Update of LoKI-B simulation tool with electron density growth by electron … · Update of LoKI-B simulation tool with electron density growth by electron-impact ionizations Duarte

Update of LoKI-B simulation tool with electron densitygrowth by electron-impact ionizations

Duarte Nuno Barreto Gonçalves

Thesis to obtain the Master of Science Degree in

Engineering Physics

Supervisor: Prof. Luís Paulo da Mota Capitão Lemos Alves

Examination Committee

Chairperson: Prof. João Pedro Saraiva BizarroSupervisor: Prof. Luís Paulo da Mota Capitão Lemos AlvesMembers of the Committee: Prof. Vasco António Dinis Leitão Guerra

Dr. Antonio Tejero Del Caz

September 2017

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Acknowledgements

I would like to acknowledge the guidance of my supervisor Prof. Luís Lemos Alves. The opportunity to

be part of the project KIT-PLASMEBA and liberty to explore the subject further, resulted in a gratifying

and interesting work.

I thank the team of this project, for even the brief discussions helped me understand and consolidate

the study. I genuinely thank Antonio Tejero that was always available to discuss and explain matters

whenever I asked.

A special thanks to my family and friends for their support through life.

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Este trabalho foi financiado pela Fundação para a Ciência e

Tecnologia, através do Projecto PTDC/FIS-PLA/1243/2014

(KIT-PLASMEBA) e pelas bolsas BL136/2016_IST-ID e

BL150/2017_IST-ID.

This work has been supported by the portuguese Fundação para a

Ciência e Tecnologia, under Project PTDC/FIS-PLA/1243/2014

(KIT-PLASMEBA) and grants BL136/2016_IST-ID and

BL150/2017_IST-ID.

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Resumo

Os plasmas de baixa temperatura têm vindo a ser cada vez mais usados em aplicações industriais,

sendo que nos últimos anos houve um desenvolvimento de aplicações ambientais e biológicas. Ex-

iste portanto uma necessidade tecnológica para melhorar a previsibilidade do comportamento destes

plasmas. Neste contexto, o projecto KIT-PLASMEBA pretende desenvolver novas ferramentas, uma

das quais um programa computacional LisbOn KInetics (LoKI), que contém um modelo de resolução

numérica da equação de Boltzmann para eletrões (LoKI-B).

O objectivo deste trabalho é introduzir um novo tratamento das ionizações por impacto eletrónico,

contribuindo para o desenvolvimento do LoKI-B. Para este fim, criou-se uma nova rotina de ionização,

onde foram incluídos dois modelos de crescimento de densidade eletrónica, bem como um operador de

ionização que usa uma secção eficaz diferencial de ionização. De forma a integrar completamente esta

rotina no código LoKI-B, foi realizado um acoplamento com a rotina de colisões eletrão-eletrão.

As previsões do primeiro coeficiente de ionização de Townsend melhoraram significativamente para

Árgon, sendo que para Azoto molecular as previsões do LoKI estão agora dentro das incertezas ex-

perimentais. Foram efectuadas verificações com outro código de resolução numérica da equação de

Boltzmann para eletrões, onde se comprovou a viabilidade do trabalho efectuado. Uma análise dos

vários operadores colisionais de ionização, permitiu descrever os mecanismos pelos quais a ionização

por impacto electrónico influencia a função de distribuição dos eletrões.

Palavras chave: plasmas de baixa temperatura, ionização por impacto electrónico,LoKI-B, modelização cinética

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Abstract

Low-temperature plasmas have been extensively used in industrial applications, with developments in

environmental and biological applications being made in recent years. There is then a technological

need to improve the predictability on the behaviour of these plasmas. To this end, the project KIT-

PLASMEBA aims at the development of new tools, one of them being the kinetic code LisbOn KInetics

(LoKI), which contains an electron Boltzmann equation solver (LoKI-B).

The goal of this work is to introduce a new description of electron-impact ionizations, supporting the

development of LoKI-B. To this end, a new ionization routine was created in which two electron density

growth models were included, as well as a non-conservative ionization collisional operator that uses a

differential ionization cross section. In order to seamlessly integrate this routine with LoKI-B, a coupling

with the electron-electron collisions routine was made.

Predictions for the first Townsend ionization coefficient improved significantly for the case of Argon,

and into experimental data uncertainty in the case of molecular Nitrogen. Comparisons against another

electron Boltzmann equation solver, verified the accuracy of the present work. An analysis of the various

ionization collisional operators allowed a description of the various mechanism for which electron-impact

ionization influences the electron distribution function.

Keywords: low-temperature plasmas, electron-impact ionization, LoKI-B, kineticmodelling

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Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Low-temperature plasma reactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Electron-impact ionizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Low-temperature plasma modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Electron Boltzmann equation solver . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Objectives and original contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Theoretical Formulation 9

2.1 The electron Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 The particle distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Normalization under the adiabatic approximation . . . . . . . . . . . . . . . . . . . 10

2.2.2 Spherical-harmonics expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Fourier expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.4 The electron energy distribution function . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Electron-impact ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Energy sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Differential ionization cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.3 Ionization collisional operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.4 Electron density growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 The two-term electron Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.1 Derivation of the electron Boltzmann equation terms . . . . . . . . . . . . . . . . . 22

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2.4.2 Isotropic and anisotropic components of the electron of Boltzmann equation . . . . 26

2.4.3 Temporal growth with DC electric field . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.4 Temporal growth with HF electric field . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.5 Spatial growth with DC electric field . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Conservation and transport equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.1 Drift-diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.2 Particle balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5.3 Energy balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Computational Approach 37

3.1 Solving the Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Discretization of the electron Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 Linear terms on the electron energy distribution function . . . . . . . . . . . . . . . 39

3.2.2 Terms with integrals on the electron energy distribution function . . . . . . . . . . . 40

3.2.3 Terms with derivatives on the electron energy distribution function . . . . . . . . . 40

3.3 Solving the non-linear electron Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . 41

3.3.1 Convergence over the ionization rate or first Townsend coefficients . . . . . . . . 41

3.3.2 Iterative algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.3 Coupling with electron-electron collisions . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Numerical verification of the conservation equations . . . . . . . . . . . . . . . . . . . . . 46

3.4.1 Particle balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.2 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Results 51

4.1 Comparison between energy sharing modes . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Comparison between electron density growth models . . . . . . . . . . . . . . . . . . . . 54

4.3 Benchmarks against BOLSIG+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Validation of the first Townsend ionization coefficient . . . . . . . . . . . . . . . . . . . . . 56

4.4.1 The use of the equal energy sharing mode . . . . . . . . . . . . . . . . . . . . . . 58

5 Prospective 61

A Discretization of the electron Boltzmann equation 67

A.1 Linear terms on the electron energy distribution function . . . . . . . . . . . . . . . . . . . 68

A.1.1 Time variation term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.1.2 Space variation term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.1.3 Inelastic/superelastic collisions term . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.2 Terms with integrals on the electron energy distribution function . . . . . . . . . . . . . . . 69

A.2.1 Ionization collisional operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A.3 Terms with derivatives on the electron energy distribution function . . . . . . . . . . . . . 70

A.3.1 Space variation term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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A.3.2 Rotational collision term - continuous approximation . . . . . . . . . . . . . . . . . 71

A.3.3 Elastic collisions terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.3.4 Electric field Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

B Verification of the discrete particle balance equation 73

B.1 Time variation term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

B.2 Space variation term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

B.3 Terms with derivatives on the electron energy distribution function . . . . . . . . . . . . . 74

B.4 Inelastic/superelastic collision terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

B.5 Ionization collisional operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

B.6 Particle balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

C Verification of the discrete energy balance equation 79

C.1 Time variation term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

C.2 Space variation term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

C.3 Terms with derivatives on the electron energy distribution function . . . . . . . . . . . . . 79

C.4 Inelastic/superelastic collision term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

C.5 Ionization collisional operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

C.6 Energy balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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List of Figures

2.1 Graphical representation of the energy sharing between the primary and the secondary

or the scattered electron. According to the product electron’s energy, the two regions,

signalled by the solid and dashed-dotted lines, identify whether it is a scattered or a sec-

ondary electron. In red (green) is the primary electron energy interval that can produce a

secondary (scattered) electron with energy u = 40eV . . . . . . . . . . . . . . . . . . . . . 15

