transport properties of charged particles in low pressure

Transport Properties of Charged

Particles in Low Pressure Plasmas

Yunchao Zhang

A thesis submitted for the degree of

Doctor of Philosophy

of The Australian National University

August 2016

Declaration

This thesis is an account of research undertaken between February 2013 and August 2016

at The Research School of Physics and Engineering, College of Physical and Mathematical

Sciences, The Australian National University, Canberra, Australia. I hereby certify that,

except where acknowledged in the customary manner, the material presented in this thesis

is, to the best of my knowledge, original and has not been submitted in whole or part for

a degree in any university.

Yunchao Zhang

August 2016

iii

iv DECLARATION

Acknowledgements

The past three and a half years are and always will be one the most wonderful times in my

life. Foremost I would like to give sincerest thanks to my supervisors, Christine and Rod,

who gave me the opportunity to be a part of SP3 family and guide me in the wonderland

of plasma physics. Your creative thinking and insights into science broaden my horizons. I

have learnt a lot from the many entertaining and fruitful discussions with you. Thank you

for inviting us to your music hub, an auditory feast of Jazz, which is greatly appreciated.

This work would not have been possible without the help of Andrew, who taught me

many useful techniques for electronic testing, Stephen, who made excellent mechanical

components for experiments, and Peter, who designed amazing structures for vacuum

maintenance. Many thanks to the talented staff of the school electronics and computer

units who always solved technical problems when they occurred.

I would like to thank Wesley and Sam for instructing me in building diagnostic probes,

Kazunori and Robert for giving insightful comments on experimental and theoretical prob-

lems, Rhys for sharing his expertise in programming, and Amelia for helping improve my

presentation skills. I really enjoy the many conversations with Craig, Andy, Antoine, Alex,

Thomas, and Ashley, and wish to thank you for your companionship and help. I also wish

to thank Uyen, Karen, Liudmila, and Suzie for their efforts in administration which make

our schooldays much more cheerful.

I am very fortunate to have been a member of University House community, where we

play cards, go shopping, and share food like at home. Many thanks to all my friends for

enriching my life in Canberra.

I dedicate this thesis to my parents, who raise me up and give me the courage to face

any challenges in life, and to my wife, who is always there for me. Your love means the

world to me.

v

vi ACKNOWLEDGEMENTS

Publications

This thesis has resulted in the following publications in peer reviewed journals.

Yunchao Zhang, Christine Charles, and Rod Boswell

Density measurements in low pressure, weakly magnetized, RF plasmas: new exper-

imental verification of the sheath expansion effect

Frontiers in Physics, Revised, (2017).


Cross-field transport of electrons at the magnetic throat in an annular plasma reactor

Journal of Physics D 50, 015205, (2017).


A polytropic model for space and laboratory plasmas described by bi-Maxwellian elec-

tron distributions

The Astrophysical Journal 829, 10, (2016).


Effect of radial plasma transport at the magnetic throat on axial ion beam formation

Physics of Plasmas 23, 083515 (2016).


Thermodynamic study on plasma expansion along a divergent magnetic field

Physical Review Letters 116, 025001 (2016).


Measurement of bi-directional ion acceleration along a convergent-divergent magnetic

nozzle

Applied Physics Letters 108, 104101 (2016).


Characterization of an annular helicon plasma source powered by an outer or inner

RF antenna

Plasma Sources Science and Technology 25, 015007 (2016).


Approximants to the Tonks-Langmuir theory for a collisionless annular plasma

Physical Review E 92, 063103 (2015).


Principle of radial transport in low temperature annular plasmas


vii

viii PUBLICATIONS


Transport of ion beam in an annular magnetically expanding helicon double layer

thruster


Abstract

This thesis discusses transport phenomena of charged particles in low pressure plasmas

which are of particular interest to electric propulsion systems.

Electrons of low collisionality behave nonlocally and their thermodynamic interpreta-

tion should be revisited as traditional thermodynamic concepts are based on the collision-

dominated local equilibrium. The polytropic process is adapted to nonlocal electron trans-

port during plasma expansion. A conservation relation between electron enthalpy and po-

tential energy is derived from nonlocal electron energy probability functions and verified

by previously published measurements in a laboratory helicon double layer thruster. Anal-

ysis of the experimental data shows that although the electron transport along a divergent

magnetic field is an adiabatic process, it yields a polytropic index of 1.17, which is less

than the classic adiabatic index of 5/3. A theoretical perspective of how nonlocal electron

energy probability functions determine the polytropic index is investigated through three

different bi-Maxwellian distributions. The polytropic index increases when the electron

energy probability function becomes more convex and decreases when more concave. The

polytropic index of 5/3 corresponds to a Heaviside distribution and is an element of a

set of polytropic indices for systems governed by nonlocal particle dynamics. Considering

interrelations between the solar wind and laboratory plasmas, a new scenario is hypothe-

sized for the thermodynamic behavior of the solar wind: although the solar electrons give

a polytropic index less than 5/3, their actual transport might be adiabatic.

Ion beam experiments are carried out in the Chi-Kung reactor implemented with a

cylindrical plasma source (cylindrical plasma thruster) or an annular plasma source (an-

nular plasma thruster). The cylindrical plasma thruster can be operated under a high

magnetic field mode and a low magnetic field mode. In the high field mode, a bi-directional

ion beam travelling in opposite directions is respectively measured in the converging and

diverging parts of a magnetic nozzle, exhibiting a very different scenario from the classic

one-directional nozzle flow of compressible gases. No ion beam is detected for the low field

mode although an axial potential drop exists in the plasma source, for which a correlation

between ion beam formation and radial plasma transport at the magnetic throat is re-

vealed. The annular plasma thruster provides an enhanced degree of freedom in terms of

electron heating by using either an outer antenna or an inner antenna. Electron transport

in the annular system is characterized and compared for the two opposite antenna cases.

An annular ion beam is observed downstream of the plasma source for the outer antenna

case while not for the inner antenna case. It merges into a solid structure (with the central

hollow filled) in the diffusion chamber and a reversed-cone wake is formed behind the inner

tube.

ix

x ABSTRACT

Transport behavior of an annular plasma is greatly changed from a cylindrical plasma

due to the occurrence of an inner wall boundary. Depending on the presence of ion-neutral

collisions or not, collisional modeling and collisionless modeling are respectively developed

to better understand radial transport of unmagnetized charged particles across annuli.

The electrons are in an equilibrium state and assumed to be governed by the Boltzmann

relation (equivalent to a Maxwellian equilibrium). The collisional ion transport is described

by three mobility governed models: a low field electric field model, an intermediate electric

field model and a high electric field model. The collisionless ion transport is studied using

the Tonks and Langmuir theory and the solution is expressed in terms of the Maclaurin

series approximant and Pade rational approximant. The annular modeling is applied to

argon plasmas and discussed for different Paschen numbers and annular geometries.

Contents

Declaration iii

Acknowledgements v

Publications vii

Abstract ix

1 Introduction 1

1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Plasma approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Sheaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Particle and power balance . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Representation of Particle Transport . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Electron dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Ion dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Scope of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Apparatus 9

2.1 Cylindrical and Annular Plasma Sources . . . . . . . . . . . . . . . . . . . . 9

2.2 Diffusion Chamber and Accessories . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Probe Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Emissive probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 Langmuir probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.3 RF compensated Langmuir probe . . . . . . . . . . . . . . . . . . . . 15

2.3.4 Retarding field energy analyzer . . . . . . . . . . . . . . . . . . . . . 17

2.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Polytropic Revisit of Nonlocal Electron Transport 21

3.1 Enthalpy Relation for Nonlocal Electrons . . . . . . . . . . . . . . . . . . . 21

3.2 Polytropic Relation in Helicon Double Layer Thruster . . . . . . . . . . . . 23

3.3 Generalized Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Bi-Maxwellian distributions . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.2 Novel hypothesis for solar wind . . . . . . . . . . . . . . . . . . . . . 31

3.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xi

xii CONTENTS

4 Ion Beam Experiments 35

4.1 Cylindrical Plasma Thruster . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Bi-directional ion acceleration . . . . . . . . . . . . . . . . . . . . . . 35

4.1.2 Magnetic field induced transition . . . . . . . . . . . . . . . . . . . . 39

4.1.3 Radial plasma transport . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Annular Plasma Thruster . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.1 Annular ion beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.2 Outer/Inner antenna cases . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.3 Wake Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Plasma Modeling across Annuli 59

5.1 Electron Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Collisional Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2.1 Ion mobility coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2.2 Electric field based models . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.3 Modeling results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Collisionless Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.1 Tonks-Langmuir Theory . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.2 Approximant methodologies . . . . . . . . . . . . . . . . . . . . . . . 74

5.3.3 Modeling results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6 Conclusions 87

Appendix A Mathematical Deduction of Enthalpy Relation 89

Appendix B Cross Sections of Electron-neutral Collisions in Argon 91

Bibliography 93

Chapter 1

Introduction

An increasing number of spacecraft have been using electric thrusters for primary or aux-

iliary propulsion [1–3] where plasmas act as either the propellant, e.g., Hall effect thruster

and helicon double layer thruster [4], or as the thermal substance to heat propellant

gases, e.g., resistojet thruster and arcjet thruster [5]. Compared to cold-gas and chemical

thrusters, the electric thruster has a major advantage of high specific impulse with long

life operation which is typically required to achieve demands of orbit transfer or attitude

control in geosynchronous and deep space missions [6]. In order to achieve a long operation

time and a high energy transfer efficiency of electric propulsion systems, the working gas

is injected into the generator chamber at a relatively low flow rate and ionized to form

a low pressure plasma by power supplies. This thesis discusses transport phenomena of

charged particles in low pressure plasmas.

1.1 Basic Concepts

1.1.1 Plasma approximation

A gas discharge qualifies as a plasma when the charged particles behave in a collective

manner and are electrically quasi-neutral on average [7], i.e., the electron density is equal

to the ion density ne = ni = n where n is defined as the plasma density. In order to

guarantee spontaneous existence of significant opposite charge densities, the volume of a

plasma must be larger than the Debye sphere whose radius (Debye length) is defined by:

λD =

(ε0Teen

) 12

(1.1)

where ε0, Te and e are the vacuum permittivity, electron temperature (in the unit of volts)

and electron charge, respectively. Derivation of formula 1.1 uses the Poisson’s equation

and assumes the potential difference within the Debye distance being much less than

the electron temperature as a first order approximation [8]. Consequently, the potential

difference across a bulk plasma region should be no greater than the electron temperature,

∆φ . Te. If the dimension of a structure is similar to the Debye length or the potential

difference across it is greater than the electron temperature, e.g., sheaths in a plasma,

the quasi-neutrality is violated and space charge appears. For any control volume in the

plasma, the electron flux and ion flux flowing through its surface must be equal to ensure

1

2 INTRODUCTION

that no net charge builds up. If the two charged fluxes transport along the same direction,

they are known as the “ambipolar” flow [8, 9].

For a weakly ionized plasma, the electron temperature is much greater than the ion

temperature and neutral gas temperature, Te Ti ∼ Tg. Since the thermal motion of

electrons is faster than that of ions, their displacement do not synchronize and the light

electrons oscillate around the heavy ions at a fundamental characteristic frequency, the

electron plasma frequency. Plasma oscillations are damped in time by atomic collisions,

which are categorized into two scattering groups: elastic scatterings including long-range

Coulomb collisions (electron-electron, electron-ion and ion-ions pairs) and short-range po-

larization collisions (electron-neutral and ion-neutral pairs), and inelastic scatterings in-

cluding ionization, excitation and ion-neutral charge exchange. The present study focuses

on weakly ionized electropositive plasmas in parent noble gases. In this case the collision

process is dominated by short-range collisions between charged particles and neutrals while

the effect of long-range Coulomb collisions is small. The charge exchange is in the form of

resonant charge transfer which can be treated as an elastic collision due to their conserved

kinetic energy [9]. Additionally, the effect of neutral depletion [10, 11] is neglected and the

gas atoms are considered to be uniformly distributed in the background, ng = const. Here

the low pressure plasmas refer to the regime of low collisionality where the mean free path

of charged particles is comparable or larger than the characteristic length of the plasma

system, i.e., the Knudsen number of charged particles satisfies Kn & 1.

1.1.2 Sheaths

For an immersed floating object in the plasma, the electrons have a larger velocity com-

pared to ions and are more easily lost onto the floating wall. As a result, a non-neutral

region, known as the floating wall sheath, is formed at the plasma boundary which acceler-

ates the ions and decelerates the electrons to balance charged particle fluxes. According to

the Bohm sheath criterion [12], the ions are first accelerated through a presheath to reach

a critical velocity (Bohm velocity) to enter the sheath, and the sheath-presheath interface

defines the threshold position of quasi-neutral plasma approximation. The Bohm velocity

is determined from the electron temperature and given by:

uB =

(eTemi

) 12

(1.2)

where mi is the ion mass. For a plasma with Maxwellian electrons, the potential drop

across a floating wall sheath is about 4.7Te and typical sheath widths are a few Debye

length.

When a wall sheath has a large potential drop such that the electron density approaches

zero in most regions, e.g., a large negative voltage biased at a probe tip, it becomes a

high-voltage wall sheath or an ion sheath. Then the ion current density flow onto a wall

is governed by the Child law of space-charge-limited current [8]:

Ji =4

9ε0

(2e

mi

) 12 V

32s

w2s

(1.3)

1.1 BASIC CONCEPTS 3

which presumes that the initial ion kinetic energy εi0 = 1/2 ·miu2B at the sheath edge is

small compared to the potential difference Vs across a sheath width ws, i.e., the electron

temperature is small compared to the potential drop Te Vs. Combining this result

with the ion current density derived from the Bohm criterion Ji = ensuB (where ns is the

plasma density at the sheath-presheath interface) yields the width of a high-voltage wall

sheath:

ws =(2)

12

3λDs

(2VsTe

) 34

(1.4)

which can extend from tens to about a hundred Debye lengths [8].

A non-neutral region can also exist in the plasma interior and separate from the wall,

known as an electrostatic double layer [13, 14]. A double layer is a combination of an

ion sheath and an electron sheath, and quasi-neutrality is satisfied at its two edges. The

fundamental difference between a wall sheath and a double layer is shown in the phase

space of charged particles: a wall sheath contains “trapped” electrons and “free” ions, while

a double layer normally consists of two “trapped” and two “free” groups. A recent double

layer study [15] has adapted a five-group model which includes a “counter-streaming”

electron group reflected from the wall sheath on the upstream side of a double layer.

An experimental double layer can be created under various conditions such as an interface

between two plasmas [16, 17], additional electron source [18] and plasma expansion [19–22].

Based on the discovery of a current-free double layer (CFDL) in magnetically expanding

plasmas, a novel type of plasma thruster, the helicon double layer thruster (HDLT), is

invented and has advantages of no electrode, no neutralizer and no moving part [4]. The

thrust of a HDLT is imparted by the momentum of a high-velocity ion beam accelerated

through the CFDL and the magnetic force due to diamagnetic drift of electrons [23].

1.1.3 Particle and power balance

In order to ignite a plasma, free electrons in the gas are accelerated by an external electric

field and cause impact ionization to generate new electrons and ions. This process repeats

during the electron avalanche [24] until enough charged particles are generated to make

a plasma. For a steady-state plasma discharge, the number of newly created charged

particles from volume ionization should be equal to that lost to the boundaries. This

global particle balance is related to the principle of “L-p” similarity for an unmagnetized

plasma: the electron temperature (representing the average kinetic energy of electrons

and positively correlating with the ionization rate) is relatively constant in the bulk region

and negatively correlates with the Paschen number Pas, which is defined as the product of

system length L and neutral gas pressure pg. In a magnetized plasma, the parallel-to-field

transport of charged particles shares many similarities with the unmagnetized case, while

the cross-field transport presents distinct behaviors due to magnetic force.

Since the plasma breakdown voltage is lower for a radio-frequency (RF) power system

compared to a direct-current (DC) system [25], most low pressure plasmas are sustained

using a RF power supply. The external RF power is transferred to the system through

electron heating mechanisms, including collisional ohmic heating, collisionless stochastic

heating and wave-particle interaction heating [8, 9, 26], and these energized electrons give

rise to impact ionization to sustain the plasma state. The absorbed power is lost from the

4 INTRODUCTION

outward energy flux carried by the charged particles, governed by a global energy balance.

1.2 Representation of Particle Transport

1.2.1 Boltzmann equation

For low pressure plasmas, the concept of fluidic continuum becomes invalid and transport

phenomena of charged particles can be better analyzed from the kinetic perspective. The

state of a particle species is described by the distribution function f (v, r, t), defined as the

number of particles per unit volume in the six-dimensional phase space of particle velocity

vector v and position vector r at a specific time t. Its dynamic evolution is given by the

the Boltzmann equation:

∂f

∂t+ v · ∇rf + a · ∇vf = Scol (f) (1.5)

where on the left hand side (LHS): ∇r and ∇v are the gradient operators with respect

to the position vector and velocity vector, and a = (eE + ev ×B) /m (where m is the

particle mass) is the acceleration due to the Lorentz force consisting of electric force eE

(where E is the electric field) and magnetic force ev ×B (where B is the flux density of

magnetic field); on the right hand side (RHS): Scol (f) is the collisional term representing

change rate of the distribution function due to collisions.

Although a full solution of the Boltzmann equation is currently unavailable, the dy-

namics of electrons and ions can be simplified by considering the dominance of isotropy and

anisotropy in their motions. Since the drift velocity of electrons is small compared to their

thermal velocity, the electron distribution function (EDF) is nearly spherical (isotropic) in

velocity space and the perturbation theory can be used [27]. Contrarily, anisotropy plays

an important role in ion dynamics and the drift motion cannot be neglected. In this case

the perturbation theory loses validity in solving the ion distribution function (IDF) and

the Boltzmann equation is transformed using the moment methods [28].

1.2.2 Electron dynamics

A common approach to solve the Boltzmann equation is the two-term approximation based

on spherical harmonics in velocity space [29, 30], given by the sum of a scalar (isotropic

term) and an anisotropic vector component:

f (v, r, t) = f0 (v, r, t) +v

v· f1 (v, r, t) (1.6)

where v is the modulus of velocity vector. The anisotropic term represents the anisotropic

drift motion of electrons and is small compared to the isotropic (thermal) part. Hence the

main interest of electron dynamics is the isotropic component of the EDF. Additionally,

this study focuses on the steady-state (time-independent) plasma and the time variable is

neglected.

The isotropic EDF can be recast in energy space using the electron energy probability

1.2 REPRESENTATION OF PARTICLE TRANSPORT 5

function (EEPF):

fpe (εe, r) = fde (εe, r) ε− 1

2e = 2π

(2e

me

) 32

fe0 (ve, r) (1.7)

where me, ve, εe = 1/2 · mev2e and fde are the electron mass, electron velocity, electron

kinetic energy and electron energy distribution function (EEDF), respectively. The EEPF

has been widely used to characterize electron dynamics as it is experimentally measurable

by a Langmuir probe based on the Druyvesteyn theory [31, 32]: fpe ∝ d2Ie/dV2bias where

Ie and Vbias are the collected electron current and the corresponding biased voltage at the

probe tip, respectively. The macroscopic quantities of electron pressure, electron density

and electron temperature can be derived from the EEPFs:

pe (r) =2

3

∫ ∞0

ε32e fpe (εe, r) dεe (1.8)

ne (r) =

∫ ∞0

ε12e fpe (εe, r) dεe (1.9)

eTe (r) =pe (r)

ne (r)(1.10)

Interestingly, when the electron energy and momentum relaxation paths are larger than

the scale of a potential drop along the magnetic field or in the absence of a magnetic field,

the electrons move across the potential structure φ (r) without encountering short-range

elastic and inelastic electron-neutral collisions. In this case the kinetic energy term εe in

the EEPF is replaced by the mechanical energy εe − eφ (r):

fpe (εe, r) = fpe [εe − eφ (r)] (1.11)

which is known as the “non-locality” of EEPFs [33]. This generalized relation, combining

the two variables of kinetic energy εe and spatial position r into one variable of mechanical

energy εe−eφ (r), has been adopted for both space plasmas [34] and low pressure laboratory

plasmas [35]. When the pressure of neutral gases increases and the effect of electron-neutral

collisions becomes dominant during electron transport, the “local” collision rate should be

included in the expression of EEPF. It should be noted that the EEPF does not present

non-local behavior across magnetic field lines and its spatial evolution is affected by the

magnetic filter effect [36, 37].

1.2.3 Ion dynamics

Most physical insights of ion dynamics are included in the first few moments of the Boltz-

mann equation [28]. For transport phenomena, the zeroth-order and first-order moments

are of main interest. This study focuses on the steady-state behavior of plasmas and the

partial time derivative is neglected. The zeroth-order moment equation gives the particle

conservation relation:

∇r · (niui) = νizni (1.12)

where ui and νiz = ngµiz is the ion drift velocity vector and ionization rate (with negligible

volume recombination for low pressure plasmas), respectively.

6 INTRODUCTION

The first-order moment equation shows the convective motion of ions. In the presence

of ion-neutral collisions, the macroscopic ion flux Γi = niui is described in terms of

mobility and diffusion [28]: the mobility represents a momentum balance between the

electric field and passive collisions; the diffusion represents a balance between the density-

gradient-induced dispersion and collisions. It should be noted that, to include the effect of

ion inertia, a complex high-order approximation should be applied to the moment equation

and here it is neglected for simplicity. The ion flux vector is given by:

Γi = niK(2)i ·E−D

(2)i · ∇rni (1.13)

where K(2)i and D

(2)i are the second-order tensors of ion mobility coefficients and diffusion

coefficients, respectively. Specific expressions of their tensor terms depend on the regime

of electric field strength [38] and the presence or absence of a magnetic field [39]. For

an unmagnetized plasma, the non-diagonal terms are zero and the diagonal terms corre-

spond to the drift direction and the other two perpendicular directions. For a magnetized

plasma, the non-diagonal terms reflect coupling effects among those directions due to the

occurrence of magnetic force.

In the absence of ion-neutral collisions and magnetic fields, the ion motion can be well

approximated by a free-flight scenario: newly created ions at the high potential side are

accelerated along the electric field (resulting from decreasing potentials) and contribute

to the ion density at the low potential region. It can be seen that this scenario is from the

Lagrangian specification while the moment methods from the Eulerian specification. Ne-

glecting the thermal motion of ions, the free-flight process is a one-dimension phenomenon

along the potential path and described by:

δni =νizn

∗dr∗

ui

(r∗

r

)c, ui =

(2eφ∗ − 2eφ

mi

) 12

(1.14)

where r is the spatial position along the potential path. The superscript asterisk rep-

resents the location of ion generation, and the exponent coefficient value of c = 0, 1, 2

corresponds to the result in Cartesian, cylindrical and spherical coordinates, respectively.

Equation (1.14) represents the ion density produced at r, i.e, δni, by the flux created at

r∗, i.e., νizn∗dr∗. Integration of equation (1.14) gives the well-known “Tonks-Langmuir

theory” [40] which has been widely used in low pressure plasmas [41, 42]. The free-flight

scenario provides a very useful approach to characterize ion dynamics in laboratory ex-

periments: ion distribution measurements at a specific position can reflect the upstream

ion groups, e.g., for ion beam detection.

1.3 Scope of Thesis

Chapter 1 reviews some basic concepts and different representations of particle transport in

the context of low pressure plasmas. Chapter 2 shows the experimental setup of Chi-Kung

apparatus and different electrostatic probes for plasma diagnostics. The main narrative

of this thesis is divided into three parts, corresponding to chapters 3 to 5, as stated below.

Chapter 6 summarizes the main results and their applications.

1.3 SCOPE OF THESIS 7

Polytropic revisit

Plasma expansion, typically along a divergent magnetic field, plays an important role in

physical systems spanning astrophysical phenomena [43–45] down to many low pressure

discharges of interest in electric propulsion systems [1–6] and materials processing [8, 9, 46].

Under these conditions the charged particles are characterized as nearly collisionless (of

low collisionality) due to the long mean free paths. A particle group dominated by colli-

sions is considered to be under local thermodynamic equilibrium (LTE) which is the basis

of traditional thermodynamic concepts [47]. However a collisionless or low-collisionality

particle system, defined here as “non-LTE”, behaves in a manner fundamentally different

to that of a LTE system [35, 48]: the mechanical energy of a non-LTE particle is conserved

along its transport path while that of a LTE particle is dissipated through collisions.

Consequently, traditional thermodynamic relations based on LTE should be revisited for

non-LTE plasmas [49]. Chapter 3 discusses the interpretation of polytropic relation (which

is widely used to characterize processes involving energy exchange between a system and

its surrounding environment [47, 50]) for electrons governed by nonlocal EEPFs, using

previous experimental measurements in the HDLT [51, 52] and a theoretical model based

on bi-Maxwellian electron distributions.

Ion beam

A potential drop structure is closely associated with plasma expansion [53, 54] and can

accelerate the ions to form an ion beam. Directional ion beams are very useful to achieve

a greater momentum transfer for plasma thrusters [6] and a more precise pattern for

materials fabrication [55]. Previous studies have investigated factors that influence ion

beam formation including double layer formation [19–21], effect of magnetic field [56, 57],

gas type [58, 59] and variable RF [60, 61]. Chapter 4 investigates ion beam formation in

a cylindrical plasma thruster and in an annular plasma thruster based on the Chi-Kung

reactor. It should be noted that the annular configuration has attracted increasing interest

over the past few years [62–64], with potential applications ranging from pre-ionization for

annular Hall effect thrusters [65, 66] to plume simulation for re-entry spacecrafts [67, 68].

