Scattering of Electrons by Atomic2 2 6 2

Systems with Configuratiofta (IS) (28) (2P) (3S) (3P)

by

Michael Joseph Conneely

Submitted to the University of London in partial fulfilment of the

requirements for the degree of Doctor of Philosophy

I R. H.O. LIBRARY 1OLA;

No. M Z . BfjCo«\

40, TIE' All j *70

ProQuest Number: 10107270

All rights reserved

INFO R M A TIO N TO ALL U SER S The quality of this reproduction is dependent upon the quality of the copy submitted.

In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed,

a note will indicate the deletion.

uest.

ProQuest 10107270

Published by ProQuest LLC(2016). Copyright of the Dissertation is held by the Author.

All rights reserved.This work is protected against unauthorized copying under Title 17, United States Code.

Microform Edition © ProQuest LLC.

ProQuest LLC 789 East Eisenhower Parkway

P.O. Box 1346 Ann Arbor, Ml 48106-1346

To Helen

ABSTRACT

The close-coupling approximation has been used to compute cross sections for a wide range of processes (electron impact, photodetachment and photoionization). Errors in previous formulations have been pointed out and corrected. The results are compared with previous calculations and experiments and are correlated to recent work on the effects of configuration inter­action.

A description of the computer code used is also given.

ACKNOWLEDGEMENTS

I would like to record my gratitude to my supervisor Professor Kenneth Smith for suggesting the topic of this thesis and for his invaluable guidance throughout the work.

I am indebted to Royal Holloway College for financial support and to the Atlas Computer Laboratory, Chilton, Berkshire and to the University of Nebraska Computer Centre, Lincoln, Nebraska, U.S.A. for use of their facilities.

In addition, I would like to acknowledge the hospitality of the Institute of Computational Sciences, The University of Nebraska where the latter part of this work was carried out. I am also grateful to Dr. L, Lipsky for helpful conversations.

Contents

Chapter I (Introduction)a) Continuum atomic processes in astrophysical problemsb) The Close-coupling Methodc) The Fesbach Formalismd) Configuration InteractionChapter IIa) Wave functions for the (np)^ systemb) Antisymmetrization of the wave functionc) Electron impact cross sectionsd) PhotoionizationChapter IIIa) Photoionization of N and N**"b) Photoionization of Pc) Photoionization of Sij. S, C£d) Photodetachment of Si , S , C&"Chapter IVA description of the code and the numerical methods usedReferencesBound PapersTrial Wave Functions in the Close-Coupling Approximation Photoionization of Atoms with Configurations (IS) (2S)

(2p)G(3S)2(3P)S

List of TablesPage

Table I

Table II

Table III

Table IV

Table V

Table VI

Table VII

Table VIII

Table IX

Table X

Table XI

Partial wave contributions to the excitation cross sections for 0"**N+, 0++Partial wave contributions to theexcitation cross sections for P'*'Partial wave contributions to the excitation cross sections for S+Partial wave contributions to the excitation cross sections forComparison of collision strengths for various (3p)^ ions between present and previous calculationsParameterization of the resonance series in the photoionization of N CP)(a) L=0(b) L=1(c) L=2Parameterization of the resonance series in the photoionization of N CD)(a) L=1(b) L=2(c) L=3Photoionization cross sections for N ( S, P) above all thresholdsPhotoionization cross sections for ( P, ^D, S)Parameterization of the resonance series in the photoionization of P ( P)Ca) L=0 Cb) L=;l Cc) L=?2Parameterization of the resonance series in the photoibnization of P ( D)Ca ) L= 1 Cb) L=2 (c) L=3

52

56

58

60

61

118

121

124

126

128

130

List of Tables (continued)

Table XII

Table XIII Table XIV Table XV

Table XVI

Table XVII

Table XVIII

Table XIX

Table XX

Table XXI

Table XXII

Photoionization cross sections for P ( S) D, P) above all thresholdsPhotoionization cross sections for A&Photoionization cross sections for SiParameterization of the resonance series in the photoionization of S( P)(a) L=0(b) L=1(c) L=2Parameterization of the resonance series in the photoionization of S( D)Parameterization of the resonance series in the photoionization of S( S)Photoionization cross sections for S (3p S) above all thresholdsParameterization of the resonance seriesin the photoionization ofCA(^P)(a) L=0(b) L=1(c) L=2

2Photoionization cross sections for CZ( P) above all thresholds

_ 4Photodetachment cross sections for S. ( S) and CA"(^S)

— 2Photodetachment cross sections for S ( P)

Page133

135136 138

140

142

143

144

146

147

149

s

List of Figures

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Scattering potentials for electrons incident of carbon and silicon.

partial wave contribution^to the ^S-^D excitation cross section of S showing re­sonance effects.Photoionization cross section of N(^S) and comparison with experiment.

partial wave contribution to the photo­ionization cross section of N { P) showing resonance series.2 eP partial wave contribution to the photo­ionization cross section of N( P) showing resonance series.2 pD partial wave contribution to the photo­ionization cross section of N( P) showing resonance series.2 oP partial wave contribution to the photo­ionization cross section of N ( D) showing resonance series.

partial wave contribution to the photo­ionization cross section of N ( D) showing resonance series.^F® partial wave contribution to the photo­ionization cross section of N ( D) showing resonance series.+ 3N ( P) photoionization cross section and comparison with Dalgarno et.al. and Armstrong et.al.

Page50

51

79

80

81

82

83

84

85

86

Photoionization cross sections of P( S) and 88 N( S) .2 oS partial wave contribution to the photo- 90 ionization cross section of P( P) showing resonance series.

List of Figures (continued)

Figure 13

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19

Figure 20

Figure 21

Figure 22

Figure 23

Figure 24

P partial wave ionization cross resonance series2 eD partial wave ionization cross resonance series2 eP partial wave ionization cross resonance series,

partial wave ionization cross resonance series,^F® partial wave ionization cross resonance series,

contribution to the photo­section of P ( f) showing

contribution to the photo­section of P( P) showing

contribution to the photo­section of P( D) showing

contribution to the photo­section of P ( D) showing

contribution to the photo- section of P ( D) showing

Comparison of the present results on the photoionization eross section of S. with the experimental observations of Rich.

partial wave contribution to the photo ionization cross section of S( P) showing resonance series.^P° partial wave contribution to the photo ionization cross section of S ( P) showing resonance series.

partial wave contribution to the photo ionization cross section of S( P) showing resonance series.^P° partial wave contribution to the photo ionization cross section of S( D) showing resonance series.

partial wave contribution to the photo ionization cross section of S( D) showing resonance series.

Page91

92

93

94

95

Comparison of three computed values (in the 97dipole length approximation) of the photo­ionization cross section of A&.'

99

- 101

- 102

- 103

- 104

- 105

;o

List of Figures (continued)

Figure 25

Figure 26

Figure 27 Figure 28

Figure 29

Figure 30

Figure 31

Figure 32

Figure 33

F partial wave contribution to the photo­ionization cross section of S( D) showing resonance series.Total photoionization cross section of S(^S) showing a single resonant series.3S( P) total photoionization cross section.

2 eP partial wave contribution to the photo­ionization cross section of Cl( P) showing resonance series.^P® partial wave contribution to the photo­ionization cross section of Cl { P) showing resonance series.2 eDa partial wave contribution to the photo- ionization cross section of CJt ( P) showing resonance series.Comparison of Robinson and Geltman's cal­culation with the present calculation of the photodetachment cross section.of S^—Comparison of Robinson and Geltman's pre­diction with the present calculation of the photodetachment cross section of S .Comparison of Robinson and Geltman*s results with the present results for photodetachment of CA".

Page106

107

108 110

111

112

114

115

116

n

THE IMPORTANCE OF CONTINUUM ATOMIC PROCESSES IN ASTROPHYSICAL PROBLEMS

The interaction of electrons and photons with matter in astrophysical sources covers a very wide range of conditions from that of interstellar clouds, where the atomic concentra­tions are very low (~8 atoms/cm^), to that of the interior

25 3of stars where the density is very high (~10 atoms/cm ). The need for photoionization and electron excitation cross-sectional values is apparent whenever these processes are significant for the energy balance of a particular system and situation.In this introduction instances where the need for photoionization and electron excitation arises, will be indicated.Planetary Atmospheres

The nonsphere of the Earth is produced mainly by the ionization of the neutral particle constitients of the atmo­sphere by solar ultraviolet radiation, leading to the production of free electrons and positive ions. The electrons initially possess a broad range of kinetic energies. As they slow down by collisions, the electrons cause excitation of the neutral particles and the resulting luminosity is an important component of the dayglow. The excitation processes

e + O C’ p> — » e + o I'D )e % o e n e + O C ’s:) .

which lead to emission of the oxygen red and green lines, together with the fine-structure transition.^

e + O — 9 Ç + O C * / j 0 — (1 .2)

IZ

are amongst the most efficient energy loss mechanisms in the neutral atmosphere.

At high altitudes the electron gas cools more efficiently in collisions with positive ions than with neutral particles and the ,ion gas temperature also uses above the neutral particle temperature.

The rate at which the electron gas loses heat to a positive ion mixture of 0^ He^ and is given approximately by ^JP = S y f n(O )

4 I'l w (H^) ....(1.3)where T, is the positive ion temperature.

The thermal electron gas is removed by recombination process, and the recombination of electrons and positive ions is an important source of heating of the neutral particle atmosphere.

Electrons can also be removed by attachment processes leading to the formation of negative ions at low altitudes.The Planetary Nebulae

Planetary nebulae are clouds of ionized gas surrounding certain very hot stars. The emitted light gives the nebulae a pale green disk like appearance resembling that of the planets Uranus and Neptune, hence the name planetary.

The model usually employed in the analogous of such stellar objects is one of a thin spherical shell surrounding a black body radiating at a high temperature T. The intensity of the radiation reaching the inner surface of the shell can be expressed by

h - IhJf' — 14 r ’- C*- ....(1.4)

where is the radius of the central star and y the inner shell radius of the nebula.

The black body intensity is thus diluted by thé geometrical factor of Rg^/4r^ and since this is very small ( 10 thenebula is exposed and radiation whose density is 10^ timesthat for thermal equilibrium at temperature T.

The primary physical process that occurs in the nebula itself is photoionization due to the absorption of stellar ultraviolet radiation i.e.

4* A — A 4 ^and occurs in atoms in the ground state. The electrons are then recaptured leaving the atoms in highly excited levels from which they cascade to lower levels emitting lines of allows ed transitions. While this recombination occurs primarily in hydrogen and helium, it also has been observed in heavier elements such as oxygen, carbon, and nitrogen.

Collisional excitations are responsible for the strongest lines in most planetary nebulae. These lines are due to the

3forbidden transitions which arise from the collisional excitation of the metastable levels that lie a few electron volts above the ground level. After the electrons have been excited to the metastable levels by inelastic collisions they cascade back to a lower level with the emmission of a forbidden quarter of the magnetic dipole or magnetic quadrupole type, or by a collision of the second kind,

3The number of collisional excitations /cm / sec. from1a lower level n ' to an upper level n , F^^l depends on the

'4

ion density in level n, the electron density Ne, the electron temperature Te, the excitation potential X^^l, of the upper level, the statistical weight of the lower level and the collision strength Q (nn ) of the particular ion and the transition involved i.e.

"4. D/ n ^ /kTc.

v^n-------------- --- (1.5)

where Î2 (n n ), or the collision strength^ is defined by

a(»->') . TTIA)„ Krt --

where Q (n n^) is the cross-section and K ^ = w h e r e V is the velocity of the incident electron. Equating the number of collisional excitations to the number of radiative de-excita­tions enables one to derive an expression, which involves Teand Ne, for the intensity of a forbidden line. F rom the intensity

5 6 -lines Te and Ne may be determined ' .Interstellar Gas Clouds

To obtain information about the abundance of elements in the interstellar medium, it is necessary to study their degree of ionization. The steady state requires that the ionization

I IIrate equal the recombination rate. Denote by N and N the number densities of some ion in two necessive stages of ioniza­tion. The radiative recombination rate n (rec) per unit volume is given by

(/itc ■) z oi (1.7)

!•>

where the recombination rate a is a function of electron temperature Te and is given by

«6 = a s T 3/ g

Iwhere g. denotes the statistical weight X. the ionization J 1potential,Aj (v) the absorption cross section of the jth excited state of the atom in the lower stage of ionization andIIg the statistical weight of the ground state in the next state

of ionization. If it is assumed that ionization is produced only by radiation of intensity I (v) the photoionization rate is

= A-ir I k )I h5 ts/' r ---(1.9)

where A^(v) is the absorption co-efficient for the ground state of the lower state of ionization and is the ionization limit, Thus the ionization equation can be written in the form

N" Ne = _PN ' ot (1.10)

The ionization rate T may be significantly altered by phenomena of auto-ionization.

Its influence on f has been discussed by Burgess et alin their investigation of the lines of neutral aluminum.

pStromgen , who developed the interstellar ionization in its modern form, has shown that the abundance ratio of

Il»

could be deduced from the ratio Call/^^^ by

Nc« /N Htt - C N.l'i-A I [ +f^) +(£) NalJ^ . . (1.11)

which leads to the curious results that the abundance ratiois about 0.03 compared to its value of 0.7 in the sun andother stars. It is possible that the photoionization rates

9are in error because transitions in Na and Call from their ground states to auto-ionizing states have not been taken into account. Possible transitions of this sort are shown in the diagram

ip* 3S

2 p % S - ’ P**

zp 'S xP^3S * 5

3 ? ^ 3 d

3 p % d V 5 ‘P

3/ 'ssp^4-s *5

N Ca IIGolgberg and Goldberg and Dupre^^iave drawn attention to severalastrophysical consequences of auto-ionization. The inverse process called delectronic recombination has been shown by Burgess^^to play an important role in solar corona.

Other atomic processes of importance in the cooling inter- sellar gas(a) atom molecule collisions11

H + H2(J=0)->H + (J=0 )(b) molecule-molecula collisions

(1.12a)

H2(J=0) + + H2(J=2) (1.12b)

17

(c) Excitation of the fine structure levels of Si"*" and c ’*’ .

Stellar AtmopsheresThe stellar atmosphere problem may be defined in terms

12of the transfer equation^ d. % / = - K/ [ I k - Cr)] (1 13)

d rthe equation of radiative equilibrium.

y (1.14)and the equation of hydrostatic equilibrium

j. - C ....(1.15)where B is the Planck function, J^the mean intensity

- 4TT J* ^ and being the flux,

Pg is the gas pressure, Pg=NKT whereVN is the total number of particles per unit volume and N= N .

For a given value of Pg and T one may calculate using the Boltzmann and Saha equations and calculate the absorption coefficient.

(PgT)=^N^A^(v) (1.16)assuming all the absorption cross section A. to be known. For principle one may solve the foregoing equations and hence obtain a complete model for the atmosphere. The unknowns Te and g and chemical composition have to be adjusted so as to obtain agreement between the calculated and observed star spectrum.

To explain the observed absorption in the solar atmosphere 13 suggested that th

negative hydrogen ion H .

13Wildt suggested that the excess could be attributed to the

12

Vardya^^ has compiled bound-free absorption co-efficients of CA and other negative ions which are an important source of opacity in the atmospheres of cool stars.

hi/ + A” — » A €Also significant contributions might arise from the bound-free absoprtion of ions

4* ^

In order to explain absorption further in the red one must include the absorption due to the free-free transitions.

14 4- e 4r Vll/ --=7 H 4- eThe absorption co-efficients due to free-free transitions

in H were evaluated by T. Ohiiora and H. Ohmura^^. Similiar work has been done recently for nitrogen and oxygen by Mjols- ness and Euffel.^^

19

The Close-Coupling ApproximationIn order to calculate the scattering cross section, within

the framework of non-relativistic wave mechanics it is necessaryto approximate the Schrodinger equation. One of the mostuseful approximation schemes is to expand the overall wavefunction of the projectile plus target in terms of the completeset of eigenstates of the target Hamiltonian. The method,

17called the close coupling approximation , was first introduced18 19 20 21by Massey and Mohr , and has been since shown by Fesbach ' '

to give rise naturally to resonance of the closed channel type.In order to make the method numerically tractable only a fewof the lower stationary states are retained in the expansion.Apart from approximations inherent in the choice of eigenstatesfor complex atoms this is the only approximation made in themethod.

The Hamiltonian for an electron colliding with an atomic system having N electrons and nuclear charge Z, neglecting magnetic and relativistic effects is

^ N . . . . a . ™

H.iii = - icv;- * ....(1.17b,where

fîj - 1 1; ' I (1.17c)

Since spin orbit coupling is neglected the total orbital angular momentum and total spin are separately conserved, the calculation

Zo

may be simplified by using a representation which is diagonalin L, S and Jf, The unsymmetrized wave function for the N+1

2 2electron system may be written

....(1.18)where X denote the co-ordinates % * » » » %î+li.e. the space and spin variables of the Kth electron. We now expand the continuum wave functions f j w*)in terms of a central fold type function.

y ® " ’’ "tf-r ■ '‘-«IHence

(1.20)

The angular and spin parts of are now coupled tothose of 2S***) to give a new basis functionwhere F denotes the complete set of quantum numbers.

r ; f L SH^ 7r ]Since

» 4 « T L M,y ( S r i I s «s) '/'(p.- 5 ,«h3 • • • • (1-21)

we have ^

^W4|) ( i h ) ____ (1.22)r*-!where

F ( r „ v , ) - lr «I.MT . ' ' '

% 4 -Sj *MT

2t

In the asymptotic region we require

F / " ~ ....,1.74,The S-matrix is defined by the relationship

= Z S p p* A p i .... (1.25)

Hence

F.C) - r A l V b ' ê * S r . é " " )(1.26)

9p X kpT - J ^,*ir - 4- ÜAQ [ - J 3Kp 0 3 *• ' (1.27)

We define a new radial function by the transformationVr,.-

a . J r 1where •'k/J “ ( T + S J ....(1.29)

28)

With this definition the F^g will be real everywhere and will have the asymptotic form

~ C Sw|j F)W(%( + >0 (1.30a)

K y < OConsequently

- I (1.31)-- Z " G,.,— • viv ^ r. r. «I

*Ci V* 4- Tm. ü-«« V r<« 4-* "aT T^T *îv 4- a A T* +-Vi 3aFor the systenf initially in state thh wave function is

r XNfD = ^ • ' w * . — d. 32)Finally we construct a properly antisymmetrized wave function

W4I-K . ^

(1.33)

zz

Where X - 25 ' “Ï ^ * * * '2f^K-i î i c n * 2T #Vfi

The S-MatrixThe elements of the S-matrix are defined in equation (i,25)

in terms of the amplitudes of the ingoing and out going waves. It also may be obtained from the R-matrix

S = ^ — (1.34)3 - i R,

The fact that it is unitary and symmetric means that it can be diagonalized by a real orthogonal matrix U

U S U - G .... (1.35)where the eigenphase shifts Ija are real. The same matrix 0 will also diagonalize the R-matrix

\J flu r i r ^ ___ (1.36)The transition matrix is defined by

T = S - T ___ (1.37)The total cross section for the transition L£-S£^LjSj is defined by i ,

crC*-: ; . . . . d . 3 8 )

The Radial Equations ^We consider the integral

lii = j J(n«')cIXv., ^ (1.39)

where N | — (1.40)Equation (1.39) can be written in terms of the unsymmetrized functions as

l i i - icM*o Xh *»*o L E 1K ^ j x • • • • ( 1 . 4 1 )

23

using the symmetry with respect to interchange of variable. The atomic eigen functions satisfyJ x„,) V'(V,' X w ) = O ____(1.42)

Thus equation (1141) becomes using eq. (1.42)

I,j . { " z F , f yJ r r, + ] F ( j ' rF ^ o C f A r ....(1.43)-Io (r )where the potential involves direct and exchange inter­

action.We now consider variation of the function F about the

exact solution which satisfy the boundary conditions,%F,.j ....(1.44)% K ' b Cos©4 , r ®o

The corresponding jvariatlou in I is% I =. J / [ & F V F 4 F V S F ] c r ^

* 2 ^J^.... (1.45)using the boundary conditions (1.30) and (1.44) we obtain

hi-- a J ^ F V r W f 4 i S k>0 *• (1.46)

Thus the variational principle (1.47)

for arbritary variations of F subject to the appropriate boundary conditions, leads to the c<^pled equations

48)for the radial function F. These equations will be discussed further in Chapter 2.

Thé Fesbach Formalism

The formal properties of the solution to equations (1.48)can be conveniently discussed using the projection operator

20 21ofrmalism introduced by Fesbach ' . We write our coupledequations formally as

(1.49a)(1.49b)

P [ H - E ](P+GL) = 0

where P projects onto the open channel subspace and Q projects onto the closed channel subspace.The operators P and Q satisfy the relations

? + 6L = » , ?■*■=?, gO- = gi. Pet=oWe now formally solve e-q. (2 7b) for Q y

s - a. H P y — (1.50)Equation (1.49a) then gives

P [ H - P H C c - - ^ _- q^ ” '' - 6 ] P V = 0 ....(1.51)The term P(H-E)P in equation (1.51) is just the close coupling operator obtained by retaining only the open channels in the expansion. The remaining term is just an optical potential. Denoting the eigenfunctions of the operator QHQ by we have

....U.52,The operator QHQ has in general a discreet spectrum plus

a continuum starting from the lowest threshold in the closed channel subspace. It is the discreet spectrum which corresponds physically to an electron bound in the field of an excited atom

or ion, that gives use to closed channel resonance solutions of equation (1.51).

The optical potential is now expanded using the andconsequently equation (1.51) becomesp r n - 2 ....a.53)

n f w — ^We wish to consider the behaviour of the solution in the

neighbourhood of an eigenvalue of QHQ. Equation (1.53) is written as

B- (1.54)

"here , _ y p w a l i n X ^ j R H P

Equation (1.54) can be formally solved to yield

where G and \jj are the Greens function and regular solutionof the operator H^-E. Multiplying eq. (1.55) on the left by^^^GlHPwe obtain< S q H T Y Z C ^ 3»«Q.HTVt>_________

* I - -1- G(.HP6- ....(1.56)^ -Cjj »

and substituting in eq. (1.55) gives

PJg +C "" — ^5

whereA . =

(1.57)

(1.58)is the shift in energy caused by the interaction with the continuum.