2.2 Fits to Opal, Peterson and Beaty’s experimental data [1] for Argon (4.6a), and Helium

(4.6b), using function 2.12 (OPB) and function 2.14 (GS). . . . . . . . . . . . . . . . . . . 18

3.1 Flowchart of the EBE solver in LoKI. In blue we present the original routine if secondary

electrons were not included. On the right hand side, we present all the various steps of

the ionization routine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Flowchart of the coupling between the ionization and the electron-electron collisions rou-

tine. In red are the steps done within the ionization routine. In green are the steps done

within the electron-electron collisions routine. . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 Experimental data on the differential ionization cross section on the secondary electron

energy for molecular nitrogen, and for three different primary electron energies [1]. . . . . 52

4.2 Plot of EEDFs calculated in LoKI for Argon with DC E/N = 1000Td and the electron

density spatial growth model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Plot of EEDFs calculated in LoKI for Argon with DC E/N = 1000Td and the energy

sharing using a SDCS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 Comparisons of the EEDF, calculated with LoKI and BOLSIG+, for Argon with DC E/N =

1000Td. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 Comparison between first Townsend ionization coefficient calculated with BOLSIG+ and

LoKI, using the exponential spatial growth model and adopting different energy sharing

modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.6 First Townsend ionization coefficient as a function of the reduced electric field. LoKI’s

simulations use the exponential spatial growth model or conservative ionization collisions

(secondary electrons not included). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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4.7 Comparison between LoKI’s calculated first Townsend ionization coefficient with equal

energy sharing mode, "one electron takes all" mode, and experimental data for Nitrogen

SST discharges [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A.1 The energy grid and the various notations of quantities used on the discretization. . . . . 68

A.2 Numerical and graphical representation of an energy sum. The summed energy is repre-

sented in light red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

A.3 Representation of the midpoint quadrature rule. . . . . . . . . . . . . . . . . . . . . . . . . 69

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Chapter 1

Introduction

1.1 Motivation

By applying an energy source to a gas, by heating or with an electric field for instance, it is possible to

free electrons from atoms and molecules. These electrons can then accelerate in an electric field, collide

with other atoms and further ionize the gas. The relative density of electrons and ions increases, until

the gas electric properties change, and a plasma is formed. There are various types of plasmas, with

varying thermodynamic conditions or ionization degrees. Since a plasma ionization degree is closely

related to its temperature [3], a weakly ionized plasma is also regarded as a low-temperature plasma.

1.1.1 Context

Man made low-temperature plasmas (LTPs) can be formed through gas discharges. A gas discharge

is a term used to describe the passage of electric current in a gaseous medium. Since even at room

temperature there is some degree of ionization, an electric field can give energy to charged particles

that can become energetic enough to scatter with other particles, further ionizing atoms and triggering

chemical reactions.

Historically, man has been observing gas discharges in nature, for instance with lightnings, Aurora

Borealis or St Elm’s fire. The study of gas discharges commenced with the first devices that used electric

discharges produced from static electricity. The development of batteries, allowed the construction of

devices that produced spark discharges, and even continuous arc discharges using the electrochemical

batteries developed by Volta. In the 19th century, Faraday produced a direct current glow discharge

by applying an electric field in an evacuated tube [4]. On the second half of the century, Townsend

studied these plasmas developing the Townsend discharge theory, the groundwork for the modern study

of plasmas [5]. In the 20th century, Langmuir developed the Langmuir probe to determine electron

temperature and density, and electric potential in a plasma [6]. This prompted a rapid development of

diagnostic and applications through the rest of the century. Low-pressure LTPs have since been used in

the processing of materials, and high-pressure LTPs have been used in recent years for environmental

1

and biological applications- For a more detailed summary of the history and applications of LTPs the

reader is requested to see [7].

The processing of materials has been one of the most rewarding applications of LTPs. Indeed, the

number of papers published in plasma-related topics for semiconductor processing is high in comparison

to other scientific plasma domains, such as fusion research [8]. Other growing fields of technological

applications are: plasma processing for flat-panel displays, silicon photovoltaics, plasma source ion-

implementation, plasma polymerization and coating, and others. There are environmental applications

related to the dilution of pollutants in the atmosphere [9], as well as plasma based CO2 conversion [10].

The ability to produce gas discharges at high pressures allowed researchers to explore biological ap-

plications, from treatment of seeds [11] to instrumentation and biomaterial sterilization [12]. Medical ap-

plications have become a new trend in recent years, however with doubts in terms of cost-effectiveness

[7].

These new technological applications demand a new degree of predictability based on fundamental

modelling. This is the aim of the KInetic Testbed for PLASMa Environmental and Biological Applica-

tions (KIT-PLASMEBA), which embodies a web-platform (KIT) with state-of-the-art kinetic schemes, and

a MATLAB R© kinetic code (LisbOn KInetics, LoKI) with a modular structure, embedding a Boltzmann

solver (LoKI-B) and a chemistry solver (LoKI-C) for different gases/gas-mixtures. LoKI provides the

combined chemical and transport description of plasma charged/neutral species, both in volume and

surface phases, for user-defined mixture compositions, pressure, radial dimension and excitation con-

ditions. One of the main tasks of this project is the development of LoKI-B, of which this work has

contributed to.

1.1.2 Low-temperature plasma reactivity

Low-temperature plasma ionization degree is low ne

ne+N < 0.1, with their electron density ranging from

ne = 108 − 1013 cm−3, and their neutral gas particle density N = 1015 − 1019 cm−3. When an electric

field is applied to this system, energy is primarily transferred to the lighter particles, the electrons, and

in turn they distribute the energy between the heavier particles, ions and neutrals, through scattering

events. Because of this, LTPs are usually in non-thermal equilibrium, with their ion and neutral particle

temperature remaining relatively low Ti ≈ Tg ≈ 300 − 600K, while their electron temperature can be

very high Te ≈ 103 − 105K. This, in junction with the fact that the plasma is weakly ionized means that

electrons will be responsible for various important collisional processes.

A single gas can have a plethora of species according to the particles’ states [13]. Usually, atomic

gases will be on a number of electronic excited states, while molecular gases can be in additional ro-

tational and vibrational states. Interactions between these species can lead to chemical reactions, with

some of them being excitation/de-excitation, attachment/detachment, ionization/recombination, excita-

tion transfer and charge transfer. These reactions happen through collisional processes such as atomic

2

or molecular collisions, with electrons or heavy particles, or through radiative processes in the case of

dipolar transitions and electronic de-excitations. The usual information necessary to provide a descrip-

tion of the heavy particles relates to each species’ density and corresponding reaction rates. Some of

the rate coefficients are due to reactions caused by electron collisions. So in order to calculate these, a

description of the electron energy distribution function (EEDF) is helpful.

Due to the profusion of heavy particles, electron-neutral collisions will have a big contribution to

the electron behaviour. They can be classified as elastic, inelastic or superelastic collisions. There are

also coulomb collisions, electron-electron and electron-ion, which can become important if the ionization

degree of the LTP approaches its highest limit. All of these effects can be condensed in the collisional

term of the electron Boltzmann equation (EBE).

Since most of the energy is being transmitted to the electrons, they are essential to the maintenance

of the plasma. Without an electric field, there will be an equilibrium between the production and loss

of charged particles in the gas, through ionizations and recombinations, respectively. With increasing

electric field, the production/loss equilibrium can be perturbed, and electrons are free to leave the plasma

region, that is, there are electron transport losses around the plasma boundaries and to the electrodes.

However, the increase of the electric field, and consequently of the mean electron energy, allows for

new processes that account for electron losses, maintaining the discharge. There are various electrode

effects, such as thermionic, photoelectric, positive-ion or metastable impact emission, that can provide

these electrons. Nonetheless, for high enough electric fields, the ionization of neutral gas particles

becomes one of the main mechanisms for charged particle production. There are various sources of

neutral particle ionizations, which can be more or less relevant depending on the gas constitution and

physical conditions. Some of them are the associative ionization, Penning ionization, chemionization,

photo-ionization, associative detachment, and collisional ionization by heavy particles. However, in

electric discharges, electron-impact ionization is still one of the main sources.