Annular modeling

An annular plasma is of theoretical interest as its transport behavior is greatly changed

from a cylindrical plasma. The cylindrical case has a central point of maximum plasma

density and zero electric field across the radial dimension, but this critical point disap-

pears in an annulus and is replaced by an inner boundary. For an annular plasma, the

density peak position becomes a variable and the radial transport changes from one di-

rection (solely outward) for a cylindrical plasma to two directions (both outward and

inward). A theoretical study about transport properties of annular plasmas is useful to

understand systems such as a probe within a plasma column [40, 69], an inner quartz tube

in a plasma source [62, 63] and a central electrode in a plasma jet [70, 71]. The axial

transport of charged particles in an annular plasma is similar to that in a cylindrical case

as implementation of an inner cylinder into a cylindrical plasma does not change the axial

boundary condition; the axial governing equation is the same except for the value of a

8 INTRODUCTION

coupled source term [72, 73]. The azimuthal dynamics is negligible due to central symme-

try of an annulus. Hence the transport phenomena in an annular plasma are dominated

by the radial dimension; the axial and azimuthal motion can be considered as “frozen”

dimensions and neglected. Chapter 5 develops modeling across annuli for collisional and

collisionless plasmas in terms of ion-neutral collisions.

Chapter 2

Apparatus

The present experiments are carried out in the Chi-Kung reactor, to which has been

attributed the discovery of a current-free double layer (CFDL) and ion beam formation in

magnetically expanding plasmas [19, 74], and to the invention of the helicon double layer

thruster (HDLT) [4]. Section 2.1 exhibits the structure of a standard cylindrical plasma

source and a newly configured annular plasma source. Section 2.2 shows the assembly

of the diffusion chamber and different accessory components for vacuum maintenance.

Section 2.3 introduces four electrostatic probes for plasma diagnostics which have been

previously developed in Space Plasma Power and Propulsion (SP3) lab, and shows how

they are built for the experiments reported in this thesis.

2.1 Cylindrical and Annular Plasma Sources

The Chi-Kung reactor, as shown in figure 2.1, consists of a helicon plasma source and a

contiguously attached diffusion chamber. A top (left) solenoid and an exit (right) solenoid

are placed around the plasma source, by mounting onto an earthed aluminium supporting

structure (figure 2.1(a)), to provide the static direct-current (DC) magnetic field. Each

solenoid has a double-coil-wound arrangement and the supplied current is equally divided

into two parallel coils. A 31-cm long, 13.7-cm inner diameter, 0.65-cm thick cylindri-

cal source tube is located inside the supporting structure and made of Pyrex glass (fig-

Figure 2.1: (a) Photo and (b) schematics of Chi-Kung reactor, showing major components andpositioning of a diagnostic probe.

9

10 APPARATUS

VTh

RTh

Cload

CtuningLd

Rd

A

A′

B

B′

Figure 2.2: Thevenin-equivalent circuit for matching the RF power supply to a plasma dischargeusing an L-type matching network.

ure 2.1(b)). Its top end is terminated to an earthed aluminium plate and sealed with an

O-ring; the exit end fits within the support plate of the diffusion chamber equipped with an

O-ring. For the CFDL experiment, an additional 1-cm thick glass plate is positioned be-

tween the top aluminium plate and source tube [19]. A 18-cm long double-saddle antenna

constructed from a number of 1.2-cm wide, 0.17-cm thick copper elements, surrounds the

source tube with its close-to-chamber endpoint located 9 cm from the source-chamber in-

terface (z = 0 cm). Eight rectangular Fiberglass spacers are fixed at the antenna corners

to prevent the antenna surface from direct contact with the source tube which could cause

localized hotspots.

The antenna is driven at a constant radio-frequency (RF) frequency of 13.56 MHz by a

power supply. The circuit diagram for matching a plasma discharge to a RF power supply

is shown in figure 2.2. The Thevenin equivalent [75] of a power supply consists of an

RF voltage source VTh and an internal resistance of RTh = 50 Ω. A standing-wave-ratio

(SWR) meter and a power meter are connected along the transmission line (not shown in

the diagram) to monitor the power transfer. A match box consisting of a variable load

capacitor Cload and a tuning capacitor Ctuning is used to adjust the matched condition of

maximum power transfer, i.e., the external impedance seen by the power supply terminals

A−A′ is equal to RTh [75]. A simplified circuit model of the antenna-discharge coupling

Figure 2.3: (a) Photos of front view [top pane] and back view [bottom pane] of the annularplasma source. (b) Schematic of the annular Chi-Kung reactor.

2.2 DIFFUSION CHAMBER AND ACCESSORIES 11

Figure 2.4: (a) Photos of inner antenna [top pane] and its positioning into the inner tube of theannular Chi-Kung reactor [bottom pane]. (b) Schematic of the annular Chi-Kung reactor withboth outer and inner antennae, which are independently operated for the present experiments.

system (seen by the match box terminals B −B′) includes an equivalent inductor Ld and

a resistor Rd. In this matching circuit an L-type network is formed by the combination of

Cload, Ctuning and Ld.

The cylindrical plasma source can be reconfigured to an annular geometry by inserting

a 42-cm long, 5-cm outer diameter, 0.25-cm thick Pyrex glass tube sealed at the internal

vacuum end, into the original source tube through the central port of an earthed aluminium

flange sealed at the top end of the source (figure 2.3(a)). The sealed end of the inner tube

is located at the source-chamber interface (z = 0 cm) to give the source a totally annular

geometry, and the plasma is created between the inner and outer tube walls (figure 2.3(b)).

This reconfiguration allows another antenna to be placed inside the inner glass tube.

A 17.0-cm long multi-loop antenna, fabricated from a 0.47-cm diameter water-cooled cop-

per tube, is used and its antenna head point is positioned at z = −9 cm. The antenna

is connected to a match box through the open end of the inner tube, and four ceramic

rings are installed along the copper tube to avoid its direct contact with the inner tube

(figure 2.4(a)). The exposed part of the inner antenna located between the match box

and inner tube is wound with earthed copper foils to prevent RF leakage from the trans-

mission line. An L-type matching network, similar to that used for the outer antenna

case (figure 2.2), is used for the inner antenna case. For this study, the two antennae are

independently operated to sustain the plasma discharge, i.e., only one of the RF power

systems in figure 2.4(b) is running.

2.2 Diffusion Chamber and Accessories

The diffusion chamber (figure 2.5(a)) consists of a 30-cm long, 32-cm diameter aluminium

chamber and a backplate, both of which are grounded to a clean common earth. A slot

covered by a transparent glass window is mounted onto the left-hand-side (LHS) wall of

the diffusion chamber for visual inspection of plasmas. There are several circular ports

manufactured on the lateral wall of the diffusion chamber to allow the assembly of different

accessory components. The neutral gas flow (e.g., of argon) is fed into this reactor through

12 APPARATUS

Figure 2.5: (a) Diffusion chamber of Chi-Kung reactor, and backplate equipped with the “vacuumslide”. (b) Structure of a diagnostic probe.

the right-hand-side (RHS) port close to the plasma source, and the flux rate is adjusted

by a mass flow controller. In order to remove gas molecules from the volume and keep a

constant gas pressure, the large rear port close to the backplate is connected to the inlet

of a turbomolecular pump via a straight pipeline and an elbow; at the exhaust end of the

turbomolecular pump, the gases are evacuated via a vacuum hose connected to a primary

pump.

The gas pressure in the reactor is measured by a Baratron gauge through one outlet

of a tee fitting mounted under the bottom port, and the other outlet is connected to a

Convectron gauge which monitors the pressure when the system is pumped down from

the atmosphere to a low vacuum condition or vice versa. When there is no gas, the

turbo/primary pumping system ensures a high-vacuum base pressure of 3× 10−6 Torr in

the reactor monitored with an ion gauge placed on the top port; hence a high purity of

working (argon) gas is guaranteed for low pressure plasma experiments. The LHS ports

on the side wall are mainly used to insert diagnostic probes. The backplate of the diffusion

chamber is equipped with a slide (sealed over an O-ring), called the “vacuum slide” [76]

which can be slid along a 20-cm long slot (cut into the backplate) under vacuum conditions

by rotating the wheel handle on the LHS. The vacuum slide allows the positioning of a

probe along both axial and radial directions without breaking vacuum.

Figure 2.5(b) shows the basic structure of an electrostatic probe, used as a main

plasma diagnostic and has the advantages of simple construction and accurate spatial

positioning. Its head comprises a functional component (e.g., a planar tip of a Langmuir

probe) for parametric measurements, and a ceramic or metallic support secured to a 0.64-

cm diameter metal shaft with a set-screw collar (or a Swagelok fitting). The wire leads

from the head are enclosed in the shaft to shield the measured signal from pickup of

background electronic noises. The other end of the shaft is connected to a cylindrical cap

with a Swagelok fitting, and the inside electrical wires are soldered to BNC connectors

mounted on a terminating plate which is sealed to the cap via an internal O-ring to form

a vacuum seal. The BNC connector acts as an interface between the probe head and the

external circuit. A feedthrough surrounds the shaft and its front (vacuum) side is mounted

onto the port of the diffusion chamber or of the vacuum-slide interface of the backplate

2.3 PROBE DIAGNOSTICS 13

through a flange. This feedthrough structure allows linear and angular displacing of the

probe head under a controlled vacuum condition by moving and rotating the shaft.

2.3 Probe Diagnostics

2.3.1 Emissive probe

Four types of electrostatic probes are used here: the emissive probe, the planar Langmuir

probe, the RF compensated Langmuir probe, and the retarding field energy analyzer.

Their operation is controlled with the LabVIEW software during experiments. The probe

head collects a voltage signal or a current signal by interacting with the plasmas, and

an electronic circuit is subsequently used to transmit and process this raw signal. A data

acquisition (DAQ) board samples the pre-processed signals and load them into a computer

for further data processing. Since a probe could perturb its local surroundings and affect

the plasma condition to some extent, a second (witness) probe is always implemented to

check the reliability of the measurements.

The head of an emissive probe (EP) is shown in figure 2.6(a) where a 0.2-mm diameter

tungsten filament is bent around a cylinder to form a U-shape and the middle arc acts

as the functional tip [77]. The two ends of the filament are separately inserted into

two 0.8-mm diameter bores of a ceramic tube, and joined to the stripped end of copper

wires through mechanical connection (any gap is filled with additional tungsten wires,

resembling a “bundled cable”). The top side of the ceramic tube is covered with a layer

of high temperature ceramic adhesive to ensure that the electrical wires in the bore are

not affected by the plasma and only the functional tungsten filament is exposed. The EP

measures the plasma potential φ using the floating potential method [78] for which the

circuit diagram is given in figure 2.6(b).

When the tungsten filament is heated by a DC current that is generated by a power

supply isolated from the earth (acting as a current source), thermionic emission occurs at

the filament and an electron flow is driven from the filament to the plasma, referenced as

the electron emission current. For weak or zero electron emission, the electron collection

current moving from the plasma to the filament surface is balanced by the ion collection

(a) (b)

filament

Iheat

R1 Vf

to DAQ

R1

Figure 2.6: (a) Head of an emissive probe (EP). (b) EP circuit diagram. R1 = 120 Ω.

14 APPARATUS

Iheat [ A ]0 0.5 1 1.5 2 2.5

Vf[V

]

0

5

10

15

20

25

30

35

40

φ

Figure 2.7: Floating potential Vf as a function of heating current Iheat. The dashed line locatesthe infection point corresponding to the plasma potential φ.

current. As the electron emission is enhanced, the electron emission current becomes large

compared to the ion collection current and is approximately equal to the electron collection

current. At this stage, the strong electron emission eliminates the sheath surrounding the

filament and pushes the floating potential Vf to equal the plasma potential. Figure 2.7

presents the floating potential measured at the probe tip as a function of the heating

current Iheat. The flat part in the low heating current range (between 0 and 1.8 A) gives

the floating potential of an inserted non-emissive object in the plasma and the inflection

point (at 2 A) corresponds to the plasma potential.

2.3.2 Langmuir probe

The head of a Langmuir probe (LP) is shown in figure 2.8(a) where a 1.9-mm diameter

(dp) nickel disc is mounted perpendicularly to the axis of a ceramic tube. The back side

of the disc and the hollow behind are covered with ceramic adhesive and only the front

side of the disc is interacting with the plasmas [77]. This tip is joined to an open-barrel

(a) (b)

planar disc

Rm

+−

ampRmIi

to DAQ

Figure 2.8: (a) Head of a planar Langmuir probe (LP). (b) LP circuit diagram. Rm = 10 kΩ.


connector which crimps the stripped end of an electrical wire (housed inside the 1.4-mm

inner diameter ceramic tube) to form an electrical connection.

When the probe is biased sufficiently negatively, the electrons are repelled from the

disc region and only the ions are collected to give the ion saturation current Isat which is

proportional to the ion density ni [79]. By varying the biased voltage Vbias, the floating

potential Vf is identified at the zero crossing point of the current-voltage (I-V) characteris-

tic, where the electron collection current is equal to the ion collection current. The plasma

potential φ corresponds to the knee of the I-V characteristic; since the electron current

dominates in the positive Vbias range, the electron temperature Te can be determined from

the inverse slope of the logarithmic electron current [31].

For a RF plasma, the sheath in front of the probe tip oscillates and the electron

current could be distorted [80, 81]. In this case only the ion saturation current can be

reliably obtained at a large negative biased voltage and here Vbias = −95 V is used. The

circuit diagram for LP measurements is given in figure 2.8(b) where the biased voltage

is provided by the DC voltage source. Due to sheath expansion in front of a negatively

biased probe tip, the actual ion collection area is effectively enhanced and the ion density

is derived from the ion saturation current using Sheridan’s method [82, 83].

Isat = 0.55AseniuB (2.1)

where a constant coefficient of 0.55 is used which takes into account the curvature of the

sheath edge; e and uB are the electron charge and Bohm velocity, respectively. The sheath

expansion area around the tip As is related to the physical area of a tip Ap = πr2p (where

rp = 1/2 · dp is the tip radius) by:

AsAp

= 1 + aηbp (2.2)

where ηp = (φ− Vbias) /Te. The coefficients a and b are given by:

a = 2.28

(rpλD

)−0.749, b = 0.806

(rpλD

)−0.0692where λD is the Debye length.

2.3.3 RF compensated Langmuir probe

To obtain a reliable measurement of the electron current component Ie in the I-V charac-

teristic, the RF compensated Langmuir probe (CP, figure 2.9(a)) consisting of RF chokes

and reference electrode is used to suppress the signal distortion caused by sheath recti-

fication in front of the probe tip in RF plasmas [51, 80, 81]. The supporting structure of

the probe head comprises a glass pipette tube and an attached ceramic tube with their

interface being secured using ceramic adhesives. Four RF chokes are housed inside the

pipette tube and joint in series by soldering and mechanical tungsten wrapping [84]. The

first two chokes resonate at the fundamental frequency (13.56 MHz) and the rest two at

its second harmonic (27.12 MHz). Each choke inductor is connected with a small variable

capacitor in parallel to make the resonant frequency occur exactly at the two critical fre-

16 APPARATUS

(a) (b)

cylindricaltip

C1

referenceelectrode

L1 L1 L2 L2

Cv Cv Cv Cv

Rm ampRmI

′′e

to DAQDt Dt

Figure 2.9: (a) Head of a RF compensated Langmuir probe (CP). (b) CP circuit diagram.C1 = 4.7 nF, Rm = 100 Ω. L1 and L2 are RF chokes resonating at 13.56 MHz and 27.12 MHz,respectively. Each choke is connected with a variable capacitor Cv in parallel. Two analog differ-entiators (Dt) are used to obtain the second derivative of electron current with respect to time,I

′′

e = d2Ie/dt2.

quencies where the signal magnitude is attenuated by about −85 dB. The first choke is

terminated with a 0.25-mm diameter (dp) nickel/tungsten wire which is put through the

top (thin) end of the pipette to form a wire tip out of the pipette with a typical length

(lp) of 3 mm to 6 mm. The tip orientation is arranged perpendicularly to the sweeping

plane determined by the vacuum slide to maximize its collecting area and obtain accurate

spatial measurements in the sweeping plane. A reference electrode is placed close to the

probe tip by twisting a long nickel/tungsten wire around the pipette top for which a metal

tube is sometimes used to enhance the electrode area. The electrode lead is connected to

the probe tip through a 4.7 nF capacitor housed inside the pipette.

The Druyvesteyn theory [31, 32] shows that the second derivative of the I-V charac-

Vbias [ V ]0 20 40 60 80

d2I e/d

V2 bias[A

rb.unit]

-10

-5

0

5

10

Vbias [ V ]0 20 40 60 80

ln(E

EPF)[A

rb.unit]

-8

-7

-6

-5

-4

-3

-2

-1

0

1

( a ) ( b )

Figure 2.10: (a) Second derivative of electron current Ie with respect to biased voltage Vbias. (b)Logarithm of electron energy probability function (EEPF).


teristic is proportional to the electron energy probability function (EEPF):

fpe (Vbias, r) =2me

e2Ac

(2e

me

) 12 d2Ie

dV 2bias

(2.3)

where Ac = πdplp is the collecting area of the cylindrical probe tip. Figure 2.9(b) exhibits

the circuit diagram for CP measurements. As the second derivative of the electron current

is obtained through two analog differentiators, i.e., with respect to time t, the sweeping

signal of biased voltage must be a linear function of time (generated by a triangular voltage

source) such that d2Ie/dt2 ∝ d2Ie/dV

2bias [51, 80]. Figures 2.10(a) and 2.10(b) show the

second derivative of the I-V characteristic and the logarithmic scale of the EEPF, respec-

tively. The electron parameters of pressure, density and temperature can be calculated

from the EEPFs as detailed in chapter 1, subsection 1.2.2.

2.3.4 Retarding field energy analyzer

The head of a retarding field energy analyzer (RFEA) is shown in figure 2.11(a), consisting

of a metal lid with a 4.0-mm diameter central hole, an attached diagnostic structure,

and a support plate welded to a short shaft which is connected to the main shaft using

the Swagelok fitting [76]. The diagnostic structure is constructed as follows: first, a

plate having a 2.0-mm diameter orifice positioned coaxially to the central hole on the

front lid; then, four grids constructed of nickel meshes spot-welded to small copper rings

and arranged in a sequence of “earthed grid”, “repellor grid”, “discriminator grid” and

“suppressor grid”, a collector plate made of nickel; finally, a clamp plate covering on the

back. These components are separated by Mica insulators with holes in the four corners,

through which the screws fasten the clamp plate to the lid. The first four insulators have

a 2-mm diameter orifice in the centre to allow the ions move onto the collector plate.

Four wire leads are soldered to the repellor grid, discriminator grid, suppressor grid and

collector grid, and are put through the support container and attached shaft to connect

to the external circuit.

(a) (b)

O E R D S C

Vr

+− +

−

Vd Vs

Rm

Vc

ampRmIc

to DAQ

Figure 2.11: (a) Head of a retarding field energy analyzer (RFEA). (b) RFEA circuit diagram.O: (earthed) orifice plate, E: earthed grid, R: repeller grid biased at Vr = −80 V, D: discriminatorgrid biased at a voltage Vd sweeping from 0 V to 80 V, S: suppressor grid biased at Vs = −18 V,C: collector biased at Vc = −9 V. Rm = 10 kΩ.

18 APPARATUS

Vd [ V ]0 10 20 30 40 50 60

Ic[A

rb.unit]

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Vd [ V ]0 10 20 30 40 50 60

IEDF

[Arb.unit]

-0.01

0

0.01

0.02

0.03

0.04

0.05

( a ) ( b )

φ

φbItot

Inloc

Idel

Figure 2.12: (a) Collector current Ic as a function of discriminator voltage Vd. (b) Ion energydistribution function (IEDF), fitted by the sum of a low energy Gaussian function (dashed line)and a high energy Gaussian (solid line). The solid line in (a) is the integral curve of the highenergy Gaussian function in (b).

The circuit diagram to obtain the I-V characteristic of a RFEA is presented in fig-

ure 2.11(b). The four grids are biased at different voltages such that the RFEA works as a

high-pass energy filter of ions: the repellor grid is biased at Vr = −80 V to repel the elec-

trons; the discriminator grid voltage Vd sweeps from 0 V to 80 V to set the cutoff energy

level; the suppressor grid and the collector are respectively biased at Vs = −18 V and at

Vc = −9 V to suppress the effect of secondary electrons produced by ion bombardment at

the collector plate. Along the drift direction of ions, the measured collector current Ic is

a function of the discriminator voltage Vd [74] given by:

Ic (Vd) = Aoe

∫ ∞ud

uifi (ui) dui (2.4)

where Ao and fi are the orifice area and the distribution function of drifting ions, respec-

tively; ui is the drift velocity of ions and its lower limit ud satisfies eVd = 1/2 ·miu2d where

mi is the ion mass. Substituting ui by the ion kinetic energy εi = 1/2 ·miu2i and setting

gi (εi) = fi (ui) in formula (2.4) yield:

Ic (Vd) =Aoe

mi

∫ ∞eVd

gi (εi) dεi (2.5)

where gi (εi) represents the energy distribution of ion species falling through the sheath

in front of the RFEA orifice [85], referenced here as the ion energy distribution function

(IEDF), and is obtained from the first derivative of formula 2.5:

dIcdVd

= −Aoe2

migi (eVd) (2.6)

When the RFEA faces an ion beam, the measured I-V characteristic is shown in

figure 2.12(a) and the corresponding IEDF exhibits two peaks which can be fitted by two

Gaussian functions as given in figure 2.12(b). The low energy Gaussian function (dashed

2.4 CHAPTER SUMMARY 19

line) centres at the local plasma potential φ and the high energy Gaussian function (solid

line) at the beam potential φb. Their respective integrals yield the local ion current Iloc

which represents the background ion group, and the nonlocal ion current Inloc representing

the beam ions. The integral of the overall IEDF gives the total ion current Itot which is

equal to the sum of Iloc and Inloc. Additionally, in order to characterize the shape of an

ion beam, the collector current is normally sampled at a chosen high discriminator voltage

(about the beam potential) for different spatial positions [86], defined as the delimited

current Idel representing the ion group with a mechanical energy higher than the chosen

energy level. When there is no ion beam or the RFEA-orifice orientation is perpendicular

to an ion beam, the IEDF presents a single peak at the plasma potential and can be solely

fitted with the low energy Gaussian function.

2.4 Chapter Summary

This chapter shows the experimental setup of Chi-Kung reactor. The cylindrical plasma

source can be modified into an annular configuration by inserting a glass tube, which allows

the positioning of an inner antenna. The diffusion chamber provides multiple interfaces for

vacuum-maintenance devices and diagnostic probes. The structure and data interpretation

of four electrostatic probes are presented.

20 APPARATUS

Chapter 3

Polytropic Revisit of Nonlocal

Electron Transport

This chapter focuses on the polytropic behavior of electrons in low pressure plasma expan-

sion where the electrons are governed by nonlocal electron energy probability functions

(EEPFs) [33, 35], represented in the form of fpe (εe, r) = fpe [εe − eφ (r)] where εe, φ and r

are the electron kinetic energy, plasma potential and spatial position vector, respectively.

It should be noted that the polytrope has also been applied to ions in the solar wind

studies [87, 88]; for many laboratory systems, the ions can be approximated as a cold

species without thermodynamic behavior [8, 22]. Since the mean velocity of a plasma flow

is normally small compared to the electron velocity, the comoving frame attached to the

flow can be considered as a stationary frame for electrons on first-order approximation

and this setting is used by default unless otherwise specified.

Section 3.1 derives the enthalpy relation for an adiabatic system governed by nonlocal

EEPFs. Section 3.2 studies the polytropic relation for electrons using previous EEPF data

in a laboratory helicon double layer thruster (HDLT) measured by Takahashi [51], and

shows that the use of traditional thermodynamic concepts based on collision-dominated

local thermodynamic equilibrium (LTE) can lead to very erroneous conclusions regarding

the thermal conductivity for non-LTE plasmas governed by nonlocal particle dynamics.

Section 3.3 focuses on a new theoretical perspective of how nonlocal EEPFs determine

the polytropic index of electrons through three different bi-Maxwellian distributions, and

hypothesizes a new scenario of electron transport in the solar wind by considering the

interrelation between the solar wind and laboratory plasmas. Additionally, the energy

conversion mechanism behind ion acceleration is briefly discussed in sections 3.2 and 3.3.

3.1 Enthalpy Relation for Nonlocal Electrons

When electrons move nonlocally along decreasing potentials, the total mechanical energy

of electrons is conserved with the electrons bound back and forth within the potential

structure [8], hence their transport is a self-consistent adiabatic process. Electron enthalpy

he is defined using formula:

dhe = dqe +dpene

=dpene

(3.1)

21

22 POLYTROPIC REVISIT OF NONLOCAL ELECTRON TRANSPORT

z [ cm ]-30 -20 -10 0 10 20 30

r[cm]

-20

-10

0

10

20

z [ cm ]-30 -20 -10 0 10 20 30

B[G

]

0

40

80

120

160( a ) ( b )

CP

-28.6 -9

top

pumpgas

exitRF solenoidsolenoid

Figure 3.1: (a) Helicon double layer thruster experiment, showing major components, the RFcompensated Langmuir probe (CP) and magnetic field lines. (b) Magnetic flux density B on thecentral axis.

where the heat term dqe is omitted due to adiabaticity. The electron pressure pe and elec-

tron density ne are obtained from the EEPFs using pe = 2/3 ·∫∞0 ε

3/2e fpe (εe − eφ) dεe and

ne =∫∞0 ε

1/2e fpe (εe − eφ) dεe, respectively. Integrating formula (3.1) along the potential

path yields:

he =

∫ −eφ−eφ0

23

∫∞0 ε

32e f′pe (εe − eφx) dεe∫∞

0 ε12e fpe (εe − eφx) dεe

d (−eφx) + he0 = eφ− eφ0 + he0 (3.2)

for which limεe→∞ ε3/2e fpe (εe − eφx) = 0 has been used when applying integration by

parts to the numerator. φ0 and he0 are the plasma potential and electron enthalpy at a

reference position, respectively. The mathematical deduction of equation (3.2) is detailed

in Appendix A.

Rearranging equation (3.2) yields a conservation relation:

∆he + ∆ (−eφ) = 0 (3.3)

which shows that the electrons transfer their enthalpy into the potential energy in an

adiabatic process. It should be noted that this conservation relation is a typical form of

the Bernoulli integral [49, 87] where the macro (convective) kinetic energy of the plasma

flow is omitted due to the approximation of a stationary comoving frame as stated earlier.

