from) a neutralatom or an ion: in the former case the electron does not feel the presence of the atomexcept when it is close to the atom. In contrast, the ion exerts a strong attractiveCoulomb force, which is of quite long range. The electron is attracted and accel-erated to the target ion even at a very long distance from the ion.

Another typical feature of excitation cross-sections of ions is explicitly shown inFig. 3.12; this is for the resonance transition of sodium-like argon. The Coulomb-Born approximation gives the smooth curve. The curve with the rich structure isthe result of the close-coupling calculation. This structure is produced by reso-nances, which will be discussed in Section 3.5. Although the cross-section ofneutral atoms is not fully free from resonance structure, resonance is far moreconspicuous in the cross-section of ions. The result of an experiment is also givenby the crosses. This experiment is performed by a method called the merged beammethod: an electron beam is merged with an ion beam, sometimes in a straightpart of a ring accelerator, and the relative speed of these two beams is adjusted tochange the excitation energy. In this example, the electron beam energy is 27.15 eV,and the ion energy is varied. Again, the number of electrons with relevant energyloss is counted.

FIG 3.11 Excitation cross-section for (a) 1 s —> 2p and (b) 1 s —> 2s transitions of neutral hydrogen z—l and hydrogen-like ions with nuclearcharge z. The abscissa is the collision energy in threshold units. Near the excitation threshold, the scaled cross-section, zVM(w), stronglydepends on z, while in high-energy regions the scaled cross-sections tend to be independent of z. (c) Excitation and ionization cross-sections from the ground state of neutral hydrogen, z—l, and hydrogen-like ions. Excitation cross-section: the results of the Born orCoulomb-Born approximation: for z — 1, for z — 2, and for z — oo. : More accurate cross-section for1 —> 2 of neutral hydrogen corresponding to (a) and (b). : approximation leading to eq. (3.29). Ionization cross-section:for z—l, for z — 2, for z — oo. (Quoted from Fujimoto 1979a, with permission from The Physical Society of Japan.)


Until now, we have looked at excitation corresponding to optically allowedtransitions. We now imagine a situation in which an electron with very high energypasses by a target atom or ion. This atom or ion feels a pulsed electric field. Thispulse may have some similarities to a half-cycle of the light wave, the frequency ofwhich coincides with that of the absorption line of the transition. Then, this atomor ion can absorb this light wave and thus be excited. We may therefore expectsome correlation between the magnitude of the cross-section and the absorptionoscillator strength. When the electron energy is higher than 5-10 times the exci-tation threshold, the cross-section can be approximated by

Energy (eV)

FIG 3.12 Excitation cross-section for transition (Is22s22p6)3s28 -> (ls22s22p6)3p2Pof a sodium-like argon ion. The result of the Coulomb-Born approxi-mation is shown with the smooth line. The close-coupling calculation givesa result which is rich in resonance structure. The result of an experiment isshown with crosses with uncertainty bars. (Quoted from AMDIS and fromBadnell et al, 1991; copyright 1991, with permission from The American PhysicalSociety.)

54 ATOMIC PROCESSES

where u denotes the energy of the incident electron in threshold units. Thisasymptotic cross-section value is called the Bethe limit. As eq. (3.24) explicitlyshows the cross-section value is proportional to the absorption oscillatorstrength of the transition. We also note that the z scaling becomes valid inthis energy range as seen in Fig. 3.11(a). Figure 3.11(c) shows several examplesof the excitation cross-sections of neutral hydrogen and hydrogen-like ion fromthe ground state. The cross-sections to low-lying excited levels are takenfrom rather simple approximate calculations. Excitation cross-sections to high-lying levels are scaled according to the oscillator strength. See Table 3.1(b)and Fig. 3.4.

Excitation of optically forbidden transitions, for which fp,q = Q and thereforeeq. (3.24) vanishes, is by no means negligible. An example is shown in Fig. 3.11(b).In this case the absolute value of the cross-section for ls^2s is rather small ascompared with that of Is —> 2p, and with the increase in the energy a cross-sectiondecreases more rapidly than that for the optically allowed transition. Note thateq. (3.24) has a logarithmic dependence on energy, which is another characteristicof the cross-section for optically allowed transitions. Another example is seen inFig. 3.13 for neutral helium. Figure 3.13(a), which is for excitation 1 :S^2 :S,

FIG 3.13 (Continued)


FIG 3.13 Examples of measured or calculated excitation cross-sections of neutralhelium for optically forbidden transitions, (a) For transition 11S^21S.Results of theoretical calculations with various sophistications are given withthe curves, and those of several experiments are shown with the points. (Quotedfrom AMDIS.) (b) 11S^23S. (Quoted from Fujimoto 1979b; copyright 1979,with permission from Elsevier.)

includes the results of several experiments, which are the energy loss measurements,and several calculations. Even for this important transition (see Fig. 1.4), theagreement among the data is rather poor. The reader can imagine what thesituation would be for the cross-section for some particular transition of less-studied atoms or ions. Figure 3.13(b), for 1 :S^2 3S, is another example forneutral helium. This is for a transition with a change in multiplicity, i.e. singlet totriplet: see Fig. 1.4. This transition is made possible only by electron exchange,

56 ATOMIC PROCESSES

which we mentioned above. This process takes place at rather low energies.Thus, the cross-section decreases rapidly with the increase in energy. Note,however, that the cross-section value just above threshold is significant, or evenlarge, as compared with the cross-section of the optically allowed transition,Fig. 3.10(b).

As we have seen above, with an increase in the energy of the incident electron,the cross-section for optically forbidden transitions decays much faster than thatfor optically allowed, or electric dipole, transitions. At sufficiently high energy,cross-sections only for optically allowed transitions would survive. This is thereason why the energy loss spectrum of Fig. 3.6, which reflects excitation andionization by electron impact at a fixed energy, 2.5 keV in this example, is virtuallyequivalent to the photoexcitation and photoionization spectrum, which is due tothe electric dipole transitions.*

Instead of the cross-section, the collision strength is sometimes used.

Klein-Rosseland relationship and deexcitation cross-section

Deexcitation is the inverse process to excitation (see Fig. 3.1), and thereforethe former process is related to the latter by the principle of detailed balance.Figure 3.14 shows schematically the excitation and deexcitation processes in theenergy-level diagram. Let E be the energy of the incident electron before excitationand e be that of the scattered electron after excitation. The corresponding speed ofthe electrons far from the target are v and v', respectively. Therefore, E=mv /2and e = mv'2/2. The total number of excitation events p —> q in a plasma per unitvolume and unit time, the excitation flux, by electrons within energy width dE isgiven by

* From comparison of Fig. 3.13(a) with Fig. 3.10(b), the reader may doubt that the opticallyforbidden transition, l ' S — > 2 ' S , may not entirely be negligible in comparison with the opticallyallowed transition, 1 ! S—>2 ! P, at this energy of 2.5 keV. However, this energy loss spectrum is forelectrons with very small scattering angles. Scattered electrons by optically forbidden transitions haverelatively large scattering angles, so that in Fig. 3.6 the energy loss peak for the 1 'S —> 2 'S transition isfound to be 1/200 of the optically allowed transition peak.

In the deexcitation process, E and e change their roles, and the correspondingdeexcitation flux is given by


FIG 3.14 Schematic diagram for explanation of the relationship between theexcitation cross-section and the deexcitation cross-section.

In thermodvnamic equilibrium both the fluxes should be ecmal:

where the electron energy distribution is given by the Maxwell distributionfunction, eq. (2.2a), and the population ratio n(q)/n(p) is given by the Boltzmanndistribution, eq. (2.3). We note the relationship E=E(p,q) + e. Then, eq. (3.26)reduces to

We may call this relationship the Klein-Rosseland formula. In terms of the col-lision strength this relationship is expressed as

Excitation and deexcitation rate coefficients

The excitation rate coefficient is obtained with an equation similar to eq. (3.19).The Maxwell distribution, eq. (2.2a), is assumed:*

* When E is measured in units of eV, kTe in eq. (2.2a) is also in eV. This is also the case in eq. (3.28)except for \fE in the integration. This factor comes from v, the speed of the incident electron,and should be in units of \fj. Thus, the resulting number should be multiplied by 4.0 x 10~10

(=1/1.6 x 10~19[C]) to obtain the rate coefficient value.

58 ATOMIC PROCESSES

As we have seen already, the excitation cross-section is a complicated function ofenergy, with a particular energy dependence for a particular transition, so that nogeneral expression for the excitation rate coefficient is available. Even the Bethelimit, eq. (3.24), results in a rather complicated function. In the discussions in thefollowing chapters, however, it is sometimes useful to make an order-of-magnitudeestimate of various quantities. For these purposes, we employ a very crudeapproximation: \nu in eq. (3.24) is replaced by a constant unity. An example ofthis approximation is shown in Fig. 3.11(c) by the thick dashed line for theexcitation cross-section ls^2p. Then eq. (3.28) reduces to

It is readily shown that, by use of the Klein-Rosseland relationship, the deexci-tation rate coefficient is related to the excitation rate coefficient as

with eq. (2.7) for Z(p). In thermodynamic equilibrium, therefore, the principle ofdetailed balance actually holds,

and the Boltzmann distribution, eq. (2.3), is established:

The deexcitation rate coefficient is likewise obtained from the deexcitationcross-section by

with

or

and the excitation rate coefficient is given by eq. (3.31).

3.4 lonization and three-body recombination

lonization cross-section and rate coefficient

As has been shown in Fig. 3.2, with an increase in the principal quantum number,or in energy, of discrete states, the "shape" of their wavefunctions changes gra-dually, and, with a further increase in energy, this change continues smoothlyacross the ionization limit to the continuum state wavefunctions. This observationsuggests that the process of excitation, in which a negative-energy or discrete-stateelectron is produced, has features much in common with those of ionization, inwhich a positive-energy or continuum-state electron is produced. In other words,an ionization process may be regarded as a continuation of the excitation process.Figure 3.15 illustrates this point for the example of neutral hydrogen. For aparticular incident energy, e.g. E=9R, cross-sections from the initial state p= 1are calculated for excitation to final states q = 2,3,4, and 5 and for ionization. Thenegative abscissa is the energy of the final state q of the atom, or of the electron inthe atom, for excitation, and the positive abscissa is the energy of the ejectedelectron for ionization. In the ordinate, for excitation, the cross-section valuesdivided by (2/q3) are plotted. This factor | d(l/q2)/dq | is the energy width allo-cated to level q in units of R (see eq. (1.5)). Therefore, the plotted quantity is across-section value averaged over this energy width, or the cross-section value perunit energy interval (R), In this figure the energy R is regarded as a unit energy.Compare these cross-section values in Fig. 3.15 with those in Fig. 3.11.* Forionization, plotted in Fig. 3.15 is the "cross-section" a\,E'(E) of producinga positive-energy (£"') electron as the final state. This quantity is also the cross-section for unit energy width (R). The usual (conventional) ionizationcross-section is given as an integration of this "cross-section" over the energy E'. It

* For example, at £ = 9R=122 eV and for q = 2 in Fig. 3.15, the real cross-section value is2.8 x (2/23)™o = 0.7™2,. This is consistent with the cross-section aiSi2s + v\s,if at u = 9_R/[(3/4)_R] = 12in Fig. 3.11(a) and (b). The numerical value of 0.77ra0

2 is 6.1 x 10~21 m2, with 7ra02 = 8.79 x 10~21 m2;

this is consistent with the (Is — 2s, 2p) cross-section at 122 eV in Fig. 3.11(c).

Then, the deexcitation rate coefficient is expressed as

In expressing eqs. (3.28) and (3.31) the effective collision strength is sometimesused:

IONIZATION AND THREE-BODY RECOMBINATION 59

60 ATOMIC PROCESSES

FIG 3.15 Cross-sections for excitation and ionization, showing the continuationproperties between these processes. Shown are the cross-section values dividedby 2/n3 for excitation and the partial cross-section <Ji^E>(E) to produce acontinuum electron having energy E' for ionization. See inset. (Reconstructedafter McCaroll, 1957.)

is seen that the "cross-section" values* are consistent with the ionization cross-section in Fig. 3.11(c).

The first point to be noted in Fig. 3.15 is that, for excitation, with an increaseof q, the "cross-section" values tend to a finite value. This indicates that the realcross-sections are approximately proportional to q~3 for very large q toward theionization limit. This is consistent with eq. (3.24), because the oscillator strengthfp>q tends to be proportional to q~3 for large q as seen in Table 3.1 and Fig. 3.4.The second point is that, with the increase in the energy of the final state ofthe target electron, the excitation "cross-section" continues smoothly to theionization "cross-section". The third point is that from the comparison of the"cross-sections" for different incident electron energies E it is obvious that both

* Since the energy dependence of the production cross-section in Fig. 3.15 is approximatelyproportional to (l+£'/R)~3 (see eq. (3.13) and Fig. 3.8, and remember that our incident electronenergy is high so that the collision processes have features in common with the radiative processes,photoionization in this case) it is straightforward to obtain the conventional ionization cross-sectionfrom the threshold value of the cross-section. For E = 9R, for example, the threshold value of 1.3 (wao/R)leads to cr l jC (9.R) = 0.65mJo = 5.7 x 1CT21 m2 on the assumption of the exact minus third powerdependence. See the ionization cross-section in Fig. 3.11(c) at E= 122 eV.


the excitation cross-section and the ionization cross-section, in conventionalterms, have a similar dependence on energy, or they have similar "shapes" asfunctions of incident energy. We see this point in Fig. 3.11(c) for excitation andionization from the ground state of neutral hydrogen and hydrogen-like ions. It isnoted, however, that we confine our discussion here to the high incident energyregion where the Born approximation is valid. For lower energies, the "shape" ofthe cross-sections could be appreciably different.

It is worth noting that the "shape" of the cross-sections in Fig. 3.15 is almostexactly the same as eqs. (3.9b) and (3.13).

We now remember the discussion concerning eq. (3.15) that, for the absorptionoscillator strength to the continuum, the differential oscillator strength, eq. (3.10a),or the absorption cross-section, concentrates in the rather narrow energy regionabove the ionization threshold. See Fig. 3.8. In fact, Fig. 3.15 indicates that theproduced continuum electrons are actually concentrated in the energy region justabove threshold. Thus, for sufficiently high incident energy the ionization cross-section, in conventional terms, is expected to be proportional to fftC and have asimilar energy dependence to that of the excitation cross-section, i.e. (\nu/u),where u is now understood to be the energy of the incident electron in units of theionization potential.

The above discussions suggest that for ionization we may employ a cross-section formula similar to that for excitation. We find, however, the followingformula better approximates the actual cross-sections:t

* Excitation and ionization cross-sections in the vicinity of theionization threshold

In the above discussions of the excitation and ionization cross-sections, we havebeen mainly concerned with the region of high incident energy. We saw in Fig. 3.15

t These formulas are due to Dr E. Baronova.

where u is the energy E in threshold units, i.e. u = E/x(p). It may be interesting tonote the following: in Fig. 3.11(c), for high-energy, the excitation cross-section<TI^(E) and the ionization cross-section a\>c(£) have similar magnitudes. This isrelated to the fact that the oscillator strengths are/i>2 = 0.416 and/i,c = 0.435 (seeTable 3.1), and that the threshold energies are E(\, 2) = (3/4)x(l).

The ionization rate coefficient S(p) may be obtained from a similar equation toeq. (3.28). We adopt the following formula corresponding to eq. (3.35):1"

Figure 3.16 shows a comparison between the above simple approximationsand experimental or theoretical cross-sections. The small open circles showthe experimentally determined ionization cross-section. The dash-dotted line isthe approximation, eq. (3.37), to this cross-section. From the slope of this line in thelinear plot, the value of <TQ is determined by eq. (3.38). The horizontal dash-dottedline is the result of this procedure; in this plot all the excitation cross-section,eq. (3.36), should lie on this line. The thick lines indicate the calculated cross-section values to high-lying excited levels, which are known to be most reliable atpresent. It is seen that the cross-sections deduced from the above arguments agreewith the calculation within 20%. The thin lines are the cross-sections convenient tobe used in the CR model which follows in the next chapter. In this figure the resultof a close-coupling calculation for excitation to p = 5 is shown with the opencircles. These values are judged less reliable than the classical calculation shownwith the thick curve. The reason is too involved to be described here.

We suppose that we are exciting and ionizing neutral hydrogen by a beam ofelectrons, the energy of which is slightly higher (by more than A£" as assumedbelow) than the ionization threshold, R, If we assume that the number of electronsraised to high-lying excited states within the energy interval — A£"< E< 0 (A£" issmall and positive) is equal to the number of electrons raised to the continuumstates within the energy interval 0 < E< A£", we come to the conclusion that thefollowing relationship should hold.

starting from the excitation threshold. We assume this approximation in thenarrow energy region considered here. An example is given in Fig. 3.16 for cr\tp(E)f o r p > 5 with the dash-dotted line. The parameter value <TQ is determined below.The ionization cross-section is known to have the energy dependence o-lc(£)oc(E— R)1'121. We now approximate this cross-section by a linear function

the continuation from the excitation process across the ionization limit to ioniza-tion, the production of the continuum states. We now consider the opposite case,i.e. excitation and ionization in the energy region of incident electrons in thevicinity of the ionization threshold. From the discussions at the beginning of thepreceding subsection, we may expect that a continuity feature also manifests itselfin excitation cross-sections to high-lying states and the ionization cross-section inthis low-energy region. As an example we take the ground state of neutralhydrogen for the initial state. It is known that excitation cross-sections to veryhigh-lying levels have finite values at the excitation threshold. For simplicity, weapproximate the cross-section to have a constant value

62 ATOMIC PROCESSES


FIG 3.16 Excitation cross-sections and ionization cross-section in the energyregion close to the ionization limit for transitions from the ground state ofneutral hydrogen. The experimental ionization cross-section (small open circles)is approximated by the dash-dotted line, eq. (3.37). From the slope of this linecombined with eq. (3.38) the excitation cross-sections are determined, eq.(3.36), and are represented by the horizontal dash-dotted line. The thick linesshow result of a classical calculation; the large open circles show results of aclose-coupling calculation; thin lines show approximations for the purpose ofpractical calculations of excited-level populations. (Quoted from Fujimoto andMcWhirter 1990; copyright 1990, with permission from The American PhysicalSociety.)

In the case that the target is ions, the power of 1.127 of the ionization cross-section tends to 1 for large z, and the above approximation should become evenbetter.

Three-body recombination and rate coefficient

The inverse process to ionization is three-body recombination. The "cross-section" of this process is rather complicated because this process involves twoelectrons. We deal with this problem in Appendix 3B. Here we derive directly therate coefficient a(p).

In thermodynamic equilibrium the principle of detailed balance should hold:

64 ATOMIC PROCESSES

Therefore, we have

where we have used the Saha-Boltzmann relationship, eq. (2.7a).

*3.5 Autoionization, dielectronic recombination, and satellite lines

In the preceding four sections we have exhausted the processes shown in Fig. 3.1.If we deal only with neutral hydrogen and hydrogen-like ions for ion (z — 1),ignoring the lower ionization stage ions, the foregoing discussions would be suf-ficient. However, if we include helium-like ions and still lower ionization-stageions in our discussion, we must take into account other important processes,autoionization and dielectronic recombination.

We start with the ground state of ion z, denoted by lz, as shown in Fig. 3.17(a). Inthese processes, the angular momentum of the incident electron, or the continuum-state electron, and that of the final-state electron are important. Remember that thecontinuum wavefunction is also specified by the angular momentum quantumnumber as noted in Sections 3.1 and 3.2. We include explicitly the angularmomentum quantum number in the following description. Excitation by electronimpact to an excited state p, denoted bypz, may be expressed as

where El and el' are the energy and the angular momentum of the incident electronand the scattered electron, respectively. We now take the excitation ls^2p ofhydrogen-like ion as an example. Its cross-section is given in Fig. 3.11(a). Supposethe energy of the incident electron decreases toward the excitation threshold. Sincethe target is now an ion, the excitation cross-section tends to a finite value. See alsoFigs. 3.10-3.12. We further decrease the energy below the excitation threshold,Ez(\,p). Remember that we are now dealing with the ion in an excited statepz; thision has the same charge as the ground-state ion \z, so that this state pz is accom-panied by states of the ions in the adjacent lower ionization stage in a similar way tothe ground state ion \z. These states are denoted with (p,nl')z_i and shown inFig. 3.17(a). In our present example, hydrogen-like level 2p is accompanied byhelium-like states (2p,«/'). Since these states consist of two excited electrons they arecalled doubly excited states. From our foregoing discussions about the continuitybetween the continuum states and discrete states, we may expect that a processsimilar to the excitation takes place in this case. This process is dielectronic captureto one of the doublv excited states of the lower ionization stase ion:

This ion has a core electron in the excited state p and another excited electron instate nl'\ this latter electron is sometimes called the spectator electron. Since both

AUTOIONIZATION, DIELECTRONIC RECOMBINATION 65

FIG 3.17 Schematic diagram depicting dielectronic capture, autoionization, anddielectronic recombination, (a) Schematic energy level structure and transi-tions; (b) the relationship between the excitation cross-section and thedielectronic capture cross-sections.

of the electrons are excited the energy of this level is situated above the groundstate Iz, and it is embedded in the continuum states of lz + e. This ion is ener-getically unstable. The level position is below the "ionization limit" pz, and thedielectronic capture, eq. (3.42), takes place only when the incident energy Ematches the energy of this doubly excited level. Let r&(p,«/') be the rate coefficientfor the dielectronic capture. Then it may be expressed in terms of the capturecross-section a&(p,«/') and its energy width 8E of the final state (see Fig. 3.17(b))

The details of the energy dependence of the cross-sections are immaterial. Thispoint is understood from the discussion below. a^(p,nl') is understood as the

66 ATOMIC PROCESSES

averaged value over the energy width SE, and only the quantity <Jd(p, nl')6E hassignificance. From our discussion about the continuation of the states across theionization limit to the continuum states, we may expect a relationship betweenthe excitation, eq. (3.41), and the dielectronic capture, eq. (3.42). We expect thecontinuation of the excitation cross-section for a positive excess energy (e > 0)down to zero energy, and further to negative energy (e < 0), to arrive at the die-lectronic capture cross-section, eq. (3.43):

where we have explicitly written the transition for the cross-sections, and (• • •)means the integration of the cross-section over the width of the doubly excited state.In this equation the energy width dE corresponds to the number of levels dn inthis energy width. Since we are considering a capture process to one of the discretestates nV the "width" dn should be equal to 1. We further assume hydrogen-like energy for the «/' level and adopt the threshold value of the excitationcross-section. We then have

This process may be understood as disintegration of the unstable state ion(p,nl')z_i. Sometimes, this process is called the Auger decay. Let Aa(p, «/') be itsprobability.

In thermodynamic equilibrium, both processes should be related by theprinciple of detailed balance,

Thus, we are able to calculate the dielectronic capture rate coefficient from thethreshold value of the excitation cross-section; see Fig. 3.11(a) or Fig. 3.12, forexample.

The inverse process to dielectronic capture is autoionization,

or


where we have used eqs. (2.7), (2.2a), (3.43), (3.44a), and (3.25). It is noted that inthe Saha-Boltzmann coefficient Z(p,nl'), the ionization potential \(p,nl') ismeasured from the ground state ion \z, so that it is negative.

Besides autoionization, the ion in the doubly excited state (p,«/') may decayto a normal or singly excited state through a transition of the core electron(Fig. 3.17(a)),

This transition is called the stabilizing transition, and its probability is denoted asAr((p,nl')^ (!,«/')).

The series of processes of eq. (3.42) followed by eq. (3.48) is called dielectronicrecombination. This recombination is accompanied by emission of a photon ofeq. (3.48). This transition may be regarded as a radiative transition of the coreelectronpz—> \z under the influence of the spectator electron «/'. The frequency ofthis photon is very close to that of the "parent" transition of pz^lz, and itsspectral line is called the satellite line lying close to the parent line.

Until now we have implicitly assumed n to be very large, n ;$> [p], where [p]denotes the principal quantum number of the core electron p. When n is small andclose to [p], the above approximate discussion becomes inadequate. For example,in the case of n equal to [p] no distinction can be made between the core electronand the spectator electron. We have to treat these doubly excited states as theyare. In such cases the frequencies of the stabilizing transition lines, or the satellitelines, are separated appreciably from the frequency of the parent line. Figure 3.18shows an example of the lithium-like satellite lines associated with the parenthelium-like line (1 :S—2 :P) of iron ions. All the satellite lines for n = 2 and « g 3are seen.

FIG 3.18 Experimentally observed spectrum from highly ionized iron in atokamak plasma. The lines labeled w, x, y, and z are the "parent" helium-likeion lines. Other lines are the lithium-like satellite lines and the beryllium-likesatellite line (/3). (Quoted from Bitter et al, 1981; copyright 1981 withpermission from The American Physical Society.)

Figure 3.19(a) shows an example of the dielectronic recombination rate coeffi-cients from the ground-state hydrogen-like boron ion to various K/' states of ahelium-like ion. In this example, only the 2p state is included as the core electron/?in eq. (3.50). Figure 3.19(b) shows the total dielectronic recombination ratecoefficient which is the sum of eq. (3.50) over all the nl' states.

Figure 3.20 shows the total recombination rate coefficient of the hydrogen-likeboron ion to the helium-like ion; the dashed curve is the sum of the rate coefficientfor the radiative recombination as discussed in Section 3.2 (the sum of the ratecoefficients similar to those in Fig. 3.9) and that for the dielectronic recombinationas given in Fig. 3.19(b). For temperatures lower than 4 x 105 K, or the reducedtemperature of Te/z

2~2.5 x 104 K, the former recombination is dominant, andfor higher temperatures the latter becomes dominant. This is because, fordielectronic recombination to take place, the doubly excited states, (2p,«/') in thiscase, which lie rather high in energy, should be populated by energetic electrons.See Fig. 3.19(b). This figure also includes the effective recombination rate coef-ficient for finite electron densities. Under these conditions, we cannot separate theradiative recombination and the dielectronic recombination, as will be discussed inChapter 5.

For ions having more than two electrons, e.g. lithium-like ions with threeelectrons Is22s, doubly excited levels of beryllium-like ions Is22p«/ lie justabove the ground state Is 2s. Dielectronic recombination for these ionsIs22s + e^ Is22p«/^ Is22s«/+ hv in this example, can be quite substantial evenat low temperatures.

Resonance contribution to excitation cross-section

As we have noted above, any of the excited levels of an ion is accompanied byseries of doubly excited states. We now take an example of the hydrogen-like 3plevel; this level forms the ionization limit of the doubly excited helium-like (3p, ri)

where we have used eq. (3.46). The effective rate coefficient for dielectronicrecombination from \z into (1, K/')Z-I is given as

The population density of the doubly excited state is given from the balance ofthe processes of dielectronic capture, autoionization, and stabilizing transition

68 ATOMIC PROCESSES

or

FIG 3.19 Dielectronic recombination rate coefficient, eq. (3.50), for a hydrogen-like boron ion. (a) Breakdown into eachterm; (b) the total rate coefficient. Several results of different calculations are given. (Quoted from Fujimoto et al., 1982.)

70 ATOMIC PROCESSES

This series is nothing but an excitation process ls^2s. This additional processshould be added to the direct excitation which was discussed in Section 3.3; or the

FIG 3.20 The effective recombination rate coefficient from hydrogen-like tohelium-like boron. In the limit of low density ( ) this is virtually thesum of the radiative recombination rate coefficient, similar to eq. (3.19) (seeFig. 3.9) for all levels p, and the dielectronic recombination rate coefficient,Fig. 3.19(b). For finite densities, the collisional-radiative (effective) recombinationrate coefficient (see Chapter 5) is given. (Quoted from Fujimoto et al., 1982.)

levels. Here we omit the angular momentum quantum number in designating thespectator electron. An ion in the ground state Is may dielectronically capture anelectron to one of these doubly excited levels (3p,«). This doubly excited helium-like ion could return to the ground Is state of the hydrogen-like ion by auto-ionization, but it could also autoionize to leave an electron in a singly excitedhydrogen-like level, 2s for example. Thus the following series of processes can takeplace:

(dielectronic capture)

(autoionization).


FIG 3.21 The excitation cross-section of hydrogen-like neon (z=10) forthe ls^2s transition. The underlying almost-constant cross-section corres-ponds to the curve in Fig. 3.11(b) for z—>oo just above the threshold. Theenergy range of this figure corresponds to u= 1 — 1.19 in Fig. 3.11(b). Thecontributions from the resonance, eq. (3.51), are expressed as sharp peaks.(Quoted from Aggarwal and Kingston, 1991; copyright 1991, with permissionfrom The Royal Swedish Academy of Science.)

excitation cross-section should have a contribution from this series of processes.This contribution is called the resonance contribution. The above picture based onthe two-step mechanism is too simplistic, and a realistic treatment of this processshould involve re-diagonalization of the doubly excited states and the underlyingcontinuum states. As a result, the cross-section shows very complicated sharpstructure, which is named resonance.

Figure 3.21 shows an example of calculations of excitation cross-sections thatinclude resonance contributions; this is for excitation ls^2s of hydrogen-likeneon (z=10). The abscissa ranges from the excitation threshold (15R) to thethreshold of the 1 s —> 3 / excitation, 8 8. 9R, and the ordinate is the collision strengthwhich was introduced by eq. (3.25). The underlying flat near-horizontal dashedline corresponds to the cross-section given in Fig. 3.11(b): at the threshold thecollision strength value of 0.0065 in Fig. 3.21 corresponds to zVij^l)/™}} =0.43, which agrees very well with the cross-section for z —> oo in Fig. 3.1 l(b). Thesharp resonances are due to the doubly excited intermediate states (3s, ri), (3p, ri),and (3d, ri) with « = 3 at around 19R, n = 4 at around 83-84J? and n = 5 at around85.5R, and so on. Resonances due to (41,41') are also seen at around 88R. We havealready seen in Fig. 3.12 an example of the resonance structures of excitation

72 ATOMIC PROCESSES

cross-sections. In these examples, the resonance contribution, when averaged overenergy, is rather minor. In some other cases the contribution is very large, e.g.more than an order larger than the underlying cross-section.

*3.6 Ion collisions

Until this point, we have exhausted the major atomic processes, radiative tran-sition and collisional transitions due to electron impact, which are important inplasma spectroscopy. In many practical situations, the spectroscopic properties ofthe ensemble of ions in a plasma are controlled by these processes, and they areenough to describe various phenomena. However, in some other situations, wehave to include collision processes other than electron impact. We briefly surveyion collision processes in this section.

Excitation-deexcitation

Roughly speaking, excitation or deexcitation by electron impact is caused by thetime-dependent electric field exerted on the target ion by the incident electron asnoted in the paragraph in p. 53 leading to eq. (3.24). This is especially true at highenergies. The incident ion also exerts an electric field on the target with the signreversed, and could induce a transition in this target ion, too. In this context, theessential feature is the temporal variation of the field, which is directly related tothe velocity of the incident particle. Thus the important parameter is the velocityrather than the energy as implicitly assumed in eq. (3.24). We recognize two points:

1. The ion mass is much larger than the electron mass; ions are some 103 or 104

times heavier. If an ion is to have the same velocity, or a similar effect, as anelectron has on the target, its energy has to be larger than that of the electron bythis factor. Figure 3.22 shows an example of cross-sections by ion collisions. Thisis for excitation of neutral helium: He(ls2 :S) + H + -> He(ls 2p :P) + H +. This isto be compared with Fig. 3.10(b); when the abscissa of the former figure is reducedby M/m = 1.66 x 1CT27/9.11 x 1CT31 = 1.82 x 103 (see Table 1.1), both cross-sections are found to be nearly the same. This is especially the case for higherenergies. In many cases, however, the ion temperature is nearly the same as ormuch lower than the electron temperature. These facts suggest that ion collisionsare rather ineffective for inducing transitions like this example.

2. Even so, for transitions between closely lying levels, ion collisions could beeffective. This can be understood from eq. (3.24); for electron impact, u, thecollision energy normalized by the energy difference between the two levels, tendsto be very large and the cross-section becomes small. For ions, in terms of velocityrather than energy, the effective u could be small. This is the case for transitionsbetween different / levels with the same n in helium-like and other ions with simpleenergy structure. Figure 3.23 shows an example of the cross-sections for the3s —> 3p transition of neutral hydrogen. For energies of practical interest, the cross-section for proton collision is much larger than that for electron impact. For higher

ION COLLISIONS 73

FIG 3.23 Cross-section for transition 3s 3p of neutral hydrogen by protoncollisions and that by electron impact. (Quoted from Sawada, 1994.)

energies the energy scaling by the factor 1.82x 103 is almost exact. In reality,however, for neutral hydrogen the levels are split into fine structure, and transitionsbetween these fine-structure levels should be considered. The calculation shown inFig. 3.23 is based on the Born approximation with the fine structure neglected.

FIG 3.22 Excitation cross-section for the transition 11S^21P of neutral heliumby proton collisions. This corresponds to Fig. 3.10(b) for electron impact.(Quoted from Ito et al., 1993, with permission from JAERI.)

74 ATOMIC PROCESSES

Charge exchange collisions

When an ion approaches an atom, it attracts the atomic electrons, and it mayeventually capture one or more electrons from the atom after its collision with thatatom. This process is called charge exchange or charge transfer. The thick solidcurve with the open circles in Fig. 3.24 gives an example of the cross-sections forcharge exchange; this is for one-electron capture He + H+ —> He+ + H. Thecaptured electron may not be in the ground state. The dashed and dash-dottedcurves in Fig. 3.24 show the cross-sections for production of excited-statehydrogen, 2p and 2s, respectively. In this case, the production of excited states is

FIG 3.24 Charge exchange cross-section for He + H+ —> He++ H. — o — o —: totalcross-section; : charge exchange cross-section producing excitedatoms H(2s). : the same for H(2p). —•—•—: resonant chargeexchange cross-section H+ H+^H+ + H. (Rearranged from Ito et al, 1994,with permission from JAERI.)

ION COLLISIONS 75

rather minor. In the case of a multiply charged ion, the situation could be dif-ferent. Figure 3.25 shows the cross-sections for O6+(ls2) + He^O5+(ls2«/) +He+. In (a) the total charge transfer cross-section is shown with x. The cross-section for producing nl= 3s is shown with •. for 3p with •. and 3d with ». Thesum of these three cross-sections for n = 3 is given with A. Figure 3.25(b) shows thedistribution among n at the collision energy of 60 keV: o for ns, n for «p, and o fornd. The symbol + is for n = 2 including the ground state 2s (the collision energy

FIG 3.25 Charge exchange cross-section for O6+(ls2) + He -> O5+(ls2n/) +He+. (a) Total charge transfer cross-section ( x ). Cross-section for producingnl= 3s («, o), 3p (•, n), and 3d (», o). The sum of these three cross-sections forn = 3 (A). The open symbols connected with the lines are for « = 4.(b) Distribution of the product ions among n at the collision energy of60 keV. KS (o), «p (n), and nd (o). +: for n = 2 including the ground-state2s (the collision energy is 9 keV). (Quoted from Watanabe, 1998.)

In this example, x* = 1-66 au or 45.2 eV. On the assumption of the hydrogeniclevel structure the principal quantum number for this energy is «* = 3.3. Thestrong selectivity in Fig. 3.25 is thus explained. In the case of the example ofFig. 3.24, the above model gives the favored principal quantum number n* = 0.66.Thus, the dominant product is ground-state hydrogen.

In some other cases the target atom may have the same ion core as the pro-jectile ion. Charge exchange in such a case is called resonant charge exchange. Anexample of the cross-sections is shown in Fig. 3.24 with the dot-solid curve:H + + H ^ H + H + . The "shape" of the cross-section has two characteristic fea-tures. The cross-section values in the low-energy region are quite large and showlittle energy dependence. In higher-energy regions, the cross-section valuedecreases sharply with an increase in energy. The critical energy roughly corre-sponds to the relative speed of the colliding particles which is equal to the speed ofthe atomic electron in its classical orbit. In this example, this energy is given fromeq. (1.3) and the proton mass in Table 1.1. This is 4.9 x 104 eV, which is consistentwith the critical energy in Fig. 3.24.

Appendix 3A. Scaling properties of ions in isoelectronic sequence

Hydrogen-like ions

Various atomic parameters of hydrogen-like ions change according to the nuclearcharge z. We call this the z scaling. Some of the scaling laws have already beenintroduced in the text. We summarize these properties in this appendix.

where z is the charge of the projectile ion and %B is the ionization potential of thetarget atom electron. Here all the quantities are measured in atomic units (au). Inthe above example, z = 6 and XB = 24.5 eV/27.2 eV = 0.90 gives Rcv = 6.5. Thefavored energy of the final state of the ion is given by

is 9 keV). In this example production of excited-state ions is dominant, and theproduced states are quite selective.

The above selective nature of charge exchange collisions is qualitativelyexplained by the "classical overbarrier model". In this model it is assumed that theoptical electron is resonantly transferred from the target atom to the projectile ionwhen the "barrier", which is due to the superposition of the Coulomb potential ofthe atomic ion core and that of the projectile ion, becomes lower than the energyof the atomic electron. This occurs at the internuclear distance

76 ATOMIC PROCESSES

SCALING PROPERTIES OF IONS 77

In the following, we express the quantity for a hydrogen-like ion z in terms ofthe corresponding quantity for neutral hydrogen z=l ; the latter quantity hassuffix "H". The energy of atomic levels is scaled as

This scaling applies to all energies, e.g. the ionization potential x(p), eq. (1.1). Thefrequency of a transition line has the same scaling,

The oscillator strength is independent of z, as is understood from its definition,eq. (3.2), and the scaling of the atomic radius, eq. (1.2), together with eq. (3A.2),

This is also consistent with the sum rule, eq. (3.5), which gives the number ofelectrons, i.e. 1. The transition probability, eq. (3.1), scales from eq. (3A.2) as

From eq. (2.12) the B coefficient scales according to

The photoionization cross-section is expressed as eq. (3.10a), and from eqs. (3A.2)and (3A.3) we have

From Milne's formula, eq. (3.17), we have

where the speed of the plasma electrons to be captured is measured according tothe scaling

Equation (3A.6) is also seen in eq. (3.18). It is interesting to see that eq. (3A.7) isconsistent with eq. (1.3), the electron speed in the Bohr orbits. From the expres-sion of the Maxwell distribution, eq. (2.2), with eq. (3A.7), we have

where we have adopted the scaling for the electron temperature

JjJ1 may be called the reduced electron temperature.

78 ATOMIC PROCESSES

With this scaling of temperature, the radiative recombination rate coefficient,eq. (3.19) or eq. (3.19a), is scaled as

* Non-hydro gen-like ions

In Chapter 1, it has been pointed out that the energy level structure of ions havinga small number of electrons can be regarded as a modification of that of hydrogen-like ions. For instance, the level energy is expressed in terms of the effectiveprincipal quantum number, eq. (1.7a). Other quantities are also scaled accordingto z, the effective core charge felt by the optical electron at a large distance fromthe core. Instead of "H" we use "1" to denote the starting point in this isoelec-tronic sequence, i.e. the neutral atom. The parameter values for z = 1 which give agood scaling for large z may be different from the actual values for the neutral

The excitation cross-section, eq. (3.24), except for the energy region near threshold(see Fig. 3.11), is scaled as

where for the electron energy we have used the threshold units u, which is inde-pendent of z if we adopt the scaling for the energy of the incident electron,

in accordance with eqs. (3A.I) and (3A.7). It is noted that eq. (3A. 11) is notconsistent with eq. (1.2). The collision strength as defined by eq. (3.25) is scaled as

With the scaling eq. (3A.9), the integration, eq. (3.28), for the excitation ratecoefficient gives the scaling

The deexcitation rate coefficient and the ionization rate coefficient follow thesame scaling. The Saha-Boltzmann coefficient, eq. (2.7), is scaled as

where the energy and the temperature are scaled according to eqs. (3A.I) and(3A.9), respectively. The three-body recombination rate coefficient is given fromeq. (3.40) and scales as

In the case of Aa > AT, the dielectronic recombination obeys the same scaling asthe radiative recombination, eq. (3A.10). With an increase in z, Ar tends to belarger than Aa, and the dielectronic recombination becomes less important ascompared with the radiative recombination.

*Appendix 3B. Three-body recombination "cross-section"

As noted in the text, the three-body recombination process involves an ion andtwo electrons, and the likelihood of this process to take place cannot be expressedby a cross-section which has been defined for two-particle reaction processes. Inthis appendix, we derive a rate coefficient in terms of a kind of cross-section.

We consider ionization from level p of atom (z — 1) to ion z, and the inversethree-body recombination. We start with the Klein-Rosseland formula, eq. (3.27),

where the factor [(z — l)/z]2 has been neglected. The dielectronic recombinationrate coefficient is given by eq. (3.50), so that it obeys rather complicated scalings

The autoionization probability, eq. (3.47a), is independent of z:

The following scaling properties concerning the doubly excited levels, (p, nl),are valid for helium-like ions. As in Section 3.5,/> denotes the core electron and nlthe spectator electron. The dielectronic capture rate coefficient, eq. (3.43), has thesame scaling as that for the excitation rate coefficient, eq. (3A.14),

and the transition probability scales as

The oscillator strength then follows the scaling

THREE-BODY RECOMBINATION "CROSS-SECTION" 79

atom. We designate a level with quantum numbers nl. The energy differencebetween the levels with different n, or the ionization potential of a level, followsapproximately the scaling of the hydrogen-like ions, eq. (3A.1). A similar situationapplies to various quantities for different n levels. The energy difference betweendifferent / in the same n is scaled as

80 ATOMIC PROCESSES

for excitation and deexcitation cross-sections (Fig. 3.14). In the present case theupper level is denoted by u. The cross-section for deexcitation u —>/> is given by

We assume that the upper level consists of a group of levels, ut, having virtually thesame energy and common excitation-deexcitation cross-sections. Then we have

We now assume that these upper levels belong to the continuum states, havingenergy E' as measured from the bottom of the continuum, or the ground state ofion z. Figure 3B.1 shows schematically the relationship of the energies. Let gu(E')be the density of states per unit energy interval. The Klein-Rosseland relationshipextended to this situation is written as

where dE' is the energy width within which the group of the upper levels are con-tained. It is obvious from the derivation of this equation that the "cross-section"ffP,u(E') has the dimension of [m2!"1]. This is a cross-section for production ofupper levels in a unit energy interval. Instead of the continuum states, we assumethe free states. Then, the quantity gu(E') is nothing but the "statistical weight" as

FIG 3B.1 Schematic energy relationship for three-body recombinationz + e(£") + e(e) -^pz-i + e(£).

THREE-BODY RECOMBINATION "CROSS-SECTION" 81

given by eq. (2.5a). Thus, the "deexcitation" cross-section is expressed as

where g(p) has been suffixed with z — 1 to make clear that level p is an atomic level,and ge = 2. As is seen in Fig. 3B.1 the energies are related to each other:E' + e = E~x(p)- We now consider "deexcitation", or more exactly, recombina-tion; The number of target "atoms" is given by nzf(E')dE', where f(E') dE' is theenergy distribution of the continuum electrons. These target atoms are acted uponby the electrons nj'(e) de with a cross-section that describes the likelihood of thisreaction to take place. The number of recombination events in unit volume perunit time Mm^3s^1l is siven bv

where use has been made of eq. (2.7). It is readily seen that this equation iseq. (3.40), which we introduced from the thermodynamic equilibrium relationship.

It is noted that the last integral is the ordinary ionization cross-section, aptC(E),as pointed out in Section 3.4 with regard to Fig. 3.15. Equation (3B.7) thenreduces to

We now shift the origin of the energy from the ground state of ion z to the positionof the atomic level p. Then, we have

where we have usedintegrand is rewritten as

v (e) = ^J2eJm, From eq. (2.2a) it is readily seen that the

We may call the above cross-section in the integrand the three-body recombinationcross-section. By substituting eq. (3B.4) into this equation we have

82 ATOMIC PROCESSES

References

The discussions in Section 3.5 are mainly based on:Beigman, I.L., Vainshtein, L.A., and Syunyaev, R.A. 1968 Sov. Phys.-Uspekhi

11,411.

The tables and figures are taken or based on:Aggarwal, K.M. and Kingston, A.E. 1991 Phys. Scripta 44, 517.AMDIS (Data and Planning Information Center, National Institute for Fusion

Science, Toki).Badnell, N.R., Pindzola, M.S., and Griffin, D.C. 1991 Phys. Rev. A 43, 2250.Bethe, H.A. and Salpeter, E.E. 1977 Quantum Mechanics of One- and Two-Electron

Atoms (Plenum, New York; reprint of 1957).Bitter, M., von Goeler, S., Hill, K.W., Morton, R., Johnson, D.W., Roney, W.,

Sauthoff, N.R., Silver, E.H., and Stodiek, W. 1981 Phys. Rev. Letters 47, 921.Fisher, V.I., Ralchenko, Y.V., Bernshtam, V.A., Goldgirsh, A., Maron, Y.,

Vainshterin, L.A., Bray, I., and Golten, H. 1997 Phys. Rev. A 55, 329.Fujimoto, T. 1979a /. Phys. Soc. Japan 47, 265.Fujimoto, T. 1979b /. Quant. Spectrosc. Rad. Transfer 21, 439.Fujimoto, T. and McWhirter, R.W.P. 1990 Phys. Rev. A 42, 6588.Fujimoto, T., Kato, T., and Nakamura, Y. 1982IPPJ-AM-23 (Institute of Plasma

Physics, Nagoya).Goto, M. 2003 /. Quant. Spectrosc. Radial. Transfer 76, 331.Ito, R., Tabata, T., Shirai, T., and Phaneuf, R.A. 1993 JAERI-M, 93-117 (Japan

Atomic Energy Research Institute, Tokai-mura).Ito, R., Tabata, T., Shirai, T., and Phaneuf, R.A. 1994 JAERI-M, 94-005 (Japan

Atomic Energy Research Institute, Tokai-mura).Liu, X.-J., Zhu, L.-F., Jiang, X.-M., Yuan, Z.-S., Cai, B., Chen, X.-J., and Xu, K.-Z.

2001 Rev. Sci. Instr. 72, 3357.McCarroll, R. 1957 Proc. Phys. Soc. (London) A70, 460.Sawada, K. 1994 Ph.D. thesis (Kyoto University).Watanabe, H. 1998 Ph.D. thesis (Kyoto University).

POPULATION DISTRIBUTION ANDPOPULATION KINETICS

where V is the volume of the plasma which we observe and dfi is the solid anglesubtended by our optics, e.g. when we use a condenser lens it determines dfL Sincewe assume A(p, q) to be known, $(p, q) is determined by n(p), apart from geo-metrical factors. Thus, the features of the spectrum like those in Figs. 1.3, 1.5, and1.7, or the distribution of the line intensities over the spectrum, which is the centralproblem of plasma spectroscopy, reduces to the problem of the population and itsdistribution over various excited levels like those in Figs. 1.4 and 1.6. In thischapter we investigate how, in a plasma, the populations are formed in excitedlevels, and what are the general characteristics of the population distributions inrelation with the nature of the plasma.