1.1.3 Electron-impact ionizations

The reduced electric field E/N , which relates the applied electric field with the gas density, is a defining

quantity. The electron mean free path is inversely proportional to the density of the heavy particles,

which is approximately the density of the gas. As a result, the quantity E/N is proportional to the

applied electric field times the electron mean free path, and it serves as a measure on the amount of

energy gained by the electrons between collisions. The higher the reduced electric field, the greater

the energy gained by electrons between collisions, and higher the probability of having electron-impact

ionizations.

One of the most striking effects of electron-impact ionizations is the effect on the electron density,

defining its spatial or temporal profile depending on the discharge type. The density profile will then

affect the way electrons behave and respond to the electric field.

One example is the case of uniform direct current (DC) electric fields, where electrons travel from

the cathode to the anode, scattering with plasma particles. Some of these electrons will ionize atoms,

3

producing ions and additional electrons. These secondary electrons can, themselves, make ionizing

collisions, and the process repeats itself. The electron density will then increase from the cathode to the

anode, meaning that an electron density gradient is present and electrons suffer another force contrary

to the electric field one. In the end, there can be an equilibrium between all of these factors, and a

steady state Townsend (SST) discharge is achieved. This is the basis of an electron density spatial

growth model.

Another example is the case of uniform high-frequency (HF) electric field. Here, electrons have zero

mean velocity, since they oscillate in response to the electric field. As a result, these electrons will ionize

atoms, and the electron density will increase uniformly in time with a mean ionization frequency. The

increase in time of the electron density also reduces the electron mobility (see sec. 2.5.1). This is the

basis of the temporal electron density growth model.

Another effect is the redistribution of the primary electron energy. After the collision, the available

energy is that of the primary less the ionization potential. This energy will be shared between the two

electrons produced by the ionization. The loss of kinetic energy to ionize the atom, plus the redistribution

of this energy between two electrons, results in a reduction of the mean electron energy.

This shows that, besides the production of electrons, electron-impact ionization can have an im-

portant role on the EEDF. Thus, the implementation of electron-impact ionizations can have dramatic

impacts on the calculation of plasma’s coefficients, especially for high electric fields.

In order to further develop LoKI-B, it is necessary to upgrade the treatment of electron-impact ion-

izations, with the introduction of two electron density growth models, as well as a non-conservative

ionization collisional operator using a differential ionization cross section. Some of the improvements

would be an increased accuracy of the predictions of the EEDF for higher E/N fields, it would allow for

further options on electron density profiles to better tailor simulations to certain discharges, and it would

also increase the accuracy of the chemistry solver by providing better predictions of reaction rates.

1.2 State of the art

1.2.1 Low-temperature plasma modelling

There are various methods to model low-temperature plasmas. Some of them are the kinetic, fluid,

global and particle-in-cell (PIC) models. Each of them has their own strengths, and more often than not,

hybrid models are implemented.

Kinetic models are implemented by describing particle kinetics using the Boltzmann equation. Ideally,

one would solve a Boltzmann’s equation for each type of particle on the plasma. However, to solve an

equation for electrons, each type of positive and negative ions, and each type and state of the neutral

particles, becomes infeasible. Instead, electrons, the most energetic particles and catalysts to various

reactions, are chosen to be described using an EBE. By solving the EBE we obtain the electron velocity

(or energy) distribution function. Using this function it is possible to calculate transport coefficients and

4

reaction rates that can then be used on fluid or global (zero-dimensional) models. Computational codes

that solve the EBE are also called EBE solvers, which is the case of LoKI-B, on which the work of this

thesis was developed.

Global or zero-dimensional models are usually applied when plasma chemistry is complex. In these

models, particle balance and/or power balance equations are used to calculate the time or spatial evo-

lution of each of the plasma’s species. In order to do this, they need reaction rate coefficients that need

to be introduced or calculated elsewhere. These models are usually less computationally demanding

than the others, and so, they are suited to be implemented in a hybrid model by connecting it to a kinetic

model. This is the reason why in LoKI, the EBE solver LoKI-B works in junction with a chemical solver

LoKI-C, which implements a global modelling of the plasma particles.

A related brother of the kinetic modelling approach is the PIC model. This type of modelling can

ultimately give the most precise prediction out of the various models. In this model, a large number of

particles are simulated, with each of their electrodynamic and collisional effects taken into account. PIC

modelling allows to calculate effects that are not accessible to the other approaches. However, this type

of modelling is also the most computationally expensive, while also having other numerical problems, for

example statistical noise [14].

Fluid models are implemented by calculating Boltzmann equation’s moments. There are various

ways of applying these models, for example by varying the number of moments to be solved. Some

hybrid models use fluid modelling in junction with EBE solvers.

For more on low-temperature plasma modelling the reader is requested to see [15].

1.2.2 Electron Boltzmann equation solver

The development of this work was done in LoKI’s EBE solver. Although this implementation will have its

effect on the chemical solver, the relevant overview of the state of the art concerns EBE solvers.

EBE solvers solve the EBE equation by discretizing it and then solving a linear system of equations.

In the case of LoKI-B, a steady state homogeneous plasma is being simulated. Several approximations

are used both on the collisional operator and on the variation of the electron distribution function. Each

solver will have its choice of the most relevant reactions and how these are treated on the collisional

operator. The variation of the distribution function includes a term due to the effect of the electric field,

however, the treatment of the temporal and spacial variation of the distribution varies between EBE

solvers.

When dealing with electron-impact ionizations, EBE solvers include: the variation of the distribution

function due to the increase of the electron density, and the ionization collisional operator.

Variation of the electron density

For the variation of electron density the most common approaches are: the electron density exponential

temporal growth, the electron density exponential spatial growth and the expansion in electron density

5

gradients.

The temporal growth model assumes that electron density increases exponentially with a mean ion-

ization frequency. This model is used in simulations when an electron density gradient is not expected,

such as in HF discharges.

The spatial growth model assumes that electron density increases exponentially with a constant

spatial frequency the first Townsend ionization coefficient. This model is used when an exponential

spatial electron density profile is expected, such as in SST discharges.

The expansion in electron density gradients makes no previous assumptions on the profile of the

electron density.

There are also EBE solvers that do not include electron density growth by electron-impact ionizations.

These models treat electron-impact ionizations as a conservative inelastic collisional process, in which

the primary electron loses the ionization energy.

Ionization collisional operator

In terms of the ionization collisional operator there are four types of treatments frequently used by the

LTP community.

Two of them use a non-conservative ionization collisional operator in which the energy sharing be-

tween the two electrons produced on the ionization is pre-defined. They assume either an equal energy

sharing, with both product electrons having half of the available energy, or an "one electron takes all"

type of sharing, in which secondary electrons are introduced at zero energy. Altough there is not much

of a discussion of which energy sharing mode should be used, there can be significant differences be-

tween them. These models are implemented on the EBE solver BOLSIG+[16], which will be used in this

work for benchmark purposes.

Another treatment would be the non-conservative ionization collisional operator in which energy shar-

ing is defined using a differential ionization cross section on the secondary electron energy. The exper-

imental data on differential ionization cross section is scarce, meaning that empirical expressions on

these cross sections are necessary. However, this description between product electrons is still closer

to reality compared to the aforementioned collisional operators.

At last, there is the treatment of electron-impact ionizations as a conservative inelastic collision.

Here, the secondary electrons are not accounted for, with the primary electron losing only the ionization

energy. With this mode, there is no increase on the electron density. This operator can be used when

electron density growth by electron-impact ionizations is not significant, which is related to low E/N

fields.

In LoKI’s previous treatment of electron-impact ionizations, the conservative inelastic collisional oper-

ator was used. No electron density growth model was assumed, and for high E/N fields the predictions

started to deviate from the expected values. Hence, there was a need to provide a new implementation

that considered secondary electrons.

6

1.3 Objectives and original contribution

One of the goals of the project KIT-PLASMEBA is the development of a state-of-the-art EBE solver,

LoKI-B. One of the priorities was to upgrade the treatment of electron-impact ionizations. The previous

treatment limited the range of working conditions, causing predictions to deviate at higher E/N . This

work was able to satisfy this need with the implementation of an electron-impact ionization routine in

LoKI-B, constituting a new tool for Grupo de Eletrónica e Descargas em Gases of Instituto de Plasmas

e Fusão Núclear (IPFN) of Instituto Superior Técnico.

The work in this thesis allowed for:

• The inclusion of two electron density growth models, a spatial and a temporal.