Equation (3.3) is a generalized result independent of the specific form of nonlocal EEPFs,

and its differentiation with respect to the plasma transport path z yields:

dpedz

+ neEz = 0 (3.4)

where Ez is the electric field along z direction. Equation (3.4) has a consistent form

of momentum balance with a LTE system, but the pressure term here is an effective

parameter determined by nonlocal motion of electrons rather than local collisions. It

should be noted that, for astrophysical electrons covering large distances, the electron

enthalpy will be partially consumed to overcome the gravitational barrier.

3.2 POLYTROPIC RELATION IN HELICON DOUBLE LAYER THRUSTER 23

−Vbias [ V ]-60 -50 -40 -30 -20 -10 0

ln(E

EPF)[A

rb.unit]

-5

-4

-3

-2

-1

0

1

2

3

z = −9 to 7 cm

−Vbias [ V ]-60 -50 -40 -30 -20 -10 0

ln(E

EPF)[N

orm.unit]

-5

-4

-3

-2

-1

0

1

2

3( a ) ( b )

Figure 3.2: (a) Logarithm of EEPFs as a function of additive inverse of bias voltage on theLangmuir probe −Vbias at each 2 cm from z = −9 cm to 7 cm. (b) EEPFs normalized at −Vbias =−30 V. The solid curves represent the measurements in the plasma source (z < 0 cm), and thedash-dotted curves in the diffusion chamber (z > 0 cm). EEPF data taken from Refs. [51, 52].

3.2 Polytropic Relation in Helicon Double Layer Thruster

The nonlocal performance of EEPFs for a plasma expanding along a divergent magnetic

field has been observed in the HDLT experiment based on the Chi-Kung reactor [52],

and here a new analysis on these measurements is carried out to investigate the related

thermodynamic properties. The experimental system operates at a low argon pressure of

3 × 10−4 Torr and the discharge is sustained by a constant radio-frequency (RF) power

of 13.56 MHz and 250 watts. A static magnetic field, whose field lines are represented by

solid curves in figure 3.1(a), is generated by two direct-current (DC) solenoids situated

around the plasma source. The calculated magnetic flux density on the central axis (B =

Bz) is given in figure 3.1(b). Under these experimental conditions a current free double

layer (CFDL) has been observed in the divergent region of the magnetic field at about

z = −5 cm [19, 51], formation of which guarantees neutrality on both the upstream and

downstream sides without the need of an additional second plasma [13, 14, 22].

The EEPFs on the central axis (figure 3.2(a)) were obtained in Refs. [51, 52] using a

RF compensated Langmuir probe (CP, consisting of a clean, 3-mm long, 0.25-mm diam-

eter nickel wire to collect electrons) with the tip orientation arranged perpendicularly to

the axial direction to maximize the collection area. These EEPF curves show a convex

distribution having a high temperature bulk (reflected by the flatter slope) in the low

energy range and a low temperature tail (reflected by the steeper slope) in the high en-

ergy range [51, 52, 89], which can also be described as a bi-Maxwellian distribution with a

depleted tail. Since no bump structure is observed in the high energy range of the EEPF,

there is no or a negligible electron beam existing along the potential drop [80, 90], suggest-

ing that the electrons are created and transported from the plasma source and the effect

caused by possible secondary electrons generated in the diffusion chamber is negligible.

Figure 3.2(a) shows the EEPF data at each 2 cm from z = −9 cm (the leftmost curve)

to 7 cm (the rightmost curve), as a function of the additive inverse of bias voltage on


-10 -5 0 5

φ[V

]

30

35

40

45

50

55

60

-10 -5 0 5

Te[V

]

0

2

4

6

8

10

12

z [ cm ]-10 -5 0 5

ne/n

e0

0

0.2

0.4

0.6

0.8

1

1.2

30354045505560

Te[V

]

0

2

4

6

8

10

12

φ [ V ]30354045505560

ne/n

e0

0

0.2

0.4

0.6

0.8

1

1.2

( a )

( b )

( c )

( e )

( d )

Figure 3.3: Axial profiles of parameters from z = −10 cm to 7 cm on axis: (a) plasma potentialφ (open circles, with error bars of ±1 V), (b) electron temperature Te (open triangles, with errorbars of ±11%), and (c) normalized electron density ne/ne0 (open squares, with error bars of±8%). Profiles of (d) electron temperature (open triangles) and (e) normalized electron density(open squares) along decreasing plasma potentials, where, for data clarification, the error bars arenot shown.

the Langmuir probe. As the Druyvesteyn method [8, 31, 32] is used for data acquisition,

the electron kinetic energy εe at a specific potential level is represented by the difference

between the plasma potential φ and probe bias voltage Vbias, i.e., εe = eφ− eVbias, where

the potential here refers to the DC component in the plasma potential and the sheath

rectification effect [80] is removed using the CP. Rearranging this yields −eVbias = εe− eφwhich is a representation of the absolute energy state of electrons, and hence the dia-

gram of EEPF as a function of −Vbias reflects the electron energy distribution in a global

manner.

It can be seen that all the curves share a similar shape but the amplitudes vary by

about 16% (i.e., an amplitude uncertainty of ±8% in fpe (εe − eφ)), probably due to the

common fluctuation of measurements taken over a number of days. This is less than the

estimated experimental error of approximately 20%. In figure 3.2(b) at−Vbias = −30 V the

3.2 POLYTROPIC RELATION IN HELICON DOUBLE LAYER THRUSTER 25

Te [ V ]0 2 4 6 8 10

ne/n

e0

0

0.2

0.4

0.6

0.8

1

z = −10 to 7 cm

Te [ V ]5 6 7 8 9 10 11

∆he&

e∆φ

[eV]

-30

-25

-20

-15

-10

-5

0

z = −10 to 7 cm

( a ) ( b )

Figure 3.4: (a) Correlation data between electron temperature Te and normalized electron den-sity ne/ne0, represented by open circles with vertical error bars of ±8% for electron density andhorizontal error bars of ±11% for electron temperature, and the polytropic curve with an indexof γe = 1.17 represented by the solid line. The upper and lower limit curves from the three-sigmarule, i.e., with γe = 1.17 ± 0.06 (3σ), are given as the two dotted lines. The dash-dotted lineand dashed line represent the processes with a polytropic index of 5/3 (adiabatic curve in a LTEsystem) and unity (isothermal curve in a LTE system), respectively. (b) Correlation data betweenelectron temperature Te and relative plasma potential ∆φ, represented by open circles with verticalerror bars of ±1 V for relative plasma potential and horizontal error bars of ±11% for electrontemperature. Relative electron enthalpy ∆he as a function of electron temperature Te (solid line).

curves are normalized to unity, by which the variation caused by experimental fluctuation

can be reduced, and present a stunning consistency. They do show a consistent nonlocal

behavior where the low energy electrons are continuously depleted along the potential

drop (discussed in detail in Ref. [52]) and the high energy ones can overcome the potential

barrier and escape from the plasma source to neutralize the ion beam current. The reliable

EEPF curves having no distortion parts and their good consistency along the system show

that the effect of possible instabilities is negligible in these measurements. Additionally,

previous measurements of the EEPF [91–93] across the radius show a consistent pattern

with those on axis and only a change of slope in the EEPF towards the boundary due to

the magnetic field configuration, also confirming the absence of any electron beams.

Axial profiles of the plasma potential φ (referring to the steady state DC component if

not specified), the electron temperature eTe = pe/ne and the normalized electron density

ne/ne0 (where ne0 is the density value at z = −10 cm at start of the divergent magnetic

field) are given in figures 3.3(a) to 3.3(c) using the EEPF data from figure 3.2(a). A rapid

potential drop occurs from z = −9 cm to −2 cm due to the CFDL, where the electron

temperature decreases from 9.3 V to 6.9 V (depletion of the hot, low energy ranges of

the EEPFs results in an overall “cooling” of the electrons) and the electron density drops

by about 2/3. This phenomenon is well identified in figures 3.3(d) and 3.3(e) where the

electron temperature and density are plotted along decreasing plasma potentials. In the

diffusion chamber (z > 0 cm) these parameters are relatively constant. The uncertainties

of these parametric measurements are also shown in figure 3.3 by the error bars. The

plasma potential obtained from the Druyvesteyn theory [31, 32, 51] has a small uncertainty


of about ±1 V which is caused by the real-time noise in CP measurements. The electron

temperature is estimated to have a maximum uncertainty of ±11% by propagation of

the amplitude uncertainty of ±8% in fpe (εe − eφ). The electron density gains the same

uncertainty of ±8% as the amplitude uncertainty in fpe (εe − eφ).

The correlation between the electron temperature and electron density obtained along

the divergent magnetic field from z = −10 cm to 7 cm (direction indicated by an arrow)

is shown in figure 3.4(a), and the data (open circles, with error bars shown) can be fitted

by a polytropic relation:TeTe0

=

(nene0

)γe−1(3.5)

where Te0 = 9.3 V is the electron temperature at z = −10 cm. The polytropic index γe is

found to be 1.17± 0.02 by carrying out a regression analysis (similarly to that described

in Refs. [94, 95]) for the experimental data shown in figure 3.4(a). The polytropic curve

with γe = 1.17 is shown as a solid line and the three-sigma rule for data fitting, i.e., with

an index of 1.17 ± 0.06 (3σ), is used to give the respective upper and lower limit curves

(two dotted lines) which well cover the measured data.

Polytropic curves with an index of 5/3 (dash-dotted line) and unity (vertical dashed

line) representing the adiabatic and isothermal processes in a LTE system are also given

in figure 3.4(a). Since the electron temperature decreases along the potential drop (from

figures 3.3(a) and 3.3(b)), a polytropic index greater than unity is obtained. If the electrons

were locally governed by Maxwellian statistics resulting from the assumption of LTE, a

polytropic index of 1 < γe < 5/3 would indicate the existence of thermal conduction from

the surroundings into the system [47]. The non-Maxwellian electrons (figure 3.2), which

behave nonlocally and are characterised by their convex EEPFs, have a polytropic index

of 1.17 satisfying 1 < γe < 5/3 but their evolution across the potential drop is obviously

a self-consistent adiabatic process. This phenomenon shows that it is inappropriate to

assign a LTE interpretation for electrons governed by nonlocal EEPFs. It should be

noted that the polytropic index of 1.17 is a typical case belonging to a group of nonlocal

EEPFs characterized by the “convex shape” similar to those in figure 3.2, with the electron

temperature decreasing along the potential drop (as shown in figures 3.3(a) and 3.3(b))

and, hence, a polytropic index being greater than unity. The value of the polytropic index

depends on the specific shape of the EEPFs (detailed in the next section), increasing when

the EEPF curves become more convex and decreasing when less convex.

The electron enthalpy can be expressed as a function of the electron temperature by

combining formula (3.1) and the polytropic relation (3.5):

he =

∫d (eTene)

ne=

γeγe − 1

eTe (3.6)

where he = 6.9eTe for the present results with γe = 1.17. Figure 3.4(b) shows that the

relative electron enthalpy ∆he = 6.9eTe − he0 (solid line) is consistent with the relative

plasma potential ∆φ = φ − φ0 (open circles, for data clarification the error bars are

not shown) as a function of the electron temperature, where he0 = 6.9 × 9.3 eV and

φ0 = 57.3 V are the reference electron enthalpy and plasma potential at z = −10 cm. These

experimental results well verify the enthalpy relation (3.3) and that the corresponding

3.3 GENERALIZED RESULTS 27

electron evolution is an adiabatic process without thermal conduction into the system.

The effect of magnetic pressure and ionization (which acts as a drag to the momentum)

is negligible compared to the electric field under the present experimental conditions,

otherwise the slope of enthalpy line in figure 3.4(b) could not fit the potential data.

In these laboratory plasmas, the ions behave quite differently from the electrons and

can be treated as a cold species without thermal effect [9, 40] for which only the drift

(convective) kinetics needs to be considered. In the absence of ion-neutral collisions, the

momentum equation of an ion swarm drifting at a velocity of u along a magnetic field is

given by:

ud (miu)

dz= eEz = − dpe

nedz(3.7)

which shows that the ion momentum is determined by the electron pressure. Its integration

gives the conservation relation of mechanical energy, i.e., ∆(1/2 ·miu

2)+∆ (eφ) = 0 whose

validity has been verified by previous experiments in the same experimental apparatus [51,

74]: the ion beam energy in the diffusion chamber measured by a retarding field energy

analyzer (RFEA) is consistent with the potential drop in the plasma source. Combining

this integration with electron enthalpy relation (3.3) yields:

∆

(1

2miu

2

)+ ∆he = 0 (3.8)

These results show that the electron enthalpy acts as a source for ion acceleration by

passing energy through a potential structure, which can be used to better understand

plasma behaviors in related laboratory thruster systems [20, 21, 74].

3.3 Generalized Results

3.3.1 Bi-Maxwellian distributions

Although the enthalpy relation is independent of the specific form of nonlocal EEPFs in an

adiabatic process (as demonstrated in section 3.1), the polytropic relation does depend on

it and their correlation is investigated using the example of a bi-Maxwellian distribution:

fpe (εe − eφ) =

c · exp(− εbeTe2− εe−eφ

eTe1

), εe − eφ < εb

c · exp(− εbeTe1− εe−eφ

eTe2

), εe − eφ > εb

(3.9)

where c, εb, Te1 and Te2 are defined as the scale coefficient, break energy, first electron

temperature for low energy range and second electron temperature for high energy range,

respectively. For the present demonstration, the parameters are set as follows: a break

energy of εb = 20 eV, a first electron temperature of Te1 = 10 V and three different

values for the second electron temperature Te2 = 5 V, 10 V and 20 V which result in

“convex”, “linear” and “concave” EEPF curves as a function of the electron mechanical

energy εe − eφ (figures 3.5(a), 3.5(b) and 3.5(c), respectively). For simplicity the scale

coefficient c is chosen for each case to have fpe (0) = 1, and the initial maximum potential

is set to zero φ0 = 0 such that the remaining range satisfies φ ≤ 0. The EEPFs at different

potential locations are represented by the curves with right shifting origins (vertical dashed


0 10 20 30 40

ln(E

EPF)[A

rb.unit]

-6

-5

-4

-3

-2

-1

0

0 10 20 30 40

ln(E

EPF)[A

rb.unit]

-6

-5

-4

-3

-2

-1

0

εe − eφ [ eV ]0 10 20 30 40

ln(E

EPF)[A

rb.unit]

-6

-5

-4

-3

-2

-1

0( c )

( a )

( b )

φ = −30 V

φ = −20 V

φ = −10 V

φ = −10 V

φ = −20 V

φ = −30 V

φ = −30 V

φ = −20 V

φ = −10 V

ǫb

ǫb

ǫb

Figure 3.5: Three cases of bi-Maxwellian EEPFs: (a) convex case (Te1 = 10 V, Te2 = 5 V, c =exp (4)), (b) linear case (Te1 = Te2 = 10 V, c = exp (2)), and (c) concave case (Te1 = 10 V, Te2 =20 V, c = exp (1)), with a break energy of εb = 20 V. The EEPF curves at different potentiallocations have their origins indicated by vertical dashed lines.

lines) and their nonlocality is reflected by the amplitude consistency.

The plasma parameters of electron pressure pe, electron density ne and electron tem-

perature Te are calculated from the bi-Maxwellian EEPFs and given by:

pec

=

[2

3(eφ+ εb) + eTe1 + eTe2

](eφ+ εb)

12 (eTe2 − eTe1) exp

(− εbeTe1

− εbeTe2

)+π

12 (eTe1)

52

2exp

(φ

Te1− εbeTe2

)erf

[(eφ+ εbeTe1

) 12

]

+π

12 (eTe2)

52

2exp

(φ

Te2− εbeTe1

)erfc

[(eφ+ εbeTe2

) 12

](3.10)

nec

= (eφ+ εb)12 (eTe2 − eTe1) exp

(− εbeTe1

− εbeTe2

)+π

12 (eTe1)

32

2exp

(φ

Te1− εbeTe2

)erf

[(eφ+ εbeTe1

) 12

]


-40-30-20-100

pe[A

rb.unit]

10-1

100

101

102

103

-40-30-20-100

ne[A

rb.unit]

10-2

10-1

100

101

102

φ [ V ]-40-30-20-100

Te[V

]

0

5

10

15

20

25( c )

( a )

( b )

eφ+ εb = 0

eφ+ εb = 0

eφ+ εb = 0

Figure 3.6: (a) Electron pressure pe, (b) electron density ne, and (c) electron temperature Teas a function of potential for three EEPF cases: convex case (open circles, Te1 = 10 V, Te2 =5 V, c = exp (4)), linear case (open squares, Te1 = Te2 = 10 V, c = exp (2)) and concave case(open triangles, Te1 = 10 V, Te2 = 20 V, c = exp (1)).

+π

12 (eTe2)

32

2exp

(φ

Te2− εbeTe1

)erfc

[(eφ+ εbeTe2

) 12

](3.11)

eTe =pene

(3.12)

These expressions are valid for potential values satisfying eφ ≥ −εb. For eφ < −εb, the

electrons are located in the Maxwellian region (as illustrated by the EEPF at φ = −30 V

in figure 3.5) and corresponding results are simply given by the Boltzmann-type relations:

pec

=π

12 (eTe2)

52

2exp

(φ

Te2− εbeTe1

)(3.13)

nec

=π

12 (eTe2)

32

2exp

(φ

Te2− εbeTe1

)(3.14)

Te = Te2 (3.15)


ln (ne/ne0 ) [ Norm.unit ]-2 -1.5 -1 -0.5 0

ln(pe/p

e0)[N

orm.unit]

-2.5

-2

-1.5

-1

-0.5

0

ln (ne/ne0 ) [ Norm.unit ]-2 -1.5 -1 -0.5 0

γe

0

0.2

0.4

0.6

0.8

1

1.2

1.4

( a ) ( b )

γe = 0.805

γe = 1

concave case

linear case

γe = 1.18convex case

Figure 3.7: (a) Correlation data between logarithm of normalized electron density ln (ne/ne0)and logarithm of normalized electron pressure ln (pe/pe0) in potential range of −15 V < φ < 0 V,and (b) plolytropic indices given by ratio of log (pe/pe0) to log (ne/ne0) at different data points, forthree EEPF cases: convex (open circles, Te1 = 10 V, Te2 = 5 V, c = exp (4)), linear (open squares,Te1 = Te2 = 10 V, c = exp (2)) and concave (open triangles, Te1 = 10 V, Te2 = 20 V, c = exp (1))cases. Their respective polytropic relations are fitted with a polytropic index of γe = 1.18 (solidline), 1 (dashed line) and 0.805 (dash-dotted line).

It should be noted that for other forms of EEPFs, analytical expressions of these plasma

parameters may not be available and numerical approximation should be used, e.g., for

the measured EEPFs in section 3.2. The results of pe, ne and Te versus φ are respectively

given on figures 3.6(a), 3.6(b) and 3.6(c) for the “convex”, “linear” and “concave” EEPFs.

The electron pressure and density decrease most dramatically for the convex EEPFs (open

circles) and to a lesser degree for the concave EEPFs (open triangles). For eφ < −εb the

three cases show a constant temperature, while for eφ > −εb the electron temperature

decreases for the convex EEPFs and increases for the concave EEPFs.

Correlation between the logarithm of normalized electron density log (ne/ne0) and

logarithm of normalized electron pressure log (pe/pe0) where the subscript “0” indicates

parametric values at the position of zero plasma potential, is shown in figure 3.7(a) for the

three EEPF cases in the potential range of −15 V < φ < 0 V (data from figures 3.6(a) and

3.6(b)). This data can be fitted by the polytropic relation log (pe/pe0) = γe · log (ne/ne0)

where γe is the polytropic index for electrons, with results for the convex EEPFs fitted by

a polytropic curve with an index of γe = 1.18 (solid line), the linear EEPFs with γe = 1

(dashed line) and the concave EEPFs with γe = 0.805 (dash-dotted line).

Figure 3.7(b) shows the polytropic index γe as a function of log (ne/ne0) given by the

ratio of log (pe/pe0) to log (ne/ne0) at different data points on figure 3.7(a), and for each

EEPF case γe [log (ne/ne0)] presents a relatively constant value which is consistent with

their respective fitted indices represented by horizontal lines. It should be noted that the

polytropic relation is an approximation method used to describe a thermodynamic process

rather than an exact model, and hence a perfect fitting between the sampled data and

polytropic curve is not always expected. The polytropic index depends on the specific

shape of EEPFs, increasing when the EEPF curves become more convex and decreasing


when more concave.

Additionally, a typical case of convex nonlocal EEPFs with a rectangular shape is

constructed as follows: the nonlocal probability function is constructed using the Heaviside

function (denoted by H) to give fpe (εe − eφ) = c · H (εmax + eφ− εe) where c is the

scale coefficient and εmax is the maximum electron kinetic energy at the initial plasma

potential (set to be zero, hence φ ≤ 0). The nonlocal Heaviside-type EEPFs present a

sequence of depleted rectangles along decreasing potentials, which is the geometric limit

of convex EEPFs. The corresponding pressure, density and temperature are given by:

pe = 2c/5 · (εmax + eφ)5/2, ne = 2c/3 · (εmax + eφ)3/2 and eTe = 3/5 · (εmax + eφ), which

result in an exact polytropic index of γe = 5/3. Hence, for an adiabatic process governed

by nonlocal EEPFs, multiple polytropic index values can be achieved, as illustrated above

using bi-Maxwellian EEPFs with γe = 1.18, 1, 0.805 and rectangular EEPFs with γe =

5/3. Use of traditional thermodynamics based on LTE would misinterpret the adiabatic

processes with γe < 5/3 as additional heat being brought into the system. The classic

adiabatic index of 5/3 for LTE systems is only an element in the set of polytropic indices

for non-LTE adiabatic systems.

3.3.2 Novel hypothesis for solar wind

Astrophysical plasmas have been studied for many decades including numerous works

on the solar wind [96–98], a typical representation for stellar winds [45, 99, 100]. Direct

space measurements of the solar wind have been achieved using probes aboard satel-

lites [94, 101, 102]. Additionally, progress in the understanding of space phenomena has

been linked and attributed to advances in theoretical modelling and experiments for lab-

oratory plasmas [103]. Hence an investigation of interrelation between the solar wind and

laboratory plasmas can contribute to a better interpretation of physics for both systems.

Electrons in magnetically expanding low pressure plasmas [52, 92] and in the solar wind

share similarities since they: 1) are well confined along magnetic field lines, 2) are nearly

collisionless due to the long mean free path in the low pressure condition, 3) have a thermal

velocity greater than the drift (convective) velocity of the plasma flow due to their small

mass compared to the ion, and 4) are closely associated with a potential drop along the

divergent magnetic field.

Most interestingly, the polytropic index of 1.17 obtained in the HDLT experiment

(section 3.2) is consistent with that identified for the solar wind [104–106]. Since the elec-

trons in the solar wind and laboratory plasmas share important similarities, a reasonable

hypothesis is that their polytropic relations are likely governed by the same principle of

nonlocal EEPFs, i.e., although the electrons in the solar wind present a polytropic index

less than 5/3 as previously reported [94, 102], their actual transport could be an adiabatic

process. Characterization of EEPFs along the acceleration direction of the solar wind is

currently unavailable and additional space measurements are still needed to verify this

hypothesis.

The ions, however, show different behaviours in the solar wind and laboratory plasmas

but this does not affect the electron similarities stated above. In laboratory plasmas, as

the internal kinetic energy of ions is negligibly small compared to their convective kinetic

energy, they can be treated as a cold species without thermal effects and the ion pres-


sure has a negligible momentum contribution [22, 53, 54]. Contrarily in the solar wind,

the thermal kinetic energy of protons (as a major component in positively charged par-

ticles) is comparable to the convective kinetic energy and the proton pressure should be

included when considering proton acceleration [94, 107]. The ion dynamics in the labora-

tory plasma has been discussed in section 3.2 and the electron enthalpy is shown to be the

source of ion acceleration. However, a direct laboratory experiment to simulate proton

transport in the solar wind is currently unavailable and continuous space measurements

along the acceleration direction of the solar wind are also limited by the circular satellite

orbits. Thermodynamic parameters of the solar protons are mainly obtained by collecting

temporal measurements from probes on satellites [94, 101, 102].

Previous studies [104–106] have reported that the solar protons can be characterized

by a polytropic relation with an index ranging from ∼ 1.1 (near the solar corona) to

∼ 1.5 (close to the Earth) along the solar wind expansion. Similarly to the electrons, a

polytropic relation connecting the proton temperature Tp and the proton density np is

given by Tp/Tp0 = (np/np0)γp−1 where γp is the polytropic index for protons. It should

be noted that the proton thermal state can be influenced by additional heating effects

from the Alfven wave [98, 108] and turbulence [107, 109], and hence the adiabaticity is

generally not held for the evolution of protons in the solar wind. The momentum equation

for protons is given by:

ud (mpu)

dz= eEz −

dppnpdz

− Gmp

z2(3.16)

where z and u are the distance from the center of the sun and the drift velocity of the solar

wind, respectively. The right-hand-side terms represent contributions from the electric

field, proton pressure and solar gravity, respectively. Substituting the electric field by the

gradient of the electron pressure (equation (3.4)) and using the polytropic relations for

electrons and protons yield:

∆

(1

2mpu

2

)= − γe

γe − 1e∆Te −

γpγp − 1

e∆Tp +Gmp∆

(1

z

)(3.17)

where on the right hand side: the first term represents the contribution from electron

enthalpy; the second term is given by subtracting the external heating energy (e.g., caused

by the Alfven wave and turbulence) from the proton enthalpy; the third term shows the

energy consumption by the gravitational potential.