4.1 Collisional-radiative (CR) model

Rate equation

From now on we confine our consideration to hydrogen-like ions (and neutralhydrogen) with nuclear charge ze, except when otherwise stated, z = 1 meansneutral hydrogen. We use the term ions to denote atoms as well as ions. (In thisand following chapters, we sometimes use the terms atoms and ions for the pur-pose of distinguishing ions in two successive ionization stages.) We assume thestatistical populations among the different angular momentum states within thelevel with the same principal quantum number. Its validity is examined inAppendix 4A. We thus adopt the simplified energy-level diagram like Fig. 1.6,where p or q represents the principal quantum number.

Suppose we observe a plasma with a spectrometer to resolve the light it emits intoa spectrum (see Figs. 1.3,1.5, and 1.7 for example) and deal with one of the spectrallines which corresponds to the transition p —> q. We make two assumptions:

1. The plasma is optically thin, i.e. all the photons emitted by ions (atoms) in theplasma leave the plasma without being absorbed inside the plasma.

2. The plasma is isotropic, i.e. the emitted photons are unpolarized and theirangular intensity distribution is isotropic.

Then the observed line intensity $(p,q) (this quantity is called the radiant flux orthe radiant power in radiometry, and has units of [W]) is given by the product ofthe upper-level population n(p) and the radiative transition probability A(p, q);

4

84 POPULATION DISTRIBUTION AND POPULATION KINETICS

A change in the population of a discrete level is brought about by spontaneousradiative transitions and transitions induced by electron impact as examined in thepreceding chapter. See Fig. 3.1. We omit other transition processes, e.g. photo-ionization and transitions induced by ion collisions. This is because our objectivein this chapter is to study the most fundamental features of the populations of ionswhich are immersed in a plasma. Then the temporal development of the popu-lation in level p in ionization stage (z—1), nz_i(p), is described by the rateequation

where we have assumed that the plasma is homogeneous and the spatial transportof the ions does not affect the population dynamics. We use the convention thatwhen we are considering level p, summation over "q<p" means summation overlevels q that lie energetically below level p. In the following, the quantities likeC(q,p)ne and A(q,p), having units of [s"1], are called the rate or the probability,and those like C(q,p)nen(q) and u(p)n2

enz, having units of [m~3 s"1], are called theflux. The first line expresses the excitation flux into this level, or the populatingflux into this level, from the lower-lying levels, the third line the populating fluxfrom the higher-lying levels and the last line that by direct recombination. Thesecond line represents the flux of population out of this level, or the depopulatingflux. Equation (4.2) is coupled with similar equations for other excited levels, forthe ground-state atom density nz_\(\~) and for the ion density nz.

Although a problem remains about how many levels should be included in theset of equations, which will be addressed toward the end of this chapter and inChapter 9, these equations could be solved numerically. Examples of the solutionsare given in Appendix 4B. Solving the set of rate equations under a specific plasmacondition would give us a solution. However, this solution is valid only under thisparticular condition. Instead, it would be more valuable for us if we have abroader perspective. In this case, we can have a more general picture about thepopulation distribution and the population fluxes, so that we will be able topredict what the situation would be under certain different conditions. Thus, wetake an alternative approach, which is more general though approximate. As willbe shown below and in Appendix 4B this approximation is quite accurate for themajority of problems of practical interest.

COLLISIONAL-RADIATIVE (CR) MODEL 85

Relaxation time

Before discussing eq. (4.2) in detail, we examine a simple system described by arate equation. Figure 4.1 shows the situation: a constant flux of water A is cominginto a bucket which has a hole at the bottom. The change in the amount of water «in the bucket is expressed as

which is readily solved as

Responding to the perturbation incurred at t = Q, this system returns to its sta-tionary state with the rate of outflow B. Its characteristic time is defined as therelaxation time tr\= \/B.

We define likewise the relaxation time for n(p) from eq. (4.2) as

FIG 4.1 Illustration of the bucket model of the rate equation.

where the flux out of the bucket through the hole is assumed to be proportionalto K. For steady state the solution is

where the subscript 0 has been added to denote steady state. Suppose, at t = 0, n isgiven as «(0) = «0 + An(0), where An(0) may be positive or negative. The rateequation reduces to

or


Examples of the relaxation times are given in Appendix 4B. Here, we firstexamine each term in the r.h.s. of eq. (4.4) for excited levels,/? > 2. In the followingdiscussion we proceed in the order-of-magnitude argument. We assume hightemperature, i.e. kTe > x(2), where x(2) is the ionization potential of level p = 2,(Te/z

2 > 4 x 104 K) except otherwise stated. First, we consider transitions betweenexcited levels, excluding transitions to and from the ground state. In the excitationrate coefficient, eq. (3.29), under this condition, the exponential factor may beapproximated to 1, because the energy difference between the excited levels ismuch smaller than the thermal energy of electrons. Its relation to the deexcitationrate coefficient, eq. (3.31), is then approximately given by

or

We now compare the two dominant rate coefficients, i.e. excitation C(p,p+l)and deexcitation F(p,p—l), From eqs. (4.7) and (4.5) we have

These relations substituted into eq. (4.5) lead to the approximation

See Fig. 3.4. The energy separation between the adjacent levels is given as

These inequalities mean that, among many collisional excitation and deexcitationprocesses originating from level p, the dominant ones are the transitions to theadjacent-lying levels (p ± 1).

As has been noted in Section 3.1 the oscillator strength may be approximatedfor p > 1 by

It was shown in Section 3.1 that fp,p+i^>fp,p+n for «>2 (see eq. (3.6a) andFig. 3.4). It is obvious that E(p,p + 1) < E(p,p + n) for n > 2. Similarly it is seen inFig. 3.4 that fp-i,p^>fp-n,p, and E(p—\,p)<E(p—n,p) for «>2. An importantconclusion is thus reached:


Figure 4.2 shows these relationships; eq. (4.9) is valid for high temperatures, andeq. (4.8) is surprisingly accurate for very high temperature.

We now consider the ionization term in eq. (4.4). From eq. (3.35a) the ioni-zation rate coefficient is approximately given as

FIG 4.2 Comparison of the rate coefficients for the two dominant collisionaldepopulating transitions, C(p,p+l)/F(p,p—l), against Te for neutral hydrogen.At high temperatures eq. (4.9), C(p, p+l)>F(p, p—\~), is valid, while at lowtemperatures eq. (4.36), C(p, p+l)<F(p, p—T) is valid. The approximateexpression for Byron's boundary, eq. (4.56), is shown with the open circles.

From eq. (3.15) we can approximate eq. (4.10) to

We conclude that, for p^>2,

We will see later that, in reality, eq. (4.11) is valid even for p = 2.

POPULATION DISTRIBUTION AND POPULATION KINETICS

We now turn to the radiative decay rate in eq. (4.4), or the natural lifetime.Equation (3.8) with eqs. (1.4) and (1.4a) and with the approximation gbb=lleads to

We adopt the factorization

Then we obtain the approximation for large p:

with

with

Figure 3.5(a) compares this approximation with the exact radiative decay rates.The above discussions lead to an order-of-magnitude estimate of the relax-

ation time,

and the formula for large n

where we have used eqs. (1.4) and (1.4a). By approximating ln/> ~ 0.7 ' ^ fp, which isreasonably good for p ~ 3 ~ 30, we have

88

We now compare the relaxation times of eq. (4.4) for/? > 2, or eq. (4.14a), witheq. (4.15). S(l) is given approximately by eq. (3.35a) with p=\, or eq. (4.10)multiplied by the exponential factor exp( — z2R/kTe), Thus, it is obvious thatS(l) < S(p) for p > 1 even for the present high temperature. This inequality,together with eq. (4.11), leads to S(l) <C C(p,p + 1). We may conclude that, even ifthe radiative decay terms in eq. (4.14) were absent, we have £ri(l) 3> tri(p) withp > 2. In many cases, the radiative decay terms make a significant contribution, oreven predominate over the collisional terms for low-lying levels or for low density.It is thus concluded that, under usual conditions, £ri(l) is longer than tri(p) withp > 2 by many orders of magnitude:


We consider the relaxation time of the ground-state atom population. This isgiven by an equation similar to eq. (4.4) with the Y,F terms and the XL4 termsabsent. We suppose that the atoms are acted upon by a plasma in the first sense.As we will see in the next section, unless the density of the plasma is very high,a significant fraction, or even almost all, of the excitation flux out of the groundstate returns to this state by radiative decay. Thus the EC terms could be elimi-nated in effect. As a rough approximation the relaxation time of the ground-statepopulation may be given as

The ion density nz is expected to have a relaxation time constant of similarmagnitude to tr.

Imagine a situation in which an ensemble of atoms in the ground state is putinto a plasma or an electron gas having a certain Te and ne. They begin to beexcited and ionized. Appendices 4B and 5A treat this problem taking the exampleof neutral hydrogen. The temporal development of the excited-level populations(Appendix 4B) and of ionization (Appendix 5A) will be examined later in detail.Figure 4B.1 shows an example. Low-lying excited levels are virtually isolated fromother levels under the condition of this figure, so that the only dominant couplingsare the influx of population from the ground state, the first term of the first line ofr.h.s. of eq. (4.2), and the radiative out-flux, or the ^2(q<P)A(p,q) term in thesecond line. This situation is exactly the same as the bucket model of Fig. 4.1. Itspopulation follows the simple exponential behavior as given by eq. (4.3) withA«(0) = — «0- The relaxation time is given by eq. (4.14), where the radiative term isdominant. For level/? = 2, the relaxation time given from eq. (4.13), ?ri(2)~ 1.4 x10~9 s is consistent with this figure. On the other hand, the populations of higher-lying levels couple strongly with each other, and their temporal development israther complicated. It is found, however, that the relaxation of these populationsas a whole is given by the relaxation time of a certain critical level, which has thelongest relaxation time among the excited levels. In this example, this level isp = 4-5 and ?ri(5) ~ 3 x 1CT8 s. This level will be called Griem's boundary level inthe later part of this chapter.


Depletion of the ground-state atoms is very slow, with the relaxation time ofIK 1CT4 s in the example of Fig. 4B.1. This is to be compared with the overallrelaxation time of the excited-level populations, t~ 3 x 10~8 s. Thus it is seen thateq. (4.16) actually holds. It is concluded that, much before the ground-statepopulation changes appreciably, excited level populations relax to their stationary-state values that are defined by the ground-state population at this instance.

Appendix 4B also presents the opposite situation in which the system startsfrom ions and they begin to recombine with electrons at t = Q. See Fig. 4B.4. Thetemporal development of excited-level populations is very different from the firstcase. However, a similar feature is found that the overall relaxation time of theexcited-level populations is determined by the longest relaxation time among thelevels, just like the above, i.e. t^2 x 1CT7 s, and it occurs much before the iondensity changes appreciably, i.e. IK 1CT2 s. Equation (4.16) with the l.h.s. replacedby the relaxation time of the ion density actually holds. It is thus concluded thatexcited level populations relax rather quickly to their stationary-state values thatare defined by the ion density at this instance.

We also see in Appendix 4B that the total number of atoms in excited levels ismuch smaller than the ground-state population in the first case, or the ion densityin the second case.* We thus conclude that, except for very extreme situations, wemay assume

Now we suppose in Fig. 4B.1 that, after the excited-level populations havereached the stationary-state values, or ?> 3 x 1CT8 s, the ground-state populationbegins to change slowly; this may be due to its depletion by ionization or for someother reasons. Then, we expect that all the excited-level populations would followclosely the change in the ground-state population, almost exactly in parallel. Sincethe relative magnitudes of the excited-level populations to the ground-statepopulation is very small, the time derivative values in eq. (4.2) for excited levels aremuch smaller than that for the ground state. We further remember the inequalityeq. (4.16), along with eq. (4.4). These two facts indicate the following. In the rateequation (4.2) for excited levels, the magnitude of the influx terms, i.e. the sum ofthe first, third and fourth lines, and that of the out-flux terms, i.e. the second line,are quite large. Even so, their absolute magnitudes are almost equal and their signsare opposite, so that they almost cancel each other to leave a very small value forthe time derivative. On the other hand, for the ground-state atoms both the termsare very different in magnitude, resulting in a time derivative value determined bythe leading term. In the example of Fig. 4B.4, we reach a similar conclusion in theopposite situation, too.

* In the second case the sum of the excited-level populations diverges if we assume an infinitenumber of excited levels. See Fig. 4B.5. In Chapter 9, it will be shown that this assumption is unrealistic,and, in a plasma, the number of discrete levels is finite.


Except for cases in which «(1) or our plasma in the first sense (Te and ne)undergoes a very rapid change so that the excited-level populations do not haveenough time to finish their relaxation, we can expect that, at a certain time, theexcited-level populations have already reached their stationary-state values thatare given by nz_ as well as by Te and ne at that instance. In the second case, theexcited-level populations would be given by nz, Te, and ne at that instance. We callthese situations the quasi-steady state (QSS). In Appendix 4B we will examine thevalidity condition of this approximation to hold.

while the time derivative should be retained for the ground-state population«z_i(l) and the ion density nz. This is the mathematical expression of our quasi-steady state (QSS). This is equivalent to formulating our problem as follows. Oursystem is divided into two subsystems: the populations of all the excited levelsconstitute the first subsystem which is "sandwiched" by the second subsystemwhich consists of the ground-state population and the ion density.

The set of coupled linear equations, eq. (4.2) with eq. (4.18), for the firstsubsystem can be expressed in matrix form:

The dimension of the matrix, or the number of levels, should be limited at someappropriate values, say 10, 20, or even 100. As we noted after eq. (4.2), thisproblem will be addressed later on. The elements of the square matrix on the l.h.s.and the column matrices on the r.h.s. are terms on the r.h.s. of eq. (4.2) and arefunctions of Te through the collisional rate coefficients, and of ne.

It is obvious from the structure of eq. (4.19) that this equation is readily solvedfor p> 2 as the sum of the two terms, each of which is proportional to n(l) and nz,respectively. We assume the solution in the form

Quasi-steady-state solution

The above considerations suggest that, in the coupled rate equations (4.2), to retainthe small time derivative for excited levels does not make much sense. Rather wemay approximate the time derivative of the excited-level populations to zero:


with the Saha-Boltzmann coefficient,

Equation (4.20) indicates that an excited-level population is a sum of two popu-lations, the first term being proportional to the ion density nz and the second tothe ground-state atom density «(1). (The words ion and atom are used here toindicate ions in two successive ionization stages.) Figure 1.9 shows this situationschematically.

We substitute eq. (4.20) into the coupled linear equations (4.19). We thenobtain two sets of coupled equations, one for r0(/>) and another for r\(p). Thesolutions rQ(p) and r\(p) are called the population coefficients and they are func-tions of Te and «e. It is rather straightforward to obtain solutions by matrixinversion. Table 4.1 shows several examples ofr0(p) and r^p). In this calculation,we adopted the accurate transition probabilities and the rate coefficients ratherthan the various approximations introduced in the previous chapter and in theearlier part of this chapter.

The second subsystem, «(1) and nz, needs another treatment. This problem willbe considered in Chapter 5.

We consider two extreme situations which may appear somewhat unrealistic:

1. Our plasma has no ions, nz = 0, and the solution is r^p) = 1 for certain excitedlevels. (This latter statement is seen to be actually realized in Table 4.1 (a) forlow-lying levels under certain conditions.) Then, we have n(p)/Z(p) = «(1)/Z(1),which is equal to eq. (2.3). Thus we have the Boltzmann distribution ofpopulations for these levels p with respect to the ground-state atom population.

2. Our plasma has no ground-state atoms, «(1) = 0, and we have rQ(p) = 1 forcertain levels. (See Table 4.1(b). The latter relation is seen to hold for manylevels in a large range of Te and «e-) Then, we have n(p) = Z(p)nzne, i.e. eq. (2.7).Thus we have the Saha-Boltzmann distribution of populations with respect tothe ion and electron densities, i.e. this level is in LTE.

Thus, rQ(p) and r\(p) are a measure of the populations of level p with respect tothose in thermodynamic equilibrium.

It is useful to note here another salient feature of the population coefficients.On the r.h.s. of eq. (4.2), for sufficiently high «e the terms of radiative transitions,A(p, q) and (3(p), become small in comparison with the terms of the correspondingcollisional transitions, F(p,q)ne and a(p)ne, respectively. By using eq. (3.31) werelate C(p,q) with F(q,p), and eq. (3.40) relates S(p) with a(p). The two sets ofcoupled equations for r0(p) and r\(p) are combined into one set of equationsfor [TQ(P) + ri(p)]. These equations are satisfied with unity substituted for each[ro(p) + ri(p)]; that is,

TABLE 4.1 (a) Population coefficients r0(p) and (in parentheses) r^p) for neutral hydrogen. Te= 1 x 103 K: The indices give thepower of 10 by which the coefficient values must be multiplied.

plg«e

— > — oo1214161718192021222324— > oo

2

2.28"19(5.29"22«e)2.89"19(5.29"10)5.68"19(5.29"8)2.84"18(5.29"6)1.12-17(5.29-5)7.30"17(5.29"4)8.34"16(5.27"3)1.20"14(5.03"2)9.24~14(3.46~1)2.28"13(8.41"1)2.66"13(9.81"1)2.71"13(9.98"1)2.71"13(LOO)

3

8.86"10(2.or22«e;1.13~9(2.01~10)2.24"9(2.01"8)1.16-8(2.01~6)4.95"8(2.03"5)4.24"7(2.13"4)5.27"6(2.9r3)1.85~5(4.39~2)2.33"5(3.4r1)2.39-5(8.4Q-1)2.40-5(9.81~1)2.40"5(9.98"1)2.40~5(1.00)

4

) 2.40"6(1.37"22«e)3.08~6(1.37~10)6.22~6(1.37~8)3.55~5(1.37-6)1.97~4(1.39-5)2.25~3(1.54-4)1.10~2(2.49~3)1.60~2(4.22~2)1.67~2(3.34~1)i.es- s s-1)1.68-\9.65-1)i.es- .sr1)i.es- .ss"1)

1.021.312.741.971.558.931.531.671.691.691.691.691.69

5

-4(l.H-22«e)

-4(i.ir10)~4(1.11~8)-3(1.12~6)-2(1.13-5)-2(1.26-")"1(2.08"3)-\3.56-2)"1(2.82"1)-\6.98-1)- s.is-1)-\8.29-1)"HS.Sl"1)

3.06"4.04"9.43"1.45"4.43"6.21"6.71"6.79"6.80"6.80"6.80"6.80"6.80"

7

3(9.43"23«e)3(9.42"u)3(9.37"9)\8.23-7')1(5.84"6)\5.12-5)l(8.Q6~*)\\.3r2)l(\.Q9~l)1(2.69"1)\3.14--1)\3.20-1)\3.20~l)

2.203.071.207.739.029.399.489.499.499.499.499.499.49

10

"2(8.67"23«e)"2(8.60"u)"1(7.82"9)"1(2.11"7)"1(1.02"6)"1(8.22"6)-\l.28~*)"1(2.18"3)-\\.13-2)"1(4.28"2)" S.OO"2)" S.OS"2)~1(5.09~2)

15

7.45"2(7.67"23«e)1.32"7.37"9.79"9.92"9.95"9.96"9.96"9.96"9.96"9.96"9.96"9.96"

1(7.22"11)1(2.23"9)\\.93-8)\S.56-8)1(6.82"7)Hi.oe"5)l.Sl"4)l S"3)

1(3.54"3)1(4.13"3)1(4.20"3)1(4.21"3)

TABLE 4.1(b) Population coefficients r0(p) and (in parentheses) r^p) for neutral hydrogen. Te= 1.28 x 105 K: The indices givethe power of 10 by which the coefficient values must be multiplied.

plg«e

— > — oo1214161718192021222324— > oo

6.86"6.87-6.90-6.98-7.09-7.3Q-7.87-9.15-9.8r9.9r9.92-9.92-9.92-

2

\\.93-22ne)l(\.93-w)1(1.92-8)\\.90-6)Hl.S?"5)\1.79~*)\\.46-3)Hs.ss-3)1(7.7Q-3)^s.or3)Hs.os-3)^s.os-3)^s.os-3)

7.33-7.34-7.38-7.5r7.71-8.35-9.44-9.88-9.98-9.99-9.99-9.99-9.99-

3

\8.55-23ne)Hs.ss-11)\8A6~9)^s.is-7)\7.6\-6)1(5.72-5)1(2.36-4)\6.\\-4)1(8.02-4)\S.29~4)1(8.32-4)1(8.32-4)1(8.32-4)

4

7.77-1(6.17-23«e)7.79-1(6.14-11)7.84-1(6.03-9)s.oe-Hs.sr7)8.56~\4.15-*)9.47-1(1.66-5)9.88-1(4.65-5)9.98-1(L09-4)1.00(1.4Q-4)1.00(1.45~4)1.00(1.45~4)1.00(1.45~4)1.00(1.45~4)

5

7.99-1(5.08-23«e)8.02-1(5.04-11)8.09-1(4.9Q-9)s.sr^s.sg-7)93r\\.85-6)9.83-\5.Q8-6)9.97-1(1.25-5)9.99-1(2.82-5)1.00(3. 63-5)1.00(3. 74-5)1.00(3.76-5)1.00(3.76-5)1.00(3.76-5)

7

8.23-1(4.08-23«e)8.27-1(4.02-11)8.4Q-\3.76-*)9.44-1(1.36-7)9.86-\3.5r1)9.97-1(7.55-7)i.oo(i.7r6)1.00(3.75-6)1.00(4.8Q-6)1.00(4.95-6)1.00(4.97-6)1.00(4.97-6)1.00(4.97-6)

10

8.39-\3Ar23ne)

8.46-1(3.32-11)8.89-1(2.42-9)9.90~\2Al~s~)9.98-1(4.88-8)1.00(9.6r8)1.00(2.1Q-7)1.00(4.54-7)1.00(5.8Q-7)1.00(5.98-7)1.00(6.0Q-7)1.00(6.0Q-7)1.00(6.0Q-7)

15

8.47-1(2.84-23«e)8.63-1(2.59-11)9.7r1(5.89-10)9.99-1(2.65-9)1.00(4.93-9)1.00(9.34-9)1.00(2.0r8)1.00(4.32-8)1.00(5.5Q-8)1.00(5.67-8)1.00(5.69-8)1.00(5.69-8)1.00(5.69-8)


for anyp and any Te. It is seen in Table 4.1 that this is actually the case. Figure 4.3shows r0(p) and r^(p) in the high-density limit for a wide range of Te. Equation(4.21) is seen to hold.

From now on in this chapter, our objective will be to investigate in detail theexcited-level populations under a variety of plasma conditions, or Te and «e. Thestructure of the excited level population, eq. (4.20), is schematically depicted inFig. 1.9. We call the second population the ionizing plasma component and the firstpopulation the recombining plasma component. This structure suggests that itwould be natural for us to examine the characteristics of each component of thepopulations first. Once we obtain these characteristics, we may combine thesecomponents to obtain the actual population.

The present method of treating eq. (4.2) with the assumption of eq. (4.18) issometimes called the collisional-radiative (CR) model*

For hydrogen-like ions with nuclear charge z, in which we are primarilyinterested, various atomic parameters scale against z, as we have seen inAppendix 3A. Plasma parameters scale accordingly: Te scales according to z2 asseen in eq. (3A.9). Scaling for «e is derived in Appendix 5B. According to thesescaling properties, as has already been noted in Chapter 1, we adopt Ts/z

2 asthe reduced electron temperature and «e/z

7 as the reduced electron density. The

* The term collisional-radiative model is used in various senses, but in this book we adopt thisdefinition.

FIG 4.3 Population coefficients r0(p) and r^p) for several/>'s in the high-densitylimit.


oblique line drawn in Fig. 1.2 is a scale which depicts this scaling law for the plasmaparameters. As will be shown in Appendix 5A, these scaled quantities make therate equation (4.2) approximately independent of z. The reason why eq. (4.2) isonly approximately so is the breakdown of the z-scaling of the excitation andionization cross-sections as seen in Fig. 3.11. For higher energies the scalingbecomes better, so that for higher temperatures this approximation becomes better.In the following, we express various relationships by using these reduced parameters.

Another important exception is the ionization-recombination relationship.This point will be examined in Chapter 5.

4.2 Ionizing plasma component

In this section we study the ionizing plasma component of excited level popula-tions in detail. According to eq. (4.20) the ionizing plasma component of thepopulation of level p is defined as

See also Fig. 1.9. As an example we take neutral hydrogen (z— 1) in a plasma (inthe first sense) with Te= 1.28 x 105 K with varying ne. Table 4.1(b) shows r^(p).Figure 4.4 shows the excited-level populations, eq. (4.22), as functions of ne.The reader can locate these plasmas on the «e — Te plane of Fig. 1.2, and draw a

FIG 4.4 The ionizing plasma component of the excited-level population against ne

[m~3]. The ordinate is the logarithm of n^(p) divided by the statistical weight.Calculation is for neutral hydrogen with «(l)=lm~3 for Te=1.28 x 105 K.(Quoted from Fujimoto, 1979b; with permission from The Physical Society ofJapan.)

IONIZING PLASMA COMPONENT 97

FIG 4.5 Population distribution of the ionizing plasma component of the excited-level populations. The ordinate is the same as Fig. 4.4, and the abscissa is thelogarithm of the principal quantum number of the level. Calculation is forneutral hydrogen with «(1) = 1 m~3 for Te= 1.28 x 105 K. The approximationseqs. (4.24) and (4.31), are compared with the calculation. The dash-dotted linesindicate the boundary pG between the corona phase and the saturation phase asgiven from the comparison of the collisional and radiative rates, eq. (4.25).(Quoted from Fujimoto, 1979b; with permission from The Physical Society ofJapan.)

horizontal line. Since the populations are proportional to «(1), we have assumedn(l) = 1 [m~3] in Fig. 4.4. The ordinate is the logarithm of the population ofexcited levels divided by the statistical weight, and the abscissa is the logarithm of«e [m~3]. This figure is approximately correct for hydrogen-like ions with theabscissa replaced by lg(«e/z

7).* Figure 4.5 shows the population distribution overexcited levels for several ne's. It is noted that the abscissa is the logarithm of theprincipal quantum number of the level. This plot is different from the conven-tional Boltzmann plot, in which the abscissa is the energy of the level.

* We follow the convention that \gx means Iog10x, and Inx is logex.

f 1 1


Figure 4.4 suggests that, according to the «e dependence of the populations, wecan define two regions of ne for each of the excited levels. The first region is lowdensities where n\(p) is proportional to ne. We say that this level is in the coronaphase in this region. In the higher-density regions the slope of the population issubstantially less than unity, and in the highest-density region it is zero. We saythat the level is in the saturation phase in these regions. Although the region of thecorona phase and that of the saturation phase can be clearly defined, the transitionbetween them is rather gradual and the boundary cannot be sharply defined; if wetake level p = 5 as an example, then the boundary is about «e= 1017-1018 m~3.Figure 1.10(a) depicts schematically these phases and the boundary in the «e—pplane.

In the low-density limit all the levels are in the corona phase. In Fig. 4.4 withan increase in ne, deviation from the linear dependence begins, starting from thehigher-lying levels. In other words, with the increase in «e the boundary betweenthe corona phase and the saturation phase in the energy-level diagram comesdown. This means that, for a certain «e, lower-lying levels are in the corona phase,while higher-lying levels are in the saturation phase. See Fig. 1.10(a). Figure 4.5includes the boundary between these phases in the population distribution; dis-cussions about this boundary will be given later. This figure clearly indicates thatthe characteristics of the population distribution are different between these twophases. The lowest excited level p = 2 persists in the corona phase up to«e~ 1021 m~3. When this level enters into the saturation phase all the populationsbecome independent of «e (Fig. 4.4).

Since we have solved the coupled equations (4.19) with nz = Q, we can examinethe magnitude of each term on the r.h.s. of eq. (4.2). We now take levelp = 5, called simply level 5, as an example of the excited levels for the purpose ofillustrating various properties of the populations and various phenomena con-cerning the population kinetics against a change in «e. We consider how thepopulation of level 5 is formed. Figure 4.6 shows the breakdown of the con-tributions from various transitions concerning level 5: Figure 4.6(a) shows thedetails of the influx, or the first and third lines in eq. (4.2). The fourth line isabsent for an ionizing plasma. Each flux is normalized by the total populatingflux into this level. The hatched area shows the radiative transition and theblank area the collisional transition. The number denotes the principal quantumnumber of the level from which this transition originates. For instance, the blankarea with "1" means the direct excitation flux from the ground state to level 5.Figure 4.6(b) shows the details of the out-flux, or the contributions from theterms in the second line in eq. (4.2), normalized by the total depopulating fluxfrom this level. The total populating flux and the total depopulating flux are equal,of course. The meanings of the blank and hatched areas are the same, but thenumber denotes the level on which this transition terminates. For instance, thehatched area with " 1-4" indicates the radiative decay from level 5 to the lower-lying levels.

IONIZING PLASMA COMPONENT

FIG 4.6 Breakdown of the (a) populating and (b) depopulating fluxes concerninglevel p = 5 into individual fluxes. This condition corresponds to Figs. 4.4 and4.5. They are normalized by the total populating or depopulating flux. Theblank area indicates the collisional transition and the hatched area the radiativetransition. Numerals indicate the principal quantum number of the level(a) from which this transition originates and (b) on which this transitionterminates. Neutral hydrogen with Te= 1.28 x 105 K. (Quoted from Fujimoto,1979b; with permission from The Physical Society of Japan.)

Corona phase

In the density regions lower than «e ~ 1017 m~3, level 5 is in the corona phase. SeeFigs. 4.4, 4.5, and 1.10(a). As we have already seen in Fig. 4.6(a) the dominantcontribution to the populating flux comes from the direct excitation from theground state with the remaining about 20% contribution coming from the cascade,or the radiative transitions from the still higher-lying levels which have been

99


populated from the ground state. In the next subsection it will be shown that thecascade contribution should be about 20%, depending little on the principalquantum number of the level. The depopulation is by radiative decay. Thus in thecorona phase the corona equilibrium is valid: i.e. the excited level is populated bythe direct excitation from the ground state and it is depopulated by the radiativedecay. This is the reason for our nomenclature of this phase.

Figure 4.7 shows a sketch of the dominant populating and depopulating fluxesamong the levels. The blank and hatched arrows indicate, respectively, the colli-sional transition and the radiative transition, and the width of an arrow is pro-portional to the magnitude of this flux. The fluxes concerning level 5 are shown inmore detail than those for other levels. This figure is for ne= 1012 m~3, where allthe levels with p < 20 are in the corona phase, and it is seen that the coronaequilibrium is actually valid for these low-lying levels. Figure 1.10(a) includes thesimplified picture of the population kinetics.

FIG 4.7 Sketch of the dominant populating and depopulating fluxes, or flows ofelectrons, in the energy-level diagram. Numbers at left are the principal quantumnumber of the levels and "c" means the continuum states, or the ion, z. Thepositions of the levels are not to scale. The blank and hatched arrows indicate,respectively, the collisional and radiative transitions. Fluxes to and from levelp = 5 are given in more detail than for other levels. The width of an arrow isproportional to the magnitude of the flux. All the levels are in the corona phase.At right the total flux of ionization starting from the ground state is given withthe arrow and the number in it indicates the collisional-radiative ionization ratecoefficient. For neutral hydrogen with Te= 1.28 x 105 K and «e= 1 x 1012 m~3.(Quoted from Fujimoto, 1979b; with permission from The Physical Society ofJapan.)


The population in this phase is given from eq. (4.2), with the neglect of thecascade contribution, by

We now consider the population distribution over the excited levels. In order tomake our discussion simple, we assume the levels to be high-lying, i.e. />;$>!. Fromeq. (3.6a) or Fig. 3.4 it is seen that the oscillator strength j\^p is approximatelyproportional to p~3 (fi,p^- 1.6 xp~3 from Table 3.1(b)). Since, in eq. (3.29), wecan neglect the dependence of the excitation energy E(l,p) on p for the presenthigh temperature, C(l,p) is proportional top~3. We have already seen in eq. (4.13)and Fig. 3.5(a) that ^2(q<P)A(p,q) is approximately proportional to p~4'5. Itfollows from eq. (4.23) that KI(/>) is approximately proportional to p~3/p~4'5 orp1'5. Therefore we have

Figure 4.5 compares this approximation with the results of the numerical calcu-lation. For the low density, this approximation is good for high-lying levels thatare in the corona phase. For low-lying levels including p = 2, the above variousapproximations become poor, and the populations deviate from eq. (4.24).Figure 1.10(a) includes the approximate population distribution, eq. (4.24).

*Cascade contribution

In the above discussion we may express the excitation rate coefficient as

where CQ is a constant. See eq. (3.29). Then eq. (4.23) with eq. (4.13) reduces to

The populating flux to level p by cascade is given as ^2q>p n\(q)A(q,p), which maybe approximated by the integration,

We use eq. (3.8) to yield (gbb is assumed to be 1)


We define the cascading contribution as

The value of C is found for p = 4, 6, and 10 to be 0.187, 0.190, and 0.193,respectively. If we take into account the cascade contribution in eq. (4.23b) fromthe still higher-lying levels, (, would be slightly larger than 20% for these lower-lying levels; this is in accordance with Fig. 4.6(a).

Transition from the corona phase to the saturation phase - Griem's boundary

In Fig. 4.4, with an increase in «e, level 5, for example, makes a transition from thelow-density region, the corona phase, to the high-density region, the saturationphase, at about «e= 1017-1018 m~3. It is seen in Fig. 4.6 that at about this «e atransition takes place both in the populating flux (a), and in the depopulating flux(b) as well. We will examine whether these simultaneous transitions are a merecoincidence or not. First we look at the depopulating flux. This transition is thechange of the dominant terms in the second line of eq. (4.2) from the radiativedecay to the collisional transitions. At this «e we have

We have seen that the dominant transition in the r.h.s. terms is C(p,p + l)«e. (Seeeqs. (4.6), (4.9), and (4.11); see also eq. (4.14).) Thus, we may conclude that thetransition in the depopulating flux out of level p takes place when

We note that, at this «e, level (/>—!) is still in the corona phase (see Figs. 4.4and 1.10(a)), and its population is given approximately by eq. (4.23);

holds. This boundary «e is given approximately from eqs. (4.7) and (4.13) as

With regard to the populating flux, Fig. 4.6(a) indicates that the dominant fluxin the lower-density region is the direct excitation from the ground state as wehave seen, in the high-density region it is the excitation from the adjacent lower-lying level, level 4 in the present example. Thus, at this transition the followingequation holds:

For large p the value of «e given by eq. (4.28) is very close to that given byeq. (4.26). Thus we may conclude that in the populating and depopulating fluxes,Fig. 4.6(a) and (b), both the transitions should take place at about the same ne,and that these transitions give the transition of the level from the corona phase tothe saturation phase.

For a given ne, eq. (4.25) or (4.26) may be interpreted as giving the boundarylevel between the lower-lying levels which are in the corona phase and the higher-lying levels which are in the saturation phase. This boundary level is denoted asp®.See Fig. 1.10(a).

The boundary ne or the boundary level pG which gives the above transition iscalled Griem's boundary. This nomenclature is offered in honor of the workerwho proposed a critical «e in his discussion of LTE. This problem of the validity ofLTE will be discussed in detail in Appendix 5C of the next chapter. His criterion isessentially the same as the present eq. (4.25) and thus eq. (4.26). By substitutingappropriate values in G and H we obtain for Griem's boundary

gives even more appropriate numerical values. Figure 1.10(a) includes thisboundary with the label "GRIEM" attached.

In Fig. 4.5 the boundary level pG, as given by eq. (4.25), or given from thecomparison of the total collisional depopulation rate and the radiative decay ratein the second line of eq. (4.2), is plotted with the dash-dotted line. Figure 4.8 showsthe sketch of the dominant fluxes of electrons among the levels at ne = 1018 m~3,where Griem's boundary level is given by the dash-dotted line.

By substituting eqs. (4.7), (4.13), and (4.23a) into eq. (4.27a) we have


Instead of this equation the following simple expression

or


Saturation phase - ladder-like excitation-ionization

Figure 4.6 shows that, for level 5, in the «e regions higher than Griem's boundarythe dominant populating process is collisional excitation from level 4 and thedominant depopulating process is collisional excitation to level 6. Here weremember eq. (4.9), i.e. collisional excitation is more likely than deexcitation, andeq. (4.11), i.e. ionization is rather minor. The above feature is also seen in Fig. 4.8for the levels lying above Griem's boundary. It is concluded that, for a level in thesaturation phase, the dominant populating flux to this level is the collisionalexcitation from the adjacent lower-lying level and the dominant depopulating fluxis the collisional excitation to the adjacent higher-lying level. The upward flux ofpopulation by stepwise excitation is thus established in eq. (4.2) with eq. (4.18):

We name this mechanism of multistep excitation, ladder-like excitation. Since thechain of this excitation flux results in ionization, we may call it ladder-likeexcitation-ionization. Within the approximation of eq. (4.30), the magnitude ofthis flux is independent of p, then we have ni(p)<yiC(p,p+\}~1. By using theapproximation (4.7), we have Ki(/>)oc/>~4, or

FIG 4.8 Sketch similar to Fig. 4.7 except that « e = l x ! 0 l s m . Theboundary level pG between the corona phase and the saturation phase, as givenby eq. (4.25), lies between levels p = 3 and 4, as indicated by the dash-dottedline. The ladder-like excitation-ionization flux through a high-lying level, levelp=lO, is shown with the filled arrow. (Quoted from Fujimoto, 1979b; withpermission from The Physical Society of Japan.)


We may call this distribution the minus sixth power distribution. Figure 4.5 com-pares this approximation with the results of the accurate numerical calculation.This simple approximation is surprisingly good. In Fig. 1.10(a), the upper half ofthe region higher than the boundary "GRIEM" shows the features of this satur-ation phase. In real plasmas in which the ionizing plasma component is dominantthe temperature is high, and the boundary "BYRON" and the region lower thanthis boundary are absent, as will be discussed in the following subsection.

With a further increase in «e no significant change is seen in Fig. 4.6(a) and (b),but the Ke-dependence of the population in Fig. 4.4 begins to show a new featureat about «e= 1021 m~3. We note that, with this increase in «e, Griem's boundarycomes down, and that at «e~1021 m~3 it reaches the first excited level p = 2,Figure 4.9 shows a sketch of the dominant fluxes at «e= 1022 m~3. Except for thedownward arrow 2 —> 1 all the arrows are collisional. Roughly speaking, the chainof the populating-depopulating fluxes of eq. (4.30) starts from the ground stateand it controls the populations of all the excited levels.

In this highest-density region, the ladder-like excitation-ionization mechanismcontrols the populations of all the excited levels, i.e.

If we eliminate «e from this relationship it becomes independent of «e. This is thereason why all the excited level populations in Fig. 4.4 are independent of «e-

FIG 4.9 Sketch similar to Fig. 4.7 except that «e= 1 x 10 m . All the excitedlevels are in the saturation phase and the mechanism of the multistep ladder-likeexcitation-ionization, eq. (4.32), is established starting from the ground state.Only the downward arrow (2 —> 1) is the radiative transition. (Quoted fromFujimoto, 1979b; with permission from The Physical Society of Japan.)

for large/?. Equation (4.33) is derived rather straightforwardly from the relation (4.5)with eq. (4.7), for the distribution eq. (4.31). The proof of eq. (4.34) may be givenon the basis of eqs. (3.6a) and (3.29). Of course, the exponential factor in the latteris neglected. It is seen in Figs. 4.6 and 4.9 that these relationships actually holdapproximately.

So far, we have assumed high temperatures so that the approximation (4.5) isvalid in the above discussion. See Fig. 4.2. For excitation from the ground state,p = l, the exponential factor in eq. (3.29) cannot be neglected even on the presenthigh-temperature assumption. Equation (4.7) and therefore eq. (4.31) cannot beapplied to p = 1. If the temperature becomes low, this neglect of the exponentialfactor may not be justified even for excited levels. If the exponential factor isretained for excitation from excited levels,/? g 2, eq. (4.31) is no longer valid. Evenin this case, so far as eq. (4.9) holds, eq. (4.30) should still be valid, but, in this case,the population distribution should include the exponential factor besides the p~6

factor. Instead of n(p)/g(p) we take ri(p);by doing so the exponential factor isabsorbed in [Z(p)/Z(l)] as shown in eq. (4.20). Figure 4.10 gives a plot of r^(p) forseveral cases of neutral hydrogen as well as of hydrogen-like ions. This figure alsoshows the two lines representing (p~6 x 2) and (p~6/2). It is seen that in the rangeof ne/z

7 g 1022 m~3 and Te/z2 g 3 x 104 K, the approximation


The population distribution is given by eq. (4.31). Figure 4.5 shows that thepopulation distribution of all the excited levels is given approximately by eq. (4.31).

If we examine Fig. 4.6 and Fig. 4.9 in more detail, we find that the relationship(4.30) is in fact a rather poor approximation to the actual populating and depo-pulating fluxes. For example, eq. (4.30) accounts only for 60% of the total fluxesfor level 5 and only 45% for level p = 20. Still, eq. (4.31) describes the populationdistribution in Fig. 4.5 rather well. This puzzling situation is resolved when wenote that, if eq. (4.31) holds, other balance relations are valid:

is valid within a factor of 2.We now consider the minus sixth power law, eq. (4.31), from another view-

point. The density of atomic states in a unit energy interval is proportional tog(p)/AE(p)Ap=l, or to p5. See eq. (1.5). If we regard the population flux of theladder-like excitation-ionization as the flux of electrons in the energy space, and ifwe require that the "speed" of this flow be finite, the population distributionshould be n(p)/g(p)<xp~5. This is inconsistent with eq. (4.31) or (4.35). We haveto think about the possibilities that something is wrong in our above discussions

and


leading to these equations. One possibility is the approximation of C(p,p +1)based on the optically allowed transition, eq. (3.29) or eq. (4.7), C(p,p+ l)oc/>4.This approximation may not be valid for very large p. In our numerical calcula-tion, the cross-sections for these transitions are based on the impact parametermethod calculation which should take into account the optically forbidden transi-tions as well, but it is not certain whether they are very accurate for very large p.If, in the numerical calculation, we use the rate coefficients derived from theMonte Carlo calculation for classical electron orbits, we obtain a distributionni(p)/g(p)°tp~5 f°r levels lying higher than p~2Q, For lower levels which areimportant in practical situations thep~6 distribution is found to be valid. We mayconclude that the present results are valid for excited levels of practical interest.

Low-temperature case

In practical situations we are unlikely to encounter the low-temperature case ofionizing plasma. However, for the purpose of keeping our theory transparent, weexamine this case briefly.

FIG 4.10 The population coefficient r\(p) against p. - - - - - - ; z=l , «e/z7 =

1023 m~3, and Te/z2 = 3.2 x 104 K; : z=l , «e/z

7=1023 m~3, andre/z

2=1.28x 105 K; : z = 2, «e/z7=1023 m~3, from top to bottom

Te/z2 = 8 x 103 K, 1.6 x 104 K, 3.2 x 104 K, and 5.12 x 105 K; : z = 2,

«e/z7= 1021 m~3, and Te/z

2= 5.12 x 105 K; : z = 2, «e/z7= 1019m-3, and

Te/z2 = 5.12 x 105 K; : (p~6 x 2) and (p~6/2). (Quoted from Fujimoto

and McWhirter, 1990; copyright 1990 with permission from The AmericanPhysical Society.)

-5


FIG 4.11 Population distributions r±(p) for several ne's. Neutral hydrogenwith Te= 1 x 103 K. The boundary between eq. (4.9) and eq. (4.36) is given bythe dotted line, and another boundary, pG, as given by eq. (4.25) is given by thedash-dotted lines. (Quoted from Fujimoto, 1979b; with permission fromThe Physical Society of Japan.)

Table 4.1 (a) shows the population coefficients r^p) for Te= 1 x 103 K. Thepopulation distributions are shown in Fig. 4.11, which corresponds to Fig. 4.5.The reader may locate these plasmas in Fig. 1.2. In Fig. 4.11 the ordinate is thelogarithm ofr^p) instead ofnl(p)/g(p), since the latter quantity has a too strong/i-dependence that is due to the exponential factor as discussed above. Weremember that ri(p) = 1 means that level p is in thermodynamic equilibrium withthe ground state, as has been noted near the end of the preceding section. As isseen in Table 4.1 (a) and in this figure, in the high-density limit, this equilibrium isactually established for the low-lying (p < 6) levels. In this limit the high-lyinglevels (p > 6) are still controlled by the ladder-like excitation-ionization flux,showing ri(p)<xp~6. The breakdown of the ladder-like excitation, eq. (4.30),for the low-lying levels results from the breakdown of eq. (4.9), which is due tothe presence of the exponential factor in eqs. (3.29) and (3.31). Figure 4.2 showsthe ratio C(p,p+\)/F(p,p— 1) against Te for several p's. In low temperatures therelationship opposite to eq. (4.9)

instead of eq. (4.30). This results in the thermal (Boltzmann) distribution betweenthe levels/? and (p — 1), and therefore among all the levels 1 <p < 6 in Fig. 4.11.It may be interesting to note that the population values of these excited levels aretoo low to be realistic; «i(2) ~ 1CT68«(1) in this example.

With a decrease in ne the balance relation, eq. (4.37), ceases to hold, startingfrom the lowest-lying level p = 2, because the radiative decay takes over the col-lisional deexcitation on the r.h.s. of eq. (4.37). Figure 4.11 contains this boundaryPG as determined by eq. (4.25). Levels p <po, are in corona equilibrium. For levelsPa<P<6, the Boltzmann distribution (same r\(p) values for different p's-seeTable 4.1 (a)) is still valid. In Fig. 1.10(a), the region lower than the boundary"BYRON" depicts the feature of these lower-lying levels at low temperature.The ladder-like excitation-ionization mechanism dominates the populations ofthe levels p>6, since Te = 1 x 103 K is high enough for these higher-lying levels.See Fig. 4.2.