These allow users to better tailor their simulation to the corresponding discharge. The spatial

growth model will in the future allow to calculate new parameters such as the longitudinal diffusion

coefficient [17]. These models also paved the way for the introduction of other non-conservative

collisional processes, such as attachment and recombination;

• The implementation of a non-conservative ionization collisional operator that uses a differential

ionization cross section, consistently deduced from a total ionization cross section;

This allows the energy sharing of product electrons to be defined based on measured data. It also

allows for an easy future update of these cross sections when experimental data, or theoretical

predictions, improve.

• An analysis of the aforementioned ionization collisional operator by comparing it with other ap-

proaches commonly used on the LTP community;

These are the equal energy sharing, and the "one electron takes all" ionization collisional opera-

tors. This allows for a deeper understanding of the treatment of electron-impact ionization mech-

anisms. The importance of the energy sharing mode is highlighted. Some of the flaws of these

treatments are also identified.

• Other contributions that help in LoKI’s development:

These are the identification of discretization errors through an analytical verification, the validation

with experimental data, the coupling with the electron-electron collisions’ routine, and also the

development of suplementary documentation.

1.4 Organization of the thesis

This document begins with an introduction that contextualizes the reader on the importance of this work,

followed by a description of other approaches to this problem by the LTP community.

In the next chapter a theoretical formulation is done. Within this chapter, the basic theory of the kinetic

treatment of gas discharges is described in accordance with LoKI-B approach. A general description

of the EBE is done, followed by the explanation of the two-term classical approximation, the adiabatic

7

approximation, and the Fourier expansion of the electron distribution function. Afterwards, the various

mechanisms of the electron-impact ionization used in this work, as well as approximations used in other

EBE solvers are described. After this, the two-term EBE is derived with emphasis on electron-impact

ionizations, though other important mechanisms are included. In the end, a physical contextualization

of the effect of electron-impact ionizations is done using conservation and transport equations.

On the third chapter, the computational approach is explained. First, a description of the discretiza-

tion process is done, and then the procedure to solve the non-linear EBE, for each electron density

growth model, is detailed. In the end, an analytical verification of the discretized terms, both with particle

and energy balance, is done.

The fourth chapter shows the main results, accompanied with a characterization of flaws and possible

solutions. A comparison between energy sharing modes is done and an explanation provided for the

differences on EEDFs. A comparison between electron density growth models is done using results

obtained on the theoretical formulation. Then follows a verification using benchmarks against BOLSIG+.

And in the end, validation with experimental data for two different gases is done.

The last chapter includes suggestions for future work.

8

Chapter 2

Theoretical Formulation

2.1 The electron Boltzmann equation

In low temperature plasmas many properties and reactions are dictated by the electron kinetics. An

electron distribution function allows the calculation of important reaction rate coefficients as well as

plasma transport coefficients.

Following a procedure analogous to the kinetic theory of gases, we can introduce an electron distri-

bution function F (~r,~v, t)d~v such that,

F (~r,~v, t)d~v = the number of electrons per unit volume

at a time t with velocities between

~v and ~v + d~v.

This function has its domain in the six-dimensional phase space (~r,~v). If there are no collisions, an elec-

tron on the elementary phase space volume d~r d~v centred at (~r,~v) at a time t, will be on the elementary

phase space volume d~r ′ d~v ′ centred at (~r + ~vdt,~v + ~adt) at a time t + dt, with ~a being the electron’s

acceleration. According to the Liouville’s theorem, d~r ′ d~v ′ = d~r d~v, which means that

F (~r + ~vdt,~v + ~a dt, t+ dt) = F (~r,~v, t).

If collisions are accounted for, this equations can be written as,

F (~r + ~vdt,~v + ~a dt, t+ dt) = F (~r,~v, t) +

(∂F

∂t

)coll

.

Expanding the left side in first order of dt, and subtracting the first term of the right side to it, we arrive

at an equation of motion for the electron distribution function,

∂F

∂t+ ~v · ∂F

∂~r+ ~a · ∂F

∂~v=

(∂F

∂t

)coll

,

with(∂F∂t

)coll

being the rate of change of the electron distribution function due to electron collisions with

other electrons, ions and neutral particles.

9

Collisions can be classified as elastic, in which there is only an exchange of kinetic energy between

the electron and the target particle, or inelastic, in which the kinetic energy of the two particle system

(electron and target particle) changes. Furthermore, inelastic collisions can be divided in conservative

and non-conservative. Here conservative means that there is the same number of electrons before

and after the collision, for example an electron that causes an electronic excitation in a neutral particle.

Non-conservative refers to a gain or loss of electrons during the collision, for example attachment or

ionization. We will not consider attachment, so we will use an elastic collisional operator I(F ), a con-

servative inelastic/superelastic collisional operator J(F ), and a non-conservative ionization collisional

operator JI(F ) .

By expanding the total derivative of F (assuming that the distribution function may have an explicit

time dependence, there can be an electron density gradient, and electrons will be under the influence of

an electric field) we arrive at an electron Boltzmann equation,

∂F

∂t+ ~v · ∇~rF −

e ~E(t)

me· ∇~vF = I(F ) + J(F ) + JI(F ) . (2.1)

2.2 The particle distribution function

In order to solve the electron Boltzmann equation some approximations are made to the electron distri-

bution function.

2.2.1 Normalization under the adiabatic approximation

The electron distribution function F (~r,~v, t) is normalized such that

ne(~r, t) =

∫F (~r,~v, t)d~v,

with ne being the electron particle density.

In the case where the electron’s mean free path is much shorter than the distance at which strong

density variations can occur (the case of relatively low pressures and away from boundaries) the spatial

dependence of the distribution function can be identified with that of the electron density. The distribution

function may be written as,

F (~r,~v, t) = ne(~r, t)F (~v, t),

in which F is a probability distribution function normalized to one,∫F (~v, t)d~v = 1.

This approximation can be improved by including higher order terms in a density gradient expansion,

see [18] and [19, pp 351]. With this adiabatic approximation, fluid equations can be derived using the

set of equations obtained from the EBE and the two-term spherical harmonics expansion, (see sec. 2.5).

10

2.2.2 Spherical-harmonics expansion

Since Legendre polynomials Pl(cosθ) 1are orthogonal and form a complete basis set, it is possible to use

them for a series expansion, namely to write the angular dependence of F (~v) as a linear combination of

Legendre polynomials [20, Ch. 10.4, Ch. 12].

For the sake of simplicity, we assume that the gas discharge takes place between two infinite parallel

electrodes, hence there is azimuthal symmetry meaning that it is not necessary to recur to the associated

Legendre polynomials.

In this case, the expansion goes as

F (~v, t) =

∞∑l=0

F l(v, t)Pl(cosθ),

in which the angle θ is defined as the angle between the velocity vector and the polar anisotropy-

direction. This way, the l = 0 term accounts for the isotropic, or symmetric, part of F and the other

terms are higher degrees of anisotropy. Note that the different coefficients F l of the expansion depend

only on the absolute value of the velocity.

If the distribution function is near-isotropic, an expansion of up to two terms is enough. The physical

argument to this approximation, is based on the fact that the electron mass (me) is much lower than

the neutral particle’s mass (M). Since me/M is very small, elastic electron-neutral collisions produce

large directional velocity changes, but relatively small electron energy losses. Thus, these collisions

randomize any directed electron motion, meaning that F is nearly isotropic on the velocity space. The

electric field and the electron density gradient may also influence the anisotropy, so for low E/N and

weak density gradient we can assume that we remain on the near-isotropic picture.

~F (~v, t) ≈ F 0(v, t) + F 1(v, t) cos θ = F 0(v, t) + ~F 1(v, t) · ~vv. (2.2)

v

θ

x

y

z

Because we chose to place the z axis along the anisotropy direction, the polar angle is defined as

the angle between the anisotropy vector ~F 1 = F 1~ez and the velocity vector ~v. The product F 1 cos θ is

then equivalent to ~F 1 · ~v/v. The term with l = 1 will be called anisotropic part of F since higher order

terms will be ignored.

The Legendre Polynomials satisfy the orthogonality relation∫Ω

Pl(cos θ)Pm(cos θ)dΩ =4π

2l + 1δlm . (2.3)

With this relation it is possible to derive the two-term Boltzmann equation in a few steps. However

we will expand F up to an arbitrary order, then truncating to the first order the resulting equations. In

order to do this two more relations will be used,

1We will refer to the Legendre polynomials simply as Pl instead of Pl(cos θ).