24O'Malley and Geltman have shown that for a two electronsystem, that asymptotically equation (1.57) becomes

lo

;ysrem, max: asyiupuuujL a.-i.j._y

= U It.' v /p .) (4 r i - 4.+ du/)

® -p /, , r k % < ' • • • •

i «PSPJ^where _p ,

. .. (1.60)£- S'^S

In equation (1.59) the resonance position Ef can be seen to be £f- “ ^5 *** . The background phase shift is denoted

by i<| and the resonant part by .The partial wave cross section can be expressed in terms

of the phase shift ~ by

*1 .... (1.61)

In the neighborhood of an isolated resonance this gives

CTj, e ^ /9t*l »j (1.62)

where ^ r "* » û - — Cot Y! , The Breit-25Wigner one level formula follows immediately by putting?^^=0

Configuration InteractionWe now consider an alternative model for describing resonant

or autoionizing states proposed by Fano?^'^^ A review of theore­tical models which predict and interpret resonance phenomena has

17been given by SmithWe consider a discreet state (|) interacting with one continuum and assume the normalization

= I (1.63)The expectation value of the Hamiltonian is

^ 4 Hl<t> > r<C = Vg ....(1.64)

^ VV-.IM\\Pg-> = e ‘ S C e '-e ')

27

where it is assumed for simplicity that the sub matrix in the continuum states has been prediagonalized. Each energy value E within the range considered is an eigenvalue of the matrix and its corresponding eigenvector we write as

- a J d B bg' Ÿ f '----------- ----(1.65)Then pre-multiplying the Schroedinger equation — G ) IPg® Ûby (|) and and integrating yields, having used equations .... (1.62)

Bp a. bg' = B CL____________ ____(1.66)I/,/ a •+ £ bt' - E be>® 28 The second of equations (1.66) can be solved formally to give

(1.67)

where 0(E) depends on the contour of integration round theIsingularity at E=E and P means that the principle value of the

integral is taken.If the states are represented by a wave function with

the asymptotic form"Vf f ^ r + %/E ) J (1.68)

where is a slowly varying potential phase shift, thenthe total wave function has the asymptotic form

bg' "'t'g'----------------- ----(1.69)

where ••••d-’O)Now upon substitution of equation (1.69) into the first of equa­tions (1.66) one obtains

I 1/ (1.71)t.a -4- P + 7(E)Kfqf J £ - 6 'E o

Z B

ThusZ ( « ) - C -

IVj/’-where the shift of the level due to interaction with the

(1.70)

... (1.71)

continuum is defineda l e ) * p r ^ '

J E - E 'The resonance with is

P = 2TÎ |V e |\- 2T îK 'l't|H /f> IThe normalization condition ^ | ^gives

a ' ....(1.72,Use of equation (1.67) yields

be' = P ^CÊ - E'-) ___(1.73)TTV É £ ' E 'The transition matrix element from source initial state z to

a resonant final state is

where ^ - ^ 4. ? J ’Hence we obtain that the ratio of the absorption cross sectionto the modified continuum to that of the unperturbed continuum

(T* . , .... (1.75a)

d e m o u i t J L e u e u i i t - x i i u u i u u u u i i c i l . u j . u h c j

J- . --3 L " K Ÿ g / T / i ? ! ' -

where £ e “ Cot " f —f/" .... (1.75b)X I - ly*.

’ ' l R ÿ l T f ? * > v Ç 7 T 7 7 > -

In the case of more than one continuum say it is possible to find linear combinations of these states such that the discreet state (|) only interacts with one of these

24

combinations say . The form of the photoionization crosssection in this case is

(T = cr« l l L i ÿ + cr* ....(i.76)where the smooth background is due to the states orthogonal to TPi

The Rydberg Series LinesThe case of several discreet lines superimposed upon a

continuum and converging to a series limit at some excited state of the atomic ion is one which is frequently encountered in absorption studies in the far ultraviolet. In terms of auto­ionization effects the Rydberg series of lines can be treated as a set of discrete states which experience configuration inter­action with the continuum. This case has been treated in detail

37 28by Fano and Fano and Cooper . Because of the second orderinteraction among the discrete states arising from their couplingwith the continuum, the discreet states ^ and their energies

will be perturbed by a matrix analogous to g (5 )

J — ....11.//)The states are replaced by new states ^ which in turn

are modified by an admixture of states from the continuum $ = * p y (1.78a)* e - e '9^ = < I ....(1.78b)

IT Vg ,, < Ir I i > (1.78c)

= z v I

30

27Fano and Cooper showed that in the interval between successiveresonances the same line shapes occur for the single discrete

, level and a single continuum. In particular the matrix elementvanishes once in each successive interval. The shape

parameter^ is approximately constant along a series and thewidth decreases in proportion to the spacing between theresonances.

29Mies has extented Fano's resonance theory to includethe interaction of many resonances with many continua, as well as overlaping resonances, from a different approach.as overlaping resonances. Shore^^ has also treated this problem

31

Chapter 2

Wave Functions for the (hp)* systemWe wish to construct from the one-electron wave functions ( f.) *^.1 the wave function of the

2-electron system.^ CP« We havetJ 'S

■‘LSrVHs - /- ‘....(2.1)

and .. /. ft )^ z u M ('« | i /" I " isMjiLSn..H, - p ....12.2)

These two functions differ in the fact that in the firstcase the first electron is in a state with angular momentum il and in the second case it is in a state with angular momentumIil . The wave function for the two electron system,

can be obtained by composing an antisymmetric combinationof the above functions.

W . ? f Ÿ ....(2.3)Noting the fact

€ ♦ € - LM i V lu r (_l)

Li/* i / * ' z (-,) y* I s Ms --- ('4)We see that

^ ceiEl) - ( < (J— ^ ^ L Hg, STherefore we obtain^ , _i (ip u , K ) + (E/ ( J“ uMtSrtj Vx I -WiSft» ‘•'*‘-*'**....(2.5)

where■4* r gU’ii«lt-HtVi f I (2.6)

3%

IIn the case of two equivalent electrons and noting that

....,2.7,

we obtain

"îuH.SH^ = O L+S oJot (2.8)

and the normalization factor is equal to h and not ^//2~,We now wish to write the wave function i '

of a group of n equivalent electrons in terms of a linear combination of the functions srt (S C ) £ L S )

** IIcorresponding to different initial terms L S of the configuration ^. Amongst the . states j" £. 1* | obtained

according to the rules of addition of angular momenta will be some which are forbidden by the Pauli exclusion principle.Only a linear combination of these functions will satisfy the Pauli principle. We write

(2.9)The co-efficients S ) are called the fract­

ional parentage co-efficients : In the case of two equivalentelectrons we have from the previous discussion that all the c.f.p.’s are equal to unity.

In the case of three equivalent electrons we will add a2third it electron to the configuration % and construct the

function ^ *^5 ) according to the rules

33

of addition of angular momenta. The function is anti symmetric in electrons 1 and 2 but not with respect to the third electron. We now couple electrons 2 and 3 and obtain

^ ^ (6QLC%')6.Sl I £,t£lL"s")L^y' * v W (g,.

where the transformation co-efficient ....(2.10)

(jHlLCi')JtLi\i, H ( l“s“)LS)

T* (is+') W(^, i S £ > s' s " J (2.11)The function C^i ■ C b is also built accordingto the rules of addition of angular momenta from the functions

s" 4 " % " ' - Only* II II

those functions for which L + S is even are antisymmetric with respect to transposition of electrons, 2 and 3. Consequent­ly we require combinations of the functions ^^3^

which do not contain functions %. . C S* t ) )with odd values of L + S , This condition is fulfilled pro­vided

S i.: = odel (2.12)

2 3Consider the P term of the p configuration. The forbidden2 3 1 1terms of the P configuration are P, D, S. Putting L=l, S=^

II II

L =0 S =1 in the previous equation we get£(p'*p|?f‘ 's) - i (p**piir'*p) + ÿ (p'*PU P'‘D)- o

(2.13)II IISimiliarly putting L =1 and S =0

I(r’'p|ip'‘s)*i(p'*rn'*‘’p) - ÜS (p *p« p'*i>) = o*> ^ (2.14)

34

We also must have Cnormaltzattohl

Z = /A's' (2.15)

These equations determine the c.f.p.'s up to a phasefactor of -1. The procedure may be. repeated for the add 4S terms. The results are in the following table.

D0

'D -1 1/2~

y-2

The sign of each row in the table was chosen following Racah^^. Although the sign of each row is at bur disposal the relative signs of the entries in each row is hot.

The same method may also be applied to the configuration of the c.f.p.'s of the configuration are known. In

(2.16)Recoupling & and we see that the c.f.p.*s must

satisfy the equationsV fsV SL| i SL)C^"sLii«" sV)6 ' V = o (2.17)

3ST

n IIfor L + s =odd where(s"i." S L | s " t” i S I . ) r / (zC*0(ZCU)& '*<)(Z&"'*0

* W (L " iL ^ : l - ’O W ( s ’Vs£ :S'S’"J ...(2 .18 )As before this system of equations do hot fix the phases

of the eigehfunctions of the different terms.We apply the foregoing euqtions to the term of the p"

configuration. We chose L =l,s''*=0, L =2 S =0, L=2,S=01 1 4 2 2and L S = S, D, and P. Consequently

fp‘"D|}p''^5Xp'‘'SllP‘ ‘0) 2\/3 W(ziai -Oi) WCo i oi = M.®) +

(p** 'DH p' *®)Cp'*D W p' 'b) V30 W(2« 2« ' 2‘) W (0 t 0i = i o) +

fr" 'Dll p’ ‘PXp'^PI) p' 'd) w C 2 > 2" '0 WC« i 01 •• i ®) = o

(2.19)Osing the previously obtained values for ( p 'ts j J p ' t S*J

and evaluating-the Racah co-efficients W(abed:ef) equation (2.19) yields.

, / 3( Y 'Ol] P (2.20)

Using equation (2.15) we obtain.(p‘"p/ip’*p)--f ÿ , (p"' w y o

(2.21)Similarly we obtain the rest of the c.f.p.'s for the

p^ configuration

■ s : ' .D ■ ; v__ 1 ■ =7 5 ' 1

2/T~ 2. ' ; / 3"'.' ■ ■ . 1 .D • • • 2 ' '2'-'.'

1

36

Note: we have chosen (P S| }p^ ^P)“ ,tl to agree with Smith^^33et al, where as Saraph et al, choses the value -1.

Ill, Sobelman^^ has published c.f.p.'s at variance with31the above. He has used the relation given by Racah between

the fractional parentages of the terms of an almost closed shell ^ and those of i.e.

However for i.e. p^ in the case we are considering,the above equation leads to difficulty and a change of sign

31may be necessary which obviously was not taken into account.5However we may use it for the configuration P . and we obtain

3 1 1P D S1/ 1/

/~375 /T~ /I5which agrees with ref. (33). For (P ^P|}P^ S) Smith^^ et al. used the value which is incorrect.

If we denote the antisymmetric wave function for ($fl) electrons by L*. and introduce the vector coupledfunction

Y \ " - ' X S j t .«I "k )

V Y C p ’ijS,- x o y \ , V M

(2.23)then

(2,24)

37

p.."’For our one-electron orbital* . we use theanalytic SCF functions of Roothan and Kelly^^ or Clementi^^.

?r 'where M is the number of terms taken to represent an orbital. The functions are normalized such that

38

Reduction of the integrals Considerable simplicfication of equation (1,39) is

achieved if we impose the condition

= 0 fn <•= I (, 27)However, without this constraint F would contain a componentof the bound p9 orbital. The unconstrained continuum orbital could then be written

p = F + oC? (F \ P) r o (2.28)

Consequently we write"'P (Q. "f’*'LS)

Substituting this into equation Cl.39) yields uponusing equation (1.42) and the symmetry properties of the wave function we may write

^ 4 if;: + L': 4 ifH * L;-

32where we employ the notation of Smith Henry and Burke except that we omit the subscripts which label the particular solution vector we are refering to since the equation®are independent of the boundary conditions imposed upon them. We find

Ci- = < *"") I HN„-E I >

* = p u ['£ c4. - H^ (2.29)

where , % cV p ’= ^ 3; fC3*;*0C^L;+0

L; >i-j -Li^) <•) (2.30)

39We have used the fact that

KJHw+i = + + Z r .cX otZ%

I-Y = J'P«. >; W,^ f| = - H < T . f , i«!r..l|H„-Ejÿjr“ s'i„-)>

- [cÏ0-Voczt.;+OC2 S.+ocifij+1 yai-j - K ^ z s j p ’l/S.-/?p’^A')

C ?* 1} p'-'ij Sj ) wc%- 4 S; : s r f I Pj- 001 A.) IO 01 AO)

Equations (2.30) and (2.32) are exactly the same as equations(19) and (22) of Smith ejk al.

We now wish to evaluate the terms linear' in C. This has32been incorrectly done by Smith et . These authors separated

off a p-electron from $ using the co-efficientsof fractional parentage, which of course is only valid for equivalent electron. Consequently, the exchange interactions with the core electrons were not accounted for. We writeif = Cj < • w f * ' L O >

4 Cj (2.33)

We take qiLS) a I LS > ;.‘!*:.VA4)

2# 2* Cx+'M !where is a antisymmetrization operator. This functionis normalized. We note that if is a function anti­symmetric in m variable and is already normalized, then

^ M ! w !is completely antisymmetric and normalized. Furthermore

40

< A „ 4 1 ^ | A h 'K> = (2.35)

provided that C ^ ~ OWe also write

X„«) 5 I n «p" C S.L; ) i; SL >- Ÿ6] -‘2-3Sa)

where ^y CO A • *».- ■*« *» ES.t.) SL)% 2(1; '‘v 4 mslS H,) , ?('*’ «‘Otjpfz*Y Cpc*(^|s~‘ M f ^ t; ) % ft I %/Qr, 1 (2.36b)

We obtain, for the first matrix element in equation (2.33) since the right hand side is antisymmetric, upon interchanging t

- N*. y?M- < A .. <p(if W <P6^h y f 0 zp'f A )fA ;... I•J7U, ' rZ'

Using equation (2.31) we see that the integral over By and E will give integrals J F| which we set equal to zero

Isince we require the F s to be orthogonal to all bound orbitals.Consider the matrix element of the 1-electron operator

H Since[ A ^ , = 0

we can take it through the operator and operta on the function in the right hand side of the matrix element, which is already antisymmetric, so consequently it gives us a factor MI

4f

Therefore the matrix element of the one-electron operator in equation (2.36) becomes

X c4„cp64*i'Oc^(j'/*«')cpCzp’ 'iff-N.i i-^yy^ (2.37)

We cannot mix the p-electrons on the R.H.S. with Is or 2s electrons, or the Is with the 2s electrons as the orbitals are orthogonal. The y s are already antisymmetric so operatingon the R.H.S. give a factor of 2 \ * . Usingequation (2.24) we obtain

< TfCT; &W..) $ t «s’«'V'hs) >

We consider now the matrix element of the 2-electron operator

I ^ “ I Cp(24* !■>«•) cp(2p* \i--- Ntl LS )^• W+i* I

(2.39)We have <► M

oC-* .... (2. 40)

4Z

y —We next consider the contribution of ^ < to the

oC-S ^matrix element. As before we get

I . - r .2 4'in !>...•> I J, t... I * A H♦ 1 (p!zr If " '

Since both sides of the matrix element are antisymmetric in co-ordinates 5 to N we can replace r by

Fq <,4., As before

A N

Aw+, — 7 A (,,.3 • A ( , » - A is y 2\ • • (*'*')!

Consequently equation (2,37) becomes(<|+| ) ^ c Tj i f ' 1 Ç lîV i} I r 1 CpCzf I 5 • t s ) ^

= 1 1 * ,)^ 3 ? Z \ ' s ; ( p h s / f p V s ' ; p 3 C 2 p . + , ; C 2 t V i J c ^ t ; . f i n î

* 2^ (- n * Y r' p' s. )C f f ji*, c " op j X=)K-0 g,oo/hoW6i-*g|’L; •LX)WCit'ri'i (2.38)

We now investigate the contribution of *7^™0 ^ « I W + l j

to the matrix element. As before we obtain 2. l I i i

X (p f3*) CjP Zf IS — WW LS)^As before when we let A^ operate on the R.H.S. of the matrixelement we obtain a factor of NÎ. Also if we try to anti-symmetrize the R.H.S. with respect to particles 3 and 4 and 5through N+1 we get a factor of 21*(qtl)Î Since both sides

43

of the matrix element are antisymmetric in particles 1 and 2 the summation can be replaced by a factor of 2. Thus equation (2.39) becomes

= (1")' %,,! ff ’'VslîA,s,j^ af„CP,.F. R.P.f)

- Z (o),.".!',.) R/P.;,P.pRO^ *

There will also be a similiar contribution from the 2s electrons and in general the contribution from the core will be

Z f 7 > C « ' * O j 5 , C P . ' r F , W - Z # $ ,yJt c C-oo*«t Pv

Therefore the term linear in C becomesIf. 2T C,. J F. rfr* where

V , ( 0 . (TO -' f ( n i l l L r S O [ ( - £ J

Wt’.cÆwJ ® / A '- ÿ / p . ' / P . / . J . 3?.ftt'.Ofei,•■0]I X L.s,»>\s,|i,!.3

^ f 11 oO I ^ o ) C I J Ô0 1 Xo) \A/ Cl • 1 * * L 3 W (I L I LJ ' X3y y ( K p K o

' .... (2.41)37This is the expression given by Smith, Conneely and Morgan

32and differs from equation (24) of Smith et al in as much as

44

the above includes terms which arise from exchange with the core. These terms are absent in reference (32) since was not properly antisymmetrized. We now have to evaluate the term guadratic in C

(2.43), I cwhere is the energy of the rtf L^

configuration evaluated with the wave functions of the np^ conf iguration.

The Hamiltonian may be broken up into single particle operators and 2-particle operators since

H.., = X H w - X T:»

The single particle operators give a contribution

M l * ~ »J (2.44)

(2,45)The matrix element of the®l®ctrostatic interaction between

groups of inequivalent electrons may be evaluated by the methodsq oof de Shalit and Thalmi

(a) The elctrons in each closed subshell will give a contribution

(2.46a)

4?

Cb) From each pair of closed subshells there will be a contribution ,

(c) From the n&^^^ open shell there will be a contribution(q + , f 0 [;2 m'<' h( V (ggûû|**’ /{’M « 7 W n 0 3

. ... (2.46C)IIfor each n I closed subshell. This is independent of thetotal angular momenta of the open shell.

(d) From the np*^^ open subshell we will have the con­tribution

^ = <K) r"*'LS I J fTj 1 hp"LS> J- 1 fn

since

* I •'P* '(l-{ S, ) P p(L*S&) L S >where we use the é“*J. symbols^^ instead of Racah W-co-ef f icients since they are more symmetrical.Consequently

» ) V A f [' i l t i l

” L pp(« 'V-s I i^^,lp'ivi-'rp(‘U .u‘>

46

We note that\ Ip#^ r ? Ls| T '(«■! s/) p p (L,S,) L S >

where A 0 ïTv ^ .

Thereforeg - iC iü.) 2 CC*'isl!r\; !)cr*7sljp\- p)CrU;iyi;h'J

7- t.tjE/> I I . ,

■E ( A s d l i ' V ; ; ; ) VA « . * ' X 2 i - i * o { t ' / k i f i'' L i l l

-11 ; i , K B . î r i t J ) i f ; B - ’ ‘ O '." ) ( " p ' ' ) .

We now use the Biedehharn-Elliott sum rule

= S LjL'i I ) / L; q A >o c • n . . I I A L; 1 1 I I L. ISo finally we obtain '

S= ’ ¥ ’ . 2 1LîLjL'S;SA

. ( r 'l ,; ,It r ' l 's X rh jS .Iff 'T s 'l V c iw c il^ ) [ t ' ^ f

4 1 ' i O i t X U ) IP^Cnr'-Jl+L;4L:-*U

C2.46d)

Equations (2,46 a,b,c,d) qiye us E^^^ The above ex­pressions are equivalent to that given by reference (37), but

32differs substantially form that given by Smith et al . Further­more the expressionE r 1 J E ^ S " E '> i^ (2 .4 7 )

4 7

2is calculated using the same methods, where K1 is the energy of the incident electron in Rydbergs and are the quantumnumbers of the lowest term of the target ion.

In order to obtain better agreement with experiment for the positions of the levels of a Rydberg series converging on one of the excited terms we use the experimentally determined

E ,or theoretical values given byreferences (35) and (36).

However these value should not be used in equation (2.47) since they are inconsistent with bur method of calculating

®N+1.

Derivation of the Radial Equations Application of the Kohn variational principle (equation

(1.47) yields, after analysis equivalent to that of reference (32), the equations satisfied by the radial equations F. • where the subscript ^ refers to the ^ solution of the vector function F.

j à C2.48)

"■“Z j - • i l l * . - - 4 #The direct potential Vij 6 defined by equation (2.30) and the integral operator W,** by equation (2.32) . The x

are unknown Lagrange multipliers as we require the radial functions FjJ for s?waves to be orthogonal to the Is2s and 3s subshells, and for r. I to be orthogonal tothe and np subshells. The numerical solution of these equation^is discussed in Chapter 4.

ResultsThe results for the cross sections for electron scatter­

ing by carbon, nitrogen and oxygen have been discussed by Smith, Conneely and Morgan^^. Henry et al^^ have pointed out that the peaks in elastic cross sections in the results for carbon and nitrogen are due to a low energy shape resonance which is supported essentailly by the angular momentum barrier term which corresponds to i#l. This feature is absent for the corresponding (3p)^ cases iae. for silicon and phosphorus.The reason for this is that the effective potential that our electron with angular momentum equal to unity incident on phosphorus sees, cannot support ànbound state. In Figure 1 we plot the effective potential Y,* against f for carbon and phosphrous.

y/ = ^ ,?.v„r

where ^\\ is given by equation (2.30).

In Table I we present the partial wave contributions toI ++the excitation cross sections Q(n,n ) for 0+, N and O . We

35have used the theoretical values for the energy differences between the terms. Henry et al^^ have used experimental energy differences for the terms and furhtermore have used the Hartree- Fock energy for the lowest term (i.e. equation 2.47) which we

have pointed out to be inconsistent. We note that the dominant contributions to the cross sections come from the p-waves.This is not the case for (3p)^ ions.

We have tabulated the partial wave contributions to theexcitation cross sections for p’*’, and as well as thetotal collision strengths in Tables 2,3, and 4. In this casethe mbst important contributions to the cross sections comefrom the d-waves. For these ions we have used the experimentalenergy differences between the terms. Table 5 gives a compar-sion of collision strengths for various ions between the presentresults and tther calculations. We see that the agreementbetween the various calculations is poorer for the C3p)^ systemthan for the (2p)^ system where excellent agreement was ob~

37tained . This may be due to the large number exchange-terms, which we have treated exactly, i.e. for a particular set of coupled equations there may be as many as 33 distinct exchange terms, all of which we have taken into account. We have cal­culated the collision strength well above all thresholds as near threshold our method of fitting our solutions to the asymptotic form of Burke and Schey breaksdown. Below the high­est term of course our model will predict an infinite number

3 0of resonances. In figure 2 we present the D partial wave contribution to the transition in s'*" which shows a numberof thses resonance profiles.

11

S!

d CM

G10 HS 3W H1 d “=5-

cr>Ljl

en

oeu

0>

00

iù

a>-a:>oKILÎzUJ

zoceHOLü_JLU

r— r - îw (p 10 i— 1— I— I— I- o10 M —

S NI 2*’D il '(Qp-S.)