Example

In many practical situations we encounter plasmas which show the populationdistributions characteristic of the ionizing plasma component. Examples are low-pressure discharges, especially the positive column of a glow discharge. Figure 4.12shows an example of the experimentally determined populations in an argonpositive column plasma. By changing the discharge current, the authors changedne over three orders of magnitude, while they kept Te almost constant at 5 x 104 Kby adjusting the filling pressure. Figure 4.12(a) shows the ne dependence of popul-ations of several excited levels of neutral argon. The reader can locate this plasmaon Fig. 1.2 by drawing a short horizontal line. Note that we are dealing withneutral species, z=l. If we compare this figure with Fig. 4.4 we recognize closesimilarities between them: the density dependences of the populations in Fig. 4.12(a)appear to correspond to those of Fig. 4.4 for p > 3 and 1018 < «e < 1022 m~3. It isnoted that, in this experiment, the lowest-lying excited levels, corresponding top = 2 in Fig. 4.4, are not observed. This is because the population is determinedfrom the emission line intensity observed in the visible region of the wavelength. Ifour assumption is correct, we may conclude that level 4p'[3/2]1; corresponding top = 3 in Fig. 4.4, is in the corona phase in the lowest-density region while all theother levels are in the saturation phase, and further that all the populationssaturate completely for «e > 1019 m~3. The difference in the boundary «e values byone or two orders of magnitude between the two figures may be explained from


holds for low-lying levels. The boundary between eq. (4.9) and eq. (4.36) is given inFig. 4.2 for various p's as the ordinate value of 1 and by the dotted line in Fig. 4.11for this particular temperature. The boundary "BYRON" in Fig. 1.10(a) is thisboundary. A detailed discussion about this boundary will be given in Section 4.4.In this high-density limit the lower-lying levels are in the balance relation

FIG 4.12 Population distribution of excited argon atoms in a positive columnplasma. T e ~5x 104 K. (a) Dependences on ne. This figure corresponds toFig. 4.4 for neutral hydrogen, (b) Population distribution among the levelscorresponding to Fig. 4.5. (Quoted from Tachibana and Fukuda, 1973; withpermission from The Institute of Pure and Applied Physics.)

RECOMBINING PLASMA COMPONENT 111

the following two points: (1) we are comparing different atomic species, hydrogenand argon; (2) in the experiment, two of the four lowest-lying levels are meta-stable, and the resonance lines originating from the other two levels, which cor-respond to the Lyman a line in the case of hydrogen, are subjected to heavyradiation trapping, which will be dealt with later in Chapter 8. These facts result ina much reduced effective transition probability from the lowest-lying levels, whichis not taken into account in the theory and tend to reduce Griem's boundary ne forthe lowest excited level.

Figure 4.12(b) shows the population distribution over the excited levels for«e= 1 x 1019 m~3, where all the levels are in the saturation phase. Since the levelshave various energy values, we take the effective principal quantum number,eq. (1.1 a), as the abscissa. They follow the minus sixth power law except for thelowest-lying levels. Its deviation is in the opposite direction to those in Fig. 4.5.This inconsistency may be explained from the fact that the temperature of theexperiment of Te = 5 x 104 K is not sufficiently high for the exponential factor tobe neglected entirely.

4.3 Recombining plasma component - high-temperature case

In this section we study the recombining plasma component of the excited-levelpopulations. See Fig. 1.9. According to eq. (4.20) this is defined as

We further assume high temperature, i.e. Te/z2 is much higher than 1.5 x 104 K.

This boundary temperature is different from that in Section 4.1. This boundary isrelevant for the recombining plasma; as we will see later, in this range of tem-perature, collisional excitation from an excited level is more probable than colli-sional deexcitation from this level. See Fig. 4.2. We take neutral hydrogen in aplasma with Te= 1.28 x 105 K as an example for the purpose of illustration. Thereader may remember the horizontal line drawn on Fig. 1.2 for the plasmas ofSection 4.2.

Table 4.1(b) gives the result of calculation of the population coefficient r0(p),A salient feature is that, for high densities, virtually all the levels have r0(p)= 1,indicating thermodynamic equilibrium populations, or LTE populations, as wehave seen earlier. For lower densities the r0(p)'sdeviate from 1, but the degree ofthis deviation is rather small. Figure 4.13 shows the population of several levels asfunctions of ne, where the population per unit statistical weight, nQ(p)/g(p), hasbeen further divided by the ion and electron densities, nzne. The population nQ(p)in this figure is regarded as n0(p)/z

3for hydrogen-like ions with nuclear charge z.

This is an approximation, but the degree of approximation is better than that forthe ionizing plasma, and even exact in the low-density limit. In the followingdiscussions of the recombining plasma component, we sometimes call[n0(p)/g(p)nztie]simply the "population". This quantity is almost exactly


FIG 4.13 The recombining plasma component of excited-level populationsagainst «e [m^1]. The ordinate is n0(p) per unit statistical weight g(p) furtherdivided by the ion and electron densities, nz and ne, respectively. Calculation isfor neutral hydrogen with Te=1.28 x 105 K. (Quoted from Fujimoto, 1980a;with permission from The Physical Society of Japan.)

FIG 4.14 Population distribution of the recombining plasma component of theexcited-level populations. The ordinate is the same as Fig. 4.13, and the abscissais the logarithm of the principal quantum number of the level. Calculation is forneutral hydrogen with re=1.28x!05 K. The Saha-Boltzmann populationgiven by eq. (2.la) is shown with the solid line. The dash-dotted lines indicatethe boundary between the CRC phase and the saturation (LTE) phase as givenby eq. (4.25). (Quoted from Fujimoto, 1980a; with permission from ThePhysical Society of Japan.)

proportional to r0(p) except for the small /^-dependent exponential factor in eq.(2.7) under this high-temperature condition. See also eq. (4.38). Figure 4.14 showsthe population distributions for several electron densities. In Fig. 4.14 the popu-lation in LTE, or the Saha-Boltzmann population distribution given by eq. (2.la),i.e. r0(p) = 1, is shown with the solid line. As has been noted the actual populations


in the high-density limit agree with this equilibrium population distribution. Itshould be noted that the range of the ordinate of these figures is very narrow. Allthe populations lie within a range of a factor of 1.5 for the whole of the range of ne

and/?; this is actually seen in Table 4.1(b): the smallest r0(p) is 0.69. Therefore, wemay conclude that, in the high-temperature case, the recombining plasma com-ponent is rather close to the Saha-Boltzmann value for low densities (for very hightemperatures it is even larger than that - see Fig. 5.10 later; see also Appendix 5C)and tends to it at high densities.

Figure 4.15 is similar to Fig. 4.6: the breakdown of the populating (a) anddepopulating (b) fluxes for level p = 5. It should be noted that Fig. 4.15(b) is

FIG 4.15 Breakdown of the (a) populating and (b) depopulating fluxesconcerning level p = 5 into individual fluxes. This condition corresponds toFigs. 4.13 and 4.14. Other explanation is almost the same as that for Fig. 4.6.Part (b) of this figure is identical to Fig. 4.6(b). (Quoted from Fujimoto, 1980a;with permission from The Physical Society of Japan.)


identical to Fig. 4.6(b), since the depopulating processes are common to theionizing plasma component and the recombining plasma component. It is seenthat, in the higher-density regions, higher than 1018 m~3, every depopulating fluxin (b) is almost exactly balanced in (a) by the corresponding populating flux of itsinverse process. In other words, the principle of detailed balance is actually rea-lized. This is consistent with the fact that the populations are almost exactly equalto their Saha-Boltzmann values in these higher-density regions.

In Fig. 4.15, we find two puzzling features:

1. For lower-density regions both the populating fluxes and the depopulatingfluxes are radiative, i.e. the radiative recombination and the radiative cascadefor the populating process in (a) and the radiative decay for the depopulatingprocess in (b). No relationship like the principle of detailed balance is expectedamong the rate constants of these processes. Still the populations are very closeto those given from thermodynamic equilibrium as we have seen.

2. The transition of the populating mechanism in Fig. 4.15(a) from the radi-ative processes in lower densities to the collisional processes in higher densitiestakes place at the density approximately equal to that in Fig. 4.15(b) for thetransition of the depopulating mechanism, i.e. Griem's boundary given byeq. (4.25).

We will investigate whether these features are a mere coincidence or they are akind of necessary phenomenon stemming from a deeper basis.

CRC phase

Figure 4.16 is a sketch of the dominant fluxes of electrons in the energy-leveldiagram for «e= 1012 m~3. This figure together with Fig. 4.15 shows that, in lowdensity, the populating fluxes to all the levels are the direct radiative recombi-nation plus the cascade contributions from the higher-lying levels. The depopu-lating process is the radiative decay. We name this situation the capture radiativecascade (CRC) phase. The population is given from the above balance relation by

We approximate the populations of the higher-lying levels q to their Saha-Boltzmann values, i.e. nQ(q) ~ Z(q)nzne (eq. (2.7a)). We rewrite eq. (4.39) as

The recombination rate coefficient (3(p) is given in Fig. 3.9 and approximatelyby eq. (3.19a). In the present high-temperature case, we adopt, for large p, the


FIG 4.16 Sketch of the dominant populating and depopulating fluxes, or flowsof electrons, in the energy-level diagram, corresponding to Figs. 4.13-4.15.«e = 1 x 1012m~3. Virtually all the levels are in the CRC phase. At right the totalflux of recombination reaching the ground state is given by the downwardarrow with the figure in it giving the collisional-radiative recombination ratecoefficient. Other explanation is almost the same as that for Fig. 4.7. (Quotedfrom Fujimoto, 1980a; with permission from The Physical Society of Japan.)

where Khas been given by eq. (4.12a). See also eq. (2.7). The second term in thenumerator of eq. (4.39a) is expressed by use of eq. (3.8) with the approximationgbb = 1. By using a technique similar to that used in deriving eq. (4.12) we obtainthe approximation

For the denominator we adopt the approximation

approximation for the exponential integral, eq. (3.21),

It is thus concluded that the near-Saha-Boltzmann populations in the low-density regions in Table 4.1(b) or Fig. 4.14 are the result of the intricate rela-tionships between the radiative recombination rate coefficient and the transitionprobabilities. It is further noted that the relative contributions in the numerator ineq. (4.39) from the radiative recombination, eq. (4.41), and the cascade, eq. (4.42),is approximately 2:1. This is in accordance with the accurate calculation shown inFig. 4.15(a).

Figure 1.10(b) gives a summary of the recombining plasma component.The left-side region indicates the CRC phase where the simplified populationkinetics is depicted. The population distribution at high temperature is given inparentheses.

Transition from the CRC phase to the saturation phase

In Figs. 4.13 and 4.14, with an increase in «e, populations make a transition fromthe near-Saha-Boltzmann populations to almost exact Saha-Boltzmann popula-tions. This transition is more clearly seen in Fig. 4.15 as transitions from theradiative processes to the collisional processes both in the populating anddepopulating fluxes. The latter phase is called the saturation phase. Figure 4.17shows a sketch of the dominant fluxes in the energy-level diagram at «e =10

17 m~

3.

For levels lower than p = 5 the feature of the population kinetics is approximatelythe same as in Fig. 4.16, indicating that these levels are still in the low-densityregion, or in the CRC phase. Levels higher than p = 6 have entered into the high-density region, or the saturation phase, so that the situation is completely dif-ferent. In this example the boundary level which divides the low- and high-lyinglevels lies between p = 5 and 6.

The transition from the CRC to saturation phases in the depopulatingmechanism is the same as that for the ionizing plasma as given by eq. (4.25), orwith the neglect of minor terms,


With the above approximations, eq. (4.39) yields

the near-Saha-Boltzmann population, or

Among the populating fluxes the dominant processes at lower densities (orlower-lying levels) in the CRC phase are, as we have seen above, the directradiative recombination and the cascade. At higher densities (higher-lying levels)in the saturation phase they are mainly the collisional deexcitation from the


FIG 4.17 Sketch similar to Fig. 4.16 except that «e = 1 x 1017 m 3. The boundarylevel pG between the CRC phase and the saturation (LTE) phase, given byeq. (4.25), lies between p = 5 and 6, as indicated by the dash-dotted line. Thedownward recombination flux through a sufficiently high-lying level, levelp = 10, is 1% of the net recombination flux as shown with the downward arrow.(Quoted from Fujimoto, 1980a; with permission from The Physical Society ofJapan.)

higher-lying levels. See Fig. 4.15(a). Thus the transition takes place when themagnitudes of these radiative and collisional fluxes become equal:

It is to be noted that the higher-lying levels q are already in LTE at «e at which thislevel p makes the transition from the CRC phase to the saturation phase, so thattheir population is given by n0(q) = Z(q)nzne. A similar procedure has already beenemployed in approximating eq. (4.39). We further note the principle of detailedbalance, Z(q)F(q,p) = Z(p)C(p,q), eq. (3.31a). Then we rewrite the right-handside of this equation (4.45) as

Equation (4.45) is then rewritten as

This equation is, within the approximations of eqs. (4.41), (4.42), and (4.12),identical with eq. (4.44), which was for the transition in the depopulating


mechanism. Thus it has been shown that, for both the populating and depopu-lating mechanisms, the transition takes place at almost the same electron density.The boundary is thus given by eq. (4.25) or (4.29a). Figure 1.10(b) includes thisboundary, as labeled "GRIEM".

Saturation (LTE) phase

For higher densities the levels are in the saturation phase. At the beginning of thissection we have already seen that, at high density, for an excited level, thepopulating fluxes from the higher-lying levels (deexcitation) and the continuum(three-body recombination) are almost exactly balanced by the respective inversedepopulating fluxes (excitation and ionization, respectively, see Fig. 4.15.). If thestill higher-lying levels are in LTE and their populations are given by the Saha-Boltzmann equation, eq. (2.la), the above balance relationship should result inthe LTE population of this level. It is further noted from the discussion aroundeqs. (4.5)-(4.11) that the most dominant depopulating flux is the excitation to theadjacent higher-lying level, rather than ionization as is sometimes assumed. Themost dominant populating flux is accordingly deexcitation from that adjacentlevel to this level, rather than three-body recombination, again as is sometimesassumed. Roughly speaking, therefore, the dominant flow pattern of the popu-lation is (p+ 1 ) — > / > — > ( / > + 1). This feature is seen in Fig. 4.15, and in Fig. 4.17 forlevels/? > 6. Figure 4.18 shows a sketch of the dominant fluxes in the high-densitylimit. The flux has been further divided by nzn^, and its magnitude is given with

FIG 4.18 Sketch similar to Fig. 4.16 but for the high-density limit, where themagnitude of a flux has been divided by nznl and given by figures. All theexcited levels are in the saturation (LTE) phase. (Quoted from Fujimoto, 1980a;with permission from The Physical Society of Japan.)

We may continue the string of this reasoning along p, (/>+!), (p+2),..., to reachvery high-lying levels, denoted by, say, r. As has been discussed already, theatomic characteristics of these negative-energy discrete levels continue smoothlyacross the ionization limit to low-energy continuum states. Since we assumethermodynamic equilibrium for the "free" electrons, it would be natural to assumethat the Maxwell distribution continues smoothly across the ionization limit to thenegative-energy discrete levels. In other words, these levels are in thermodynamicequilibrium with the continuum, or they are in LTE, and their populations aregiven by the Saha-Boltzmann values, nQ(r) = Z(r)nzne, eq. (2.7a). See also theparagraph at the close of Section 2.1. If we now trace back the above reasoningdown to level p, then we arrive at the conclusion that this level should be in LTE:

and the resulting population ratio is the Boltzmann distribution, eq. (2.3) oreq. (3.32)


the numbers. In this figure (and also in Fig. 4.15(a)) it is seen in fact that thedirect contribution from the three-body recombination is substantially smallerthan the deexcitation. Therefore, the dominant relationship of population balanceis given by

Therefore, the saturation phase may be called the LTE phase. In Fig. 1.10(b), theupper region higher than the boundaries "GRIEM" and "BYRON" correspondsto this phase, and the relationship, eq. (4.46), is depicted schematically.

The above discussion concerning the continuation of the Maxwell distributionof the "free" electrons to the negative-energy discrete levels may appear lessconvincing. This is partly due to our assumption of "free" electrons for the con-tinuum state electrons, and partly to the ambiguity in treating very high-lyinglevels. The first assumption is obviously wrong for low-energy electrons, since theinteraction of an electron with the ion cannot be neglected in comparison with itskinetic energy. Remember that the Coulomb force is strong and of long range. Thesecond point is also problematic since we implicitly assume that the principalquantum number of levels can increase indefinitely. We then immediatelyencounter the difficulty that the total statistical weights of the levels, or the statedensity of the levels for a finite energy width, or the LTE populations, diverge.These points will be addressed and an adequate resolution will be introduced inSection 9.4. Then, the above arguments gain sound footings.

At the close of Section 4.1 an important relationship was introduced:

It is easily seen that, for Te/z2^> 1.5 x 104 K, the deviation of r0(p) from 1 is very

small except for, say, p = 2. See Fig. 4.3. This is another explanation of the LTEpopulation, eq. (4.47).

Equation (4.48) has been derived in the limit of high density. It should benoted, however, that this equation is also valid at much lower densities. As hasbeen shown in Figs. 4.13 and 1.10(b), for example, an excited level is in thesaturation phase for densities down to Griem's boundary, and r0(p) continues totake its high-density-limit value until that boundary. In Table 4.1(b), level p = 5,for example, has r0(5) = 1 — 5~6 = 1.00 from the high-density limit down to«e~1017 m~3 within a 10% deviation. This boundary is nothing but Griem'sboundary as clearly seen in Fig. 1.10(b).

4.4 Recombining plasma component - low-temperature case

In this section we examine the low-temperature case which is more important inpractical situations. We assume the temperature to be low, i.e. Te/z

2 -c 1.5 x 104 K.As an example we take neutral hydrogen with Te = 1 x 103 K. The reader can drawanother horizontal straight line in Fig. 1.2.

Table 4.1 (a) shows r0(p). Unlike the high-temperature case of Table 4.1(b),r0(p) is, in many cases, much smaller than 1. Only very high-lying levels haver0(p)~ 1 in high densities. Figure 4.19 shows the populations of several excitedlevels against ne, where, as before, the population per unit statistical weightnQ(p)/g(p) has been further divided by the ion and electron densities nzne. Thisquantity, [«o(/")/ g(p)nzne], is proportional to rQ(p); see eqs. (4.38) and (2.7). Foran excited level, we may divide the range of ne into four regions according to thedependence of its population on ne: the low-density limit (for «e < 1012 m~3 in thecase of level p = 5 taken as an example), the gradual increase in the population, orr0(p), with an increase in ne (1012<«e< 1016 m~3 for level 5), the steep increase(1016<«e<1019 m~3), and finally saturation («e>1019 m~3). The first threeregions are named altogether as the capture-radiative-cascade (CRC) phase, andthe last region as the saturation phase. This nomenclature follows that of the high-temperature case.

On each curve in Fig. 4.19, the closed circle indicates ne at which the dominantdepopulating process changes from the radiative decay to the collisionaldepopulation, as given by eq. (4.25). It is obvious from this figure that thisboundary ne gives the transition from the CRC phase to the saturation phase.


which is valid in the limit of high density where all the radiative transitions can beneglected. Another approximate relationship was noted for high temperatures andhigh densities (Fig. 4.10):

It is thus concluded that, in the limit of high density, we have

RECOMBINING PLASMA COMPONENT

FIG 4.19 The recombining plasma component of excited-level populations forthe low-temperature case against ne. The ordinate is n0(p) per unit statisticalweight g(p), further divided by the ion and electron densities, nz and ne,respectively. Calculation is for neutral hydrogen with T e = l x l 0 3 K. Theboundary ne between the low-density region (CRC phase) and the high-densityregion (saturation phase) as given by eq. (4.25) is shown with the closed circles.(Quoted from Fujimoto, 1980b; with permission from The Physical Society ofJapan.)

Figure 4.20 shows the population distribution over the excited levels for severalelectron densities; Fig. 4.20(a) corresponds to Fig. 4.14, and Fig. 4.20(b) is anotherplot, the conventional Boltzmann plot, of the populations. The boundary levelgiven by eq. (4.25) is shown with the dash-dotted lines. The excited levels aredivided into two groups: the levels lying lower than this boundary and the higher-lying levels. From the arguments in the preceding paragraph it is obvious thatthe former group is in the CRC phase and the latter is in the saturation phase.See Fig. 1.10(b). When we compare Figs. 4.19 and 4.20 with the correspondingfigures in the high-temperature case (Figs. 4.13 and 4.14), we recognize that thepopulation characteristics of these two cases have almost nothing in common.Specifically,

1. In the present low-temperature case, the populations in the CRC phase aremuch less than their Saha-Boltzmann values, eq. (2.7a), and populationinversion, i.e. larger nQ(p)/g(p) values for larger p, is established for all theexcited levels.

2. In the saturation phase the populations of higher-lying levels tend to theSaha-Boltzmann values, which was also the case for the high-temperature

121

FIG 4.20 Population distribution of the recombining plasma component for7~e= 1 x 10 K. The ordinate is the same as Fig. 4.19 and the abscissa is (a) thelogarithm of the principal quantum number of the level, and (b) the ionizationenergy of the level. Part (b) is the conventional Boltzmann plot. The Saha-Boltzmann population given by eq. (2.la) is shown in (a) with the solid-and-dashed curve, and in (b) with the solid line. In (a), the Saha-Boltzmanndistribution at p#, where/IB = 7.25 as given by eq. (4.56), continues smoothly tothe line, given by eq. (4.54). The boundary pG between the CRC phase and thesaturation phase, or eq. (4.25), is given with the dash-dotted lines and theboundary />B given by eq. (4.55) is shown with the dotted line. (Quoted fromFujimoto, 1980b; with permission from The Physical Society of Japan.)


case. However, those of the several lowest-lying levels (j> < 7) never reach theSaha-Boltzmann values even in the limit of high electron density.

We will examine below these features in detail.

CRC phase

Figure 4.21 shows a sketch of the dominant fluxes of electrons in the energy-leveldiagram in the low-density limit. The first point to note is the relative magnitudesof the fluxes of radiative recombination into various levels. In comparison withthe high-temperature case, Fig. 4.16, the relative magnitudes are quite different:they have a much weaker dependence on the levels. This difference has alreadybeen noted at the close of Section 3.2 and in Fig. 3.9. For the recombination ratecoefficient, eq. (3.19a), we adopt the approximation to the exponential integral,eq. (3.22), which is the opposite case to eq. (4.40), i.e.

This is valid for low-lying levels. Equation (3.19a) then reduces to

FIG 4.21 Sketch of the dominant fluxes of electrons in the energy-level diagram.Neutral hydrogen with Te=l x 103 K. The low-density limit, where all thelevels are in the CRC phase. The recombination flux through a high-lying level,level p= 10 taken as an example, is shown with the solid arrow. (Quoted fromFujimoto, 1980b; with permission from The Physical Society of Japan.)


Therefore, /3(p) is proportional top~l and Te~°'5, and these dependences are quitedifferent from the high-temperature case in which /3(p) otp~2'5Te~

1'5. Figure 3.5(b)illustrates this point; at this low temperature for low-lying levels the p^1-dependence is seen, while for levels />;$> 10, the ^-dependence is stronger. Thisweak /i-dependence is the reason why the higher-lying levels are more heavilypopulated in the low-temperature case than in the high-temperature case. Figure4.22, corresponding to Fig. 4.15, shows the breakdown of the populating anddepopulating fluxes for level 5. In the low-density limit, in comparison with the

FIG 4.22 Breakdown of the (a) populating and (b) depopulating fluxes concern-ing level/? = 5 into individual fluxes, corresponding to Figs. 4.19 and 4.20. Theexplanation is the same as for Fig. 4.15. (Quoted from Fujimoto, 1980b; withpermission from The Physical Society of Japan.)

leading to the population inversion. In Fig. 4.20(a) this approximation is com-pared with the result of the numerical calculation. Figure 1.10(b) contains thisdistribution in the region of the CRC phase.

With an increase in ne, very high-lying levels first enter into the saturationphase (Figs. 4.19 and 4.20). For collisional transitions concerning these levels, theenergy differences of the important transitions (excitation and deexcitation to theadjacent levels; see the discussions around eqs. (4.5)-(4.11)) are much smaller thanthe electron temperature (multiplied by Boltzmann's constant). So, the situation issimilar to the high-temperature case. Thus, the arguments in the preceding sectionconcerning the transition of the population from the CRC phase to the saturationphase are valid; that is, Griem's boundary, eq. (4.25) or (4.29a), gives the tran-sition, and the levels lying higher than this boundary are in the saturation phaseand thus in LTE. This feature is actually seen in Fig. 4.20(a) and (b). Figure 1.10(b)includes the boundary "GRIEM" and the population kinetics in the upper part ofthe region of ne higher than this boundary.

In the present case of low temperature, the Saha-Boltzmann populations,n(p)/g(p)nzne= Z(p)/g(p),for these levels are substantially higher than theirpopulations in the CRC phase. As a result, with the increase in «e, the downwardradiative cascading fluxes from these levels to lower-lying levels in the CRC phaseincrease substantially. This is the reason why, in Fig. 4.22(a), the relative con-tribution from the cascade increases with «e.

The features described above may be understood in a different way: Thelowering of Griem's boundary level with ne may be regarded as if it is the loweringof the ionization limit down to this level. The higher-lying levels are engulfed bythe "continuum", and the threshold for "radiative recombination" is lowered toGriem's boundary level, resulting in an increase in the "radiative recombination


high-temperature case, the cascade contribution is more important, about 50% ofthe total populating flux; this is a consequence of the heavier populations in thehigh-lying levels. The depopulating flux is the radiative decay. The populationbalance is given by eq. (4.39). It may be shown by a technique similar to the oneleading to eq. (4.12) and eq. (4.42) that the ratio of the contributions from thedirect radiative recombination, the first term on the r.h.s. of eq. (4.39), and thatfrom the cascade, the second term, is about equal for any level, being independentof p. (In this derivation, however, the population distribution of higher-lying levelsis assumed to be n0(q)/g(q)<xq, instead of eq. (4.51) below; this assumption ismore accurate numerically as seen in Fig. 4.20(a).) Thus, for the purpose ofderiving the population distribution among the levels we may neglect the cascadecontribution in eq. (4.39) to reach the approximation

or


rate". Then, it is natural that, with the increase in «e, all the populations of the low-lying levels increase at almost the same degree; this is actually seen in Figs. 4.19and 4.20 as parallel movements of the curves and the points, respectively, of thelow-lying levels in the CRC phase; in Fig. 4.19 all the curves move almost inparallel for low densities, and in Fig. 4.20 the populations of low-lying levels moveupward in parallel. The population inversion is conserved. It should be noted thatthe above argument has nothing to do with the lowering of the ionizationpotential, which will be discussed in Chapter 9, or with the merging of levels asdiscussed in Chapter 7.

With a further increase in «e, the boundary between the CRC and saturationphases gradually comes down. See Fig. 1.10(b). Figure 4.19 and 4.20 show that, at«e~ 1017-1018 m~3, level 6, and higher-lying levels have entered into the saturationphase. Figure 4.22(a) shows that, with this increase in «e, the contribution from thecollisional deexcitation to level 5 begins to increase; in particular, that from theadjacent higher-lying level 6 is dominant. On the other hand, the dominantdepopulating mechanism is still the radiative decay for « e<3 x 1018 m~3. Thepresence of this intermediate density region is characteristic of the low-temperature recombining plasma. This persistent cascading nature of thedepopulating process is the origin of the nomenclature of the capture-radiative-cascade phase. The population balance in this highest «e region of the CRC phaseis given approximately by

Since the upper level (/>+!) has entered into the saturation phase, nQ(p+l) isalmost proportional to nzne. Thus, nQ(p) is approximately proportional to nzn^.This explains the steep slope in Fig. 4.19 of nQ(p) in this highest-density region ofthe CRC phase.

Saturation phase - Byron's boundary

As seen in Fig. 4.22(b), with a further increase in «e, the depopulating processchanges from radiative decay to collisional depopulation. This transition of thedepopulating mechanism is given from eq. (4.25) and the boundary «e value isshown in Fig. 4.19. The boundary between the lower-lying levels in the CRC phaseand the higher-lying levels in the saturation phase is shown in Fig. 4.20 with thedash-dotted line. We have already noted this boundary at the beginning of thissection. This boundary is also shown schematically in Fig. 1.10(b). Note that theactual boundary «e is higher by about two orders of magnitude in this low-temperature case. See eq. (4.59) later.

Figure 4.19 shows that, with the above increase in «e, level 5, for example,enters into its high-density limit, or into the saturation phase at «e~ 1019 m~3.Figure 4.23 is a sketch of the dominant fluxes of electrons in the energy-level


FIG 4.23 Sketch similar to Fig. 4.21 except that «e = 1 x 1020m 3. The boundarypG given by eq. (4.25) is shown with the dash-dotted line and/?B given by eq. (4.55)with the dashed line. (Quoted from Fujimoto, 1980b; with permission fromThe Physical Society of Japan.)

diagram for ne = 1 x 1020 m~3, where levels p > 4 are in the saturation phase. Thisfigure and Fig. 4.22 for level 5 indicate that, in this phase, the dominant popu-lating process is the collisional deexcitation from the adjacent higher-lying level 6.The dominant depopulating process is the collisional deexcitation to the adjacentlower-lying level 4. This feature still holds in the higher «e regions as seen inFig. 4.22 and Fig. 4.24 for the high-density limit. This feature is contrasted to thehigh-temperature case in the preceding section, Figs. 4.15 and 4.18, where thedominant depopulating flux was excitation to the adjacent higher-lying level. Thisnew feature is common to many lower-lying levels (Fig. 4.24).

For these low-lying levels the exponential factor of the excitation rate coeffi-cient, eq. (3.29), is no longer negligible, and eq. (4.36), F(p,p—1)> C(p,p+l),holds instead of eq. (4.9), F(p,p—T) < C(p,p+l), See Fig. 4.2. In other words, forthese levels the energy separation between the adjacent levels (proportional to p~3,see eq. (1.5)) becomes quite significant for the electrons whose average energy is ofthe order of kTe. The population balance is therefore

Thus, the ladder-like deexcitation flow of electrons is established in the energy-level diagram (Fig. 4.24). It is noted that eq. (4.53) is nothing but eq. (4.33), and


FIG 4.24 Sketch similar to Fig. 4.21 except that this is for the high-density limit.The magnitude of the effective recombination flux has been divided by nzr%, andshown with the figure in the arrow at right. (Quoted from Fujimoto, 1980b;with permission from The Physical Society of Japan.)

therefore, we have a population distribution similar to eq. (4.31), the minus sixthpower distribution,

for low-lying levels for which eq. (4.36) holds. Figure 4.20(a) shows that thisapproximation is good for high density and for low-lying levels, except for thelowest-lying levels for which some of the above approximations break down.Figure 1.10(b) depicts these features: in the density region higher than Griem'sboundary and for low-lying levels, shown are eq. (4.53) for the population kineticsand eq. (4.54) for the population distribution.

For higher-lying levels for which eq. (4.9) is valid instead of eq. (4.36), thesituation is different. As is seen in Fig. 4.24 and expected from the discussion inthe preceding sections, the dominant depopulating process from such a level p isthe collisional excitation to the adjacent higher-lying level (/H-l). The balanceof the population is approximately given by eq. (4.46), n0(p+l)F(p+l,p)ne=n0(p)C(p,p + l)«e, as was the case in the high-temperature case. Thus, eq. (4.47),n0(p) = Z(p)nzne, holds, and the discussion leading to this equation shows thatall of the high-lying levels for which eq. (4.9) holds are in LTE. This is actuallyseen in Fig. 4.20 to be the case.

We note that exp(o) is nothing but p, so that at Byron's boundary, pB, as given byeq. (4.56), the slope of the Saha-Boltzmann distribution is —6. This means that theSaha-Boltzmann populations in the higher-lying levels continue smoothlyat Byron's boundary to the minus sixth power distribution for the low-lyinglevels, eq. (4.54). We can thus approximate the whole population distribution: forP >/"B, eq. (4.47) is valid, and for p <p#, as a linear extrapolation of eq. (4.47), theminus sixth power distribution, eq. (4.54), is valid. This approximation is com-pared in Fig. 4.20(a) with the numerical calculation. Note that the approximation,eq. (4.56), gives/?B = 7.25 in the present case of Te/z

2 = 1 x 103 K, while the actualp^ is between p = 6 and 7.

At the beginning of Section 4.2, the boundary temperature between the high-temperature case and the low-temperature case was introduced as Te/z

2~1.5 x 104 K. This is the boundary temperature derived from eq. (4.56) with /?B = 2.For temperatures higher than this, eq. (4.9) is approximately valid for all of theexcited levels, and the discussions in the preceding section are valid to a large


For high enough density, the boundary level p between these high-lyinglevels and the low-lying levels is given from the boundary between eq. (4.9) andeq. (4.36):

Figure 4.2 shows C(p,p+l)/F(p,p—l) against Te for several levels. For the tem-perature of the present example, 103 K, this boundary lies between/? = 6 and 7 asshown in Figs. 4.20, 4.23, and 4.24. With the approximate rate coefficient,eq. (3.29) along with appropriate approximations like eq. (3.7), fp,p+\—p/4,and eq. (1.2), E(p,p+l)~2z2R/p3,together with the Taylor expansion of the

exponential factor and the approximation [(/>—1)//>]6~ 1 — (6/p), we arrive at anapproximate expression for the principal quantum number of this boundary

We differentiate the Saha-Boltzmann populations in this figure with respect to a:

The subscript B denotes "Byron et al.", since this expression was first derived bythese authors. Byron's boundary, as expressed as the boundary temperature, isgiven in Fig. 4.2. It is seen that this approximation is good for large p, as isexpected from the approximations. Figure 1.10(b) includes the boundary bearingthe sign "BYRON", as determined by eq. (4.55) or (4.56).

The Saha-Boltzmann population, eq. (4.47), for the high-lying levels as shownin Fig. 4.20(a) has negative slopes in this figure. We define for the abscissa

In this case, however, r^p) in the high-density limit is much larger than eq. (4.35)for all the levels as seen in Table 4.1 (a) and Fig. 4.3, resulting in smaller r0(p)values than the high-temperature case. This high-density limit value of r0(p)continues to lower densities, until ne reaches Griem's boundary. See Figs. 4.19 and1.10(b) and Table 4. l(a).

Secondly, Table 4.1 (a) and Fig. 4.3 suggest that the relative magnitudes of thehigh-density limit values of rQ(p) and r\(p) have a relationship with/>B. From thediscussions in the preceding sections, it may be concluded that for levels lyinghigher than Byron's boundary, rQ(p) is larger than ri(p) and can be close to 1, and


extent. This boundary temperature has yet another basis, which will be given inthe next chapter.

Griem's boundary

We have defined Griem's boundary as the boundary level, or the boundary «e, atwhich the radiative decay and the collisional depopulation are equally probable.In deriving the numerical formula, eq. (4.29), we relied on the simple approx-imation for the excitation rate coefficient, eq. (4.7). When Te is low and ne is high,Griem's boundary lies below Byron's boundary; then Griem's boundary should bedetermined by the deexcitation rate coefficient rather than the excitation ratecoefficient as assumed in deriving eq. (4.29). In this case, instead of eq. (4.7) witheq. (4.5), another formula approximates better the deexcitation rate coefficient:

This formula is deduced from a Monte Carlo calculation for classical electronorbitals.

By combining this with the approximation for the radiative decay probability,eq. (4.13), we obtain

where we have simplified the expression by ignoring the very weak dependenceon Te. Here, we have assumed z = 1. We have restricted our discussion to neutralhydrogen because the z-scaling of eq. (4.58) may not necessarily be valid, as under-stood from the excitation cross-sections in Fig. 3.11. In this low-temperature casethe deexcitation rate coefficient is determined largely by the excitation cross-section values near the threshold, which do not follow the z-scaling. Note theKlein-Rosseland relationship, eq. (3.27).

Griem's boundary pG as given by eq. (4.59) is about a factor of 2 larger thanthat given by eq. (4.29a) in the «e region of practical interest. This point should beremembered when Fig. 1.10(b) is applied to low-temperature plasmas.

We note here two points: First, as in the case of high temperature, rQ(p) in thehigh-density limit is given by

SUMMARY AND CONCLUDING REMARKS 131

for the lower-lying levels vice versa. This is actually seen to be the case fromFigs. 4.3 and 4.2.

Examples

We now show some examples of the population distributions determinedexperimentally for recombining plasma. Figure 4.25(a) shows an example of theBoltzmann plots of measured populations for neutral hydrogen in an expandingplasma jet. Te and «e are estimated to be 0.15 eV (1.7 x 103 K) and 2.6 x 1019 m~3,respectively. The reader can locate this plasma in Fig. 1.2. The solid line shows theSaha-Boltzmann population distribution given by eq. (2.7a). This line has beenfitted to the experimental populations of high-lying levels. We obtain from eq. (4.59)pG ~ 4.1 and from eq. (4.56) />B ~ 5.5, which are given in the figure with the dash-dotted line and the dashed line, respectively. Also shown are the calculatedpopulations by the collisional-radiative model for the recombining plasma com-ponent.

This distribution illustrates well the characteristics of the low-temperaturerecombining plasma as discussed in the present section. For levels p>pv thepopulation is given by the Saha-Boltzmann value, for pG <p<p# the levels are inthe flux of the ladder-like deexcitation and n(p)/g(p)otp~6, and for p<pG thelevels are in the CRC phase and population inversion is established. Of course, thesecond group has only one level p = 5 in this example, and the minus sixth powerdistribution is not obvious. Note that the distribution in this figure is rather closeto that in Fig. 4.20(b) for «e= 1020 irT3.

Another example is a similar population distribution for neutral helium in asimilar plasma. Figure 4.25(b) is the result with re~0.13 eV (1.5 x 103 K) and«e ~ 8.6 x 1018 m~3. In this case, pG ~ 4.8 and pB ~ 5.9.

4.5 Summary and concluding remarks

So far we have examined the characteristics of the ionizing plasma component andthe recombining plasma component of excited level populations within the fra-mework of the collisional-radiative model. In each component the excited levelswere divided into the phases, depending on «e. Figure 1.10, which was introducedin Chapter 1 for the purpose of giving the reader some ideas about the theory to bedeveloped in the later chapters, shows the schematic "phase diagram" of theexcited levels for the ionizing plasma component (a), and for the recombiningplasma component (b). Two boundaries are given: Griem's boundary divides thewhole area into the low-density region and the high-density region and, in thehigh-density region, Byron's boundary divides the levels into low-lying levels andhigh-lying levels. In each area or phase, given are the name of the phase, approxi-mate population characteristics and a schematic diagram of the populationkinetics of level p with which we are concerned. In many practical situations inwhich the ionizing plasma component is dominant, Te is rather high. Then,


FIG 4.25 (a) Boltzmann plot of the measured and calculated populations ofneutral hydrogen in an expanding plasma jet. The ionization limit is 13.6 eV.(Quoted from Akatsuka and Suzuki, 1994; copyright 1994, with permissionfrom The American Physical Society.) (b) Boltzmann plot of the measuredpopulations of neutral helium. The ionization limit is 24.5 eV. (Quoted fromAkatsuka and Suzuki, 1995, with permission from IOP Publishing.) BoundariespG (eq. 4.59) and/?B (eq. 4.56) are given by the dash-dotted line and the dottedline, respectively.

SUMMARY AND CONCLUDING REMARKS 133

Byron's boundary, as is seen from eq. (4.56), lies far below p = 2, and does notappear in the present phase diagram for p>2. Therefore, the phase diagram ofFig. 1.10(a) usually lacks Byron's boundary and the "saturation (TE)" phase. Seethe discussion of the low-temperature case in Section 4.2.

Now, we have reached the point where we can interpret the spectra in Fig. 1.5.These are of the resonance-series lines (transitions terminating on the groundstate) of singly ionized helium, or hydrogen-like helium with z = 2 (Fig. 1.6). Inother words, we try to locate these plasmas on the n&— Te plane of Fig. 1.2, andidentify the intensity distributions in terms of the ionizing plasma component,Fig. 1.10(a), or the recombining plasma component, Fig. 1.10(b), or even acombination of them, Fig. 1.9. We have to note two adverse situations here:

1. The spectrometer used in this observation was not calibrated for its relativesensitivity over the wavelength range. Thus, the observed signal intensity fordifferent lines may not directly reflect the number of photons accepted by thespectrometer. However, since the wavelength range is rather narrow, we mayassume constant sensitivity over the observed lines.

2. The plasmas may violate one of our assumptions on which we have based ourforegoing discussions: that is, in the present case the emission lines may sufferradiation reabsorption within the plasma. The population of the ground-state ions(virtually equal to «e in the present case) is so high that the photons emitted byexcited-level ions can be absorbed by them before leaving the plasma. This effect isapproximately proportional to the absorption oscillator strength. Remember thatthis quantity is a measure of the ability of the ions to absorb the light of thistransition. See Table 3.1(b) and Fig. 3.4. Therefore, the line 1 <—2, and 1 3 to acertain extent, may suffer an apparent decrease in the observed intensity. Thisproblem will be treated in detail in Chapter 8.

With the above reservations, especially the second one, in mind, we try tointerpret the spectra. Since, these plasmas are known to have «e about 1020 m~3,the reduced density n^/z1 is about 1018 m~3; this is the density to be referred to ininterpreting these spectra in terms of the theory developed in this chapter. SeeAppendix 5B. Figure 1.5(a) reveals that, with the increase in the principal quan-tum number p of the upper level, the observed line intensity, $(p, 1) (eq. (4.1)),decreases sharply. Note the unusually thick profile of the 1 <— 2 line. This isinterpreted as due to the saturation of the detector near the line center. If wecorrect for this saturation effect the peak would be at around the asterisk markedon this figure. If we ignore the small v dependence in eq. (4.1), the relative popul-ation n(p)/g(p) is given from $(p, 1) divided by A(p, l)g(p), which is approxi-mately proportional to p~ , as seen from eqs. (3.1) and (3.6a) and Fig. 3.4. Theresulting population distribution which decreases sharply with/? is found to be con-sistent with the one in Fig. 4.5 at n^/z1 = 1018 m~3. Thus we may conclude that thisplasma reveals only the ionizing plasma component of the excited-level popula-tions and that this plasma is located at around «e/z

7— 1018 m~3 in Fig. 1.10(a).Obviously, the recombining plasma component in eq. (4.20a) or Fig. 1.9 is


virtually absent. Figure 1.5(b), shows that the populations n(p)/g(p) show smallvalues for small p and a broad maximum at around p = 5-7. This distributionmay be interpreted as a recombining plasma component, or this plasma is locatedsomewhere in Fig. 1.10(b). This distribution is close to that in Fig. 4.20 with«e/z

7= 1018 m~3. The ionizing plasma component in eq. (4.20a) or in Fig. 1.9 isabsent.

The important point here is that, even in the present adverse situation as notedabove, we can make a definite conclusion, even if it is qualitative, about the natureof the plasma from its emission line spectrum. This is because the overall patternof the population distribution is very much different for the ionizing plasmacomponent and the recombining plasma component as clearly demonstrated inFig. 1.5.

In Appendix 8A, the spectra of Fig. 1.5 are interpreted further with theradiation reabsorption effect taken into account. As Fig. 8 A.I (a) shows, thespectrum of Fig. 1.5(a) leads to the population distribution of the ionizing plasmacomponent with ne/z

7 = 1018 m~3 or a little lower. Figure 8A.l(b) shows that theplasma of Fig. 1.5(b) is the recombining plasma component of ne/z

7 = 1018 m~3.Our above speculations will be thus justified. The final identification of theseplasmas on Fig. 1.2 will be postponed until Chapter 8.

Figure 1.3 is also interpreted as a spectrum of a recombining plasma. Theintensity distribution pattern of the series lines shows a typical characteristic ofa recombining plasma. The continuation of the series lines to a continuum isalso a signature of a recombining plasma. This latter problem will be treated inChapter 6. In Fig. 1.3, the observed emission lines terminate on excited levels andthe radiation reabsorption effects should be minor. However, the sensitivityproblem still remains and our conclusion here has to be qualitative again.

In this chapter we based our theory on the assumption that excited levels arespecified by the principal quantum number as depicted by Fig. 1.6. This assump-tion is equivalent to assuming the statistical population distribution among thedifferent-/ levels having a common principal quantum number, as depicted inFig. l.ll(b). However, it is obvious that this assumption is not valid at lowdensities. In Appendix 4A we examine this problem. It is shown that, in the low-density region, lower by some two orders than Griem's boundary, our aboveassumption breaks down. Therefore, our conclusions deduced in this chapterabout the populations in the corona phase of the ionizing plasma component andin the CRC phase of the recombining plasma component should be treated withcare in applying them to real plasmas at low density.

*Appendix 4A. Validity of the statistical populations among thedifferent angular momentum states

As noted in Section 3.6, heavy particle (ion) collisions can be important ininducing a transition between two levels having a small energy difference or evenbetween nearly degenerate levels. Figure 3.23 shows for the transition 3s —> 3p, as

VALIDITY OF THE STATISTICAL POPULATIONS 135

an example, a comparison between the cross-sections by electron impact and byproton collisions. These are the results of calculations in the Born approximation.For higher energies of practical interest, both cross-sections have the energydependence of \/E, and the magnitude of the cross-section for proton collisions islarger than that for electron impact by about three orders. The reason was given inSection 3.6. We thus have to take into account the former collision processes inconsidering the population distribution among the different / levels.

A collisional-radiative model for neutral hydrogen has been constructed withdifferent / levels resolved. Figure 4A. 1 shows an example of the populations of theionizing plasma components ni(p)/g(p) forp = 4s, 4p, 4d, and 4f against a changeof «e, where the proton density is assumed equal to ne and the electron and iontemperatures are 10 eV. The ground-state atom density is «(1) = 1 m~3. For lowdensities, the populations (per unit statistical weight) among the different / levelsdiffer by three orders of magnitude. In this density region each of the levels is in itsown corona equilibrium and their populations are proportional to ne. With anincrease in «e, the population of the 4f level begins to increase with respect to thecorona equilibrium value. This increase is found to be due to the substantialincrease in the populating flux by cascading transition from the 5g level. With a

FIG 4A. 1 An example of the populations (per unit statistical weight) of the ioni-zing plasma component with the different / levels resolved for neutral hydrogenwith Te= 10 eV and «(!)= 1 m~3. (Quoted from Sawada, 1994.)

Figure 4A.2 shows this critical density with the solid line, where Te= 10 eV isassumed. This figure contains the critical density which is determined from thecalculated cross-sections like those shown in Fig. 3.23. The open circles give thedensity at which the deviation of the populations becomes smaller than 20%.

The actual energy-level structure of hydrogen-like ions follows the LS couplingscheme, and an L level (except for S levels, of course) splits into two / (= L ± 1/2)levels owing mainly to the spin-orbit interaction: i.e. fine structure. Sampsonconsiders a similar problem to the above in this case and gives the critical density

This critical density is given in Fig. 4A.2 with the dotted line.As we have seen in Fig. 4A.1, in the lowest-density regions, each of the dif-

ferent / levels are in corona equilibrium. Figure 4A.2 shows with the open squaresthe upper limit of density below which corona equilibrium is valid for all of the/ levels.

*Appendix 4B. Temporal development of excited-level populations andvalidity condition of the quasi-steady-state approximation

In the present chapter we introduced the quasi-steady-state (QSS) approximationand examined various characteristics of excited-level populations on the basis of


further increase in «e, proton collisions tend to make all the populationsn\(p)jg(p) equal; the 4f population comes close to the 4d population, and the 4spopulation comes close to the 4p population. The most persistent level is 4d. Whenthe radiative decay of 4d becomes dominated by the collisional (by proton)transition 4d —> 4p, the 4d population tends to the 4p population, and all the 41levels are equally populated, i.e. the stastical populations. For other n levels thesituation is found to be similar.