11

(2l + 1)Pl cos θ = (l + 1)Pl+1 + l Pl−1, (2.4)

(2l + 1) sin2 θ∂Pl∂ cos θ

= l (l + 1) (Pl−1 − Pl+1) . (2.5)

The two-term approximation is very useful when trying to calculate the mean value of a certain

velocity-dependent quantity. The mean value of a scalar function is non-zero only when integrating over

the isotropic part. And a vectorial function, such as ~v, has non-zero mean value only when integrating

over the anisotropic part. This becomes obvious if we consider the orthogonality relation 2.3 and express

scalar and vector functions using the Legendre polynomials.

2.2.3 Fourier expansion

We will assume that the plasma of a Direct Current (DC) or an Alternate Current electric field varying

with angular frequency ω, or a combination of both,

~E(t) = ~E0 + ~E1 cos(ωt).

In this work we will only use alternate currents of High Frequencies (HF), that is a frequency between

3MHz and 30MHz.

If we go to the complex plane we can treat this expansion as a two-term Fourier expansion,

~E(t) = ~E0 + ~E1ejωt.

It is expected that the electrons respond to the electric field also in a harmonic way. This response

can be studied using a Fourier expansion of the distribution function F (~v , t).

F (~v , t) =∑m

Fm (~v ) ej mω t ≈ F0(~v) + F1(~v)ejωt.

The reasons why this expansion converges well enough with only the first two terms in the case of

high frequencies, are different for isotropic and anisotropic terms of the distribution function [21].

So expanding now also with the Legendre polynomials

F (~v , t) =∑l

∑m

F lm(v)Pl ej mω t

≈[F 0

0 (v) +F 0

1 (v) ej ω t]P0 +

[F 1

0 (v) + F 11 (v)ej ω t

]P1 (2.6)

≈ F 0(v) + F 10 (v)P1 + F 1

1 (v)P1 ej ω t.

On equation 2.6, F 01 (v) can be approximately set to zero, assuming that: the energy gained by

the electron in a half cycle of oscillation is very small compared with its thermal energy, and that the

frequency ω is much larger than the relaxation frequency of the isotropic term ω >> me

M νe. In other

words, the relaxation time of the isotropic part of the distribution function is long enough so that it cannot

12

change, in a significant way, in half cycle of the applied electric field. The same goes for higher orders

of the Fourier expansion.

In the case of the anisotropic part, after the Boltzmann equation is employed and the orthogonal

properties of the Legendre and Fourier terms are used, terms of the second order in Fourier (f l2) will

exist only for Legendre terms with l ≥ 2. However we assume a near-isotropic distribution function (see

sec. 2.2.2) in which only the first two terms of the spherical harmonics expansion are considered. Be-

cause of this, the Fourier expansion up to two terms in good enough.

This expansion can be directly substituted on the two-term Boltzmann equations, where most terms

are linear on ej ω t. However, when dealing with the electric field term, we encounter what appears to

be a second order term, the multiplication of E(t) ≈ Eej ω t and F 1(v, t) ≈ F 10 (v) + F 1

1 (v)ej ω t. In reality

we are dealing with physical quantities, that vary with <(ejωt), and as a result the product EF l is the

product of the real parts and not the real part of the product.

Writing E(t) = E0 + E1 cos(ωt) then

< [E(t)]<[F l(t)

]= < [E(t)]<

[ ∞∑m=0

F lmej mt

]=

=

∞∑m=0

E0<[F lm]

cos(mωt) +

∞∑m=0

E1<[F lm]

cos(ωt) cos(mωt) =

=

∞∑m=0

E0<[F lm]

cos(mωt) +

∞∑m=0

E1<[F lm]

2[cos(ωt(m+ 1)) + cos(ωt(m− 1))] ≈

≈ E0

<[F l0] + <[F l1] cos(ωt)

+ E1

Fourier order 1︷︸︸︷<[F l0]

22 cos(ωt) +


2 cos(2ωt) +


2cos(0)

=

= E0

<[F l0] + <[F l1] cos(ωt)

+ E1

<[F l0]

cos(ωt) +<[F l1]

2

,

and back to the exponential notation we have,

< [E(t)] <[F l(t)

]≡ <

[E0

(F l0 + F l1e

j ω t)

+ E1

(F l0e

j ω t +F l12

)]. (2.7)

2.2.4 The electron energy distribution function

In this subsection we will derive terms of the electron Boltzmann equation 2.1 and introduce some of the

most commonly used collisional operators. When treating each term we will use the electron distribution

function F (~r,~v, t). But our computational application uses energy-dependent functions f l(u, t), so that

a change in variables is necessary. These are probability functions, here normalized to one, that can

also be expanded in Fourier series with coefficients f lm(u). The l = m = 0 order term, noted f(u), is the

so-called electron energy distribution function (EEDF), which has the normalization condition∫ ∞0

f(u)√udu = 1. (2.8)

13

We also prefer to use reduced quantities, identified with the subscript R. A reduced quantity is just

the original quantity divided by the gas density N , for example the relative populations δj , the reduced

frequency wR, the reduced electric field ER and others.

To aid this change we use the following relations and variables:

u[J ] = u[eV ]e =mev

2

2⇔ v2 =

2ueme

F 0(v) 4π v2dv = f(u)√u du

γ ≡ meN ne

4π e√u

νij(v) = σij ni v

δj =njN

Cα =〈να〉N

(2.9)

The energy units are eV . The variable γ will appear on every term of the EBE and as such is ignored

in the end. The variable δj is the population of the j particles relative to the gas density N . The cross

section σij refers to the cross section of the inelastic/superelastic collision that results in a change from

the gas specie i to the gas specie j. The quantity να is the collision frequency of the collision α, defined

by the cross section of the respective collision, multiplied by the electron velocity and the density of the

target particle. The reaction rate coefficients Cα are defined as the mean frequency of the collision α

divided by the gas density.

2.3 Electron-impact ionization

With increasing E/N , electrons take up high enough energy so that their production by electron-impact

ionizations become significant. Without accounting for these secondary electrons, computational mod-

els’ predictions can deviate significantly from the experimental data [22].

In some numerical codes, electron impact ionizations are treated with a conservative inelastic col-

lision operator in which the primary electron simply loses the ionization energy. This means that the

creation of secondary electrons is not accounted for.

In order to include secondary electrons it is necessary to introduce a non-conservative ionization

collisional operator. Since this operator is non-conservative, it is necessary to introduce an electron

density growth term that accounts for the addition of the new secondary electrons. The various types of

non-conservative ionization operators and growth models are discussed in this subsection.

2.3.1 Energy sharing

In the first step, an incident electron collides with a neutral particle, atom or molecule. In the second

step there is the production of an ion and of two electrons,

e− + A −→ A+ + 2e−.

14

As we can see in the previous expression, the electrons produced are indistinguishable. However, for

bookkeeping purposes, it is customary to name the faster of the two electron as scattered and the

slower as secondary. The initial electron will also be distinguished from the produced electrons and will

be called the primary electron. In terms of energy we have,

0 ≤ usec ≤ usca ≤ ε− VI ,

ε = VI + usca + usec , (2.10)

with ε the primary electron energy, usca the scattered electron energy, usec the secondary electron en-

ergy, and VI the ionization energy or ionization potential (in eV ). Figure 2.1 resumes this information.

20

40

60

80

20 40 60 80 100

u(e

V)

ε(eV )

u=ε−V I

u=ε−V I

2

Energy of primary electron

Ene

rgy

of p

rod

uct

elec

tron

Secondary electrons

Scatteredelectrons

Figure 2.1: Graphical representation of the energy sharing between the primary and the secondary or

the scattered electron. According to the product electron’s energy, the two regions, signalled by the

solid and dashed-dotted lines, identify whether it is a scattered or a secondary electron. In red (green)

is the primary electron energy interval that can produce a secondary (scattered) electron with energy

u = 40eV .

2.3.2 Differential ionization cross sections

In order to characterize collisional processes it is necessary an energy distribution and an angular dis-

tribution of the intervened particles. This can be done with differential cross sections, which are function

15

of the primary electron’s energy. In an ionizing collision, the most complete description is done using a

triple differential cross section in usec, the secondary electron solid angle Ωsec, and the primary electron

solid angle Ωp,

d3σI(ε, VI)dusec dΩsec dΩp

.