TABLE IPartial wave contribution to th.e

excitation cross sections for O"*" and o"*"

çz

Ion K

N+3.2

LStt2pC

2pO

2d®

2d°2j,e

Total

2pO

2d®2go

Total

2 s®2pô

2dS

2p®Total

QC^P, D)

.01735

.16525

.00322

.32403

.10370

.00043

.613980(3p^lg)

.06617

.01358

.00002

.07977 0 ( D, Ig)

.04074

.04076

.08325

.03210

.196 85D)

QC^Df 3p)

.04285

.40782

.00795

.80086

.25591

.001061.51563

Q C^S-^P)

1.77379

.36440

.00058 2.13877

Q C^S-^D)

.44254

.44276

.90431

.34867 2.13826

Q ( D> F)

TABLE I(continued)

S3

0++ .2 2p®2pO

2d®

2d°

2p®2p0

Total

2d§

Total

2ge

2pO

2d®

2p°Total

3pO

.02061

.08902

.00312

.16582

.12068

.00039

.399643 1

Q(^Pf SI

,03443

.02346

.00005

.05794QC^Dflsl

.02269

.01647

.04969

.02813

.11698

,00090

.56721

.05230

.22594

.00792

.42090

.30632

.001001.01438

1.09661

.74729

.001671.84557QC^S.^D)

.28468

.20667

.62348

.352951.46778OC^D/^S)

.00062

.34830

TABLE I (continued)

54

3pe

Total

3pO

3j,e

Total

Ipe

3pG

3pO

lD°

IpO

IpO

3po

Total

.07185

.00060

.58056 Q(*S, 2p)

.17406

.02518

.00002

.19928Q(^D,2p)

.03507

.03564

.03135

.08057

.09199

.03211

.06041

.03711

.00328

.00401

.00925

.07723

.49386

.04967

.00042

.39901Q(^P,*S)

.38354

.05550

.00005

.43910Q(^P,^D)

.11179

.11359

.09993

.25684

.29323

.10836

.19286

.11829

.10147

.01280

.02946

.246181.57421

TABLE I (continued)

1Collision Strengths, Q(n,n )

n " 0++ 0+0 (1 ,2) 3.29286 2.54556 1.535100 (2,1) 3.29285 2.54553 1.536230(1,3) .42768 .36906 .526920(3,1) .42775 .36911 .526920(2,3) .42767 .29355 1.905380(3,2) .42765 229555 1.90540

ÇL

TABLE IIPartial wave contributions to the excitation cross sections of P^

Q Cn, n^ ) ^P-^D

^P-^S

^D-^S

LSvr/k .3 .4 .6 .8 1.0.979 .406 .209 .112 .063

2pO .272 . 195 .117 .077 .053

2d® .125 .089 .054 .035 .023

2d° .468 .332 ,195 .127 .088

2p® 2.192 1.594 .950 .588 .358

2p° .002 ,002 .002 .001 .0012gO .mm4 .018 .025 .028 .030

Total 4.052 2.636 1.552 .969 .6162p? .092 .065 .038 ,824 .016

.650 .501 .302 .181 .1052pO .001 .002 . 003. .00.4 .004

Total . 743 .568 .343 .209 .125.162 .018 . 021 .005 ,001

2pO .229 .154 .093 .066 ,0512j e .423 .227 .107 .073 .059

2pO .374 .301 ,218 .168 .134

.130 . 117 .101 .090 .081Total 1.318 ,877 ,5i0 , 402 ,326

5*7

TABLE II Cconti.nued>

üil,2l 10.940 9,49.0 8.381 6.977 5.5440(1,31 2.006 2.045 1.852 1.505 1.1250(2,3) 1.444 1.399 1,401 1.445 1.498

s s

TABLE XIX.Partial waye contrlbutiona to

the excitation cross section for

Q(n,n^) LSïï/k ,3 ,4 .6 ,8 1,0,035 *022 ,007 .001 ,000

.934 ,690 .434 ,300 .218

3.986 2.805 1.694 1.154 ,807

,00.8 .012 ,019 ,025 .031Total 4.96.3 3,529 2.154 1.480 1,056

G" P 3pe 286 .212 .133 ,091 .065

1.557 1.384 1.025 .733 .506

^F® ,001 ,003 .005 .008 .010Total 1.844 1.599 1.163 .832 ,581

^D-Zp IpO ,320 .172 ,079 . .051 .038

^P® .189 .113 ,061 ,040 .030

^P° .456 .201 .056 .019 .010

G P .119 ,073 .041 ,029 . .022

,983 ,583 ,299 ,187 .125

,201 ,139 .088 .063 ,049

,604 .376 .182 .096 .051

S512 .345 .216 .157 1122

S9

TABLE III Ccontxnued)

IpO .153 .189 .197 .163 .120

.045 .039 .032 .029 .026

4 ° 2.144 1.029 .406 ,232 .160

V .119 .093 .072 .059 .050Total 5.854 3.325 1.729 1.125 .8020 (1 ,2) 5.956 5.646 5.169 4.736 4.2240(1,3) 2.213 2.558 2.791 2.662 2.3240(2,3) 9.623 8.861 8.031 7.474 6.932

TABLE IVPartial wave contributiona to the excitation

cross sections for Ci

60

Q (n n )

^P-^S

LSTT/k .3 .4 .6 .8 1.0

2pe 1.015 .771 .507 .359 .2572pO .468 .350 .225 .359 .1172dB 1.097 .814 .529 .384 .289

.008 .007 .007 .008 .009

.002 .002. .003 .004 .004Total 2.590 1.945 1.271 .912 .675

2pO .022 .013 .00.5 .003 .0022d0 .345 .276 .186 .134 .0992pO .000 .000 .00.1 .001 .00.1

Total .367 .289 .192 . .138 .102

.173 .113 ,064 .045 ,039

2pO .458 .280 .145 .092 .064

1.156 .703 .316 .167 .0962ÿO .157 .134 .107 .092 .080

.015 .019 .019 .018 .017Total 2.005 1.295 .693 .453 .332

0 (1 ,2) 6,993 7,002 6.863 6.566 6.0750 Cl,3) .991 1.040 1.03.6 9994 ,918

0(2,3) 1.94,2 1,902 1.711 1.571 1,484

41

TABLE yComparison of collision strengths

Cn,n ) for C3P)3 ions

ION n nl *Present Ref 1431 Ref C44i** Re# 14!P+ 1 2 10.94 6.31

1 3 2.01 1.122 3 1.44 1.11

s"*" 1 2 5.96 3.06 2.021 3 2.21 1.28 0.382 3 9.62 6.22 12.7

CA++ 1 2 3.372 3.1891 3 1.847 1.9672 3 5.743 6.63

Ar++" 2 1.008 1.4321 3 0.326 0.6452 3 3.651 4.920

.3 except for Ci respectively

and A^^^ where it equals ,5 and 1.0

• .0005 in these calculations.

Photo ioni z a t i on

We now wish to use the continuum wave functions defined by equation (2.48) to calculate photoionization cross sections. The derivation we give is essentially that of Henry and Lipsky

We consider a photon of energy hv impinging on a target system of N electrons and Z nuclear charge with quantum number L ^ , M ^ , S ^ , a n d energy described by the wave function

m ( * N . If hv is greater than the lowestionization potential of the target, ionization can take place and the final states can be described by wave functions

Ms whereP ’S. ,1 ^ ^

stands for the channel of the emitted electron. The cross section42in the dipole velocity form is

....U.49)where

CÜ 2 O is the statiscalweight since we assume the target is unpolarized.

- ' A C B c ; '

"^S/1 M represents an outgoing spherical wave inchannel F plus incoming spherical waves in all channels, normal­ized per unit energy range. This condition will be met if we take X, c H H to be of the form

é3

r. T . -i . . . J &.= h'i*’ (#(7 5*., J N ) 2

l~N

T . r é , ~ ‘' ^ n

r' Qwhere

6 (r %N_, î„ ) = Z'p^'' ^ ^ 1"** $ a r 5 p

Here Kp is the wave number of the electron ejected in the Fth channel and is related to the incident frequency and the Tth ionization potential Ip by the Einstein relation

■*'p = ....(2.51)Application of the commutation relation

r r Hi - i t pL - • j " - (2.52)to equation (2.49) yields the dipole length form of the cross section

\ ' l è - ' ‘ £ K 4 j r r: "^ 1 ' (2.53)

The total cross section will be

^ “ Z "n p (2-54)

Consequently we are interested in evaluating the matrix element

% “ ClLo+'XXSo+b ! V s ' r i n ^ ^Where _ :

^ * r = z ^ yr.,*,js, (2.55>

64-

. Z , 0 - » R V r ' ' ' 4 u

P iIf T M ti% denotes the close coupling wave functionswith the boundary conditions given by equations ( 1,30) then

\will have the asymptotic behaviour of equation (2.50). Thus

' C%L.+')C?s.+i) V r "

> ....C2.56)Since

(I 3 ' O l p " = (ifR^) r r * ’

* ( k O ' )rr' ( R (i - ) f n " » i jjf'*'? ^,ip

(2.57)Since we have —— . , , -Iv'lZ L 0 » R ’V ; p . ( R f i * R ' 5 " ) p p " + p V = i"-P* P** Y •*P' M tand since ^ P P

ipY- - Tr”p' ,then the two imaginary sums over F and F cancel each other.So we define the 3-script matrix

A p y - ^ X v * = C ^ r r ' n '

>*■ * ) P fr" ____(2.57)

S j C W ' O..A-'l.I'p"

6S

where M M,

LS = fFT p'

(2.59)

P,-; -f (.’i j ) = 4-(j, i)and Iv]

P (f%) , p. ■K)/ris the 'j of the previous section and

\V‘- £ C/'—))(/:'X (Sp t /"r MyT"rl S 4 s )

.... (2.60)The radial functions used in are different from those usedin ^ and will be designated by primes when adistinction must be made, We may note

c p = A k... g ( i s MV ? 2'

V. (p ( ‘'c If I i ' - '^‘-rV= A ^ _ _ > 0 sMu)Cp'f,s'hO

Thusr, Hrte y CP6 I'O cpas' I-s o(p r / — 2*. (IS/-O Î CG-0 \

r H|i 'y/W p » ^ _ rt,-,« ~I'’ /— (

X ' I ’ X i C«9 1 7 ' n ^

X CLr' (/ r' - V I L M ) D r i /"r' ^ ,

where we again have the antisymmetrization operator of equation C2.35).

6 6

We want to evaluate matrix elements of the forms = < I »r I \T‘ yX ('p-' I ; " ) I £ ' I A p , T("'I") I»;

This is a sum over permutations of products of one electron integrals of the form

<n 'i'x **e' i *"4 M ;) t<h'£'I n« > ‘à 4 1'. . . (2. 64a)

and one of the type

I *<»'«■ t m / i f t " ' 1 "X

X (^'l M , ' / W | 100 !•< o) (2.64b)

wheref . = T tli

the gradient operator andÛ % f ± y ^ r T >1 -s

for the dipole operator. The Clebsh-Gordon coefficient,(f 10 01 4 0 ) is zero unless

l ± \

which is the usual one electron selection rule for dipole radiation. This means that an S-electron can only couple to a

67

p-electron and a p-electron can only couple to an s- or ad-electron. If an s-electron from the L.H.S. of equation(2.63) couples to a p-electron on the R.H.S. there will onlybe three s- electrons remaining on the L.H.S., whereas therewill be at least four s- electrons on the R.H.S. Such terms

w Iwill vanish because they contain inner products between /q S and s . Only a p-electron from the L.H.S. can couple to the R.H.S. through the dipole operator. It can be coupled to the outgoing electron provided that

= 0 Orv 2and it can be coupled to one of the s-electrons if Ofor them there will be four uncoupled s- électrons on each side of equation (2.63). Using the facts and the properties of the antisymmetrization operator and also noting that both sides are completely antisymmetric in co-ordinates S through N we

5 2 \ z \ p ' r t f , ' M r '

X CV i A ' I Ms) C I rtp- H.".)

I S. fis.)

d. 65) ^But

Aç = 0 " P '« "- A ( I - P,sr - p4s")

X

and

4 # O

6S

The integral in equation (2.65) juay be written

< ^ 4 cp'cis'Mt) ) I

X <^(iiM',2) cp(2^|5,(t) fp'p^^

- ( r „ + PtOqpo^'iv^ 1 3 » y^^'(?r

- (?3, -t ?4 < ) I » cfCzs 13 » >

.... (2.66)using equation (2.64) and the fact that ^

Cp(n>ll.2) = C i i ^we see that equation (2.66) may be written as

fl-C ii.'is’ lisas ’) H,-« + C o o o l ^ p ' o )

0< rt-rtol V rt-Hr') (<2p'/ f*, Pp.l ,

Therefore

I , <,>■»■ |.s.x>'<Sr'lxr>‘'"

X fp.' £ ( I I »,-"r l!^o)Clr f.'M ,„.M qin)Mp'Alp'

« (Lp. I rtp. |LoM,)(S^ J, /,/ ”s.->“r|S."«ô)CSr'|:f'r'A ,-/-/|Srts)• « • • \,z # u y / /

where

i v ' ('f'‘xS.lif’" ‘-.'V'Yl l o o U r i»)^^ jp'l (;,^.lF^.p>

- \ 4 5 4 4 X 1 ^ ^ I f,. I ' >> - % % W < z p ; c. / joj Î2S hs 2S> O s zs |lV2V> I (2,68)

69

But

^ C>5p' ( Mr' I \ Mr' "/^r'l^ ^A ' - ^and also S,»

21 C\ M.-Hp' M-Mg \ ép. M-rtp )CLp- (p> Mp' M- M,v | L M )

Mp’ X ( L p - I hç' H ,-M p '| L, Mg)

*■ I L L.fW(,43

X C l Lo ~ M M o ) I M,-M ) C2.69)

= i- o'*” [ 1 I ; fp' L, }

so ^ j^ -- V T <riSZS I'S 2 0 (^p l^ r ) ^ S .s \ ,M jJ ^ ^ L 4 l) (E tg - t0 3 ï

>C-i F ‘''‘T 2. 2-tf-M Mg |i 0 )2 1 # ,., W(LLp. n f p » . . (2.70)Substituting back into equation 2.58 we obtain

f '* = c z & 'ô (2 s .*o [ c i s 2 V | .s 2 0 ^ a p '|v p y ' 'J ^

% CzLH) Z 3',-.',-, r'r” r"nr'r" ' '

'' '^M . rtc C *-*-0 "'M M, I I n , - M ) ^* (2.71)H My

where 3f'p ‘ ( l Y " ’ V > - o, ' C <r c

' - / Iand ^Lg “ H Hp |; I M . - M ) = 3 A ( L I L ^ )

nPo

10

whereA ( L | L-Û ) - ^LLô ^ ^ L t-|

= ounless

[l - 1 i _F L g < ( L f l )rso the final expression for P is

9-1-1 zP*’ " 3 ACLILp) ll<'i»'«'lis2s>Vzp’ l2p> ’ J

K (2 L - H ) ^ ^ , Ap'pi. 3 r "7p- '* r ' ' --- (2.72a)with.

2 : Z ( C n ool^p'D)

X < 2 p ’ I f,o 1 'S> -4-iiilihl5:I><Zp I f,.| 2S?3\ .... (2 . 73a)We now make some approximations to equation (2.73a). Since

{IS z V h S 2 V > -<is’ |i5)><2s' U O ~<lb'UsX2i'|'S>we can neglect the second term in the above expansion since

\ \ \the overlap of the IS and 2S radial functions of the atom I■with the 2S and IS of the ion will be negligible. Also noting that '

the first term inside the square brackets in equation (2.73a) will have a similar factor we may write

A -- ^ c A U S o l ] f '% 'S r O O 'O o l^ ’p'0) iA/(lLp'll -.VLg) -' p'^ [<^P‘ f P, fp'l ^'p'> .... (2.73b)

1 f,„ 1 ZS> j

7/

The effect of the last term in equation (2.73b), called the s-wave core relaxation term by Henry and Lipsky^^ is very small.For example we find thât the photoionization cross section of

2nitrogen at K ^=.3 Rydberg equals 11.98 and 12.18 megabarn when we exclude and include the s^wave term. In view of this we have not antisymmetrized with respect to the Is,2s, and subshells in our treatment of the C3p)^ configuration. For this case we write the s-wave core relaxation term as

‘b o .0 ! * > > > < 3p 1 f . o h s )r'U \ * P n ' > I MO I " / 12. 73c)The factor containing overlap integrals of the atomic wave functions and those of the residual ion in equation (2.72) become in the (3p)9 case

Pf)'' 3 (2.72b)

^ 7 .

Chapter 3

In this chapter we present calculations of photoionization cross sections using equations (2.27) and (2.73) for various atoms and ions of the (2p)^ and (3p)^ configuration. In all previous calculations of these systems, except that of Henry^^ for atomic nitrogen, it has been assumed that the coupling between final state channels is negligible. As we have seen in Chapter I it is this coupling that give use to structures in the photoionization cross sections. Since these line shapes have been the subject of considerable theoretical and experimental investigation in recent years it is interesting to compare the line shapes predicted from perturbation theory, c.f. equation (1.76), with our ab initio calculations. We notice from the tables and graphs in this chapter that the photoionization cross sections in the dipole length and dipole velocity approximations do not agree and that the agreement is worse for the (3p)^ system than the (2p)^. The two approximations should agree exactly if our wave functions for the initial and final state were eigen­functions of the same Hamiltonian-which they are not. The wave function for the initial state is obtained from a Hartree-Fock calculation and the final state wave function of course is obtained from Kohn's variational principle i.e. equation (1.47).HI

4 2There are three terms in this configuration i.e. S, D and2 4P. For N ( S) we consider only the transitions

^ Kv + -— ■» 4 6.43 (...(3.1)■ We note that üotk channels are open so we do hot get anystructure in the cross section

73

In Figure 3 our results are compared to the experimental resultsof Comes and Elzer^^'^^. Our present calculations do not reflectthe resonant structure that is evident in the experimental resultsbecause we have not coupled together the and states ofthe ion. A Rydberg series of levels has been observed by Car-rol^^ in the region 694-612°A and the have attributed the seriesto transitions from the ground state of the nitrogen atomto the Rydberg terms US M p P

We now consider photoionization of the ^P° term. In orderto obtain the total cross section for this term we need the

2 e 2 e 2 econtributions from three partial waves C S , P , D ).In Figure 4 we show the contribution of the partial

wave.

k v 4 N C r ) ' 3 N X ' D ) * ^ « 1ixl"* ( ' S ) t e s • • • ■

If we decoupled these two channels we would get the usual Rydberg series converging on the threshold. With coupling these states show up as resonances corresponding to excited states of the atom *S S . Thesestates have a very short life time as they decay into the ad­jacent continuum. From eguation (1.75) wo see that at the resonant energy the cross section takes the form

The cross section has a zero ininimum at as is evidentfrom the graph. Of course experimehters measure photo transmiss­ion and do not see this zero as they only see the total cross

74

section.The excess transition probability due to the discreet

state is simply •o

f -•O

' i I f . ‘

" % I 4 ^ . ' " '

where we subtract of the background contribution and neglect îîj|S as it is an odd function.

We now discuss the case

kv + M(^P) — 7^ / + fe d.

(3.5)For sufficiently low energy the third channel is closed and we have two continuum states and one discreet state. This gives rise to a series of resonances converging on the n ' C D) thres­hold, The cross section now takes the form (c.f. equation(1.76))

% '-'a I — rl )

This is because we can form linear combinations of the openchannels such that the discreet state interacts with only oneof them. The other then gives the background whichis cpniinupus. We pee from Figure 5 that ihere is a big jump

1in the cross section at threshold where the D channel opens up.

75-

However If we average tiie oscillator strength of these transitions2f/

i.e. add the quantity ^n+l'^^n^ to the background cross sect­ion for each pair of resonances, we see that the total oscillator strength is a continuous function of energy. The dashed line in Figure 5 shows the 'smoothed out' cross section. The aboveremarks merely imply that our calculations are self-consistent.

2 eThe remaining partial wave we have to consider is D (Figure 6) .

U + ^ N T p I f e d

4* é d-+ fe I- NX'S)

.... (3.7)

We see that we have three Rydberg series of resonances converg-1 + 1 ing on the D state of N and one series converging on the S

state. At energies below the threshold there are four seriesof discreet states superimposed on one continuum. Equation

29C3.6> does not apply in this case. Mies gives the formula

0 -. Z ^ / 6 - owhere thessummation is over the number of interacting resonances. However we have used equation (3.6) in curve fitting our results to extract the quantitiesWe expect this formula to hold if the résonances are well

76

separated. The results are tabulated in Tables 6. We also27have calculated the correlation co-efficient

f ' - i t e r . » ---(3,9)where is the continuum state generated by. autoionizationand is the continuum state generated by direct photon ab­sorption from the ground state. Since the infinite sequences of resonances are due to the residual Boulcomb interactions in the final state their positions can be predicted to fit a Rydberg series

-XE oo-r (3.10)

where is the series limit, d the quantum defect, and n the'principle quantum number' of the autoionizing state, d is a slowly varying function of energy as can be seen from the tables

The resonant phase shift can be parameterized in terms of E^,fand the background phase shift (c.f. equation 16%).The^ and r that are extract from parameterizing the phase shift agree very well with those obtained by curve fitting our cross sections to equation (3.6). This provides another check on the consistency of our calculations.

Curve Fitting Frooedure . We wish to minimize the quantity

■ W )I + f ] (3.11)

77

where is the cross section we calculate and E. is the energy at which it is calculated. N is the number of points to which we wish to fit our formula for the cross section. We normally take N to be about 15 and we chose the E^ such that they well define the profile. We let

+the equation

o-,- = o; r/becomes upon multiplying across by the denominator

cr ^ <sr d E ; - a ;

— y — ^ ' 4" ^ 0^ E . k

/...C3XI2)We now make a further change of variables

3:, = -ar 2< r^ Q ^ ~ £ f

c + (1%.

+ ....C3U3)In terms of these variables equation (.3.11) becomes

A = ^ C^; - OJ X/ - * 2. ■^5 ““The extremum conditions

gives us the following matrix equ&tion' V» "v ^ ÊV" "A [6V]EÔ-*' É‘<J“‘Eo- EV E*cr = êV^? ËÔ- N E Ë* .Xjj ivËRr Êô- E Ê» £VH U v j (3.14)

78

N— ^ ^ ^ 2.where ^ . Equation (3.14)together with equation [3.13) gives us the parameters we require. We may improve on these first approximation parameters ^ by making use of differentials.

We let

' 13.15)

where q _ 9i' ? , . 9 *= ^and where the are obtained from equation (3.14). We nowsolve for the We have

= ZfT, - [ f j *✓ W

7 0st <f .... C3,16)which gives us a system of linear equations for the .

This method works very well in most cases i.e. we can obtain three pààce agreement between the calculated cross section and our parameterized cross section. The position E^, and width r, of the resonance is also bbtained by parameterizing the phase shift according to equation (1.60), and they agree very well with those ^iven by the above method.