For the solar wind near the corona, the electrons and protons can be assumed to

have similar temperatures (Te = Tp = T ) as they originate from an equilibrium state

within the corona [105, 106]. Considering the quasi-neutrality of a plasma, the two charged

species are governed by the same polytropic relation and the combined effect of electron

and proton pressure (sum of the first two terms) in equation (3.17) can be replaced by

−2γe/ (γe − 1) · ∆T . Consequently, with the polytropic index ranging from 1.1 to 1.2

as suggested by previous studies [106], a decrease of 1 V (∼ 104 K) in the temperature

corresponds to a work of 12 eV to 22 eV done by the combined pressure. Since the solar

wind is mostly accelerated in the space range between ∼ 1.3 R (where R is the solar

radius) and ∼ 50 R to reach a typical velocity of ∼ 400 km/s (∼ 0.85 keV) and the

gravitational barrier additionally consumes ∼ 1.5 keV, the total work needed from the


combined electron and proton pressure is ∼ 2.35 keV corresponding to a temperature

decrement of ∼ 1.2 MK to ∼ 2.3 MK along the solar wind which is consistent with

previous studies [48, 106]. This brief calculation shows that the temperature of electrons

and protons decreases dramatically in the acceleration range of the solar wind due to their

low polytropic indices.

3.4 Chapter Summary

This chapter mainly revisits polytropic relation of electrons in low pressure laboratory

plasmas and in the solar wind where the electrons behave nonlocally. For an adiabatic

evolution of electrons along a potential path, a conservation relation between the elec-

tron enthalpy and plasma potential is found and it is independent of the specific form of

nonlocal EEPFs. This relation is verified by on-axis EEPF measurements in a laboratory

HDLT. The experiment reveals that although the transport of electrons is clearly an adia-

batic process, the electron temperature and density can be related by a polytropic relation

with an index of 1.17, i.e., less than 5/3, showing fundamental difference in interpreting

thermodynamic behaviors of non-LTE and LTE particles. The electrons are shown in non-

local momentum equilibrium under the electric field and the gradient of electron pressure.

In laboratory plasmas, the ions can be treated as a cold group whose dynamics is simply

determined by the conservation of mechanical energy, and electron enthalpy is shown to

be the source for ion acceleration.

Correlation between the polytropic relation and the shape of EEPFs in an adiabatic

process is investigated using three typical cases of bi-Maxwellian EEPFs with convex,

linear and concave shapes. Polytropic indices greater than unity, equal to unity, and

less than unity are respectively obtained, suggesting that the polytropic index increases

when the EEPF becomes more convex and decreases when more concave. The classic

adiabatic index of 5/3 for LTE systems is only an element in the set of polytropic indices

for non-LTE adiabatic systems governed by nonlocal particles. Since electrons in the solar

wind and laboratory plasmas share important similarities, they could be dominated by

the same principle of nonlocal EEPFs; although the electrons in the solar wind have a

polytropic index less than 5/3, their actual transport might be adiabatic. The ions in

the two plasma systems behave differently due to distinct internal thermal effects. Ion

acceleration in laboratory plasmas is solely determined by the electron pressure, while

proton acceleration in the solar wind results from combined effects of electron and proton

pressure and solar gravity. A brief calculation for the electron and proton temperature

change in the solar wind gives results consistent with previous studies.

Chapter 4

Ion Beam Experiments

Aiming at a better understanding of the mechanisms behind ion beam formation, this

chapter reports on experiments carried out in a cylindrical plasma thruster and in an

annular plasma thruster based on the Chi-Kung reactor (detailed in chapter 2). Since

a magnetic field guides the charged particle flux along field lines due to the magnetic

force [53, 54], a convergent-divergent field configuration is akin to a Laval nozzle [22, 110]

and used for the present experiments.

For the cylindrical plasma thruster (section 4.1), formation of ion beams travelling in

opposite directions is respectively measured in the converging and diverging parts of a

magnetic nozzle. This bi-directional scenario is unique to plasma flows and quite different

from the classic uni-directional nozzle flow for compressible gases. Further experimental

investigations reveal that ion beam formation is not a one-dimensional phenomenon and

correlated with radial plasma transport in the plasma source. For the annular plasma

thruster (section 4.2), an annular ion beam is observed when the plasma is sustained by

an outer antenna surrounding the outer source tube, and a wake region [111–113] exists

just downstream of the inner tube. The annular configuration for a plasma source provides

an enhanced degree of freedom in terms of electron heating location by inserting another

antenna inside the inner source tube. Ion beam condition and cross-field behavior of

electrons are compared between the two antenna cases.

4.1 Cylindrical Plasma Thruster

4.1.1 Bi-directional ion acceleration

The present experiment is carried out in the Chi-Kung reactor (figure 4.1) with a cylindri-

cal plasma source which is terminated with an aluminium earthed plate rather than the

glass plate used for the helicon double layer thruster (HDLT) experiment [19]. Argon gas

is fed to the system at a constant gas pressure of 5.0× 10−4 Torr monitored with a Bara-

tron gauge. A double saddle antenna operating at a constant power of 310 watts and at a

constant RF of 13.56 MHz is used to sustain the low pressure plasma. The exit solenoid is

solely used (the top solenoid is not shown for clarity) to generate a convergent-divergent

magnetic nozzle whose field lines are calculated from the Biot-Savart law and represented

by solid curves in figure 4.1. In this case the geometry of the magnetic nozzle is determined

from the winding arrangement of coils in the solenoid and independent of the current. A

35

36 ION BEAM EXPERIMENTS

z [ cm ]-30 -20 -10 0 10 20 30

r[cm]

-20

-10

0

10

20

RF

-9

solenoidexit

pumpgas

RFEA_s

LP / EP /RFEA_c / CP

Figure 4.1: Cylindrical Chi-Kung reactor implemented with a convergent-divergent magneticnozzle, showing major components and diagnostic probes. The calculated field lines are plottedwithin the reactor geometry.

Langmuir probe (LP), a compensated Langmuir probe (CP), an emissive probe (EP) and

a retarding field energy analyzer (denoted as “RFEA c”) are separately put through the

vacuum slide mounted on the back plate of the diffusion chamber to allow positioning of

the probes along both the axial and radial directions without breaking vacuum (except

when changing the probe). Another energy analyzer, denoted as “RFEA s”, is inserted

via the top aluminium plate terminating the plasma source.

A current of 9 A generated from a direct-current (DC) power supply is transmitted

into the exit solenoid, defined here as the “solenoid current”, and due to the double-coil-

wound arrangement of the solenoid a current of 4.5 A flows in each coil. The magnetic

flux density on the central axis (where Bz = B) ranging from z = −25 cm to 10 cm, is

measured by a gaussmeter. The data, represented by open squares scaled with the right

labelled y-axis in figure 4.2(a), show a maximum of 200 Gauss at z = −9 cm, i.e., location

of the magnetic throat, and a symmetric decrease to tens of Gauss in the top region of

the plasma source and in the diffusion chamber. Calculated results from the Biot-Savart

law are given by the solid line and consistent with the measurements.

The on-axis profile of ion (plasma) density ni, measured by the LP and calculated using

Sheridan’s method [82, 83], is represented by open circles scaled with the left labelled y-

axis in figure 4.2(a) and well follows the the magnetic flux density B with a maximum of

about 3 × 1010 cm−3 at the magnetic throat at z = −9 cm. This configuration is similar

to that generated in previous studies of electrodeless helicon thrusters [23], where the

maximum ion density corresponds to the location of plasma generation which is further

verified by measuring the axial profile of plasma potential φ using the EP, represented by

open circles in figure 4.2(b): a peak value of about 41 V is measured at z = −9 cm and

decreases along both axial directions; the larger potential decrease of ∼ 10 V measured

from z = −15 cm to −25 cm compared to ∼ 5 V from z = −3 cm to 10 cm results from

the closer proximity of the grounded source end plate at z = −31 cm.

In order to fully characterize ion transport and acceleration along both directions of

the magnetic nozzle, two RFEAs are positioned face-to-face (figure 4.1) with the orifice of

RFEA c facing the plasma source and the orifice of RFEA s facing the diffusion chamber.

Under the present experimental conditions, placement of RFEA c in the axial range of

z > 1 cm in the diffusion chamber or RFEA s in the range of z < −18 cm in the source-

4.1 CYLINDRICAL PLASMA THRUSTER 37

z [ cm ]-25 -20 -15 -10 -5 0 5 10

ni

[

cm−3]

×1010

0

0.5

1

1.5

2

2.5

3

3.5

4

z [ cm ]-25 -20 -15 -10 -5 0 5 10

φ[V

]

0

10

20

30

40

50

B[G

auss]

0

50

100

150

200

( a ) ( b )

magneticthroat

Fig. 4.3 (a)

Fig. 4.3 (b)

Figure 4.2: (a) Right labelled y-axis: on-axis magnetic flux density B generated by a currentof 9 A supplied into the solenoid, measured using a gaussmeter (open squares) and calculated byBiot-Savart law (solid line). Left labelled y-axis: axial profile of ion (plasma) density ni measuredby LP (open circles). The vertical dashed line shows the location of source-chamber interface atz = 0 cm. (b) On-axis profiles of plasma potential φ measured by EP (open circles), beam potentialφb obtained by RFEA c (solid diamonds) and RFEA s (solid triangles), and plasma potential φobtained by RFEA c (open diamonds) and RFEA s (open triangles).

top region has a negligible perturbation (less than a few percent) on plasma parameters,

determined by moving one RFEA on axis and using the other one as a witness probe. For

these regions, the local plasma potential measured by the RFEAs show similar results to

those obtained by the EP, with a maximum deviation of about 3 V and an ion beam is

simultaneously detected by both RFEAs: the beam potential φb measured by RFEA s in

the range from z = −25 cm to −18 cm in the plasma source, represented by solid triangles

in figure 4.2(b), and those measured by RFEA c from z = 1 cm to z = 10 cm in the

diffusion chamber, represented by solid diamonds, are in very good agreement with the

EP-measured maximum plasma potential at the magnetic throat (z = −9 cm). Other

high-field experiments [23] have also shown that the plasma density and potential profiles

are defined by the magnetic flux density profile.

Examples of the ion energy distribution function (IEDF, with respect to the discrimi-

nator voltage Vd and its amplitude being normalized for clarification), obtained by RFEA c

at z = 7 cm in the diffusion chamber and by RFEA s at z = −25 cm in the plasma source

(both locations being 16 cm away from the magnetic throat as indicated in figure 4.2(b)),

are given as solid lines in figures 4.3(a) and 4.3(b). At both positions, the curves present a

similar two-peak distribution with a beam potential value of about 40 V with an ion beam

energy of about 10 eV. Magnetic field intensity influences ion beam strength, as shown

by the IEDF measurements (represented by dash-dotted lines) at both RFEA locations

with a smaller current of 5 A supplied into the solenoid: a two-peak IEDF is still observed

where the beam potential is unchanged (compared to the current case of 9 A) as previously

observed in similar systems [114], but exhibits a lower magnitude at the beam potential.

These experimental results provide clear evidence of a bi-directional ion acceleration

condition where an ion beam, with a zero convective velocity at the magnetic throat, is


Vd [ V ]0 10 20 30 40 50 60

ln(IEDF)[N

orm.unit]

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Vd [ V ]0 10 20 30 40 50 60

ln(IEDF)[N

orm.unit]

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

( a ) ( b )

RFEA c in chamberRFEA s in source

Figure 4.3: Normalized IEDFs obtained by (a) RFEA s at z = −25 cm, r = 0 cm in the plasmasource and (b) RFEA c at z = 7 cm, r = 0 cm in the diffusion chamber, for solenoid current casesof 9 A (solid line) and 5 A (dash-dotted line). Each IEDF curve is normalized by its amplitude atthe low energy peak position.

formed and simultaneously travels “forward” into the diffusion chamber and “backward”

in the closed region of the plasma source. Consequently, the ion generation should be

localized in the throat region of the magnetic nozzle to supplement this bi-directional

particle loss carried by accelerated ion fluxes. Ion “swarm” acceleration along one di-

rection of the nozzle similarly to a compressible gas flow in a Laval nozzle has been

described [53, 54, 110, 115]. Plasma dynamics in a magnetic nozzle can be greatly affected

by detail of ion creation. By using a one-dimensional fluidic approach to describe a fully

magnetized plasma expansion along a magnetic nozzle, Fruchtman [22] predicted that ion-

ization (acting as a mass addition term to the nozzle equation) could cause bi-directional

ion fluxes. In the simple case of no magnetic field, ion generation is the source of free-fall

ions moving along decreasing potentials as detailed by Tonks and Langmuir [40].

Interestingly, the present plasma source configuration generating bi-directional ion

acceleration could provide a compact and simplified system for deorbiting space debris

where one ion beam targets the space debris and the opposite ion beam prevents spacecraft

drift. A few studies on the “ion beam shepherd” technique involving two plasma propulsion

systems implemented onto the spacecraft have been recently conceptually described [116,

117] in the active field of space debris mitigation. It should be noted that the experimental

nozzle configuration used here is not fully “symmetric” due to the geometric expansion at

z = 0 cm (marked by the vertical dashed line in figure 4.2). To illustrate this, the radial

profiles of normalized ion saturation current Isat (measured by the LP at Vbias = −95 V) at

z = −17 cm in the source-top region and at z = −2 cm in the source-exit region are shown

in figures 4.4(a) and 4.4(b), respectively. Both profiles are symmetric around the central

axis but a single-peak profile is observed in the exit region likely due to plasma expansion

from the source-chamber interface into diffusion chamber. This asymmetry suggests that

additional radial and azimuthal effects could play an important role in governing plasma

dynamics in a magnetic nozzle [23, 118].


r [ cm ]-6 -4 -2 0 2 4 6

Isat

0

0.2

0.4

0.6

0.8

1

1.2

r [ cm ]-6 -4 -2 0 2 4 6

Isat

0

0.2

0.4

0.6

0.8

1

1.2

( a ) ( b )

z = −2 cmz = −17 cm

Figure 4.4: Radial profiles of normalized ion saturation current Isat measured by LP at (a)z = −17 cm (open triangles) and (b) z = −2 cm (open circles). The two profiles are normalizedby the maximum ion saturation current at their respective axial positions.

4.1.2 Magnetic field induced transition

Further IEDF measurements with different magnetic field intensities (by varying the

solenoid current) have identified a magnetic field induced transition occurring at a trig-

gering solenoid current of about 4.5 A. Above this threshold value, the plasma discharge

is defined as the “high field mode” and a stable ion beam condition exists as shown by

the current cases of 9 A and 5 A in figure 4.3; below this threshold, the plasma is in

the “low field mode”. Figure 4.5(a) shows the IEDF curve (measured by RFEA c) at

z = 7 cm, r = 0 cm for the low field mode created by a solenoid current of 3 A, ex-

hibiting a single-peak distribution in with a small energetic tail (resolved in the inset),

i.e., no ion beam being detected. Since the IEDFs show a similar distribution pattern

for different solenoid currents within each mode, the typical solenoid current cases of 9 A

and 3 A are used respectively to represent the high field mode and the low field mode

unless otherwise specified. Previous studies have reported a similar transition occurring

in a constant-divergent magnetic nozzle with a triggering mechanism shown to be the ion

magnetization by using different source geometries [119]. It should be noted that a change

in conductivity of cross-field electrons could also result in a mode transition [120].

A detailed description of ion transport from the plasma source into the diffusion cham-

ber needs two-dimensional modeling of plasma dynamics in the magnetic nozzle and is

related to a coupled ionization term between the radial and axial dimensions, which is

beyond the scope of this study. Here the transport of energetic ions is simply reflected

by tracing the on-axis nonlocal ion group accelerated from the plasma source, which is

represented by the nonlocal ion current Inloc derived from integral of the high energy

Gaussian function fitted to a two-peak IEDF (as detailed in chapter 2, subsection 2.3.4)

for the high field mode in figure 4.3. Correspondingly, a local ion current Iloc representing

the background ion group around the RFEA is given by integral of the low energy Gaus-

sian function. For the low field mode (figure 4.5(b)), the IEDF can only be fitted with a

Gaussian function (dashed line) to the low energy part which yields the local current. It


Vd [ V ]0 10 20 30 40 50 60

ln(IEDF)[N

orm.unit]

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

z [ cm ]0 4 8 12 16

Inloc[A

rb.unit]

0

0.2

0.4

0.6

0.8

1

1.2

35 40 45 50

0

0.1

0.2

0.3

( a ) ( b )

low field mode

energetic tailhigh field mode

low field mode

Figure 4.5: (a) Normalized IEDF measured by RFEA c at z = 7 cm, r = 0 cm for low field mode(open triangles), fitted with a Gaussian function (solid line) to the low energy part. Inset resolvesthe energetic tail for clarification. (b) On-axis profiles of nonlocal ion current Inloc for high fieldmode (open circles) and low field mode (open triangles).

should be noted that the absence of an ion beam does not mean there is no accelerated

ions but their fraction is small compared to the background ions. The nonlocal component

contributed by the small energetic tail is given by subtracting the local current (Iloc) from

the total ion current Itot (integral of the overall IEDF). The on-axis profile of nonlocal ion

current obtained from z = 1 cm to 15 cm is given in figure 4.5(b) and shows higher values

for the high field mode (open circles) compared to the low field mode (open triangles). The

ion beam of the high field mode decays along the axial direction due to plasma expansion

in the diffusion chamber and ion-neutral collisions.

Compared to the high field mode, the on-axis density profile for the low field mode,

represented by open triangles in figure 4.6(a) exhibits a maximum of about 6× 1010 cm−3

at z = −15 cm, close to the middle of the RF antenna rather than at the magnetic

throat (z = −9 cm). The profile of plasma potential is given in figure 4.6(b) and shows

a consistent trend with the respective density profile, with a maximum of about 43.5 V

at z = −15 cm. If the ion transport was considered as a one-dimensional phenomenon,

an ion beam should be observed in the diffusion chamber for both field modes due to the

potential drop in the plasma source. The unexpected absence of an ion beam in the low

field mode suggests that radial plasma transport could play a role in ion beam formation

along the magnetic nozzle and will be compared between the two field modes.

4.1.3 Radial plasma transport

An important feature of radial plasma transport in the source region is the cross-field

behavior of electrons [9, 121–123] which is characterized by the CP-measured electron en-

ergy probability function (EEPF, represented by fpe (εe, r) where εe and r are the electron

kinetic energy and spatial position vector, respectively). It should be noted that, similarly

to the IEDFs, the EEPFs show a similar distribution pattern for different solenoid currents

within the high field mode or low field mode, and hence the solenoid current cases of 9 A


z [ cm ]-25 -20 -15 -10 -5 0 5 10

ni

[

cm−3]

×1010

0

1

2

3

4

5

6

7

8

z [ cm ]-25 -20 -15 -10 -5 0 5 10

φ[V

]

0

10

20

30

40

50

( a ) ( b )

Figure 4.6: For the low field mode, on-axis profiles of (a) ion density ni (open triangles) measureby LP, and (b) plasma potential φ (open triangles) measured by EP.

and 3 A can still be used to represent the respective mode. The EEPFs are obtained at

the magnetic throat at z = −9 cm (logarithmic scale on figure 4.7) and, for each mode

the curves are normalized by the maximum amplitude of on-axis data (r = 0 cm). The

reasoning behind this axial position choice is to minimise the effect of axial drift of par-

ticle fluxes on the radial profile. For the high field mode, the axial gradient of both the

plasma density and potential are zero (figure 4.2) and the drift would be approximately

zero. For the low field mode, there is no measurable ion beam (figure 4.6) and hence the

drift influence should also be small.

For the high field mode, the on-axis EEPF curve, represented by the solid line in

figure 4.7(a), shows a convex shape in the main energy range and beyond this an energetic

tail starting at εe ∼ 35 eV which is masked by the experimental noise at the highest energy

part. The EEPF measured at r = −6 cm (dashed line), close to the lateral wall of the

source tube, presents a flattened curve shape compared to the on-axis data and its upper

energy limit increases to about 80 eV. Figure 4.7(b) shows the EEPF measurements for

the low field mode. The on-axis EEPF presents a convex shape with an energetic tail over

εe ∼ 45 eV similarly to the 9 A case in figure 4.7(a), and the EEPF at r = −6 cm exhibits a

decrease in amplitude in the low energy range compared to the on-axis data and the upper

energy limit remains unchanged. The electrons are heated near the outer wall (due to skin

heating effect near the RF antenna [64, 91, 124]) for the high magnetic field case with more

high energy electrons located at r = −6 cm than at r = 0 cm (figure 4.7(a)), while this

heating mechanism is not identified for the low magnetic field case (figure 4.7(b)).

The radial profile of electron (plasma) density at the magnetic throat (z = −9 cm) is

calculated from the EEPF data using ne =∫∞0 ε

1/2e fpe (εe, r) dεe. The high field mode’s

density profile (open circles in figure 4.8(a)) exhibiting a maximum of about 2.5×1010 cm−3

at r = −5 cm, combined with the hollow plasma potential profile (obtained from the zero

crossing of the second derivative of current-voltage traces) presenting a peak of 46.5 V at

the same position in figure 4.8(b), indicates that the plasma is sustained in the inductively

coupled mode [4, 125]. Both the density and plasma potential profiles for the low field mode


εe [ eV ]0 10 20 30 40 50 60 70 80

ln(EEPF)[N

orm.unit]

-6

-5

-4

-3

-2

-1

0

1

εe [ eV ]0 10 20 30 40 50 60 70 80

ln(EEPF)[N

orm.unit]

-6

-5

-4

-3

-2

-1

0

1

( a ) ( b )

Low field modeHigh field mode

Figure 4.7: EEPFs for (a) high field mode and (b) low field mode, measured by CP at z = −9 cm,r = 0 cm (solid line) and r = −6 cm (dashed line). The EEPF curves are normalized by themaximum amplitude of the data measured at r = 0 cm for each mode.

(open triangles) show a peak on axis, with a density of about 6×1010 cm−3 and a potential

of 42.5 V, likely due to an additional wave mode [4] which is consistent with a higher

density measured for the low field mode compared to the high field mode. Experiments

carried out in the annular Chi-Kung reactor (detailed in the next section) also support this

scenario as no density rise in the inner region was observed for the low solenoid current

case in the annular system. The maximum values in the potential and density profiles at

about r = −5 cm in the high field mode suggest that a higher ion production rate occurs

in the edge region compared to the central region, and this is consistent with the EEPFs

in figure 4.7 where more high energy electrons are located at r = −6 cm (dashed line)

compared to r = 0 cm (solid line). Hence the ions are well confined in the central region

by a peripheral “potential barrier” and not lost onto the lateral wall. This ion confinement

scenario does not exist in the low field mode where centrally-peaked density and potential

profiles result in radially outward motion of ions.

The average kinetic energy of electrons is represented by the electron temperature

eTe = 2/ (3ne) ·∫∞0 ε

3/2e fpe (εe, r) dεe. Its radial profile for the high field mode, represented

by open circles in figure 4.8(c), keeps relatively constant in the region of −3 cm < r < 0 cm

and starts to increase at about r = −4 cm. The increase in electron temperatures is

consistent with the results in figure 4.7(a) where the EEPF curve is flattened at r = −6 cm

compared to that at r = 0 cm. Similar phenomena have been observed in the helicon

double layer thruster (HDLT) experiment [91] and magnetic filter experiment [36]. The

low field mode also presents increasing electron temperatures along the radial direction

(open triangles) but with a smaller gradient compared to the high field mode, as it is solely

caused by the decrease of EEPF amplitude in the low energy range (without extending the

upper energy limit) which effectively increase the average electron energy (figure 4.7(b)).

The high field mode exhibits a lower electron density and a larger electron temperature

compared to the low field mode (figures 4.8(b) and 4.8(c)), which is in agreement with the

energy balance in the discharge. The total energy lost per electron-ion pair lost from the


-6 -5 -4 -3 -2 -1 0

ne

[

cm−3]

×1010

0

2

4

6

8

-6 -5 -4 -3 -2 -1 0

φ[V

]

0

10

20

30

40

50

r [ cm ]-6 -5 -4 -3 -2 -1 0

Te[V

]

0

5

10

15

20( c )

( a )

( b )

Figure 4.8: Radial profiles of (a) normalized electron density ne, (b) relative plasma potential∆φ (reference value being on-axis plasma potential), and (c) electron temperature Te, obtained byCP at z = −9 cm for high field mode (open circles) and low field mode (open triangles).

system (which is the sum of the collisional energy loss per electron-ion pair created, the

electron energy lost to the wall, and the ion energy lost to the wall) positively correlates

with the electron temperature [8]. Additionally, the Bohm velocity is proportional to the

square root of electron temperature. Hence for a specific input power, the plasma with

a higher electron temperature (the high field mode) is expected to have a lower density

compared to the plasma with a lower electron temperature (the low field mode). On the

other hand, whether the strong magnetic confinement of the high field mode plays a role

in decreasing its density difference from the low field mode needs a complete modeling of

the magnetic nozzle and the plasma reactor, and will not be discussed here.

The electron temperature has a positive correlation with the ionization rate and hence

its behavior is a good indicator of particle balance within the plasma cavity [8]. Since

the electron Larmor radius is much smaller than the ion Larmor radius, the electrons

can be considered as independent micro-discharges along different magnetic field lines

while the ions are less magnetized and move across the magnetic field more freely. Each


z [ cm ]-40 -30 -20 -10 0 10 20 30

r[cm]

-25

-15

-5

5

15

25

exit

gas

RF

-9

EP / RFEA /

LP / CP

pump

solenoidRF

Figure 4.9: Annular Chi-Kung reactor implemented with a convergent-divergent magnetic nozzle,showing major components and diagnostic probes. The calculated magnetic field lines are plottedwithin the reactor geometry. The outer antenna and inner antenna are independently operated forthe experiments.

electron group should guarantee the balance of ions created nearby and ion flux loss

in both parallel-to-field and cross-field directions. The parallel-to-field balance can be

explained by the principle of L-p similarity [126, 127] where “L” is the length of magnetic

field lines terminated by the source wall and “p” is approximately a constant for weakly

ionized plasmas (with negligible neutral depletion [10, 11]). The electron temperature is

negatively correlated to “L × p” and hence it is larger on the short magnetic field lines

close to walls than that on the long field lines in the central region.

The cross-field particle balance behaves differently under the two field modes. For

the high field mode, electrons gain energy from skin heating near the source tube wall

(figure 4.7(a)) which results in enhanced impact ionization. Correspondingly the plasma

density and potential profiles (figures 4.8(a) and 4.8(b)) exhibit maximum values (with

a very definite peak in potential) at about r = −5 cm and the newly created ions will

move in both outward and inward radial directions. For the low field mode, the ionization

behavior changes (more ions being produced in the central region) with both the density

and potential profiles decreasing from the central axis to the edge, thereby generating

one-directional ion loss in the outer region (at about r = −5 cm) towards the source wall.