For our purposes, we only need a cross section describing the ionization collision as a function of the

secondary electron’s energy. By integrating the triple differential cross section in Ωsec and Ωp we get,

dσI(ε, VI)dusec

≡ qIsec(ε, usec),

in which the differential cross section qIsec keeps the parametric dependence on VI .

This cross section is also referred to single differential cross section (SDCS), or simply as differential

ionization cross section. It gives an energy distribution for the secondary electron and also for the

scattered electron with relation 2.10.

Integrating the differential ionization cross section on the secondary electron’s energy (between 0

and (ε− VI)/2 see Figure 2.1), one gets the total ionization cross section,

σI(ε, VI) =

∫ (ε−VI)/2

0

qIsec(ε, usec)dusec. (2.11)

The total ionization cross section will also be referred with an explicit function of the energy σI(ε), keeping

the parametric dependence on VI .

For more information in energy distributions of secondary electrons produced by electron impact ion-

ization the reader is requested to see [23, Ch. 10].

When calculating the collisional operators, LoKI reads the values of the corresponding cross section

for each primary electron energy. In the case of the differential cross sections, it would be necessary

to discriminate the values of the cross section as also a function of the secondary electron energy. On

top of being demanding in computational resources terms, the available experimental data of SDCS is

scarce and usually measured for only a few different primary electron energies.

The most comprehensive measures were taken by Opal, Peterson and Beaty in 1971 for a number

of gases [24]. They measured the double differential cross section (that is on the energy and angle of

the secondary electron), having integrated it later to obtain the SDCS. Although there are some con-

cerns with the normalization errors (20-30%), and with the low values for extreme forward and backward

angles, their shape is reliable [23]. By comparing secondary electron energy distributions for different

primary electron energies, Opal, Peterson and Beaty found out that a function of the form

qIsec(ε, usec) =C(ε)

1 +(usec

w

)2.1 (2.12)

describes the data for most gases well enough if an appropriate choice for w is made [24]. Here C(ε)

is a normalization constant, and by using relation 2.11 it is possible to get an expression for it. The

integration is accurate enough if the exponent in equation 2.12 is substituted with the value 2.0. Then

16

the expression for the differential cross section becomes,

qIsec(ε, usec) =σI(ε, VI)

w arctan ε−VI2w

1

1 +(usecw

)2 . (2.13)

Using equation 2.10 it is possible to get also the equivalent differential cross section for the scattered

electrons,

qIsca(ε, usca) =σI(ε, VI)

w arctan ε−VI2w

1

1 +(ε−VI−usca

w

)2 .

Since expression 2.13 has an explicit dependence on the total ionization cross section, and because it

satisfies exactly equation 2.11, we have diminished possible normalization errors and allowed for using

the most recent data on the total ionization cross sections.

In LoKI’s ionization routine, this differential cross section is used when writing the ionization collisional

operator. For each measured gas the parameter w is set according to the original estimates [24]. For

most gases, the value for w is comparable to the ionization energy. These values are a result of the least

squares fit of the function 2.12 to available data.

Differential cross section for heavy noble gases

For most gases with available measurements (He, N2, O2, Ne, H2, NO, CO, H2O, NH3, C2H2 and CO2)

it was possible to get a good fit to function 2.12. However, for Ar, Xe and Kr this was not the case. These

are heavy noble gases with strong emission features for which some of the measured events may be

from auto-ionization or from electron "shakeoff" following the removal of an inner-shell electron, causing

a change in the shape of the differential cross section [24].

In order to address this issue, we tested another semi-empirical function for the differential cross

section. This model was proposed by Green and Sawada [25], following the results obtained by Opal,

Peterson and Beaty, and aimed at improving older theoretical models. These models have generally

used the Born-Bethe Approximation, in association with the so-called generalized oscillated strength

and empirical distortions to arrive at a differential cross section for the secondary electron. Based on

these approximations, they examined the form

fΓ(usec) =Γ2

(usec − T0)2

+ Γ2,

which is similar to the Fourier transform of an exponentially damped oscillatory wave function. This

form can then be multiplied by a primary electron-energy-dependent amplitude A(ε), rendering the

SDCS,

qIsec(ε, usec) = A(ε)Γ2

(usec − T0)2

+ Γ2. (2.14)

Here, Γ and T0 are functions of the primary electron energy. Following the same procedure as before,

17

the SDCS 2.14 can be integrated as in equation 2.11 leading to an expression for A. The full set is:

A =σI(ε)

Γ

1

arctan(Tm−T0

Γ

)+ arctan

(T0

Γ

) ;

Γ = Γsε

ε+ Γb;

T0 = Ts −1000

ε+ 2VI.

with Γs and Ts fitted parameters and where for most gases (excluding the heavy noble gases) Γb is the

ionization energy. The fits were made to Opal, Peterson and Beaty’s data [1].

The improvements of this differential cross section with respect to equation 2.13 were not significant(

see Figure 2.2). For Argon, a heavy noble gas, the Green and Sawada form 2.14 is slightly better than

2.13, yet giving results that remain within the experimental data uncertainty. For Helium, a light noble

gas, both function 2.13 and 2.14 are equally reliable. Also, function 2.14 is more complex which leads

to longer computation times when implemented in the ionization routine. Because of these reasons, the

differential cross section form 2.13 is preferred.

eVsecu0 10 20 30 40 50 60

/eV

2 c

m-2

0 1

0× Iσ

0

200

400

600

800

1000

OPB and GS fit Ar

=500 eV∈Exp Points Ar

OPB form

GS form

(a)

eVsecu0 10 20 30 40 50 60

/eV

2 c

m-2

0 1

0× Iσ

0

20

40

60

80

100

120

140OPB and GS fit He

=500 eV∈Exp Points He

OPB form

GS form

(b)

Figure 2.2: Fits to Opal, Peterson and Beaty’s experimental data [1] for Argon (4.6a), and Helium (4.6b),

using function 2.12 (OPB) and function 2.14 (GS).

2.3.3 Ionization collisional operator

In the electron Boltzmann equation, the total derivative of the electron distribution function is equal to the

rate of change of the distribution function due to electron collisions. The latter term yields the so-called

collisional operator, which was divided into several components, one of them being the non-conservative

ionization collisional operator.

Holstein [26] showed that the rate of change of the electron energy distribution function, due to

18

ionizing collision, can be written as

JI(u)

γ=

∞∫2u+VI

ε qisec(ε, u) f(ε) dε+

2u+VI∫u+VI

ε qisca (ε, u)f(ε) dε− uσI(u) f(u). (2.15)

This collisional operator is composed by three terms. The first term accounts for secondary electrons

that enter the distribution function with energy between u and u + du. The second term accounts for

scattered electrons that enter the distribution function with energy between u and u+du. The third terms

accounts for electrons that leave the distribution function at the energy u due to ionizing collisions.

In the first term, due to secondary electrons J1I (u), there is the product of the secondary electron

differential cross section by the electron energy distribution function. This gives the frequency of having a

secondary electron with energy u produced by a primary electron with energy ε [√εqisec(ε, u)], multiplied

by the probability of having a primary electron with energy ε [√εf(ε)dε]. This expression is integrated

between 2u + VI and ∞, corresponding to the energy domain of primary electrons that can produce a

secondary electron with energy u (see Figure 2.1).

In the second term, due to scattered electrons J2I (u), there is the product of the scattered electron

differential cross section by the electron energy distribution function. Similarly to the first term, the

integration is made over the values of the primary electron energy (between u+VI and 2u+VI ) capable

of producing a scattered electron with energy u (see Figure 2.1).

In the third term, due to primary electrons J3I (u), there is the product of total ionization cross section

by the electron energy distribution function, that is, the probability of having an ionizing collision that

makes primary electrons with this energy to leave the energy distribution function (hence this term is

negative).

Conservative ionization collisional operator

In some cases an approximation can be made in which the ionization collisional operator is written as

a conservative one, similarly to other inelastic operators. This approximation is good enough for low

values of E/N [22]. This was the case of the previous ionization collisional operator in LoKI, which was

written as,JI(u)

γ= (u+ VI) σI (u+ VI) f (u+ VI)− uσI(u) f(u). (2.16)

This first term of this operator accounts for scattered electrons entering the distribution function with

energy between u and u + du, produced by a primary electron with energy u + VI . The second term

accounts for primary electrons with energy u that leave the distribution function due to ionizing collisions.