In figures 7, 8 and 9 we present the and partialwave contributions to the photoionization cross section of the 2D state of nitrogen. The parameters for the resonances shown in these graphs are tabulated in Table 7. We note from figure 9

a

CURVE C U R V EE X P E R I M E N T A L

pi POLE LEMCTH Oipoie VELOCITY

80

Çsl dO QlOH^3BHi g,

w

o

cro

ùJ

Q-

Ll_

crroHsaBHL

o

83

-mI»CO

r-VDiX w

if

o

ii>

L11% WJ’jyyJtggSgBgJgau-' ■-:3=-in»aj3ar*swcesai»37»- •vaNijia5awecss«=o;r.irK=aV84

11

«»V3<rII_j

C»CLvz

%3VJ

^ /V ai^H93^hix s ,

\

.0

cS"

H X"

+ ^ jO 01OHÎ3\λX

Tr

7^■ y II 1«l ■)■ - ■CO

( 8,-O' ) 2om

Kî

60

0T0NS3ÜW1 q, 4 N“ü"ro

10m

oUi

O VUa»

r~\Q-

m u o

Ll

JD

S7

that the series ( O) F is missing or is too

narrow to show up in our calculation. The partial wave con­tributions to the photoionization cross section of nitrogenabove all thresholds are tabulated in Table 8.+N

No experimental results have been reported for photoioniza­tion cross sections for N^, so it is not possible to evaluatethe accuracy of our results as we have done in for NC^S) infigure 3. However in figure 10 we compare our results to a

51similiar calculation by Dalgarno et al and to a calculation 52by Armstrong et al who used a modified Burgess-Seation approx­

imation .The cross sections in tabular form are given in table 9.

PAtomic phosphorus has the same configuration as atomic

nitrogen so it is intersting toccompare our results for the two systems. In figure 11 we compare the results for NC^S) and P(^S). We note the phosphorus cross sections are approx­imately four times larger than those for nitrogen. This is because the atoms are larger. Using hydrogenic functions we see

f --x* --

where f' dénote the average distance from the nucleus. Further­more we note that the slop^of nitrogen in figure 11 is more gentle than that of phosphorus. This can be explained qualita-

o

vO

tively in the ^ollowin# , The cross section apart from geometrical factors will depend on matrix elements of the type

^ P». p I r I 6 cl and ^ ^ . We draw thefunctions in the following diagram

As the energy of Fed increases i.e, the wave length decreases, we see that the Bp wave function cancels out faster thar Pip.This explains the difference in the graph,

2 e 2 e 2 eThe S , P , and D partial wave contribution to photo--2ionization cross sections of P C P) are given in figures 12, 13

and 14, and the parameters for the resonances shown in these graphs are given in table 10. The corresponding figures and graphs for nitrogen are figures 4, 5 and 6 and table 6. The various quantum numbers involved in a photoionization process

3for an atomic system with initial configuration P have been tabulated by Smith^^, The results for photoionization of PC^D) are given in figures 15, 16 and 17 and table 11. We note

Ho cncPMSSyMLL I

JUi

o

ll.1

mo

Û-

U_

UJOà e ,rt,!™ o » » u c *m ii» is « ie B K $ n £ îi» a « s a 3 e a s w rt «--"iK:

CM<4 a s

V#

I—

1 0 H

LU

UU 0

4g@am.wz*y.=

44

. d dO 010HS3ÜH1 S4- f

m

vO

2iL

-in*llvO

w

Ons

o_+ d jO GlOHSayHl <3

(D

«53-

CM

7

mo

(ÙQ

s

CJQ

O(T.

>CD(TUJZUJzoCE5LUd

8UJ3

J Lo<e o oOJmo B I-01

(DiL

II

-inj

j O 0 1 0 H S 3 B H X 0|

Oiô o«s*

•010<M

-01 ‘1-0

u

that the widths, qf the resonances in phosphorus are greater than those in nitrogen. This is because the coupling between the channels is greater in the (3#)^ case. In table 12 we give the partial wave contributions to the photoionization cross sections of phosphorus above all thresholds.

Total absorption cross sections have been measured by 53Kozlov et ai. , while calculations have been carried out by

54 55 56Vainshtein and Norman , Burgess et ai. , Peach and Manson57and Cooper , According to Peach, the absorption of radiation

by aluminum in the ground state ep ^P° may be the cause of the abrupt change in the solar continuum radiation at 2085 °A. In figure 18 we compare our computation of the photoionization cross sections for the 3p ^P° level with the results of Peach and Burgess et We recall that whereas our model is thefull Hartree-Fock treatment for a single configuration, the other models involve extrapolating measured quantum defects into the energy region of interest in the collision problem, as well as employing the Coulombic asymptotic form of the free electron orbital in the matrix element for all T. Within our model the ^P° state of aluminum can phdtoionize into two final states

and^D® with the ejected electron having angular momentum 0 and 2 respectively, with the residual ion having the config­uration of magnesium.

At energies just above the photbionization threshold the algorithiri used to compute the asymptotic solution to the close coupling equations breas ^own. From figure 18 we see that the

97

o' lu

o

Jo H19hï31 BlOdia

98

maximum predicted by the quantum defect method if it existed at all In the close^couplinq approach. In the neighborhood of 0.05 Ryd, the three c&lcuations are in reasonable agreement.At higher energies, one would expect the quantum defect method to become unreliable due to it being an extrapolation procedure and therefore the long tail of the close-coupling approximation at about twice the previous calculated values should be the better values. The Peach minimum at 0.125 Ryd. is probably due to severe cancellation of the matrix elements as previously discussed. In table 13 we present the cross sections in tab­ular form.

SiPhotionization cross sections for the ^P® and the excitedterms of the ground state configuration have been measured

by Rich. In figure 19 we compare the measured photoionization cross sectionsof the ^ level of Si with our calculations. Previous càlculations have been performed in either a hydrogenic approximation or using the quantum defect method of Burgess and Seaton^^ (who calculated p-)-es and 3p- ed amplitudes in Si"*") .As remarked by Rich one would expect the quantum defect method to be inaccurate since it requires extrapolation of poorly behaved experimentally observedcquantum defects of bound levels, the extrapolation naturally getting worse the further one goes beyond the ionization threshold.

From figure 19 we see that there is excellent agreement between the observations of Rich and the calculations from first

010HS3UH1

c:1 1 ^

aioHS3UHi g a

aioHsaynx sdç

. , 3 %,iW3 0/ !$ P N0I103S SSOiJD NOIlVZINOIOlOHdÇ a l ~

too

principles, as carried out in the close^coupling approximationfor photoionization from the ground state of silicon. Justabove threshold as previously mentioned, we have also plottedRich's single energy point for photoionization from the first

1 eexcited state D , which is about 10% below the value predictedby our calculations. Finally we present the cross section for

l ephotoionization from the S excited term of Si ground config­uration for which we have nothing to compare with. The import­ance of figure 19 must be stressed! in view of the very few laboratory experiments carried out on these reasonably complex structures. It is clear evidence of the quantitative correct­ness of the close-coupling results presented in this paper. In table 14 we have tabulated the partial wave contribution to the photoionization cross sections above all thresholds.S

In order to compute the photoionization cross section of 3 ethe P state of sulphur we need three partial wave contributions

C S°, and . The residual ion having configurationhas three states that give rise to structures in the photoioniza­tion cross sections. In figures 20 to 22 where we present theindividual partial wave contributions to the photoionization

3cross section of S( P) we demonstrate the profound influence the excited states of the residual ion have on the cross sections by comparing with the results obtained when those states are neglected Crepresented by dashed lines on the graphs). In figure 27 we have attempted to sum the three partial wave con­tributing to photoionization from the ground state. The para­meters fE>® the various series of resonances in each partial

o

11

k.

I III

(03

<n&

Ll

âO 0 1 0 H S 3W H 1

W D ei-oi '1 ^

fo4.^s jO 0 n 0 H S 3 W.H 1 d

{i v4

I

\/uia ‘i)

n r r

1 w % i9 iW .* '- iC S is 3 = -« « e r . t- s * ,< r ; \ % w : a k f W , . w r r t f j m r 'a 5 A a ( ^ ( ^ : i z ^x ic a rT ;

los

sh

Lc

do aio 3sgiiH,i j oOJ

Ob

(=otim

rY

L O

OD•ro

6rf :Y:

iO

G

UJO

t -

I -01 ‘ IJDoN

4'.2r<*açBgC2%* j 'W: ' r#wc3@*a5*-'3T:Wwmml'»2V«a#Tv*T, •

~t“

106

S JO 010HS3WH1

O

10HS3WH1 dg

Ll .

m

ICL

t£ wU.

>too-JW ' > : tu I»J,o •A. '5 ;

CO

CÜoz<oCvl(!)zh-a-jowz

yog

tMro

cvl

+ S jO 010HS3WHJL 4.

f s 30 aioHsawHi a

œtyvamxtssrxiifxra

/OŸ

wave are tabulated in table 15.1Tbe results for the D state of sulphur are similar and

are presented in figures 23 to 25, and in table 16, Weenote that the partial wave which also contributes to the photo-

Iionization of S ( S) has only one series of resonances whereas+ 2we expect two series converging on the S C P) threshold sincehi/ + S(’o) 'P S*Cap)’ ^D -+ &&

4 t CLThe results for S(^S) are presented in figure 26 and

table 17. The partial wave contributions to the photoionization cross sections above all thresholds for the three terms of sulphur are given in table 18.

Recently new Rydberg series of Chlorine have been observedin absorption in the 600-1500 °A region by Huffman ^ al^^.The series are due to transitions from the ^P° ground state to(3p)^ ns and t^P)^ nd and converge to the C^P)^ states P,and ^S. In figures 28 to 30 and in table 19 we present graphs

1and tables of resonance series converging on the excited Dand states of €£"*'. Unfortunately ref, C61) was a preliminaryreport so we cannùt compare the observed positions of theseresonances with our calculated values, as we have done for

37oxygen . Line shape parameters, namely on q-values, havebeen determined theoretically and experimentally only for simple

2 7systems such as Helium . It is hoped that the values presented

JO cnOiVSS’i'HJ. g

o

W

110

111

-a-

o

Ou-

cvw

U _

s

+TD do gnoHS3ïHx ç

oaaLL.

jO QlOHSaSHi d

11?

hexe should sexye aa a ui,de to experimenters on what toexpect in more complicated systems such as the ones we haveconsidered. In table 19 we present the partial wave contribut-

2ions to the photoionization cross sections of CZ P above all thresholds.The Negative ions

The photodetachment cross sections of electrons from the negative ions of silicon, sulphur and chlorine have been cal­culated by Robinson and Geltman whose tabulated values we com-' pare with the present calculations in figures 31 32 and 33.These authors use a central-field model i.e. a single radial equation, in which a parameter is adjusted in the potential to yield the observed binding energies of the negative ions.In other words the method is somewhat empirical, like the quantum defect method, in which experimental observations are used to impose constraints on the model. On the other hand the close- coupiing approach is strictly an ab initio calculation. Con­sequently it is remarkable that the two techniques are in close agreement for photodetachment from the ground state of silicon (figure 31).

We have also computed photodetachmentccross sections for S and a ,and we compare them with those given by Robinson and Geltman in figures 32 and 33. Calculations on C£ have also been reported by Moskvin^^ and Cooper^^. We note that our agreement with reference C62) is poor for C&^. Berry et al gives a value of 15 "5 x lo'^^ cm for the photodetachment cross

et

wo am

MS'

r\Wro

e>Ü.

jO lN3MHDVi300i0Hd

SP01 'N0I1D3S S S O & O l N 3 m D V 1 3 a 0 1 0 H d ' ‘j O

117

2section et K =;,QQ64 ed we see from figure 33 that our values are outside the error limits of this experiment.

In the photodetachmerit calculations we have used the bind­ing energies tabulated in reference (62) and we have represented our initial states by analytic Hartree-Fock wave function given by Clementi^^. It is possible that our results could be improved by using wave functions other than the Hartree-Fock type to represent our initial states, since for negative ions these wave functions are overdamped asymptotically.

H B

ACN

gS'•Hsmo§-H-P(dNo•HsPi

sH>

i

OJ+

w04“

Q CO

%f

%+

0COCNIPjCN

+

43

g +J3 u4J 0 C m 0 0â°rQ

Idr—} I(N

fe=l N

H

ro OH OH r>kO o CO ro oO 00 <Ti <Ti (Ti (Ti oiH o o O O o rHO1 o1 o1 O1 O1 o1 o1

in ro 00 o (TiKO o <Tl r" r Os o-O (Ti <T| <T (T (T\ <T|rH O o o O O o

H iH rH rH 1—1 rH rH

M CN ro m ro ro roO O o o o o oH I—1 rH rH rH rH rH:X X X X X X Xm rH in o 00 'ïf (Ti<yi O in l£> 00 rsro (Ti o (T <Ti <Tiéin iH <T CH rH rH

o o o o O O o

in 00 00 O O CT(T Oh KO Ch rH H O«531 in in in

OH ro in o rH oo o rs in 00OnI 1 1 lO1 KO1 ID1 VD1

0 «531 in in in1 1 f 1 1 f 1o O o o o o orH I—1 rH rH I—1 rH rHX X X X X X XrH OH rH in 1—1 "4» in00 OH ro o CM oOH O in o O orH in OH rH <T KO

o CNJ rH œ 00 OH00 OH VO (T ID rH inr> ro in KO i> 00 00rH OH OH OH OH m OH04

*d T) Td T5ro in lO O'* 00 as

p»CNfi0tnOU4->•Hiz;M-lOfi0 •H -M fd N -H§■H51

H>m

g

030+

ro04-

Pi Qro

4- 4*13tPi

S

CN

PiCN

%f:

e 4J3 ü+j 0 fi 44 td 0s «

Oto4*(d

rHICN

h(dtoCN

Cd

H

un ro ro 00 to*00 «5}i ro CN CN<Ti CT m CT (T G1» CN CN CN CN CNCN • • • • •

CN in KO to, to* to*00 00 00 00 00 00on as Ch Ch Ch Ch• • •

ro o 00 lO UO uoro ro ro roCT> <T» Ch Ch Ch Ch

H1 CN

1CNI

ro1

ro1

ro1

O O O o o orH 1—1 I—1 rH rH HX X X X X XLf) CTl uo lO oo VDin CN uo to* rHo KO lO 1—1 CNuo ro rH Ch lO 'If

si? CN KO Ch rH CNo 1—1 rH rH CN (NrH rH rH rH 1—1 rH*to* 00 ro UO KO KOCN uo UO UO UOto* to* to* to* to* to*rH rH rH rH I—1 rH

O to* to* 00 I—1 rHlO CN CN to* UO roro rH O Ch Ch ChCN CN CN rH 1—1 H

0 in UO KO KO KO

O o O O O OrH I—1 rH rH I—1 rHX X X X X XrH 00 O uo 1—1 OrH vo to* o rHlO Ch Ch rHro CN rH 00 uo

i> UO to* Ch CN ChCh to* Ch i-H Ch roCN to* Ch rH rH CNO O O rH rH iH

T3'd TJ T5 T i T i T iuo lO to* 00 Ch

19

m(N

Ti W Tî tr> t(ü 0 0 0 0+ + + + +Pi0 roCh0

u +p S•rl% +44 Pi0 CNfi +0‘AP Pitd CNN■ri gfi0 +•r(0p0f!Pi

.0*w>H>HfilmCEh

s +!3

43

td +o

CN

H cn

(d

HM

4 ,KO CN r4 to* O UO ro 00 Ch roro 00 Ch to- rH CN KO Ch CN to*Ch Ch r* 00 to* 00 KO to* KO toVD KO VD VD VD VD KO VD KO KOo O o O O O O O O O01

O1

o1

O1

O1

O1

O1

O!

O1

O1

ro ro to* O Ch O to* ro ro CNKO Ch r4 uo Ch KO 00 KO uo VDo o O rH Ch H Ch 1—1 Ch rHr4 o r4 O o O o o o OI—1 CN r4 CN rH CN rH CN H CN

1—1 r4 H rH 1—1 1—1 rH rH H rH

•H ro ro ro ro CN uo roO O o O o o O O o or4 r4 H rH rH rH rH rH H rHX X X X X X X X X Xo UO to* UO Ch 00 Ch Ch ro00 KO rH o 00 ro 00 VD

VD to- Ch KO to* 'f o oro to* rH CN TM rH rH rH ro

o O O O O O o o o o00rH

rHO '=f to VD 'f to to 00UO 00 to Ch to O 00 o to oCN CN CN CN CN ro CN ro CN ro

rH 1—1 to o ro KO CN rHVD 00 rH KO ro KO Ch O ro ro« • « • • • rH • uo •UO uo o CN I—1 • to • KOro CN rH CN CN to 'f CN CN

D KO KO KO KO KO to KO to toO o O O O O O O o oI—1 rH rH rH rH rH H rH rH rHX X X X X X X X X X00 to 00 Ch 00 to 00 o 00 CNto Ch uo Ch o KO uo KOCh uo to ro ro o VD ro VD rHKO CN ro H CN to rH 'If Ch

rH "'f to uo Ch ro 00 VD UO oo to ro KO to Ch O rH 00 ChCN CN to to Ch Ch rH rH rH rHO o o o o o 1—t rH rH rH

rH

m ■d m •d co T i ra T) m •d'f uo uo KO KO to to 00 00 Ch

T5HOÆM054êco

Htnfi■HthPIOÜ

uo KO to 00o ro O 00 toto VD KO uo uoCN CN CN CN CN•

CN rH 'fCh ro UO KO too rH rH rH rHQ O O O oCN CN CN CN CN

UO rH O CN ro00 to KO VD uoo O O O o

H CN CN CN CNO O O O orH rH rH rH rHX X X X Xuo KO uo CN VDCN Ch Ch rH CNro to KO CNrH VD ro CN rH

VD Ch H to Chro to VD ro CNKO to 00 VD Ch«if 'f

H UO 1—1 to KOro KO rH O toro ro ro CN

CN 1—1 o OrH CN KO Ch 00rH KO VD 00CN CN CN CN CNI 1 1 I 1'fI r «5f

I I IO O O o orH rH rH rH HX X X X Xto rH O 00 rHCh CN to toro o o ro Ch

CN rH vo ro

Ch Ch UO Ch rouo 'f to Ch to00 ro UO VD torH CN CN CN CN

T i T i Td T iUO KO to 00 Ch

CNSmofio■HPfdNfiO•HSostd

HH>HI

oïd «d OXD 0 + +Pi Q ro H

+ 4-S S

PiCN

QCNS4-M

B P P ü P 0 fi P P 0S ®

CN

Pt=IcNtdD

td

m

uo ro ro 00 to00 ■'f ro CN CNCh Ch oo Ch Ch ChCN CN CN CN CN CN

CN UO vo to to toCO 00 00 00 00 00Ch Ch Ch Ch Ch Ch• • • •

CN Ch uo ro ro CNUO uo uo vo ro ro

'f

ro ro ro roo o o o o orH rH rH rH rH rHX X X X X Xo H O 00 vo voCN CN 00 vo CN 'îfro rH uo uo Ch oto UO CN rH 00 VD

vo ro uo 00 CN rHCN Ch to VO VD -sfto vo VO VD VO VD

'

Ch Ch ro to vo 00Ch 00 vo to o <30uo uo uo UO uo

rH H to CN ro Chto o CN Ch 00 voIf uo ro ro rorH rH rH rH rH rH

UO uo UO VD VD VOI 1 1 1 I 1O o O O O OrH 1—1 I—1 rH rH I—1X X X X X XI—I rH VOro

00 o UOvo to^ ChCN rH 00

rH o o p os I—Iuo

to UO to Ch CN ChCh to Ch rH Ch roCN to Ch rH rH CNO o O rH rH rH

ISLt

■o rduo •d

KO"dto 'd00 "dCh

CN

PHH>mI

e pp u p 0 p p p 0 p Q OI

QH

t Jw T ith ’dP 00 00 0,0tn + + +OUP Pi a co•rf ro rH rH% + + +P % ^ %0 f c

0 ^P o0 CN■HPP Q ,N CN•HP g0-H +0P >O P:PJp,

P

r—IICN*5"t= |CN Pb

b

P

MM

KO CN rH to O in ro 00 Ch roro 00 Ch to rH CN KO Ch CN toCh Ch to 00 to 00 KO to KO toVD © © KO © © KO © KO ©o O o O O O O o O oo1

o1

o1

O1

O1

O1

o1

o1

o1

o1

00 ro to O Ch O to ro ro CNKO Ch rH LO Ch KO 00 © lO ©o o o rH Ch 1—1 Ch rH Ch 1—1rH o rH O o O o O o OP CN rH CN p CN rH CN rH CN

mIo

CNIO I—I

O

00

oo

'f'fKO

mIo

mIo

X X X X<hinro

Ch 00to oroin rH

roIo

torHrHp

CNIO'fO

roIo

nIoj_iX X X X00too ro

ChCN

rooCbm0000

IoI—IXro<hCNro

I—I to VD

ChVDto toro voto to

o

in roKO to to to

mtoto

©

roChto

LOO00

T5rHOÆm0g

POtnP•HtnU0ê8

LO © 'f to 00O ro o 00 toto KO © lO LOCN CN CN CN CN

CN rH 'fCh ro LO © too 1—1 rH 1—1 rHo o o o oCN CN CN CN CN

CN CN ro coCN o © ro CN'vT Ch o rH CNCO ro 'f ■îf

f lOO o o o o1—1 1—1 rH 1—1 rHX X X X Xo 1—1 00 rHro ro CN toLO Ch 00 1—1 roin CN rH rH to00LO H LO ro rHrH Ch 'f CN ChCN o o o ChCN CN CN CN rH

ro 00 o © ChLO ro o CN LOrH ro •'f 'frH rH rH 1—1 rH

LO Ch o LO 'f o LO rH Ch Ch00CN

to KOCh

© O©

O o rH o CN 00 O ro 00 LO Ch © LO roe « • « • • . , t LO

'îf ro CN Ch KO lO to LO CN KO • • • • •ro LO1 1—1 1

CN KOI rH CN1 rH rHï O1 o1 O1 O1 o1D KO © KO © © to KO to to TP 'f1 1 1 1 1 1 1 1 - 1 I 1 1 1 1O O O O O O O O O O O o O o 1,0rH rH rH 1—1 rH rH rH rH rH 1—1 rH 1—1 —C-H rH rHX X X X X X X X X X X X X X X

00 to 00 Ch 00 to 00 o 00 CN to 1—1 o rH rH'f to Ch LO •<=f Ch o © LO KO Ch CN to to toCh LO to ro ro o KO ro © rH ro O o Ch Ch© CN ro rH CN to rH Ch CN rH ro ro

rH to LO Ch ro 00 © LO O Ch Ch LO ro roO to ro KO to Ch o 1—1 œ Ch LO to to toCN CN to to (h Ch rH 1—1 H rH 00 ro LO to toO O o o O o rH rH rH I—I rH CN CN CN CN

œ •d m "d ra Tl m Tl 0 •d TI «d 'd «d ■dw - LO LO KO © to to 00 00 Ch LO KO to 00 Ch

IZ2

CN

P0tnOUP-HSPOPO-HppN•HP-S8OS

HH>

i

d tn d tn

0 0 0 0+ 4" + +Pi

roQ

%

kCN

QCN

4Dg P P U P 0 P P P 0

s °

p

rHICN3P ,t=|CN

Pb

ab

P

t7*

mu

LOI—1 KO lO to 00 enCN to 00 rH CN enLO LO lO © © KOO O o o O O

CN CN en o o 00Ch CN CN enen 'f 'f Ifo O o o O orH rH rH rH rH rH

O Ch Ch 00 Ch Ch00 Ch Ch Ch Ch ChCh Ch Ch Ch Ch Ch

H CN CN CN CN CNO O O O O OrH 1—1 rH rH rH HX X X X X XlO LO LO 00 toCh lO KO Ch rH CNro 00 LO O1—1 to ro CN 1—1 rH

00o CN CN LOLOen "«f

KO O o O o Oo O o O o O• • •

Ch I—1 Ch LO to 00to en to Ch 1—1Ch en m 'f 'f LOCN en en en 00 en

LO Ch Ch rHTP © CN Ch O 0000 Tf en en en CNrH rH H rH 1—1 rH

0 lO LO LO lO LOO O o O o OrH rH I—1 1—1 rH 1—1X X X X X X© lO to o rH 1—1rH rH LO en'f LO CN en © 1—1Ch © CN rH 1—1

rHCh m 'f CN 00

LO © 00 en 00 toisf to 00 rH 00 00CN to Ch rH H CNO O o rH 1—1 rH

d d d d d d«q* lO © to 00 ch

zi

(24

TABLE yLIIPartial wave contributions to the photo ion i z a ti on

cross sections of N (^S/ ^D, P)Dipole Velocity Dipole Length

L So f

.025 9.799 10.318

.05 9.967 10.631

.1 10.205 11.142

.15 10.330 11.512

.2 10.365 11.914

.25 10.332 11.914

.3 10.243 11.982

.35 10.112 11.982

.4 9.949 11.924

.5 9.553 11.676

.6 9.103 11.304

.8 38.148 10.3601.0 7.221 9.3271.5 5.269 6.9592.0 3.865 5.163

2pe ^D® 2^e V 2d ^ 2p,e

.15 1.030 0.000 5,291 1.063 0.000 6.264

.20 1.019 0.000 5,468 1.078 0.000 6.645

.25 .996 0.000 5.534 1.075 0.000 6.878

.30 .971 4.159 5.550 1.1.067 3.313 7.038

.35 .944 2.963 5.528 1.053 3.436 7.138

TABLE y m .(continued!