The radiative decay rate from the nd level is approximately given by

The rate coefficient for the nd —> np transition by proton collisions is approxi-mated by

Remember that the cross-section was inversely proportional to energy. The criticaldensity is thus given as

QUASI-STEADY-STATE APPROXIMATION 137

FIG 4A.2 The boundary level or boundary ne above which statistical populationscan be assumed. Proton density and temperature are assumed equal to theelectron density and temperature (10 eV). Q: from the numerical calculation.

: eq. (4A.3). : eq. (4A.4). In this figure, the boundary below whichcorona equilibrium is valid for the individual / levels is shown with D • (Quotedfrom Sawada, 1994.)

this approximation. If our plasma (in the first sense) is stationary, QSS is not anapproximation but is exact. If, on the other hand, the plasma, in the first sense aswell as in the second sense, is transient, so that its properties change in time, QSS isan approximation and may not be justified under unfavorable conditions. In thisappendix we examine the validity condition for this approximation.

Ionizing plasma

As an example of an ionizing plasma, we consider an ensemble of hydrogen atomsin an electron gas or a plasma with Te = 10 eV and «e = 1018 m~3. The reader maylocate this plasma on Fig. 1.2. Neutral hydrogen atoms with «(!)=! m~3 areimmersed in this plasma, and at t = 0 these atoms begin to be acted upon by thisplasma, i.e. excited and ionized. If «(1) is kept constant, after the initial transientthis system should reduce to the high-temperature ionizing plasma which wastreated in Section 4.2. Figure 4B.1 shows the temporal development of the excited-level populations (per unit statistical weight), ni(p)/g(p). The horizontal dashedlines indicate the population values in the final stationary state with «(1) = 1 m~ .


FIG 4B. 1 Temporal development of the excited-level populations for «(1) = 1 m 3,re=10eV, and «e=1018m~3. The dashed lines show the steady-state values,and the dash-dotted lines show the result of eq. (4B.3). Q: the time at which thetransition of eq. (4B.4) takes place. The assumption of constant n(l) is valid untilt ~ 1CT4 s, the relaxation time of the ground-state atom density. (Quoted fromSawada and Fujimoto, 1994; copyright 1994, with permission from The AmericanPhysical Society.)

Figure 4B.2 shows changes of the population distribution with time. Thestationary-state values are shown with the closed circles. This distribution approxi-mately corresponds to the one in Fig. 4.5. At early times of t< 10~9 s, all thepopulations increase linearly with time and »i(/>)/£(/>) is approximately propor-tional to/>~5. For t> 10~9 s, with the course of time, level 2 reaches its final valuefirst, level 3 second, and then level 4. At t~ 1 x 10~7 s, all the higher-lying levelsp>5 reach the final values simultaneously. We define the transient time ttr(p) atwhich, in Fig. 4B.1, the transient population reaches 63% of its final stationary-state value. In Fig. 4B.3, ttr(p) for each level is shown with crosses. We define theresponse time ?res as the largest among the ttr(p)'s. We regard our system as beingin the initial transient in t < ties, and at t ~ ?res the stationary state is reached.

At the early times of t < 1CT9 s, all the levels are populated by the direct exci-tation from the ground state and depopulation is virtually absent. This is just thesituation of eq. (4.3) with An(0)= -n0, or n\(i) = n0(l - e~50 ~~rt, n0Bt = At;

t<^,\/Bat the start of filling the bucket with water, the flow of water out from the hole at


FIG 4B.2 Population distributions for several values of t. •: the steady-statevalues, n: the level at which the transition of eq. (4B.4) takes place. (Quotedfrom Sawada and Fujimoto, 1994; copyright 1994, with permission from TheAmerican Physical Soceity.)

the bottom can be ignored. The population is thus simply accumulating virtuallywithout depopulation:

where n(l) = 1 m~3. The fact that nl(p)/g(p) is proportional to p~5 is understoodfrom the p dependence of C(l, p) oc/i;/, <xp~3, eq. (4.23a), and from g(p) = 2p2,

Figure 4B.3 shows the depopulation rate by electron impact and that byradiative transitions: on the r.h.s. of eq. (4.4) they are the first three terms and thelast term, respectively. For lower-lying levels the latter radiative depopulation rateis dominant and for higher-lying levels the former collisional depopulation ratesare dominant. Levels 4 or 5 is the boundary between these levels, i.e. Griem'sboundary, pG. (In Figs. 4.4-4.8, pG for «e= 1018 m~3 was between p = 3 and 4.This difference is due to the slight difference in Te and slightly different cross-section data used in the numerical calculations.)

For a lower-lying level than/?G we can neglect the collisional depopulation andeq. (4.2) is approximated by


FIG 4B.3 A: the radiative decay rate; o: the collisional depopulation rate; and n:the total depopulation rate, or the relaxation time tri(p), eq. (4.4), which isreferred to the r.h.s. ordinate. For Te = 10 eV. +: the time at which thetransition of eq. (4B.4) takes place, x: the time at which the population comesclose to the steady-state value. (Quoted from Sawada and Fujimoto, 1994;copyright 1994, with permission from The American Physical Society.)

This is readily solved as

The dash-dotted curves in Fig. 4B.1 indicate eq. (4B.3). Slight differences in thefinal stationary state come from the neglect of the cascading contributions ineq. (4B.2). See p. 101. For these levels, ttr(p) is given by [^2q<pA(p, q)]~l. See Fig.4B.3 and eq. (4.14).

Figures 4B.1-4B.3 show that levels higher than/?G have a common ttr(p). Fort > 1 x 10~9 s, populations of excited levels become substantial, and the populatingprocess of levels lying higher than/?G changes from direct excitation, eq. (4B.1), tothe excitation from the adjacent lower-lying level, which is the dominant popu-lating flux in the final stationary state, as we have seen in Figs. 4.6, 4.8, and 4.9.


In Figs. 4B.1 and 4B.2 the time of this transition or the level which is undergoingthis transition is shown. At the time of this transition for level p, the relationship

holds. It may be interesting to note that this equation is exactly the same aseq. (4.27). At this time level (/>—!) is still in the initial phase, eq. (4B.1), so thatwe have

This is readily solved for t:

By combining eqs. (4B.3) and (4B.7) we have thus reached a conclusion: tri(p)defined by eq. (4.14) gives ttr(p) for levels p <po, while for levels p >po it gives atransition from the initial phase, eq. (4B.1), to the ladder-like excitation, not ttT(p).Figure 4B.3 indicates that the latter statement is correct within a factor of 2.

As we have seen above, all the levels lying higher than the level that isundergoing the transition, eq. (4B.4), are in the ladder-like excitation mechanismand their populations are determined by the population of the lower end of theexcitation ladder. This is because the relaxation time tri(p) foip >PG is shorter forlarger p as shown in Fig. 4B.3. Therefore these populations appear in Fig. 4B.1 asstrictly parallel and in Fig. 4B.2 proportional to p~6. With a further increase intime, the level undergoing this transition comes down. From our discussions in thetext, we recognize that the lowest level that undergoes this transition is approxi-mately equal to Griem's boundary level, pG. It is noted that this level has thelargest tr\(p). This is the reason why tTi(pG) gives the common ttr(p) for the levelsP>Po and why these levels enter into the final stationary state at this time (seeFig. 4B.3). We thus conclude that ties is given by tTi(pG).

Until now we have assumed that, in the time scale of tr\(p), the ground-statepopulation «(1) is constant. In Appendix 5B we discuss the temporal developmentof n(l) and nz. Equation (3.35a) indicates that the ionization rate coefficient in thepresent example is of the order of 10~14 m3s^1. The effective depletion ratecoefficient of the ground state atoms is of similar magnitude. The ground-state

where we have used the relation for large p. With the aid of eq.may be approximated toor


population is thus depleted with a time constant of 1CT4 s. Thus, our assumptionof constant n(l) in the time scale of Figs. 4B.1 and 4B.2 is justified.

Recombining plasma

Protons ofnz=l m~3 are immersed in an electron gas or a plasma with Te = 0.1 eVand «e= 1018 m~3. The reader may locate this plasma on Fig. 1.2. At t = 0 theybegin to be acted upon by the plasma and to recombine. Figures 4B.4 and 4B.5show transient characteristics of the populations. The dashed lines in Fig. 4B.4and the closed circles in Fig. 4B.5 show the final stationary-state values, whichapproximately correspond to the distribution in Fig. 4.20(a). In Fig. 4B.5, theSaha-Boltzmann, or LTE, populations, eq. (2.7a), are shown with the squares.Figure 4B.6 shows ttr(p) (crosses) determined from Fig. 4B.4 in a similar mannerto the ionizing plasma case. This figure also includes the collisional and radiativedepopulation rates and tr\(p).

We now examine the transient populations. Figure 4B.7 shows the rate coef-ficients for radiative recombination and three-body recombination. It is noted that

FIG 4B.4 Temporal development of the excited-level populations for nz= 1 m 3,Te = 0.1 eV, and «e = 1018 m~3. The dashed lines show the steady-state values,and the dash-dotted lines show the result of eq. (4B.8). The assumption of con-stant nz is valid until ?~ 1CT2 s, the relaxation time of the ion density. (Quotedfrom Sawada and Fujimoto, 1994; copyright 1994, with permission from TheAmerican Physical Society.)


FIG 4B.5 The population distribution for several values of t. •: the steady-statevalues, n: the LTE populations given by eq. (2.1 a). (Quoted from Sawada andFujimoto, 1994; copyright 1994, with permission from The American PhysicalSociety.)

the former rate coefficients are almost the same as those in Fig. 3.5(b). For levelsp>4 the latter dominates over the former, which in turn is dominant forp<4. Inthe early times of t< 1CP10 s the population distribution in Fig. 4B.5 directlyreflects Fig. 4B.7, suggesting that the dominant populating process is the directrecombination, radiative, or three-body process. Remember that in Fig. 4B.5 thepopulation has been divided by the statistical weight. The linear increase in thepopulations with time is seen in Fig. 4B.4. In this figure the dash-dotted lines show

It is interesting to note that population inversion is established for/? > 4 because ofthe dependence of a(p) <xp6. It is seen in Fig. 4B.4 that, with an increase in t, thepopulation of levels/? > 4 deviates upward from eq. (4B.8), and they reach the finalQSS values starting from very high-lying levels. The last point may be regarded assuggesting that ttr(p) is given by tri(p). However, ttr(p) is found to be appreciablylonger than tri(p) as seen in Fig. 4B.6. The above two behaviors are explained fromthe fact that these high-lying levels, when their populations come close to thestationary-state or LTE values, are strongly coupled to each other by collisionaltransitions. See Figs. 4.22-4.24. As a result, a level, when its population is lower


FIG 4B.6 A: the radiative decay rate; O: the collisional depopulation rate; andD: the total depopulation rate; or the relaxation time tri(p), eq. (4.4), whichis referred to the r.h.s. ordinate. For Te = Q.l eV. x : the time at which thepopulation comes close to the steady-state value. (Quoted from Sawada andFujimoto, 1994; copyright 1994, with permission from The American PhysicalSociety.)

than the LTE value, receives additional populating flux from the neighboringhigher-lying levels, or it is "pulled" upward toward its LTE value. This level, p,reaches its final value only when deexcitation flux from this level to the adjacentlower-lying level (p — 1) is balanced by the excitation flux from (p — 1) to p. Inother words, when the population of a level is close to the LTE value, it is "pulled"downward by the lower-lying levels which are still far from LTE. These levelscannot be considered as independent levels. This is the reason of the appreciabledeviation of ttr(p) from tr\(p).

Among the lower-lying levels, but still higher than pG, there is Byron'sboundary level p#. In the present example, />B lies between p = 6 and 7. See alsoFigs. 4.20-4.24. Between the levels lower than/>B (and still higher than/>0), col-lisional coupling becomes only downward, eqs. (4.36) and (4.53), and the popu-lation of a level is controlled by that of the adjacent higher-lying level. It mightthus be assumed that ttT(p) for these levels, as in the ionizing plasma case, is givenby ttT(pv). This turns out not to be the case. This is because lower levels havelonger tri(p)'s and their populations lag behind n0(pB),

In Fig. 4B.4 the slope of the population of level p = 2 first deviates down-ward from the linear relationship, eq. (4B.8) with a(p)ne replaced by (3(p), at


FIG 4B.7 Recombination rate coefficients. O: radiative recombination, A: three-body recombination multiplied by «e=1018 m~3. (Quoted from Sawada andFujimoto, 1994; copyright 1994, with permission from The American PhysicalSociety.)

t~2x 10~9 s, which corresponds to tr\(2); see Fig. 4B.6. If the cascading contribu-tions from levels/? > 2 were absent, the population would have reached a final valueat this time. In fact, the cascading contributions are substantial and even domi-nant in later times. See Fig. 4.23. Depending on the time dependence of popula-tions of the levels that contribute to the cascading population of level 2, itspopulation increases with time. A similar, but less prominent, feature is seen withlevel 3. This is the reason why these lower-lying levels cannot reach their sta-tionary-state values until Griem's boundary level po = 6, which has the largesttri(p), reaches its stationary-state value; see Fig. 4B.6. We thus again conclude that?res is given by tA(pG).

In Chapter 5 and Appendix 5A it turns out that the effective recombinationrate coefficient for the present example is of the order of 10^16m3s^1, so that nz

changes appreciably only in 10~ s. Thus, during the transient of the excited-levelpopulations discussed above, nz is virtually constant.

Validity condition of QSS

We have seen that both for the ionizing plasma and the recombining plasma theoverall response time of the excited-level populations, tres, is given by tr\(pG), the


FIG 4B.8 The relaxation time of Griem's boundary level pG as calculatednumerically are given by the lines. The response time tTes as determined from thetemporal development of the populations, O: ionizing plasma of Te = 10 eV andx : recombining plasma of Te = 0.1 eV. +: recombining plasma of Te= 10 eV.

(Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permissionfrom The American Physical Society.)

longest relaxation time among the excited levels. Figure 4B.8 compares tTes, asdetermined from Figs. 4B.1 and 4B.4 and from similar calculations in the broadrange of «e, and tr\(pG), as determined numerically.

First we assume «e and Te are fixed, so that tTes is constant. We have shownthat, under the condition of constant n(l) or nz, if the transient population of levelpG, which determines the response time tres, reaches its stationary-state value, allthe excited levels enter into the final stationary state.

We now examine the case of ionizing plasma in which «(1) changes with time.This change may be due to its depletion by ionization or for some other reasonslike spatial transport. The temporal development of the population of level pG isapproximately expressed by

where N(f) is the ion density, in this case, at t, and Z(p) is the Saha-Boltzmanncoefficient, eq. (2.7). In the QSS approximation, the population of po is expressedwithin this approximation as

In our example of Figs. 4B.1-4B.3, the change in «(1) is assumed to be only due toionization. Even if we include the depletion of «(1) at a later time, the requirement(4B.13) is well satisfied, and QSS is valid.

Next we examine the temporal change in nz for a recombining plasma. Thedominant populating flux of level pG is deexcitation from the adjacent higher-lyinglevel. We now take into account "the time lag" of this population. Then thepopulation of pG is approximately given as

This relation indicates that the parameter (ties/T) gives a measure of deviationof the CR population from its actual value. Thus the validity condition of QSSis that


where n(i) and N(t) stand for n(pG) and n(l), respectively, at t. In theQSS approximation, the time derivative is set equal to zero, eq. (4.18), and thepopulation is given as

We assume that the temporal change of N(f) is expressed in terms of a timeconstant T as

where NO is the initial value and T is positive for decreasing N(f) and negative forincreasing N(f). Under the condition of t, \T\ ;$> tres, eqs. lead to

If N(f) is again expressed by eq. then eqs. lead toand

for t, | T\ > tres. The validity condition of QSS is again given by eq.

(4B.9)-(4B.11)

(4B.11), (4B.14) AND (4B.15

(4B.13).


When «e changes with time, we may start with eq. (4B.9) or eq. (4B.14). Insteadof N(f) in eq. (4B.9), or N(T- tres) in eq. (4B.14) the factor ne now changes. Wemay proceed with our discussion in a similar way to the above cases. We thenreach the same conclusion as for the previous cases. The change in pG or tres withthe change in ne, which we have neglected so far, has little effect in the abovereasoning. This is because n(pG) is almost linearly dependent on «e.

The case of a temporal change in Te is not straightforward because the collisionrates have nonlinear dependence on Te. First we examine an ionizing plasma andstart with eq. (4B.9). Instead of the change in N(f), C(l,pG) changes this time.When we note that tres is very weakly dependent on Te (see Fig. 4B.8), we canproceed with our discussion in much the same way as for the previous cases. Thevalidity of QSS would then be \ties[dC(l,pG)/dt]/C(l,pG)\ < 1. The excitation ratecoefficient is approximately given by eq. (3.29). Except for very high temperature,the temperature dependence is mainly determined by the exponential factor, andthe T,T1/2 factor may be neglected. Then the validity condition is written as

For a recombining plasma, except for the case of pG<pv, i.e. the case of lowtemperature and high density, the QSS value of n(pG) is approximately given by itsLTE value,

References

The discussions of this chapter are based on:Fujimoto, T. 1979a /. Phy. Soc. Japan 47, 265.Fujimoto, T. 1979b /. Phy. Soc. Japan 47, 273.Fujimoto, T. 1980a /. Phy. Soc. Japan 49, 1561.Fujimoto, T. 1980b /. Phy. Soc. Japan 49, 1569.Fujimoto, T. and McWhirter R.W.P. 1990 Phys. Rev. A 42, 6588.Kawachi, T. and Fujimoto, T. 1995 Phys. Rev. E 51, 1440.

Appendix 4A is based on:Sawada, K. 1994 Ph.D. thesis (Kyoto University).

Appendix 4B is based on:Sawada, K. and Fujimoto, T. 1994 Phys. Rev. E 49, 5565.

This equation suggests that the validity condition for QSS would be given by

REFERENCES 149

Figure 4.12 is taken from:Tachibana, K. and Fukuda, K. 1973 Japan. J. Appl. Phys. 12, 895.

Figure 4.15 is taken from:Akatsuka, H. and Suzuki, M. 1994 Phys. Rev. E 49, 1534.Akatsuka, H. and Suzuki, M. 1995 Plasma Sources Sci. Tech. 4, 125.

5

IONIZATION AND RECOMBINATION OF PLASMA

In the preceding chapter, we developed our theory of the excited level populationson the basis of the quasi-steady-state approximation to the solutions of the rateequations, eq. (4.2). The rate equation for the ground-state atom density, «(1), andthat for the ion density, nz, have been left as they were. Since we have obtained thesolutions for the excited level populations we substitute them into these rateequations. Equation (4.2) for p = 1 reduces to

These quantities express the effective rate of ionization and recombination of theplasma (in the second sense) which is immersed in a plasma (in the first sense).They include the ionization or recombination processes via excited levels. Thequantity SCR is called the collisional-radiative (CR) ionization rate coefficient, andO?CR the CR recombination rate coefficient. Appendix 5A shows how SCR and «CRare established in the temporal development of the plasma. Like the populationcoefficients rQ(p) and ri(p), these coefficients are functions of «e and Te. It is to benoted that, in the expression of eq. (5.3) for SCR, only ri(p) appears and rQ(p) isabsent, and vice versa for aCR of eq. (5.4). This is an important feature of ourformulation as schematically depicted in Fig. 1.9: the ionizing plasma component

After rearrangments of the terms this equation is rewritten as

with

COLLISIONAL-RADIATIVEIONIZATION 151

of the populations gives the effective ionization flux of the plasma SCRn(l)ne (seeFigs. 4.7-4.9; the upward arrow on the right-hand side indicates the magnitude ofSCR) and the recombining plasma component gives the effective recombinationflux aCR«z«e (Figs. 4.16-4.18 and Figs. 4.21, 4.23, and 4.24; the downward arrowon the right is «CR except for Figs. 4.18 and 4.24, in which the rate coefficientdivided by ne is given). These features suggest that, from a measurement of n\(p),we may determine ScRn(l)ne, and from nQ(p) we determine aCRnzne. This pointwill be discussed later on in this chapter.

In many practical situations we encounter the cases in which the overallpopulations summed over all the excited levels are very small in comparisonwith [«(!) + «J. We saw examples of this situation in the preceding chapter. SeeFigs. 4.5, 4.14 and 4.20.* The apparent difficulty of divergence of populations ofan infinite number of Rydberg levels for a recombining plasma will be resolved inChapter 9. Thus, we may assume

5.1 Collisional-radiative ionization

Figure 5.1 shows an example of SCR for neutral hydrogen for Te= 1.28 x 105 K,corresponding to the case which we treated in Section 4.2 in the discussion ofthe ionizing plasma component of the population. The reader may rememberthe horizontal line drawn in the ne-Te plane of Fig. 1.2 when he or she studied

* In Fig. 4.20 in the high-density regions of «e > 1022 m~3, a large population of «(2) > nz couldoccur, thus violating this statement, so long as we assume the recombining plasma component. However,in many actual plasmas, «(1) can be larger than n2 by many orders of magnitude, and our assumptionbecomes valid. Rather, if the excited-level population becomes larger than [«(!) +nj, this plasmawould constitute an interesting new field to be investigated, which is outside of the scope of this book.

In the following discussions we are mainly concerned with the high-temperaturecase, in which Byron's boundary does not play any role. The reasons are givenbelow. The low-temperature case will be treated in Section 5.3 only with regard tothe CR recombination rate coefficient; this case is very important in treatingactual plasmas.

Since the problems of ionization-recombination and ionization balance of aplasma are particularly important in considering ions in various ionizationstages rather than neutral atoms, we put some emphasis on hydrogen-like ionswith nuclear charge z rather than neutral hydrogen. For ions with a large z, thetemperature range of interest tends to be higher. The reader will understandthis point later in this chapter. So, we present approximate formulas whichfit better the result of numerical calculations for ions of high-temperatureplasmas.

152 IONIZATION AND RECOMBINATION OF PLASMA

Section 4.2. Figure 5.1 shows the breakdown of the net ionization rate coefficient SCRinto components: the magnitude of ionization fluxes from various levels is shown foreach level from which this final step of ionization takes place. See also Figs. 4.7-4.9.Figure 5.2 shows an example of SCR for a hydrogen-like ion with Te/z

2 = 6 x 104 K.In both the cases, starting from the low-density limit, SCR increases with ne, and itfinally reaches a constant high-density-limit value. We will investigate why SCRchanges this way by examining the details of the ionization process.

Region I (ne < nf)

We begin our discussion with the low-density region. The upper bound of thisregion nf will be given later. As we have seen in Section 4.2, in the limit of lowdensity, all the excited levels are in the corona phase as shown in Fig. 1.10(a), andall the excitation fluxes originating from the ground state p=\ eventually returnto it through the chain of the radiative cascade decays. See Fig. 4.7. Thus, the fluxof electrons into the continuum states comes only from the ground state, i.e. directionization. This is also seen in Fig. 5.1.

* In this footnote, we justify Fig. 5.1 as an expression of SCR which was defined by eq. (5.3).Insteadof eq. (5.1), we start with

We substitute our solution of the excited-level populations, eq. (4.20) or (4.20a), into eq. (1), andrearrange this equation into two terms, each proportional to «(1) and nz, respectively. Then, we have

where we have adopted the convention that «[(!) = n(l), and r0(l) = 0. The first term on the r.h.s.divided by n(l)ne is nothing but SCR shown in Fig. 5.1. We now show that the second term agrees withour definition of aCR by eq. (5.4).

We sum eq. (4.2) over all the excited levels with/; > 2. The excitation and deexcitation fluxes amongthe excited levels cancel each other and the remaining quantities are

Again, we express «(/>) on the r.h.s. in terms of n(l) and nz. Owing to our assumption of QSS oreq. (4.18), the above eq. (3) vanishes, so that each of the coefficients of n(l) and nz should vanish. Forthe coefficient of nz we have

It is straightforward to see that the second term on the r.h.s. of eq. (2) agrees with the r.h.s. of eq. (5.4),of course, with the sign reversed. It is also interesting to see the coefficient of n(l) in eq. (3) also leads tothe expression of eq. (5.3).

The above reasoning is due to Professor M. Otsuka.

COLLISIONAL-RADIATIVE IONIZATION 153

FIG 5.1 Breakdown of the collisional-radiative ionization rate coefficient SCRinto the individual ionization processes. The numerals indicate the principalquantum number of the level from which the final step of ionization originates.For neutral hydrogen with Te= 1.28 x 105 K. See also Figs. 4.7-4.9. (Quotedfrom Fujimoto, 1979b, with permission from The Physical Society of Japan.)

The superscript 0 means the low-density limit. This expression is of a similarfunctional shape to eq. (3.35a), but fits better to the actual rate coefficient in thetemperature range considered here, as seen in Figs. 5.2 and 5.3. Note that, ineq. (3.35a), the factor (z2 R/x(p))7^4 is simply 1 for the ground state.

With an increase in «e, Griem's boundary pG comes down (see Fig. 1.10(a)),and the levels lying higher than this boundary enter into the saturation phase. Forthese levels the multistep ladder-like excitation-ionization mechanism is estab-lished, and, roughly speaking, the excitation flux from the ground state into theselevels eventually results in ionization. See Fig. 4.8: this figure is for «e= 1018 m~3

and 3 <po < 4. As Fig. 5.1 shows, with the increase in «e, the contribution to SCRfrom ionization from very high-lying levels increases. Both figures show that thecontributions from levelsp > 10 are substantial at ne= 1018 m~3. We may expressthe increasing ionization rate coefficient as

We adopt the approximattion


FIG 5.2 Collisional-radiative ionization and recombination rate coefficientsversus the reduced electron density, «e/z

7, for hydrogen-like ions. The reducedelectron temperature is Te/z

2 = 6 x 104 K. All the quantities are scaled againstthe nuclear charge z. Approximations according to the formulas in the text,eqs. (5.10), (5.13), (5.18), and (5.24), are compared with the numericalcalculation. The scaled ionization ratio [z4«z/n(l)j is also shown. The range of«e/z

7 is divided into three regions: I, II, and III. (Quoted from Fujimoto, 1985,with permission from The Physical Society of Japan^)

The discussions in the preceding chapter, especially that in p. 125-126, suggestthat the lowering of Griem's boundary may be regarded, in effect, as loweringof the ionization limit. Those discussions were concerning a recombining plasma.Here, however, for the ionizing plasma, similar arguments can be made: i.e. theionization limit comes down to Griem's boundary level. The ionization cross-section, eq. (3.35), may be modified, and if we return to the approximation ofeqs. (3.29) and (3.35a), we have the approximation

COLLISIONAL-RADIATIVE IONIZATION 155

FIG 5.3 Comparison of approximate formulas and numerical calculation for thecollisional-radiative ionization and recombination rate coefficients in the low-density limit and in the high-density limit. Sg^ and a~R are related by eq. (5.23).(Quoted from Fujimoto, 1985, with permission from The Physical Society ofJapan.)

with

with pG given by eq. (4.25) or eq. (4.29b). In Fig. 5.2 we compare this approx-imation with the numerical calculation for Te/z

2 = 6 x 104 K, where we haveadopted eq. (4.29b) for pG_

Under the condition ofpo^> 1, the oscillator strength, eq. (3.6a), is approximatedto/i^ ~ l.6p~3 (Table 3.1(b)) and/o is given in this approximation as/o — 0.8/>Q2.Since the excitation rate coefficients have a slightly different dependence on Te

from that of S(l), we adopt an approximate expression starting from eq.


With a further increase in ne, the boundary pG further comes down andreaches p = 2 at n£°, which is the upper boundary of region I. See Fig. 1.10(a).Note here that, since we assume high temperature, Byron's boundary does notappear in this discussion. This density is approximately given from eq. (4.29a) forpG = 2 as

Regions II and III (ne > nf)

At about this boundary nf, SCR approaches the high-density-limit value as seen inFigs. 5.1 and 5.2. (See also Fig. 4.4.) As we have seen in Section 4.2, in the high-density regions of present interest, the multistep excitation-ionization ladder isestablished among all the excited levels starting from the ground state, as seen inFig. 4.9. Virtually all the excitation fluxes leaving the ground state contribute tothe net ionization flux.

In these high-density regions, all the excited levels are in the saturation phase(Fig. 1.10(a)), and the interrelationship between the population coefficientsro(p) + ri(p)= 1> ecl- (4-21), holds. In eq. (5.3), if we neglect the radiative decayterms,* which is justified in the highest-density region as seen in the next section,we have

* In region II, A(q, 1) in eqs. (5.3) and (5.4) is neglected in comparison with, say, C(q, q + l)«e, thatis, the radiative decay is neglected in the depopulating processes. However, A(q, 1) is still larger thanthe corresponding competing collisional rate F(q, l)«e. See Section 5.2 later.

This boundary is shown in Fig. 5.2.

where we have used eq. (3.31a). The superscript oo means the high-density limit.When we remember that, at high temperature, r0(q) is approximated to (1 —p~6)(eq. (4.48)), or very close to 1, we have

This situation is actually seen in Fig. 4.9. Among the fluxes originating from theground state the dominant flux is the excitation to the first excited level p = 2,and the following approximation is found to reproduce well the numericalcalculation.

CR RECOMBINATION - HIGH-TEMPERATURE 157

FIG 5.4 Collisional-radiative recombination rate coefficient aCR for neutralhydrogen. The boundary density «^°+ given by eq. (5.20) with p = 2 is shownwith open circles, and that by eq. (5.25) with closed circles. (Quoted fromFujimoto, 1980a, with permission from The Physical Society of Japan.)

This approximation is shown in Fig. 5.2 with the horizontal line and also inFig. 5.3 as converted to a£?R (see later). This simple expression is good to within afactor of 2 in the temperature range of Fig. 5.3. In Fig. 5.2, in the density rangebeyond n£° the approximation, eq. (5.10), is extrapolated to connect to the high-density limit value, eq. (5.13).

5.2 Collisional-radiative recombination - high-temperature case

Figure 5.4 shows «CR, the collisional-radiative recombination rate coefficient forprotons and electrons to form neutral hydrogen atoms in the ground state against«e for several temperatures. The reader may identify the parameter range of thisfigure in the ne-re plane of Fig. 1.2. Note that we are now treating the case ofz=l. Figure 5.2 shows an example of «CR for fully stripped ions to formhydrogen-like ions. Figure 5.5 shows, for the high-temperature case of Te= 1.28 x105 K treated in Section 4.3, the breakdown of the fluxes of the final steps ofrecombination reaching the ground state, or each term of eq. (5.4), normalized bythe total recombination flux. The hatched areas show the contributions from theradiative transitions, i.e. direct radiative recombination, f3(l)nz «e (denoted by j3),and the spontaneous transitions from excited levels (the upper level is indicated bythe numeral).


FIG 5.5 Breakdown of the collisional-radiative recombination flux into the indi-vidual fluxes of the processes reaching the ground state, normalized by the totalcollisional-radiative recombination flux. See eq. (5.4). The hatched areas showthe radiative transitions and the blank areas the collisional transitions. On theright-end ordinate, the corresponding breakdown of the collisional-radiativeionization rate into the individual rates starting from the ground state (seeeq. (5.3)) is shown for the high-density limit. Note that this breakdown is differentfrom that shown in Fig. 5.1. For the relationship, see the footnote in p. 152. Seealso Fig. 4.9. Neutral hydrogen with Te = 1.28 x 105 K. (Quoted from Fujimoto,1980a, with permission from The Physical Society of Japan.)

Region I (ne < n£°)

Figures 5.2-5.4 give the low-density limit «CR, which is common to hydrogen-likeions and neutral hydrogen. In this limit (Fig. 4.16) the effective recombination ratecoefficient is given as

In the present high-temperature case, /3(p) is approximately proportional to p 2'5

for/? > 2 (Section 3.2; see Fig. 3.9 and also eq. (4.41)), and the summation of f3(p)over these excited levels gives a contribution comparable with (3(1) as actually seenin Fig. 5.5. Note that the cascading contribution as a whole is equal to S(/,>2)/3(/>).The argument of the exponential integral, eq. (3.19a), of /3 (1) in the temperaturerange of Figs. 5.2 and 5.3 is of the order of 1, and the Te dependence of /3(1) isneither T~1-5 (for extremely high temperature, eq. (3.21) or eq. (4.41)) nor T~°-5

(for low temperature, eq. (3.22) or eq. (4.50)) in these intermediate cases.

This approximation is shown in Figs. 5.2 and 5.3.Figure 5.5 shows that, with an increase in ne, the relative contribution from the

cascade increases slightly. This is due to the slight increase in the "populations",[Ko(/0/g(p)nzne], of the excited levels that have entered into the saturation (LTE)phase (Figs. 4.13, 4.14 and 1.10(b)), resulting in a slight increase in aCR in Fig. 5.4.This increase, however, is barely noticeable. This increase is more pronounced forlower temperatures in Fig. 5.4 and also in Fig. 5.2. On the assumption thatthe populations of the levels in the CRC phase are very low in comparison with theSaha-Boltzmann value in the saturation (LTE) phase (which is incorrect forthe case of Te= 1.28 x 105 K and higher but is correct for lower temperatures)the effective rate coefficient may be expressed as

The idea of this approximation is that the levels lying higher than Griem's boundaryare strongly collisionally coupled, directly or indirectly, with the continuum, andthese levels are the origin of the cascading transitions to the lower-lying levels.This picture may be interpreted in another way: on p. 125-126 we regarded thelowering of Griem's boundary level as an effective lowering of the ionization limit.In the present context, the threshold energy of the radiative recombination islowered to the energy of Griem's boundary, and the transitions nQ(p)A(p, q) forq<Pa<P m&y be regarded as an effective enhancement of radiative recombina-tion to q. We may therefore express eq. (5.16) as lowering of the threshold energyfor the radiative recombination to the energy of Griem's boundary level:

See Fig. 3.9. We find that, in the range of Te shown in Fig. 5.3, the followingapproximation is rather good


It may be understood that the exponential factor accounts for the effective"increase" in the ion density due to the inclusion of the higher-level populations:the second term on the r.h.s. of eq. (5.16). We note that, since j3(p) is proportionalto p~2'5 for p^>l, the direct recombination terms Y>pf3(p) for p>pG^>\ areextremely small as compared with, say, (3(1). From eq. (5.17), we have

It is found that eq. (5.18) is a good approximation in the temperature rangeconsidered here rather than the low-temperature case.* An example for compar-ison is seen in Fig. 5.2.

* At extremely high temperature, the populations in the CRC phase are even higher than the LTEvalues (see Fig. 5.10 later), and with an increase in «e they decrease to their LTE values. As a result,OCR decreases slightly. In such a case eq. (5.18) does not apply, of course.


With a further increase in ne, Griem's boundary comes down to reach the firstexcited level p = 2 at n£° as given by eq. (5.11). For higher densities all the excitedlevels have the LTE populations or their high-density-limit values.

Region II (nf <ne < nf+)

Figure 5.5 indicates that, in densities higher than n f , the final steps of recombina-tion are still mainly radiative transitions up to a certain «e. Since, in this region, thepopulations of virtually all the excited levels are in LTE with respect to the iondensity (eq. (4.47) or eq. (4.38) with eq. (4.48); see Figs. 4.13 and 4.14) the effectiverate coefficient of recombination is, in this approximation, independent of ne, asindicated in Fig. 5.2 by the horizontal line. In this figure this feature is notconspicuous, but in Fig. 5.4, the «CR curve for Te = 3.2 x 104 K, and also that for8 x 103 K, show a plateau which corresponds to this region.

Figure 5.5 shows that, with a further increase in «e, the final steps of recombina-tion change from radiative processes to collisional processes at about 1023 m~3 i.e.the radiative recombination (ft) is taken over by the three-body recombination (a)and the spontaneous transitions to the ground state by the corresponding colli-sional deexcitations. This transition in the final steps is expressed by

An interesting feature is that these equalities, which define the upper boundaryKoo+ Qf region ii, appear to hold at almost the same ne, whether it is for recom-bination, eq. (5.19), or it is for deexcitation, eq. (5.20), and for the latter, almostbeing independent of p. We now examine this point below.

The transition probability A(p, 1) is given by eq. (3.1) with eq. (3.6a), and thedeexcitation rate coefficient F(p, 1) is given by eq. (3.31) with the approximation(3.29). We assume level p to be high, so that the energy difference E(l,p) isapproximated to z R. Then, eq. (5.20) gives for the boundary ne

where G is given by eq. (3.30). This boundary ne is independent of p because boththe radiative transition probability and the collisional rate coefficient have theoscillator strength fif in common. For the temperature of Fig. 5.5, eq. (5.21) gives«e/z

7 ~ 8 x 1022 m~3'.In eq. (5.19) we adopt for (3(1) the approximation eq. (3.19a), where the Gaunt

factor gbf has been assumed to be 1. For the three-body recombination, the ratecoefficient is related to the ionization rate coefficient by eq. (3.40), and the latter isgiven by eq. (3.35a). Then, eq. (5.19) reduces to


We remember that the oscillator strength/1>c is 0.435 (Table 3.1(b)) and that, forthe temperature range of Figs. 5.3 and 5.5 the exponential integral multiplied bythe exponential factor is of the order of 1. Then eq. (5.22) gives ne which is veryclose to that given by eq. (5.21). In the present example of Fig. 5.5, the differenceis 16%. This is the reason why the transitions from the radiative processes to thecorresponding collisional processes, eqs. (5.19) and (5.20), take place at nearly thesame ne for all the final steps of recombination.

The above arguments suggest that in Fig. 5.5 the relative magnitude of theradiative flux at lower density should be equal to the corresponding collisionalflux at high density. This is only approximately true; e.g. for (3(1) and a(l) thedifference is a factor of 3. This inconsistency is due to our crude approximationsadopted above.

Region II is limited by the boundary «^°+ given by eq. (5.19) or eq. (5.20). Weadopt eq. (5.21) for the definition of nf+:

In Fig. 5.4 the boundary nf+ given by eq. (5.20) for p = 2 is shown by the opencircles. In Fig. 5.2 the approximation of the constant «CR continues up to this nf+.

Note here the difference between nf and nf+. The former boundary comesfrom the comparison between the radiative decay A(2,1) with the collisionaldepopulation rate J^ C(2,p)ne at high temperature. We saw that this ne givescomplete saturation of the ionizing plasma component of the populations. See Figs.4.4 and 1.10(a). The latter boundary comes from A(2,1) = F(2, l)«e. Figure 4.2shows that, for high temperature, C(2,3)/F(2,1)~ 100. This factor gives thedifference between n£° and K^°+. This figure suggests that the situation is verydifferent for low temperatures, which turns out to be true as we will see in the nextsection. It may be worth noting an interesting coincidence: if we put pa = 1 ineq. (4.29a), this equation gives ne which is very close to the present nf+. On thebasis of this coincidence, we may extrapolate Griem's boundary in Fig. 1.10(b)to p = 1. Then, pG = 2 gives nf and po=l gives nf+. It is noted that nf and nf+

are almost independent of Te for high temperature.

Region III (ne > nf+)

In this highest-density region all the terms of radiative transitions in the rateequation, eq. (4.2), or in eq. (5.4), are small in comparison with the terms of thecorresponding competing collisional transitions, and they can be entirely neglected.As we have seen in Section 4.1, the interrelationship between the populationcoefficients r0(/>) + r1(/>) = 1, eq. (4.21), is valid. It is straightforward to see thateqs. (5.3) and (5.4) lead to another interrelationship:


Then we obtain from eq. (5.13)

This is the approximation which has been compared in Fig. 5.3. Figure 5.2 alsoshows the comparison of this approximation with the numerical calculation.Figure 5.6 shows (a~R/»e) calculated for neutral hydrogen. Equation (5.24) isalso plotted on this figure. The small discrepancies may be attributed partly to thedifference of the scaled excitation cross-sections near the threshold energies forions which are considered here, and those for neutral atoms as seen in Fig. 3.11.

Figure 5.5 shows the high-density limit of the breakdown of the final recom-bination fluxes. On the right-end ordinate, for the ionizing plasma, the breakdownof the initial steps, excitation and ionization, is shown. See eq. (5.3). Thisbreakdown is different from that in Fig. 5.1. It is seen that the principle of detailed

FIG 5.6 The high-density-limit values of the collisional-radiative recombinationrate coefficient divided by «e for neutral hydrogen. The approximation (5.24) isplotted for the high-temperature case of Te Si 1.5 x 104 K. For low-temperaturethe approximation eq. (5.27) is given. (Quoted from Fujimoto, 1980a, withpermission from The Physical Society of Japan.)

CR RECOMBINATION LOW-TEMPERATURE 163

balance is actually realized between the individual elementary processes in ioni-zation and in recombination.

5.3 Collisional-radiative recombination - low-temperature case

In this section we examine the recombination process in the low-temperature case(Te/z

2<1.5xl04K).Figure 5.7 is for neutral hydrogen with Te= 1 x 103 K, which corresponds to

Fig. 5.5 for the high-temperature case, and shows the breakdown of the final stepsof recombination into the contributions from the individual transitions to theground state. The reader may remember the horizontal line drawn in Fig. 1.2 whenhe or she studied Section 4.4. In spite of several substantial differences fromFig. 5.5, with an increase in «e, the transition from radiative processes to collisionalprocesses at about nf+ ~ 1023 m~3 appears similar. This boundary, eq. (5.20), forp = 2 is shown on the curves in Fig. 5.4 with the open circles. As noted above, theweak dependence of this boundary on temperature is the result of the smalltemperature dependence of F(2,1) in eq. (5.20) as seen from eq. (3.31) witheq. (3.29). We notice from Fig. 5.4, however, that, for the present low tempera-tures, «CR has already reached its high-density-limit value, which is proportionalto ne, at much lower densities. This is in contrast to the high-temperature case andmay appear puzzling. This point will be considered later.

In the limit of low density, all the excited levels are in the CRC phase, and thetotal recombination rate coefficient is given by eq. (5.14). In this case, as has been

FIG 5.7 Similar to Fig. 5.5 except that this is for Te = 1 x 103 K, low temperature.(Quoted from Fujimoto, 1980b, with permission from The Physical Societyof Japan.)


noted with regard to Fig. 4.21, the relative importance of recombination intoexcited levels as compared with that into the ground state is much higher incomparison with the high-temperature case, Fig. 4.16. See also j3(p) in Fig. 3.5(b).This feature is reflected in Fig. 5.7 as the small contribution (about 25%) from thedirect radiative recombination (denoted with j3) as compared with Fig. 5.5, whereit was about a half the total recombination flux. At low temperature, the down-ward flux of electrons by radiative transitions in the energy level diagram througha very high-lying level is important: Figure 4.21 shows that the radiative cascadingflux through level p =10, taken as an example of high-lying levels, is about 25% ofthe total recombination flux. This means that about half of the total recombi-nation flux is into levels 2 <p < 9, See Fig. 3.5(b).

With an increase in «e very high-lying levels enter into the saturation phase, orthe LTE phase, as seen in Fig. 1.10(b), and they have populations substantiallyhigher than the low-density limit as seen in Figs. 4.19 and 4.20. The cascadingfluxes originating from these levels accordingly increase. This is reflected in Fig. 5.7as the gradual decrease in the magnitude j3 with the increase in ne. It is notedthat its absolute magnitude is actually f3(l)nzne/nzne (see eq. (5.4)) and is constant,and its apparent decrease indicates the relative increase in the cascading fluxoriginating from these high-lying levels; this results in the substantial increase in«CR seen in Fig. 5.4. As has been described in the preceding section concerningeqs. (5.16) and (5.17), we regard the lowering of Griem's boundary level as theeffective lowering of the ionization limit, and the fluxes n0(p)A(p, q) for q <pG <pare regarded as an effective enhancement of radiative recombination to q. Thisfeature is reflected in Fig. 5.7 as the almost constant relative contributions fromlevels p = 2, 3, and p>4 plus the direct recombination, up to «e~1018 m~3.Remember that, in this region, nQ(p) for levels of p<po has almost commondependences on ne as seen in Figs. 4.19 and 4.20. In the present case, since thetemperature is low, the relative increase in aCR is large as compared with the high-temperature case. See eq. (5.17).

Figure 5.4 shows that, with the above increase in «e, aCR enters into its high-density limit at around «e~ 1018 m~3 for Te= 103 K; this density is much lowerthan nf+ as shown with the open circle. This is obviously a puzzle as mentionedat the beginning of this section: Radiative processes are still dominant in Fig. 5.7and they depend strongly on «e, and yet aCR does not depend on «e. In order toresolve this puzzle we first look at the sketch of the dominant fluxes of electrons inthe energy-level diagram in the high-density limit, Fig. 4.24. Levels above Byron'sboundary, p>l, are in LTE, and they are so strongly coupled with each otherand, therefore, with the continuum states that the downward fluxes into theselevels are almost balanced by the inverse upward fluxes from these levels (Fig. 4.24and also Figs. 4.15,4.17, and 4.18). In contrast to this, the downward fluxes amongthe levels lying below Byron's boundary, p < 6, are not balanced; this is because ofthe relationship F(p,p — 1) > C(p,p + 1), or eq. (4.36), for these levels. Thus, themultistep ladder-like deexcitation flux is established, which eventually reaches theground state, resulting in recombination. In other words, the electrons which

CR RECOMBINATION - LOW-TEMPERATURE 165

originate from the continuum states and reach the levels in LTE (p>pv) canreturn to the continuum states easily, while those which have crossed Byron'sboundary downward and left these high-lying levels can no longer return to theselevels and thus to the continuum states. They simply flow down, finally reachingthe ground state, completing the process of recombination.

Now we look for the critical process that determines the magnitude of «CR, theeffective rate of recombination, in this high-density limit. As we have seen thethree-body recombination to the ground state never plays a dominant role. See inFig. 5.7 that the contribution from a is virtually absent. As we have seen already,any collisional process involving the levels lying above Byron's boundary, p >p^,is almost balanced by the respective inverse process, so that none of them candetermine the rate. A collisional deexcitation flux between levels lying lower thanByron's boundary, p <PB, e.g. n(3)F(3,2)ne in Fig. 4.24, is found almost exactlyequal to the effective recombination flux shown on the right of this figure.However, this process does not determine the recombination flux: it simply dumpsall the population fluxes coming from the adjacent higher lying level 4 to theadjacent lower lying level 2, i.e. they are in the middle of the ladder-like deexcita-tion chain. Thus, it is concluded that the magnitude of the recombination flux isdetermined by the flux crossing the boundary between the levels in LTE and thosein the ladder-like deexcitation chain. This is Byron's boundary level /?B- In thepresent example, /?B — 6-7. In this sense, this level has been called the bottleneck.It is found that this level has a minimum of Z(p)nztieF(p,p — 1), which shouldbe approximately equal to the effective recombination flux.