These results are consistent with the larger electron temperature gradient measured in the

outer region for the high field mode compared to the low field mode in figure 4.8(c).

The above discussions show that ion beam formation along a divergent magnetic nozzle

is not a one-dimensional phenomenon solely determined from the axial potential drop and

it is correlated with the radial confinement of ions at the magnetic throat. For the high

field mode, high energy electrons are located near the source tube wall with enhanced

ionization due to skin heating effect; the radial profiles of plasma density and potential

create a peripheral potential barrier confining the ions. Additionally, the high magnetic

field intensity for the high field mode is positively correlated to the ion confinement in

the central region. Consequently an axially focused ion beam forms. For the low field

mode, the ionization mechanism changes and more ions are created in the central region,

resulting in outward motion of ions from the central region towards the radial edge. In

this configuration no ion beam is detected downstream of the plasma source although an

axial potential drop (similar to that for the high field mode) exists. These results could

4.2 ANNULAR PLASMA THRUSTER 45

z [ cm ]-10 -8 -6 -4 -2 0 2 4 6 8 10

φ[V

]

0

10

20

30

40

50

z [ cm ]-10 -8 -6 -4 -2 0 2 4 6 8 10

φ[V

]

0

10

20

30

40

50

( a ) ( b )

r = −4 cm r = 4 cm

Figure 4.10: (a) Along r = −4 cm axial profiles of plasma potential φ measured by EP (open cir-cles) and by RFEA (open diamonds), and beam potential φb measured by RFEA (solid diamonds).(b) Along r = 4 cm axial profiles of plasma potential φ measured by EP (open triangles) and byRFEA (open squares), and beam potential φb measured by RFEA (solid squares). The plasma issustained by the double saddle antenna (outer antenna) surrounding the outer source tube.

be applied to improve the propellant efficiency of electric thrusters and to optimize the

system design of focused ion beam devices.

4.2 Annular Plasma Thruster

4.2.1 Annular ion beam

This section investigates how the insertion of an inner tube into the plasma source, which

can be considered as an enlarged diagnostic probe or an immersed power electrode, affects

ion beam formation and its transport away from the annular source region. The study is

carried out in the annular Chi-Kung reactor (figure 4.9) constructed by inserting a 5-cm

diameter glass tube into the 13.7-cm diameter cylindrical source tube, with its sealed end

being located at the source-chamber interface (z = 0 cm) to make the source a totally

annular geometry (as detailed in chapter 2, section 2.1). The experimental conditions are

chosen to be the same as those for the high field mode of the cylindrical plasma thruster

in the previous section: a constant RF power of 310 watts operated by the double saddle

antenna surrounding the outer source tube, an argon gas pressure of 5.0× 10−4 Torr and

a convergent-divergent magnetic field generated by the exit solenoid supplied with a 9 A

current which results in a maximum of 200 Gauss at z = −9 cm. Four electrostatic probes

are used for diagnostics, an EP, a RFEA, a LP and a CP, which are separately mounted to

the vacuum slide of the backplate for two-dimensional measurements. Additionally, this

annular configuration allows another antenna to be housed inside the inner tube as shown

in figure 4.9 and related plasma phenomena are detailed in the next subsection.

The axial profile of plasma potential φ measured by the EP along r = −4 cm on the

negative side and r = 4 cm on the positive side, represented by open circles in figure 4.10(a)

and open triangles in figure 4.10(b), presents a maximum plasma potential of about 33 V


r [ cm ]-10 -8 -6 -4 -2 0 2 4 6 8 10

φ[V

]

0

5

10

15

20

25

30

35

40

z [ cm ]0 1 2 3 4 5 6 7 8 9 10

φ[V

]

0

5

10

15

20

25

30

35

40

( a ) ( b )

Figure 4.11: (a) Radial profiles of plasma potential φ (open markers) and beam potential φb(solid markers), measured by RFEA at z = 1 cm (circles), 3 cm (triangles), 5 cm (diamonds)and 7 cm (squares). (b) Axial profiles of plasma potential φ (open circles) and beam potential φb(solid circles) along the central axis. The plasma is sustained by the double saddle antenna (outerantenna) surrounding the outer source tube.

at the magnetic throat (z = −9 cm) in the plasma source decreasing to about 20 V in

the diffusion chamber. An auxiliary source-facing RFEA mounted through the side wall

of the diffusion chamber (not the primary RFEA put through the vacuum slide on the

backplate) is used as a witness probe to detect whether the EP positioned in the plasma

source dramatically perturbs the plasma or not. The results from the witness probe keep

constant when the EP is moved from the source into the diffusion chamber, showing that

the presence of the EP in the source has little influence on the plasma and the probe gives

reliable measurements. Considering a reduced radial dimension of the plasma source, the

RFEA measurements are only taken downstream in the axial range of z > 1 cm in the

diffusion chamber and not in the source region to minimize perturbations.

The RFEA-measured beam potential φb, represented by solid diamonds along r =

−4 cm and solid squares along r = 4 cm, shows a relatively constant value of about 30 V

and it is consistent with the upstream plasma potential measured by the EP in the plasma

source. The plasma potential obtained from the RFEA (open diamonds and squares) in

diffusion chamber also well correlates with that from the EP with a maximum discrepancy

of about 2 V. These results show the formation of an annular ion beam accelerated out

of the source region and verify the central symmetry of plasma transport in the system.

The radial profiles of plasma potential and beam potential obtained by the RFEA

placed every 2 cm from z = 1 cm to 7 cm in the diffusion chamber, are given in fig-

ure 4.11(a). The plasma potential profile at z = 1 cm (open circles) behind the source-

chamber interface (z = 0 cm) has a similar shape to that in the annular source region

and after this position it keeps constant in the inner region of the diffusion chamber.

The beam potential data (solid markers) change from a two-peak profile to a flat pro-

file as the ion beam propagates into the diffusion chamber, indicating the existence of

a wake just downstream of the inner tube. The increase of beam potential on axis, as

shown in figure 4.11(b), suggests that the beam ions from the plasma source fill the wake


Figure 4.12: (a) Radial profiles of delimited current Idel = Ic (Vd = 30 V), measured by RFEAat z = 1 cm (open circles), 3 cm (open triangles), 5 cm (open diamonds) and 7 cm (open squares).(b) On positive side: colormap of delimited current Idel and magnetic field lines (solid curves).On negative side: boundaries of annular ion beam with an integration percentage χ of 60% (opencircles), 75% (open squares) and 90% (open triangles), determined from equation (4.1). Contoursof plasma potential (vertical solid curves) and electric field pointers in the annular plasma source.The plasma is sustained by the double saddle antenna (outer antenna) surrounding the outer sourcetube.

region [111, 112] due to spatial expansion of the annular ion beam.

In order to characterize the shape of the annular ion beam, a constant voltage is chosen

to define the delimited current using Idel = Ic (Vd = 30 V) which is the RFEA collector

current at the discriminator voltage of 30 V. This technique has been previously used

for ion beam characterization in the HDLT experiment [86] and interpreted as the ion

population with a mechanical energy higher than the chosen energy level. The radial

Idel profile located along the axial direction, as shown in figure 4.12(a), changes from a

two-peak shape to a single-peak shape, behaving similarly to the beam potential profile

in figure 4.11(a) with an ion beam merging location between z = 5 cm and 7 cm. The

maximum value of delimited current at z = 7 cm is half of that at z = 1 cm and the

average magnitude also decreases due to spatial expansion and ion-neutral collisions. The

boundary of the ion beam is derived from the threshold current integration method:

χ =

∫ R−R q [Idel (r)] dr∫ R−R Idel (r) dr

, q [Idel(r)] =

Idel(r) Idel(r) ≥ I∗

0 Idel(r) < I∗(4.1)

The boundary location corresponds to the radial position satisfying Idel(r) = I∗ where

I∗ is the threshold current determined by the integration percentage χ. This procedure

is similar to the idea of Lebesgue integration [128] by which the integration percentage χ

accounts for the contribution of the delimited current from maximum to minimum. For a

given χ, the ion beam is defined by the highest current regions which fits the main body

of the radial profile. The boundary locations of the ion beam derived from equation (4.1)

are presented on the negative side for different χ values in figure 4.12(b) and the colormap

of delimited on the positive side. The inner edge of the annular ion beam moves inward


-3.3-3.9

-4.5

r [ cm ]

-5.1-5.7

-6.380

60

40

εe [ eV ]

20

0

-1

-2

-3

-4

-5

0

ln(EEPF)[N

orm.unit]

-3.3-3.9

-4.5

r [ cm ]

-5.1-5.7

-6.380

60

40

εe [ eV ]

20

0

-1

-2

-3

-4

-5

0

ln(EEPF)[N

orm.unit]

( b )( a )

Figure 4.13: EEPFs for (a) outer antenna case and (b) inner antenna case, measured by CP atz = −9 cm. The EEPF curves are normalized by the maximum amplitude of the measurementsfor each antenna case.

along the axial direction and merge on the central axis, showing the beam ions accelerated

out of the annular aperture fill into the central region downstream. The ion beam changes

to a solid structure after the wake region which is identified to be a reversed cone with a

half opening angle of about 30 and a length limit at about z = 6 cm. When χ decreases,

the outer boundary shrinks radially and the inner boundary broadens axially, making the

ion beam into a thinner annulus.

If the beam ions were only influenced by the magnetic force (the magnetic field line are

presented on the positive side in figure 4.12(b)), their trajectories would not extend into

the central region as the axial component of the ion velocity remained dominant [129].

Difference between the plasma potential profile at z = 1 cm and the following flat profiles

in figure 4.11(a) suggests that the electric field in the plasma source could be a key factor

determining the inward motion of the ion beam. A divergent electric field, represented by

pointers in figure 4.12(b), is identified in the annular source from the potential contours

(vertical solid curves). The electric field near the inner tube (within r = −5 cm) has an

inward component which accelerates the beam ions towards the wake region.

4.2.2 Outer/Inner antenna cases

A multi-loop antenna is positioned inside the inner source tube and provides an additional

RF heating location. The original outer antenna (surrounding the outer source tube) and

this inner antenna are independently operated to power the plasma discharge through

electron heating in the skin depth (i.e., skin heating effect) without activating any wave

mode [92, 130–132], defined here as the “outer antenna case” and the “inner antenna

case”, respectively. For this comparative study, the experimental conditions of RF power,

magnetic field and gas pressure for the inner antenna case are set to be same as those used

for the outer antenna case reported in the previous subsection.

In order to verify the location of electron heating under the two antenna cases, fig-

ure 4.13 exhibits the CP-measured EEPFs at different radial positions at the magnetic


-6.3 -5.7 -5.1 -4.5 -3.9 -3.3

ne

[

cm−3]

×1010

0

0.5

1

1.5

2

-6.3 -5.7 -5.1 -4.5 -3.9 -3.3

∆φ[V

]

-10

-5

0

5

10

r [ cm ]-6.3 -5.7 -5.1 -4.5 -3.9 -3.3

Te[V

]

0

5

10

15

20( c )

( a )

( b )

Figure 4.14: Radial profiles of (a) electron (plasma) density ne, (b) relative plasma potential∆φ (reference value being maximum plasma potential for each antenna case), and (c) electrontemperature Te, obtained by CP at z = −9 cm for outer antenna case (open circles) and innerantenna case (open triangles).

throat (z = −9 cm). For clarity the EEPF data is normalized by the maximum amplitude

for each antenna case. In the outer antenna case (figure 4.13(a)), the upper energy limit

of the EEPF is about 50 eV at r = −3.3 cm and increases to about 70 eV at r = −6.3 cm

where the curve shape is also flattened, suggesting that more high energy electrons are

located close to the outer tube wall. In the inner antenna case (figure 4.13(b)), an inverse

scenario occurs where the upper energy limit shrinks from about 60 eV at r = −3.3 cm

to about 30 eV at r = −6.3 cm, and hence more high energy electrons are located in the

vicinity of the inner tube wall. These results are consistent with the expected location of

electron heating for both antenna cases [64, 92, 124].

The radial profile of electron (plasma) density at the magnetic throat (z = −9 cm),

derived from the EEPF data and given in figure 4.14(a), shows a maximum at about

r = −5 cm for the outer antenna case (open circles) and at about r = −4 cm for the inner

antenna case (open triangles). The plasma potential profile, represented by the relative

value in figure 4.14(b) to resolve the potential variation (for which the reference value is


z [ cm ]-25 -20 -15 -10 -5 0 5 10

ni

[

cm−3]

×1010

0

0.5

1

1.5

2

z [ cm ]-25 -20 -15 -10 -5 0 5 10

φ[V

]

0

10

20

30

40

50

( a ) ( b )

Figure 4.15: Axial profiles of (a) ion (plasma) density ni measured by LP and (b) plasma potentialφ measured by EP, along r = −4 cm for outer antenna case (open circles) and inner antenna case(open triangles)

chosen as the maximum plasma potential for each antenna case), presents a consistent

trend with the respective density profile. For the outer antenna case, the electron tem-

perature (open circles in figure 4.14(c)) increases radially from about 11 V at the inner

wall to the maximum of about 16 V at the outer wall, consistent with the broadening of

the energy range in figure 4.13(a). For the inner antenna case, the electron temperature

(open triangles) decreases from about 16 V at r = −3.3 cm to the minimum of about 8 V

at r = −5.5 cm and then slightly increases to the outer wall by about 1 V.

Since both antenna cases show a decrease of electron density and temperature away

from electron heating locations, the “magnetic filter” effect [37] plays an important role

in governing radial transport of electrons. The magnetic nozzle acts as a filter across the

radial dimension at the magnetic throat (z = −9 cm) and presents some difference to that

reported in previous studies [36, 64, 133]: a) the magnetic flux density increases from the

inner tube to the outer tube, i.e., an asymmetric filter across the gap; b) due to the sudden

expansion from the source region into the diffusion chamber, there is an abrupt change

to the length of magnetic field lines from those completely bounded in the plasma source

to those partially terminated at the wall of the diffusion chamber, and the threshold is

the last magnetic field line to touch the source-chamber interface (marked by the green

dash-dotted line in figure 4.9) passing the position at z = −9 cm, r = −4.1 cm. It can

be seen that there are also magnetic field lines interrupted by the inner tube, with the

threshold being the line to touch the lateral wall of the inner tube (marked by the red

dashed line in figure 4.9), but these interrupted field lines have no direct influence on the

magnetic filter at z = −9 cm.

For the inner antenna case, a small rise of electron temperature ranging between r =

−5.7 cm and −6.3 cm is unexpected. It is noted that this is not caused by the broadening

of the energy range for the outer antenna case, but by the decrease of low energy electrons

in the EEPF (as shown in figure 4.13(b)) which effectively increases the average kinetic

energy of electrons. Since the electrons located in this region are not directly affected


Vd [ V ]0 10 20 30 40 50 60

IEDF

[Norm.unit]

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

r [ cm ]-12 -10 -8 -6 -4 -2 0

Inloc[A

rb.unit]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

( a ) ( b )

Figure 4.16: (a) Normalized IEDFs measured by RFEA at z = 3 cm, r = −4 cm for outerantenna case (solid line) and inner antenna case (dash-dotted line). (b) Radial profiles of nonlocalion current Inloc at z = 3 cm for outer antenna case (solid circles) and inner antenna case (solidtriangles).

by electron heating from the inner antenna and can be considered as independent micro-

discharges along magnetic field lines, one possible scenario for this phenomenon is due to

parallel-to-field particle balance: according to the principle of L-p similarity [126, 127], the

electron temperature would increase along shorter magnetic field lines (i.e., characteristic

discharge length) near the outer tube wall.

The axial profiles of plasma density (measured by the LP) and potential (measured by

the EP) along r = −4 cm are compared for the two antenna cases in figure 4.15. For the

outer antenna case, the density (open circles in figure 4.15(a)) exhibits a maximum at the

magnetic throat (z = −9 cm), while for the inner antenna case, the maximum occurs at

z = −18 cm close to the middle of the inner antenna. The plasma potential (in 4.15(b))

shows a consistent trend with the respective density profile; the inner antenna case presents

a similar potential profile to the outer antenna case in the axial range between z = −9 cm

and 10 cm, but from z = −9 cm back to −18 cm the potential continues to climb to a value

of about 44 V and then slightly decreases. Interestingly, the inner antenna case results

are similar to those performed on the central axis for the low field mode in the cylindrical

plasma thruster (figure 4.2), as these two conditions have their plasma generation in the

central region of the plasma source.

It should be noted that the measurement along r = −4 cm does not follow the mag-

netic field line but passes through curved field lines. Although this axis cannot represent

the transport path of magnetized electrons during the plasma expansion, it does reflect

the acceleration path of weakly magnetized ions. Figure 4.16(a) illustrates the RFEA-

measured IEDF measurements at z = 3 cm, r = −4 cm where a two-peak distribution

occurs for the outer antenna case (solid line) and the related ion beam phenomena have

been studied in the previous subsection. Only a single-peak IEDF is identified for the

inner antenna case (dash-dotted line) with a negligibly small energetic tail. Additionally,

there is no ion beam detected for all radii under the inner antenna case, and an auxiliary


r [ cm ]-12 -10 -8 -6 -4 -2 0

Iloc[A

rb.unit]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

r [ cm ]-12 -10 -8 -6 -4 -2 0

µiz

[

cm3·s−

1]

×10-8

0

0.2

0.4

0.6

0.8

1

1.2

( a ) ( b )

Figure 4.17: Radial profiles of (a) local ion current Iloc obtained by RFEA and (b) ionizationrate coefficient µiz calculated from formula (4.2), at z = 3 cm for outer antenna case (open circles)and inner antenna case (open triangles).

experiment by pushing the RFEA into the plasma source also confirms the absence of an

ion beam (despite possible perturbing effects caused by the probe).

For the outer antenna case, the ion beam strength represented by the radial profile

of nonlocal ion current (solid circles in figure 4.16(b)), shows a peak around r = −2 cm

and becomes weak near r = −9 cm likely due to ion beam detachment from the magnetic

nozzle [4, 115]. For inner antenna case (solid triangles), the nonlocal ion current profile

presents a low magnitude approaching zero. One scenario for the presence and absence

of an axial ion beam is to compare the generation and loss areas of ions across the radial

dimension in the annular plasma source: for the outer antenna case, the ion generation

area (near the outer tube wall) is larger than the ion loss area (close to the inner tube wall),

and hence more ions are supplemented into the axial dimension to form an ion beam; for

the inner antenna case, the generation area (near the inner tube wall) is smaller than the

loss area (close to the outer tube wall), and hence less ions can be accelerated along the

axial direction. This is consistent with the axial density magnitude of the outer antenna

case being higher than that of the inner antenna case as shown in figure 4.15(a), and

with an auxiliary experiment using an increasing input power which shows a continuously

enhanced energetic tail in the IEDF.

The radial profile of local ion current at z = 3 cm is given in figure 4.17(a) where the

outer antenna case (open circles) shows a peak on the outer side of the diffusion chamber

at r = −9 cm, forming a density conic in the plasma plume which has been observed

in previous studies [93, 134], while the inner antenna case (open triangles) presents a top

region around r = −5 cm that is closer to the inner side. Measurements of total ion cur-

rent made with a RFEA facing the radial cylindrical wall (with ion beam effect removed)

show a consistent shape with the local ion current data for each antenna case. This phe-

nomenon is related to the variation of electron energy distributions at the magnetic throat

(figure 4.13), and a result of local ionization enhancement caused by the energetic electrons

which transport along magnetic field lines from the plasma source into the diffusion cham-


z [ cm ]0 4 8 12 16

Inloc[A

rb.unit]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

z [ cm ]0 4 8 12 16

Iloc[A

rb.unit]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

( a ) ( b )

Figure 4.18: On-axis profiles of (a) nonlocal ion current Inloc and (b) local ion current Iloc,obtained by RFEA for outer antenna case (circles) and inner antenna case (triangles).

ber [93]. Hence it is constructive to compare the ionization rate coefficient µiz between

the two antenna cases, which is derived from the EEPFs using:

µiz =

(2e

me

) 12 1

ne

∫ ∞0

σizεefpe (εe, r) dεe (4.2)

where e = 1.6022 × 10−19 C and me = 9.1094 × 10−31 kg are the electron charge and

electron mass, respectively. σiz is the ionization cross section and for argon plasmas, the

result formulated by Phelps [135] is used (Appendix B). Figure 4.17(b) shows the radial

profile of ionization rate coefficient which has its peak position at about r = −8.5 cm for

the outer antenna case and at about r = −6 cm for the inner antenna case, consistent

with the respective local ion current profile in figure 4.17(a). Since the inner antenna case

has less energetic ions accelerated out of the plasma source and more local ions generated

in the diffusion chamber, the IEDF (measured in the diffusion chamber) would present a

high-magnitude low energy part and a low-magnitude energetic tail as shown in figure 4.16.

4.2.3 Wake Region

A region of interest is the wake cavity in the diffusion chamber (as identified in figure 4.12)

just downstream of the sealed wall of the inner tube, which is not directly connected to

the electron heating location (figure 4.9) and the charged particles can only fill in through

cross-magnetic-field transport. This region can be seen as a simplified plume simulation

for re-entry spacecrafts in the plasma wind tunnel.

The nonlocal ion current on axis from the closed end of the inner tube into the diffusion

chamber is given in figure 4.18(a) where the outer antenna case (solid circles) shows an

vaulted profile with a peak at about z = 5 cm which corresponds to the length limit of the

wake region (figure 4.12), while the inner antenna case presents a negligibly low magnitude

that is consistent with its absence of an ion beam (figure 4.16). The contribution of the

low energy background ions is represented by the local ion current in figure 4.18(b), and


-10 -8 -6 -4 -2 0

r [ cm ]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

D[

cm2·s−

1]

×105

-10 -8 -6 -4 -2 0

r [ cm ]

0

2

4

6

8

10

12

14

Te[V

]

0

10

20

30

40

50

60

B[G

auss]

( a ) ( b )

Figure 4.19: At z = 3 cm, (a) Right labelled y-axis: radial profile of magnetic flux densityB calculated from Biot-Savart law (solid line). Left labelled y-axis: radial profiles of cross-fielddiffusion coefficient of ions Di (open squares) calculated from formula (4.3), cross-field diffusioncoefficient of electrons De calculated from formula (4.4) for outer antenna case (open circles) andinner antenna case (open triangles). (b) Radial profiles of electron temperature Te, obtained byCP for outer antenna case (open circles) and inner antenna case (open triangles).

the profiles of both antenna cases show a similar trend and a broad top region further

from the plasma source around z = 10 cm to 12 cm. Hence, for the outer antenna case,

transport of beam ions into the wake region enhances the wake filling process.

The filling of background ions is considered to be driven by the same mechanism for

both antenna cases due to their similar profiles in figure 4.18(b). Since the plasma potential

has a constant profile around the wake region with negligible radial electric field (as shown

in figure 4.11), the transport of ions into the wake region is dominated by diffusion effect

due to variation in particle concentration [8, 136]. The ion temperature is approximately

equal to the gas temperature in partially ionized plasmas [8], i.e., Ti = Tg = T , and the

two antenna cases have the same cross-field diffusion coefficient of ions [137]:

Di = αi ·Di0 =ν2mi

ν2mi + ω2i

· 3 (π)12

8ngσmi

(eTgmi

) 12

(4.3)

where Di is the isotropic (unmagnetized) diffusion coefficient obtained using the Chapman-

Enskog approximation [138], and the scale factor αi reflects the magnetic field effect on

ions. ωi = eB/mi is the ion gyrofrequency and the argon ion mass is equal to mi =

39.95 u (u = 1.6605 × 10−27 kg is the atomic mass unit). νmi is the ion-neutral collision

rate given by νmi = σmivinng where σmi is the momentum-transfer cross section for ion-

neutral collisions (including elastic polarization collisions and resonant charge exchange)

and σmi = 10−18 m2 is normally applied to argon plasmas [8]; vin = (8/π · eTg/mR)1/2 is

the mean velocity for ion-neutral collisions due to their thermal motions, wheremR = mi/2

is the reduced mass for ions in parent gas; ng = pg/(eTg) is the neutral density given by

the gas law, and for the present experiment pg = 5× 10−4 Torr and Tg = 0.026 V (room

temperature). It should be noted that the ion temperature can be considerably higher

than the gas temperature in many cases, e.g., Ti = 0.1 V, and this change will not affect


ǫe [ eV ]-5

-4

-3

-2

-1

0

1

ln(EEPF)[N

orm.unit]

z [ cm ]0 2 4 6 8 10

Te[V

]

0

2

4

6

8

10

εe [ eV ]0 20 40 60 80

-5

-4

-3

-2

-1

0

( a ) ( b )

outer antenna case

inner antenna case

Figure 4.20: (a) EEPFs measured by CP at z = 5 cm, r = 0 cm for outer antenna case andinner antenna case. The EEPFs are normalized by the respective maximum amplitude for eachantenna case. (b) On-axis profiles of electron temperature Te for outer antenna case (open circles)and inner antenna case (open triangles).

the major conclusions in the following discussions. Formula (4.3) shows that the ion

diffusion coefficient is positively correlated with the ion temperature and negatively with

the magnetic flux density. The radial Di profile at z = 3 cm is represented by open squares

in figure 4.19 scaled with the left labelled y-axis, and shows an increasing profile along

the radial direction corresponding to the decreasing magnetic flux density (scaled with the

right labelled y-axis).