2p* V V V

.40 .916 2.910 5.479 1.034 3.409 7.189

.50 .857 2.780 5.322 .989 3.312 7.179

.60 .799 2.632 5.120 .936 3.177 7.068

.80 .687 2.324 .664 .822 2.856 6.6681.0 .589 2.032 4.208 .712 2.525 6.1731.5 .403 1.442 3.227 .490 1.818 4.9382,0 .284 1.039 2.489 .343 1.318 3.916

2g0 2pe ZoG 2g0 2p0.15 0.000 2.948 0.000 0.000 3.012 0.000.20 0.000 2.976 0.000 0.000 3.137 0.000.25 0.000 2.958 0.000 0.000 3.200 0.000.30 .784 2.921 6.086 .973 3.233 7.761.35 .765 2.871 6.088 .950 3.241 7.835.40 .696 2.811 6.057 .921 3.229 7.923.50 .696 2.675 5.899 /863 3.161 7.918.60 .651 25527 5.666 .808 3.053 7,753.80 .566 2.231 5.116 .705 2.779 7.187

1.0 .492 1.957 3.402 .435 1.829 4.9852.0 .250 1.031 2.568 .315 1.347 3.829

f 2.6

TABLE IXPartial wave contributions to the photoionization

cross sections of N+ t"pf S).Dipole Velocity Dipole Length

; ; . 3pO . V

.025 1.923 4.537 2.146 5.443

.05 1.928 4.557 2.157 5.478

.10 1.862 4.453 2.082 5.326

.15 1.797 4.276 2.023 5.174

.20 1.733 4.139 1.956 5.002

.30 1.612 . 3.875 1.829 4.726

.40 1.499 3.625 1.707 4.440

.50 1.394 3.389 1.591 4.167

.60 1.296 3.169 1.483 3.909

.80 1.123 2.773 1.289 3.438

.0 .976 2.432 1.123 3.025

.5 .700 1.774 .805 2.218

.0 .516 1.321 .592 1.655

IpO IpO 1D° ^F°

.025 .397 1.088 4.482 .442 1.088 5.827

.05 .396 1.085 4.519 .442 1.086 5.892

.10 .378 1.039 4.402 .443 1.041 5.769,15 .362 .994 4.285 . 406 997 5.643

.20 .347 .952 4.169 .389 954 5.515

.30 .319 .872 3.943 .358 875 5.259

127

TABLE IX(continued)

,40 .294 .799 2.725 ,331 .803 5.005,50 .271 .734 3.516 ,305 .736 4.756.60 .251 .675 3.319 .283 .676 4.516.80 ,217 .573 2,956 .243 .573 4.063

1,0 .188 .489 2.635 .211 .488 3.6531.5 .137 .340 1.992 ,151 .336 2.8072.0 .103

IpO.245 1.527 .112 .241 2.177

.025 6.459 7.812

.05 6.489 7.859

.10 6.274 7.619 .

.15 6.063 7.381,20 5.858 7.146.30 5.467 6.692.40 5.101 6.259.50 4.759 5.852,60 4.443 5.471,80 3.879 4.784

1.0 3.398 4.1921,5 2.480 3.0552.0 1,852 2.275

p.Ci

IflIœ5Pum0g•H■pcüN•Hg•H5504

X

i

wCN

0404

04+>

ItJ■p eu g mtd (Dâ «

fdm

0) CD

+ +iH

Q CQ I1—1 1—1 CN

+ + p-t.04 04 : t=|oi

, fdf b

Ab

(d

H

"4M (P on 0404 co LOO o o o

co 1—1 in coo I—1 1—1 c?»a\ <p m 00

H I—1 r4 1— I

"4 CN en eno O o o1—1 iH r4 r4

X X X X04 in cr> ■"d*O CD inen 1—1 'd" TM

o o o Oo o o O

040 " <M O04 « # •• CN CN en

CP O' CN enO' 1—1 en

CN CN CN CN

roI I I Io o o oiH 1—1 r—1 I—fX X X XCD CD CN oen I—1H CD en CN

en O co 00CD 00 en CDen ir> CD O'r4 1—1 r—1 1—1

co en CO coin CD O' 00

üî 'Xi Td(D 0) <ü

+ + +A Sro

+O,+04

04

04

0404

04+

XIg

r—1 en 1-4 CD RMen en en CN lO'RM RM RM

en CN 1—1CN en en RM eno o o O o

•CD en LO lO 00en CD O 00 enen 00 00 O'

4 rH CN CN enO o O o oH 1-4 r4 r-4 rHX X X X Xin CD rH O' 0000 CN en 00 RMr—) H O' LO CN

04 lO RM CN RMCN LO 00 CN CO

r4 1-4

LO 00 1-4 lOCN CD enm en RM RM CN

CDRM

00 00 en OCD CD RM 00 1-4en en en en en

1 en RM RM RMo o o o o1—1 1—1 1-4 I—1 1-4X X X X XlO 1-4 r4 CD O'en LO CN en enCN 1—! en lO en

00 RM 00 en CNb* o CN o lOiH RM lO co coo o o o o

izg

•«?}*n3IT) vo rJ00

12.9

ftCN

COboUACOOCMmogo

■r4-PfdN-H

O•HO-P

âCM

ÜX

HPII

iti4-1 (U g MH td 0)s°fdTd CO Td triTd b

0 0 0 0 0+ +CM P

m

+ +CM CM

CN

CMCNCM

+à

+m

CM

43b+

fd

1-4ICN

t=|CN0

b

Ab

fd

Cr*

w

r4 CO O' RM CN oO' CN CN O CO RMO O o O O OO o o O O

in o CO CN in RMo RM cn in 00 tncn r-4 00 r4 00 cn

CNI

r-1

r-4 r4I Io o

O 'cnRM

CNIoroIo

1-4

X X X XCOO' o CO

RMRM

COIO1-4

X XCOO '

toin00CO

o o o O o o

o o o CO RM COo CN o 00 in O'O' CO RM CN 1—4

f'" CN m 00 in CNin r-4 r-4 O' CN cnCN r-4

r-400 O' in RM

M RM RM RM in RMO O o o o or-4 r-4 r-4 r-4 r-4 r-4X X X X X Xcn CN in O' O' cno CO RM CD cn COO' CN CN r4 CO r-4

CO CO CO O' CN 00r-4 00 CN r-4 RM OCN CO RM in in CDO o O o o O

m td CO Td CO COin in CD CO O' 00

Tdr-4CO0u

gU]

§tn-r4Cr>PIo

CO 00 CDro cn cn1-4 o o

RM O' cnO' CD O'r-4 r-4 1—1' <i

1—1 CO cnCN CD O'r-4 O o

4 1—1 r-4o o Or-4 r-4 r-4X X X00 cn 00in CN CN00 CN r-4

in O <nr-4 CN r-4ro CD CDr-4 r-4 1—!

1—1 in cn00 r-4 COr-4 r-4 r-4

O' CO COr-4 CN COCO cn 00r-4

COI roI RMIo O or-4 t-4 r-4X X Xm r-4 oO' m CDr-4 r-4 00

CO CO r-400 00 O'CN in CDr-4 r-4 r-4

TdO '

Td00 Tdcn

o'CN 0 Td Td0 0 0W

b + + +p0XI CM CMa, n iHco0Æ + +CM CM CMCM fO CMb CN0•H t4Jfd -o—N Q•H CNG '—0•H fM04J +0XCM

fdHX

I

O

R3 u4J 0g mfd 0

(d +fd

r4ICN3É=|CNfdb

Ab

fd

ü*

H

I 30

1—I ro 1—1 CD CNro ro ro CN inRM RM RM RM RM

RM RM en CN r-4CN ro ro RM roO o o O o• * •

ro en en O roCN O' CD CD iniH o o O o

M CN CN CN roO O o O o1—1 r4 I—1 r-4 r-4X X X X Xco RM o ro t'­O' r-4 RM CD en00 RM CN r-4 en

O' in CD ro I'-in 00 O' r' M'1-4 r4 1-4 1-4 1-4

CN CD ro r-4 OCN r4 r4 r-4 r-4

CN H CD O enro in ro co coen o 1-4 CN CN1

r41

t-41

r-4I

r-41

n RM RM RM1 ro 1 1 1o 1 o o o1-4 o r4 r-4 r-4

HX X X XXin iH r4 CD t'­en in CN en en%CN H en in ro

00 RM 00 en CNO' O CN o inîH RM in CDO O o O O

TdRM Tdtn Tdco TdO' Td00

0QCN +

CM0 CObu0 CNjbCM CM0 roo — -PÎ +CM CMMH t0

kC3 CN0■H t-Ptd QN CN-Hbo■H0 CM-P rooÆCM CM

+>

X P!'HXHPICQ<Eh

Td tp Td tp 0 0 + + o

CNCMm+

CM

f

g 4J3 Ü-p 0 b M-i (d 0 b Qa

(dfd

I—I ICNCPPr"#|CNfdb

td

H

tzi

C' O CN CNRM ro CN ro b-ro ro CO CO ro

m CO 00 CN CNCN RM r-4 CO RMro CN CN ro ro•

cn cn r' r-4 00m CN o cn incn 00 00 00 00

4 CN CN CN roO o O o o1—1 r4 r-4 r-4 1—1X X X X X

r-4 r-4 cn ino cn O t-4 RM1—1 r-4 CN r-4 RM

<n RM CN cn roRM ro CN r-4r-4 r-4 r-4 r-4

ro cn ro RM CNro cn in in b-00 CO in o mr-4 r-4 CN

CO ro ro CD cnro CD CN RM or-4 CN ro r-4 r-4

CNI mI cnI rmI RMIo o o o or-4 t-4 t-4 r-4 r-4X X X X X00 RM CD ro b-ro <n RM ro 00RM CN CN CO in

CN b- ro o CDm O oo ro b-ro in in CD CDo o o o o

m CO CO <T>

Q(NWbpAwfMMHOg■H-PfdN•Pbo•HSêCM

ü

HX

IQCM

CM+X

O

tdb

g +Jb u -p 0 b cp td 0s °

+td

b

HICNTd 0 Td Td 3

0 0 0 0 0 p-v, t=|cN+ + + btd>-NCM Q m

ro rH rHXb

+ + +CM CM CMiQ tdCN b

Ü*

HP

rH ro b» RM CN o ro 00 CDb- CN CN O ro RM ro en enO o o O o o rH o oO1 o1 0

1O1 01

m o ro CN tn RM RM b- eno RM en un 00 in b- CD ben rH 00 rH 00 en H H rH* * '

CN RM CNl> CD brH H rH rH rH rH

■ ■ ■

CN CN CNSI CN CN ro CN ro 1 1 11 1 1 1 1 1 O O OO O o o O O rH rH 1—1H rH rH rH rH rH TdrH X X XX X X X X X 0

X b- o rob ' rH b- rH ro 00 0 rH o CNo Ib RM en o en 0 • • •> • • • • • p CN CN rHen rH CN en rH rH P! 1 1 1-P

WrH

rH ro RMo O O o O O b CN en b0o o O o O O ro RM rotPb•iHo b - CO rH 00 00 tP CD CD ro

00 CO CD CD en tn P rH b - tn0

CN co rH RM ro in > 00 00 en

RMI RMI

en en o ro rH CN 00 O CNCD CD ro o RM CD H CN CN

in CN CD ro RM rH

RMI RMI mI RMI enI roIO o o O O orH H H rH rH iHX X X X X Xen CN tn b b eno CD RM CD en rob CN CN rH 00 rH

ro CO CD b CN 00rH 00 CN H RM oCN ro RM lO tn CDO o O O o o

0 Td 0 Td 0 0tn tn CD CD b 00 TdRM TdLO

RMfo o o

H H HX X Xtn rH ob tn coH iH 00

ro 00 rH00 roCN tn corH rH rH

TdCO

liZ

t 35

TABLE .XIIPartial. wave contributiona to the photoibnization

cross section of P t‘5, ^D,

Dipole Velocity Dipole Length2

Kf 4p

.015 32.019 54.352

.1 29.114 50.462

.15 25.994 45.874

.2 22.875 41.016

.3 17.149 31.546

.4 12.439 23.457

.6 6.204 12.198

.8 2.996 6.1221.0 1.451 3.0472.0 .284 .289

> 9 ? 9 9 9 9D P F P D F

.3 1.488 5.226 11.161 2.721 9,136 24.960

.4 1.167 3.816 9.702 2.176 6v@65 22.788

.6 .733 2.011 6.822 1.345 3.802 17.524

.8 .476 1.082 4.281 .825 2.129 12.0851.0 .326 .603 2.357 .528 1:209 7.4772.0 .113 .121 . .007 .169 __159 ... .213 . . .

,2(2 2q .2,g . 2 p . . . ..... "

D

.3 1.574 5.632 12.828 2.701 10.895 26.937

.4 1.193 4.558 10.817 2.109 9.182 24.047

TABLE XII (continued)

2s 2p 2d 2s 2p 2d.6 .650 2.700 7.333 1.193 5.711 17.946.8 .359 1.467 4.563 .664 3.174 12.164

1.0 .211 .770 2.541 .381 1.677 7.4442.0 .049 .112 .066 .059 .200 .235

/3S"

2pO

TABLE XIIIPartial wave contribution to the photoionization

2cross sections of AiC P)

Dipole Velocity Dipole Length

K / 2ge ,2pe 2ge 2^eI. -

.01 .849 14.373 1.805 21.992

.02 .609 14.236 2.159 21.833

.03 .582 13.301 2.065 20.439

.04 .557 12.427 1.978 19.132

.05 .534 11.669 1.892 18.216

.075 .482 9.808 1.717 15.183

.10 .438 8.295 1.565 12.891

.15 .370 5.959 1.324 9.353

.20 .318 4.307 1.142 6.847

.30 .246 2.280 .879 3.734

.40 .197 1.215 .691 2,038

.50 .163 .643 .548 1.099

.60 .137 .332 .442 .586

.70 .117 .163 .365 .312

.80 .102 .073 .311 .164

.90 .089 .027 .272 .082

I3Ù

3pe

Id®

TABLE JCIVPartial wave contributions to the photoionizatipn

cross sections of S)

Dipole Veiocity Dipole Length2Kg 3„o ' 3_o..... 3 o 3^0 ■ 'P D P D

.01 8.989 18.867 13,035 29.587

.02 9.193 19.636 13.404 30.964

.03 8.841 19.124 12.936 30.315

.04 8.495 18.335 12.469 29.212

.05 8.156 17.698 12,006 28.334

.06 7.826 17.072 11,551 27.460

.08 7.195 15,858 10.667 26.734

.10 6,604 14.697 9.829 24.053

.15 5.313 12.056 7.965 20.125

.20 4.373 9.808 6.454 . 16.779

.30 2.805 6.342 4.258 11.245

.60 .939 1.523 1.381 3.075

.80 .530 .522 .753 1.2251.0 .339 .145 .464 .444

1 Ô P lyO IpO V lp.0

.01 2.020 3.756 14.701 3.327 3.882 27.185

.02 2.133 3.850 16.419 3.553 3.966 30.609

.03 2.070 3.641 16.139 3.460 3.740 30.315

.04 2.010 3.348 15.855 3.369 3.424 29.995

/?7

Ige

TABLE XIV

(continued)

IpO V Ipb IpO IpO IpO,

.05 1.952 2.803 15.566 3.280 3.182 29.651

.075 1.818 2.637 14.836 3.071 2.685 28.705

.10 1.699 2.233 14.106 2.876 2.223 27.676

.15 1.492 1.618 12.676 2.529 1.575 25.505

.2 1.324 1.193 11.335 2.292 1.161 23.567

.3 1.062 .659 8.890 1.793 .617 19.455

.6 .570 .119 3.617 .871 .103 9.395

.8 .403 .034 1.690 .579 .029 5.0951.0 .300 .006 .684 .422 .005 2.579

IpO IpO.01 25.653 38 .353.02 26.713 40 .726.03 26.227 40 .749.04 25.761 40 .864,05 25.243 40 .747.075 23.912 40 .214.10 22.556 39 .336.15 19.864 36 .904.2 17.301 34 .866.3 12.775 27 .278.6 4.309 10 .275.8 1.912 3 .659

1.0 .819 1.918

0mmRM

COa

m+

nJ

<o

g 4J3 ü+J 0 b mh (d 0

2(0 40 3 CNn 3d 0 0w h*H + + t=lcNd tofdm %) 0Qw RM CN0 dbü ro ro0 fd 4-•ri a a D+J ro roId •wN■rl + +d W m0 A•H t t e>0d 00 w

rok fd

X

fd

HM

CN o 00b en enCN CN CNO o o

en RM brH r—1 f—IO O O

CN b CN00 rH CNo o o

3 0 Td

0 0 0'-r-'

+ + +

O Oo aCN CN

ro ro'a aro ro

m

fO o o

O0aro

1—1 CN CN trH en biH o O 0aroCO RM 00 RMro H O• • • aen en o 1—1 ro

03+

M RM RM1 1 >o o Or-4 iH I—1X X Xen co enro RM iHin rH 1—1 X

rH b COro CN LfOCN b eno O o

Td Td Tdro RM m

+

+j 0 b MH fd 0 b Qa

rHI

CN3

^ It=|CNfdo

+(dfdb

X

fd

MM

b O RM en bin 1—1 00 en coCN ro CN CN CN

■ ’«

b en CN CD CN00 00 00 en enen b en b <niH o1 1—1 o1 r-4

-4 CN CN CN CN

O o O O OH iH (H H iHX X X X X

00 b CN RM oH co CD uo CDLO rH CN ro iH

O o o o O

CN CN CN rH lOb o co 00 RMb o 1—1 m b in

CN b rH IfO roCO b 00 00 coCNI

1—1 I

CN1

CN1

ro1

0 RM RM RM1 RM 1 : 1 1O I o o O1—1 o

rHr-4 rH iH

X X X X Xen en rH LO bCN en ro O lOb RM RM RM rH

00 ro CN rH rorH o RM RM CDCO œ 00 en enrH 1—1 1—1 rH I—1

0co TdRM

0b TdlO 0

00

1 3 S

amu0MHr—Ia44Og•HX(dN■Hg•H0 -M1

X

S

I

Td m Td tn Td0 0 0 0 0+ + + + +<omRM

ro

ro

aro

RM

CM

ro

P

aro

W CO

aroco+

X

aCN

roaro

CO

f

co

§ u-P 0 b 44 fd 0â “

CN3

"ëlCNo'"

fd

X

fd

tJ*

P-r

M

RM co rH CN rH RM ro CNro b RM CD O CD 1—1 mCN CN ro ro RM ro CN rH

enRM rH O CN RM en CN Oro en rH in ro 00 CN rorH en CN en ro en rH rH•>O o O o o rH rH rH

CD LO r—i m RM RM rH bO CD ro en o b Td ro b00 ro CN o O o rH I—1 O• oCD • Æ!m

0u CN bX O b

rH rH rH rH rH r-i X rH oaCNb RM rH

O o O O O O 0 CN CNO o o O O O tn CD CD

b rH rH•Htn

CD LO en rH RM b U in CDCD rH ro 00 m en 0 00 ro• , • » • • > • •

CD b ro CO ro ro b rH rH

roI RMI roI RMI roI RMIo O o O o o1—1 rH rH rH rH rHX X X X X Xo ro rH uo CN 1—1CD ro ro b rH oCD en laO rH 1—1 ro

00 CN O CO b 00ro LO CD RM en tn1—1 CN CD b 00 eno O O O o o

Td m Td m Td 0RM CD LO b co 00

ro H o o CN CN CN- CNen O b CN o 00 ro enen CO ro b in CO en 00

RM RMr fo o

X XcoCNtn

enr4œ

CNcoRM

enRMen

Tdco Tdb

(39

llf.0

Q

b44rHbw44O§

-H■PfdN•H

§-Hsâ(Xi

H

i

Td 0 Td0 0 0+ + +P aCN

ro

RM

CN

mam

fP

aen+m

aroœ+

o

4J 0b 44 (d 0â °

fdb

Xb+fd

b

1—I ICN

3

I=|CNfd

b

X

fd

Hu

co CD cotn 00 RMîH iH iH

HI

Oo

RMI

OO

RMI

CNI

OO

inIo O OrH rH 1—1X X Xro en CNb RM rHCN rH 00

rH 00 bro RM CDCD 00 enrH I—1 rH

0 0 0

Td0

en0

Td0

ro+

PiHRM

ro

CO

b CNCN eniH rHO o

rM 00 rH CN 00UO uo CD rH RMo o O (N CNO o O

ro enCN oRM RM

roIo

roro

00RMCN

roo00

RM uo co Tduo

RMI

O O o + + + + 1—1 o1—1 rH rH

P a X1—1

X X X CN CN00 X

CD uo b CD o00 b en ro ro » roroRM 1—1 ro a a 1 1

I—1RM

CN 00 00 1—100 b 00 00 RM# » « ro 1—1ro RM UO f • •1—1 rH rH U") LT)