Since both energies are defined, the total ionization cross section is used, and no term accounting for

secondary electrons is considered.

Note that 2.16 can be obtained from 2.15 by taking

qisec(ε, u) = σI(ε)δ(u),

which corresponds to the assumption that the secondary electrons are created with energy zero, whereas

the scattered electrons end up with energy ε−VI . When the previous expression is used in 2.15 it yields

19

an extra term〈νI〉 δ(u)

N√

2eme

corresponding to the introduction of secondary electrons at energy zero, which is neglected for a con-

servative ionization collision operator.

Non-conservative ionization collisional operator with predefined secondary electron energy

Some electron Boltzmann equation solvers use non-conservative ionization collisional operators with

predefined value for the secondary electron energy. By predefining this energy, there is no need for

using a secondary electron energy distribution. As a result, the creation of secondary electrons can be

accounted for without the need of a differential ionization cross section.

The two most common models are: "equal energy sharing", where both the scattered and the sec-

ondary electrons have the same energy (usec = usca = (ε−VI)/2); and "one electron takes all", in which

the secondary electron is introduced at zero energy (usec = 0, usca = ε− VI ).

For the "equal energy sharing" case the collisional operator can be written as [16],

JI(u)

γ= 4 (2u+ VI) σI (2u+ VI) f (2u+ VI)− uσI(u) f(u).2 (2.17)

Here the first term accounts for the scattered and the secondary electrons entering the distribution

function at an energy between u and u + du, produced by a primary electron with energy ε = 2u + VI ;

whereas the second term accounts for primary electrons with energy u that leave the distribution function

due to ionizing collisions. Note that 2.17 can be obtained from 2.15 by taking

qisec(ε, u) = σI(ε) δ(u− ε− VI

2

).

In the case of "one electron takes all" the ionization collisional operator can be written as [16],

JI(u)

γ= (u+ VI) σI (u+ VI) f (u+ VI)− uσI(u) f(u) + δ(u)

∫ ∞0

εσI(ε) f(ε) dε. (2.18)

Here the first term account for the scattered electrons, that enter the distribution function with energy

between u and u + du, produced by a primary electron with energy u + VI ; the second term accounts

for the primary electrons that leave the distribution function due to ionizing collisions; and the third term

accounts for secondary electrons that enter the distribution function with zero energy. Note that 2.18

corresponds to the non-conservative form of 2.16.

These two predefined energy sharing modes will be used mostly for benchmark purposes.2The value 4 may lead to some confusion since an ionizing collision produces two electrons, not four. This value comes from

a change of variable. Assume that we have an electron with energy ε, belonging to the electron distribution. The number of new

electrons Ne produced by this primary electron with energy between ε and ε+ dε, after an ionizing collision is given by

BNe(ε) dε = 2 εσI (ε) f(ε) dε,

in which B is a variable that ensures the correct units for the expression. To write this result as function of the product electrons’

energy u, such that ε = 2u+ VI , we obtain

BNe(ε) dε = 2 (2u+ VI)σI (2u+ VI) f(2u+ VI) 2du = 4 (2u+ VI)σI (2u+ VI) f(2u+ VI) du .

20

2.3.4 Electron density growth

Because of electron-impact ionizations, the number of electrons in the system will increase. In the

electron Boltzmann equation, the rate of change of the distribution function due to ionization is accounted

for in the non-conservative ionization collisional operator. As a result, an explicit growth of the electron

distribution function occurs through the growth of the electron density.

Two models were used to describe the change in the electron density: an exponential spatial growth

and an exponential temporal growth.

Exponential spatial growth

It is possible to have electron ionizations on the volume of the gas. If we assume that these events

cause an exponential spatial growth of the density one can use the first Townsend coefficient α as a

constant spatial frequency. In this way, the density will grow in space in the opposite direction of the

applied electric field

ne(~r, t) ≈ ne(z, t) = ne(t) eα z, (2.19)

or equivalently

α =∇rnene

· ~ez.

This density growth model is usually implemented when simulating Steady State Townsend (SST)

discharges maintained with a DC electric field [27, sec. 3.4].

Exponential temporal growth

Another model is used when the mean electron drift velocity is zero, in which case there is no electron

density gradient. As a result, the electron density increases exponentially as a whole in time with a

growth constant 〈νI〉, corresponding to the mean electron ionization frequency,

∂ne (~r, t)

∂t= 〈νI〉 ne (~r, t) . (2.20)

The temporal electron density model was used to simulate Pulsed Townsend (PT) discharges by

Tagashira et al [27, sec. 3.5] using the formulation of one dimensional continuity equation of electrons

under uniform field of Thomas [28]. For HF discharges this growth model is also appropriate since there

is no electron density gradient created by electron impact ionizations.

2.4 The two-term electron Boltzmann equation

In this subsection we will derive each term of the EBE using the expanded electron distribution function

in order to get the equations for its isotropic part and for its anisotropic part.

21

2.4.1 Derivation of the electron Boltzmann equation terms

Here we will derive each term of the EBE using the previous approximations and expansions, and

changing variables to adopt a description in the electron kinetic energy.

Time dependent term

On the Boltzmann equation 2.1 there is a partial derivative in time of the distribution function. After the

spherical harmonics and Fourier expansions we end up with

F (~r , ~v , t) ≈ ne(~r, t)F (~v, t) ≈ ne(~r, t)[F 0(v) + F 1

0 (v)P1 + F 11 (v)P1 e

j w t]

The explicit writing of the partial derivative in time gives, assuming an electron density exponential

temporal growth (using 2.20),

∂F (~r , ~v , t)

∂t=

∂

∂t

[ne(~r, t)F

0 (v) + ne(~r, t)F10 (v)P1 + ne(~r, t)F

11 (v)P1 e

j w t]

=∂ne(~r, t)

∂tF 0 (v) +

∂ne(~r, t)

∂tF 1

0 (v)P1 +∂[ne(~r, t) e

j w t]

∂tF 1

1 (v)P1

= 〈νI〉 ne(~r, t)F 0 (v) + 〈νI〉 ne(~r, t)F 10 (v)P1 + (〈νI〉+ j w) ne(~r, t)F

11 (v)P1 e

j w t. (2.21)

Changing to a kinetic-energy description on equation 2.21 and using the relations in 2.9 we have

∂F (~r , ~v , t)

∂t= N

neN v

me

4πe

[√u

1

ne

∂ne∂t

f (u) +√u

1

ne

∂ne∂t

f10 (u)P1 +

√u

(1

ne

∂ne∂t

+ j ω

)f1

1 (u)P1 (u) ej w t]

= γ

u CI√2eume

f (u) + uCI√

2eume

f10 (u)P1 + u

CI√2eume

+ jωR√

2eume

f11 (u)P1 e

j w t

,in which CI is the ionization rate coefficient.

Spatial dependent term

Assuming a density variation along the Z direction, the gradient term is

∇~r ·[~v F (~r,~v, t)

]≈ ~v · ∇~r [ne(~r, t)F (~v, t)] = ~v · ~ez

∂ne(~r, t)

∂zF (~v, t) = v

∂ne(~r, t)

∂zF (~v, t) cos θ.

22

Using the two term approximation,

∇~r ·[~v F (~r,~v, t)

]≈v ∂ne(~r, t)

∂zF (~v, t) cos θ

=v∂ne(~r, t)

∂z

[F 0(v) +

(F 1

0 (v) + F 11 (v)ej ω t

)cos θ

]cos θ

=v∂ne(~r, t)

∂z

[F 0(v) cos θ +

(F 1

0 (v) + F 11 (v)ej ω t

)cos2 θ

]and by writting cos θ and cos2 θ in terms of the Legendre polynomials

=v∂ne(~r, t)

∂z

[F 0(v)P1 +

(F 1

0 (v) + F 11 (v)ej ω t

)(2P2 + P0

3

)]≈v ∂ne(~r, t)

∂z

(F 1

0 (v) +F 1

1 (v)ej ω t

3P0 + F 0(v)P1

)≈v ∂ne(~r, t)

∂z

(F 1

0 (v)

3+ F 0(v) cos θ

)

If we assume an electron density exponential spatial growth (using 2.19), the gradient term becomes

∇~r ·[~v F (~r,~v, t)

]= v αne(~r, t)

(F 1

0 (v)

3+ F 0(v) cos θ

),

and by changing to a kinetic-energy description and using the relations in 2.9 we have

∇~r ·[~v F (~r,~v, t)

]= γ

[αR u

(f1

0 (u)

3+ f(u) cos θ

)].