Now we look at Fig. 4.23, the sketch at ne= 1020 m~3. At this density Griem'sboundary is between p = 3 and 4, and «CR still takes the high-density limit value,as seen in Fig. 5.4. The major difference of the features in this figure from those inFig. 4.24 is that the fluxes concerning the levels lower than Griem's boundary areradiative instead of collisional, including the final steps of recombination reachingthe ground state. This is also seen in Fig. 5.7. However, the features around thecritical level p^, which determine the recombination flux or aCR, are exactly thesame as in Fig. 4.24. This is the reason why aCR has already reached its high-density-limit value at this density. This is because />B lies above PQ. Thus we con-clude that, with the increase in «e, when Griem's boundary comes down and reachesByron's boundary level, aCR enters into its high-density limit. This transitionoccurs at «e~ 1018 m~3 for Te= 1 x 103 K. This critical density as determined by

mately given by

is shown in Fig. 5.4 with the solid circles, where approximations of eq. (4.29b) forPo (slightly different numerically) and eq. (4.56) for/>B have been adopted.

As noted above the effective recombination flux a£?R«z«e should be approxi-

a

We may approximate n0(pB) by Z(pB)nzne, Instead of eq. (4.8) with eq. (4.7) whichis good only for high temperature, we adopt another approximation, eq. (4.58), forneutral hydrogen at low temperature:

FIG 5.8 Collisional-radiative ionization and recombination rate coefficients forneutral hydrogen (z= 1) against Te with «e as a parameter. (Quoted from Gotoet al., 2002; copyright 2002, with permission from The American PhysicalSociety.)

We then obtain

This is compared in Fig. 5.6 with the result of numerical calculations.

Figure 5.8 shows SCR and «CR against Te with «e as a parameter. This is forneutral hydrogen. It may be interesting to note that both coefficients have almostthe same value at about Te= 1.5 x 104 K, and that this is approximately true forany «e values. For higher temperatures ionization is faster than recombination,and for lower temperature vice versa. This is the second reason why we chose thetemperature of Te/z

2= 1.5 x 104 K as the boundary between the low-temperature


IONIZATION BALANCE 167

case and the high-temperature case. However, we have to be careful when we treations. From the scaling law given in Appendix 5B and already adopted in this andprevious sections, SCR decreases as z increases while «CR increases. This point willbe discussed in more detail in the next section.

This balance is established, in principle, when the plasma is stationary, so thatthere are no temporal changes, and homogeneous, so that spatial transport ofplasma particles does not affect the ionization balance.

Ionization ratio

In the following we call [nz/n(l)] the ionization ratio; it is different from theionization degree [nz/{n(l) + nz}]. In the preceding sections we have examined thecharacteristics of SCR and «CR in detail. The ionization ratio is determined by thesecoefficients. In this section we assume the high-temperature case. This is because,as shown later, in considering ionization balance of ions, the relevant reducedtemperature tends to be higher for increasing z. Figure 5.2 shows an example of theionization ratio, together with these collisional-radiative rate coefficients, for thecase of hydrogen-like ions with nuclear charge z in a plasma of Te/z

2 = 6 x 104 K,the high-temperature case.

5.4 lonization balance

Plasma (in the second sense) in ionization balance is important for two reasons:

1. In certain practical situations, plasmas can actually attain this balance, at leastapproximately.

2. Many plasmas encountered in laboratory and astrophysical observations arefar from this balance, but these plasmas are better understood with reference tothe idealized situation, i.e. ionization balance.

lonization balance is defined by

Region I (ne < n£°) In the limit of low «e, the collisional-radiative rate coeffi-cients are given by eqs. (5.6) and (5.14). The ionization ratio is given by

where the superscript 0 means the low-density limit. This situation has beencustomarily called corona equilibrium. It should be noted that the meaning of thisterm is different from that of the corona phase of the ionizing plasma componentof populations at low density (Section 4.2).

With an increase in «e, both rate coefficients increase, i.e. eqs. (5.8)-(5.10) forSCR and eqs. (5.16)-(5.18) for aCR. The ratio [nz/n(l)] changes accordingly. If we


adopt the approximations (5.10) and (5.18), then we obtain

In this region, both rate coefficients are, in the presentapproximations, constant, as shown in Fig. 5.2. In this case, SCR has increased andreached the high-density-limit value, while «CR is still close to its low-density-limitvalue. Thus, the ionization ratio takes the maximum value, as seen in Fig. 5.2.

Both rate coefficients take the high-density-limit values,and are related by eq. (5.23). The ionization ratio is given by the Saha-Boltzmannequilibrium value,

In Fig. 5.2, the approximation of eq. (5.31) is given with the dash-dotted line.We now examine an important consequence of the scaling law of various

quantities against nuclear charge z. We have seen in Appendices 3A and 5B, that,in order for the rate equations (4.2) to be independent of z, Te should scaleaccording to z2, eq. (3A.9), and ne to z7, eq. (5B.1).* In this sense, the quantitiesTe/z

2 and ne/z7 are called, respectively, the reduced electron temperature and the

reduced electron density. This scaling law means that the horizontal lines drawnfor z = 1 in Fig. 1.2, which represents the parameter range of Fig. 5.2, for example,should be shifted for z > 1 parallel according to the oblique scale line shown in thisfigure. The reader can try this shifting procedure by taking an example of z, sayz = 30. A consequence of this scaling is that the collisional-radiative ionization andrecombination rate coefficients scale according to eq. (5B.5) and eq. (5B.5a),respectively, i.e. according to z~3 and z. The ordinate of Fig. 5.2 follows thisscaling. We now suppose that, by keeping Te/z

2 and ne/z7 constant, we change z.

This procedure is shown in Fig. 1.2, where we follow the oblique scaling line fromz= l to z = 30. As a result of the above scaling, the ionization ratio scalesaccording to z~4. This is the reason why the quantity [nzz

4/n(l)] has been plottedin Fig. 5.2. For the constant reduced temperature of this figure, Te/z

2 = 6 x 104 K,the ionization ratio has a strong z-dependence: for z = 1 (neutral hydrogen) theionization ratio in the low-density limit is about 104, almost completely ionized.

* As seen in Fig. 3.11, the excitation cross-section values near the excitation threshold do not followthe z scalings. Therefore, the scaling in the present context is necessarily approximate, especially forlow temperatures.

Region II

Region III

It is straightforward to see that, for low «e or large po (Fig. 1.10), when the expo-nential factor is expanded, the terms in the square brackets are approximated to\kTe/z

2R + 6.8//>QJ. This indicates that, starting from the low-density limit, withan increase in «e, the second term increases. Therefore, the ionization ratio increases.This reasoning explains the increase in the ionization ratio shown in Fig. 5.2.


For z = 10 (hydrogen-like neon) it is about 1, and for z = 26 (hydrogen-like iron) itis 2 x 10~2, only 2% ionized. This point can also be understood from Fig. 5.8.Te and ne in this figure are to be understood as the reduced temperature anddensity, Te/z

2 and ne/z7, respectively. This figure is, of course, approximate for

ions. With an increase in z starting from 1, SCR shifts downward by z3, while «CRshifts upward by z. Then, for a certain reduced temperature, Te/z

2, the ionizationratio changes drastically, as we have seen already.

The particular temperature at which the ionization ratio is 1 is of importance;this temperature is called the optimum temperature, Teo. The temperature inFig. 5.2 is approximately the optimum temperature for z= 10. This temperaturehas the significance that, for an ionization balance plasma at low density, with achange in Te, intensities of the emission lines from excited levels have theirmaximum values at around this temperature; this point will be discussed later inthis section. Figure 5.9 shows the optimum temperature Teo/z

2 against z at«e/z

7 = 1016 m~3 or in region I (the thick line). The thin line shows an approx-imation based on approximate expressions for S^R and o^R similar to those givenby eqs. (5.7) and (5.15); z2R/kTeo~ 13.5-41nz. Note that the optimum tem-perature at z = 10 is consistent with Fig. 5.2. The slight decrease of reo/z

2 with theincrease of z= 1 to 2 is due to the different ionization cross-section values near thethreshold for the neutral atom and for ions as seen in Fig. 3.11(c).

This point is again understood from Fig. 5.8. With the increase in z, thereduced temperature at which both the coefficients become equal in magnitude, orat which they cross each other, increases.

FIG 5.9 Reduced optimum temperature for low density, at «e/z7=1016 m

(thick line). The thin line shows the approximation z2R/kTeo~ 13.5 — 41nz.(Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permissionfrom The American Physical Society.)


The above features are the important exception of the scaling law which wasalready mentioned in Chapter 1 and Section 4.1.

Excited-level population - high-temperature case

The population of excited levels is expressed by eq. (4.20) as a sum of the ionizingplasma component and the recombining plasma component. See Fig. 1.9. We callthis n(p) the total population. This equation may be rewritten as

FIG 5.10 Total population, ^IB(^), for a plasma in ionization balance against/?(thick lines), and its recombining plasma component, r0(p) (thin lines), with«e/z

7 as a parameter. The power of ten is given. The ground-state population£>(l)iB is given on the left-end ordinate. For hydrogen-like iron (z = 26) withTe/z

2= 5.12 x 105 K. The horizontal dotted line is regarded as the upper boundof the approximate Saha-Boltzmann population, b(p)= 1+0.1. (Quoted fromFujimoto and McWhirter, 1990; copyright 1990, with permission fromThe American Physical Society.)

Figure 5.10 shows b(p) of hydrogen-like iron (z = 26) in ionization balance forTe/z

2 = 5.12 x 105 K for several ne's. Note in Fig. 5.9 that this very high temperatureis rather close to the optimum temperature for z = 26. The ground-state iondensity in ionization balance, 6iB(l), is given by the solid circles on the left-end

For the purpose of simplicity, we define b(p) as the relative population withrespect to its Saha-Boltzmann value. Then eq. (5.32) becomes


ordinate, and bi^(p) of excited levels are shown with the thick lines. The subscriptIB means the ionization balance. The reduced density ne/z

7 is given as a para-meter. The power of ten is attached to each curve. If the reader tries to identify theparameter range of this figure in Fig. 1.2, not the scaled ones but the real ne andTe, he or she will find them extremely high. Also shown with the thin lines are therecombining plasma component, rQ(p). The horizontal dotted line indicatesb(p) = 1 ± 0.1, i.e. the range of approximate Saha-Boltzmann populations. In thisvery high-temperature case, the recombining plasma component of the populationof levels in the CRC phase, or in the case of «e/z

7 = 1012m~3 levels/? lying lower thanpG ~ 20, is close to the Saha-Boltzmann value, or even slightly larger than that.This is slightly different from Fig. 4.14 for Te/z

2= 1.28 x 105 K, where thepopulation was slightly smaller. Note that radiative recombination strictly followsthe z scaling. These near Saha-Boltzmann populations of the high-temperature, low-density recombining plasma component have been already discussed in Section 4.3.The addition of the ionizing plasma component, ri(p)bi^(l), results in total popu-lations substantially larger than the Saha-Boltzmann values as seen in Fig. 5.10.They are about one order larger, as shown below.

We now consider the relative magnitude of the ionizing plasma component andthe recombining plasma component in ionization balance plasma. See Fig. 1.9. Inthe low-density limit the ionization ratio is given by eq. (5.29). The ionizingplasma component is given approximately by eq. (4.23)

The recombining plasma component is given by eq. (4.39), and with the neglect ofthe cascading contribution (see Fig. 4.15(a)) we have

We now compare the magnitudes of these components for large p, say p of theorder of 10. On the assumption of high temperature we finally arrive at

where (9(0.1) means "the order of 0.1". Here we have utilized the approximaterelationships, eq. (4.41) or /?(/?) oc/?~2'5, eq. (4.23a) or C(l,/?)oc/?~3, and 5*(1)~C(l, 2) (see the cross-sections in Fig. 3.1 l(c)). This last relationship comes from theoscillator strengths, /i,2—/i,c (Table 3.1(b)). See also Fig. 4.7. We have alreadyseen an example of this conclusion in Fig. 5.10.

Thus, we have come to an important conclusion: In the low-density limit,

1. the recombining plasma component is close to the Saha-Boltzmann equilibriumvalue, and

2. the ionizing plasma component is larger than that by about one order ofmagnitude.


These properties are independent of Te so long as the temperature is high. Inparticular, the second property is salient. In the case of neutral hydrogen, forexample, at Te = 8 x 103 K, which is even outside the range of high temperature,a numerical calculation shows that the ionization ratio nz/n(l) is 3 x 10~5 and"oGOMGO is 0.1-0.7 for^ > 5. At Te= 1 x 106 K, the ionization ratio is 2 x 107,more than 10 orders of magnitude higher, yet nQ(p)/ni(p) is 0.2-1. A similarargument can be done for different ions. Suppose we fix the reduced temperaturere/z

2 at a certain value and change z. As we have seen in the previous subsection,for Te/z

2 = 6 x 104 K the ionization ratio [nz/n(l)] changes by almost six orders ofmagnitude for the change of z from 1 to 26. Still, eq. (5.35) is valid for all the ions.Figure 5.10 is also an example of this statement.

It is sometimes assumed that the excited level population in a plasma is givenby eq. (5.33), with the neglect of the contribution from the recombining plasmacomponent. The above arguments show that, so long as the plasma is in ionizationbalance, this assumption is approximately correct.

With an increase in ne, it is seen in Fig. 5.10 that the recombiningplasma component (ro(/>)) tends to the Saha-Boltzmann value; this is the problemwhich we have examined in Section 4.3. The total population, eq. (5.32a) oreq. (4.20), tends to its Saha-Boltzmann value, too, starting from high-lying levels.This feature is related to the problem of the establishment of LTE, which isimportant in practical plasma spectroscopy. This problem will be discussed inAppendix 5C.

* Excited-level population - low-temperature case

Figure 5.11 shows the population distribution of neutral hydrogen for Te =4 x 103 K. The reader may identify the parameter range of this figure in Fig. 1.2.This temperature is, from the practical viewpoint, unrealistically low for a plasmain ionization balance. In this figure, however, an important feature is manifestedwhich is characteristic of the low-temperature case.

We note that, at low density, the total population, which is the sum of theionizing plasma component and the recombining plasma component, of high-lying levels, e.g. for/? ~ 10-20 at «e = 1012m~3 at which/?G~ 20, is very close to theSaha-Boltzmann value. There appears to be no obvious basis for this to happen.We now examine this point.

At these low temperatures, the rate coefficients for excitation and ionizationfrom the ground state are determined by the cross-section values near the thresholdfor each process. We have established the interrelationship between these cross-sections values for excitation to very high-lying levels and for ionization, eqs. (3.36)-(3.38). As its consequence, we have an approximate relationship between the ratecoefficients,


FIG 5.11 Similar to Fig. 5.10 but for neutral hydrogen with Te = 4x 103K.Other explanations are similar to those for Fig. 5.10. The thick dotted curveshows the populations in slightly recombining plasma for ne= 1016m~3 withb(l) = 9 x 106 instead of bi#(l) = 1-65 x 107. This condition happens to give6(3)= 1. (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, withpermission from The American Physical Society.)

for large p. The ionizing plasma component of the population of level p in thecorona phase is given by eq. (5.33). By using eqs. (5.36) and (5.29) we obtain

We now consider £/3(g). For lower-lying levels q we adopt the approximation(4.50) for (3(q), Then we have

where q\im means the upper limit of q below which the approximation (4.50) isvalid. This limit may be 5, 10, or 20, depending on Te. See the example of the/^-dependences of (3(p) in Fig. 3.5(b), where q\im is around 10. See also Fig. 3.9. Forlevels lying above q\im another approximation, eq. (4.41), is valid so that (3(q)<xq~2'5, and the summation converges rapidly. Therefore, S is of the order of 3.

We are concerned with a higher-lying level p, which may be/>~ 10-20, wherethe radiative decay rate is approximated by eq. (4.12):

we conclude that ni(pf of eq. (5.37) is about (2/3)Z(p)nzne, or n(p)bIB(l')~2/3.In the same approximations as the above it may be shown that the recombiningplasma component is not far from one-third of the Saha-Boltzmann value, i.e.n0(p)° ~(l/3)Z(p)nztie. This is the reason why the total population at low temper-ature and low density is close to the Saha-Boltzmann value.


with

Here, \np = ft q ! dq is of the order of 3. By noting that the Saha-Boltzmanncoefficient is written as

Excited-level population - high-density case

In the limit of high density, or in region III defined in Section 5.2, the followinginterrelationship holds:

The ionization ratio is therefore given by the Saha-Boltzmann relationship

We remember

In this limit, the factor n(l)/[Z(l)nzne] is unity. From another interrelationship,

for all the excited levels. This equation, together with eq. (5.42), indicates that, inthis high-density limit, we have the Saha-Boltzmann equilibrium (LTE) popula-tions with respect to the ion density for all the levels including the ground state.This conclusion is independent of Te. This situation is nothing but the completeLTE as defined in Section 2.1.

We now examine the above discussion in more detail. Figure 4.3 gives the high-density-limit values of rQ(p) and r\(p) for severalp's against temperature. We haveseen above that the r.h.s. of eq. (5.32) reduces to rQ(p) + ri(p), leading to the LTEpopulations for all p's. Therefore, Fig. 4.3, for a particular temperature, directlygives the ratio of the recombining plasma component and the ionizing plasmacomponent of the total population for each level. As we have noted toward theend of Section 4.4 (p. 130-131), Byron's boundary/IB determines which of ro(p) orr\(p) is dominant in the total population; for levels lying above /?B therecombining plasma component is dominant, and for levels below, the ionizingplasma component is dominant. Figure 1.9 is a schematic illustration of thissituation. Of course, all the total populations are LTE populations.

Figure 1.7 is the spectrum of a hydrogen plasma virtually in the high-densitylimit (ne ~ 2 x 1023 m~3, higher than K^°+), and the emission line intensities as givenfrom the LTE populations are shown with the dotted lines, where line broadening,which will be discussed in Chapter 7, and the effect of radiation reabsorption,which will be discussed in Chapter 8, are taken into account. As Fig. 4.3 indicates,for Tc=\ eV, except for level p = 2, which has a contribution from theionizing plasma component of about 10%, the population of other levels virtuallyconsists only of the recombining plasma component. Note in passing that the LTEpopulations for p<6 in Fig. 4.11 of the very low-temperature (Te= 103 K)ionizing plasma component is understood from Fig. 4.3.

Figure 5.12 shows an example of the population distribution in lower density«e = 1020 m~3, intermediate between regions I and II, i.e. pG ~ 2. Two salientfeatures are seen:1. Even for this low density, the total population is rather close to or slightly larger

than the LTE population for all the excited levels including p = 2. The ground-state atom density n(l) is larger than the Saha-Boltzmann value by more thanthree orders of magnitude.

2. The relative contribution from the recombining plasma component and theionizing component is quite different from that in the high-density limit. Forp = 2, for example, in the high-density limit the relative contribution was 86% to14% (see Fig. 4.3), while at the density of Fig. 5.12 it is 1% to 99%. This meansthat, with the decrease in «e, the first term in eq. (5.32) decreased by two ordersof magnitude, then the second term increased to compensate this decrease.

This interesting but puzzling feature belongs the problem of establishment of LTEpopulations. We will discuss this problem in Appendix 5C.


we have


We note here an important characteristic of excited level populationsin ionization balance plasma. We remember our discussions above and look atFigs. 5.10-5.12. For high temperature, in the low-density limit, where eq. (5.35) isvalid the total populations are larger than the Saha-Boltzmann values by anorder, or bi^(p) ~ O(10) in terms of eq. (5.32a). For low temperatures, the totalpopulations tend to be close to the Saha-Boltzmann values even in low densities.See Fig. 5.11. An important conclusion is that, in ionization balance plasma excitedlevel populations tend to be larger than the Saha-Boltzmann values in low densities.The only exceptions are that, for extremely low temperature and density, veryhigh-lying levels can have slightly smaller populations than the Saha-Boltzmannvalues. See Fig. 5.11 with «e=1012 m~3. With an increase in «e all the totalpopulations come exactly to the Saha-Boltzmann values, or bi^(p) = 1. Thisproblem will be further discussed in Appendix 5C.

Ionizing plasma and recombining plasma

So far in this chapter, we have assumed that ionization balance is established bet-ween the "atoms" n(l) and the "ions" nz. In many practical situations this assumption

FIG 5.12 Boltzmann plot of the total populations of neutral hydrogen in anionization balance plasma. Te= 1 x 104 K, «(1) = 3.2 x 1022 m~3 and n& = nz =1 x 1020 m~3. The contribution from the ionizing plasma component is dominantfor/? = 2 and 3, while the recombining plasma component is dominant forp > 3.


is not correct; rather the ionization ratio is far from the ionization balance. If theplasma is time dependent, or if the plasma is spatially inhomogeneous so thattransport of the plasma particles affects the ionization-recombination, then ioni-zation balance is not established.

It should be noted that, in the above discussions of the excited level popula-tions in a plasma in ionization balance, both the contributions from the ionizingplasma component and the recombining plasma component could be substantial,or even comparable in magnitude; i.e. at low density and high temperature theformer contribution is larger than the latter by about an order, and at high densityand rather low temperature both the components are necessary for the levels tohave the LTE populations, eq. (5.43). Now, we come to an important conclusion:If the plasma is far from ionization balance, either of the components tends topredominate over the other, and we may neglect the other contribution entirely in thetotal population. We introduce here the concepts of an ionizing plasma and arecombining plasma: an ionizing plasma is defined by [nz/n(l)] <C [KZ/K(!)]IB, wherethe subscript IB means ionization balance. In this case, «(1) is highly "over-populated", so that the ionizing plasma component is likely to be dominant in thetotal population for many excited levels. On the contrary, a recombining plasma isdefined by nz/n(l) > [«z/«(l)]iB- In this case, nz is highly "overpopulated", so thatthe recombining plasma component will be much larger than the ionizing plasmacomponent, which could be neglected entirely.

Consider typical examples of actual plasmas. The first is a hydrogen (z= 1)positive-column plasma in a low-pressure discharge with Te = 5x 104K, nz =«e= 1017 m~3 and «(!)= 1022 m~3. See Fig. 1.2. This is the condition which cor-responds approximately to a helium-neon gas laser discharge. This plasma hasnz/n(l)= 10~5, which is to be compared with [nz/n(l)]i# = 103. This overpopula-tion of «(1) by eight orders of magnitude is the result of the transport or diffusionof ions to the discharge cell wall, owing to the spatial inhomogeneity of the plasma.In order to take this effect into account the terms describing the spatial transportof the ions and atoms should be added to the r.h.s. of eq. (5.2). The recombinationterm aCRnzne is, in this case, smaller than the ion diffusion loss term by eightorders of magnitude. The ionization term balances with this diffusion loss term, inthis case. Thus, this is a typical example of an ionizing plasma. Remember that theexample cited in Section 4.2 (Fig. 4.12) was a positive-column plasma. In fact, Te

of that plasma was 5 x 104 K and «e was 1017-1020 m~3. Similar situations arerealized for hydrogen and impurity ions in a magnetically confined plasma, and ina plasma heated by a shock wave during the process of ionization relaxation.

Figure 5.13 shows an example of an ionizing plasma, where [nz/n(l)] is smallerthan [«2/K(l)]iB by two orders of magnitude. For the lower-lying levels than/? = 7,the ionizing plasma component is predominant, while for the higher-lying levelsthis is not the case. The reason why the recombining plasma component becomesdominant in this ionizing plasma is that, with an increase in/?, the ionizing plasmacomponent decreases very rapidly, r^p) <xp~6, while the recombining plasma com-ponent, r0(p), stays almost constant (~ 1). It should be noted that the total


population for low-lying levels is larger than the Saha-Boltzmann values. This situa-tion of b(p) > 1 for low-lying levels is the common feature for ionizing plasmas.

An example of a recombining plasma is an afterglow plasma of neutralhydrogen, Te = 2 x 103 K, and nzn&= 1018 m~3. See Fig. 1.2. This condition gives[«Z/«(!)]IB ~ 1CT27, or K(!)IB 1045 m~3, which is too large to be realistic. Since n(l)is "underpopulated" by more than 20 orders of magnitude, the ionizing plasmacomponent is almost completely predominated over by the recombining plasmacomponent. The examples cited in Section 4.4 (Fig. 4.25) were the flowing after-glow plasmas. There are many kinds of plasmas which fall in this class of recom-bining plasma. For instance, the plasma produced by illumination of intense laserlight on a solid target sometimes shows this characteristic. A plasma surroundinga star which emits strong ultraviolet radiation, and being photoionized by it isanother example. Highly ionized impurity ions diffusing from the central plasmato the outer region of a magnetically confined plasma are also an example of therecombining plasma. Divertor plasmas sometimes fall into this class.

FIG 5.13 Similar to Fig. 5.12 except that the plasma is slightly ionizing, nz/n(\) ~KT2[«Z/«(1)]IB. re = 4 x 104 K, n(l)= 1 x 1019 m~3and ne = nz=lx 1020 m~3.Even in this ionizing plasma, the populations of the high-lying levels aredominated by the recombining plasma components.


lonization flux, recombination flux, and emission-line intensity

In Chapter 4 we examined various features of the populations of excited levels, andin this chapter we have studied the features of ionization and recombination of aplasma in the second sense. Both the ionizing plasma component, eq. (4.22), andthe ionization flux, eq. (5.2), are proportional to the ground-state atom density,n(l). Both the recombining plasma component, eq. (4.38), and the recombinationflux, eq. (5.2), are proportional to the ion density, n2. See also Fig. 1.9. Theseobservations suggest that from the measurement of an excited level population, or ofan emission line intensity, we can infer the magnitude of the flux of ionization or thatof recombination, or even both.

Figure 5.14 shows an example of the relationships between these quantities.Suppose we have neutral hydrogen atoms of «(1) = 1 m~ in a stationary plasma of«e= 1018 m~3 with varying temperatures. Shown are the ionization flux and thecorresponding emitted photon numbers of the Balmera a(p = 2-3) transition perunit time and volume. It is seen that these two quantities have a similar depen-dence on temperature. Figure 5.15 shows the proportionality factor[5>cR«(l)«e/»i(3)^4(3,2)], or the number of ionization events per Balmer a photonemission. Figure 5.14 also shows similar quantities for a recombining plasma: for

FIG 5.14 The ionization flux and the Balmer a line intensity, or the number ofphotons emitted by excited atoms, originating from the ionizing plasma compo-nent; «(1) = 1 m~3 and «e = 1018 m~3. The recombination flux and the line inten-sity from the recombining plasma component; nz=\ m~3 and «e= 1018 m~3.For the situation where n(l) = nz=l m~3 is assumed, the sum of the lineintensities is given with the solid curve. (Quoted from Goto et al., 2002;copyright 2002, with permission from The American Physical Society.)


protons of KZ = 1 m 3 in a plasma of ne = 1018 m 3 the recombination flux and thecorresponding emitted photon numbers are shown. Again both quantities have asimilar temperature dependence. Figure 5.15 shows the proportionality factor[aCRnzne/n0(3i)A(3,2)], or the number of recombination events per Balmer aphoton emission. It is interesting to note that, for recombination, the efficiency ofproducing photons per recombination event has weak dependences on Te and ne inthe parameter range of this figure.

We now return to a plasma in ionization balance in the present context.Figure 5.16 shows an example. The total number density of atoms and ions isassumed constant, i.e. The atom density

FIG 5.15 (a) Proportionality factor for the number of ionization events perBalmer a photon emitted, (b) Similar quantity for the recombining plasma.(Quoted from Goto et al., 2002; copyright 2002, with permission from TheAmerican Physical Society.)


FIG 5.16 A plasma in ionization balance, where n(l) + nz= 1 m 3 is assumed.The densities, n(l) and nz, are shown. Also shown are the magnitude of the fluxof ionization-recombination and the number of the emitted Balmer aphotons with its breakdown into the rates coming from the ionizing plasmacomponent and the recombining plasma component. (Quoted from Goto et al.,2002; copyright 2002, with permission from The American Physical Society.)

and the ion density against a change in temperature, as calculated from SCR andaCR (Fig. 5.8), are given by the thin solid lines. For low temperatures the atomsare dominant and for high temperatures the ions are dominant. In this figure, alsoshown are the flux of ionization, ScR_n(l)ne, or that of recombination, aCRnzne.Since the ionization balance is defined by eq. (5.28) or ScRn(l)ne = acRnzne, bothfluxes are equal. It is to be noted that the magnitude of the ionization-recombination fluxes have the maximum at about the optimum temperature atwhich «(1) = nz holds. The reason may not be obvious, but it can be understoodfrom physical considerations; For low temperatures, the atom density is high, butthe CR ionization rate coefficient is small. See Fig. 5.8. For high temperatures, theCR ionization rate coefficient is large, but the atom density is now low. A similarargument can be made concerning the recombination flux.

At the beginning of this section we showed that, for a plasma in ionizationbalance, the relative magnitude of the ionizing plasma component is larger thanthe recombining plasma component by about an order of magnitude, eq. (5.35).The Balmer a line intensity as given by eq. (4.1), or the photon number, comingfrom the ionizing plasma component [Ki(3)J(3,2)] is given in this figure by theupper dashed line and the line intensity from the recombining plasma component[n0(3)A(3,2)] is given by the lower dashed line. The above statement is seen tohold, especially at high temperatures.


The total intensity, i.e. the intensity we observe from this plasma, is given bythe dash-dotted line. It is noted that the total line intensity moves approximatelyin parallel with the magnitude of the flux of ionization or recombination. Thus,the line intensity takes its maximum near the optimum temperature Teo. If we lookat this figure more closely, we recognize that the actual temperature of the max-imum emission intensity is slightly higher than the optimum temperature. Theionization-recombination flux takes the maximum at a still higher temperature.This problem is examined in Appendix 5D.

We now consider a plasma out of ionization balance, or an ionizing plasma ora recombining plasma. In Fig. 5.14 suppose we fix «(1) = nz=\ m~3, and the tem-perature changes. The total intensity, eq. (4.20) or Fig. 1.9, is the sum of theintensities from the ionizing plasma component (the dashed curve) and from therecombining plasma component (the dashed curve), to yield the total intensitygiven by the thin solid line. The important point to note here is that the totalintensity takes the minimum at about the optimum temperature. The situation atthis particular temperature is rather close to the situation of the ionization balanceplasma, Fig. 5.16; as mentioned above, near the optimum temperature, the intensitytook the maximum. Note the consistency of the intensities of these minimum andmaximum in these figures, respectively. In Fig. 5.14, for temperatures higher thanthe optimum temperature, the line intensity becomes high, indicating that theionization flux is large because this plasma is an ionizing plasma. For tempera-tures lower than the optimum temperature, the line intensity is again high indi-cating that, this time, the recombination flux is large for this recombining plasma.

We have now come to another important conclusion. When a plasma emitsline radiation this is an indication that this plasma is undergoing ionization orrecombination, or even both of them simultaneously in the case of ionizationbalance. A plasma out of ionization balance tends to emit intense radiation, indicatingthat this plasma is undergoing strong ionization or strong recombination. The lineintensity is thus a measure of the magnitude of this ionization flux or the recom-bination flux. In other words, a plasma in ionization balance emits the weakestradiation among the plasmas in various ionization-recombination states.

5.5 Experimental illustration of transition from ionizingplasma to recombining plasma

As has been noted in the preceding section, a plasma cannot attain ionizationbalance if it is spatially inhomogeneous or time dependent. The example ofexperimental observations of the ionizing plasma shown in Section 4.2 and that ofthe recombining plasma in Section 4.4 were both stationary. So the origin of thedeviation from the ionization balance was that these plasmas are inhomogeneousand the spatial transport of the plasma particles was important in determining theionization-recombination state of the plasma.

In this section we take an example of a time-dependent plasma, a pulsed gasdischarge. When a pulsed current (Fig. 5.17(a)) is drawn through a gas, helium in

EXPERIMENTAL ILLUSTRATION OF TRANSITION 183

FIG 5.17 A discharge plasma is produced from a helium gas of 2torr filled in adischarge tube of 5 mm inner diameter, (a) Discharge current, (b) Emission lineintensity of the neutral helium line Hel A587.6 nm (23P — 33D) measured fromthe side of the tube. See the dotted line in Fig. 1.4. The first peak is called peak Aand the second peak B. (Quoted from Hirabayashi et al., 1988; copyright 1988,with permission from The American Physical Society.)

this case, a plasma is produced, and the emission line intensity shows two peaksduring the course of time as is seen in Fig. 5.17(b). This is a neutral helium line HelA587.6 nm(23P - 33D). See the energy-level diagram of Fig. 1.4. We now call theformer and latter peaks A and B, respectively. Peak A corresponds to the timeduring the current rise. Peak B appears just after the finish of the discharge currentand the intensity decays rather slowly. From the discussions at the close of theprevious section, we may suppose that peak A indicates that the "gas" is under-going ionization during this period and its intensity is proportional to the ioniza-tion flux, and that peak B and the subsequent emission indicate the recombinationflux. We observe the emission line spectra from these plasmas, as shown in Fig. 5.18(a) and (b) for peak A and peak B, respectively, and deduce the excited levelpopulations according to eq. (4.1). We plot the population distribution in a graphsimilar to Figs. 4.5 and 4.20(a). Figure 5.19(a) is the result. Peak A clearly shows theminus sixth power law, and the distribution at peak B looks similar to Fig. 4.20(a)at high densities. Figure 5.19(b) is another plot, the Boltzmann plot like Fig. 4.20(b),of the peak B populations.

The minus sixth power population distribution is characteristic of the high-density ionizing plasma for levels lying higher than Griem's boundary (Figs. 4.5and 1.10(a)), or it also occurs to levels lying lower than Byron's boundary in the


FIG 5.18 (a) Observed spectrum at peak A. (b) at peak B. The series lines of23P — «3D and, in (b), the recombination continuum are seen. See also Chapter 6.(Quoted from Hirabayashi et al,, 1988; copyright 1988, with permission fromThe American Physical Society.)

high-density, low-temperature recombining plasma (Figs. 4.20(a) and 1.10(b)).From the fact that (1) the minus sixth power distribution in Fig. 5.19(a) seems toextend to very high-lying levels, and (2) peak A appears when the plasma is in thebuild-up process from a gas during the current rise, we may conclude that theplasma at peak A comes from an ionizing plasma and the emission intensityindicates the magnitude of the ionization flux.

From the facts that peak B appears in the period of plasma recombination andthat the population distribution matches the low-temperature recombiningplasma, we may conclude that the plasma at peak B is a recombining plasma andthe emission intensity indicates the magnitude of the recombination flux.


FIG 5.19 Excited-level population distribution for «3D levels for peak A andpeak B (O). See Fig. 1.4. The calculated result is shown with +. (a) Plot similarto Figs. 4.5 and 4.20(a). (b) The Boltzmann plot of the population distributionfor peak B. Note that the experimental populations extends to the continuumstates. See Chapter 6. (Quoted from Hirabayashi et al., 1988; copyright 1988,with permission from The American Physical Society.)


We now construct a collisional-radiative model for helium. The ground-stateatom density is given from the filling pressure of the gas, i.e. n(l :S) = 6 x 1022 m~3.An excited-level population is given by eq. (4.20), and for peak A we take only thesecond term, the ionizing plasma component, and for B only the first term, therecombining plasma component. By fitting the calculated populations to experi-ment (see Fig. 5.19), we obtain Te and ne for these plasmas. The peak A plasma hasTe of 4-5 x 104 K and ne ~ 1020 m~3 and the peak B plasma has Te = 5.1 x 103 Kand ne= 1.25 x 1021 m~3. It is noted that the ion density is virtually equal to theelectron density under the present condition. The reader may locate these plasmason Fig. 1.2. We are concerned here with neutral helium, z = 1.

We can thus explain the temporal development of the excited-level populationin Fig. 5.17(b) as follows. During the current rise, Te is high so that energeticelectrons ionize the gas; this means that the situation similar to Figs. 4.9 and 4.12is actually realized. Therefore, the ionizing plasma component dominates over therecombining plasma component, and the population is high, indicating a largeflux of ionization. Figure 5.20(a) shows the calculated total population distributionin this plasma including both components. The above conclusion is clearly seen.After the current takes the maximum the population begins to decrease. Until thistime a plasma with a sufficient ionization ratio has been formed. The total numberof ionization events during this initial stage, or the resulting ion density of this plasmaat its peak, may be inferred from the area under the peak A. During the currentdecay, there is no need for the plasma electrons to ionize the gas further so that Te

decreases to about 2 x 104 K, close to the optimum temperature. An ionization stateclose to the ionization balance is established, and the intensity shows a minimum,being consistent with Fig. 5.17(b). When the current ceases, there is no mechanismto sustain high Te, and it drops very rapidly. With this decrease in Te the recom-bining plasma component increases, and the ionizing plasma component becomessmall by many orders of magnitude, as is seen in Fig. 5.20(b). Thus, the intensity ishigh, again suggesting a high recombination flux. From the area under peak B wemay infer the total number of recombination events during this afterglow. If weadopt the proportionality factor for the hydrogen Balmer a line, Fig. 5.15, for ourcase of He (23P — 33D), the number of ionization events during peak A is almostequal to the number of recombination events in peak B. We could even deduce thetime history of ne. Unfortunately, however, the intensity is measured only rela-tively (Fig. 5.17(b)), so that we cannot determine the absolute value of ne.

It is noted that the situation realized in this experiment is similar to thesituation of Fig. 5.14, where both the ground-state atoms and ions are presentsimultaneously and temperature changes over a wide range. In the present example,we start with a high temperature during the current rise leading to the strongemission intensity; during the current decay the temperature decreases giving theminimum of the intensity; and the further temperature decay after the finish of thecurrent gives rise to the strong emission again. The reader will understand that thisfigure is quite universal, i.e. the condition of n(l) = nz= 1 m~3 is not a strongconstraint in understanding real situations.


FIG 5.20 Calculated distribution of the total population, eq. (4.20), for «3D levelstogether with the ionizing plasma component and the recombining plasmacomponent resolved, (a) For peak A; (b) for peak B. (Calculation by M. Goto.)


Appendix 5A. Establishment of the coUisional-radiative rate coefficients

In Appendix 4B we examined the transient characteristics of the excited-levelpopulations approaching the final stationary state. We worked out the conditionsunder which the QSS approximation is valid, or we can use the populationcoefficients, r0(p) and r^p). We likewise have to justify eqs. (5.1)-(5.5) for thedescription of ionization and recombination of our plasma in the second sense.

We take the example of the ionizing plasma treated in Appendix 4B. Thetemporal development of excited-level populations has been shown in Fig. 4B.1.As has been noted, at early times the level populations are simply accumulating.This means that the ground state has a depopulating flux, while it has no returningpopulating flux. Figure 5A. 1 shows the temporal development of the net depletionrate coefficient from the ground state. It starts with the sum of the excitation ratecoefficients to all the levels and the ionization rate coefficient. This corresponds tothe first two terms of the r.h.s. of eq. (5.3). With the course of time the excited levelpopulations develop, and they finally reach the stationary state. The returning fluxto the ground state gradually increases, so that the net depletion flux decreases.

FIG 5A. 1 Temporal development of the effective rate coefficients for depletion ofthe ground-state population and that for production of ions, corresponding toFig. 4B.1. Both the rate coefficients tend to SCR after all the excited levels cometo QSS. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, withpermission from The American Physical Society.)

ESTABLISHMENT OF CR RATE COEFFICIENTS 189

This process corresponds to the development of the third negative term in ther.h.s. of eq. (5.3). At about t= l(T7s, or at tres, this transient state is over, and thenet depletion rate coefficient tends to the collisional-radiative (CR) ionization ratecoefficient, SCR.

This figure also includes the net production rate coefficient of ions. It startswith the direct ionization rate coefficient. With the accumulation of the excited-level populations it also tends to the CR ionization rate coefficient.

We remember that the time constant of depletion of the ground-state atoms is1CT4 s in this example. So, the CR ionization rate coefficient is established muchfaster.

We may draw two conclusions:

1. The CR ionization rate coefficient lies somewhere between the ionization ratecoefficient and the sum of the rate coefficients for excitation and ionization, allfrom the ground state. The actual value of the rate coefficient depends on ne andTe of the plasma.

2. As in the case of the excited-level populations the response time ties gives thetime for the validity of using SCR to describe the effective ionization rate of theplasma.

FIG 5A.2 Temporal development of the effective rate coefficients for depletionof ions and for production of the ground-state population, corresponding toFig. 4B.4. Both the rate coefficients tend to aCR after all the excited levels cometo QSS. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, withpermission from The American Physical Society.)


Figure 5A.2 shows a similar plot for the recombining plasma treated inAppendix 4B. Corresponding population developments have been given inFig. 4B.4. In the present figure, the effective depletion rate coefficient of the ionsand the effective production rate coefficient of the ground-state atoms are shown.If we include recombination into very high-lying levels, at the start, the formerquantity diverges, because, as is seen in Fig. 4B.7, the three-body recombinationrate coefficient becomes very large for large p. The production rate coefficient ofthe ground-state atoms is the direct radiative recombination rate coefficient (seeFig. 4B.7). This corresponds to the first two terms in eq. (5.4). Note that, for theground state, under our present condition the tree-body recombination rate ismuch smaller than the radiative recombination rate. With the course of time boththe net rate coefficients tend to the collisional-radiative (CR) recombination ratecoefficient. At about t = tres the CR recombination rate coefficient is established.Thus, in this case again, the response time for excited-level populations gives thevalidity of using aCR to express the effective recombination rate of the plasma.

We again remember that the depletion time constant of the protons is 10~2 s inthis example, much longer than the time for establishing the CR recombinationrate coefficient.

Appendix SB. Scaling law

In Appendix 3A we have seen the scaling properties of atomic parameters ofhydrogen-like ions against the nuclear charge z. The speed and energy of plasmaelectrons also scaled by the scaling of the atomic parameters. Therefore, Te scaledaccording to z2: Te/z

2 is the reduced temperature. In Chapter 4, we introduced therate equation and the collisional-radiative model, and investigated the populationsof excited levels. In the present chapter we examined the processes of ionization-recombination and the ionization balance of a plasma. In this appendix, weinvestigate the scaling law which enables us to scale various quantities whichappear in the CR model for hydrogen-like ions with respect to those for neutralhydrogen. We also point out the limitation of the scaling laws.

We want the rate equation, eq. (4.2), to become independent of z. We adopt thescaling laws of the rate coefficients as introduced in Appendix 3A into thisequation. Then it is obvious that the electron density should scale as

Then, the time scales as

Unfortunately, this scaling is inconsistent with the scaling for the bound electron,eq. (1.4), i.e. z~2 scaling. It is readily seen that, if we adopt the above scalinglaws, eq. (4.2) becomes independent of z except for the last line representing

LOCAL THERMODYNAMIC EQUILIBRIUM 191

recombination. If we further require that the recombination follow the scaling, theion density should scale according to

If we adopt this scaling the conservation of particles is violated in the recombi-nation processes. The scalings of the collisional-radiative coefficients are obvious:

We consider the ionization balance, eq. (5.2):

or eq. (5.28). This leads to the same scaling as eq. (5B.3). This is the reason why weused the quantity [z4nz/n(l)] in the discussion of ionization balance. See eq. (5.30)and Fig. 5.2, for example.

We now turn to recombination of the plasma, eq. (5.2) with eq. (5.5):

In this case the conservation of particles, nz, should be valid. Then, the time forrecombination scales according to

being inconsistent with eq. (5B.2). These inconsistencies concerning recombina-tion lead to an interesting exception of the scaling law for Te as discussed inSection 5.4 and shown in Fig. 5.9.

We again note another limitation of our scaling law; as Fig. 3.11 shows, thecross-section values for excitation and ionization near the threshold do not followa simple scaling law, so that our results, eqs. (5B.4), (5B.4a), (5B.5), and (5B.5a),become less valid for low temperatures.

*Appendix 5C. Conditions for establishing local therniodynaniic equilibrium

In Chapter 2, we introduced local therniodynaniic equilibrium, which is abbre-viated to LTE. As we have seen in Chapter 4 the populations in high-lying


Rydberg levels are strongly coupled with the continuum state electrons, and theytend to be in thermodynamic equilibrium with the continuum electrons moreeasily. Partial LTE is the state that these levels are actually in thermodynamicequilibrium, or that these levels have Saha-Boltzmann populations. Low-lyinglevels, however, may depart from this equilibrium relation. There is thus thelower-bound level above which the higher-lying levels are in LTE. In the case thatthis lower-bound level comes down to the ground state, this situation is calledcomplete LTE. We have considered this problem already in the text. In thisappendix, we treat this problem further, with the aim of providing numericalexpressions for conditions for establishing partial and complete LTEs.

Partial LTE

We start with the expression for the excited-level populations, eq. (5.32a),

where b(p) is the population normalized by its Saha-Boltzmann value. Seeeq. (5.32). We now define LTE for level p to be

Our problem is to find the lower boundary level which satisfies this definition in aparticular plasma in the first sense, and to a certain degree, in the second sense. Wehave calculated rQ(p) and r\(p) for neutral hydrogen in Chapter 4, as given inTable 4.1, and examined the properties of the excited-level populations for a widerange of «e and Te. Similar calculations have been done for hydrogen-like ions,z = 2 or ionized helium, and z = 26 or 25 times ionized iron. We use the reducedelectron temperature and the reduced electron density

respectively, in this appendix.We first consider recombining plasma. The definition of a recombining plasma

in Section 5.4 is expressed as 6(l)<6iB(l) in eq. (5C.1). Here, IB stands forionization balance. We look at Figs. 5.10-5.12: these figures show examples of theexcited-level populations as given by r0(p) and by eq. (5C. 1) with b(l) = 6iB(l). Wecalled the former situation a purely recombining plasma, and the latter was anionization balance plasma. The populations of our recombining plasma lie some-where between these two limiting distributions, depending on the instantaneousvalue of 6(1). The former plasma is nothing but the recombining plasma com-ponent which we examined in detail in Section 4.3 and Section 4.4. The latter hasbeen treated in Section 5.4, and will further be considered later on. Figure 1.10(b)for a purely recombining plasma suggests that the region of TQ(P) ~ 1 is bound byGriem's boundary and Byron's boundary: the former is concerned mainly with ne


and the latter with Te. Figures 5.11 and 5.12 also give the idea of how r0(p) tendsasymptotically to 1. Figure 5C. 1 shows the lower boundary level />R with TJ as aparameter as determined from the numerical calculations ofr0(p) for z =1,2, and26. The "dip" near O«2-3 x 105 K is a rather exceptional situation for hightemperature; r0(p) — 1 holdsevenfor very low«eas discussed in Section 4.3. Since thissituation violates the spirit of LTE, we ignore this dip. In fact, for very low density,even our assumption of the statistical distribution among the different / levels isno longer valid (see Fig. 4A.2), so that this special situation should be taken withsome care. The region in the left-bottom corner, i.e. />R cannot become small forlow temperature, comes from Byron's boundary. In this figure are plotted thenumerically fitted expressions for the conditions of LTE; the thin solid curve is

and the thin dotted line is

FIG 5C.1 The principal quantum number of the lower boundary level />R forestablishment of partial LTE in recombining plasma. The thick curves are theresult of numerical calculation. z= l ; z = 2; z = 26. Thethin lines are numerical formulas. eq. (5C.3); eq. (5C.4). (Quotedfrom Fujimoto and McWhirter, 1990; copyright 1990, with permission fromThe American Physical Society.)