CN rH RM1—1 CN uo a 00 uoro o CDen b RM '— en en

ro roI Io o

X X00co

enroenI—I

TdCD

p

ub44r4544O§

"H4JfdN•H

§■r4O4JOg

ü

Td tn Td b> 0 0 0 0 + + 4-P a

CN

m

a

à

CN

+U1

Rt*a.coœ+

g -p b ü ■P 0 b 44 fd 0â «

b

f-4I

CNtdt=|CNfdb

A

fd

tr*

m

en CNH CN O O

tn enb bo o

bf»RM

bCN00

\k\

bCOCD

CNIOr4

00min

coIoX Xr4enCN

CDcoRM

inb00

00coco

coCN

co

enRMLO

LOenCD

bolO

LO lOI Io op4 I—IX X

coCN

RMLOen

TdLO

TdCD

w

gIwë§HSNMIoEHSaHHS

I

Td M Td0 0 0'-T-'+ +Q aCM CN

ro

RM

roaro

Wt

a4+W

aro

œ

aroW4-:>f:

ob

g -P P ü ■P 0 b 44 0 0s«

Xb

4-(d

CNII-5*P-,

(db

b(d

tn

m

/ V 2 -

CDUO

CD00

CDRM

RM 00 Huo UO CDO o OO o o

r4 1—1 r-4

N CN CN1 1 1O O O1—1 r4 HX X Xuo RM CDuo ro r4RM CN p4

O o OO o O

RM ro enb RM uoCN CN ro

1—1 1-4 CNro UO r4CD1 CD1 UO1

M RM UOO o oH r4 1-4X X Xro en CNb RM r4CN r4 00

H 00 bro RM UOCD 00 enr4 1-4 r-4

0RM

mtn 0CD

M

TABLE XyiIIPartial wave contributions to the photoionization

3 1 1cross sections of SC P, D, S)

Dipole Velocity Dipole Length

L ^S° . ^P° ^D° . ; .s:° : , ^D°O O t

D

1S

.3 2.907 8.616 14.049 5.053 13.314 26.242

.4 2.637 7.029 13.170 4.739 11.154 25.895

.6 1.928 4.226 10.949 3.618 7.004 23.493

.8 1.264 2.335 8.495 2.414 4.022 19.7511.0 .781 1.237 6.052 1.491 2.216 15.2622.0 .121 .099 .286 .191 .141 1.248

IpO IpO IpO IpO IpO 4 °.3 2.822 8.064. 11.894 3.695 14.129 22.638.4 2.158 7.628 11.376 3.016 14.129 22.638.6 1.249 6.329 9.813 1.967 12.856 21.684.8 .763 4.800 8.021 1:325 10.385 19.334

1.0 .495 3.336 6.144 .919 7.554 10.0802.0 .103 .328 .272 .151 .698 1.681

IpPO IpO

.3 23.807 37.182

.4 22.378 38.588

.6 18.858 37.774

.8 15.138 33.7481.0 11.415 27.6352.0 .832 2.805

Td m0 0a + +

CN Q wiH rH0■b-H RM RMp0 a arH ro rou '— '—+ +WH PI PI0 U ub0 f•rH-P0 wN CN•Hb0■HOX a0 CNXa lO

aroX '—HX PIUHPI +m< >Eh X

-p 0 b MH 0 0

fdb

CM3p-",G=|(N0b

fd

tn

F-h

H

CDCDmboCD

CN inRM mo oI—I I—I

CNIOX00ro

roIo

XiH I—IRM

enr4RM

CDCNpHI

roIo

XCDmro

oRMpH

U]RM

bORM

RMCNI—II

roIoXen

cooen

g -P3 ü+j 0 b M-i fd 0â “

a

Td0+

fdb+i-ib

ro

RM

aCN

aCNinam

d+

RM 1a a CNro ro >5*H- +Pî PI t=|CNU U 0bf t

fd

tn

H

CD O LOo 00 enco ro ro

lO rH CN00 O orH CN CN

CN CD rom LO «D00 CD LO

H 1—1 CNO O orH rH 1—1X X Xb o bro CN RMrH I—1 00

en (—! RMLO RM bRM b RM

1—1 1—1

CD ro OCD ro enCN ro rH

ro 1—1 CNLO o enro ro ro

0 ro roo o orH rH 1—1X X XrH H" 00RM 00 enLO CN rH

LO 00 LOb CN CDro co bo o o

TdRM TdLO Tdco

Td0bb•H4Jboü

5XH

I

Td 0 Td en Td0 0 0 0 0+ + + + +

a oro H+ +Pî PI U O

QCN

aCNPIU

X

mPIu

X 0 b X 0 0 b Qa

0

CNarHICN3hr*t=|CN0b

X

en

H b b 00 en 00œ CN ro CD RMro RM CD b COo O o o O

Td 0 Td Td TdRM RM m CD b

co

oœ00tH

TdRM

COCN ro m in RM 00 CNCD 00 ro RM RM CN roCN 1—I ro ro ro • •• • • • • 1—1 iH

CN ro CD 1—1 o 00 00m ro RM CD b 1—1 RMI—1 o iH rH 1—1 r—1 1—1

* ’ ’ ■

enRM RM

1—1 iH iH I—1 1—1 (N oO o

‘ ■

H •H rH CN CN H CNO O o O O O O1—1 1-1 I—1 iH 1—1 i-1 iH

X X X X X X Xro b CN CD o iH enRM ro CN CN CD 1—1 rHiH iH 1—1 b lO I—1 00

00 enRM 1—1

O O O O O 00 00iH 1—1

ro CD ro 00 CN lO oro O LO ro O RM en• « • . • O oro ro o o 1—1 « •iH iH 1—1 1—1 1—1

1—1 lOen 1—1 o en ro CD RMb ro LO en b • •LO 001—1 in ro CN ro CN 1—1

ro0 RM ro 11 1 RM RM RM 1 Oo o 1 1 1 o 1—1iH 1—I O O O 1—1

1—1 H I—I XX X XX X X oo CD o RMiH RM lO CD lO RM b• . • • • CN •ro CN CD LO CN • iH

enCN

Tden

TABLE XX - Partial wave contributions to the

2photoionization cross sections of CJl C P)

Dipole Velocity Dipole LengthL S ^D^o o f

2p .3 2.612 7.474 12.397 3.791 13.424 23.339.4 2.225 7.248 12.151 3.303 13.612 23.942.6 1.422 6.499 11.159 2.194 13.169 23.782.8 .826 5.461 9.801 1.315 11.710 22.404

1.0 .460 4.253 8.131 .754 9.598 19.9132.0 .039 .476 .805 .055 1.235 3.288

f47

TABLE 21Photodetachiïient cross sections of Cal S and

1 LCbJ Ci"C S®)

ta) S. 4 e LSn= P

Kf2 ^L % *L.005 10.986 7.151 .15 31,461 41.028.01 14.745 9.867 .17 31.611 43.036.015 18.996 13.378 .19 31.555 44.578.02 22.987 17.667 .21 31.332 45.706.025 24.071 20.541 .23 30.978 46.477.03 24.871 20.541 .25 30.519 46,950,035 25.499 21.734 .3 29.053 47.137.04 26.021 22.836 .4 25.373 44.651.045 26.476 23.876 .5 21.280 39.609.05 26.887 24.875 .6 17.163 33,278.07 28.299 28.661 .7 13.337 26.973.09 29.476 32.248 .8 9.701 20.848.11 30.409 35.581 .9 7.289 16.501.13 31.069 38.535(b) C&" LSI

*L % ^L.01 5.468 4.494 .40 17.518 27.750.02 7.294 6.204 .50 17.675 30.322.03 8.741 7.436 .60 17.535 32.027

luî

TABLE 21(continued)

%.05 10.010 9.294 .70 17.244 33.146.07 11.074 10.801 .80 16.921 33.962.09 11.927 12.162 .90 16.544 34.510.11 12.663 13.454 1.10 15.770 35.230.13 13.321 14.708 1.30 14.908 35.510.15 13.918 15.936 1.50 13.822 35.230.17 14.460 17.139 1.70 12.346 33.488.19 14.951 18.314 1.90 10.373 30.181.21 15.392 19.449 2.10 8.005 25.112.23 15.784 , 20.545 2.30 5.573 19.013.25 16.129 21.591 2.50 3.506 13.109.30 16.804 23.982

(4 9

TABLE 22Partial wave contributions to the Photodetachment

cross sections of S C P°)Dipole Velocity Dipole Length

K / V 2pe V 2gO 2pe 2d"

.01 .000 4.462 .031 .000 2,478 .013

.02 .000 5.645 .109 .000 3.395 .068

.03 .000 6.289 .360 .000 4.024 .177

.05 .000 6.267 .916 .000 4.867 .721

.07 .000 6.811 1.778 .000 5.401 1.131

.09 .001 6.682 4.217 .001 5.753 3.782

.11 .034 6.527 6.221 .030 6.037 6.243

.13 .112 6.422 7.325 .106 6.337 7.762

.15 .222 6.371 8.118 .219 6.674 8.981

.17 .344 6.361 8.738 .351 7.046 10.047

.19 .452 6.378 9.238 .462 7.444 11.011

.21 .621 6.412 9.645 .942 7.857 11.895

.23 .867 6.426 10.007 1.493 8.486 12.759

.25 1.058 61494 10.356 1.868 8.685 13.649

.30 1.430 6.579 11.206 2.549 9.643 15.974

.40 1.890 6.623 12.701 3.379 11.097 20.414

.50 2.103 6.519 13.739 3.755 11.923 23.729

.60 2.159 6.337 14.171 3.830 12.298 25.530

.70 2.106 6.093 13.941 3.706 12.373 25.867

.80 1.973 5.833 13.168 3.447 12.309 25.068

1 ^ 0

TABLE 22 (continued!

V 2d"

.90 1.782 5.506 11.983 3.100 12.168 23.4821.10 1.302 5.001 9.3861 2.268 11.679 19.8661.30 .888 4.336 7.226 1.484 10,797 16.9221.50 .491 3.534 5.562 .895 9.377 14.5582.00 .104 1.438 2.296 .212 4.447 7,967

I S I

Chapter 4

The code can be logically divided into four sections and the transfer of control within the program may be described by the following diagram

MAIN

SECTION 1Sets up the system of eq­uations (2.48) Computes rele­vant parameters and co-efficients

SECTION 2Generates lin­early indepen­dent solutions of this system and integrates them to the 2 match points

o' ■

SECTION 3Solves eg. (39) of Ref. (32). Evaluates eigen phases and cross sections

SECTION 4Evaluates eq­uations (2.49) and (2.53) using the continuum wave functions of the previous sec­tion.

We now describe the function of each routine in the code and the COMMON variables read in or computed there.

MAIN Reads in IREAD, IWRITE the logical input andoutput tape numbers and calls the sectiors in turn,

SECTION 1SUBROUTINE MAINl

The remainder of the data which we now describe is read in this subroutine.

IPH -ve { 0 1

call EXITimplies more data cards to be read implies no more data cards to be read

LRGL Total orbital angular momentum of systemSPN Total spin angular momentum of systemNPTY Parity of systemW1 Energy of incident particle (in Rydbergs)H Basic mesh size

isa

IDEBUG(8) { 0 no print out in intermediary steps in the calculation.

1 Prints out result of intermediary calculationIDEBU6 (8)= -1 if we require the 4th section to

be enteredIPH { o Read in more data

1 No more data to be inputed.10 Value of the nuclear chargeNST Number of states to be included in the expansionNELT No. of radial equations allowed (maximum=5)IQ No. of electrons in np subshell of targetIPl No. of closed subshells

The code assumes that the orbitals of the atomic electrons are accurately represented by analytic SCF functions i.e.T.(D -

We need IP=IP1 + 1 of these for the Ls, ^S...np orbitals of the target and also 2 more such orbitals for the (np)^^^ system, namely and or tor the 4thsection since the photoionization cross section is proportional to I Ÿ; I f I . The is merely the (np)^+l system

|S3

and tke ^ is, described by the continuum wave functions that are computed in sections 1 to 3.

Total number of orbitals - IP + 2 + IP2

EN (NST) Energies of the target states in atomic unitsPTL Ionization potential of the (np)^^^ ^o^o system

IRl Number of mesh steps in the first change in meshsize. IR1*H is also one of our matchiig points for the functions. The integration from the origin outwards is terminated at this point.

IR2,IR3,IR4 Number of steps from the origin to regions wherethe mesh size is doubled i.e. the interval size between IRl and IR2 is 2*H etc.

IRA Number of steps from the origin to RA.1RS First match points^IR5*H: The IRI's satisfy

the inequalitiesIR5<IR1<IR2<IR3<IR4<IRA

where each IRl is at least 2 larger than the previous.

The functions are integrated from the origin as far as IR1*H. The values of the functions at IR5 and IRl are stoed in the matrix Bl. The solutions generated in the asymptotic region are integrated inwards as far as IR5*H. The values of these functions at IRl and IR5 are also stored in Bl.

fS4

ITAPE

LOSO 1CNORM

Logical output tape number of temporary storage of the independent solutions until they are re­called in MAIN4.The orbital and spin angular momentum of the (np)^^^ system under consideration in photoionization calculations. x-i 2.

l^p) 0 (IS J where ^ denotes the radialwave functions of the (np)^^^ L^S^ system. For the (3p)^^^ system we neglect the contributionfrom the IS and only include [ (3p* I as'ss'fîs 3s f ]

R2SDR2SE !»p i L” dr

(kr r

The following variables are also computed in MAINlIRAINNANBIFAIFBNMU

NMÜSOL(NMU)

NFEQN(NMU)

IRA + 1Number of channels Number of open channels Number of closed channelsNumber of distinct wave numbers in open channels Number of distinct wave numbers in closed channels Number of Lagrange multipliers needed to make F^'s orthogonal to atomic orbitals.Number of the atomic orbitals involved in the NMUth Lagrange multiplier.Number of the radial equation involved in the NMUth Lagrange multiplier.

fse

L2P(N) Angular momentum of the outgoing electron inIL2CN) }AL2 (N) a particular channelILl(N) Orbital quantum number of the target in a part­

icular channelSPNl(N) Spin quantum number of the target in a particular

channelWP(N) Wave number squared for each chnnelKABOVE(N) Number of the channel above thresholdKBELOW(N) Number of the channel below thresholdABOVE(N) Distinct particle energies above thresholdBELOW(N) Distinct particle energies below thresholdR(IRAl) Values of the independent variable *r' at which

the functions are computed. Note: R(1)=E andR (2)=H.

RA Value of 'r' beyond which exponential potentialsare neglected.

RB Value of 'r' at which asymptotic expansion is used.HX(IRA) Mesh sizes over the range of integration:

IRT(N,N) Used to locate the potentials V.. (IRA). SinceNfhHOV^j is symmetric only ^ of them are stored.

EXCQ Excess nuclear charge (=g-number of atomic electrons)ETAAB(NST) excess nuclear charge

}ETABIW(NST) divided by distinct wave numbers

TAU(30) Needed in subroutine CLEBSH

/S6

HF Final mesh size (= 16*H)NT0T=N + NE Total number of second order differential

equations.NT0TNA=NT0T+NANANA=N + NA N2T0T=2*NT0T

NIN Number of independent solutions in the insideregion

N00T Number of independent solutions in the outer region.CALLS SETLSF, VIJOFR, SUBEX, GENPl.SUBROUTINE VIJOFR

The following arrays are calculated in this subroutine.N(N-fl) , /A ^

A( 2 , IRAI) Direct potentials - If.r'- r * J

when is defined by equation (2.30).,^^,N(N+T) I V/

2 , 2) Co-efficient of of a s f o r A - 2if such a term occurs in the direct potential,zero otherwise.

V(N,IRAI) Eq nation (2.41)NV {=0 If no. terms are in the equations

=1 If the equations contain a term.NVJ(N) {=0 If no. terms are in a particular equation

=1 If equation has a V. term.CALLS YSUB, CLEBSH, RACAH.SUBROUTINE CLEBSH

Calculates the Clebsh-Gordon co-efficient D=(ABOO/CO) SUBROUTINE RACAH (A,B,C,D,E,F,G)

Calculates Racah co-efficient G=W(ABCD;EF)

IS7

SUBROUTINE YSUB ;Numerical evaluation of Y^ (R .P r)The function is stored in the array B (IRAI)

FUNCTION FG03B(A,B,C,P#Q,R,X,Y,Z)Evaluates r ABQ 1

i CPX V i.e. t ( RYZ J in Ref.

the Wigner 9-j symbol as defined (39).

SUBROUTINE RSUB( I,J> K,L,X)L=1 calculates X=R|<(P.P .P;. P . )% 1 1 ]L=2 calculates X=Ry.(P.P.P.P.)K 1 3 3 1CALLS YSUB

SUBROUTINE SETLSPTabulated LS term values for the (np)^ configuration and

stored in the arrays AL(3) and ASPN(3).SUBROUTINE CFP (q,L^,S^,L^,S^,C)

Tabulated values of C= CP ^4^4^ i.e. fractionalparentage co-efficients for the (np)^ configuration.SUBROUTINE SUBEX,

Determines the number of distinct exchangeterms and computes the following variables:

NE Used to count the exchange terms, its final valuebeing the number of distinct exchange terms NE.

IXl(NE) Temporary array used in subroutine.1X2(NE) Value of for the NEth exchange term1X3(NE) The particular involved in the NEth exchange term1X4 (NE) The particular Fj involved in the NEth exchange term.

I S S

AX1(N,NE) Th.e co-efficients: of the exchange termsAX2 (NE) A C ^ + 0AX3 (NE,N) f Z A + O

CALLS CFP, RACAH, CLEBSH, FG03B.SUBROUTINE GENP l

— " ^ *1

Calculates ENPl ® E[C where E= ^•f fc where L^S^ are the quantum numbers of the lowest

term and the actual expressions for the E's are obtained from equation (2.46). The wave functions of the (np)^ system are used in evaluating b f L S | .CALLS CLEBSH, RACAH, CFP, RSUB.

SECTION 2SUBROUTINE MAIN2

CALLS INTEGRTSUBROUTINE INTGRT

Generates NT0T independent solutions of the system of homo­geneous equations Z " B Z where Z= is a columnvector which has NT0T elements. A further (NV+NMV) independent

IIsolutions of the inhomogeneous system Z =BZ + G where G involves the terms and the Lagrange multipliers are generated by setting the co-efficients C a n d e q u a l to unity in turn and the re­mainder equal to zero. These solutions are integrated from E to H using the Runge-Kutta method. This involves subroutines STRSLl RUNGKTl RUNGE and SUBG and the array, Z(N2T0T), ZPRIME (N2T0T). Knowing Z(o) and Z(H) we now can use the Numerov method

/S9

to integrate IRl steps and we store the solutions at the points corresponding to ÏR5 and IRl in the matching matrix Bl.

dr>- ^ 'J d ' (4.1)

This gives us

+ 6 ; (NZ(r) + G.(r) + j^ (4.3)

V - _ 1F 3 (NT0T, NT0T) \Z 'jr s . . L 6; (0 1 )F6(NT0T,NIN) L^'4 1% J -* J

r i — % (r ")F7(NT0T,NIN) l2 *J J J

F8(NT0T,NIN) y^T B; V 6 .(f j ) - 4 < -Hj|^ d c)

F1(NT0T,NIN) Z.(r) J - I, . . • N i NF2 (NT0T/NIN) j - ~ N % N.

F4(NT0T,NIN) = R.H.S. of Numerov formula

2 F 6 - F7 4- F &

F1(NT0T,NIN)= F3[{NT@T,NT0t 3~^4(NT0T,NIN)

/éo

The integration from the asymptotic region r=rg is now begun by calling STRSL2 in this re^on potentials which involve exponential terms are negligible so the equations take the form

N^ U{( F. i « I, ••• N

df-L j.i J J (4.4)

where

- L — r

46The asymptotic form of Burke and Shey for the functions is assumed

. i X r 'P

K " ?i: iic - p ,

-V c o . C i , r M K V V > 4 ' ]y rv

^ r J ^ i T _ p ,

••••(4«6)Substituting these F^'s into the asymptotic form of the diff­erential equations gives us a recurrence relation for the

i K :Kolp > rp and Op . In order to solve these

• 5 * arecurrence relations o( p and must Be specified, i.e.

/è/

2*NA + NB constants are unknown. Consequently 2*NA+NB linearly independent solutions can be generated by setting

M C NC

AlC NC

_ I „ 3.uA*\ Nc

and solving the recurrence relations 2.NA+NB times (NC counts the number of solutions).

Substitution of the co-efficients0 ^ back into equation (4.6) enables us to calculate the F^*s at r_ and (r_-HF) . These

Z i5 csolutions of the homogeneous system of N coupled equations areintegrated in wards using the Numervov method and the asymptoticform (4.5) of the potentials to the point EL where the exchangepotentials might be expected to begin contributing

At this point the exchange potentials are explicitly takeninto account. We define a further NE linearly independent solutionsof the homogeneous system of NT^T coupled equations by setting

—Xthe co-efficients of f in the exchange terms each equal to unity in turn and the rest equal to zero. A further NV+NMU independent sclutions- of the inhomogeneous system are generated by setting C and theJiJL *s equal to unity in turn as in the inner region.These NOUT=0NA+NB+NE+NV+MMV solutions are integrated inwards to the point corresponding to IR5. The values of the functions at the points corresponding to IR5 and IRl are stored in the matrix Bl.

/6Z

Since the mesh size is halted at the points corresponding to ÏR4 IR3 IR2 IRl the following arrays are required for inter­polation of F6 F7 and F8 in these regions F9(NT0T,N2UT),F10(NTOT,NOUT),F11CNT0T,NOUT)Subroutines called by INTGRT.STRSLl,RNGKTl, MATINV, UFLSPR, SETF3 SETCAE, SETF8, STRSL2,

SETF3A, SETDA SUBROUTINE UFLSPR

Machine language subroutine written to supress programe interupt messages caused by underflows oh the IBM 360-65. SUBROUTINE RSTSUP

Machine language subroutine which terminates the under­flow supressor programe.SUBROUTINE STRSLl ,

Computes F. (é) ; • The resultsare stored in Z(N2T0T). Use is made of the fact that

r r^’"' y ^ r1 r -9 o A r o

The NIN linearly independent solutions in the inner region are generated by setting the co-efficients of the powers of equal unity in turn and setting the remainder equal to zero. SUBROUTINE SUBG

Computes the right hand side of the differential equation at 2 . Result is stored in G(NT0TyFUNCTION RUNGE

RUNGE-KUTTA formula.