The cross-field diffusion coefficient of electrons [27, 29] is given by:

De =2e

3mene

∫ ∞0

αe ·ε

32e fpe (εe, r)

νmedεe =

2e

3mene

∫ ∞0

ν2meν2me + ω2

e

· ε32e fpe (εe, r)

νmedεe (4.4)

where the scale factor αe represents the magnetic field effect on electrons; νme = ngσmeve

is the electron-neutral collision frequency, where σme is the effective momentum-transfer

cross section for electron-neutral collisions (including elastic collisions, inelastic excitation

and ionization collisions) and ve = (2εe/me)1/2 is the electron velocity. σme is a function

of the electron energy εe, i.e., σme = σme (εe), and its formula is given in Appendix B. It

should be noted that under the present experimental conditions the electron-ion collision

frequency is less than one percent of the electron-neutral collision frequency, and hence

the contribution of electron-ion collisions is neglected in formula (4.4). ωe = eB/me is

the electron gyrofrequency; since αe contains the term νme, it cannot be taken out of the

integral. The De data is presented in figure 4.19(a) and scaled with the left labelled y-axis,

where the inner antenna case (open triangles) shows a slightly higher magnitude than the

outer antenna case (open circles). The continuous increase of the diffusion coefficient in

the outer region (beyond r = −9 cm) is due to the decreasing magnetic intensity nearby.

Although the electrons are better confined by the magnetic field than the ions ωe ωi, the electron diffusion coefficient is greater than the ion diffusion coefficient due to a

large difference in their internal thermal motion reflected by Te Ti. The short-circuit


effect, by which the electrons are transported radially through an equivalent “electric

circuit” formed by the magnetic field and the diffusion chamber wall [136, 139, 140], is

not necessarily needed for cross-field transport of electrons under the presumed condition

of room-temperature ions. This mechanism could be important as the ion temperature

increases and the ion diffusion is enhanced. Here the short-circuit effect is neglected for

simplicity, the electrons move inward through electron-neutral collisions which dissipate

kinetic energy of electrons and result in an overall cooling behavior [141], reflected by the

electron temperature profile in figure 4.19(b) which has a minimum level within the wake

region and shows consistent results for both antenna cases. It should be noted that the

low temperature region beyond r = −9 cm is also due to collisional cross-field diffusion of

electrons similarly to the wake, as the magnetic field lines nearby cannot trace back into

the plasma source.

Along the central axis behind the inner tube, a quasi-Maxwellian electron distribution

occurs in both antenna cases, as shown in figure 4.20(a), and results in a constant axial

profile of electron temperature in figure 4.20(b) at about 5.5 V which is consistent with

the inverse slope of the logarithmic EEPF curves [31] in figure 4.20(a), i.e., the plasma

wake is a “quiescent” region dominated by quasi-Maxwellian electrons.

4.3 Chapter Summary

This chapter presents the ion beam experiments carried out in the Chi-Kung reactor

consisting of a cylindrical plasma source (cylindrical plasma thruster) or an annular plasma

source (annular plasma thruster). Both configurations are implemented with a convergent-

divergent magnetic nozzle. For the cylindrical plasma thruster under the high field mode,

bi-directional ion acceleration is observed along the magnetic nozzle with a zero convective

velocity at the magnetic throat, and the ion beam strength is positively correlated to the

magnetic field intensity. Axial profiles of plasma density and potential follow the magnetic

flux density profile. Ion generation should be localized in the throat region of the magnetic

nozzle to balance the bi-directional particle loss and is considered to be an important factor

in determining this novel plasma flow scenario. When the magnetic field intensity is further

decreased, a mode transition occurs and the plasma discharge changes into the low field

where the location of maximum plasma density and potential is close to the middle of

the antenna rather than at the magnetic throat. No ion beam is identified in this mode

although decreasing potentials do exist along the axial direction. Ion beam formation

along a divergent magnetic nozzle is not a one-dimensional phenomenon solely determined

from the axial potential drop, and it is correlated with the radial plasma transport which

shows different behaviors between the two field modes. The high field mode has a better

confinement of ions at the magnetic throat compared to the low field mode, and the

energetic ions accelerated out of the plasma source are more axially focused to form the

ion beam.

When the annular plasma thruster is operated in the outer antenna case, an annular

ion beam forms out of the plasma source and transforms to a solid structure (with the

central hollow filled) in the plume, resulting in the formation of a reversed-cone plasma

wake just downstream the inner tube. The inward motion of ion beam is considered to be


caused by the divergent electric field in the source-exit region. This annular configuration

allows the creation of a low pressure argon plasma with an antenna housed inside the

inner source tube. The electron heating location in the annular plasma source, close to

the outer tube wall for the outer antenna case and to the inner tube wall for the inner

antenna case, is clearly shown from changes in the fraction of energetic electrons and in

the shape of EEPF curves across the annulus at the magnetic throat. The radial behavior

of electrons is affected by the magnetic filter effect, and the transport properties present

opposite trends for the two antenna cases. The absence of an ion beam condition in the

inner antenna case (with the occurrence of an axial potential drop in the plasma source) is

explained in terms of radial loss of ions in the plasma source and enhanced local ionization

in the diffusion chamber. The plasma fills in the wake region through cross-field transport.

The filling process of ions is enhanced by the ion beam effect in the outer antenna case,

and the transport of background ions is due to free diffusion for both antenna cases. The

collisional diffusion likely plays an important role in the cross-field transport of electrons

towards the plasma wake and results in electron cooling during this process. The wake is

a quiescent region mostly filled with quasi-Maxwellian electrons for both antenna cases.

Chapter 5

Plasma Modeling across Annuli

This chapter studies radial transport properties of electropositive, unmagnetized, weakly

ionized plasmas (from noble gases) across annular geometries. The plasma is classified

into the collisional regime and collisionless regime. Here the term “collisional” refers

particularly to the ion-neutral collisions including the elastic polarization scattering and

resonant charge transfer; this reference is used by default in this chapter unless otherwise

specified. The long range Coulomb scattering (including ion-ion, ion-electron and electron-

electron collisions) can be neglected for weakly ionized plasmas. Particle transport in a

collisional plasma is well described in terms of “diffusion” and “mobility”. This represen-

tation method has the advantages of connecting kinetic and fluidic theories [28, 30], and

unifying unmagnetized and magnetized plasmas into a simple algebraic form [8, 136]. For

a collisionless plasma, the concepts of diffusion and mobility lose validity and a kinetic

approach for ion motion should be used.

Section 5.1 briefly discusses the electron equilibrium in low temperature plasmas. Sec-

tion 5.2 characterizes the ion transport in a collisional annular plasma with three electric

field dependent models: a low electric field (LEF) model, an intermediate electric field

(IEF) model and a high electric field (HEF) model. In section 5.3 the collisionless ion

transport is studied using the Tonks and Langmuir theory [40] and the solution is ex-

pressed in terms of the Maclaurin series approximant and Pade rational approximant. In

both sections 5.2 and 5.3 the collisional model and collisionless model are used to obtain

numerical results for the argon plasma.

5.1 Electron Equilibrium

Due to the large mass difference between an electron and a neutral, the light electron expe-

riences a fast momentum transfer and a suppressed energy transfer during electron-neutral

collisions. This process contributes to Maxwellianization of electrons in low temperature

plasmas over a wide pressure range. On the other hand, for a very low pressure plasma

with fewer electron-neutral collisions, the Maxwellian distribution of electrons has also

been shown to be largely valid and this phenomenon is known as the “Langmuir para-

dox” [142–144]. In this sense the major interest of particle transport is to analyze the

non-equilibrium ions (as detailed in sections 5.2 and 5.3). For simplicity, the electron

transport is briefly discussed for the plasma regime where both electron-neutral and ion-

neutral collisions contribute to the momentum transfer of charged particles. The electron

59

60 PLASMA MODELING ACROSS ANNULI

flux vector Γe = neue and ion flux vector Γi = niui where n and u represent the density

and velocity vector (subscripts “i” and “e” denoting ions and electrons, respectively), can

be expressed as the sum of drift (mobility) due to the electric field E and diffusion due to

the density gradient ∇rn (where ∇r is the gradient operator with respect to the position

vector):

Γi = niK(2)i ·E−D

(2)i · ∇rni (5.1a)

Γe = −neK(2)e ·E−D(2)

e · ∇rne (5.1b)

where K(2)i,e and D

(2)i,e are the second order tensors of mobility coefficients and diffusion co-

efficients, respectively. Their non-diagonal terms are zero for unmagnetized plasmas. The

electron and ion fluxes are governed by the local ambipolarity, Γi = Γe (non-ambipolarity

may arise in the magnetized case [121, 136]), and the electrical neutrality is held within

the bulk plasma, ni = ne = n. Combining equation (5.1b) and (5.1a) yields:

n(K(2)e + K

(2)i

)·E +

(D(2)e −D

(2)i

)· ∇rn = 0 (5.2)

For low temperature plasmas, the electrons have a more profound thermal motion

than the ions, i.e., Te Ti, and respond more quickly to the electric field due to the

small inertial mass. The mobility and diffusion efficiency is much higher for electrons

than for ions, and the diagonal elements of the coefficient tensors satisfy Ke Ki and

De Di [8, 136]. Then equation (5.2) is further simplified to give:

nK(2)e ·E + D(2)

e · ∇rn ∼= 0 (5.3)

which is equivalent to neglecting Γe in equation (5.1b). This shows that the overall flux

of electrons is small compared to the induced motion by either drift or diffusion, i.e., the

electrons are in an equilibrium state. When the electrons are in in a Maxwellian equilib-

rium with a constant electron temperature Te across the plasma region, their mobility and

diffusion coefficients are connected by the famous “Einstein relation” [28, 29]:

D(2)e = Te K(2)

e (5.4)

Substituting this relation into equation (5.3) yields the Boltzmann relation [8, 28]:

n = n0 exp

(φ− φ0Te

)(5.5)

where n0 is plasma density at a reference potential φ0. Conveniently, the maximum plasma

potential is chosen as the reference potential and set to be zero, i.e., φ ≤ 0.

5.2 Collisional Modeling

5.2.1 Ion mobility coefficient

Differently from the electrons, the ion flux Γi cannot be neglected for equation (5.1a) as the

heavy ions are not in equilibrium. As for an unmagnetized annular plasma no azimuthal

5.2 COLLISIONAL MODELING 61

flux exists and, on first approximation, a constant plasma flow is assumed along the axial

direction, the radial dimension can be separately solved at the cost of some loss in accuracy.

The radial ion flux is given by:

Γir = −Dirdn

dr+ nKirEr (5.6)

where r is the physical radial position and its subscript notation indicates the radial

component of vectors. Combining it with the Boltzmann relation yields:

Γir = − (Dir + TeKir)dn

dr= −Dfr

dn

dr, (5.7a)

uir =Γirn

= −Dfrdn

ndr=Dfr

TeEr (5.7b)

where Dfr = Dir + TeKir is defined as the effective transport coefficient.

When the ion drift velocity is small compared to the ion thermal velocity, the diffusion

coefficient Dir and mobility coefficient Kir are connected by the linear Einstein relation

Dir = TiKir and Dfr = (Ti + Te)Kir ≈ TeKir. When the drift velocity is comparable to

or larger than the thermal velocity, Dir and Kir violate the linear Einstein relation and

a nonlinear generalized Einstein relation (GER) should be used [145]. In this case the

ion diffusion coefficient becomes a complicated electric field dependent parameter [28, 146]

and was normally neglected in previous studies [8, 147]. This study follows a consistent

path by neglecting the diffusion effect, i.e., the ions behave in an ion mobility governed

manner, and the effective transport coefficient is rewritten as:

Dfr = TeKir (5.8)

Hence the radial ion flux Γir (5.7a) is determined by Kir. The ion mobility coefficient Kir

exhibits different expressions in regard to the comparative role of internal thermal effect

and electric field in ion-neutral collisions, for which three electric field dependent mobility

models are used: a low electric field (LEF) model, an intermediate electric field (IEF)

model and a high electric field (HEF) model. For weakly ionized plasmas, the neutral

depletion and gas heating [10, 148, 149] are negligible; the neutrals are homogeneously

distributed in the background and their temperature is approximately equal to the ion

temperature, Tg = Ti = T .

In the LEF model (first introduced by Schottky [150]) the ion-neutral collisions are

dominated by the thermal motion of ion and neutral particles and the electric field effect

is negligible. The ion mobility coefficient is independent of the electric field and given

by the first Chapman-Enskog approximation [138]. When the electric field strength is

increased the dependency of the mobility coefficient on the electric field appears. At the

upper limit of the electric field strength described by the HEF model (firstly introduced

by Godyak [147]), the ion-neutral collisions are dominated by the strong electric field

and the thermal effect becomes negligible (cold gas limit). The ion mobility coefficient

is inversely proportional to the square root of the electric field strength [38, 147]. The

IEF model considers both the thermal effect and electric field effect, and an “effective

ion temperature” is given to connect the two effects [145, 146] and obtain the mobility


coefficient. For a bounded plasma, the LEF regime corresponds to the central region of

maximum plasma density and the HEF to the near-wall presheath which accelerates the

ions to the Bohm velocity [12, 151]. The presheath width extends as the gas pressure

decreases [152], and hence the LEF and HEF regimes dominate the high pressure and low

pressure plasmas, respectively.

Since the deduction of ion mobility coefficient is complicated and can be found in the

literature on particle physics (e.g., [28]), only a summary of the important results is listed.

In the LEF model Kir is solved by the first Chapman-Enskog approximation [138]:

Kir =3 (π)

12

8

e

ngσ∗mi

(1

mieTg

) 12

(5.9)

where e and mi are the electron charge and ion mass, respectively. σ∗mi is a cross section

averaged over a distribution of the ion-neutral collisional energy εc:

σ∗mi =1

2 (eT )3

∫ ∞0

σmi (εc) e− εceT ε2c dεc (5.10)

where T is a characteristic temperature for the ion-neutral collision energy distribution and

σmi (εc) is the momentum transfer cross section. Since σmi depends weakly on εc in low

temperature plasmas, it is approximated to be a constant (hard sphere collision) [8, 72].

In this case the distinction between σ∗mi and σmi disappears, σ∗mi=σmi.

In the HEF model Kir is inversely proportional to the square root of the electric field

strength [24, 38]:

Kir = ξH

(e

mingσmi|Er|

) 12

(5.11)

where the factor ξH = 2/π1/2 is chosen from the Smirnov model [24, 38]. It should be noted

that ξH slightly varies depending on the specific mobility theory. The absolute value of Er

is used in the above formula as the electric field can be either positive or negative within

an annulus (always positive in a cylinder).

For the IEF model, both the thermal effect and electric field are included by defining

an effective ion temperature Tif :

3

2eTif =

3

2eTg +

1

2miu

2dr (5.12)

where udr is the electric drift velocity given by udr = KirEr, equivalent to the ion mean

drift velocity uir in a homogeneous plasma. In a non-homogeneous plasma udr should

be less than uir as the latter includes the influence of both electric field and density

gradient. Since the diffusion effect (due to density gradient) is neglected in the present

study, udr = uir is satisfied by substituting Dfr = TeKir into equation (5.7b). The first

term and second term on the right hand side of formula (5.12) represent the contributions

of neutral thermal effect and electric field to the collisional energy of ions [145, 146]. It

should be noted that in previous studies a combination of cold neutrals and warm ions [153]

and the opposite case of cold ion beam and warm neutrals [72] have also been reported.


For the IEF model, Kir is given in terms of the effective ion temperature [28]:

Kir = ξIe

ngσmi

(1

mieTif

) 12

(5.13)

where the factor ξI = 3π1/2/8 is taken from the Mason model [28, 146]. This formula

is an implicit equation of Kir and to make it explicit, two parameters are introduced: a

dimensionless electric drift velocity

udr =

(8

π

) 12 udrvin

where vin = (8/π · eTg/mR)1/2 is the mean velocity for ion-neutral collisions due to their

thermal motions and mR = mimg/ (mi +mg) = mi/2 is the reduced mass; and a dimen-

sionless electric field parameter

εr =αIrbTe

Er

where αI = 3 (2π)1/2 /16 · eTe/ (σmiPas). Formula (5.13) is rewritten as:

udr

(1 +

2

3u2dr

) 12

= εr (5.14)

which is a quadratic equation of udr. Its real root is given by:

|udr| =(3)

12

2

[(1 +

8

3ε2r

) 12

− 1

] 12

(5.15)

The final expression of Kir is given by:

Kir =(6π)

12

8

vinrbTe

[(1 + 8

3α2IE

2r

) 12 − 1

] 12

|Er|(5.16)

The effective ion temperature (5.12) and ion mobility coefficient (5.16) for the IEF

model are actually unified parameters for the LEF and HEF models. In order to show

their universal property, a dimensionless ion mobility coefficient Kir and a dimensionless

effective ion temperature Tif are firstly defined. Kir follows the definition of Kir = udr/εr

for all the three models and Tif is given as follows. Substituting udr = (π/8)1/2 · vinudrinto formula (5.12) yields Tif = Tif/Tg = 1 + 2/3 · u2dr for the IEF model. The definition

of Tif can be generalized to the LEF and HEF models: the electric field is negligible for

the LEF model and Tif = 1; the neutral thermal effect is negligible for the HEF model

and Tif = 2/3 · u2dr. These results are summarized below:

LEF model: Kir = 1, Tif = 1, (5.17a)

HEF model: Kir =4

π

(2π

9

) 14

| 1εr|12 , Tif =

2

3

(Kir εr

)2, (5.17b)


|ǫr|10

-210

-110

010

110

2

Kir

10-1

100

101

|ǫr|10

-210

-110

010

110

2

Tif

10-2

10-1

100

101

102

( a ) ( b )

Figure 5.1: (a) Kir (|εr|) curve and (b) Tif (|εr|) curve obtained by LEF model (dash-dottedline), IEF model (solid line) and HEF model (dashed line).

IEF model: Kir =(3)

12

2

[(1 + 8

3 ε2r

) 12 − 1

] 12

|εr|, Tif = 1 +

2

3

(Kir εr

)2(5.17c)

The Kir (|εr|) curve and Tif (|εr|) curve are given in figures 5.1(a) and 5.1(b) for the

LEF model (dash-dotted line), the IEF model (solid line) and the HEF model (dashed

line). Figure 5.1(a) shows that Kir is a decreasing function of |εr| for all the three models,

and figure 5.1(b) shows that Tif is an increasing function of |εr|. In both figures the IEF

curve is consistent with the LEF curve for the range of |εr| < 0.1, and consistent with the

HEF curve for the range of |εr| > 10. Hence Kir and Tif for the IEF model are unified

parameters for the other two models at their respective electric field strength limits.

5.2.2 Electric field based models

Governing equations

For an annular plasma, the ions are bounded between an inner wall at radius ra and an

outer wall at radius rb. Since the width of wall sheath is normally much smaller than the

scale of a bulk plasma, the plasma boundaries are approximately located at ra and rb.

The continuity equation for ions is given by:

dΓirdr

+Γirr− νizn = 0 (5.18)

where νiz = ngµiz is the ion generation rate. For simplicity, the ion generation process is

considered to be governed by electron impact ionization, i.e., νiz is equal to the ionization

rate, and the rate coefficient µiz for Maxwellian electrons is given by:

µiz =

(8e

πmTe3

)1/2 ∫ ∞0

σiz (εe) e−εeTe εe dεe (5.19)

where εe and σiz are the electron energy and ionization cross section, respectively.


Substituting the ion flux (5.7a) into equation (5.18) yields:

d2n

dr2+

1

r

dn

dr+νizr

2b

Dfrn+

1

Dfr

dDfr

dr

dn

dr= 0 (5.20)

where n = n/n0 (n0 is the maximum plasma density) and r = r/rb are the normalized ion

density and normalized radial position, respectively. This equation is normally used for

the LEF model due to dDfr/dr = dDfr/dEr ·dEr/dr = 0 (from equations (5.8) and (5.9)).

For the IEF and HEF models, Dfr is a function of the electric field with dDfr/dEr 6= 0 and

hence it is more convenient to express equation (5.18) with respect to the dimensionless

electric field Er = dη/dr (also satisfying Er = rbEr/Te = −dn/ (ndr)) where η = −φ/Teis the dimensionless form of plasma potential φ.(

1 +ErDfr

dDfr

dEr

)dErdr

+Err− E2

r −νizr

2b

Dfr= 0 (5.21)

Now the radial transport of ions across an annulus can be solved by adding the bound-

ary conditions at the inner and outer boundaries, which are given by making the ion mean

drift velocity uir equal to the Bohm velocity uB = (eTe/mi)1/2:

(uir)r=ra = −uB, (5.22a)

(uir)r=rb = uB (5.22b)

Replacing uir by the variable n (for equation (5.20)) and Er (for equation (5.21)) using

formula (5.7b) and dimensionless relations defined above gives:

−(

dn

ndr

)r= ra

rb

=(Er

)r= ra

rb

= −uBrbDfr

, (5.23a)

−(

dn

ndr

)r=1

=(Er

)r=1

=uBrbDfr

(5.23b)

LEF model

Dfr is independent of the radial position, dDfr/dr = 0, and equation (5.20) is reduced to:

d2n

dr2+

1

r

dn

dr+ β2Ln = 0 (5.24)

where βL satisfies β2L = νizr2b/Dfr = 2/3 · µizσmivin/u2B · [Pas/(eTg)]

2, and Pas = pgrb

(where pg = eTgng is the neutral gas pressure) is the Paschen number for neutral gases.

The boundary condition is given by substituting formulae (5.8) and (5.9) into (5.23):

−(

dn

ndr

)r= ra

rb

= −uBrbDfr

= −2σmivin3uB

PaseTg

= −ψL, (5.25a)

−(

dn

ndr

)r=1

=uBrbDfr

=2σmivin

3uB

PaseTg

= ψL (5.25b)


Equation (5.24) is a Bessel-type equation and its general solution is given by:

n = C1J0 (βLr) + C2Y0 (βLr) (5.26)

where J0 and Y0 are zero order Bessel functions of the first kind and second kind, and C1

and C2 are coefficients to be determined. Substituting the solution form into the boundary

condition (25) yields: [a1,1 a1,2

a2,1 a2,2

][C1

C2

]= 0 (5.27)

The entries in the coefficient matrix are given by:

a1,1 = βLJ1

(βLrarb

)+ ψLJ0

(βLrarb

), a1,2 = βLY1

(βLrarb

)+ ψLY0

(βLrarb

),

a2,1 = βLJ1 (βL)− ψLJ0 (βL) , a2,2 = βLY1 (βL)− ψLY0 (βL)

where J1 and Y1 are first order Bessel functions of the first kind and second kind, re-

spectively. The determinant of the coefficient matrix in equation (5.27) vanishes for a

nontrivial solution of C1 and C2, and det[a] = 0 determines the electron temperature Te.

If the ratio of C2 to C1 is given by C2/C1 = −a1,1/a1,2 = κ, solution (5.26) is rewritten

as:

n = C1 [J0 (βLr) + κY0 (βLr)] (5.28)

where C1 is determined at the peak position of the normalized radial density profile rp,

satisfying dn/dr = 0 and n = 1. As the Bessel function of the second kind κY0 (βLr)

diverges near zero, the convergence of solution (5.28) at an infinitesimal inner radius

ra → 0 needs to be tested. Define a function F (r) as:

F (r) = κY0 (βr) = −a1,1a1,2

Y0 (βr) = −βLJ1

(βL

rarb

)+ ψLJ0

(βL

rarb

)βLY1

(βL

rarb

)+ ψLY0

(βL

rarb

)Y0 [βr] (5.29)

When r = ra/rb approaches zero, J0 and J1 approach unity and zero, and Y1 diverges

faster than Y0 with Y1/Y0 → +∞. Then F (ra/rb) is rewritten as:[F

(rarb

)]ra→0

=1

βLψL

Y1(βL

rarb

)Y0

(βL

rarb

) + 1

∼ 0 (5.30)

Hence, for an infinitesimal ra, the solution (5.28) to an annular plasma is convergent due

to the adjustable ratio κ and reduced to the classic cylindrical solution n = J0 (βLr).

HEF model

Dfr is obtained using formulae (5.8) and (5.11), and its first derivative with respect to the

variable Er is given by:dDfr

dEr= −1

2

Dfr

Er(5.31)


Substituting formulae (5.8), (5.11) and (5.31) into equation (5.21) yields:

dErdr

+ 2Err− 2E2

r − 2βH |Er|12 = 0 (5.32)

where βH = 1/2 ·µiz (πσmi)1/2 /uB · [Pas/ (eTg)]

3/2. The HEF boundary condition is given

by substituting formulae (5.8) and (5.11) into (5.23):

(Er

)r= ra

rb

= −uBrbDfr

= −(πσmi

4

PaseTg

) 12

|Er|12 ⇒

(Er

)r= ra

rb

= −πσmi4

PaseTg

, (5.33a)

(Er

)r=1

=uBrbDfr

=

(πσmi

4

PaseTg

) 12

|Er|12 ⇒

(Er

)r=1

=πσmi

4

PaseTg

(5.33b)

Equation (5.32) is an Abel-type equation with no analytical solution and the boundary

value problem (BVP) for the HEF model is numerically solved. The Hermite-Simpson

method is used to solve this ordinary differential equation (ODE), and an initial solution

guess is firstly evaluated as normally done for BVPs [154]. The validity of this method

has been verified by applying it to the LEF model which has an analytical solution, and

the numerical results calculated from the BVP solver were equal to the analytical results.

IEF model

Dfr is obtained using formulae (5.8) and (5.16), and its first derivative with respect to the

variable Er is given by:

dDfr

dEr= −1

2

Dfr

Er

1− 1(1 + 8

3α2IE

2r

) 12

(5.34)

Substituting formulae (5.8), (5.16) and (5.34) into to equation (5.21) yields:1 +1(

1 + 83α

2IE

2r

) 12

dErdr

+ 2Err− 2E2

r − 2βI|Er|[(

1 + 83α

2IE

2r

) 12 − 1

] 12

= 0 (5.35)

where βI = [32/ (3π)]1/2 · µiz/vin · Pas/ (eTg). The IEF boundary condition is given by

substituting formulae (5.8) and (5.16) into (5.23):

(Er

)r= ra

rb

= −uBrbDfr

= −(

2Te3Tg

) 12 |Er|[(

1 + 83α

2IE

2r

) 12 − 1

] 12

⇒(Er

)r= ra

rb

= −(

16

3π

) 12 σmiPas

eTe

[(1 +

2Te3Tg

)2

− 1

] 12

, (5.36a)

(Er

)r=1

=uBrbDfr

=

(2Te3Tg

) 12 |Er|[(

1 + 83α

2IE

2r

) 12 − 1

] 12


⇒(Er

)r=1

=

(16

3π

) 12 σmiPas

eTe

[(1 +

2Te3Tg

)2

− 1

] 12

(5.36b)

Similarly to equation (5.32), equation (5.35) is a nonlinear ODE with no analytical

solution and the BVP for the IEF model is numerically solved using the same method as

that for the HEF model.