FIG 5C.2 The principal quantum number of the boundary level />R in the 77 — Oplane for recombining plasma. The region to the right of a curve is the region ofdensity and temperature in which the higher-lying levels than the boundarylevel are in LTE. This figure is constructed from the three cases (z=l, 2,and 26) in Fig. 5C.1, and is therefore approximate. (Quoted from Fujimotoand McWhirter, 1990; copyright 1990, with permission from The AmericanPhysical Society.)

Both conditions should be met simultaneously. It is noted that eq. (5C.3) isessentially the same as eq. (4.29) or even eq. (4.29b), and that eq. (5C.4) is the sameas eq. (4.56). The small differences, e.g. instead of the 3We in the denominator ofeq. (4.56), eq. (5C.4) adopts 2kT&, are partly due to the difference in the definitionsof LTE; it is more stringent in this appendix, eq. (5C.2), than eq. (4.55) which givesByron's boundary level. Figure 5C.2 is another plot of Fig. 5C.1: the region oftemperature and density is shown in which levels p >/>R are in LTE with/>R as aparameter.

In order for a level to be in LTE, the radiative transition processes concerningthis level should be predominated over by the competing collisional transitions andcan be neglected. Equation (5C.3), or eq. (4.25) with the equality sign replaced byan inequality sign, expresses this condition. If this condition is met the problembecomes the relationships among the collisional transition processes. We have seenthat among the collisional transitions from a level, the dominant ones are excitationor deexcitation to the adjacent level, eq. (4.6). Figure 5C.3 is a schematic diagram


FIG 5C.3 Schematic diagram of the four possible cases of the dominant populat-ing and depopulating processes of level p in which we are interested, underhigh-density conditions where the radiative transitions are neglected. (Quotedfrom Fujimoto and McWhirter, 1990; copyright 1990, with permission fromThe American Physical Society.)

for the dominant collisional populating and depopulating processes concerninglevel p which is under consideration. This figure depicts the four possible schemeswhich can be realized in dense plasmas. For the recombining plasma, in which thepopulation flux originates from the continuum state electrons, scheme (b) or (d) ispossible. In order for the level p to be in LTE, scheme (d), eq. (4.36), should beexcluded. Equation (5C.4) is the expression for this condition.

We now come to the ionization balance plasma in which b(l) in eq. (5C.1) isgiven values &IB(!)- We investigated this problem in Section 5.4; for high tempera-ture, starting from low density with b(p) ~ <9(10), with an increase in «e it tends to 1.An example is seen in Fig. 5.10. Figure 5C.4 shows a similar plot to Fig. 5C.1 forthe ionization balance plasma. The sharp lowering of pi# with the decrease in Te

for low densities is the result of the near LTE populations for low-density and low-temperature ionization balance plasma as discussed in Section 5.4. Neutralhydrogen (z = 1) and hydrogen-like helium (z = 2) show this lowering, buthydrogen-like iron (z = 26) does not. No reason can be found that the argumentsconcerning the near LTE populations in Section 5.4 do not apply for such high-zions. The computer code may not yet be refined enough in the present calculation.This lowering, however, violates the spirit of LTE as the case of the dip in therecombining plasma. We ignore these special cases again. We obtain fromnumerical fitting

196

FIG 5C.4 The boundary level for partial LTE in an ionization balance plasma,corresponding to Fig. 5C.1. The explanation is the same as for Fig. 5C.1, exceptthat the thin solid curve is for eq. (5C.5). (Quoted from Fujimoto andMcWhirter, 1990; copyright 1990, with permission from The American PhysicalSociety.)

This lower boundary is shown with the thin solid line in Fig. 5C.4. It is interestingto note that this expression is quite close to eq. (4.29); this equation gives/>G = 345eao67/?7CU18.*

Again the minor difference in the two expressions is due partly to the differencein the definitions, LTE on one hand and Griem's boundary level on the other.Figure 5C.5 is another plot of Fig. 5C.4. In this case, the lowering of/>rB for low Ghas been ignored, and approximate boundaries are determined from z=l , 2,and 26.

If we compare Fig. 5C.4 (or Fig. 5C.5) with Fig. 5C.1 (or Fig. 5C.2) for arecombining plasma, we recognize a substantial difference: at high temperaturethe addition of the ionizing plasma component makes it difficult for levels to enterinto LTE, while for low temperature the addition makes it easy for them. The firstpoint is easily understood from Fig. 5.10. The recombining plasma component

Professor Griem gives in his new book, which was mentioned in the references of Chapter 1, asimilar criterion, />« 8436° '059/tf'ns. The numerical factor was about half in his book of 1964.

IONIZATION AND RECOMBINATION OF PLASMA


FIG 5C.5 The same as Fig. 5C.2, but for an ionization balance plasma. (Quotedfrom Fujimoto and McWhirter, 1990; copyright 1990, with permission fromThe American Physical Society.)

alone is quite close to LTE, so that the additional population hinders the totalpopulation to be in LTE. In order for the population to return to the LTEvalues the ionizing plasma component should become sufficiently small. The low-temperature case is not straightforward. We start with

This equation is rewritten as

In the high-density limit we have the factor [acR/ne]/[Z(l)Sci(]= 1. As Fig. 5.4shows, with a decrease in «e from K^°+, [acR/«e] keeps its high-density limit valueuntil «e reaches eq. (5.25). We now examine SCR and r\(p) for low-lying levels,especially p = 2. Figure 4.11 shows that, in the high-density limit, the levels p <PB,or p < 6 are in Boltzmann equilibrium with the ground state, and when level 2


deviates from it for ne < nf+, levels 3 <p < 5 are in Boltzmann equilibrium withrespect to level 2. We now compare the excitation flux to level p from level (p—1)and that from q with q< (/>—!). By omitting the common factors we have, fromeq. (3.29),

In the case that level (p— 1) is in Boltzmann equilibrium with level q, we have

Therefore, the dominant populating mechanism of level p is stepwise excitation(p— 1) —> p so long as level (p—1) is in Boltzmann equilibrium with lower-lyinglevels. See Fig. 5C.3(c). At high density, starting from the high-density limit all thepopulations of excited levels are controlled by the population of the lowerboundary level in Boltzmann equilibrium. This level is level 2 down to ne at whichlevel 3 deviates from Boltzmann equilibrium with level 2.

In this low-temperature case, in the high-density region the dominant con-tribution to ionization is the ladder-like excitation-ionization as suggested by thep~6 distributions in Fig. 4.11. These populations, and thus the ionization flux, arecontrolled by the population of the low-lying level p = 2 or 3. Thus, it is naturalthat SCR behaves much the same as the population r^p), especially ^(2). Weexpect that

for p = 2, 3 , . . . , pB, Numerical calculations actually confirm that the relationship(5C.10) is quite accurate.

As seen in Fig. 4.3, for this low temperature, the LTE populations in the high-density limit [n(p)/Z(p)nzne] = r0(p) + r\(p) = 1 consists mainly of the second term,the ionizing plasma component, for low-lying levels. With a decrease in ne, thedecrease in ri(p) (Table 4.1(a)) is compensated in eq. (5C.6) almost exactly by thedecrease in SCR according to eq. (5C.10), keeping the same value of the secondterm. The decrease in TQ(P) as seen in Fig. 4.19 does not affect much the popu-lation, because the first term is quite small for these low-lying levels. See Fig. 4.3.

and

It is obvious from the statistical weights, the energy differences, and the oscillatorstrengths shown in Fig. 3.4, that we have


For temperatures around the optimum temperature, the situation is differentas we have noted concerning Fig. 5.12: for this low density, the populations areclose to LTE, while, the relative contributions from the recombining plasmacomponent and the ionizing plasma component is quite different from that in thehigh-density limit. We take level p = 2 for an example. In this temperature range,eq. (5C.10) is still valid in the density region close to nf. This is because ionizationis controlled by the ladder-like excitation-ionization starting from/? = 2 at aroundthis density. With a decrease in ne from the high-density limit, r0(2), the first termof eq. (5C.6), decreases substantially. At the same time r:(2) decreases, but thisdecrease is compensated by a decrease in SCR, as in the case of low temperature. Inthe present temperatures, however, (aCR/»e) increases (see Fig. 5.4), leading to anincrease in the second term. This substantial increase compensates, or even over-compensates, the decrease in the first term. The resulting population tends to behigher than the LTE population for lower densities as has already been noted inthe text.

We now turn to the ionizing plasma, i.e. 6(1) > &IB(!)- It is clear that, if ourplasma is purely ionizing, i.e. nz = 0, and lacks the first term of eq. (5C. 1), no LTEpopulations can exist. In many practical situations, we encounter plasmas whichare ionizing, yet the first term is substantial to a certain degree. In these cases, inFigs. 5.10 and 5.11, for example, the populations would lie somewhere above thethick curves, or the second term tends to be larger than "it should be". Thus, thesecond term tends to make the level populations larger than the LTE values,preventing the levels from entering into LTE. This extreme situation is schema-tically illustrated by Fig. 5C.3(a): the population flux coming from the lower-lyinglevel is too large, resulting in the ladder-like excitation-ionization. Under certainconditions, e.g. 6(1) is larger than £>IB(!) only by a small amount, some levels maybe in scheme (b) and they can be in LTE. An example has been shown in Fig. 5.13:this plasma is slightly ionizing, i.e. 6(1) ~ lOO^iB(l), and levels p > 10 are in LTE.Thus, the condition of LTE includes, besides ne and Te, some constraint about theoverall balance of ionization of the atoms or ions under consideration.

The conditions for LTE in the present ionizing plasma reduce to r0(p) ~ 1 and

The first condition has been taken care of already, eqs. (5C.3) and (5C.4). Inpractical situations where LTE becomes an issue in this class of plasma, i.e. highdensity and temperature, these conditions are almost always met. Thus, the mostcrucial condition is eq. (5C.11). We now remember that, for high density andtemperature, r\(p) is well approximated by eq. (4.35) or ri(p)=p~6. See Fig. 4.10.Within this approximation, eq. (5C.11) is rewritten as

This set of conditions for establishment of LTE involves several parameters. Itwould be illustrative to show the condition for certain particular cases. Table 5C. 1


TABLE 5C. 1 The critical level p\ for establishment of partial LTE in an ionizingplasma. Numbers in brackets denote powers of 10.

1 Extrapolated.

b(l)102

104

106

b(l)10103

10s

b(l)103

10s

107

b(l)10103

10s

b(l)10s

107

109

b(l)10103

10s

(a)z= l6IB(1) = 8.53[1]

2.66.4

13.2

(b)z=lMl) = l-71[0]

2.24.9

10.1

(c)z= lMl) = l-49[2]

4.29.4

21.5

(d)z= lMl) = 2.24[0]

2.04.49.8

(e)z= lMl) = 2.12[4]

7.015.636a

(0z=lMl) = l-60[0]

2.76.0

12.2

6 = 3.2x 104K, 77 =z = 26m(l) = 3.78[l]

2.97.2

15.1

e = 3.2x!04K, 77 =z = 26m(l)=1.33[0]

2.25.3

11.1

6 = 5.12x 10s K, 77 =z = 26m(l)=1.35[2]

4.39.7

22.5

e = 5.12x!05K, 77 =z = 2Ml) = 2.12[0]

2.04.5

10.1

e = 2.56x!05K, 77 =z = 2Ml)=1.88[4]

7.216.539.0

O = 1.6x 104K, 77 =z = 26m(l)=l-25[0]

2.66.5

13.7

1021 m-3

z = 266m(l) = 2.58[l]

3.27.3

16.2

1023 m-3

z = 266m(l) = l-22[0]

2.35.3

11.6

:1021m-3

z = 26Ml) = l-31[2]

4.39.8

22.6

: 1023 m"3

z = 26Ml) = 2.09[0]

2.04.5

10.1

1019m-3

z = 26Ml) = l-82[4]

7.317.041.0

1023 m-3

z = 266m(l) = l-15[0]

2.76.5

14.5

eq. (5C.12)

3.26.8

14.8

eq. (5C.12)

2.24.7

10.0

eq. (5C.12)

4.710.021.7

eq. (5C.12)

2 24.7

10.0

eq. (5C.12)

10.021.746.8

eq. (5C.12)

2.24.7

10.0


gives several examples of the lower boundary level p\. The boundary determinedfrom the numerical calculations for z= 1, 2, and 26 are compared with eq. (5C.12)

Complete LTE

In the text we examined the dependences of SCR and «CR on «e, and concludedthat nf+ gives the region in which these rate coefficients take their high-density-limit values. Thus, nf+ gives the lower boundary of complete LTE, perhaps exceptfor a small numerical factor as discussed below. This boundary was given fromeqs. (5.19) and (5.20), the comparison between the radiative and collisionaltransitions terminating on the ground state. This boundary density is larger thanGriem's boundary for/>0 = 2. In fact, in the discussions after eq. (5.21a), we haveseen that ri^+ almost coincides numerically with Griem's boundary, eq. (4.29a)extended to/>Q= 1- Figure 5C.6 shows the lower boundary ?ys for complete LTE asdetermined from numerical CR model calculations for z= 1, 2, and 26. A sub-stantial difference for different z is seen. Our numerical expression includes z

FIG 5C.6 Boundary electron density for establishment of complete LTE for anionization balance plasma. The thick curves show the results of numericalcalculation. : z = 1; : z = 2; : z = 26. Thethin curves are : eq. (5C.13) with eq. (5C.14); — .. — . .—: 10,4(2, I)/^(2.1); — . — .—: 10/3(l)/a(l). (Quoted from Fujimoto and McWhirter, 1990;copyright 1990, with permission from The American Physical Society.)

Figure 5C.6 compares the above formula with the result of the numerical calcu-lations. In this figure the boundaries determined by eq. (5.19) and eq. (5.20) withp = 2 are given, where a factor 10 has been multiplied to be consistent with thepresent definition of LTE, eq. (5C.2). The plasma density of Fig. 1.7 is barelyoutside of the above criterion for complete LTE. In this case, however, the plasmais optically thick toward Lyman lines which terminate on the ground state (seeFig. 1.6), and the effective A coefficients are substantially reduced in eq. (5.20).Therefore, the above criterion should be relaxed substantially.

Boltzmann equilibrium

A short remark is given here on the condition for the establishment of theBoltzmann distribution, eq. (2.3), of an excited-level population with respect tothe ground-state atom density. The case of an ionization balance plasma has beenexamined already. We now consider an ionizing plasma. The population kineticscheme should be case (c) in Fig. 5C.3, with the chain of this relationship continuingdown to the ground state. We have seen an example in the close of Section 4.2,Fig. 4.11. In this example, in the limit of high density the Boltzmann distributionwith the ground state is established up to p = 5. We remember that this upperboundary for the levels comes from the difference between the scheme (c) andscheme (a) in Fig. 5C.3. Thus, for this distribution to be realized, the densityshould be high and the temperature should be low, so that r^p)^ 1. When weremember that the condition for complete LTE comes from eq. (5.20), the samecondition for density, eq. (5C.13), applies to the present problem. For tempera-ture, the left-bottom corner of Fig. 5C.1 gives an approximate region for thisdistribution. More quantitatively, the upper bound given by the thin dotted line isreduced by a factor of 2, i.e. p < 5 for O = 103 K and p = 2 for O = 104 K. Thisfactor of 2 again partly comes from the present definition of LTE.

If we examine these parameters it is concluded that an ionizing plasma havinga high density, eq. (5C. 13) and low temperature as given above is too extreme to bepractically possible.

Appendix 5D. Optimum temperature, emission maximum, and flux maximum

For an ionization balance plasma we introduced the optimum temperature, i.e. thetemperature at which the ionization ratio [nz/n(l)] is unity. As Fig. 5.16 showsthe emission line intensity is strong at around this temperature (1.35eV). But if welook at this figure more closely, we realize that the temperature at which theintensity takes a maximum is slightly higher in this example of neutral hydrogen.Furthermore, the ionization-recombination flux takes a maximum at still higher


with

OPTIMUM TEMPERATURE 203

temperature (1.7 eV). At this latter temperature [nz/n(lj\ is 11.5, much larger than 1.We consider this problem now.

As in Fig. 5.16, we pose a constraint,

In the ionization balance plasma the ionization ratio is given by eq. (5.28):

First, we consider the ionization-recombination flux, ScRn(l)ne = acRnzne, andlook for a temperature at which this flux takes a maximum:

It is straightforward to rewrite eq. (5D.3), under the constraint of eqs. (5D.1) and(5D.2), as

Figure 5.8 shows SCR and «CR as functions of Te. For low density, SCR is wellapproximated by S(l), eq. (5.6), which may be given by eq. (3.35a). It is straight-forward to show that the slope of S(l) against Te is given by R/kTe, with theneglect of the small Te dependence of G. Figure 5.16 shows that the flux maximumtakes place at Te= 1.7eV. The slope at this temperature should be 13.6/1.7 = 8.However, the approximation eq. (3.35a) is too crude for the present quantitativediscussion. See Fig. 3.11(c): eq. (3.35a) corresponds to the dashed straight line.Actually, the slope of SCR is about 10-12 as seen in Fig. 5.8. The slope of «CR isabout —1. See also eq. (5.15) and Fig. 5.3. Thus eq. (5D.4) gives the ionizationratio [nz/n(l)] of 10-12. This is consistent with the value 11.5 as we have seen inFig. 5.16.

As Fig. 5.15 suggests the emission efficiency of ionization, or the number ofphotons emitted per ionization event, has a substantial temperature dependence atlow temperature: the efficiency is higher at lower temperature. The emissionefficiency of recombination has a small but opposite temperature dependence.These facts make the peak of the emission line intensity shift slightly to a lowertemperature than the peak of the ionization-recombination fluxes.

We have now to consider the problem of whether the emission maximum canbe lower than the optimum temperature or not. For this purpose, we consider, atthe optimum temperature Teo at which nz = «(1), whether the emission line intensityhas a positive slope or a negative slope. Under the same constraint, eq. (5D.1), weconsider the temperature derivative of [C(l, 3)n(l)]. We assume that, in the vicinityof Teo ~ 1.35 eV, the excitation rate coefficient is proportional to (Te/Teo)^ and the


ionization rate coefficient is proportional to (Te/Teo)a, Then, it is rather

straightforward to show that if 2/3/ (a + 1) is larger than 1, the slope of the lineintensity is positive. From the above argument and from Fig. 3.1 l(c), it is obviousthat both the slopes, a and j3, of the rate coefficients have similar magnitudes ofaround 15. Thus, the line intensity has a positive slope at Teo so that it should takea maximum at a temperature higher than the optimum temperature.

Thus, we are able to understand the slight differences among the optimumtemperature, the emission maximum, and the flux maximum.

We have examined above the example of neutral hydrogen. As we noted inSection 5.4 (Fig. 5.9), the optimum temperature, Teo/z

2, becomes high for high-zions. For z = 26, hydrogen-like iron for example, Fig. 5.9 gives Teo/z

2 ~ 2 x 105 K.In Fig. 5.8, at this reduced temperature the slopes of SCR and «CR are approxi-mately 1 and —1, respectively. Equation (5D.4) suggests that, in this case, theoptimum temperature and the maximum flux temperature coincide with eachother. We further note that j3 for the excitation rate coefficient is around 1 and a isalready noted above. Thus, the slope of the emission intensity is about null. Then,the emission maximum also coincides with the optimum temperature.

References

The discussions of this chapter is based on:Fujimoto, T. 1979a /. Phy. Soc. Japan 47, 265.Fujimoto, T. 1979b /. Phy. Soc. Japan 47, 273.Fujimoto, T. 1980a /. Phy. Soc. Japan 49, 1561.Fujimoto, T. 1980b /. Phy. Soc. Japan 49, 1569.Fujimoto, T. 1985 /. Phy. Soc. Japan 54, 2905.Fujimoto, T. and McWhirter R.W.P. 1990 Phys. Rev. A 42, 6588.Hirabayashi, A., Nambu, Y., Hasuo, M., and Fujimoto, T. 1988 Phys. Rev. A 37,77.

Appendix 5A is based on:Sawada, K. and Fujimoto, T. 1994 Phys. Rev. E 49, 5565.

Appendix 5C is based on:Fujimoto, T. and McWhirter R.W.P. 1990 Phys. Rev. A 42, 6588.

Appendix 5D is based on:Goto, M., Sawada, K., and Fujimoto, T. 2002 Phys. Plasmas 9, 4316.

CONTINUUM RADIATION

Figure 1.3, which was a spectrum of neutral helium in a recombining plasma,shows continuum radiation, underlying the series lines, or in the shorter-wavelength region, extending from the series lines. Figure 1.7 also shows a pro-minent continuum. In this chapter we investigate the characteristics of thecontinuum radiation.

6.1 Recombination continuum

We consider the radiative recombination process as schematically depicted inFig. 6.1, which is essentially the same as Fig. 3.7; an electron having energy e iscaptured by an ion in its ground state "1" in ionization stage z, and an ion (atom)in levelp is formed in ionization stage (z—1). A photon is emitted which carries thereleased energy away. Since, in a plasma, the energy of the electrons are dis-tributed over the continuum from zero to high energy, the energies or the fre-quencies of the emitted photons are distributed from the threshold to high energy(frequency). Thus, this spectrum is continuous, extending from the threshold tohigher frequencies. We call this continuum radiation the recombination continuum,

In Section 3.2, we introduced the radiative recombination cross-section,crej,(e), and its explicit expression was given by eq. (3.18) for the case in which afully stripped ion captures an electron to produce a hydrogen-like ion afterrecombination. The radiated power of the recombination continuum in

FIG 6.1 Radiative recombination of an electron having energy e with the ground-state ion z to form an "atom" in level p. A spontaneous transition q^>p is alsoshown. Level q is allocated the energy width /zAz/.

6

206 CONTINUUM RADIATION

FIG 6.2 (a) Schematic illustration of the recombination continuum and theaccompanying line emissions. The intensity of the line q—>p, fP^(i/)di/, isreplaced by P^(v)/^v. (b) Boltzmann plot of the discrete-level populations andits extension to the continuum electrons. The three points in negative energycorrespond to the three emission lines in (a) and the point in positive energycorresponds to the recombination continuum in (a).

frequency width dz/ from a plasma having ion density nz(l) and electron density«e is given by

where/(e)de is the energy distribution function of the continuum electrons and theenergy width de is equal to hdv. See Figs. 6.1 and 6.2(a). In the literature, insteadof the radiative recombination cross-section, the photoionization cross-section isgiven as standard data. By using Milne's formula, eq. (3.17), we rewrite eq. (6.1) interms of the photoionization cross-section, where the Maxwell distribution,

6.2 Continuation to series lines

The radiated power of a transition line <?—>/> is given as (see Figs. 6.1 and 6.2(a))

We consider the radiated power averaged over the frequency width Az/ thatcorresponds to the energy "territory" allocated to level q. See Figs. 6.1 and 6.2(a).Then, the middle expression of the above equation may be written as /^(z/jAz/,where P^(y) is the line intensity averaged over Az/. We now remember that thetransition probability is expressed in terms of the absorption oscillator strength,eq. (3.1), and that the latter is related to the photoabsorption cross-sectionthrough eq. (3.9):

In the same spirit as above, we replace the l.h.s. of this equation by the averagedcross-section (crpiq(y)} times Az/. Similar quantities have been considered alreadyin deriving eq. (3.9b) for the case of hydrogen-like ions. The transition probabilityis then given as

CONTINUATION TO SERIES LINES 207

eq. (2.3a), is assumed for/(e)de:

The recombination continuum radiation for recombination of a fully strippedion z to form a hydrogen-like ion (z— 1) in level p is explicitly given from eq. (6.2)and the photoionization cross-section, eq. (3.13):

with

In deriving the above equation (6.3) we have used the relation hv = e + z2R/p2, SeeFigs. 6.1 and 6.2(a). Thus, on the assumption that the Gaunt factor is unity, theslope of the recombination continuum spectrum gives Te.


If the upper level q is in LTE, its population is given by eq. (2.7) or eq. (2.1 a):

This quantity is expressed by the square in Fig. 6.2(a). This square has the samearea as the hatched area of the emission line. Equation (6.8) is identical to eq. (6.2)except for the exponential factors and the cross-sections. The difference betweendv and Az/ is trivial. As suggested from the similarities between eqs. (3.9) and(3.10a), with an increase in energy of the upper level, from the discrete levelsinto the continuum, the photoabsorption cross-section, (crpiq(v)}, would changesmoothly across the ionization limit to the photoionization cross-section apf(v).We saw an example of this smooth transition in Fig. 3.6. See also eqs. (3.9b) and(3.13), and Fig. 3.3 for the Gaunt factors for hydrogen-like ions. In an emissionspectrum like Fig. 1.3, we find isolated lines, eq. (6.5) or eq. (6.8), at low energies,the recombination continuum, eq. (6.3) or eq. (6.2), at high energies and, betweenthem, a transition from the line to continuum, or even the quasi-continuum, nearthe series limit. Thus, we may conclude that, if all the above assumptions arejustified, the emission intensity of series lines, averaged over the frequency width,changes continuously along the series lines toward the quasi-continuum, andacross the series limit to the recombination continuum.

Since the electrons of the upper levels considered above are loosely bound(E< 0) or have a slightly positive energy (E> 0) they may be affected easily by theplasma environment. Then the atomic properties established for an isolated atommay be modified in the plasma. Thus, some of the above assumptions, e.g. eqs. (3.9)and (6.6) valid for an isolated atom, may be modified in a dense plasma. Thesepoints will be addressed in Chapter 9.

Extension of the Boltzmann plot

For the purpose of determining Te as well as nz(T)ne of the plasma, level popula-tions per unit statistical weight, nz_i(q)/gz_i(q), are plotted against the energy ofthe level E(q) or x(<J)- We call this figure the Boltzmann plot. Examples haveappeared in Figs. 4.20(b), 4.25, 5.12, 5.13, and 5.19(b). If level q is in LTE, itspopulation is given by eq. (6.7); the slope of the fitted line gives Te and theintercept at x(q) = 0, the ionization limit, gives nz(l)ne.

As we have seen above the intensities of emission lines continue smoothlyacross the ionization limit to the recombination continuum. We should thusbe able to extend the Boltzmann plot into the transition region including the

By substituting eqs. (6.6) and (6.7) into eq. (6.5) we readily have

CONTINUATION TO SERIES LINES 209

quasi-continuum, and further to the recombination continuum. The quantitygiven in the Boltzmann plot is nz--i(q)/gz-i(q), which is determined from eq. (6.5).In Fig. 6.2(b) three points are plotted in the schematic Boltzmann plot; thesepoints correspond to the three emission lines depicted in (a). If level q is in LTE,eq. (6.5) reduces to eq. (6.8). Thus, nz_\(tf)/gz_\(tf) is expressed as

Likewise, a quantity which corresponds to the above quantity may be definedfrom eq. (6.2) for the recombination continuum:

In Fig. 6.2(b) a point is indicated which is calculated from the value of Pf(v) in(a) through eq. (6.10).

The above procedure is followed in analyzing Fig. 5.18(b) and the "popula-tions" of the continuum states are plotted on Fig. 5.19(b). They actually lie on thestraight line, and the determination of Te and nz(l)ne, with the recombinationcontinuum being taken into account, is more reliable than a plot only of thepopulations of the discrete levels.

We consider recombination continua terminating on the Rydberg levels. Forthe purpose of illustration we consider a hydrogen-like ion. We ignore for a whilethe accompanying series lines. As Fig. 6.3 shows, for a particular frequency v, wehave several lower levels possible for radiative recombination. The lowest level

FIG 6.3 Radiative recombination processes to several levels. pmin is the lowestlevel relevant to the photon energy hv. The final "level" may be in thecontinuum state.


pmin is given from

Figure 6.4 shows schematically the recombination continua spectrum. The totalradiated power of the recombination continua is given from eq. (6.3):

For the short-wavelength, or high-frequency, region of v > z2R/h, Fig. 6.4suggests that, in the summation in eq. (6.11), only the first term of />min=l ispredominant and other terms may be neglected. For the long-wavelength region ofv <C z R/h, the summation may be approximated by integration, so that

FIG 6.4 Radiative recombination continua for several lower levels as given byeqs. (6.3) or (6.11).

FREE-FREE CONTINUUM - BREMSSTRAHLUNG 211

where gbf has been assumed to be unity. It should be noted that this simpleeq. (6.12a) is based on the neglect of the series lines, and is a rather poorapproximation to the actual spectrum. The same criticism is true for Fig. 6.4: inthis calculation only the recombination continuum, eqs. (6.2), (6.3), and (6.11), isincluded and the accompanying quasi-continuum and the series lines are neglec-ted. However, in real spectra, as seen in Figs. 1.3 and 5.18(b), pure recombinationcontinua as shown in Fig. 6.4 constitute only a part of the spectrum. The sharpthreshold is absent.

Equation (6.14) indicates that the nature of the continuum depends on the tem-perature: at very high temperatures of kTe ;$> 3z2R ~ z2 x 40 eV, the second term inthe square brackets is larger than the first term. Thus, the Bremsstrahlung domin-ates over the recombination continuum, and vice versa at low temperatures. Forthe long-wavelength region of v <C z R/h, several terms should be taken intoaccount in eq. (6.11), and the associated series lines also contribute to the spec-trum. Even in this case, we may conclude that, at high temperatures of kTe > z2R,the Bremsstrahlung gives the dominant contribution. The reader can gain some ideasin Fig. 1.2 about the temperature range in which the Bremsstrahlung is strongerand the range in which the recombination continuum is stronger. Actually, formagnetically confined plasmas, e.g. "tokamak, helical", the Bremsstrahlungradiation is used for the purpose of determining the density of impurity ions

By replacing the summation on the r.h.s. of eq. (6.11) with eq. (6.13) we obtainthe spectral intensity distribution of the Bremsstrahlung continuum PB(i/)di/.A salient feature of the Bremsstrahlung is that the wavelength, or frequency,dependence is, except for the difference in the Gaunt factors, exactly the same asthat of the recombination continuum.

Now we combine the recombination continuum and the Bremsstrahlung. Forthe short-wavelength region of v>z2R/h, the continuum intensity is givenapproximately by

6.3 Free-free continuum - Bremsstrahlung

In Fig. 6.3, if we move the final level p up and go across the ionization limit, wehave final states in the continuum. Yet use of eq. (6.11) should be justified if wereplace the Gaunt factor gbf by gff, the free-free Gaunt factor. Then, the range ofintegration of eq. (6.12) is from —oo to 0. By assuming g{{= 1, we have

where we have collected all the z-dependences here, including those in /. For arecombination continuum to be strong, the factors on the r.h.s. of eq. (6.15)should be large: i.e. high nz and ne, and low Te. See Fig. 5.18(a) and (b). Inlaboratory spectroscopy, there are two classes of plasmas which emit strongrecombination continua. The first is the arc discharge plasmas which have ratherhigh nz and ne with temperatures about 1 eV or a bit higher. The plasma of Fig. 1.7happens to be very similar to this class. Another class is the afterglow plasmas. Inthis case the temperature is rather low, less than 1 eV, sometimes less than 0.1 eV.In such a case, even if nz and ne are rather low we can observe a recombinationcontinuum. Figure 1.3 is an example. In this case nz and «e are about 1020m~3 andlower than those in Fig. 5.18(b) by an order, but Te is about 0.15 eV and lower by afactor of 3, and the recombination continua are strong enough to be observed. Itmay be interesting to note that the r.h.s. of eq. (6.15) contains the factor z8. Seealso eq. (5B.8). This suggests that, for the same reduced temperature and density,a recombination continuum from high-z ions tends to be strong in comparisonwith that from singly ionized ions like the present examples. This may be thereason why the recombination continuum of ions is observed from the recom-bining laser-produced plasma which is quite small in volume.

Reference

A part of the discussions in this chapter is based on: Cooper, J. 1966 Rep. Prog.Phys. 29, 35.

present in the hydrogen (deuterium) plasma. In the latter range the plasmasthat actually emit strong recombination continua are limited in a rather lowtemperature range. The reason is considered below.

From eq. (6.3) the recombination continuum intensity at the threshold isapproximately given as


Spectral lines emitted by ions (atoms) in a plasma are not strictly monochromatic.Rather, they are broadened and have finite widths. They may even be shifted fromthe original positions, too. A remarkable example is shown in Fig. 1.7. Otherexamples are seen in Fig. 1.3: in the long-wavelength region, the 23P-«3D lines aresubstantially broadened as contrasted to the adjacent accompanying 2 P n Slines. This feature is the origin of the nomenclature "the diffuse series" for theformer lines and "the sharp series" for the latter. In this chapter, we examine thesephenomena. Since this subject requires theoretical treatment that is quite involvedand complicated, and which are outside the scope of this book, we restrict our-selves to a rather elementary level of discussion. In the formulation below, we useangular frequency LO instead of frequency v to express the profile of the spectralline. This choice is more natural, as will be understood below. The symbol n isused to designate a level or the principal quantum number of the level, while /> or qare used to designate a state within a level.

Doppler broadening

When an emitter of radiation of angular frequency LOO is moving with velocitycomponent v toward the observer, the radiation is detected with a frequency LO,where

BROADENING OF SPECTRAL LINES

When the emitting ions have a Maxwell velocity distribution, eq. (2.2) with m andTe replaced, respectively, by the ion mass Mi and the ion temperature T-v theobserved profile of the line has the Gaussian shape

where AwD is called the Doppler (half) width and is given by

which corresponds to the most probable speed vp. See Section 2.1. The full widthat half maximum (FWHM) is given as

*7

214 BROADENING OF SPECTRAL LINES

In terms of the wavelength, the FWHM is expressed as

where A0 is the central wavelength.Other kinds of broadening stem from changes, quasi-static or temporal, in

atomic states of the emitter ions.Photons that constitute a spectral line are emitted when the ions make a

transition from an upper level to a lower level. The energy of a photon is thedifference in energies of these levels, to each of which we assign a definite energy,for a while. Let the angular frequency of this photon be UJQ, In a plasma, ions andelectrons constituting the plasma produce an electric field, called the plasmamicrofield, at the location of the emitter ion. The atomic states of the emitter ionare perturbed by this microfield, and so are the energies. In the case of the linearStark effect, for example, the energies shift in proportion to the electric fieldstrength. Accordingly the photon energy, or frequency, is changed. So, our taskwould be to understand the characteristics of this microfield induced in collisions;the microfield may be quasi-static or temporally changing, or even a short pulse.Then, the frequency shift and thus the overall spectral line broadening wouldresult as the collection of responses of the ions to these perturbations.

7.1 Quasi-static perturbation

Holtsmark field

Suppose the temporal changes of the plasma microfield are so slow that we mayregard this field as quasi-static. We begin with the simplest model, the binaryapproximation, of this field. The assumptions are:

1. perturbers are uncorrelated among them and with the emitter, so that they aredistributed randomly in space; and

2. only the perturber located nearest to the emitter exerts the field on the emitterand no other perturbers affect it.

Let the perturber density be N, The probability that a perturber is found in asmall volume element dv is given as

The probability that no perturbers are found within the same volume is

The corresponding probability for a finite volume v is

QUASI-STATIC PERTURBATION 215

The probability that no perturber is found in a sphere of radius r centered at theemitter and that one perturber, the nearest one, is found in the spherical shell withinner and outer radii (r, r + dr) is

The average distance between the adjacent perturbers pm is given by

The probability dP(r) of finding the nearest perturber at r is expressed in termsof An,

The strength of the field at the emitter exerted by a perturber at a distance r havinga charge Ze is

The normal field strength is defined as

The probability that the emitter is subjected to field F is

with (3 = F/Fo, and J0°° WE((3)d(3 = 1. Strictly speaking, dP(F) in eq. (7.9) is not thesame as dP(r) in eq. (7.7) so that a different notation should be used. However, forthe sake of avoiding unnecessary complications, we use the same notation here.

In actual situations, in which the binary approximation above is not valid, thefield at the emitter is the vector sum of the fields produced by all the perturbers. Ifwe take this fact into account, or if we remove assumption 2 above, the field

or


distribution function W^((3) is modified, especially for small (3 where many distantperturbers contribute to the field at the emitter. The Holtsmark distributionapplies to this situation:

In this case, a slightly different (0.2%) definition of the normal field strength fromeq. (7.8) is adopted:

Figure 7.1 shows this distribution function as the curve with a = 0.The quasi-static distribution applies mainly to the fields produced by ion

perturbers. The ions repel each other by the Coulomb repulsion, and assumption 1.above will be violated, too. The Holtsmark distribution should thus be mod-ified. Taking the ion correlation into account automatically includes the shieldingof the perturber charges by the plasma particles, the Debye shielding. Calculationsof the distribution functions including these effects are reported in literatures.

FIG 7.1 The Holtsmark field distribution (for a = 0). /3 = F/F0, where F0 is thenormal field strength, eq. (7.8). Distribution functions with the ion correlationeffects being taken into account are shown for a > 0, where a = pm/Ri). Herepm is defined by eq. (7.6), and RD by eq. (7.11). (Quoted from Hooper Jr. 1968;copyright 1968, with permission from The American Physical Society.)

FIG 7.2 Linear Stark shifts of hydrogen levels of n = 2 and 3, and the Balmer a transition lines connecting these levels. Thesolid and dotted lines connecting the Stark split levels indicate, respectively, the TT and cr components, where, when observedfrom the direction perpendicular to the applied electric field, the former components are polarized in the direction of theelectric field and the latter perpendicular to it. The relative intensities and positions of these lines are given, and they arecompared with the experimentally observed Stark spectrum. In this calculation, the fine structure splitting is included for the« = 2 levels. (Calculated by M. Goto, and the experiment quoted from Mark and Wierl, 1929.)


An example is shown in Fig. 7.1, where the parameter a = pm/Ro represents thestrength of the shielding. Here, the Debye radius is given as

It is noted that Z= 1 is assumed in this calculation.Another effect appears when the emitter is an ion, instead of a neutral atom as

is implicitly assumed so far. In this case, the emitter repels perturber ions, againviolating assumption 1. The distribution functions for z= l are found slightlyshifted (by about /3^0.1) to the lower field strengths than those in Fig. 7.1.

As has been noted in the last paragraph before Section 7.1 the energies of theupper and lower states of an emitter are shifted by the Stark effect. In the case ofthe linear Stark effect, the energy shift of the states is proportional to the fieldstrength, and thus each of the Stark components of the spectral line shiftsaccordingly. An example of the linear Stark splitting is shown in Fig. 7.2 for thecase of the hydrogen (2-3) line, i.e. the Balmer a line, in a homogeneous electricfield. Since, in a plasma, the field strengths are distributed, the spectral line isbroadened. In the case of the linear Stark effect the line profile directly reflects thefield distribution function, e.g. Fig. 7.1. We will examine this problem further indetail in Section 7.4.

7.2 Natural broadening

We start with a classical electric dipole oscillator having resonance frequency UJQ asthe model of the emitter ion. The radiation emitted by this oscillator is given byexp(iw0?). If the oscillation is stationary, it is obvious that this spectral line ismonochromatic. In reality, however, the atomic system interacts with the radiationfield or the vacuum, and excited atomic states are no longer strictly stationary.Rather, they make spontaneous transitions, and the oscillator and thus theradiation decay with time. We express this damped oscillator by

We readily obtain a profile having a Lorentzian distribution

where the profile has been normalized to 1.

where 7 represents the decay rate of the energy of the oscillator. The spectral lineprofile is given from the Fourier transform,

TEMPORAL PERTURBATION IMPACT BROADENING 219

In the last paragraph before Section 7.1, we started with the assumption thatdefinite energies are assigned to the upper and lower levels of the transition. Atthis point, we remove this assumption: at least the upper level has an energy widthcorresponding to eq. (7.14).

7.3 Temporal perturbation - impact broadening

Autocorrelation Junction

We assume a stationary atomic oscillator. This oscillator may be perturbed bycollisions with plasma particles. Owing to the Stark effect, the electric fieldinduced at the emitter during a collision gives rise to a frequency shift Aw, themagnitude of which depends on the field strength and the response of the oscil-lator to the electric field. In the course of the collision event, the electric fieldincreases from zero, takes a maximum and decreases. The accumulation of thefrequency shifts over the duration of the collision results in a shift of the phasefrom the original phase in the absence of the collision. Let rj(t) be the total phaseshift accumulated from time 0 to t. Then the state of the oscillation at time t isexpressed by

The line profile is given by the Fourier transform similar to eq. (7.13),

Since the collisions are random (stochastic) and stationary, the Fourier transformcan be replaced by another Fourier transform:

where we have defined the autocorrelation function

It may be interesting to see that, if we apply the above procedure to the case ofnatural broadening, i.e. eq. (7.12), we obtain the same profile as eq. (7.14).

In the autocorrelation function, we eliminate the e1"0' factor, to have (j>(s):


The time average, eq. (7.19a), can be replaced by an ensemble average overstatistically identical atomic oscillators, which is expressed as ( • • • ) :

A: Adiabatic collisions -perturbation to the phase

We assume the situation that the duration of a collision is short compared with thetime interval between successive collisions, or the mean free time. Then, the detailsof the temporal change of the frequency shift Aw during the collision are insub-stantial. Rather, the dominant effect of the collision would come from the resultantphase shift brought about by the collision. The emitter experiencing successivecollisions is subjected to the phase shifts over these collisions. We now calculatethe autocorrelation function <j)(s). For an increase As in the time interval s we havea change in <j)(s)

where A?y is the additional phase shift induced by collisions during the timeinterval As. Since collisions are assumed statistically independent, A?y is inde-pendent of TJ. Then, we have

Let P(p, v)dpdv be the probability of collisions in unit time which induce the phaseshift 77, where p and v are, respectively, the impact parameter and the relativespeed of the collision. Here the impact parameter is defined such that the targetion is assumed stationary, and the perturber particle is incident along a straightline. The distance of this line to the target is called the impact parameter. Then,we have

where f(v) is the velocity distribution function like eq. (2.2). We now rewriteeq. (7.23) by introducing cross-sections,

By changing A to d in eqs. (7.22) and (7.23a), we integrate eq. (7.22) to obtain


By substituting eq. (7.24) into eq. (7.17) with eq. (7.19), we readily obtain

where, for the purpose of simplicity, we have omitted the velocity dependence andP(p, v). For the purpose of illustration we assume here that the phase shifts r/ arevery large and random. Then the cross-section becomes ar = 2irfp dp. The width isproportional to N, or inversely proportional to the mean free time. These factssuggest that the broadening is caused by the disruption of the phase of the atomicoscillation, or of the wave train of radiation, by collisions. We may call this thecollisional relaxation of coherence. Equation (7.27) correctly accounts for thecoherence relaxation by collisions including those with small r/. The shift is givenby the imaginary part

We now consider the case of small r/. The sign of 0-j and thus that of A are the sameas that of the averaged r/. Their magnitudes are also proportional to each other. Ifwe remember that a phase shift is the accumulation of the frequency shifts over theduration of the collision, a positive phase shift after the collision is induced bypositive frequency shifts during the collision, and vise versa. Thus, we mayunderstand that the shift of the line corresponds to frequency shifts averaged overcollisions.

We may expect that, starting from very distant collisions, or from a very largeimpact parameter, with a decrease in the impact parameter the phase shift rjinduced by a collision would increase. The impact parameter at which the phaseshift rj | is one radian is called the Weisskopf radius and denoted by p0. Collisionswith impact parameters smaller than the Weisskopf radius may be called strongcollisions. In the above illustration of the broadening and the shift, we consideredstrong collisions and weak collisions, respectively.

In this section, we have assumed an atomic oscillator. In reality, the emittedradiation is due to a transition of the ion (atom) from the upper state p to the

with

and

We have again arrived at the Lorentzian profile with the FWHM 7. In the presentcase, however, the central frequency is shifted by A.

We now examine the physical meaning of eq. (7.26). The cross-sections arewritten down from eq. (7.23); namely, for the real part


lower state q. In quantum mechanics, an energy eigenstate p evolves with timeaccording to exp[ix(p)t/K\, where x(p) is the ionization potential of this state.(Remember our convention that x(P) is taken positive.) The phase of the oscil-lation of the classical atomic oscillator eluj°' corresponds to the difference betweenthe phases of the upper state, \(p)tjh, and the lower state, \(cf)tjh. The opticalcoherence is defined as the phase correlation between the upper and lower states.The phase change induced to the classical atomic oscillator by a collision, whichwe considered above, corresponds to a phase perturbation in the upper state or inthe lower state or in both of them. In terms of quantum mechanics, the impactbroadening is interpreted as the relaxation of optical coherence between the upperand lower states.

In the above, we have considered adiabatic collisions; that is, the atomic stateof the emitter ion is perturbed during a collision and the ion returns to its originalstate after the collision.

If some other kinds of collisions give rise to relaxation of optical coherencethen these collisions also contribute to broadening. We mention them brieflybelow.

B: Collisional transition within the same level - disruption of the phase

An atomic level having an angular momentum / is degenerate, or consists ofseveral states having the same energy, with (2J+ l)-fold degeneracy. Thesedegenerate states are called magnetic substates, each specified by a magneticquantum number mj. In a collision by a perturber particle, a transition may takeplace between two magnetic substates in the same level. The wave train of theemitted radiation may continue, but the information of the phase may be lost inthe transition. This kind of collisions also contributes to the relaxation of opticalcoherence and thus to broadening.

The phase correlation among the magnetic substates is called the Zeemancoherence. Relaxation of Zeeman coherence due to transitions among the mag-netic substates constitutes an interesting area of research, but this is outside thescope of this book.

C: Inelastic transitions - termination of the wave train

In some cases, a collision may result in a change of the atomic level, excitation ordeexcitation, from the original upper level or the lower level or even both. In thiscase, the wave train terminates and the optical coherence is completely destroyed.This kind of collisions leads to broadening of the spectral line connecting theoriginal upper and lower levels.

In Section 7.2 and also in Section 7.3, we assumed eq. (7.12) to express thephenomenon of radiative decay of an excited ion. However, the spontaneoustransition is also a stochastic process like collisions. Strictly speaking, we have totreat the radiative decay in a similar way to that for collisions. If we proceed in this


way, we will find that the result is the same as the foregoing one, eq. (7.14). It maybe noted that the natural broadening is also regarded as due to the relaxation ofoptical coherence. In this case, the relaxation is equivalent to the decay: thecoherence decays along with the upper-level population.

We can conclude that, in the case of h <C 1, impact broadening is dominant, andfor h > 1 quasi-static broadening is dominant.

For impact broadening we may assume that r ~ Pm/v, where v is the averagerelative speed of the perturber. We have noted in deriving this relationship the factthat the perturbers are charged particles, and the effects are long-range Coulombfields. For the quasi-static perturbation, the electric field strength at the emitter isproportional to r~ , where r is the distance to the nearest perturber. A typical valueof r would be pm. The frequency shift in the linear Stark effect is proportional tothe electric field strength and in the quadratic Stark effect it is proportional to thesquare of the field strength. Thus, we may write

Criterion between quasi-static broadening and impact broadening

In Section 7.1 we introduced the quasi-static field which would result in linebroadening, and we have been concerned with the impact broadening in thissection. We now consider which aspect, quasi-static or impact, is more importantin particular situations.