SUBROUTINE RNGKTlConverts the system of NT0T coupled second order differential

equations to a system of N2T0T first order differential equations in preparation for the Runge-Kutta formula by making the trans­formation

z = z I +z: 2 + G, (4.7)1The are stored in the array ZPRIME(N2^0T)

The above system may be written as

z ' = f (r z , . i<=K kis known from STRSLl

.'. From the Runge-Kutta formula

z ( H ) - z k ) - - " D» ' D (4.8)

where

t., = ---- )t., . « " ■ • f . ( f " J ' , 2 *'*,^*..

■àil = (f -"V", ............ i K n l r ^

•iii -- •••• '

/é4‘

where HM=H-6CALLS SUBG, RUNGE

SUBROUTINE SETP3(IA,HA)Computes the NT0T X NT0T array F3 at the point RÇEA+1)

with H=HA .SUBROUTINE SETF8 (lA, HI, JA, K)

Computes the array F3 at the point R(IA+1) with H=Hi.K=0 implies that the integration is in the 'inner'

region i.e. number of solutions = NIN K=1 implies that the integration is being performed in

the outer region i.e. number of solutiors = N0UTJA=0 implies that the mesh size is uniform in the region

of integration1

JA=2} means that we are preparing to decrease the step sizeby a factor of 2.

SUBROUTINE MATINV (A,N,B,M,DETERM)M=0 inverts the NXN matrix A

-1M=1 computes A BDETERM= dt/k

SUBROUTINE SETF3AComputes the NXN matrix F3 in the asymptotic region.

SUBROUTINE STRSL2Fits the F^'s to the asymptotic form of Burke and Shey^^.

If a good fit is not obtained it automatically increases r and tires the expansion again. Computes F.( 3 ) , .

CALLS STRSL9.

/éS-

NC NC

SUBROUTINE STRSL9 (ALPHA, BETA, GAMMA)iK

Solves a difference equation for the co-efficientsand Op of the asymptotic expansion using oC

NC communicated to it by STRSL2.The results are stored in the arrays ALPHA, BETA, and GAMMA, Computes lAASE i.e. the number of terms in our multipole expansion SUBROUTINE SETCAE (II)

Evaluates the orthogonality integral in the 'inner' region using the trapozbdial rule at each step of integration.

S(i« 0

£ (NMU, nim) = j J, JJ r I, 1, - -

whereIU=NMVSOL (NMV)IV=NFEQN (NMV)It also evaluates the integral involved in equation (2.48).

^ fC (NIN) r ^ ( V; F. c(f ,,1 .. . MIN c I ^

SUBROUTINE SETDA (II)Evaluates the same integrals as SETCAE for the 'outer'

region. The results are stored in the arrays I(NMU,N0UT) and D(N0UT) .

SECTION 3SUBROUTINE MAIN3

CALLS SOLVE

/ 66

SUBROUTINE SOLVE. .Completes setting up the Matching matrix Bl which in­

corporates the requirement that the solutions be continuousover the entire range 0 ^ B and also the conditionsexpressed by (33) and (41) of Ref.(32), The orthogonality requirements are also expressed in the Bl matrix. The re­sultant set of linear equations may be expressed in matrixform

|4--- NV-»[ KinU

r— V-lA 'Q.oc

t>2JLtiz

ΫZi

F,, .... F%%

' ' J,

1 *

F,-( p - V

cC40

D

E EtP F

0 0 0 . ■* *

C

/ unoW,Wn

0

i I%I

0

/67

These equations are solved NA times to give the unknown co-efficients Vx, j ^ i=l, NA which are storedin the array W(M,NA). The R-matrix is then computed using equation (42) of SHB. The R-matrix is daigonalized and the eigenphase shifts computed.CALLS MATIN 3,HDIAG,GAMMAR, RMSIE SUBROUTINE MATIN3

Inverts the Bl matrix SUBROUTINE GAMMAR(ARGl,ARG2,ARG3,ARG4)

Computes the argument of the complex gamma function Qra rCaRCI * iAHOZ) s. OACte^ ^

SUBROUTINE HDIAG(H,N,lEGEN,U,NR)Diagonzlies the NXN symmetric matrix H.

SUBROUTINE RMSIGCalculates S and T matrixes aid partial wave cross sections

for the transitions L^S.^L.S.. Also computes the normalizationp 1 ] ]matrix Ap*p**of Equation (2.57) and stores it in the array AB(NA,NA,NA)

SECTION 4SUBROUTINE MAIN4

Rewinds ITAPE and reads in the solutions F which were computed at each point of the integration. Linear combinations of these solution vectors are now formed using the co-efficients computed in SOLVE and which are stored in W(M, NA). The

ISi

result is stored in the array F2 (N/NA,IRAI)CALLS PHOTO SUBROUTINE PHOTO

This subroutine evaluates equations (20) of Ref. (4) to give the photoionization cross sections in the dipole length and dipole velocity approximations. We note that the function gp(E) (equation 2.43a) is approximated by the following express­ion (equation 2.43b)

I C O .V V pi

„ (if t'n) 4- ( z / 1

- f(ipIr '

where we use the same notation as Ref. (41)CALLS INTRAPSUBROUTINE. ŒNTRAP (A,I,J,UK,RHO,IND)

Evaluates the integral f.K » o = ( A

Jq Jusing the Trapezoidal rule.

/69

REFERENCES

1. D. R. Bates, Proc. Phys. Soc. London B 6_ , 805 (1951).2. A. Dalgarno, M. B. McElroy and J.C.G. Walker, Planetary

and Space Science 331, (1967a).3. I.S. Bowen, Astrophys, J., 6^, 257, (1928).4. M.J. Seaton, Proc. Phys. Soc. A231, 32, (1955).5. M.J. Seaton, Reports Prog. Phys. 22, (1960).6. M.J. Seaton, Advances in Atomic and Molecular Physics,

Academic Press, New York, (1968).7. Burgess A, Astrophys. J. 131, (1961).8 . B. Stromgen, Astrophys. J. 108, (1948).9. L. Goldberg, Auto-ionization, edited by A. Temkin, Mono

Book Corp., Baltimore fl966).9a. Leo Goldberg, Andrea Dupree and John W. Allen, Annales D '

Astrophysique, 28 No. 3 (1965).10. A. Burgess, Astrophys. J. 139, (1964).11. M.J. Seaton, Rev. Mod Physics, (1968).12. M.J. Seaton, Lectures in Theoretical Physics IV, Boulder

(1961).13. R. Wildt, Astrophys. J. 89.' (1939).14. Vardya, Mon. N. Royal Ast. Soc. 72' (1967).15. T. Ohmura and H. Ohmura Astrophys. J. 131, 8, (1960).16. Mjosness and Ru el, Phys. Rev. 154, 98 (1967).17. Smith, K. Reports on Prog. Phys. 22.' (1966).18. Massey, H.S.W. and Mohr (C.B.O., (1932) Proc. Roy Soc (London)

A136, 289.19. FesbaCh, H. Ann. Phys. 5, 357, (1958).20. Fesbach, H. Ann. Phys. 19, 287, (1962).21. Fesbach, H. Rev. Mod. Phys. 36, 1076, (1964).

170

REFERENCES(continued)

22. K. Smith, Lecture Notes (unpublished).23. A.M. Lane and R.G. Thomas, Rev. Mod, Phys. 32, 257 (1958).24. O Malley, T.F. and S. Geltman, Phys. Rev. 137, A1344, (1965).25. G. Erect and E. Wigner, Phys. Rev. 22.' 519, (1936).26. U. Fano, Phys. Rev., 122' 1866, (1961).27. U. Fano and J.W. Cooper, Phys. Rev. 137, A1364, (1965).28. P.A.M. Dirac, Physik 21' 585, (1927).29. F.H. Mies, Phys. Rev. 175, 164, (1968).30. B. W.Shore, Rev. Mod, Phys. 39 , 439, (1967).31. G. Racah, Phys. Rev. 63.' 367, (1943).32. K. Smith, R.J.W. Henry, and P.G. Burke, Phys. Rev 147, 21

(1966).33. H. E. Saraph, M.J. Seaton and J. Stemming, (1968) preprint.34. I.I. Sobelman, introduction to the Theory of Atomic Spectra,

Moscow translated by the U.S. Department of Commerce (1963).35. C.C.J. Roothan and P.S. Kelly, Phys. Rev. 131, 1177, (1963).36. E. dementi, IBM, J. Res. Develop. 9, 2 (1965).37. K. Smith, M.J. Conneely and L.A. Morgan, Phys. Rev 177, 196

(1969).38. A. De Shalit and I. Thalmi, Nuclear Shell Theory (Academic

Press, New Yôrk), 1963.39. Rotenberg, M.R., N. Metropolis and J.K. Wooten, The 3-j and

6-j symbols. The Technology Press M.I.T., Cambridge, Mass.40. R.J.W. Henry, P.G. Burke, and A-L Sinfailam, Phys. Rev.

(in press).41. R.J.W. Henry and Lester Lipsky, Phys. Rev. 153 51-56, (1967).

m

REFERENCES(continued)

42. H. Bethe and E. Salpeter, Quantum Mechanics of Ohe and TwoElectron Systems, Academic Press Inc., New YOrk (1957).

43. S.J. Czyzak, T.K. Kruegar, H.E. Saraphyand J. Shemming,Proc. Phys. Soc. (London) 92.' 1146, (1967).

44. S.J. Czyzak and T.K. Kreugar, Proc. Phys. Soc. London 90,623, (1967).

45. M.J.Seaton, Rev. Mod. Phys. 30.' (1958).46. P.G. Burke and H.M. Shey, Phys. Rev. 126, 163 (1962).47. R.J.W. Henry, J. Chem. Phys. 4 8 , 3635, (1968).48. F.J. Comes and H.Elzer, Phys. Letters 25A, 374, (1967).49. F.J. Comes and H. Elzer, Z. Naturforsch, 23A, 133, (1968).50. P.K. Carroll, R.E. Huffman, J.C. Larrabee and Y. Tanaka

Astrophy. J. 146, 535, (1966).51. Dalgarno Henry and Stewart, Planetary and Space Science,

12, 235, (1964).52. B.H. Armstrong, R.R. Johnston and P.S. Kelly, Technical

Report No. AFWL-TK 65-17, Lockheed Missiles and SpaceCompany, (1965).

53. M.G.Kozlov, E.I. Nickonova and G. P. Startsev, SovietPhysics Opt. and Spect. 21, 298, (1966).

54. L. A. Vainshtein and G.E. Norman, Soviet Phys. Opt andSpect. 8 , 79, (1960).

55. A. Burgess, G.B. Field and R.W. Michie, Astrophys. J. 131,529 (1960).

56. G. Peach, Mont. Not. Roy, Astron. Soc., 124, 371, (1962).57. S. T. Manson and J. W. Cooper, Phys. Rev., 165, 126 (1968).58. J.C. Rich, Astrophys. J. 148, 275 (1967).59. A. Burgess and M.J. Seaton, Mônt.Not. Roy. Astron. Soc. 120,

121, (1960).60. K. Smith, Sommerfeld Memorial Meeting, Munich, Sept. (1968),

to be published.

/7ZREFERENCES(continued)

61. R.E. Huffman, J.C. Larrabee and Y, Tanaka, J. Chem. Phys.47, 856, (1967).

62. E. J. Robinson and S. Geltman, Phys. Rev. 153, 4, (1967).63. Yu. V. Moskvin, Soviet Phys. High Temp. 2' 765 (1965).64. J.W. Cooper and J.B. Martin, Phys. Rev. 126, 1482 (1962).65. R.S. Berry, C.W. Reinmann and G.N. Spokes, J. Chem. Phys.

37, 2278 (1962).

ReprirUed from

P H Y S I C A L R E V I E W V O L U M E 177, N U M B E R 1 5 JANUARY 1 9 6 9

Trial Wave Functions in the Close-Coupling Approximation*

Kenneth Smith, M. J. Conneely, and L. A. Morgan B ehlen L ab o ra to ry of P h y s ic s , U n iv ersity of N ebraska, L incoln, N eb rask a 68508

(R eceived 13 M ay 1968)

The th e o re tic a l and n u m erica l consequences of choosing d iffe ren t t r ia l w ave functions in the c lo se-co u p lin g ap prox im ation a re c o n sid e red . In p a r tic u la r , ca lcu la tio n s a re c a r r ie d out on the s c a tte r in g o f e le c tro n s by a tom ic sy s te m s w ith configu ra tions 1 ^ 2 ^ 2 p ^ , g = 2 ,3 ,4 , w h ere p ro p e rly an tissm am etrized bound configu ra tions ls^ 2 s* 2 p ? + l a r e m ixed w ith th e t r ia l continuum wave function. An a lte rn a tiv e tre a tm e n t, w hich e lim in a te s the need fo r such bound con fig u ra tio n s i s p re se n te d .

I. INTRODUCTION

In order to calculate the cross section for a scattering process within the framework of non- relativistic wave mechanics, it is necessary to approximate the Schrodinger equation. One of the most successful approximation schem es is to ejq)and the over-all wave function of the projectile plus target in term s of the complete set of (as­sumed known) eigenstates of the target Hamilton­ian, see Burke and Smith J This method has come to be known as the “close-coupling approxi­mation” because only a few of those atomic term s close to initial and final states are retained in the eigenfunction expansion. The unknown coef­ficients, , in such ah expansion are Hie solutions of coupled second-order integro-diffef- ential equations with prescribed boundary condi­tions, and are interpreted as describing the radial motion of the projectile relative to the target.

If there are N such coefficients, then the Schrodinger partial differential equation has been replaced by an N-channel problem in one dimen­sion, r . Those channels in which % oscil­lates as r — are said to be “open”, and corres­pond to continuum orbitals; the remaining (N-iV^) channels are said to be ‘‘closed” since the asso­ciated f (g will vanish exponentially as

The theory of the. scattering of electrons by hydro­

gen atoms, within this approximation, was carried out by Per cival and Seaton^ and extended to hydr ogèn- like ions by Burke, Me Vicar, and Sm ith / Two points of these papers are of interest to us here. Firstly, the eigenstates of the target Hamiltonian are known exactly; and secondly, the e^ansion coefficients were not orthogonalized with respect to the boimd target orbital. The fact that the target eigenstate is a single exactly known orbital means that antisymmetrization of the target is a non-existent problem and no errors are introduced into the cross sections because of the bound orbital. However, since only a few slates can be taken in the expansion, only a part of the full effective polarizabilities are included.A technique for improving the calculations has been suggested by Damburg and Geltman.'* These authors recommend replacing the three degener­ate n= 3 hydrogenic target orbitals with an alter­native non-degenerate trio, which results in the full effective polar izabilities being accounted for. lu other words, even in the sim plest coUision problem, me one-electron orbitals of the target are a source of discussion.

Extensive calculations for electron collisions With positive ions with configurations is ls®2s??p* have been carried out by Saraph ef uf.® They imposed the condition that the radial functions should be ortho^nal to the bound one-electron

196

197 C L O S E - C O U P L I N G A P P R O X I M A T I O N

orbitals with the same orbital angular momentum Physically, this can be interpreted as exclu­

ding the projectile from being captured, virtually, into incomplete subshells. This restriction on their total wave function was removed by super­imposing a second configuration, 0 , constructed entirely from bound orbitals. Their trial wave function in the close-coupling approximation was taken to be

P..J J J J

+ Of. (1)

problem. The same approximation was made with r e j e c t to the orbitals of the two configura­tions I s 2 ^ 9 eZ +1s22s 22 ^+ 1; however, the analytic self-consistent-field (SCF) functions of Roothaan and Kelly® were used in the numerical calculations of Smith et al.^° and of Rudd and Smith.“ This method involves the evaluation of matrix elements

(4)

using the notation of Ref. 5, where F is the complete set of quantum numbers for the system and is a vector-coupled antisymmetrized function constructed from one-electron orbitals. This same method was used by Czyzak and Krueger.® Equation (1) can be interpreted as configuration interaction: In the first term one of the electrons is in a continuum orbital, while all electrons are in bound orbitals in the (j> term. Saraph e t al. use the same orbitals for the term s Sjl^- as well as in the 0 configuration; these orbitals were calculated using a computer code written by Froese.^

The underlying reason behind choosing 0 to be constructed from the same orbitals as 0' is that the F 's are constrained to be orthogonal to these orbitals, yet without the constraint F would con­tain a component of the bound orbitals. The unconstrained continuum orbital could be written

F =F+a^P, {FP) = 0. (2)However, permitting the projectile to be captured into the 2p subshell means that there are now {q +1) equivalent electrons, not q . in the 0 term of E q.(l).

Seaton’s method, described above, for choosing the trial wave function in the close-coupling approximation in the scattering of electrons by atomic system s with incomplete p subshells has also been used by Smith e t oZ.® (SHB) for the same

in the notation of Ref. 8, who used C for the variational parameter denoted by ot here. In their evaluation of these matrix elements. Smith e t aZ.® used the orthogonality, Eq. (2), between P and F, which would not be possible if the bound orbitals of the actual + l configuration were used in 0 . Furthermore, they chose to anti- symmetrize the additional p electron in 0 as being inequivalent to the other q electrons. That is to say, the {q f l) th p electron in 0 was treated on the same footing as the F electron in 0 . This results in 0 not being fully antisymmetrized under the interchange of any pair of electrons. The purpose of this paper is to present the fully anti­symmetrized function and to examine its effect on the results of Ref. 10 and 11, as well as to compare with the results of Ref. 5.

In Sec. 2 we evaluate the matrix elements (3) and (4) by specializing the general results of Smith and Morgan, referred to henceforth as SM, to the single incomplete 0 - configuration problem. The results obtained are verified in Sec. 3 using the methods of deShalit and Talmi.^® In Sec. 4 we re-examine the problem of choosing the trial function by discarding the restriction that the close-coupling functions F must be orthogonal to the bound-state orbitals P. Finally, in Sec. 5 we discuss the numerical effects of various 0 symmetrizations on the scattering of electrons by atomic system s with configurations

n . RECOUPLING COEFFICIENT METHOD

The theory of the scatterir^ of electrons by atomic system s with any number of incomplete subshells has been developed within the Hartree-Fock, or close-coupling, approximation by Smith and Moi^an.^^ These authors assumed that the bound-electron orbitals are independent of the configuration. They impose the orthogonality condition, Eq. (2), with the result that the term s linear in a. (see Eq. (54) of SM) reduce to

k,= + (ZZ|l/r|M>) (5)

which is the many- configuration generalization of Eq. (3), and where the two term s are defined in Eqs. (56) and (60), respectively, of Smith and Morgan. The sum p runs over all the incomplete subshells included in the eigenfunction expansion. We now restrict consideration to a sii^ le incomplete p Subshell, as in Saraph e t ah, ® so that Only the p=wp term appears in Eq. (5) .

The matrix element of w ill be nonzero only when the interactii% electron is in the incomplete wp subshell. In the notation of Smith and Morgan^ p =np =b^,N p =q +1, Spl^ -S , LplJ--L, Sp^ =S{, and ip* ~ L i,

S M I T H , C O N N E E L Y , A N D M O R G A N

s in c e a l l o th e r s u b s h e l l s a r e c lo s e d . C o n se q u e n tly , th e f i r s t t e r m o f E q . (5), r e d u c e s to

~W t) ] " ,5| (S.i)s> (L.l.,L I iL.p)L). (6)x_Î

Since there is no recoupUng of the ai^ular momenta, both recoupling coefficients are unity. Equation (6) leads to a different leading term in Eq. (24) of Smith e t oZ.» in that (g +1)^replaces (N+l)% where N is the total number of target electrons.

In the evaluation of the second term of Eq. (5), we note that =N\I^ for all subshells except X=2p.In the notation of Smith and Morgan, the outermost interacting subshell in # i s o =«/>^and we must have Pi =Pp=P, which is summed over all the subshells. Furthermore, we have AP = 0, 5^= 8, L (j= Li,So and Lçf = L. For € = 0, we have v =p and t = np, which gives the spin and orbital re coupling coefficients to be

<s,. I s^> ”= < (Sp-i)Sp.. . S i , is | (Sp4)Sp.. . (S.4)S> = i

<© I e^> . . l< ïp*p)ip.. . .

which is precisely the same form as the direct recoupling coefficient [se e SM Eq. (49)]. The dots in Eq. (7) represent the quantum numbers, either spin or orbital, of the spectator subshells lying between the two interacting subshells X =Zp and X=Z%.

If € = 1, then 77=wp and £ =p, resulting in the recoupling coefficients

<SjS^>^ = ( [^ i(N ) ]S ^ ., . , i (N + l) ;S l[y (N + l) ] .^ .. . , i (N ) ;S > " (g)

<© J e > '=< [Ip, (zi)z] i-p... ![ » (ï'V y V ‘ 'These re coupling coefficients are precisely those which appear in the exchange term [see SM Eq. (40)], since the order of coupling is the same in both sets of coefficients.

Substituting Eqs. (6), (7), and (8) into Eq- (5) gives the fimction ’Vi {r), defined in Eq- (53a) [printed following Eq. (59)] of Smith and Morgan, as

r /r ) = ( « - D" |ô ^ ^ j{{ÿ ^ * i£ S |} /.% s,^ ) [ ( - | | t ) - P ^ W

2(2îp+ -5 S ,(2 < + D - ' (/plGOM^r-

+ 3 , S e , ,( /* l£ S |J ÿ « X 'S ^ )[ (2£ '-H )(2£jt 1 ) ] ! ? , / -L '8 ' i ---J. : ,;ri; ;

X ( ^ ^ 2 , ^ (11001ZO) W( 12,'Z 2^ ; I,Z ) ; Z,gZ )

X (iz^ 001ZO) [(2Z + 1)3]^ Y fr p np;r)P^p{r) | . 0 )

■ ' ' ' ii ' iz ■This differs from Eq. (24) of Smith e t al.^ in the over-all {q +1) factor replacing (N+1) and in the sum over t in the sqimre brackets, which was missing in Ref. 8. Equation (9) does agree with the results o f Shemniihg,^ which presumably were used in Refs. 5 and 6. . v v ,j r .

The incomplète: antisymmetrization # f # used in Ref. 8 a lso ,affects the matrixjelpments defined in Eq.(4). This ihaitrix ëlhnîWt is'eYal^atCd in Wo parts, ^ Aitn) hn^ fhther tlW cbniputing Eghq in

n = l - 'SM. Starting With Eq. (61) of SM, the first part can be shown to be

np Ç / Z (Z +1) \ #p = I s p = I s

199 C L O S E - C O U P L I N G A P P R O X I M A T I O N

np

p > a(s subshell)

L p subshell VA^p'^OO W ' b ”'p)a p

_ _S_ ,(P«"^£sfp%s J.)(p?"i£sjp»i V) L X .L 'S S ' * *

PAi^ 1/1» JbXIF(L.1L 1;L'2)TF(L.1L 1;L2)R Of/»"), (lo)

X l ) ( 2f. +1)]:

while the second part is s t ill given by Eqs. (62) and (62a) of SM. It is the first part which is different from the corresponding first two term s of Eq. (25) of Ref. 8, which are simply

np

One of the principal advantages of working with Eq. (10) rather than SM Eq. (72) is the fact that it becomes unnecessary to know the energies of highly excited configurations. A detailed proof that the work of Saraph e t al. and Smith e t al. when the above antisymmetrization is taken into account, are proper subsets of the formulation of Smith and Morgan is given in Morgan.^® That is to say. Refs. 5 and 8 are special cases of Ref. 12, which permits any number of configurations to be included in the trial wave function.