Due to the IEF mobility coefficient being a universal parameter over the entire elec-

tric field strength range (as stated in subsection 5.2.1), the related IEF transport equa-

tion (5.35) is also a unified result. At the lower limit of infinitesimal electric field Er → 0,(1+8/3·α2

IE2r

)1/2can be approximated by its first order Taylor expansion

(1+4/3·α2

IE2r

).

Substituting this approximation into equation (5.35) and considering αI |Er| 1 yield:

dErdr

+Err− E2

r − β∗I = 0 (5.37)

where β∗I = (3/4)1/2 ·βI/αI = 2/3 ·µizσmivin/u2B · [Pas/(eTg)]2 is equal to β2L. Substituting

Er = −dn/ (ndr) into the above equation gives:

d2n

dr2+

1

r

dn

dr+ β∗I n = 0 (5.38)

which is exactly the LEF transport equation (5.24).

At the upper limit of infinite electric field Er →∞,(1+8/3 ·α2

IE2r

)1/2is approximated

by (8/3)1/2 ·αI |Er|. Substituting this approximation into equation (5.35) and considering

αI |Er| 1 yield:

dErdr

+ 2Err− 2E2

r − 2β∗∗I |Er|12 = 0 (5.39)

where β∗∗I = (3/8)1/4 ·βI/ (αI)1/2 =

[64/(27π3)

]1/4 ·µiz (πσmi)12 /uB · [Pas/ (eTg)]

3/2 is 5%

higher than βH . Hence the IEF transport equation (5.39) is good approximation to the

HEF equation (5.32) at the high electric field limit.

5.2.3 Modeling results

The annular modeling is applied to argon plasmas. The input parameters are the Paschen

number Pas and the annular geometry ratio Rio = ra/rb, and the output parameters are the

normalized ion density n, the boundary loss coefficient LR (defined later in formula (5.40))

and the electron temperature Te. The neutral gas is assumed to have a room temperature

of Tg = 0.026 V. The argon has an atomic mass of mg = 39.95 u (u = 1.6605× 10−27 kg

is the atomic mass unit), and the ionization cross section σiz formulated by Phelps [135]

(given in Appendix B) is used for the ionization calculation (5.19). For low temperature

argon plasmas, a momentum transfer cross section (including the elastic collision and

resonant charge transfer) of σmi = 10−18 m2 is normally used for ion-neutral collisions [8].

Since σmi is a constant, Pas is inversely proportional to the Knudsen number Kn = λi/rb

where λi is the ion mean free path. It should be noted that the collisional annular model

has a validity limit of Kn . 1 − ra/rb (i.e., the mean free path should be shorter than

the system length), and is invalid for collisionless plasmas when Kn > 1 − ra/rb (the

collisionless scenario is studied in the next section).


0.4 0.5 0.6 0.7 0.8 0.9 1

Kir

10-1

100

101

0.4 0.5 0.6 0.7 0.8 0.9 110

-1

100

101

r

0.4 0.5 0.6 0.7 0.8 0.9 1

n

0

0.2

0.4

0.6

0.8

1

r

0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

( c )

( d )

( a )

( b )

Figure 5.2: For an annular geometry ratio of Rio = 0.4, radial profiles of (a) dimensionless ionmobility coefficient Kir and (b) normalized plasma density n when Pas = 0.01 Torr · cm, and radialprofiles of (c) Kir and (d) n when Pas = 1 Torr · cm, obtained by LEF model (dash-dotted line),IEF model (solid line) and HEF model (dashed line)

Figures 5.2(a) and 5.2(b) present the radial profiles of Kir and n in an annulus with a

typical geometry ratio of Rio = 0.4 at a low Paschen number of Pas = 0.01 Torr · cm (for

which λi is comparable to the annular width (rb − ra) with Kn ∼ 1/3, 1 − ra/rb = 0.6)

obtained from the LEF model (dash-dotted line), the IEF model (solid line) and the HEF

model (dashed line). Figure 5.2(a) shows that the IEF Kir profile reaches the maximum at

r ∼ 0.64 and is consistent with the LEF result due to the zero electric field there, while the

HEF profile approaches infinity. The LEF Kir profile remains constant across the radial

dimension as the electric field effect is neglected (similar to that in figure 5.1). Figure 5.2(b)

shows that the density profiles obtained from the three models are asymmetric around a

similar peak position at rp ∼ 0.64, which is related to the peak position of Kir profiles in

figure 5.2(a). The boundary-to-maximum density ratio (na and nb for the inner boundary

and outer boundary, respectively) is about 0.95 for the LEF model and about 0.8 for the

IEF and HEF models.

Figures 5.2(c) and 5.2(d) present the radial profiles of Kir and n at a high Paschen


Rio

0 0.2 0.4 0.6 0.8 1

rp

0

0.2

0.4

0.6

0.8

1

Rio

0 0.2 0.4 0.6 0.8 1

LR

0

0.5

1

1.5

2( a ) ( b )

Figure 5.3: (a) rp (Rio) curve and (b) LR (Rio) curve obtained by IEF model for different Paschennumbers of Pas = 0.01 Torr · cm (dash-dotted line), 0.1 Torr · cm (solid line) and 1 Torr · cm(dashed line). The dotted-line in (a) shows the middle position of the annulus as a function of Rio.

number of Pas = 1 Torr · cm (for which λi is small compared to (rb − ra) with Kn ∼1/300). Figure 5.2(c) shows that the IEF model has the maximum value of Kir at r ∼0.67 which is consistent with the LEF model, while the HEF model diverges nearby (not

completely shown to maintain visual clarity). Figure 5.2(d) shows that the density profiles

are asymmetric around a similar peak position at rp ∼ 0.67, and the boundary density

ratio is about 0.1 for the three models.

Figures 5.2(a) and 5.2(c) show that the Kir profile of the IEF model is consistent with

that of the LEF model in the central peak region (or LEF regime) and consistent with

that of the HEF model in the boundary region (or HEF regime). The IEF Kir profile’s

central consistency with the LEF profile covers a wider region for the high Paschen number

than for the low Paschen number, while its boundary consistency with the HEF profile

dominates a broader region for the low Paschen number compared to the high Paschen

number. The universal performance of radial Kir profiles obtained by the IEF model in

figures 5.2(a) and 5.2(c) is consistent with the IEF Kir (|εr|) curve shown in figure 5.1(a).

Since the IEF model gives more accurate results of Kir, Tif and n than the LEF and HEF

models over the entire electric field strength range, it will be used to further investigate

the radial transport properties of an annular plasma.

In order to check the particle loss out of the annulus, a boundary loss coefficient LR,

similar to that for the cylindrical and plane-parallel plasmas in literature (e.g., [8]), is

defined for an annular plasma:

LR =rana + rbnb

rb= Riona + nb (5.40)

which is a generalized boundary density ratio and can be used to estimate the value of

the maximum density n0 by considering power balance [8, 72].

The density peak position rp and boundary loss coefficient LR can be used to charac-

terize the particle distribution in an annular plasma. Figures 5.3(a) and 5.3(b) present the


Pas [ Torr · cm ]10

-310

-210

-110

010

1

rp

0

0.2

0.4

0.6

0.8

1


-310

-210

-110

010

1

LR

0

0.5

1

1.5

2( a ) ( b )

Figure 5.4: (a) rp (Pas) curve and (b) LR (Pas) curve obtained by IEF model for different annulargeometry ratios of Rio = 0.2 (dash-dotted line), 0.4 (solid line), 0.6 (dashed line) and 0.8 (dottedline).

rp (Rio) curve and LR (Rio) curve obtained by the IEF model for different Paschen num-

bers of Pas = 0.01 Torr · cm (dash-dotted line), 0.1 Torr · cm (solid line) and 1 Torr · cm

(dashed line). Figure 5.3(a) shows that rp is a monotonically increasing function of Rio

and the variation is stronger in the low Rio range. It approaches the middle of the annulus

(marked by the dotted line) when Rio > 0.8, and is greater for a high Paschen number.

This result is consistent with the geometric scenario of an annulus approaching a cylinder

when Rio → 0, and approaching a slab (or a plane-parallel geometry) when Rio → 1.

Figure 5.3(b) shows that LR is a monotonically increasing function of Rio and the vari-

ation is greater in the high Rio range. In the low Rio range LR increases faster for low

Paschen numbers, while in the high Rio range it increases faster for high Paschen numbers.

Contrarily to the rp results in figure 5.3(a), LR is greater for a low Paschen number than

a high Paschen number.

Figures 5.4(a) and 5.4(b) present the rp (Pas) curve and LR (Pas) curve obtained by the

IEF model for different annular geometry ratios of Rio = 0.2 (dash-dotted line), 0.4 (solid

line), 0.6 (dashed line) and 0.8 (dotted line). Figure 5.4(a) shows that rp is an increasing

function of Pas (consistent with the results in figure 5.3(a)) and the total increment ∆rp

over the domain of 10−3 Torr · cm < Pas < 10 Torr · cm is reduced for a larger Rio, with an

increment of ∆rp ∼ 0.08 for Rio = 0.2 and ∆rp ∼ 0.003 for Rio = 0.8. Figure 5.4(b) shows

that LR is a reversed “S” shape decreasing function of Pas (consistent with the results in

figure 5.3(b)) and the variation is stronger in the middle range between ∼ 0.01 Torr · cm

and ∼ 1 Torr · cm than at the edges. It approaches zero when Pas reaches a high value

of Pas ∼ 10 Torr · cm. Figures 5.4(a) and 5.4(b) also suggest that rp is a more robust

parameter than LR as a function of the Paschen number.

Figure 5.5(a) shows the IEF model obtained Te (Rio) curve for different Paschen num-

bers of Pas = 0.01 Torr · cm (dash-dotted line), 0.1 Torr · cm (solid line) and 1 Torr · cm

(dashed line). Te is an increasing function of Rio and a stronger variation occurs in the

high Rio range. The Te (Rio) curve exhibits a higher magnitude and increases faster for


Rio

0 0.2 0.4 0.6 0.8 1

Te[V

]

0

5

10

15

20

25

30


-310

-210

-110

010

1

Te[V

]

0

5

10

15

20

25

30( a ) ( b )

Figure 5.5: Obtained by IEF model, (a) Te (Rio) curve for different Paschen numbers of Pas =0.01 Torr · cm (dash-dotted line), 0.1 Torr · cm (solid line) and 1 Torr · cm (dashed line), and (b)Te (Pas) curve for different annular geometry ratios of Rio = 0.2 (dash-dotted line), 0.4 (solid line),0.6 (dashed line) and 0.8 (dotted line).

a smaller Paschen number, with an increment of ∆Te ∼ 12 V for Pas = 0.01 Torr · cm,

∆Te ∼ 2.3 V for Pas = 0.1 Torr · cm and ∆Te ∼ 0.9 V for Pas = 1 Torr · cm over the

domain of 0.01 < Rio < 0.9. Figure 5.5(b) presents the Te (Pas) curve for different annular

geometry ratios of Rio =0.2 (dash-dotted line), 0.4 (solid line), 0.6 (dashed line) and 0.8

(dotted line). Te is a decreasing function of Pas and the variation is more pronounced in

the low Pas range. For the range of Pas > 1 Torr · cm, the electron temperature approaches

a similar value for different annular geometries, with Te ∼ 1.8 V at Pas = 1 Torr · cm and

Te ∼ 1.2 V at Pas = 10 Torr · cm. The Te (Pas) curve shows higher values and decreases

faster for a larger Rio, with a drop of ∆Te ∼ −3.4 V for Rio = 0.2, ∆Te ∼ −3.9 V for

Rio = 0.4, ∆Te ∼ −4.8 V for Rio = 0.6 and ∆Te ∼ −7.5 V for Rio = 0.8 over the domain

of 0.01 Torr · cm < Pas < 1 Torr · cm. Figures 5.3 to 5.5 show that Te has a positive

correlation with LR in terms of the variables Rio and Pas. Additionally, the change trend

of both LR (figures 5.3(b) and 5.4(b)) and Te (figure 5.5) can be characterized using a

combined parameter Pas (1−Rio) which is proportional to (rb − ra) /λi, given by that LR

and Te are negatively correlated with Pas (1−Rio).

5.3 Collisionless Modeling

5.3.1 Tonks-Langmuir Theory

For collisionless plasmas, the ion transport scenario discussed in section 5.2 becomes in-

valid due to the absence of ion-neutral collisions. The classic Tonks-Langmuir theory [40]

has been widely used to study the collisionless plasmas [41, 42] and is currently used to in-

vestigate the collisionless ion transport in an annular plasma. The electrons are presumed

to be in Maxwellian equilibrium and governed by the Boltzmann relation as discussed in

section 5.1. The ion thermal motion is neglected, i.e., cold ions with Ti = 0, and the ion

dynamics is determined from particle balance between local ion generation (with a zero

5.3 COLLISIONLESS MODELING 73

initial velocity) and free-fall ion flux along the decreasing plasma potentials towards the

boundary. This process can be represented by the “T-L integral equation” which, for an

annular plasma, is directly given in cylindrical coordinate without repeating the deduction

details [40]:

−s e−η ±∫ s

sp

sze−c ηz (η − ηz)−

12 dsz = 0 (5.41)

The T-L integral equation connects two important parameters that characterize the ion

transport, the dimensionless potential η = −φ/Te (hence the normalized plasma density

given by n = e−η) and the dimensionless position s = (1/2)1/2 · rν(P/U)

iz /uB where the

superscript “(P/U)” of the ion generation rate νiz represents two cases:

(i) P-case: the ion generation density (the number of ions generated per second per

unit volume) is proportional to the plasma density, i.e., νiz is the ionization rate, corre-

sponding to c = 1 in equation (5.41). This case is for the plasma dominated by one-step

ionization [40, 155] while the multi-step ionization [156] is beyond the scope of this work

and not investigated.

(ii) U-case: the ion generation density is uniform and independent of the plasma density

and νiz is given by the ratio of the ion generation density to the maximum plasma density,

corresponding to c = 0. This case normally occurs when the ion generation is supplied by

an external ion source [155] or plasma supplements [8] from an extra dimension (e.g. the

axial frozen dimension).

The solution to equation (5.41) is in a function form of η = η (s) and depends on

the dimensionless peak position sp which locates the maximum plasma density and zero

plasma potential in the annulus. When sp = 0 the annular plasma is reduced to a normal

cylindrical case. Arithmetic operator “±” in the equation represents the outer radial range

beyond sp, i.e., s > sp, and the inner range within sp, i.e., s < sp, and this notation will

be used through the following content. The T-L integral equation is a nonlinear Volterra

equation of the second kind with respect to η (s) and has a kernel whose singularity is

determined along ηz = η. No solution for Volterra equations with such a kernel can be

found in the literature. Considering the monotonicity of η (s) in the range of s > sp or

s < sp where the plasma potential and radial position hold an injection relation, it is

possible to cancel the kernel’s singularity by transforming the above equation of η (s) to

a modified equation of the inverse of η (s), i.e., s (η), which is given by:

−s e−η ±∫ η

0sze−c ηz dsz

dηzd[−2 (η − ηz)

12

]= 0 (5.42)

where the singularity is removed from the integrand by variable substitution but a deriva-

tive term dsz/dηz has to be introduced. Equation (5.42) is further revised by using the

relative position x = s − sp, the square root of the dimensionless potential ρ = η1/2 and

its trigonometric relation ρz = ρ sin (θ) to give:

− (sp + x) e−ρ2 ±

∫ π2

0(sp + xz) e−c ρ

2z

dxzdρz

dθ = 0 (5.43)

which becomes an integral equation of x (ρ), equivalent to η (s) of equation (5.41), and

satisfies x = 0 when ρ = 0, i.e., zero plasma potential is defined at the position of maximum


plasma density. As sp is the only variable for the T-L integral equation, for each specific

sp there is a corresponding “solution curve” x (ρ) which can be used to calculate the radial

profile of plasma parameters such as the normalized plasma density and normalized mean

drift velocity of ions (given in subsection 5.3.3).

The domain of solution curve x (ρ) is determined by the inner boundary sa (corre-

sponding to ra) and outer boundary sb (corresponding to rb) of an annular plasma. In

order to keep a good consistency with Tonks and Langmuir’s work, the plasma boundaries

are defined at the position with an infinite electric field, given by dη/ds = ∞ which is

equivalent to dx/dρ = 0. This condition corresponds to a plasma without sheaths and is

untrue for actual physical systems. Other types of boundary conditions can be adapted

by setting related constraints, e.g., a Bohm-type boundary is located by finding the po-

sition where the mean drift velocity is equal to the Bohm velocity [8, 9]. The following

subsection applies two approximant methods, the Maclaurin approximant and the Pade

approximant, to obtain the solution curve x (ρ) of equation (5.43).

5.3.2 Approximant methodologies

Maclaurin series approximant

Since most smooth functions are well approximated using the Taylor series, which reduces

to the Maclaurin series when it centers at zero [157], the T-L integral equation for the

plane-parallel and cylindrical plasmas has been plausibly solved using the Maclaurin series

approximant [40]. Additionally, for the plane-parallel geometry an analytical solution can

be found in terms of Dawson functions [158]. Some preliminary discussion for the annular

case was done by Tonks and Langmuir in their original work [40], and the main results were

only restricted to the U-case. It should be noted that their formation of the Maclaurin

approximant for an annular plasma is actually incomplete due to the lack of even-index

terms which are omitted for the cylindrical and plane-parallel geometries but not for

the annulus. Here a complete Maclaurin approximant is firstly obtained for the annular

plasma. The unknown solution function x (ρ) is approximated by a series as follows:

x =+∞∑i=0

aiρi (5.44)

The coefficient sequence an is derived by substituting this power series into the T-L

integral equation (5.43) and zeroing the merged coefficient of each power order. The zeroth

element a0 is always zero due to x = 0 when ρ = 0. For the purpose of computational

convenience, the coefficients are represented in forms of recurrence relations as below. The

first two elements a1 and a2 are same for both the P- and U-cases, and given by:

a1 = ± 2

π, a2 =

±a1 − a212sp

(5.45)

The even-index terms a2n for the P-case are given by:

a2n =1

(2n) sp

sp

n−1∑i=1

(−1)n−1−i

(n− i)!(2i) a2i ±

(2n− 1)!!

(2n− 2)!!

n∑i=1

(−1)n−i

(n− i)!a2i−1


−2n−1∑i=1

iai

n+1−d i+12e∑

j=1

(−1)n+1−d i+12e−j(

n+ 1− d i+12 e − j

)!a2j+2d i−1

2e−i

(5.46)

and the U-case:

a2n =1

(2n) sp

± (2n− 1)!!

(2n− 2)!!

n∑i=1

(−1)n−i

(n− i)!a2i−1 −

2n−1∑i=1

iaia2n−i

(5.47)

where d e and b c represent the ceiling and floor algorithms, respectively.

The odd-index terms a2n+1 for the P-case are given by:

a2n+1 =1

(2n+ 1) sp

sp

[± (−2)n

(2n− 1)!!

2

π+

n∑i=1

(−1)n−i

(n− i+ 1)!(2i− 1) a2i−1

]

± (2n)!!

(2n− 1)!!

2

π

n∑i=1

(−1)n−i

(n− i)!a2i

−2n∑i=1

iai

n+1−b i+12c∑

j=1

(−1)n+1−b i+12c−j(

n+ 1− b i+12 c − j

)!a2j+2b i+1

2c−i−1

(5.48)

and the U-case:

a2n+1 =1

(2n+ 1) sp

± (2n)!!

(2n− 1)!!

2

π

[sp

(−1)n

n!+

n∑i=1

(−1)n−i

(n− i)!a2i

]−

2n∑i=1

iaia2n+1−i

(5.49)

The above coefficients show two important properties. First, those for the radial ranges

of s > sp and s < sp, denoted as a+n and a−n , have the sign relations of:

a+2n = a−2n , a+2n+1 = −a−2n+1 (5.50)

which are satisfied by both the P- and U-cases, and can be verified using formulae (5.46)

to (5.49). Hence a+n and a−n exhibit the same convergence or divergence performance

which is determined from the modulus series |an|. Following these notations, the outer

and inner boundary locations (sb and sa) are derived from:

dx±

dρ=∞∑i=1

ia±n ρi−1 = 0 (5.51)

where x has to be represented in the truncated form of the Maclaurin series (5.44), i.e., the

Maclaurin series approximant, for a real computation with finite bits. This equation yields

the value of the square root of the dimensionless potential ρb and ρa at respective outer

and inner boundaries, which are used to calculate the boundary locations sb = x (ρb) + sp

and sa = x (ρa) + sp.

Second, the recurrence relations (5.46) to (5.49) show that an is a function of the

dimensionless peak position (sp) of the plasma density in the annulus, and the computa-

tional results suggest that an is only convergent for large values of sp for which some

typical values are listed in figure 5.6 as an illustration. The results of a+n with sp = 1.6


Index n

0 3 6 9 12 15

an

-5

-4

-3

-2

-1

0

1

2

3

4

5

Index n

0 3 6 9 12 15

an

-5

-4

-3

-2

-1

0

1

2

3

4

5( a ) ( b )

P-case U-case

Figure 5.6: Coefficient series an of Maclaurin approximant for (a) P-case and (b) U-case withsp = 0.4 (open triangles) and 1.6 (open circles).

for the P-case (a−n can be found using relations (5.50) and is not plotted), represented

by open circles in figure 5.6(a), present a convergent sequence. The open triangles for

coefficient values with sp = 0.4 show a divergent oscillating behavior. The results for the

U-case (figure 5.6(b)) show a similar behavior. It has been found that an converges

when sp is above a lower limit of about 0.72 for the P-case and about 0.95 for the U-case.

Additionally, for a convergent coefficient series an, the recurrence relations (5.46) to

(5.49) have been shown to be a stable algorithm: when an error (e.g., due to roundoff)

was added to the initial two elements a1 and a2, the following elements obtained from the

recurrence algorithm presented a damped deviation along the index sequence.

The divergent behavior of an with respect to sp can be characterized by the radius

of convergence Rc (within which the Maclaurin series is a valid solution). Considering the

alternating performance of an shown in figure 5.6, the Cauchy-Hadamard theorem [159]

is used to determine the radius of convergence: 1/Rc = lim supn→∞ |an|1/n where “sup”

represents the limit superior. The radius of convergence is the same for the radial ranges

of s > sp and s < sp as can be seen from relations (5.50). Since an infinite index is not

approachable in a real computation, a truncated radius of convergence R∗c is defined as:

1

R∗c= sup

|an|

1n : 20≤n≤25

(5.52)

which uses a finite integer instead of infinity as an approximation. In figure 5.7 the P-case

(solid line) and U-case (dash-dotted line) show a decreasing truncated radius of conver-

gence towards zero when sp approaches zero, suggesting that the Maclaurin approximation

is not a reliable method for solving the T-L integral equation (5.43) for small values of

sp where an could become highly divergent. The smaller lower limit of sp ∼ 0.72 for

the P-case compared to sp ∼ 0.95 for the U-case is consistent with the performance of

R∗c (sp) curves where the P-case curve is higher than the U-case and hence has a broader

applicable range of sp.

Additionally, the lower limit of sp for the convergence of an is shown to be the


sp

0 1 2 3 4 5

R∗ c

0

0.5

1

1.5

2

2.5

Figure 5.7: Truncated radius of convergence R∗c for P-case (solid line) and U-case (dash-dotted

line) as a function of sp.

threshold ensuring a valid solution to the boundary condition (5.51), and this coincidence

suggests that the validity of a Maclaurin approximant depends on the convergence of its

coefficient series an. For the P-case, when sp = 0.72 a solution of ρb ∼ 1 is obtained

at the outer boundary which is close to the respective radius of convergence R∗c ∼ 1.2

identified on figure 5.7, and for the U-case a similar phenomenon is observed. As will be

shown later, the annular geometry ratio of inner radius to outer radius Rio = ra/rb = sa/sb

has a positive correlation with sp; hence the Maclaurin approximant should be used for

annuli with relatively large inner radii. It should be noted that the Maclaurin approximant

is convergent and valid for the cylindrical geometry (i.e., a zero inner radius) and for the

plane-parallel geometry, both cases having no even-index terms in the power series [40].

The recurrence relations for the two geometries are given below for reference.

For a cylindrical plasma, the even-index coefficients a2n of the Maclaurin series (5.44)

are zero and the odd-index terms a2n+1 (where a1 = 1) for the P-case given by:

a2n+1 =1

2 (n+ 1)− (2n+1)!!(2n)!!

[(2n+ 1)!!

(2n)!!− 1

] n∑i=1

(−1)n+1−i

(n+ 1− i)!a2i−1

−2n∑i=2

iai

n+2−d i+12e∑

j=1

(−1)n+2−d i+12e−j(

n+ 2− d i+12 e − j

)!a2j+2d i−1

2e−i

(5.53)

and the U-case:

a2n+1 =1

2 (n+ 1)− (2n+1)!!(2n)!!

(2n+ 1)!!

(2n)!!

n∑i=1

(−1)n+1−i

(n+ 1− i)!a2i−1 −

2n∑i=2

iaia2n+2−i (5.54)

Similarly, for a plane-parallel plasma, the even-index terms a2n are zero and the

odd-index terms a2n+1 (where a1 = 2/π) for the P-case are given by:

a2n+1 =1

(2n+ 1)

[(−2)n

(2n− 1)!!

2

π+

n∑i=1

(−1)n−i

(n− i+ 1)!(2i− 1) a2i−1

](5.55)


and the U-case:

a2n+1 =(−2)n

(2n+ 1)!!

2

π(5.56)

Pade rational approximant

In order to solve annular plasmas with small inner radii, the Pade rational approximant is

introduced to extrapolate the geometry limit determined from the Maclaurin approximant.