As we have seen in eq. (7.26) or eq. (7.27), roughly speaking, the impactbroadening width 7 is given by the mean free time r between successive collisions,

The statement that a collision with impact parameter of the Weisskopf radius p0

induces a phase shift of 1 radian may be expressed as

where we introduce the typical value of the frequency shift Aw\y for this Weisskopf-radius collision. We may further assume that

For the purpose of comparison of the relative importance of the two broadeningmechanisms, it would be natural to compare 7 with the quasi-static broadeningwidth, say Aw. We now introduce the parameter h

7.4 Examples

Hydrogen-like ions - linear Stark effect

Hydrogen-like ions, with which we have been mainly concerned in earlier chap-ters, have energy-level structures in which different / states having the sameprincipal quantum number have almost the same energy. See Fig. l.ll(b). We willignore here small energy separations from the fine structure splitting which isexplicitly shown for n = 2 states in Fig. 7.2. The one-electron system with nuclearcharge z in an electric field is analytically solved in parabolic coordinates. Thequantum numbers specifying the states are: n, the principal quantum number;KI and HI, the parabolic quantum numbers; and m, the magnetic quantum number.An example of the energy splitting of levels and the Stark splitting of the spectralline is shown in Fig. 7.2 for neutral hydrogen n = 2 and 3 levels and the Balmer aline connecting them. Suppose we apply an electric field F in the z-direction. Theenergy shift of a Stark state p is given by the first-order perturbation from theinteraction Hamiltonian, H' = er • F, and the unperturbed wavefunction f/>°,

where (p \ z \p) is the matrix element, or the expectation value, of the z-componentof the electron position vector r in the direction of the electric field. See eq. (3.3)and Table 3.1(a). Since KI + n2 < n — 1 (nl5 «2 > 0), the maximum shift is given byeq. (7.30) with (n\ — «2) replaced by (n — 1). For the purpose of a rough estimate,we assume large n and adopt for a typical frequency shift

In the above, small numerical factors have been dropped. The quantity p^N is ameasure of the number of perturber particles inside the sphere of Weisskopfradius. Thus, the criterion between the dominance of impact broadening, K3<g; 1,and that of quasi-static broadening, h3 > 1, has another interpretation.

or by using eq. (7.6)

Combining the above approximate relationships, we rewrite eq. (7.29) as


EXAMPLES 225

It may be interesting to remember that (n2a0/z) is the radius of the classical orbit,eq. (1.2). We now consider eq. (7.29). Suppose F is given by F0 of eq. (7.8) andT ~ Pm/v. With the neglect of a small factor we have

It is readily seen that, in many cases of practical interest, we have for ion per-turbers hi ;$> 1. Thus, the quasi-static picture is relevant for ion broadening. On theother hand, for electron perturbers, we have he <C 1 except for the cases of very lowtemperature or extremely high density. Thus, in many cases, the impact broadeningis predominant. We now examine each case below.

Ion broadening

As can be seen in Fig. 7.2, the Stark splitting pattern of a spectral line is rathercomplicated, and we have to distribute each of the components over the frequencyaccording to the Holtsmark distribution function WH(J$) or its modified version asshown in Fig. 7.1.

The line profile corresponding to the Holtsmark distribution is given by thefunction TH(X);

with the parameter

where n and n' are the principal quantum numbers of the upper and lower levels.The profile of TH(X), which will be called the Holtsmark profile, is shown inFig. 7.3 as y = Q; this function is normalized to 1. The far wing is proportional to(v — uj0)~

3- This is the direct consequence of the Holtsmark distribution, Fig. 7.1(a = 0). It is seen that the FWHM is AwH = &ABF0, or

It is noted that SFQ is essentially the same as Aw of eq. (7.31). This line profile ison the assumption that the Stark shifts are distributed rather uniformly over the

226

FIG 7.3 The Holtsmark profile function for ion broadening TK(x), eq. (7.33),and the Stark profile function T(x,y), eq. (7.38), of hydrogen-like ions, whereTK(x) is identical with T(x, 0). (Based on Sobelman et al., 1981.)

frequency. The Balmer a line, as shown in Fig. 1.7, approximately meets thisrequirement (see Fig. 7.2). However, the Balmer (3 line as well as the Lyman (3 line,for example, lacks the unshifted component in the Stark splitting. Thus, they showa central dip in the profile, which is the signature of ion broadening. This feature isbarely seen in Fig. 1.7.

Electron broadening

As we have seen, impact broadening is relevant to electron collisions. In this case,however, the dominant broadening mechanism is non-adiabatic, i.e. transitionsamong the Stark states in a level, similar to class B in Section 7.3. Since the per-turbation is of rather long range, collisions with large impact parameters give themain contribution, where the upper bound of the range of perturbation is limitedby the Debye shielding. The first-order perturbation is sufficient, and the mag-nitude of the perturbation is closely related with the spatial extent of the wave-functions as was the case in the ion broadening above. The resulting line profileturns out to be of Lorentzian shape, except for the line center where the positionsof the Stark split components of the line are located at different frequencies. Thebroadening parameter 7e corresponding to the FWHM of the Lorentzian profile isgiven by

BROADENING OF SPECTRAL LINES

EXAMPLES 227

where the Weisskopf radius is approximately given by

Here /(«,«') is a coefficient representing the strength of the interaction, or aparameter related with the spatial extent of the wavefunctions. The definition ofthe Debye radius here is slightly different from eq. (7.11) besides the allowance formultiply charged perturbers. The term 0.215 accounts for the contribution fromstrong collisions having impact parameter smaller than p0. For transitions withlarge values of n and a small value of n', I(n, n') is approximately given by n4/z2,

The actual broadening is a combination of ion broadening and electronbroadening. The resulting profile may be represented by modifications of theHoltsmark profile, eq. (7.33), with the parameter ^/e/SFQ,

Non-hydrogen-like ions - quadratic Stark effect

For non-hydrogen-like ions in which different / levels have different energies,the effect of an electric field is different from that for degenerate hydrogen-like ions, i.e. the quadratic Stark effect. Figure 1.4 shows an example of the

and RD is the Debye radius given by

A few examples of this profile are shown in Fig. 7.3, where the case of ^/e/SFQ = 0corresponds to the Holtsmark profile. For ^/e/SFQ ;$> 0, electron broadeningbecomes dominant, and the profile tends to Lorentzian. For moderate-densityplasmas, it turns out that the electron broadening is rather unimportant,i.e. 7e/#F0< 1- This conclusion may look inconsistent with our assumption of/2e<§; 1 below eq. (7.32). This is because of the gross underestimate of 7 ~ v/pm asan approximation of 7e as given by eq. (7.36) in these situations.

Figure 1.7 shows an example of Stark-broadened hydrogen lines in a verydense plasma. In the case of the Balmer j3 line, for example, ^/e/SFQ is about 3, andthe electron broadening is dominant. Still the central dip remains, which is thesignature of the ion broadening as noted before.

For lines of principal importance, e.g. lower members of the Lyman andBalmer series lines, detailed calculations and comparison with experiments havebeen reported. The lines in Fig. 1.7 are fitted by these detailed data. Readers arereferred to the relevant literature.


FIG 7.4 The Stark effect for neutral helium lines of (a) the 21P-41S, P, D, Ftransitions and (b) the 23P^3S, P, D, F transitions. The transition strength isexpressed as the brightness of the traces. Since the lower levels are almostunaffected, the shifts and the intensity changes are due entirely to theperturbation to the upper levels. (Calculated by M. Goto.)

non-hydrogen-like energy-level structure, i.e. neutral helium. Figure 7.4 showsan example of the Stark shift patterns of lines; i.e. for neutral helium (a) the21P-41S, P, D, F lines and (b) the 23P-43S, P, D, F lines. The first-order pertur-bation gives rise to the wavefunction mixing

where p and q denote the states, and the superscript 0 means the unperturbed stateand energy. The interaction Hamiltonian H' = er • F is the same as in eq. (7.30).Thus, the perturbation is strong between levels /<->(/± 1) having a large electricdipole matrix element ( q \ z \ p ) and having a small energy separation. SeeTable 3.1 (a) for the former (although this is for hydrogen, the radial wavefunctions

EXAMPLES 229

and the matrix elements are much the same for helium) and Fig. 1.4 for the latter.In the present example, for allowed transitions to 2P, the wavefunctions of theupper states, 4S or 4D, could be mixed into the 4P or 4F wavefunctions. Other nlevels have smaller "overlap" of the wavefunctions and are energetically too far toaffect substantially the n = 4 levels. As a result, the "forbidden transitions", 2P-4Pand 2P-4F, become partially allowed. In this figure, the transition strength isexpressed as the brightness of the trace. It is actually seen that these transitionsbecome substantially strong in finite field strengths. The intensity is determined bythe degree of mixing, eq. (7.39). The 4S wavefunction does not mix substantiallywith the larger L states because of its large energy distance from these levels. SeeFig. 1.4 and eq. (7.39). It is noted that the lower level, 2:P or 23P, is only slightlyperturbed, so that the Stark effect in this figure is virtually the perturbationsincurred to the upper levels.

The energy shift results from the second-order perturbation: the quadraticStark effect. The energy perturbation to state p is given by

As eq. (7.39a) suggests, the energy of state p is "repelled" by other states that areconnected by the dipole matrix element, in the opposite direction. See eq. (3.3).The strength of this repulsion is proportional to the square of the matrix elementand inversely proportional to the energy separation between these states. In theexample of Fig. 7.4, this feature is seen for the S states, and, to a certain extent, theP states. Note the opposite locations of the 4P state with respect to the 4D and 4Fstates for the singlet and triplet systems, as seen in Fig. 1.4. When the Stark shiftbecomes comparable to the energy separation between the states interacting witheach other, eq. (7.39a) is no longer valid, and the energy shift deviates fromquadratic. Rather the Stark effect tends to the linear Stark effect, as in the case ofhydrogen. This transition is seen with the 4:P states. The D and F states have verysmall energy separations so that the transition takes place at very small fieldstrengths, and the Stark effect virtually starts as the linear Stark effect.

We now confine ourselves to the strictly quadratic Stark effect. Since weassume that the electric field is in the z-direction, the matrix element is Fe(q\z\p).The energy shift, or the frequency shift, is given by

By adopting the normal field strength, eq. (7.8), for the typical field strength of Fwe may rewrite this equation as


where the energy separation in the denominator has been normalized by theRydberg constant multiplied by z2. Here, it is remembered that z is the effectivecore charge felt by the optical electron and that Z is the charge of the perturber.We now calculate the criterion parameter h, eq. (7.29). We assume that thedominant perturbation comes from one level, and that \(q \ z \p)\2 is of the order of(lQl-lQ3)(a0/z)2 as suggested from the integrals of the position vector as shown inTable 3.1(a). In many cases, E(p,q)/z2R is of the order of lO^-lCP2. Thus, wehave

For ion perturbers, we have hi < 1 in many cases. For high density and lowtemperature, hi could be larger than 1. For electrons he is much smaller than one.Impact broadening is important.

In the following, for both kinds of perturbers we assume impact broadening.As is easily seen, e.g. in eq. (7.26), impact broadening is approximately propor-tional to the average speed, so that electrons are the dominant perturbers. In thefollowing we consider only electron broadening.

By following a quantum calculation similar to the procedure that led toeq. (7.25), we arrive at the Lorentzian distribution which is exactly the same aseq. (7.25). On the assumption that only one perturbing state q is responsible forthe broadening and shift, the width and shift are expressed by

and the cross-sections are given by

It is readily seen from eq. (3.2) that the first factor of the r.h.s. of eq. (7.43) is\/g(q)/3 \(p\r q) \ in atomic units, which is of the order of lO1^2. See Table. 3. l(a).The multiplication factors /'(/?) and /"(/?) are shown in Fig. 7.5.

It is obvious from eqs. (7.42) and (7.44) that the effects of ion collisions areapproximately Z4//3(m/A/)1//6 times the effects of electron collisions. Thus, ion

where the parameter j3 is defined by

EXAMPLES 231

FIG 7.5 The parameters for the broadening and shift in quadratic Starkbroadening. See eqs. (7.42a) and (7.42b). (Based on Sobelman et al, 1981.)

contributions are of the order of 10% of the electron contributions. In the casethat /?;$> 1, the collision mechanism is adiabatic, class A in Section 7.3, and for/?<§; 1, it is inelastic, class C.

In Fig. 7.4 we have seen that, for the S state, the quadratic Stark effect isapproximately valid for the whole range of the electric field in this figure. On theother hand, for the P, D, and F states the splitting feature changes with increase inthe field strength. Thus, the nature of line broadening is expected to be quadraticor linear, depending on the typical field strength and the line. Since levels withn > 5 have higher / levels, i.e. G, H, . . . , levels, all of which have very smallquantum defects, broadening tends to be linear.

Figure 7.6 shows an example of experimental observations of Stark broad-ening. The spectrum is of neutral helium in a plasma produced by shock waveheating. The Stark broadening profiles corresponding to Fig. 7.4(a) and (b) areseen. The normal field strength is of the order of 107 V/m under this experimentalcondition. The singlet lines are not obvious. The triplet lines show the typicalfeatures of the Stark effects:

1. We may regard the 22P—43D, F line to be rather close to the linear Starkbroadening as noted above.

2. On the wing of this broadened intense line, the "forbidden" line 23P 43Pappears. See Fig. 7.4(b).

3. Another line 23P—43S is only slightly broadened. It is suggested for this line that/2e<l, and that /2i>l; the ion broadening is quasi-static. The asymmetricprofile reflects the frequency shift due to the quadratic Stark effect; see

FIG 7.6 An example of observed Stark broadening. This is for neutral helium lines. The triplet lines show typical featuresof Stark broadening. This figure is reproduced from photographic film, so that the ordinate is approximately thelogarithm of the actual intensities. (QuotedfromOkasakae^a/., 1977, with permission from The Physical Society of Japan.)

VOIGT PROFILE 233

Fig. 7.4(b). Unfortunately, however, this spectrum is recorded on photographicfilm, so that the ordinate is nonlinear, and is time integrated over the plasmaformation and decay. Thus, we should restrict ourselves to qualitativediscussions.

Inglis-Teller limit

For hydrogen-like ions, the FWHM is approximately given by eq. (7.35). Weconsider the situation in which we observe series lines with a small n' with varying n,e.g. the Lyman lines or the Balmer lines. With an increase in n the width increasesaccording to eq. (7.35), while the separation between the adjacent lines decreasesaccording to eq. (1.5). When the line width becomes equal to the line separation,we expect that the adjacent lines cannot be resolved. Rather, these overlappinglines form a quasi-continuum, which was mentioned in the previous chapter.Examples are seen in Figs. 1.7, 1.3, and 5.18(b). Remember here that, although thelatter examples are for neutral helium, the upper states are «3D, which are ratherclose to the hydrogen-like states as noted above. We thus expect to have a certainrelationship between the plasma density (ion density) and the principal number nof the last line that is discernible as a line. We call this limit the Inglis-Teller limit.The relationship is given as

7.5 Voigt profile

As we have seen above, natural broadening and impact broadening, or relaxationof optical coherence, led to a line profile with a Lorentzian shape; see eqs. (7.14)and (7.25). Even the Holtsmark profile is rather close to a Lorentzian profile, asseen in Fig. 7.3. We sometimes encounter situations in which the ions emitting theline radiation with the Lorentzian profile are in thermal motion, having a Maxwellvelocity distribution and producing Doppler broadening, eq. (7.2), provided theline were monochromatic. The line profile we observe in this case is thus the resultof convolution of the Lorentzian profile with the Gaussian profile.

For the purpose of simplicity we rewrite (LJ — LJQ) in eq. (7.14) or (LJ — UQ — A)in eq. (7.25) as uj. The Lorentzian profile is

where TV is in [m ]. We apply this method first to Fig. 1.7. We assign n = 6. Thenwe have NK 3 x 1023 m~3. This should be equal to ne, which is quite close to thevalue derived from the overall fitting of the spectrum. Figures 5.18(b) and 1.3 givethe last discernible line with n= 11 and n=\l, respectively. These numbers givevalues «e«3 x 1021 m~3 and 1 x 1020 m~3, respectively.


The Gaussian profile is

Convolution of eq. (7.46) over eq. (7.47) yields

with the parameter

FIG 7.7 Examples of the Voigt profiles along with the Gaussian profile (a = 0)and the Lorentzian profile (a = oo), where all the profiles have FWHM = 2.

The profile of eq. (7.48) is called the Voigt profile and a above is the Voigtparameter. Figure 7.7 shows examples of the Voigt profiles along with theGaussian profile (a = 0) and the Lorentzian profile (a = oo), all having the sameHWHM. It is seen that even for small a values the line wing is heavily affected bythe Lorentzian profile. This feature has significant consequences in radiationtransport of the line as discussed in the next chapter.

REFERENCES 235

Note the following: superposition of the Gaussian profiles with the Dopplerwidths AwDi, AwD2 , . . . results in a Gaussian profile with Doppler width AwD,where

References

The standard textbook of line broadening is:Griem, H.R. 1974 Spectral Line Broadening by Plasmas (Academic Press, New

York).

The discussions in Section 7.4 are largely based on:Sobelman, 1.1., Vainshtein, L.A., and Yukov E.A. 1981 Excitation of Atoms and

Broadening of Spectral Lines (Springer, Berlin).

The discussion in Section 7.3 about the relationship between the collision broad-ening and the coherence relaxation is experimentally verified in:Hirabayashi, A., Nambu, Y., Hasuo, M., and Fujimoto, T. 1988 Phys. Rev. A 37,83.

The figures are taken from:Mark, H. and Wierl, R. 1929 Z. Phys. 55, 156.Hooper, C.F. Jr. 1968 Phys. Rev. 165, 215.Okasaka, R., Shimizu, M., and Fukuda, K. 1977 /. Phy. Soc. Japan 43, 1708.

Superposition of Lorentzian profiles with FWHM 71; 72,. . . results in aLorentzian profile with FWHM

*8

RADIATION TRANSPORT

Until now, we have assumed that photons emitted in a plasma leave it withoutbeing absorbed. In real situations, photons emitted by atoms (ions) in transitionq top (Fig. 3.1) could be absorbed before leaving the plasma by atoms in levelp whichare present in some other locations in the plasma. In the case that the populationof atoms p is substantial, the effect of absorption cannot be neglected, or it evencontrols temporal changes and spatial distributions of n(q) in the plasma. In thecase that level p is the ground state of atoms or ions, this is especially the case. Inthis chapter we deal with the phenomena related to absorption of photons. Herethe term atom is used instead of ions. This is because, in many cases in whichradiation transport is important, atoms play the major role rather than ions.

8.1 Total absorption

Before treating the collective process of emission and absorption of photons in aplasma, we consider a simple situation: a medium containing atoms in level pabsorbs white background light by making transition p —> q. A good example ofthis situation is the Fraunhofer absorption lines in the solar spectrum.

For the purpose of simplicity we assume the situation of one dimension as depictedin Fig. 8.1. In the region x > 0 there is a uniform medium. From the —^-direction,radiation is incident on the medium at x = 0 and continues to travel in it. Let theintensity of the incident radiation at frequency v be 7° per unit frequency interval(dj/= Us"1]) per steradian. This quantity is the intensity, or the spectral radiance,introduced at the beginning of Section 2.2. Let the intensity at x be Iv(x). Theabsorption property of the medium is specified by the absorption coefficient/cjm"1] at frequency v in the absorption line profile of transition p^q. Anexample of nv is schematically shown in Fig. 8.2(a). The change in the intensity ofthe radiation in the medium is given by

FIG 8.1 Geometry of one-dimensional radiation absorption. In the region x > 0we have a uniform absorbing medium.

TOTAL ABSORPTION 237

FIG 8.2 Formation of absorption and emission lines, (a) Profile of the absorptioncoefficient, (b) Formation of an absorption line. The optical thickness near theline center increases to reach almost complete absorption, (c) A similar plot foran emission line. The black-body radiation with the excitation temperaturelimits the maximum of the emission intensities near the line center.

or

This situation is shown in Fig. 8.2(b) for the weak absorption cases. In theopposite case of rv ;$> 1, the optically thick case, the incident background radiationis almost completely absorbed

This situation is seen in Fig. 8.2(b) near the line center for strong absorption cases.The absorption coefficient is given from the absorption cross-section, intro-

duced by eq. (3.9) or (3.9a):

Since we assume a uniform medium eq. (8.la) is readily solved:

Figure 8.2(b) shows examples of the absorption line profiles. The quantity KVX iscalled the optical thickness or the optical depth and sometimes denoted as rv,

In the case of small optical thickness rv<^_\, which we call the optically thincase, we retain the first two terms of the expansion of the exponential function ineq. (8.2):

This quantity may be expressed in terms of Einstein's B coefficient and theabsorption line profile P'(v), which is normalized to 1 over the line. The line shapeof Fig. 8.2(a) is nothing but this profile.

or, corresponding to eq. (3.9),

Thus, the absolute magnitude of j'rvdv is proportional to n(p)fx,Figure 8.2 shows schematically (a) the absorption coefficient with the Voigt

profile and (b) the development of the absorption line profile, eq. (8.2), forincreasing x, normalized by the background, Iv(x)/I%. For small absorption orsmall x, the absorption profile is exactly the same as that of nv, eq. (8.3). With theincrease in x the medium becomes optically thick near the line center, and theabsorption profile becomes broader. In the case of sufficiently large x, eq. (8.5)becomes valid near the line center. Since the Voigt profile has a broad wing comingfrom the Lorentzian profile as shown in Fig. 7.7, when the medium is opticallyvery thick near the line center, the line develops broad wings where the mediumchanges from optically thick to thin. The strong absorption lines in Fig. 8.2(b)show this feature.

As a measure of the amount, or the strength, of absorption of the line by themedium, the total absorption or the equivalent width is introduced:

which is nothing but the area of the absorption in Fig. 8.2(b). This quantity ma;be expressed in terms of wavelength in units of [nm].

where KA is the absorption coefficient corresponding to nv. The merit of thisquantity is that, in practical observations, this is independent of the instrumentalfunction, or resolution, of the spectrometer. Figure 8.3 shows an example of thetotal absorption against the n(p)fx value, or the thickness of the medium. Thesecurves are called the curve of growth. In the case that the medium is optically thinover the line it is obvious from eqs. (8.4) and (8.3) that the total absorption isproportional to eq. (8.7a) multiplied by the dimension of the medium x,

238 RADIATION TRANSPORT

TOTAL ABSORPTION 239

FIG 8.3 (a) The curve of growth of the equivalent width against the increase inthe number of absorbing atoms. 2Aw1/2 is the FWHM of the Dopplerbroadening, eq. (7.3a), and the parameter a is the Voigt parameter as defined byeq. (7.49). (b) An example of the curve of growth constructed from the observedequivalent widths of the solar Ti and Fe lines. (Quoted from Wright, 1948.)

This linear dependence is seen in Fig. 8.3 in the left-bottom corner. With anincrease in the medium thickness the total absorption increases. When the mediumbecomes optically thick at the line center eq. (8.5) begins to apply there, and thetotal absorption grows more slowly than in the optically thin case (see Fig. 8.2(b)).In such a case, roughly speaking, the total absorption is determined by theseparation between the line wings at which KVX is about 1. See Fig. 8.2(b). If theabsorption profile is Gaussian, as Fig. 7.7 suggests, the total absorption increasesslowly. This corresponds to the case of a = 0 in Fig. 8.3(a), where a is the Voigtparameter, eq. (7.49). The slope is approximately 0.1 against the medium thick-ness, or the n(p)fx value. As in the case of Fig. 8.2(b), we sometime encountersituations in which the absorption profile has a Voigt profile, eq. (7.48); in this

8.2 Collision-dominated plasma

We now replace the above absorbing medium in Fig. 8.1 with a plasma which isagain assumed to be uniform. The ions (atoms) in it may emit line radiationcorresponding to the transition p <— q as well. The emission property is expressedby the emission coefficient i]v\Wm^3 sr^1 s]. Equation (8.1) is modified to includethe emission process,

RADIATION TRANSPORT240

case the Lorentzian wing becomes important at higher thicknesses of the medium;see Figs. 7.7 and 8.2(b). Then, the slope becomes 0.5, as shown in Fig. 8.3 forseveral a values. This is the case when the Lorentzian broadening is due to naturalbroadening, which is unlikely in laboratory plasmas, or Stark broadening with aconstant FWHM. In some other cases,/and x are kept constant, and n(p) varies,and further, the broadening of the line is proportional to n(p), i.e. the resonancebroadening for the case that level p is the ground state; in these cases the slope is 1at high densities. This case is not shown, though. The reason why this is so isthat, with an increase in n(p), the FWHM grows in proportion to n(p), so thatthe optical thickness at the line center is independent ofn(p). See eq. (8.7a). Figure8.3(b) shows an example of observed equivalent widths: the absorption lines areneutral metal atom lines in the solar atmosphere.

Since we assume a uniform plasma eq. (8.10a) is readily solved:

The emission coefficient is expressed as

where P(v) is the emission line profile normalized to 1. In the following we assumeP(v) = P'(y)', this is equivalent to assuming complete redistribution of frequency.This assumption is that the atom, when it emits a new photon, has lost its memoryof the frequency of the photon it absorbed. We also add to eq. (8.7) the inducedemission process to yield

We define the source function

OR

COLLISION-DOMINATED PLASMA 241

which is rewritten from eqs. (8.12) and (8.13) as

In deriving the last line we used the relationships (2.12) between Einstein's A and Bcoefficients. We express the population ratio [n(p)/g(p)]/[n(q)/g(q)] in terms ofthe temperature; see eq. (2.3). We define the excitation temperature Tex for the levelp and q populations; see eq. (2.13). Then we have

where we have used the definition of the black-body radiation, eq. (2.14). Equation (8.16)is called Kirchhoff's law. Thus, equation (8.11) is written as

We may replace KVX by TV.In many cases of practical interest, the incident radiation is virtually absent.

Then eq. (8.17) reduces to

Figure 8.2(c) schematically shows the development of an emission line withincreasing value n(p)fx. This figure corresponds to Fig. 8.2(b). It is readilyrecognized that eq. (8.18) is complementary to eq. (8.2). In the optically thin case,or rv <C 1, we have

If our plasma is optically thin over the line, the profile of the emission line is thesame as that of nv which is shown in Fig. 8.2(a). This is also true for the cases ofsmall absorption profile in Fig. 8.2(b). In this case, the total line intensity which isthe intensity, eq. (8.19), integrated over the line profile

is proportional to the upper-level population and the dimension of the plasma.This is the situation which we assumed up to the preceding chapters. It should be

q


noted that eq. (8.19a) is essentially the same as eq. (4.1). Note that we assume one-dimensional geometry here. In the optically thick case, or T^> 1, we have

Figure 8.2(c) shows schematically the development of the emission line profilewith increasing n(p)fx value. It is noted that, with the increase in the thickness, thepeak intensity increases and tends to that of the black-body radiation, eq. (8.20),at the excitation temperature, but it never exceeds that.

In actual situations, we sometimes encounter an inhomogeneous plasma - thecentral part of the plasma may have higher excitation temperature than the sur-rounding plasma. This is the case for a positive column of discharge plasmas. Insuch a case we have to integrate eq. (8.10a). For the purpose of understandingwhat happens in such a case, we consider a simple example: a plasma having twolayers as depicted in Fig. 8.4; the layer 1 plasma has thickness x\ and excitationtemperature Tl5 which is higher than T2, the temperature of the layer 2 plasmahaving thickness x2. We observe the radiation emitted by this composite plasmafrom the layer 2 side. It is easily understood that, even if another low-temperatureplasma layer on the left, or the opposite side to the observation direction, werepresent it would contribute little to the observed radiation, so that the actualplasmas can be well modeled by this two-layer model. We further assume that theabsorption coefficients are the same for both layers. This situation is realized withmany plasmas when level p is the ground-state atom. An example is a dischargeplasma including mercury in the fluorescent lamp. It is straightforward fromeqs. (8.18) and (8.17) to obtain

Iv = (TiXl-expC- KVXI}} exp(-K^2) + BV(T2)[\ - exp(-K^2)]. (8.21)

Figure 8.5 shows schematically the profile of the emission line. The first term isshown with the dashed curve, and the second term with the lower dash-dottedcurve. The resulting profile is shown with the solid curve. A central dip develops;this phenomenon is called self-reversal.

FIG 8.4 A model of the geometries of an inhomogeneous plasma.

COLLISION-DOMINATED PLASMA 243

FIG 8.5 The emission line profile from an inhomogeneous plasma. Self-reversaldevelops.

We now consider a more general case where the absorption coefficient and theemission coefficient may change along the line of sight; we integrate eq. (8.10a)to obtain

with

Self-reabsorption

Suppose we observe a plasma with an optical system, i.e. a lens and a spectrometer(Figure 8.6). Let the intensity observed at a certain frequency be I\. (If we use alens and a spectrometer, the quantity we actually measure corresponds to eq. (4.1)with units of [W]. However, we simplify the situation here and deal with thespectral radiance having units of [W m~2 sr^1 s].) We place a concave mirror on theopposite side of the plasma and focus the image of the plasma on the plasma itself.The observed intensity with this mirror installed is 72. We assume the plasma tobe uniform. Then, this situation is a special case of the situation of Fig. 8.4 andeq. (8.21), where x2 = xi, and T2= T\. In the case that the plasma is optically thinthe intensity 72 is, if we neglect the finite reflection efficiency of the mirror and theabsorption losses by the plasma container, twice I\. If the plasma is sufficientlyoptically thick, 72 is equal to I\. See eq. (8.20). In other words, we can determine


FIG 8.6 Geometry for the self-reabsorption experiment.

FIG 8.7 An example of line-absorption A^ against the increase in themedium thickness or the number of absorbing atoms. The abscissa is theoptical thickness at the line center for Doppler broadening,KOX = e2n(p)fp!qx/4mcs0AujY)\/fK', see eq. (8.35) later. The parameter a is theVoigt parameter.

the optical thickness of the plasma at this frequency from

In many cases, we cannot resolve the profile of an emission line by using, say, aspectrometer. Rather we observe the total line intensity integrated over the profile

An example of line absorption is shown in Fig. 8.7. This is for the Voigt profile;a is the Voigt parameter, eq. (7.49). It is to be noted that, with an increase in theplasma thickness, the line absorption tends to 1 for the pure Gaussian profile(a = 0). For the Voigt profile it never tend to 1. The reason is understood fromthe explanation already given in Section 8.1. It tends to a constant value, 0.586.

In more general cases when the excitation temperature and the absorptioncoefficient vary over the plasma we can readily generalize eq. (8.25).

8.3 Radiation trapping

We consider a medium, not necessarily a plasma, which consists of uniformly dis-tributed atoms having two levels p and q; level p may be the ground state. Thegeometrical shape of the medium may be an infinite slab, an infinite cylinder or ofmore complicated shapes. As the initial condition, excited atoms in level q areprepared with a certain spatial distribution. For example, in the case of a cylindricalmedium the initial excitation may be limited in the thin axial region: a pencil-shapedexcitation. This situation is actually realized when we excite the atoms with a beamof pulsed laser light or with an electron beam. At time zero these excited atoms beginto make transitions p <— q, emitting photons of this transition line. A photon maytravel over a certain distance until it is absorbed by an atom in level p at some otherlocation in the plasma. The excitation of the first atom is thus transferred by thephoton to another location. This emission-absorption process may be repeatedseveral times. As a result of this chain of processes, the spatial distribution of theexcited atoms q is redistributed in the medium. At the same time, photons that donot suffer absorption escape the plasma with a certain probability, and the excita-tion is lost with a certain rate from the medium. The spatial distribution of the upperlevel atoms and the loss of the excitation are controlled by these repeated emission-absorption processes. This phenomenon is called radiation trapping.

In the early twentieth century the phenomenon of radiation trapping, or of themigration of excitation, was considered to be analogous to diffusion. The termdiffusion is sometimes still used even now. In both cases, excitation in the presentcontext is transferred over space in steps. In each step in the latter (i.e. diffusion) ittravels by a distance of the order of the mean free path. By repeating many steps itspreads in space and finally it may escape from the medium. The diffusion coef-ficient is a parameter to quantify how fast the diffusion takes place. Later it wasrealized that radiation trapping is very different; we assume a medium having anabsorption coefficient of the line like in Fig. 8.2(a) and its optical thickness at theline center over the medium is substantially larger than unity, i.e. an optically thickmedium. We assume this situation throughout in the following discussions. Since

RADIATION TRAPPING 245

We define the line absorption in a similar way to eq. (8.24):


the emission probability has the same profile as that of the absorption coefficient,an emitted photon almost always has a frequency in the region of the line core,where the medium is optically thick. This photon is readily absorbed by an atomlocated close to the original atom. Since the optical thickness is large, even aftermany repetitions of this process, excitation can spread over only a short distance.So far, the phenomenon is similar to diffusion. In this optically thick medium,however, the far wings of the line are optically thin. In one stage of this emission-absorption chain, a photon may be emitted in the far wing, even with a smallprobability. This photon has a small chance of being absorbed in its path in themedium and escapes with a high probability. This escape is almost independent ofthe location of the photon emission. This is the reason why photons escape theoptically thick medium. It is readily understood that the phenomenon of radiationtrapping is substantially different from that of diffusion. For radiation trappingwe cannot define a quantity like the diffusion coefficient. If we calculate the meanfree path of photons it diverges. Radiation trapping is thus essentially a non-localphenomenon and needs a different treatment.

From the above argument, we can infer the emission line profile observed froman optically thick medium. As will be seen in the following subsection, the upper-level population, or the excitation temperature, is high in the central region, whereit is difficult for the emitted photons to escape, and low in the periphery region,where the photons easily escape. The profile of the emission line is formed byaccumulation of photons that succeeded in escaping the medium. Photons in theoptically thin far wing come from all the regions of the medium, so that it reflectsthe properties averaged over the medium. In the near wing where the opticalthickness is KVR^ 1, where R is the typical dimension of the medium, e.g. theradius in the case of a cylinder, the photons carry information collected over themedium. The intensity is high because it reflects the central high temperature.Near the line center, photons emitted in the central region are immediatelyabsorbed. We can observe photons coming only from the thin (thickness d) peri-phery region of the medium where the optical thickness measured from theboundary is nvd< 1. In this region the excitation temperature is low so that theblack-body radiation intensity, eq. (8.20) with eq. (8.16), is low. In other words, we"see" the region of the medium within K^^ 1 from the boundary. As a result wehave a line which develops the self-reversal profile, which is similar to Fig. 8.5;Note that the intensity of the strong peaks is closely related with the black-bodyradiation with the higher excitation temperature.

Eigenmode analysis

For simplicity we denote the population n(q) as n, n(p) as N, and A(q,p) as A. Weassume n<^N, and TV is constant over the volume. The time development of thepopulation n at location r is given by


where G(r',r)dr' is the probability that a photon emitted at r' is absorbed at r. Thusthe second term represents the production rate of the population n at r from thepopulation at r' through a single emission-absorption process. This function isgiven by

where the emission profile P(v) is defined by eq. (8.12) and KV is given by eq. (8.13).Here again we assume the complete redistribution of frequency, i.e. P(v) = P'(v).The integrand of eq. (8.28) is schematically shown in Fig. 8.8(a) for several cases of p;the area under the curve gives the transmission probability, which is schematicallyshown in Fig. 8.8(b). The induced emission, the second term of eq. (8.13), has beenneglected because we assume low excitation, n(q) <C n(p), or n <C N.

Two examples of numerical calculations of the integro-differential eq. (8.26)for excited-atom populations are given in Fig. 8.9. These examples are for acylinder with infinite length, and the initial excitation is (a) on the axis, or (b) nearthe surface of the cylinder. The line profile is assumed to be Lorentzian, eq. (7.46).The spatial profile of the relative population distribution is given for several time-spans after the start. The overall population in the medium, of course, decays withtime. It is seen that, with the course of time, the radial distribution gradually tendsto a bell-shaped profile denoted by oo irrespective of the initial population dis-tribution. Then, the decay of the population becomes exactly a single exponential.

FIG 8.8 (a) The integrand of eq. (8.28) for several cases of the traversingdistance p; (b) transmission probability.

with p = | r' — r , where T(p) is the transmission probability. This is theprobability that a photon traverses a distance p without being absorbed, and isgiven by


The eigenvalues are arranged in the order g0 < gi < g2 < • • • . In the presentexample, the eigenvalue for the lowest mode is approximately given by

where KP is the absorption coefficient at the line center and R is the radius of thecylinder. From eqs. (7.46) and (8.7a) we have

FIG 8.9 Examples of the temporal development of the relative populationdistributions of excited atoms in radiation trapping under optically thick condi-tions. An infinite cylinder and Lorentzian profile are assumed. The parameteris the time given in units of the time constant of the fundamental decay mode,T0 = (g0A)~l. (Quoted from Golubovskii and Lyagushchenko, 1975.)

The solution of the integro-differential equation (8.26) may be expressed as

Equation (8.26) reduces to an eigenvalue problem

This has been solved for several cases. An example is given in Fig. 8.10; the radialeigenfunction <pn(r) is given for the cylindrical geometry. As suggested from thisfigure, the eigenfunctions are orthogonal and normalized,


FIG 8.10 The spatial profiles of the eigenfunctions of eq. (8.29) in several lowestmodes. An infinite cylinder with radius r = 1 and the Lorentzian profile.(Constructed from the table of Golubovskii and Lyagushchenko, 1975.)

The quantity go is called the escape factor for the fundamental decay mode.Figure 8.11 shows the relative magnitudes of the eigenvalues for higher modes.The above examples of the numerical calculation shown in Fig. 8.9 are interpretedas follows. The spatial profile of the initial excitation is expressed as a superposi-tion of the eigenfunctions. Each eigenmode population decays with its characteristicdecay rate. Higher modes have shorter time constants. After a sufficiently longtime all of the higher-mode populations have died out, and only the fundamentalmode survives. Then, the shape of the population distribution no longer changes,and the population and the emitted radiation intensity decay with the commonrate g$A. For the geometry of an infinite slab, the expression of go is slightlydifferent from eq. (8.32), but the relative magnitudes of the eigenvalues for highermodes are almost the same as the case of the cylinder as seen in Fig. 8.11.

In the case of a Gaussian profile, eq. (7.47), for an infinite cylinder with radius R,an approximate expression for g0 is given as

where KO is the absorption coefficient at the line center. From eqs. (7.47) and (8.7a)we have


FIG 8.11 Escape factor of the higher modes normalized by that of the lowestdecay mode. For diffusion the first higher mode has a rate nine times that ofthe fundamental mode.

Figure 8.11 shows the relative magnitudes of the eigenvalues for the Gaussianprofile. Another formula fits more accurately the numerical calculation over awider range of optical thickness:

Figure 8.12 shows the comparison of eq. (8.34a) with a numerical calculation andalso eq. (8.34).

We have noted the essential difference of the radiation trapping phenomenonfrom the diffusion. An important consequence lies in Fig. 8.11. For diffusion, theeffective decay rate of the next higher mode is nine times that of the fundamentaldecay mode, for an infinite slab, for example. Thus, the higher modes would dieout rather rapidly. On the other hand, in the case of radiation trapping the decaytime constants of higher modes, especially in the case of the Lorentzian profile, arenot much larger than that of the fundamental decay mode: gi is only 1.6 timeslarger than g0. This means that, starting from an arbitrary initial populationdistribution it takes a long time to reach the final distribution. In Fig. 8.9, it is seenthat time t = 4/(g0A) is not long enough to reach the final profile.

Escaping photon analysis

We treat the situation as assumed in eq. (8.26). Suppose an excited atom is pro-duced at time 0. The probability of that atom to survive at time t and make a


FIG 8.12 The escape factor, eq. (8.34a), compared with eq. (8.34) (dashed line)and the result of numerical calculation (solid line, Phelps, A.V., 1958 Phys.Rev. 110, 1362).

transition during the time interval dr is given by

where A stands for, as before, the transition probability [s l]. Suppose there are n°excited atoms at t = 0. Among them ao«° atoms are assigned to emit photons thatescape the medium without being absorbed. The number of these photons cor-responding to eq. (8.36) is

Another fraction a^n0 of atoms emit photons that are absorbed once beforeescaping the medium. The number of these escaping photons from the medium at tis given by the convolution of a^A exp(—At) with p(t),

This quantity has a maximum att=l/A. Similarly, a2n° atoms emit photons thatare absorbed exactly twice before escaping. The corresponding function S2(f) isthe result of convolution of a function similar to eq. (8.38) with p(f).


FIG 8.13 Examples of the coefficients an for an infinite cylinder and Gaussianprofile. The parameter A0 is (l/k0R) with radius R. (Quoted from Wiorkowskiand Hartmann, 1985; copyright 1985, with permission from The OpticalSociety of America.)

Likewise, the number of photons that escape from the medium after beingabsorbed n times is given as

This function has a maximum at t = n/A.The overall photon number function is the sum of the Sn(f)'s:

The set of values of an depends on various parameters, e.g. the geometry and theoptical thickness of the medium, the absorption line profile, the initial spatialexcitation profile. Figure 8.13 shows an example of the expansion coefficients.

Equation (8.41) corresponds to the population function, eq. (8.29), in theeigenmode analysis. After a sufficiently long time, the latter tends to the funda-mental decay mode with the effective decay probability g$A. Equation (8.41)should decay with the same effective rate. Thus, for sufficiently large n the coef-ficients should follow

Appendix 8A. Interpretation of Figure 1.5

We have now reached the point where we can interpret on a sound footing thespectra in Fig. 1.5, which were for ionized helium.

INTERPRETATION OF FIGURE 1.5 253

It is known that the helium plasma of Fig. 1.5 has electron density about1020 m~3. See the close of Section 7.4. This density is virtually equal to the ground-state ion density, «(1). We assume «(1) = 1020 m~3, the ion temperature 7~i = 104 K,and the radius of the plasma R = Q.Q5 m.

Table 8A.I gives, for each transition, the wavelength, the FWHM of theDoppler broadening as given by eqs. (7.3) and (7.3a), where the angular frequencyhas been converted to frequency, and the FWHM of the Stark broadening as givenby eq. (7.35), where z = 2 and Z=l. It is seen that the former broadening ispredominant, especially for the lower members of the lines for which the effect ofradiation trapping is strong. We neglect the latter broadening in the followingdiscussions. This table contains the absorption oscillator strength, the absorptioncoefficient at the line center, and the corresponding optical thickness over the plasmaradius. We now assume that the effect of radiation trapping is approximatelydescribed by the effective reduction of the transition probability by the amount ofthe escape factor of the fundamental decay mode, go. We use Fig. 8.12 or eq.(8.34a) to obtain the escape factor, which is given in the last column of this table.

Table 8A.2 gives for Fig. 1.5(a) and (b) the apparent recorded intensity inrelative units, where the obvious saturation effect of the detector for the 1-2transition in (a) has been corrected for: by assuming the same observed profile in(a) and (b), the peak of the 1-2 line in (a) was enhanced by a factor of 2.1. Wecorrect for the reduction of the observed line intensity due to radiation trapping bydividing the intensity by the escape factor. The result is given in the next column.The transition probability is given by standard tables or partly in Fig. 3.5(a).Finally the upper-level population per unit statistical weight is given.

These population distributions for (a) and (b) are plotted in Figs. 8 A. l(a) and (b),respectively. Since the measurement is relative, the populations are given on arelative scale. In these figures, the calculated population distributions for theionizing plasma component, Fig. 4.5, and those for the recombining plasmacomponent for low temperature, Fig. 4.20(a), are given with the curves. Since the

TABLE 8A. 1 For each transition this table lists the wavelength, FWHM of theDoppler broadening, FWHM of the Stark broadening, the oscillator strength, theabsorption coefficient and the optical thickness at the line center, and the escapefactor for the fundamental decay mode.

1-2 30.38 7.04(11) 6.8(9) 4.16(-1) 1.47(2) 7.35 9.4(-2)1-3 25.63 8.35(11) 1.8(10) 7.91(-2) 2.3(1) 1.15 4.3(-l)1-4 24.30 8.81(11) 3.4(10) 2.9(-2) 8.2(0) 0.41 0.71-5 23.73 9.02(11) 5.4(10) 1.39(-2) 3.83 0.19 0.81-6 23.43 9.13(11) 7.9(10) 7.8(-3) 2.12 0.11 0.91-7 23.26 9.20(11) l-l(H) 4.8(-3) 1.3 0.065 0.931-8 23.15 9.24(11) 1-4(11) 3.2(-3) 0.86 0.04 0.97

Transition A (nm) 2A^1/2 (s ') A^H (s ') fi,n k0 (m ') k0R go


TABLE 8A.2 For each transition this table lists the signal peak (corrected forsaturation), the value corrected for the radiation trapping effect, and the upper-level populations per unit statistical weight.

experiment is for hydrogen-like helium, the nuclear charge is z = 2. We thus haveto compare the experiment with the calculation for reduced electron density«e/2

7 ~ 1018 m~3. See the scaling law, eq. (5B.1). It is seen that, for both cases, theexperimental population distribution fits well into the calculated distributions,being consistent with our speculation made in Chapter 5; Fig. 1.5(a) is the spec-trum of an ionizing plasma and Fig. 1.5(b) is for a recombining plasma.

In the above collisional-radiative model calculation the effect of radiationtrapping is not taken into account. For the transition 1-2, Table 8A.I suggeststhat the effective transition probability should be reduced by a factor of 10, so thatthe actual population would be larger than the present calculation by a certainamount. In view of this inconsistency, the agreement of the experimental popu-lations with the calculation is surprisingly good. It is noted that we did not adjustthe electron temperature in the calculation. It is Te = 22 x 1.28 x 105 K forFig. 8 A.I (a) and 22 x 103 K for Fig. 8A.l(b). See eq. (3A.9).

In the former case of the ionizing plasma, it is known that the overall popu-lation distribution is rather insensitive to electron temperature so long as thetemperature is high. So, we cannot say anything definite except that this plasmais a high-temperature ionizing plasma with density at around ne K 1020 m~3. In thelatter case of the recombining plasma, roughly speaking, the density is determinedat the position of the peak population, p = 5-6, and the temperature is reflected inthe population distribution of the higher levels. It is suggested that this latterplasma happens to have Te K 4 x 103 K and ne K 1020 m~3 or a bit higher.

The Balmer lines in Fig. 1.7 are broadened predominantly by Stark broad-ening, in contrast to the above. These lines are fitted with the profiles fromaccurate calculation for the Stark broadening for each line. It is found that theBalmer a line has substantial optical thickness at the line center, and the peakintensity is reduced by about a factor of three under this condition. This reductionof intensity is incorporated in Fig. 1.7.