111. OPERATOR METHOD

Although the work of Smith and Moigan had brought to light the incomplete antisymmetrization of Ref. 8, it was thought necessary to verify the expressions for the matrix elements, Eqs. (3) and (4), given in the previous section by using an alternative method. The method chosen was the antisymmetrization opera­tors Ajv of deShalit and Talmi.^^ In this method the (iST+l)-electron wave function for the bound term in Eq. (1) is

\ l s h s h p ^ ^ ^ ; L S ) = -----------— ------------- 0 (ls^ l2 )0 (2 s^ 3 4 )0 (2 /'" 5 ,. . . ,N+ 1;XS), (11)[2!2!(N+l)!(q + l)!]

where the g ' s are properly antisymmetrized for the electrons in that subshell, but effects theantisymmetrization between the su tehells. Using the commutators

[Ay, I» = 0, and 1] = 0. (12)

gives the same matrix element as Eq. (6). In order to evaluate the matrix elements of the two-electronoperator r we consider closed and open subshells separately, leading to the term s in the

a = l ^+^2® p sum of Eq. (9).

Finally, to compute the term s quadratic in or, we consider matrix elements between closed subshells, between open subshells, and then the cross term s between open and closed. For the interaction of open,I jj with closed, Zg, we find that the matrix element of the two-electron operator is independent of the total L and is given by

7(1,1,£) (13)

where and are the usual Slater integrals, and/^ and are geometric factors e ^ r e ss ib le in Racah algebra. For the matmx elements between a pair of closed subshells we find that the contribution

S M I T H , C O N N E E L Y , AND M O R G A N 200

to Eq. (3) is

(2Z+1)(2Z'+1) r 2 F ° (« /w r ) -S ( - + (14)L \0 0 0 / J

N+1The matrix elements of Hi(j% ) were determined as in the previous section, and the over-all results

Û!— 1confirmed the correctness of Eqs. (9) and (10).

IV. VIRTUAL CAPTURE

In the Introduction it was emphasized that when the close-coupling functions were orthogonalized, Eq.(2), with respect to the bound orbitals, it was equivalent to neglecting virtual electron capture into open subshells. For this reason, configuration mixing via the # term s was introduced in Eq. (1).

In this section, we shaU discuss briefly the elimination of the orthogonalization constraint on the F ' s relative to the bound oribitals of the incomplete subshells; a complete discussion w ill be given e lse ­where.^® This discussion w ill be within the framework of the techniques of Fano,^® and leads to the elimination of the so-called #- dependent term s of Refs. 5, 6, 8, and 12. Consequently, virtual capture is now implicit in the close-coupling functions F.

The principal advantage of the orthogonalization procedure was the elimination of factors in the ex­change term as a result of the vanishing of the overlap integral of F with a P. (It is clear that such an overlap does not occur in the direct terms, since both F ' s are found in the same integral. ) We now have to derive an alternative expression for SM Eq. (25), with the orthogonality constraint expressed in the form

J d r F _ (r)P^ (r) = A Uj;X)ô^ . , (15)

for the incomplete subshells P \ , but the A still vanish for the closed subshells.N + Î

We begin by evaluating the matrix elements of the one-electron operators (a) and note thata = l

only those for N - 1 contributeequaUy. If we let and M j be the subshells containing the labels Nand N + l in the distributions and q j, respectively, then using the methods of SM we can show that

t J

W , (16)

with one less integral for H^{N) and ^^(N+l) on account of F and i l being interchanged. The C factors are defined in Ref. 15. ^

A sim ilar separation of the two-electron operator can be effectedN + 1 1 N —\ jy~ 1

Uj ' w ^ i - ‘>- O’)The final matrix element is given in SM Eq. (41), and the remaining three matrix elements can be readily evaluated by the same method after using symmetry properties.

The variational principle of SM Eq. (73) is now replaced by

where 3Rx are the Lagrange xmdetermined multipliers chosen to ensure the orthogonality of the con­tinuum with # e closed subshellSj and (r) is an integral operator on F with term s from Eqs. (16) and (17). The associated Euler equations are

S x . .F . ,+ S jL /(r )+ E W = 0, (19)j % j ^ X, closed ^ ^

where each of the factors of the inhomogeneous X term is of the form

x ir)=9j >riAy, (20)

201 C L O S E - C O U P L I N G A P P R O X I M A T I O N

where S (r) are known functions^® and A ^j are unknown numbers dependent on F. If there are N V such inhomogeneous term s, then we can generate NFparticular solutions of the inhomogeneous system by setting the A 's in turn equal to unity, the remainder zero. This is a variation of the numerical method developed in Ref. 8 and is equivalent to that used by Burke and McVicar^’’’ for the electron- hydrogen-like-ion problem.

V. NUMERICAL RESULTSIn the preceding sections two changes have

been proposed in the formalism. Firstly, there is the problem of antisymmetrization, which is now properly handled in Secs. 2 and 3; and secondly, there is the alternative approach to virtual capture described in Sec. 4. Here we shall report conipütâtions carried out on the effects of antisymmetrization.

In Table I we present results for the collision strengths of electrons Incident on a variety of ions, as computed with both the partly and fully antisymmetrized ^ functions, and we compare with the results of Saraiih eZ al . We emphasise tla t the boUnd orbitals for the electron problem have' bëen used in both # and of Eq: (1). Fur­thermore, in the computation of the matrix e le ­ment, given in Eq. (4), we have written

(21)where are the quantum numbers of the lowest term, andLhe actual expression for % is obtained from Eq. (10) . Two conclusions can be drawn from Table I. Firstly, the results do depend markedly on the S3rmmetry of (j> ; and secondly, the quite different numerical procedures and 'i ;, bound orbitals of Refs. 5 and 8 lead to the same collision strengths to within a few percent. When the theoretical 'values of Roothaan and Kelly are used in Eq. (21), the results are hardly affected at all.

The large change in the collision strengths méntiOHéd above, brOughh about by the # l F ahti- symmetrization of 4> ra ises the question of its

TABLE I. C ollision S treng ths, S2(«nO.

r ION n n ' <f) (SHB) : \ ' Ref., 6

9f W 6.342 3.292 3.203' =; Is-*-I - 0 ,677 ' 0 ^ 2 8 0.424 ' 33p 0.931 0.428 -

0++ 3p 5.016 2.545 2.398^ '0.438 0.294 0.319®p ‘s 0.796 0 :369 , ; 0;84S 1

TABLE II. ,4ZaluéB of■ conserve!d qimntmnrnh^uehced b y the virtual # terin.

(Zj) R ef. Target (2S+1)jT

129

0+cN3p« Ic®

o 2pO -

0 O C v | C ' J O ) W ' 2 * C D ' ^ C O r - l r - ( E ^ " ! t ' C < l f - ( l î 5 e D ï O t M U 5 0 0 00O —li-l.-IOOOOWlQOOOOWOt-tCOOOOiHo o o o o o o o o o o o o o o o o o o o o oo’ â ® à ® ® à® S o' S ® B ®" s . S- S ® 2.

r-t'^TO;DO'-(OOtOOOOi-tOOOOCOOOON»-<'^ O O O O O O O O O O O Ô Q O * - I C < I O O O O O OO 2.0 2® 2® 2® 2® 2 ® 2° 2® 2® 2® 2-Hc5‘otoasr-tu5'co«iMS"aoS'i-iP‘s>£~w?5'ioP U5 O ® eo T-l m rj< CO O Oi O N U5 t - m r-t ÏO e>» 00O i - I O I M O O O O r ^ r - I O O . H O C O C O O ' H O O O O

® 2 9 2 ® 2 ® 2 ® 2 ® 2 ® 2 ® 2 ® 2 ® 2 ® 2

t-Olf3f-0>«f3COOt~^C<«ft'S*OOt-COrH'2'»OtDtOp ® Ifl ' » 00 CO OO N 05 4Ï3 C~ r-t N ■>0' O US r-l eO N M 03 US W O T - < r - I C 0 O O O O r - t r H O O O O e 0 U S r H < K I C > O O > - t

® 2® 2® 2® 2® 2® 2 ® 2® 2® 2® 2® 2

u s o t - i t o i > o c J i H 0 5 U s e a ' ^ o o ' ^ - < o o o o 5 co o> usU S i - H. - t ' ^ t O C i l ' i ' c O i - t U S - i ' C S Or H i H OO S O S C O N t - t OT - t e a « O U S O r - t C S r - l < M ( M O O t - t O U S C ' r ^ ( N O r - < O i - t

O o o o o o o 2,0) 2,0 2,0 2.0 2 ® 2 ® 2 ® 2

N > - H C O O S 0 5 c B ' a 5 i n u s f - ' ^ f ' U S 0 5 0 5 t D —ICOOO 05 CO 04a § S 3 S3 3 .s S 833SS8N33S3® 2® 2® 2 ® 2® 2® 2 ® 2® 2® 2 ° 2® 2

E - O O C ' O O e i l O S C O a O t - N C - O O C D O S T l t r - I O s É - C O ' ^ ' ^ Oc~co«3us i Ml >oot ' OUSaoc»s i - i <N<j j o i >c<i usp- i oa3 e.os <n 00 M M O co o o N o t-: o in ps o m m m0.2® 2® 22 2® 2® 2® 2® d-® 2® 2® 2

'PO>«S^MtOC-COUS«OeOCOO>OOi-l®t'Mtf>^-pM C 0 0 0 t ' t - ' j * 0 3 O 0 0 ’ J ' t ~ p > P S C 0 PS O t > 0 3 OO u s 0 0 © u s

, PS PS O 0 3 M r 4 M M CO PS o O M O 0 3 O OS M O O M . - I® 2 ® 2 ®‘ 2 ® 2 ® 2® 2® 2® d- ® 2 ° ® ® °OOr-l t-O’pS®C-«USr-4OT'TPO'<NC0O03!»o' 0 3 M C S O O O P 3 t > 0 3 0 0 C — C S M U S t M O O O O U S C O O O C - ^TfoOMMCMMMPSPSOOINOOi-IINMOOOO© 2.0 M 06 2® 2® 29 20 2'-' d® 2® 2® 2<O^PS^'=)*S3aO©o2 O O r - H i —l O O O O O O

r-4^*^if3«£>000 f*0>aïfHe5CàOOrd^r-lO>tD;O B"" d ® 2 2 2 ® 2

USO-MUSéoC-UÎoO?a c s o o J - ^ o o o o o u sD O M < S I i- I O O O O ' 1 * Md2 2"® 2®2':.o o e- oIN PS P- TM O PS M 00 .

aT'- ÆS-:r‘v'

''--4-'- a," i Q ’e'i; >o sdj hi :--srnr

S M I T H , C O N N E E L Y , AND M O R G A N 202

TABLE IV. P a r tia l-w a v e co n tribu tions to the e la s tic c ro s s sec tio n s . N um bers in p a re n th e se s a r e from R ef. 10.

T arg e t P ro c e s s SLtt K/= 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.8 1.0c 3p_,3p 450 1.989 2.246 2.263 2.182 2.063 1.804 1.570 1.373 1.078 0.877

(2.462) (3.005) (3.096) (2.950) (2.735) (2.215) (1.754) (1.392) (0.914) (0.658)2pO 10.623 6.429 4.157 3.001 2.429 1.829 1.496 1.275 0.996 0.829

(3.898) (3.747) (3.344) (2.804) (2.364) (1.681) (1.208) (0.884) (0.516) (0.344)ZjD» 8.564 5.257 4.258 3.703 3.317 2.779 2.407 2.129 1.744 1.493

(4.163) (4.169) (3.970) (3.629) (3.240) (2.501) (1.916) (1.488) (0.965) (0.703)‘D - 2p0 0.644 2.413 2.883 2.939 2.854 2.626 2.406 2.212 1.896 1.658

(0.893) (2.215) (2.858) (3.060) (2.994) (2.598) (2.192) (1.854) (1.397) (1.132)2pO 0.350 1.831 3.100 3.619 3.788 3.722 3.492 3.244 2.814 2.448

(0.404) (1.393) (3.194) (4.064) (4.297) (3.907) (3.253) (2.642) (0.803) (1.355)is-Is 2p0 0.026 3.143 5.852 8.684 9.727 9.972 9.540 8.754

(0.028) (3.321) (6.364) (9.124) (9.693) (9.367) (8.197) (7.146)

O 3p_3p 2p0 9.209 4.463 2.727 2.103 1.778 1.336 1.108 0.962 0.774 0.655(1.017) (1.372) (1.485) (1.530) (1.523) (1.407) (1.254) (1.096) (1.830) (0.619)

'D-'D 2p0 0.104 0.298 0.566 0.696 0.732 0.746 0.736 0.693 0.653(O.llQ (0.050) (0.23^ (0.431) (0.697) (0.800) (0.808) (0.725) (0.640)S- ‘S 2p0 2.928 5.119 6.045 6.405 6.143

(0.692) (1.643) (2.393) (3.244) (3.663)

N S- zpe 25.696 8.279 5.119 4.110 3.576 3.206 2.693 2.344(5.349) (6.648) (5.373) (4.319) (3.540) (2.956) (2.146) (1.654)

2D- 3p« 1.250 1.513 1.526 1.464 1.293 1.136(0.965) (1.607) (1.696) (1.581) (1.237) (0.956)

ijf 1.529 1.790 1.739 1.596 1.285 1.033(1.596) (2.248) (2.174) (1.899) (1.316) (0.893)

2JP 3pg 0.244 1.073 1.536 1.760 1.874 1.812( . . . ) (0.421) a.2S8) (1.675) (1.835) (1.708)0.131 0.644 0.949 1.094 1.155 1.101

( . . . ) (0.261) (0.786) (1.035) (1.100) (0.998)

influence on the positions of the autoionized levels of oxygen, as reported in Rudd and Smith, " and on the cross sections presented in Ref. 8. In Table n, we lis t those values of the conserved quantum numbers which are affected by the 4> problem.

In Tables IH and IV we present the partial-wave contributions to the excitation and elastic cross sections, respectively, as computed with the fully antisymmetrized <p, and compare them with the results of Ref. 10, which are given in parentheses. The general feature of these results is that at lower impact energies the new #> produces larger cross sections than the partly antisymmetric while at the higher energies the new results are lower. In this regard the new results are in no better agreement with the experimental total cross sections for electrons on either oxygen or nitrogen. Of course the effects of excited configurations which have been neglected here, may well have a significant influence. We note that tiie relative phases of some of the coefficients of fractional parentage used in Refs. (10), (14) and SobeT mai?® are incorrect due to incorrect use of Eq. (19) in Racah“ .

In Table V we present the results of the two partial waves in electron-0 ’ scattering which are affected by the antisymmetrization problem. For autoionized 3pe states we see that the new #> r e ­sults are consistently higher than those of Rudd and Smith by about 0 .1 ev. However, one notable feature of the new results is that we have been able to locate the low-lying resonances found by experiment, as well as to identiftr the

very narrow / ser ies. For autoionized states we see that the new p has very little affect at all. Indeed, since the theoretical results of Rudd and Smith were obtained on a GDC-6600, while the present results were computed on an IBM %stem 360/65, the difference due to (j> is indeed slight.A new feature of the results presented here is

TABLE V. P o s itio n s of au to ion ized s ta te s in a tom ic oxygen.

L S rA ssign­m en t n old # #

E x p e ri­m en t

1 ,1 , even ^Pnp 3 15.768 16.0174 15.774 17.305 17.4069 17.315 17.849 17.91036 18.121 18.21677 18.278 18.401

^Dnp 3 14.461 14.164 15.729 15.625 16.121 16.204 16.226 16.312 16.48187 16.5074 16.0875

2 ,0 ,even ^Pnp 4 17.387 17.4035 17.908 17.9036 18.135 18.1597 18.2858 18.366

^Pnf 4 17.7865 18.0946 18.2557 18.380

203 C L O S E - C O U P L I N G A P P R O X I M A T I O N

that we have been able to isolate several members of the ser ies.

In conclusion we note that the numerical results presented in this paper are based on a formalism which largely neglects polarization of the target by the impinging electron. To account for such polarization, we could include excited configura­tions shch as 1s^2s2p^ +1 and ls^2s^2tA - Isd, etc. as in SM, or we could take the dipole distor­tion of the atom produced by the incident electron into account by using the method of polarized orbitals as used by H en ry .C a lcu la tio n s based on the SM formalism are under way, and compari­son w ill be made against the results of the polar­ized orbital method. Since no rigorous criteria exist for complex targets for selecting one ap­proximation scheme over another, one can only

compare theoretical predictions with known experimental results to determine which features of the model are important before computing physical processes which have not been investi­gated experimentally.

Note added in proof: Henry et al. (unpublished) have carried out the calculations in Table IV to lower energies and have analyzed the low-energy peaks predicted here as “ shape” resonances.

ACKNOWLEDGMENT

The authors are indebted to Professor M. J. Seaton for discussions and a helpful exchange of correspondence.

♦W ork su p p o rted by the N ational Science foundation , the U. K . Science R e se a rc h Council and by th e A ir F o rc e W eapons L ab o ra to ry , K irtlan d AFB, New M exico, und^r c o n tra c t No. A F29(601)-68-C -0052 w ith Quantum S y s tW s , In c ., A lbuquerque, New M exico.

'p . G. B urke and K . Sm ith, Rev. Mod. P hys. M , 458 (1962).

% C . P e rc iv a l and M. J . Seaton, P ro c . C am bridge P h il. Soc. a , 654 (1957).

®P. G. B urke, D. D. M cV icar, and K. Sm ith, P ro c . P h y s . Soc. (London) 83, 397 (1964).

‘‘r . j . D am burg and S. G eltm an, P h y s . Rev. L e tte r s 485 (1968).

SR. E . Saraph, M. J . Seaton, and J . Shem ming, P ro c . P h y s . Soc. (London) 89, 27 (1966).

®S. J . C zyzak and T . K. K ru eg er, P ro c . P h y s . Soc. (London) 90, 623 (1967).

^C. F ro e se , C anad. J . P h y s . 1895 (1963).®K. Sm ith, R . J . W. H enry and P . G. B urke, P hys.

Rev. m , 21 (1966).®C, C. J . R oothaan and P . S. K elly , P h y s . R ev. 131,

1177 (1963).**K. Sm ith, R. J . W . H enry and P . G. B urke, P h ys.

R ev. 157, 51 (1967).^^M. E . Rudd and K. Sm ith, P h y s . Rev. 169, 79

(1968).^^K. Sm ith and L . A. M organ, P h ys. Rev. 165, 110

(1968).‘®A. deShalit and I. T alm i, N u c lea r Shell T h eo ry

(A cadem ic P r e s s In c ., New Y ork 1963), p . 470.Shem m ing, P h .D . th e s is , U n iv ersity o f London,

1965, (unpublished^.^®L. A. M organ, P h .D . th e s is , U n iv ersity of London,

1968, (unpublished).« U . Fano, P h ys. Rev. 1 ^ , A67 (1965).^^P. G. B urke and D. D. M cV icar, P ro c . P h y s . Soc.

(Londor) ^ 989 (1965).**I. I. SobeTm an, In troduction to the T h eo ry of Com ­

p lex S p ectra , (Moscow 1963), T ra n s la te d by th e F ore ig n Technology D ivision, W r% h t-P a tte rso u A FB , 1967.

»G . Racah, P h y s , Rev. M , 367 (1943).20r . j . W . H enry, Phys. Rev. 162, 56 (1967).

«3»

: PnOTOIONlZATION OF ATOMS WITH CONFIGURATIONS

M. Conneely, L. Lipsky, K. Smiih

(Royal Holloway College, Englofield Green, Egliam, Surrey, England).

The abundance of some atoms in the second row of the periodic table (Al, Si, P, S) in the sun is comparatively large, being approximately one tenth that of C, N, or 0 but ton times that of Fe [1]. Therefore, it is expected that their contributions to the solar spectrum will be significant. In particular, the photoabsorption cross sections for Al and Si are large

0.1

Fig. 1. Photoionization cross section cf of (3j>) silicon.1 — experimental; 2 — this work; 3 — quantum defect.

enough to virtually obliterate light originating in the photosphere for cer­tain parts of the spectrum. Recently, shock-tube experiments have been performed to determine these cross sections [2]. General interest in the inert gases has caused several experimental groups to measure the photoabsorp­tion of argon [3, 41. I t is therefore useful to have theoretical results avai-

0.10. 0.2 0.3 U E,Ry

Pig. 2. PEotoionization cross section ff of silicon.• 1 — this work; 2 — quantum defect.

lable for the (3p)^ elements. The quantum defect method [5] has been used extensively to calculate cross sections, but it becomes unreliable, when ap­plied to the 3p-shell. This is due to the fact that the quantum defects of the singly excited states of these (and heavier) atoms are not smooth func­tions of R, the principle quantum number, and therefore it is difficult to extrapolate to above threshold.

The procedure followed here, in calculating the'photoionization cross sections, is described in Henry and Lipsky [6J. Briefly, we calculate the wave functions for an electron scattering of a particular ion, assuming L —S

■ 619

\

coupling» using the close coupling code of Smith, Henry and Burke 17], These functions are then used to calculate the dipole matrix elements between states of an outgoing electron in a particular channel and a state of the ini-

Fig. 3. Photoionization cross section a of (3p)® silicon.1 — experimental; 2 — this work; 3 — . guantum defect.

tial atom. The analytic Hartree—Fock functions of Clementi [8] are used to represent the wave functions of the atoms and residual ions.

Some calculations have already been completed, and the results for >. silicon and argon are reproduced in fig. 1—4, together with available experi-

10 -

0 0.331

■ Fig. 4. Photoionization cross section G of argon, tt I — experimcat; 2 — tills work; 3— 3s3p* edge.

mental data. The e,xperimental data as well as the quantum defect curves for silicon aré taken from Rich [2]. The argon data was given to the authors by Ederer et al. [4], in advance of publication. The dipole length calculations agree-fairly well with experiment for energies below the excited state thresholds, whereas the dipole velocity approximation gives values Vg to /g that of exjperiment. .

R E F E R E N C E S

1. L. G c l d b e r g, E. Q. M u l l e r, L. H. A l l e r , As trop. J. SnppL, 5, 1 (1900).2. J. C. R i c h , As trop. J , (to bo published).3. J. A. R. 8 a m s 0 n, G* C. A. Technical Report, N® 63-N, (1964).4. p . E d e r e r , K. C o d l i n g , R. M a d d e n (private communication).5. B u r g e s s , S e a t o n , Rev. Mod. Phys. (19C0).6. R . J. W. H e n r y , L. L i p s k y , Phys. Rev. (1967).7. K e n n e t h S m i t h , R. J. W. H . e n r y , P. G. B u r k e , Phys: Rev., !*<»

21(1966), .8. E . C I e rn e n t i. Table of atom ic functions, I . B. M. J . Res. Develop., 9, 2 (I9C5).

V