It uses a rational function, whose numerator and denominator can be directly calculated

from the pre-calculated Maclaurin approximant [160, 161], to approximate an underlying

solution. The genuine advantage of the Pade approximant, over most other approximation

methods, is that it can still work even if the Maclaurin series is divergent. A general

analysis to evaluate the validity of a Pade approximant is a complex task, especially for

the present case where the mathematics of the T-L integral equation (5.43) has not been

fully solved. Here the validity of Pade approximant is verified by substituting the obtained

solution curve x (ρ) into the T-L integral equation and by checking the error magnitude.

The Pade approximant has the following form:

[M/N ] =

M∑i=0

piρi

1 +N∑i=1

qiρi(5.57)

where a diagonal rational approximation, i.e., M = N , is considered for this study. Its

denominator coefficients qi are calculated from the coefficient series an of the Maclau-

rin approximant following the formulae below, which can be found in the literature about

Pade approximant [160–162]:aN aN−1 · · · a1

aN+1 aN · · · a2...

.... . .

...

a2N−1 a2N−2 · · · aN

q1

q2...

qN

= −

aN+1

aN+2

...

a2N

(5.58)

and the numerator coefficients pi:

pi =i∑

k=0

qk ai−k (5.59)

where q0 = 1 and p0 = a0 = 0. The denominator and numerator coefficients for the radial

range of s > sp,q+i

andp+i

, are calculated froma+i

, and those (q−i

andp−i

)

for s < sp froma−i

. The outer and inner boundary locations (sb and sa) of an annular

plasma are derived from:

dx±

dρ=

(M∑i=1

ip±i ρi−1)×(

1 +N∑i=1

q±i ρi

)−(M∑i=0

p±i ρi

)×(

N∑i=1

iq±i ρi−1)

(1 +

N∑i=1

q±i ρi

)2 = 0 (5.60)


ρ

0 0.2 0.4 0.6 0.8 1

x

0

0.2

0.4

0.6

0.8

1

ρ

0 0.2 0.4 0.6 0.8 1

Λ

-0.2

0

0.2

0.4

0.6

0.8

1( a ) ( b )

Figure 5.8: For a P-case annular plasma with sp = 0.4, (a) solution curve x (ρ) and (b) relativeerror curve Λ (ρ) in the outer radial range (s > sp) calculated by Maclaurin approximant (dash-dotted line) and Pade approximant (solid line).

The reliability of a Pade approximant is determined by the accuracy of linear algebraic

equation (5.58). There are a number of algorithms designed to calculate the Pade approx-

imant, depending on the preferred criteria (e.g. reliability or efficiency) and amongst all,

the direct routine of “matrix inversion” is the most stable and reliable method though

at the cost of some efficiency [160, 162]. The validity of the matrix inversion method is

further enhanced by the fact that: a low order Pade rational function is normally chosen

for calculation as an excessively high order sometimes causes computational instabilities

accumulated from both roundoff and truncation errors [161], especially when the matrix

of coefficients [a] is singular in equation (5.58). Morris [162] suggested a general order

principle of N . 10 for single-precision computing, and hence N . 20 for double precision

which is defaulted for the present study. Considering the possible degeneration of matrix

[a] which is likely the case for small values of sp, the LU decomposition is used to solve

the algebraic equation (5.58) and an iterative improvement is supplemented to refine the

solution [161, 162]. An order of 7 to 12 is suggested for the diagonal Pade approximant for

this study, and their stability has been verified: when a small error was added to the first

few coefficients of Maclaurin series (an), the related Pade coefficients showed a negligible

magnitude of deviation.

To show the advantage of the Pade approximant for small values of sp, i.e., annular

geometries with small inner radii, sp = 0.4 for the P-case is used as an example where the

Maclaurin approximant is divergent (figure 5.6). The solution curve x (ρ) for the radial

range of s > sp is obtained from a Pade approximant [9/9] and given as the solid line in

figure 5.8(a) where the boundary of ρb ∼ 1 is determined by equation (5.60). Substituting

the calculated x (ρ) and dx/dρ into the T-L integral equation (5.43) yields a residual value,

and a relative error is defined to estimate the deviation of approximant solutions:

Λ (ρ) =

∣∣∣∣∣∣− (sp + x∗) e−ρ2 ±

∫ π20 (sp + x∗z) e−c ρ

2zdx∗zdρz

dθ

− (sp + x∗) e−ρ2

∣∣∣∣∣∣ (5.61)


where the superscript “∗” refers to the computational results. The validity of the Pade

approximant is verified by its error curve Λ (ρ) being within the magnitude of 10−4, rep-

resented by the solid line in figure 5.8(b). The solution curve x (ρ) obtained from the

Maclaurin approximant (dash-dotted line in figure 5.8(a)), which uses the first 25 ele-

ments (excluding a0 = 0) in the power series, presents a clear discrepancy from the Pade

approximant beyond ρ ∼ 0.7. In this case the Maclaurin approximant is not a valid method

to solve the T-L integral equation as a divergent Λ (ρ) curve is shown in figure 5.8(b) with

an error value of 100% at ρ ∼ 0.8. When sp is set to be a large value (e.g. sp = 1.6), the

Pade and Maclaurin approximants give consistent solution curves and both of their Λ (ρ)

curves approach zero.

Annular limits

The T-L integral equation (5.43) can be rewritten as:

sp

[−e−ρ

2 ±∫ π

2

0e−c ρ

2z

dxzdρz

dθ

]+

[−xe−ρ

2 ±∫ π

2

0xze−c ρ2z dxz

dρzdθ

]= 0 (5.62)

It is noted that the expression in the first square brackets has the same form for a plane-

parallel plasma and, the expression in the second brackets for a cylindrical case, i.e., the

equation for an annular plasma is a combination of a weighted plane-parallel part (by a

factor of sp) and a cylindrical part. Hence it is expected that the annular solution will

approach the cylindrical case for small values of sp, and when sp becomes large it nears

the plane-parallel case, indicating that the annular geometry ratio (Rio = sa/sb) should

be an increasing function of sp which is verified by the results in figure 5.9.

The Rio (sp) curve for the P-case, calculated by the first 25 elements of the Maclaurin

series approximant (excluding a0 = 0), is shown as the dash-dotted line in figure 5.9(a). It

has a lower limit of sp at 0.75 determined by the relation: sup Λ (ρ) : sa < s (ρ) < sb <10−3, which states that for each value of sp on the curve, its respective solution curve

x (ρ) to the T-L integral equation (5.43) has a maximum relative error smaller than 10−3

across the annulus. Hence the calculated Rio (sp) curve satisfies a well defined accuracy.

Figure 5.9(b) shows the results obtained from the Pade approximant with a diagonal order

of [9/9] to [12/12] represented by the solid line, which has the same solution accuracy of a

maximum relative error less than 10−3 for the T-L integral equation. The Rio (sp) curves

calculated by both approximants present quite consistent results for the range of sp > 0.75

(overlapped if plotted on one figure), and the Pade approximant further extrapolates the

lower limit of sp from 0.75 determined by the Maclaurin approximant to 0.3 as marked

by the vertical dashed lines in figures 5.9(a) and 5.9(b). Consequently, the lower limit of

the annular geometry ratio Rio is extrapolated from 0.35 to 0.1 by the Pade approximant

whose valid range covers most annular geometries.

The Rio (sp) results for the U-case are plotted on figures 5.9(c) and 5.9(d) where the

Maclaurin and Pade approximants give very consistent results for the range of sp > 1.05

(overlapped if plotted on one figure), and the latter extrapolates the lower limit of sp from

1.05 to 0.27, and the limit of Rio from 0.5 to 0.1. In summary, the Maclaurin approximant

has a valid range of annular geometry ratios of greater than 0.35 for the P-case and greater

than 0.5 for the U-case, and the Pade approximant extends the valid range to greater than


0 1 2 3 4 5

Rio

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

sp

0 1 2 3 4 5

Rio

0

0.2

0.4

0.6

0.8

1

sp

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

( c )

( d )

( a )

( b )

Figure 5.9: For P-case, Rio (sp) curve calculated by (a) Maclaurin approximant (dash-dottedline) and (b) Pade approximant (solid line). For U-case, Rio (sp) curve calculated by (c) Maclaurinapproximant (dash-dotted line) and (d) Pade approximant (solid line). The vertical dashed linesmark the lower limit of sp for validated solutions of the two approximants.

0.1 for both the P- and U-cases.

The results in figures 5.9(b) and 5.9(d) show that the annular geometry ratio Rio

and the dimensionless peak position sp have a positive injection relation and hence they

are equivalent representations. Rio approaches zero (i.e., cylindrical case) when sp has

a small value and it is close to unity (i.e., plane-parallel geometry) when sp becomes

large, consistent with the qualitative conclusion deduced from equation (5.62). Although

the Pade approximant can be successfully applied to most annular geometries, there is

still a small gap between the annular geometries of Rio ∼ 0.1 to 0 remaining unfilled

(figures 5.9(b) and 5.9(d)) where the Maclaurin coefficient series an becomes highly

divergent and the Pade approximant calculated from these coefficients loses validity.


5.3.3 Modeling results

General radial profiles

The solution curve x (ρ) to the T-L integral equation, obtained from the Maclaurin ap-

proximant and Pade approximant, is used to investigate the radial transport of charged

particles in collisionless annular plasmas. Radial profiles, for which the radial position is

represented by s = x + sp as given in subsection 5.3.1, of the normalized plasma density

and mean drift velocity of ions (electrons have been assumed to be in equilibrium) are

calculated as follows: the former is determined by the Boltzmann relation n = e−η (where

η = ρ2), and the latter, normalized by the Bohm velocity, is given by:

u =±√2

∫ sspsze−c ηz dsz∫ s

spsze−c ηz (η − ηz)−

12 dsz

=±√2

∫ π20 (sp + xz) e−c ρ

2z dxz

dρzρ cosθ dθ∫ π

20 (sp + xz) e−c ρ2z dxz

dρzdθ

(5.63)

where the numerator and denominator represent the ion flux and ion density, respectively.

The normalized plasma density and mean drift velocity are only functions of the dimen-

sionless peak position sp or, more practically, the annular geometry ratio Rio which is

monotonically connected to sp as shown by figure 5.9, and hence their radial profiles are

general results that are independent of the specific gas type and of the Paschen number

(Pas = pgrb).

Radial profiles of the normalized plasma density for annular geometries of Rio = 0.2

to 0.8 are given in figure 5.10(a) for the P-case, with the radial position being normalized

by r = r/rb = s/sb. The solid line for Rio = 0.2 is obtained from the Pade approximant

where the Maclaurin approximant fails as shown in figure 5.9, while the dash-dotted lines

can be obtained from both approximants which give consistent (overlapped) results. The

boundary densities at the inner and outer sides are about 0.35 to 0.5 of the maximum

density with the inner-side value being slightly higher than the outer. The peak position

of the density profile is closer to the inner boundary due to the asymmetric particle loss

where a larger loss area locates at the outer side. The density profile becomes more

symmetric as the annular geometry ratio Rio increases from 0.2 to 0.8, approaching the

plane-parallel case, i.e., the Maclaurin or Pade approximant solution for an annular plasma

of Rio = 0.8 approaches the analytical Dawson solution for a plane-parallel plasma [158].

The mean drift velocities at the inner and outer boundaries, as shown in figure 5.10(b),

have a value of ∼ 1.15 times the Bohm velocity resulting from the infinite field boundary

conditions governed by equations (5.51) or (5.60). It is noted that the boundary-to-

maximum density ratio calculated from the collisionless model (na, nb ∼ 0.5 for the Bohm

boundary condition) is lower than the density ratio calculated from the collisional model

for the low Paschen number of Pas = 0.01 Torr · cm (na, nb ∼ 0.8, figure 5.2(b)), and

higher than the density ratio from the collisional model for the high Paschen number of

Pas = 1 Torr · cm (na, nb ∼ 0.1, figure 5.2(d)). This phenomenon suggests that a transition

from the collisional model to collisionless model occurs at a threshold value of the Paschen

number. Additionally, the inconsistency of boundary-to-maximum density ratio between

the collisional and collisionless models at the low Paschen number is likely due to the

neglect of ion inertia effect in the collisional model.

Figures 5.10(c) and 5.10(d) present the results for the U-case where the normalized


0.2 0.4 0.6 0.8 1

n

0

0.2

0.4

0.6

0.8

1

rp

0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

0.2 0.4 0.6 0.8 1

u

-1.5

-1

-0.5

0

0.5

1

1.5

r

0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

1.5

( c )

( d )

( a )

( b )

Figure 5.10: For P-case, radial profiles of (a) normalized plasma density n and (b) normalizedmean drift velocity of ions u for different annular geometries of Rio = 0.2, 0.4, 0.6 and 0.8. ForU-case, radial profiles of (c) n and (d) u for Rio = 0.2, 0.4, 0.6 and 0.8. The dash-dotted linesrepresent overlapped results that can be calculated by both the Maclaurin and Pade approximants,and the solid line represents the result that is only valid using the Pade approximant.

plasma density and mean drift velocity show similar behaviors to those for the P-case. By

carefully comparing figures 5.10(a) and 5.10(c), the density in the radial range of r > rp

is found greater for the U-case by about 1% to 5% compared to the P-case for different

annular geometries, and the mean drift velocity thereby, as shown in figures 5.10(b) and

5.10(d), is greater for the P-case than the U-case due to a larger proportion of high velocity

ions in the plasma flux moving towards the boundary.

Electron temperature in argon plasma

The electron temperature Te is included in the position parameter s = 1/21/2 · rν(P/U)

iz /uB

where the ion generation rate ν(P/U)

iz and Bohm velocity uB are functions of Te. For the

P-case the ionization (generation) rate is given by ν(P )

iz = ngµiz same as that stated in

subsection 5.2.2. The value of ν(U)

iz for the U-case is the ratio of the ion generation density



-410

-310

-210

-110

0

Te[V

]

0

5

10

15

20

25

30

Figure 5.11: Te (Pas) curve for P-case argon plasmas (with sb calculated by Pade approximant)for different annular geometries of Rio = 0.2 (dash-dotted line), 0.4 (solid line), 0.6 (dashed line)and 0.8 (dotted line).

to the maximum plasma density. Since the U-case ion generation is determined by the

external ion supply or the supplements from an extra dimension (subsection 5.3.1) and

these effects are not included in the present scope, the results of electron temperature are

only given for the P-case which is governed by self-consistent particle conservation. In order

to obtain the electron temperature as a function of the Paschen number, the dimensionless

outer boundary of the annular plasma, which connects these two parameters of interest,

is given by:

sb =rbνiz√

2uB=

µiz√2uB

PaseTg

(5.64)

whose value can be calculated from the boundary conditions (5.51) or (5.60). The above

equation shows that the electron temperature Te depends on the specific gas type (reflected

by the ionization rate constant µiz) and the Paschen number Pas, differently from the

normalized plasma density n and the normalized mean drift velocity of ions u which are

general results (figure 5.10) and solely determined from the dimensionless peak position

sp, or equivalently, the annular geometry ratio Rio.

Here the results of electron temperature for argon plasmas are illustrated. The input

parameters are the Paschen number and the annular geometry ratio (Rio) which determines

the value of sb, and the output parameter is the electron temperature by solving the above

equation (5.64) where the gas temperature Tg = 0.026 V (room temperature) is used, same

as that in section 5.2 for the collisional model. Te, for which sb is calculated using the

Pade approximant, is plotted in figure 5.11 as a function of Pas ranging from 10−4 to

1 Torr · cm for different annular geometries of Rio = 0.2 to 0.8. The Te (Pas) curve has

a higher level for a larger Rio and the difference is more dramatic in the low Pas range.

In the high Pas range, Te of different Rio cases converge to a similar value of ∼ 2.4 V at

Pas = 1 Torr · cm. The curves in figure 5.11 obtained from the collisionless model, exhibit

a similar trend with those in figure 5.5(b) obtained from the collisional model, but the

two models’ results do not hold a good continuity, for which a unified theory connecting


the collisional and collisionless models needs further investigation.

5.4 Chapter Summary

In this chapter, radial transport properties of low temperature annular plasmas have been

investigated. The electrons are assumed to be in Maxwellian equilibrium and governed by

the Boltzmann relation. The ion transport across an annulus is studied for both collisional

and collisionless plasmas.

For the collisional regime, the annular modeling is given by ion mobility based forms

(the diffusion effect is neglected) whose applicable range is determined by the accuracy

of the ion mobility coefficient at different electric field ranges. The mobility coefficient

calculated by the LEF or HEF model has a good accuracy at the low or high electric

field limit, and their validity has been verified in previous studies for cylindrical and plane

parallel plasmas. The novel IEF model presents a unified mobility coefficient over the

entire electric field strength range and approaches the results of LEF and HEF models

at the respective electric field limits. Hence the IEF model gives more accurate results

compared to the LEF and HEF models over the entire electric field range. The modeling

is applied to a low temperature argon plasma in an annulus, and the IEF model obtained

radial profiles of ion mobility coefficient and plasma density join the LEF model in the

central peak region and join the HEF model in the boundary region. These profiles are

asymmetric across the annulus due to the different inner and outer loss areas. The density

peak position is an increasing function of the annular geometry ratio and Paschen number.

The annular boundary loss coefficient is an increasing function of the annular geometry

ratio due to an increased loss area, while a decreasing function of the Paschen number (or

an increasing function of the Knudsen number) due to reduced ion-neutral collisions. The

electron temperature has a positive correlation with the annular boundary loss coefficient

with respect to the annular geometry ratio and Paschen number. When the boundary loss

of ions is enhanced, the electron temperature increases to enhance ionization and satisfy

particle balance.

For the collisionless regime, the Maclaurin approximant and Pade approximant are

used to solve the Tonks-Langmuir theory for an annular plasma in the P-case or U-case.

The coefficient series of the Maclaurin approximant is represented in forms of recurrence

relations which are convenient for computation. This approximant method has been shown

to have a lower limit for the annular geometry ratio, below which it becomes divergent

and invalid. The Pade approximant is consequently introduced and shown to be a more

robust method. Though its numerator and denominator are calculated from the pre-

obtained coefficients of the Maclaurin approximant, it extrapolates the annular geometry

ratio limit to a broader range which can include annular geometries with small inner

radii and cover most annular geometries. However, there is still a small gap between the

annular geometry ratio of 0.1 to 0 (cylindrical geometry) where the coefficient series of the

Maclaurin approximant is highly divergent and the Pade approximant loses its validity.

The validity of the two approximants is tested by substituting their solution curves into the

T-L integral equation and by checking the magnitude of the error. The annular modeling

is applied to obtain general radial profiles of the normalized plasma density and mean


drift velocity of ions across annuli. Both profiles are independent of the gas type and the

Paschen number of the discharge, and present similar behavior for the P- and U-cases.

The electron temperature is calculated for an argon plasma in the P-case as a function

of the Paschen number and shows positive correlation with the annular geometry ratio

similarly to that found in the collisional model.

Chapter 6

Conclusions

• Some aspects of thermodynamics have been revisited for electrons in low pressure

plasmas. Electron transport governed by nonlocal EEPFs is a self-consistent adi-

abatic process, during which the electron enthalpy converts into potential energy,

therefore being the source of ion acceleration. The polytropic relation determined

from previously reported convex EEPFs along a divergent magnetic field in the

HDLT experiment gives an index of 1.17 less than 5/3, which could be misinterpreted

as the existence of external heating using traditional thermodynamic concepts based

on LTE. These results provide physical insights into the energy transfer mechanism

during plasma expansion and are useful for the design of electric propulsion systems.

• The polytropic model is extended to nonlocal EEPFs of three bi-Maxwellian dis-

tributions showing convex, linear and concave shapes, respectively. The polytropic

index increases when the EEPF becomes more convex and decreases when more

concave. The classic adiabatic index of 5/3 for adiabatic LTE systems is only an

element (corresponding to the Heaviside-type EEPF) in the set of polytropic indices

for non-LTE adiabatic systems governed by non-local particles. Since electrons in

the solar wind and in the laboratory share many similarities, they might be domi-

nated by the same principle of non-local EEPFs. A hypothesis is given for the solar

wind: although the solar electrons show a polytropic index lower than 5/3, their

actual transport could be adiabatic.

• Ion beam experiments are carried out in the Chi-Kung reactor implemented with

a cylindrical plasma source and an annular plasma source, both configured with a

convergent-divergent magnetic field. In the cylindrical plasma thruster, bi-directional

ion beam accelerating along both directions of a magnetic nozzle is observed and ex-

hibits a different scenario to the classic one-directional compressible gas flow in a

Laval nozzle. A magnetic field induced transition is identified between a high field

mode with an ion beam, and a low field mode with no ion beam. Ion beam forma-

tion along a divergent magnetic nozzle is not a one-dimensional phenomenon solely

determined from the axial potential drop but correlated with the radial confinement

of ions at the magnetic throat. The cylindrical plasma thruster being capable of

bi-directional ion acceleration provides a compact system for space debris removal

in orbits; correlation between ion beam formation and radial plasma transport can

be used to improve propellant efficiency and optimize ion beam devices.

87

88 CONCLUSIONS

• The annular plasma thruster is powered by either an outer antenna with electron

heating located near the outer tube wall, or an inner antenna with electron heating

close to the inner tube wall. For the outer antenna case, an annular ion beam

is accelerated out of the plasma source and combines into a solid structure (with

the central hollow filled) in the diffusion chamber. For the inner antenna case, no

ion beam is detected likely due to a larger radial loss area for energetic ions in

the annular plasma source compared to the outer antenna case. Both cases show

enhanced local ionization in the diffusion chamber, caused by energetic electrons

originating from the plasma source. A wake region is observed just downstream of

the sealed wall terminating the inner tube. The nonlocal beam ions fill in this region

as a result of a divergent electric field in the source-exit region, and move inward

more efficiently than the background ions due to free diffusion. The wake is shown to

be a quiescent region dominated by quasi-Maxwellian electrons. The outer-antenna

powered annular plasma thruster could perform as the first (preionization) stage

of a two-stage Hall effect thruster; the inner-antenna case provides an approach to

control the spatial behavior of electron energy distributions. The wake region located

downstream of the inner tube is a simplified simulation for spacecrafts re-entering

the ionosphere.

• Radial transport of charged particles in unmagnetized annular plasmas is investi-

gated using theoretical modeling for both collision and collisionless regimes in terms

of ion-neutral collisions. The solution to an annulus is complicated compared to

the cylindrical and plane-parallel geometries, as the critical position of maximum

plasma density, zero electric field and zero drift velocity of ions becomes a variable

rather than a constant at the center. The electrons are approximately in an equi-

librium state and assumed to be governed by the Boltzmann relation. The collision

regime is studied using three electric-field-based mobility models: the LEF model,

the IEF model and the HEF model. The newly proposed IEF model is a unified

model smoothly connecting the classic LEF and HEF models at respective electric

field limits, which better describes plasma transport process for a wide parametric

range of the Paschen number and geometric annular ratio, and for different regions

in a plasma such as the central region and pre-sheath.

• The collisionless regime is described using the classic Tonks-Langmuir integral equa-

tion, and solved by the Maclaurin series approximant and the Pade rational approx-

imant. The Maclaurin approximant is expressed in the form of recurrence relations

which are convenient for computation, but it loses validity for annuli with small

radii. The Pade approximant is introduced to solve this discrepancy and covers

most annular geometries, contributing to a new perspective of the Tonks-Langmuir

theory. For a specific annulus in the collisionless regime, radial profiles of normalized

plasma density and mean drift velocity of ions are general results independent of the

Paschen number and gas types. Both collisional and collisionless annular modeling

is applied to argon plasmas and can be used to predict behaviors of charged particles

in configurations such as electrodes in plasma jets and probes in plasma columns.

Appendix A

Mathematical Deduction of

Enthalpy Relation

During an adiabatic process (i.e., no heat term), the differential of electron enthalpy dpe

along the potential path −eφ is given by:

dhe =dpene

=dpe

ned (−eφ)d (−eφ) (A.1)

where pe and ne are the electron pressure and electron density, respectively, defined by

formulae (1.8) and (1.9) in chapter 1, subsection 1.2.2.

The derivative of pe with respect to −eφ is given by:

dped (−eφ)

=2

3

∫ ∞0

ε32e

dfpe (εe − eφ)

d (−eφ)dεe =

2

3

∫ ∞0

ε32e f′pe (εe − eφ) dεe

=[ε3/2e fpe (εe − eφx)

]∞0−∫ ∞0

ε12e fpe (εe − eφ) dεe (A.2)

where the second line is obtained using integration by parts.

When limεe→∞ ε3/2e fpe (εe − eφx) = 0 is assumed, substituting formula (A.2) into (A.1)

and integrating formula (A.1) yield:

he =

∫ −eφ−eφ0

−∫∞0 ε

12e fpe (εe − eφx) dεe∫∞

0 ε12e fpe (εe − eφx) dεe

d (−eφx) + he0 = eφ− eφ0 + he0 (A.3)

where φ0 and he0 are the plasma potential and electron enthalpy at a reference position,

respectively.

89

90 MATHEMATICAL DEDUCTION OF ENTHALPY RELATION

Appendix B

Cross Sections of Electron-neutral

Collisions in Argon

The cross sections, formulated by Phelps [135], are used in calculations for electron-neutral

collisions in argon plasmas. The following results are in the unit of 10−20 m2.

Elastic collision

σel =

∣∣∣∣∣∣∣6[

1 + εe0.1 +

(εe0.6

)2]3.3 − 1.1ε1.4e[1 +

(εe15

)1.2] [1 +

(εe5.5

)2.5+(εe60

)4.1]0.5∣∣∣∣∣∣∣

+0.05(

1 + εe10

)2 +0.01ε3e

1 +(εe12

)6 (B.1)

Ionization

σiz =970 (εe − 15.8)

(εe + 70)2+ 0.06 (εe − 15.8)2 exp

(−εe

9

)(B.2)

Excitation

σex =0.034 (εe − 11.5)1.1

[1 +

(εe15

)2.8]1 +

(εe23

)5.5 +0.02295 (εe − 11.5)(

1 + εe80

)1.9 (B.3)

Effective momentum transfer

σme = σel + σiz + σex (B.4)

91

92 CROSS SECTIONS OF ELECTRON-NEUTRAL COLLISIONS IN ARGON

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