1-2 15 159 4.25 4.5 47.9 1.281-3 3.2 7.4 0.74 4.6 10.7 1.071-4 1.0 1.4 0.34 5.3 7.6 1.871-5 0.3 0.38 0.18 6.5 8.1 3.951-6 - 3.4 3.8 3.221-7 - 2.0 2.1 2.841-8 - 0.9 0.9 1.83

Transition (a)

$(«,!) $/go n(n)/g(n)

(b)

$(«,!) $/go n(n)/g(n)

REFERENCES 255

FIG 8A.1 Population distributions derived from Fig. 1.5. (a) For Fig. 1.5(a).The line intensities, after various corrections, result in the relative populationdistribution plotted with the closed circles. The curves are the calculatedpopulation distributions reproduced from Fig. 4.5 for an ionizing plasma,(b) For Fig. 1.5(b). The curves are reproduced from Fig. 4.20(a) for a low-temperature recombining plasma.

References

Parts of the discussions in this chapter are based on:Holstein, T. 1947 Phys. Rev. 72, 1212.Holstein, T. 1951 Phys. Rev. 83, 1159.


Mitchell, A.C.G. and Zemansky, M.S. 1934 Resonance Radiation and ExcitedAtoms (Cambridge University Press, London).

Wiorkowski, P. and Hartmann, W 1985 Opt. Comm. 53, 217.

The figures are taken from:Fujimoto, T. 1979 /. Quant. Spectrosc. Radial. Transfer 21, 439.Fujimoto, T. and Nishimura, Y. 1985/. Quant. Spectrosc. Radial. Transfer 34,217.Golubovskii, Yu.B. and Lyagushchenko, R.I. 1975 Opt. Spectr. 38, 628.Wright, K.O. 1948 Pub. Dom. Astrophys. Obs. 8, 1.

DENSE PLASMA

When ions (atoms) are immersed in a dense plasma, they experience an environ-ment which is substantially different from those encountered in ordinary situa-tions. Since the density is high some of the plasma particles may come so close tothe ion core that they substantially perturb the original potential exerted by theion core to the bound electrons, resulting in modifications to the motions of theseelectrons. Thus the properties of the electronic state of the ions may be modified.The Stark broadening of the transition line treated in Chapter 7 falls into thiscategory. Electron collisions are so frequent that qualitatively new features mayappear in the population kinetics of the ion. Various features which result fromthe collection of collisional and radiative processes, as treated in Chapters 4 and 5,may be regarded as belonging to this category. In this chapter we review someother salient features of ions, or those manifested by them, immersed in denseplasmas.

As a measure of the criterion for a "dense plasma" we can consider severaldensities in comparison with the properties of ions: geometrical and dynamical.One geometrical quantity of the plasma would be the mean distance between theions, pm, as defined by eq. (7.6), and another would be the typical distance overwhich the Coulomb force is effective, RD, as defined by eq. (7.11) or eq. (1.31 a). Ifthe dimension of the atomic wavefunction, as given by eq. (1.2), of the ion underconsideration becomes non-negligible compared with either of the above dis-tances, the plasma may be regarded substantially dense for this ion. The criticaldensity would be of the order of 1028z2 (m~3) for the ground state. When excitedions are concerned, because of the larger electron orbit radius, the critical densitywould be substantially lower. We will see this to be actually the case later on. Thedynamical quantity would be a typical excitation rate by electron impact for thision in this plasma, in comparison with the inverse residential time of the ion, or thetransition probability. A criterion of the reduced density of ne/z

7 ~ io17~20 m~3

has been encountered already in Chapters 4 and 5 for singly excited ions. SeeFig. 1.10, for example. Another criterion, which becomes important in this chapter,concerns doubly excited ions: the autoionization probability is of the order of1013 s^1, which is almost independent of the z of the ion. Thus, the criterion in thiscase would be ne/z

3 ^ io23~26 ni~3. The reader may have some ideas about the«e regions in Fig. 1.2 where the high-density effects play a significant role.

9.1 Modifications of atomic potential and level energy

In this section, unless otherwise stated, we will consider a plasma made of protonsand electrons. Thus, ions with which we will be concerned are protons. In thisplasma some of the protons may have a bound electron, i.e. they form hydrogen

*9

258 DENSE PLASMA

atoms in discrete states. It will be shown later that the distinction between thebound discrete states and the continuum states is rather ambiguous in a denseplasma.

We first focus our attention on one of the ions having a bound (optical)electron. Since the plasma is dense and the plasma particles are in thermal motion,the chances are high that a plasma particle comes close to the ion core, even insidethe bound electron orbit, and exerts an electric field on the electron. Thus thebound electron experiences various perturbations. If we average these perturba-tions over time, the net effect would be regarded as screening or shielding of theion core charge by the plasma particles. This is nothing but the Debye shieldingintroduced by eq. (7.11) or eq. (1.31 a). The effective potential felt by the opticalelectron is thus approximated by the Debye-Hiickel potential,*

For small r, this potential behaves similarly to eq. (9.1), and at r = RQ the potentialtends to zero. This picture may be interpreted as: each proton has its own territorywith radius R0, and an electron having a positive energy migrates to the neigh-boring territories of other ions. In this model, again, the discrete state energies arepushed up.

* Since the ion core is charged positive, the electric potential is positive. However, we will beconcerned with the motion of an electron, which is attracted by the ion core. In this regard, it is moreconvenient to express the potential as negative. We adopt this convention in this chapter.

where the Debye radius or the Debye length, eq. (7.11), is

where «i is the ion density. Owing to the modification of the Coulomb potentialthe wavefunctions, as given in Fig. 3.2, and thus the energy eigenvalues, as given inFig. l.ll(b), are modified. Intuitively speaking, as a result of the "narrowing" ofthe Coulomb potential, eq. (9.1), the discrete states of the electron are "squeezed".The energy is "pushed up", and high-lying bound levels are lost into the "con-tinuum" states. Thus, the number of discrete states becomes finite.

If the temperature of the plasma is low or the density is high, the picture of"shielding" becomes inadequate. Rather, the ion sphere model is more appro-priate. It is customary to express the mean distance between the ions as RO, inplace of pm as defined by eq. (7.6). We follow this convention here. Several ver-sions of the model potential have been proposed. One of them is

ATOMIC POTENTIAL AND LEVEL ENERGY 259

FIG 9.1 Shifts of level energies in a dense plasma, (a) Positions of several levels ofhydrogen-like ions normalized by the original ionization potential of each level.The abscissa is the Debye length (times z) in atomic units. Debye-Hiickelmodel. (Quoted from Rogers et al, 1970; copyright 1970, with permission fromThe American Physical Society.) (b) Level positions of hydrogen-like Ne . Ionsphere model. (Quoted from Yamamoto and Narumi, 1983, with permissionfrom The Physical Society of Japan.)

260 DENSE PLASMA

An example of the calculations of the level energies is shown in Fig. 9.1; (a) isfor a hydrogen-like ion calculated according to the Debye-Hiickel potential,where the energy position is normalized to its original ionization potential. It isseen that, with an increase in density, or a decrease in the Debye length, the levelsare pushed up and, further, engulfed into the continuum. Figure 9.1(b) is the resultof calculation for hydrogen-like neon (z= 10) based on the ion sphere model. Twofeatures are noted in this figure:

1. The shifts of level energies are approximately parallel, so that, if we look atthis figure in such a way that the energy of the ground Is state is fixed, then theenergy level structure of the low-lying levels stays almost the same and, with anincrease in density, the ionization limit comes down. This phenomenon is calledthe pressure ionization or the lowering of the ionization potential. The magnitude ofthe lowering for the Debye-Hiickel potential is given by

FIG 9.2 (Continued)

TRANSITION PROBABILITY AND COLLISION 261

FIG 9.2 (a) Spectra from an aluminum plasma produced by laser irradiation.Plasma polarization shifts of hydrogen-like A111+ Lyman S and Lyman e linesare seen, (b) Comparison of the observed shifts of the Lyman 6 line with theresults of several model calculations. (Quoted from Renner et al., 1998; withpermission from IOP Publishing.)

In this equation we have assumed the charge of our ion to be ze. The ion spheremodel gives a different value for the lowering depending on the details of themodel.

2. Even so, there remains a small difference in the shifts of the levels. Thisdifference results in a small shift of the spectral line connecting a pair of the levels.This phenomenon is called the plasma polarization shift. An example of theexperimental observations of this shift is shown in Fig. 9.2 for a laser-producedaluminum plasma. Figure 9.2(a) shows emission spectra of hydrogen-likealuminum Lyman 6 (1-5) and Lyman e (1-6) lines. With a decrease in the distancefrom the target surface, or an increase in the plasma density, the peak of both ofthese lines shifts toward the longer-wavelength direction. Figure 9.2(b) comparesthe observed shifts with the results by several calculations.

9.2 Transition probability and collision cross-section

The shift of level energies was a result of the modification to the "shape" of thewavefunctions (Fig. 3.2) caused by a change in the potential. Another consequenceof this modification is a change in the oscillator strength (see eqs. (3.2) and (3.3))and thus the radiative transition probability, eq. (3.1). Figure 9.3(a) shows forneutral hydrogen an example of the changes in the transition probabilities against

262 DENSE PLASMA

a change in the Debye length. All of the probabilities decrease with the increase inthe plasma density.

It should be remembered here, however, that the level energies werealso modified as we have seen above. Figure 9.3(b) shows another example ofcalculations by the ion sphere model; for hydrogen the energy positions of excitedlevels with respect to the ground state are given by the horizontal positions ofsymbols (+) and the lowering of the ionization potential is given with the verticallines. Note that these results are consistent with Fig. 9.1 (a). It is seen that, with thedecrease in the Debye length, the shifts of the level energies result in a decreasein the energy separation between the levels. We now remember eq. (3.9b), thephotoabsorption cross-section averaged over the energy width that is allocated toone upper level. In calculating the quantity corresponding to eq. (3.9b) for adense plasma we may expect that the decrease in the transition probabilities,Fig. 9.3(a), or the corresponding decrease in the absorption oscillator strength,

FIG 9.3 (Continued)


FIG 9.3 (a) Changes in the radiative transition probabilities of neutral hydrogenin a dense plasma. (Quoted from Roussel and O'Connell, 1974; copyright 1974,with permission from The American Physical Society.) (b) Changes in theenergy positions of excited levels (with respect to the ground state) and thelowering of the ionization potential of hydrogen in a dense plasma. Thequantity corresponding to eq. (3.9b) is given by the vertical positions ofsymbols (+). The overall absorption spectrum depends little on the plasmadensity. Ion sphere model. (Quoted from Hohne and Zimmermann, 1982; withpermission from IOP Publishing.)

might be compensated by the decrease in the energy width. Figure 9.3(b) shows thephotoabsorption cross-section, or the absorption oscillator strength, averagedover the energy width allocated to each upper level by the vertical positions of thesymbols (+). This figure also shows eq. (3.9b), the corresponding quantity for

264 DENSE PLASMA

the low-density limit, with the smooth curves, and as its extension, eq. (3.13), thephotoionization cross-section, for energies above the "ionization limit". It isremarkable that the overall features of the absorption cross-sections for photo-excitation and photoionization depend little on the plasma density, especiallyfor plasmas with a Debye length larger than 2QaQ. This is in sharp contrast tothe strong dependence of the boundary energy between photoexcitation andphotoionization, or the lowering of the ionization potential, on the plasmadensity.

We saw that the Debye shielding modifies the atomic wavefunctions. Thesame shielding would affect collision processes, i.e. in a collision process thecharge of the incident particle, an electron or an ion, is shielded by the nearbyplasma particles, reducing the effect of this Coulomb field on the ion which isacted upon by this incident particle. Thus, the excitation cross-section, forexample, may be reduced in a dense plasma. Figure 9.4 shows an example ofcalculations for excitation of the ground-state hydrogen-like ion: (a) is forexcitation of ls^2s, and (b) is for ls^2p. Both the collision strengths, eq.(3.25), and thus the cross-sections, decrease with a decrease in the Debyelength. However, it is noticed that the plasma effect is more salient for theexcitation of 2p than for 2s. This is interpreted as: in excitation of an opticallyallowed transition, collisions with large impact parameters are relativelyimportant, while for excitation of an optically forbidden transition collisionswith small impact parameters are dominant. Even an exchange of the incidentelectron and the target electron could take place when the two electrons comeclose. Thus, the Debye shielding affects the former collisions more stronglythan the latter.

So far, we have relied on the picture of "screening" or "shielding" to expressthe effect of a dense plasma environment. However, the real situation for an ion ina dense plasma may be substantially different. The perturbation by plasma par-ticles is strongly time dependent, and sometimes even violent, for instance. It isthus questionable whether the above picture describes properly the real situations.Unfortunately, a realistic treatment of a dense plasma environment involves verycomplicated procedures, e.g. correlation and coherence between the collisionscannot be neglected and collisions involving more than two particles becomeimportant. Only a limited number of attempts have been made in this directionuntil now.

An alternative approach is to treat a transition of ions as induced by a fluc-tuating electric field originating from the thermal motion of plasma particles. Anexample of such calculations is shown in Fig. 9.5. This is for a proton-electronplasma with Te = 340 eV and ne = 1029 m~3. The transition of a hydrogen-like ion2Si/2^2P3/2 is calculated for Ne9+ or Ar17+ ions immersed in this plasma.Figure 9.5(a) shows the frequency spectrum of plasma density fluctuationsweighted by the generalized oscillator strength of the transition. The abscissa is theangular frequency normalized by the electron plasma frequency. The frequencycorresponding to the energy difference between the lower and upper levels is shown


FIG 9.4 Dependence of the excitation cross-section (collision strength) on theplasma density. Hydrogen-like ions. The abscissa is the collision energynormalized by 2z2R. (a) Is —> 2s transition; (b) Is —> 2p transition. (Quoted fromHatton et al., 1981; with permission from IOP Publishing.)

with the arrows. Figure 9.5(b) shows the temporal development of the upper-levelpopulation «(2P3/2) for the initial condition of «(2Si/2)=l and K(2P3/2) = 0 att = 0. No transitions other than 2Si/2^2P3/2 are considered in this calculation.The effect of coherent excitation is manifested in the case of Ne9+. Also shown are

266 DENSE PLASMA

FIG 9.5 Excitation of 2S i /2 —> 2P3/2 of hydrogen-like ions in a dense plasma withre = 340eV and «e=1029 m~3. (a) Frequency spectrum of plasma densityfluctuations weighted by the generalized oscillator strength of the transitions,(b) Temporal development of the upper-level populations (solid lines).Dashed lines show the solution of the rate equation. (Quoted from Kitamuraet al., 2000; copyright 2000, with permission from The American PhysicalSociety.)

the solution of the rate equation with the dashed lines; this is equivalent toassuming the Born approximation for the excitation cross-section.

9.3 Multistep processes involving doubly excited states

In Section 3.5 we noted that any of the levels of an ion (not an atom in thiscase), irrespective of whether they are the ground state or an excited state, are

MULTISTEP PROCESSES WITH DOUBLY EXCITED STATES 267

accompanied by a Rydberg series of excited levels of an ion in the adjacent lower-ionization stage. See Figs. 3.17(a) and 9.6. Suppose that the singly excited ion is ahydrogen-like excited level, say, the 2p level. This "normal" level is the ionizationlimit of the doubly excited levels of, in this case, helium-like (2p,«/') with thespectator electron «/'. As we have seen in Section 3.5 these doubly excited levelsplay an important role in the recombination process of the ground-state ion, Is inthis case, through dielectronic recombination. In a dense plasma other featuresemerge.

DL excitation and deexcitation

If an ion in a doubly excited level, say helium-like (2p,«/'), is immersed in a denseplasma, a collisional transition may take place before this ion spontaneouslydecays through autoionization, (2p, «/')—> ls + e(£7), eq. (3.45), or a stabilizingradiative transition, (2p,n/')->(ls,n//) + /"', eq. (3.48). See Fig. 3.17(a). If n islarger than 2 the core electron, 2p in this example, is more tightly bound than thespectator electron «/', so that electron collisions would induce a transition morelikely in the latter electron. As we assumed in our discussion of singly excited ionsin a plasma, we assume here again that the doubly excited ions with different /' arepopulated according to their statistical weights, so that n is enough to specify thelevel. We thus denote a doubly excited ion with a core electron p and a spectatorelectron q as (p, q). Here p or q means the principal quantum number. In thefollowing we simply write pq instead of (p, q). See Fig. 9.6. If the temperature isnot very low, the most likely collision process is excitation pq + e -^p(q + 1) + e, asin the case of a singly excited level, eqs. (4.9) and (4.30).

In Chapter 4 we divided the excited level population into two components: theionizing plasma component and the recombining plasma component. We proceedhere in a similar spirit; in the ionizing plasma in Fig. 9.6 the ground-state ion has apopulation but the singly excited ion p has no population, and in the recombiningplasma, the population of the ground state ion 1 is absent. We thus divide thepopulation of the doubly excited ions into the corresponding two components.What we are interested in here is the "ionization" process and the "recombina-tion" process through these doubly excited levels rather than the population ofthese levels itself. We may thus adopt a rather crude approximation. The formerprocess consists of the series of processes

See Figs. 4.8, 4.9, and 1.10(a). In eq. (9.5b) we omit "e" which induces the ladder-like transitions. This series of processes, eqs. (9.5a) and (9.5b), is a collectiveprocess which reduces to net excitation, 1 + e —>/> + e. Thus, in a dense plasma, the

268

FIG 9.6 Schematic energy-level diagram of the ground state 1 and an excitedstate p of hydrogen-like ions and helium-like Rydberg states Iq converging tothe ground state 1 andpq converging top. Levelpq is populated by dielectroniccapture and depopulated by autoionization, radiative transition, including thestabilizing transition and collisional excitation.

direct excitation is supplemented by this indirect excitation through the doublyexcited levels. Figure 9.6 schematically illustrates this process. We call this newexcitation process dielectronic capture ladder-like (DL) excitation.

For recombining plasma, very high-lying doubly excited levels should be inLTE with respect to the "ion" density n(p) and electron density ne, or we shouldhave n(pq) = Zp(pq)n(p)ne, where Zp(pq) is given by eq. (2.7) with respect tolevel p, i.e. the statistical weight g(p) for the ion and the ionization potentialmeasured from the position of level p. See Figs. 4.17-4.25 and 1.10(b). Thispopulation balance may be expressed as

where we have omitted "e" again. This ion may autoionize:

DENSE PLASMA


This series of processes is nothing but net deexcitation p + e —> 1 + e. Thus again,the direct deexcitation process is supplemented by this indirect multistep process,which is called DL deexcitation.

We now estimate the magnitudes of the rates for these processes. We take anexample of hydrogen-like neon (z= 10) and consider excitation and deexcitationIs—2s and Is—2p. We confine ourselves to high temperature. We define (extended)Griem's boundary level pqo for the doubly excited levels. This boundary is givenby an equation similar to eq. (4.25) with a modification that, on the l.h.s., theautoionization probability, Aa(pq), and the probability for the stabilizing transi-tion, AT(pq—> Iq), are added. See Fig. 9.6. We thus determine extended Griem'sboundary which behaves in a similar manner to that in Fig. 1.10(a). The "ionizationpotential", or the principal quantum number, of extended Griem's boundary levelis given in Fig. 9.7(a).

We first consider the DL excitation. Suppose an electron is dielectronicallycaptured by an ion in level 1 to form one of the levels pq, eq. (9.5a), for whichq > qo- See Fig. 9.6. According to the discussion developed in Chapter 4 con-cerning the ladder-like excitation-ionization, we may expect that the spectatorelectron q may be further excited in the ladder-like excitation chain and finally"ionized" to leave the core ion in level p. We remember that, for singly excitedions, the CR ionization rate coefficient is approximated by eq. (5.8). In the samespirit, the effective rate coefficient for the DL excitation CDL(1, p), may beexpressed by

It is noted that the Saha-Boltzmann coefficient in eq. (3.46), written as Z^pq)here, refers to the ground state 1, and is related to Zp(pq) by

where r&(pq) is the dielectronic capture rate coefficient as defined by eq. (3.43) andgiven by eqs. (3.46) and (3.47). Thus, we have

This may be approximated to

FIG 9.7 (a) Extended Griem's boundary level in the doubly excited levels 2sq and2pq of Ne9+ against ne. (b) Horizontal lines: the direct excitation ratecoefficient (referred to the l.h.s. ordinate) and the deexcitation rate coefficient(r.h.s. ordinate) of Ne9+ for the ls-2s transition and that for the ls-2ptransition for Te = 106 K. : approximate DL excitation or deexcitationrate coefficient given by eq. (9.10) or eq. (9.12); and — • — • —:results of numerical CR model calculations of the DL excitation anddeexcitation rate coefficients, respectively. (Quoted from Fujimoto and Kato,1985; copyright 1985, with permission from The American Physical Society.)

where o-ex(lz^>pz) is the excitation cross-section extrapolated below the excitationthreshold. See Fig. 3.11. This equation is just the extrapolation of the excitationrate coefficient from the threshold down to the extended Griem's boundaryenergy. See eq. (3.28). This energy is given in Fig. 9.7(a), and from eq. (9.10) weobtain the DL excitation rate coefficient. The result is shown with the solid line inFig. 9.7 (b). This figure shows the excitation (and deexcitation) rate coefficient forthe direct process with the horizontal lines for a particular temperature of thisfigure. Under the condition of this figure, the DL excitation rate coefficient bothfor Is-2s and Is — 2p becomes larger than that for the direct excitation in higher-density regions.

Figure 9.7(a) shows that with an increase in electron density the boundary level2s(/G comes down faster than 2pqG. This is because the former levels lack thestabilizing transition. The autoionization probability is of similar magnitude forboth the 2sq and 2pq levels. This latter statement is understood as follows; theautoionization probability is approximately given by the threshold value of thedirect excitation cross-section (Fig. 3.11: on the ordinate scale they are 0.45 and1.9, respectively, for 2s and 2p) divided by the statistical weight of the doublyexcited level, eq. (3.47) (1:3 for 2sq, and 2pq), Thus the resulting autoionizationrates are only slightly smaller for 2sq^ Is than those for 2pq^ Is.

We now turn to the dielectronic capture ladder-like deexcitation. We saw inSection 4.3 that the levels lying higher than Griem's boundary are in LTE withrespect to the density of the next ionization stage ion and electron density, eq.(4.47). At high temperature, Byron's boundary plays no role. See Fig. 1.10(b) andeq. (4.56). We assume a similar situation for the doubly excited levels: levels pqwith q>qo have LTE populations with respect to n(p) and ne, and those withq<q<3 have low populations. Then the DL deexcitation rate coefficient FDIj(p, 1)is approximately given by

Thus, the rate coefficient for DL excitation, eq. (9.10), and that DL deexcitationobey the principle of detailed balance, eq. (3.31) or (3.31a). The solid lines in

By using eq. (3.47) with eq. (9.9) we approximate eq. (9.11) as


By using eqs. (3.42)-(3.44a) we may rewrite eq. (9.8) as

272 DENSE PLASMA

Fig. 9.7(b) also show the DL deexcitation rate coefficients, which are referred tothe right-hand side ordinate.

Figure 9.7(b) includes the results of detailed numerical calculations similar tothose for the rate coefficients of collisional-radiative (CR) ionization andrecombination, as done in Chapter 5. They agree reasonably well with the aboveapproximations.

CR recombination of excited ions

In the above subsection, we assumed high temperature. We now consider thesituation of low-temperature recombining plasma, i.e. we start with level p ions ina low-temperature plasma. The electron density may be high.

In Chapters 4 and 5, we reviewed the features of the low-temperaturerecombining plasma by taking the example of hydrogen. As Fig. 1.10(b) sche-matically shows Byron's boundary, eq. (4.55) or eq. (4.56), plays an importantrole; when the electron density is high so that Griem's boundary lies below Byron'sboundary, excited levels are divided by this latter boundary. Levels lying higherthan this boundary are strongly coupled with each other and thus to the con-tinuum state electrons, so that their populations are in LTE with respect to the iondensity and electron density. Lower-lying levels than Byron's boundary are in theflow of ladder-like deexcitation. See Fig. 4.24. Another important feature was thatfor higher electron density for which pG </>B holds, the flux of the CR recombi-nation is controlled by the collisional deexcitation flux through Byron's boundarylevel, eq. (5.26). See Fig. 5.6.

The situation should be similar for doubly excited levels. For simplicity weconsider the high-density limit first. We may conclude as follows. We considerpopulations in the levels pq with q > gB. These populations may be lost byautoionization, and this loss is replenished by recombination of the ions p,resulting in DL deexcitation, as we have seen above. In the present case, the lowerend of the integration of eq. (9.12) should be replaced by E^, the energy of Byron'sboundary pq#. Another flux of electrons flows downward through Byron'sboundary pq#. This flux of electrons out of the group of levels pq with q > q# iscompensated by the influx of electrons from the ion p, again. In other words, bothof these fluxes contribute to depopulation of the ion p. It is noted that themagnitude of the second out-flux may be approximated by the CR recombinationrate coefficient for hydrogen-like ions, which should be rather close to that forneutral hydrogen as shown in Fig. 5.4. We here assumed low temperature so thatpB is large.

An approximate calculation of these processes are made for lithium-likealuminum ions (the ground-state configuration is Is22s) concerning researchwith an x-ray laser. An example of the results is given in Fig. 9.8. This figure issimilar to Fig. 9.7; the direct deexcitation rate coefficient is shown for 3d —> 2sand 3p —> 2s by the horizontal lines and extended Griem's boundary and Byron's


FIG 9.8 Deexcitation rate coefficient of lithium-like aluminum ions in therecombining phase in a dense plasma. Horizontal lines: the direct deexcitationrate coefficient for transitions between the singly excited lithium-like aluminumlevels. Thin lines: extended Griem's boundary and Byron's boundary for 3pqand 3dq doubly excited levels. the DL deexcitation rate coefficient for3d —> 2s; for 3p —> 2s; — • — • — the CR recombination ratecoefficient which is common to the low-lying levels. (Quoted from Kawachiand Fujimoto, 1990; copyright 1990, with permission from The AmericanPhysical Society.)

boundary for the doubly excited (3dn/') and (3p«/') levels are shown with thethin lines. The DL deexcitation rate coefficient is given with the thick solidand dashed lines. The CR recombination rate coefficient is given with thedash-dotted line. In this low electron temperature (remember that the reduced

274 DENSE PLASMA

temperature in the present example is re/z2~450K, where z is 10), the last loss

mechanism is predominant at higher densities. Figure 9.9 compares the calcu-lated population distribution over the singly excited levels in a Boltzmann plotby the conventional CR model and that with these additional recombinationand all the deexcitation processes included. The difference is substantial, espe-cially for low-lying levels. This figure also includes the result of experimentallydetermined populations, indicating good agreement with the calculation whichincludes all the above processes. Thus, this figure demonstrates the validity ofthe above theory.

FIG 9.9 The populations of a recombining lithium-like aluminum plasma in theBoltzmann plot. A: result of calculation by the conventional CR model.O: result of the CR model calculation with the DL deexcitation and CRrecombination processes included. 0A: experiment. (Quoted from Kawachiet al., 1999; with permission from TOP Publishing.)


Decrease and disappearance of resonance contributions toexcitation cross-section

In Section 3.5 we introduced the resonance contributions to the excitation cross-section by taking the ls^2s transition in hydrogen-like ions as an example. Seeeq. (3.51) and Fig. 3.21. See also Fig. 9.10. Figure 9.11 is another plot of Fig. 3.21;the resonance contributions, which appear as sharp peaks in the latter figure, havebeen averaged over energy, and in the present figure they appear as smooth cross-sections added on the top of the direct excitation cross-section. The energy rangeof Fig. 3.21 is from 1.02 to 1.21 keV. Contributions from the (3s, ri), (3p,«), and(3d, K) levels are shown separately.

It would be natural to assume that, in a dense plasma, the intermediate state,(3p,«) for example, may suffer electron collisions before it autoionizes. Compareeq. (3.51) with eq. (9.5): the first lines are common and, in a dense plasma, instead

FIG 9.10 Schematic energy-level diagram of hydrogen-like Is, 2s, and 3p levelswith the accompanying doubly excited helium-like levels. The doubly excitedlevels 3p« are populated by the dielectronic capture from the ground state. It isdepopulated by autoionization to the ground state and to 2s, the latter of whichresults in the resonance contribution to the excitation cross-section for Is —> 2s,stabilizing radiative transition and collisional excitation which results in theDL excitation of Is—>3p.

276

FIG 9.11 Another plot of Fig. 3.21; the excitation cross-section for Is —> 2s, plusthe resonance contributions through the doubly excited levels 3s«, 3p«, 3d«.These contributions are lost owing to the development of the DL excitationprocess in these doubly excited levels. The corresponding increase in the directexcitation of, say ls^3p, is expressed as extrapolation of the excitation cross-section below the excitation threshold down to extended Griem's boundarylevel. Similar extrapolation is given to the direct ls^2s excitation cross-section. (Quoted from Fujimoto and Kato, 1987; copyright 1987, withpermission from The American Physical Society.)

of the second line of eq. (3.51), eq. (9.5b) becomes dominant. We define extendedGriem's boundary again for the doubly excited levels, say (3p,ri). Then the flux ofdielectronic capture into the levels lying higher than this boundary will enter intothe ladder-like excitation chain to result in the DL excitation of 3p. Thus, thecorresponding part of the resonance contribution to the excitation cross-sectionIs —> 2s is lost. Figure 9.11 illustrates this situation. For a certain electron densitythe part of the resonance cross-section with energies higher than the energy ofextended Griem's boundary is lost. For each of the contributions, (3s, ri), (3p,ri),and (3d, ri), the energy positions of extended Griem's boundary levels are given,and the part of the resonance contributions at higher energies are lost. This partinstead contributes to the DL excitation of the core singly excited level. In thisfigure, the excitation cross-section Is —> 3p is extrapolated to this energy, eq. (9.10),and the excitation cross-section Is —> 2s is also extrapolated below the threshold to

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DENSITY OF STATES AND SAHA EQUILIBRIUM 277

account for the DL excitation introduced in the preceding subsection. It is notedthat, in the electron densities considered in this figure, ne < 1028 m~3, the energy-level structure and the excitation cross-section are almost unaffected. See Fig. 9.1 (b)for the former and Fig. 9.4 for the latter: RQ > 100a0 for the present example.

This density of states is inversely proportional to electron density, and is shown inFig. 9.12 in the positive energy region with the dash-dotted curves for severalvalues of ne. A strong discontinuity is seen at the zero energy. This is the result of

where we use the rydberg units of energy. In Fig. 9.12 the smooth curve in thenegative energy region and the dotted curve connecting with it, representseq. (9.13). It is obvious that eq. (9.13) diverges toward the ionization limit, or zeroenergy. This is a natural consequence of our assumption of isolated atoms whichhas an infinite number of Rydberg levels.

In the preceding chapters the continuum states of electrons were approximatedas free states. An example is the Maxwell distribution of electron energies,eqs. (2.2) and (2.2a). In this case, the density of states is given from eq. (2.5a) withge = 2 and g(l) = 1 as

See eq. (1.5). This quantity is understood as the energy width allocated to thislevel p, which is 2p2-fold degenerate. Thus, we may define the number of states in aunit energy interval, or the density of states, g(p)dp/dE, or

9.4 Density of states and Saha equilibrium

Density of states

In Chapter 2, in considering thermodynamic equilibrium, we derived the Saha-Boltzmann equilibrium relationship. In doing so, we implicitly assumed an iso-lated atom. In the case of a hydrogen atom, the statistical weight of a level withprincipal quantum number p is 2p2. Thus, the number of states of a level over oneprincipal quantum number (remember Fig. 1.11 (a)) is

The energy width corresponding to one principal quantum number is

278 DENSE PLASMA

our approximations which are inconsistent each other: i.e. an isolated atom fornegative energies and free electrons for positive energies.

We now remember that our atoms and ions are in a plasma; we assume aplasma made of protons and electrons. In reality, the electron states with negativeenergies are affected by plasma particles as we have seen in Section 9.1, and theelectrons with positive energies move in the Coulomb potentials of protons underthe influence of other plasma particles. If we intend to resolve the difficulty above,

FIG 9.12 Density of states of neutral hydrogen in the energy region close to theionization limit. connecting to in negative energy: for anisolated hydrogen atom, eq. (9.13); — • — • — for free electrons, eq. (9.14);

in positive energy, extending across the ionization limit to negativeenergies: the density of states on the basis of the ion sphere type model,eqs. (9.22)-(9.26b). (Quoted from Shimamura and Fujimoto, 1990; copyright1990, with permission from The American Physical Society.)

To help justify this reasoning, see the discussions around eq. (1.6). From the abovewe may define the density of states for this angular momentum. The density ofstates for the present E is given by the summation of that over /:

where lc is the cut-off angular momentum which is defined by eq. (9.16) withr0 replaced by R0, For sufficiently high energy, eq. (9.19) may be approximated to

We assign the energy width to this electron state


these effects should be properly taken into account. As we have noted already inSection 9.2, this poses enormous difficulties. Instead, we adopt here a rather crudemodel. We start with a model potential of the ion sphere type for an electron:

and treat the electron motion classically. To be consistent with eq. (9.1), we areusing here the convention that an electron is regarded to have a positive charge e.

We first consider an electron having a high energy E so that its trajectory isvirtually a straight path inside the sphere. Let p0 be the momentum and r0 be thedistance of the closest approach to the proton. We may define the angularmomentum quantum number semiclassically

Then we may assign to this angular momentum the number of states 2(27 + 1), i.e.the number of the directions of the angular momentum (space quantization) timesthat of the electron spin. We have the transit time

280 DENSE PLASMA

Here we have used eq. (7.6) for R0 = pm and E =pQ/2m2, We have arrived at anexpression that is exactly the same as eq. (9.14). Thus, by following the aboveprocedure we are able to obtain the "correct" density of states of free electrons.

For lower energies, we follow similar procedures; instead of the straight pathwe adopt hyperbolic trajectories for positive energies and elliptic trajectories fornegative energies. We calculate the density of states from the transit times of theelectron over the sphere.

We define the units of energy

and the dimensionless energy

Then \X\ 5//2G(o), which may be called the reduced density of states, is inde-pendent of «e. Figure 9.13 shows the reduced density of states of eqs. (9.13) and(9.14) with the thin dashed line for negative energy and the dash-dotted line forpositive energy, respectively.

For positive energy, X> 0, the electron trajectory is hyperbolic as shown inFig. 9.14(a). From arguments similar to those leading to eq. (9.19a) we obtain ananalytical expression

The result is plotted on Fig. 9.13 with the solid line for X > 0. It is noted that thistends to a finite value at the null energy. For slightly negative energies,— 1 < X< 0, the major axis of the elliptic electron orbit is so large that the orbitextends outside the ion sphere. The circular orbit is absent because its radius islarger than the ion sphere radius, R0, See Fig. 9.14(b). By following a similarprocedure we obtain the expression

with

We rewrite eqs. (9.13) and (9.14) in the form


FIG 9.13 Reduced density of states. See text for details. (Quoted fromShimamura and Fujimoto, 1990; copyright 1990, with permission from TheAmerican Physical Society.)

which is plotted in Fig. 9.13 with the solid line for —1 < X < Q . This curve con-tinues smoothly from the curve, eq. (9.24), at the null energy. At X=—\ the radiusof the circular orbit is equal to the ion sphere radius. This is seen from eq. (1.2),i.e. n = ^/Ro/a0, and the energy is given from eq. (1.1), i.e. E(ri) = —Ra0/R0, Forstill lower energies, —2<X<—1, elliptic orbits with high ellipticity extend out ofthe sphere. At the same time, elliptic orbits with small ellipticity and circular orbitstay within the sphere. See Fig. 9.14(c). Thus, the density of states in this energyregion consists of two parts,

with the former contribution

The sum of these contributions, i.e. eq. (9.26), is plotted with the solid line. Thiscontinues smoothly from eq. (9.25) at X=—\. It should be noted that the formerorbits extending out of the sphere may be regarded as continuum states. This isbecause in actual situations no solid wall is present at r = RQ. Rather, the electronarriving at RQ crosses the boundary to enter into the territory of the adjacent ion,

which is plotted in Fig. 9.13 with the dotted line. The latter contribution is given as

282 DENSE PLASMA

and it repeats this migration. For the lowest energy region of X < —2, all the orbitsstay within the sphere (Fig. 9.14(d)) and we have G(a) = I , or the density of statesis nothing but eq. (9.13), as shown with the solid line in Fig. 9.13. Again, thefunctions are continuous at X= —2.

The same plot as the above is also given in Fig. 9.12 with the solid lines. In thisfigure the energy positions of several discrete levels are given with the arrows.For «e=1025 m~3, for example, X=—\ corresponds to n ~ 7(~2v/5). For—2<X<—\, both the discrete states with periodic orbit motions and the con-tinuum states whose orbit extends out of the sphere exist for 5 < n < 7(~v/2 • 5).See Fig. 9.14(c).

The energy X= —2 may be regarded as continuum lowering or the lowering ofthe ionization potential, which has been discussed at the beginning of this chapter.In our model this is given as

Compare this result with eq. (9.4).

FIG 9.14 Trajectories of electrons with respect to the ion sphere for variousenergy regions.


In the above, we followed the semiclassical procedure, i.e. classical electronorbits, quantization of angular momenta and space quantization, and the energywidth given by the transit time. By adopting an alternative procedure, i.e. theBohr-Sommerfeld quantization to the electron orbits, we can obtain a result thatis exactly the same as the above. For details, the reader is referred to the paper inthe references at the end of this chapter.

Correction to the Saha-Boltzmann and Saha relationships

In Section 2.1 we derived the thermodynamic equilibrium relationship between the"atom" density and the "ion" density, eqs. (2.7) and (2.9). In the process ofderivation we assumed the Maxwell distribution of electrons. As we have seenabove, this "free electron states" model is increasingly worse for plasmas withhigher densities. The ionization potential also suffers lowering in these plasmas.We thus have to modify eq. (2.7). As Fig. 9.12 indicates the density of states of the"ion" should be increased in positive energies, and the continuum states withnegative energies should be included. For the "atom" only the discrete statesshould be counted. In other words, the correct procedure to calculate the "atom"and "ion" densities is: in Fig. 9.12, for a particular electron density, we take afunction of density of states and "fill" this function with electrons according to theBoltzmann distribution. We then enumerate the total number of electrons dis-tributed over the discrete states and the number of electrons in the continuumstates. Here we assume our plasma to be electrically neutral. We thus obtain thedensity ratio of "atoms" and "ions". This ratio is expressed as a correction toeq. (2.9). This correction is expressed as a correction factor O(re, «e) to eq. (2.7):

* We give here the correction factor to eq. (2.7), not to eq. (2.9). If we want to enumerate the totalnumber of "atoms" including those in excited levels, it is straightforward to do that. However, for lowtemperature, say Te = 10 K, excited-level populations are almost negligible. For high temperature, sayTe = 6 x 104 K, the ionization ratio [«z/«(l)] is about 104. See Figs. 5.2 and 5.8. In this latter case, thenumber of neutral atoms is extremely low irrespective of whether we include excited levels or not.

t This parameter is sometimes called the plasma parameter. We do not adopt this nomenclature,because we call ne and Te the plasma parameters.

where «(1) is the atom density in the ground state, «i is the "ion" density and Z(l)is the original Saha-Boltzmann coefficient without the lowering of the ionizationpotential included. Figure 9.15 shows the correction factor for several tempera-tures. For low temperatures, the correction is large as expected.*

We now define the coupling parameter1" F, which is the ratio of the averageenergy of the Coulomb potential to the average thermal energy of ions,

284 DENSE PLASMA

FIG 9.15 The correction factor to the Saha-Boltzmann relationship, eq. (2.7).The corresponding values given by Ebeling et al. (1976) Theory of Bound Statesand lonization Equilibrium in Plasmas and Solids (Akademie-Verlag, Berlin) arealso given. (Quoted from Shimamura and Fujimoto, 1990; copyright 1990, withpermission from The American Physical Society.)

It is noted that this parameter is nothing but AX/Win our model. See eq. (9.27).It may be shown that our correction factors in Fig. 9.15 are plotted with a singlecurve as a function of F as shown in Fig. 9.16. In this figure the functionexp(—F) is also plotted; this function is based on the simple assumption that theoriginal state density for free electrons is shifted by AX, eq. (9.27). See Fig. 9.12.It is seen that our correction factor is approximated very well by this simpleexpression.

In Chapter 2 we mentioned the Saha equilibrium. We noted the difficulty ofdivergence of the partition function. This was due to the summation of the sta-tistical weights over the excited levels to infinite principal quantum number. As wehave seen above, the difficulty of divergence is resolved by the present model ofthe density of states.

REFERENCES 285

FIG 9.16 The correction factor as a function of the coupling parameter. Itcompares well with the simple expression for the lowering of the ionizationpotential. (Quoted from Shimamura and Fujimoto, 1990; copyright 1990, withpermission from The American Physical Society.)

References

The discussions in Section 9.3 are based on:Fujimoto, T. and Kato, T. 1985 Phys. Rev. A 32, 1663.Fujimoto, T. and Kato, T. 1987 Phys. Rev. A 35, 3024.Kawachi, T. and Fujimoto, T. 1997 Phys. Rev. E 55, 1836.

The discussions in Section 9.4 are based on:Shimamura, I. and Fujimoto, T. 1990 Phys. Rev. A 42, 2346.

The figures are taken from:Hatton, G.J., Lane, N.F., and Weisheit, J.C. 1981 /. Phys. B 14, 4879.Hohne, F.E. and Zimmermann, R. 1982 /. Phys. B 15, 2551.Kawachi, T., Ando, K., Fujikawa, C., Oyama, H., Yamaguchi, N., Hara, T., and

Aoyagi, Y. 1999 /. Phys. B 32, 553.Kitamura, H., Murillo, M.S., and Weisheit, J.C. 2000 Phys. Plasmas 7, 3441.Renner, O., Salzmann, D., SondhauB, P., Djaoui, A., Krousky, E., and Forster, E.

1998 /. Phys. B 31, 1379.Rogers, F.J., Graboske Jr. H.C., and Harwood, D.J. 1970 Phys. Rev. A 1, 1577.Roussel, K.M. and O'Connell, R.F. 1974 Phys. Rev. A 9, 52.Yamamoto, K. and Narumi, H. 1983 /. Phy. Soc. Japan 52, 520.

INDEX

absorption coefficient 236absorption cross-section 39, 237adiabatic collisions 220autocorrelation function 219autoionization 64

Bethe limit 54black-body radiation 25, 241Boltzmann distribution 22, 26, 92, 109, 202Boltzmann plot 121, 132, 208Bremsstrahlung 211BYRON 10, 105, 109, 119, 129Byron's boundary 129, 133, 144, 164, 165, 192,

194, 272

cascade 99, 101charge exchange 74collision strength 56collisional-radiative ionization rate

coefficient 150, 189collisional-radiative model 95collisional-radiative recombination rate

coefficient 150, 190complete LIE 24, 192, 201continuum radiation 205corona equilibrium 100, 135, 167corona phase 99, 100CRC phase 114, 123, 126cross-section 41, 48, 61, 72curve of growth 238

Debye radius 218, 227Debye-Hiickel potential 258deexcitation cross-section 56dense plasma 257density of states 24, 277dielectronic capture 64, 267dielectronic capture ladder-like

deexcitation 271dielectronic capture ladder-like excitation 268dielectronic recombination 64, 67diffusion 245dipole matrix element 31Doppler broadening 213

effective collision strength 59effective principal quantum number 20, 111Einstein's A and B coefficients 26

electron broadening 226emission coefficient 240equivalent width 238escape factor 249, 253excitation cross-section 48, 66, 71, 73, 78, 264, 275excitation temperature 241exponential integral 47, 115, 123, 161

first Bohr radius 14forbidden transition 31, 54, 229, 264free-free continuum 211

Gaunt factor 37, 43, 45, 211Gaussian profile 213, 233GRIEM 10, 103, 105, 118, 119Griem's boundary 103, 111, 120, 128, 130, 131,

139, 145, 153, 159, 161, 164, 192, 196, 269,273, 276

Grotorian diagram 3, 19

Holtsmark field 214Holtsmark profile 225, 233

impact broadening 219, 223, 230impact parameter 220, 264Inglis-Teller limit 233ion broadening 225, 231ion sphere model 258ionization balance 167, 176, 180, 186, 195, 202ionization cross-section 59, 61, 62ionization flux 156, 179, 182ionization ratio 154, 167, 174ionizing plasma 137, 177, 182, 184, 188, 199, 202,

254, 267ionizing plasma component 9, 10, 95, 96, 171,

173, 175, 177, 181, 186, 253

Kirchhoff s law 241Klein-Rosseland formula 57, 79Kramer's formula 37

ladder-like deexcitation 127ladder-like excitation-ionization 104, 109line absorption 245linear Stark effect 218, 224, 231Lorentzian profile 218, 221, 233lowering of ionization potential 260, 282

INDEX 287

lowering of ionization limit 154, 159, 164LIE 24,92, 118, 143, 191

Maxwell distribution 22, 48, 57, 77, 283Milne's formula 46minus sixth power distribution 105, 129, 183

natural broadening 218, 240normal field strength 215, 231

optical coherence 222optical depth 237optical electron 14optical thickness 237, 245, 252optically allowed transition 31, 53, 264optically forbidden transition 31, 54, 229, 264optimum temperature 169, 181, 202oscillator strength 31, 34, 37, 45, 53, 77, 86, 101,

161, 264overbarrier model 76

partial LIE 24, 192, 200photoionization 42, 45Planck's distribution 27plasma 13plasma microfield 214plasma polarization shift 261population coefficient 92population inversion 121, 126pressure ionization 260

quadratic Stark effect 223, 227, 229, 231quantum cell 17, 23quantum defect 20, 231quasi-static broadening 214, 223quasi-steady state (QSS) 91, 136, 145

radiation trapping 245radiative decay 88, 100, 114, 126, 136,

140, 144radiative recombination 42, 45, 68, 78, 114, 123,

125, 145, 164Rayleigh-Jeans law 27recombination continuum 205, 211recombination flux 157, 179, 181

recombining plasma 142, 177, 182, 184, 190, 192,267, 272

recombining plasma component 9, 10, 95,111, 120, 171, 174, 175, 177, 181,186, 253

reduced electron density 15, 192, 254reduced electron temperature 15, 77, 192relaxation of coherence 222relaxation time 85, 89, 140, 142, 144, 146resonance contribution 71, 275resonance line 20, 48response time 138, 146Rydberg constant 14Rydberg level 20

Saha equilibrium 25, 284Saha-Boltzmann distribution 24, 92satellite line 67saturation phase 98, 102, 104, 109, 116, 118,

121, 125, 126scaling law 2, 15, 76, 190self-reversal 242, 246spectator electron 64, 267spectrum 3spherical harmonic 32stabilizing transition 67Stark effect 218, 228statistical populations 18, 83, 134statistical weight 19, 97, 111, 120, 277Stefan-Boltzmann law 27sum rule 34

three-body recombination 63, 79, 118, 142, 145,160, 190

total absorption 238transient time 138transit time 279transition probability 31, 40, 77, 262

Voigt parameter 234, 245Voigt profile 233, 245

Weisskopf radius 221Wien's displacement law 29

Zeeman coherence 222

plasma spectroscopy

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