Electric field term

Assuming that the applied electric field is along the zz

~E = −E~ez,

and knowing that

(∂

∂vz

)vx,vy

= cosθ∂

∂v+sin2θ

v

∂

∂cosθ,

then the electric field term for a general component of order l of the distribution function expansion is

−e ~E(t)

me· ∇~v F = ne

∑l=0

−e ~E(t)

me· ∇~v(F lPl) = ne

∑l=0

eE(t)

me

∂F l

∂vPl cosθ + ne

∑l=0

eE(t)

me

F l

v

∂Pl∂cosθ

sin2θ,

and using relations 2.4 and 2.5

−e ~E(t)

me· ∇~v F = ne

eE(t)

me

[∑l=0

∂F l

∂v

(l + 1)Pl+1 + l Pl−1

2l + 1+∑l=0

F l

v

l(l + 1)

2 l + 1(Pl−1 − Pl+1)

]=

= neeE(t)

me

[∑l=−1

(∂F l+1

∂v

l + 1

2 l + 3+F l+1

v

(l + 1)(l + 2)

2 l + 3

)Pl +

∑l=1

(∂F l−1

∂v

l

2 l − 1− F l−1

v

l(l − 1)

2l − 1

)Pl

]

For the two-term expansion 2.2 we have

23

−e ~E(t)

me· ∇~v F = ne

1

3 v2

∂

∂v

(eE(t)

meF 1(v, t) v2

)P0 + ne

∂

∂v

(eE(t)

meF 0(v)

)P1. (2.22)

Changing to a kinetic-energy description on equation 2.22, and using relations 2.7 for the Fourier

expansion, and 2.9, we obtain

∇~v ·

(−e ~E(t)

meF (~r,~v, t)

)

=γ

∂

∂u

[u3ER(t)f1(u, t)

]+ u

∂

∂u[ER(t) f(u)] cos θ

=γ

∂

∂u

[u3

(E0Rf

10 (u) + E1R

f11 (u)

2

)]+ u

∂

∂u

[E0R f(u) + E1R f(u) ejωt

]cos θ

+ (2.23)

+γ∂

∂u

[u3

(

:0E1Rf

10 (u)ejωt +

:0E0Rf

11 (u)ejωt

)]=γ

∂

∂u

[u3

(E0Rf

10 (u) + E1R

f11 (u)

2

)]+ u

∂

∂u

[E0R f(u) + E1R f(u) ejωt

]cos θ

.

The last two terms of 2.23 are zero since with HF electric field the stationary anisotropic part (f10 ) is

zero, and with DC electric field the time dependent anisotropic part (f11 ) is zero.

Elastic collision term

The elastic collisional operator may be written as [29]

I0 =me

M

1

v2

∂

∂v

[v3 νc

(ne(~r, t)F

0(v) +kB Tgme v

∂ne(~r, t)F0(v)

∂v

)],

in which νc = Nσc(v)v, with σc = 2π∫ π

0(1−cosχ)σ(v, χ) sinχdχ is the momentum transfer cross-section,

σ the elastic collision cross-section, and χ the angle between the initial and final electron trajectories.

We can see that the second term ensures that the electron distribution function goes to equilibrium with

the gas molecules at temperature Tg, when the elastic collisional operator dominates. This form was

first deduced by Chapmann and Cowling [30].


I0 = γ∂

∂u

[2me

Mu2 σc(u)

(f(u) +

kB Tge

∂f(u)

∂u

)]. (2.24)

Inelastic/superelastic collision term

The inelastic/superelastic conservative collisions between electrons and neutrals can lead to the excitation/de-

excitation of electronic, vibrational and rotational states of the gas. The operator follows the same struc-

ture for all types of collisions. The subscripts i and j denominate the electron energy levels of the

molecule, being j the most energetic one. The inelastic/superelastic term can be written as [26],

24

J0 = ne∑i,j

v + vij

vνij (v + vij)F

0(v + vij)− νij(v)F 0(v)

+v − vij

vνji (v − vij)F 0(v − vij)− νji(v)F 0(v)

(2.25)

We define ni as the density of the particles in state i, the corresponding relative population being

δi = ni/N , and u ± uij = 12me

2 (v + vij)2 as the electron energy plus or minus the energy interval

between levels i and j.


J0

γ=∑i,j

δi(u+ uij

)σij(u+ uij) f(u+ uij)− δiu σij(u) f(u)

+

+∑i,j

δj(u− uij

)σji(u− uij) f(u− uij)− δju σji(u) f(u)

. (2.26)

It is now possible to use Klein-Rosseland relation

gju σji(u) = gi(u+ uij

)σij(u+ uij),

in which gi and gj are the statistical weights of energy levels i and j respectively, to re-write equation

2.26 as

J0

γ=∑i,j

δi(u+ uij

)σij(u+ uij) f(u+ uij)− δiu σij(u) f(u)

+

+∑i,j

gigjδju σij(u) f(u− uij)−

gigjδj(u+ uij

)σij(u+ uij) f(u)

.

Rotational collision term - continuous approximation

Rotational collisions have energy thresholds much smaller than the vibrational and electronic ones.

Hence the numerical code must use an energy step smaller than the lowest threshold, which is compu-

tationally demanding. In the case of N2, O2 and H2, it is possible to use a continuous approximation for

the rotational collisional operator, with a "Chapman-Cowling" term [31],

J0rot

γ=

∂

∂u

[4Bσ0u

(f(u) +

kBTge

∂f(u)

∂u

)].

Anisotropic collisional operator

We will assume an effective collision frequency that accounts for elastic, inelastic and super-elastic

processes, to write the anisotropic collisional operator as,

I1 + J1 = −νeff ne(~r, t)F 1,

or in a kinetic-energy description

I1 + J1 = −γ u σeff(u) f1(u).

25

2.4.2 Isotropic and anisotropic components of the electron of Boltzmann equa-

tion

Using the normalization condition 2.2.1, the isotropic component of the EBE can be obtain by integrating

the equation multiplied by the first Legendre polynomial P0 using the corresponding orthogonality relation

2.3. The anisotropic component can be obtained by integrating the EBE multiplied by P1. Since we will

only use a DC or a HF electric field, we will also separate the stationary/non-stationary anisotropic parts

in each case. The different terms can be grouped as,

Zero order terms

γ4πuCI√

2eume

f(u)+γ4π

(αR u

f10 (u)

3

)+ γ4π

∂

∂u

[u

3

(E0Rf

10 (u) + E1R

f11 (u)

2

)]=

=γ4π∂

∂u

[2me

Mu2 σc(u)

(f(u) +

kB Tge

∂f(u)

∂u

)]+ 4π

[J0(u) + JI(u) + J0

rot(u)]

(2.27)

First order stationary terms

γ4π

3u

CI√2eume

f10 (u) + γ

4π

3αR u f(u) + γ

4π

3u∂

∂u(E0Rf(u)) = −γ 4π

3uσeff (u)f1

0 (u) (2.28)

First order non-stationary terms

γ4π

3u

CI√2eume

+ jωR√

2eume

f11 (u) + γ

4π

3u∂

∂u(E1Rf(u)) = −γ 4π

3uσeff (u)f1

1 (u) (2.29)

The full isotropic Boltzmann equation

We are interested mostly on the isotropic part of the electron distribution function, termed EEDF. We can

get the equation for the EEDF by rewriting equations 2.28 on f10 and 2.29 on f1

1 (u) in order to f(u), and

substituting them in equation 2.27.

First we re-write the anisotropic, Fourier order 1, equation of f11 (u). Using σIeff (u) = σeff (u) + CI√

2eume

we have,

f11 (u)

σIeff (u) + jωR√

2eume

= −∂ (E1Rf(u))

∂u

⇔ f11 (u) = −

σIeff (u)− jωR/√

2eume

σIeff (u)2

+ ω2R/

2eume

∂ (E1Rf(u))

∂u

since we need only the real part

f11 (u) = −

σIeff (u)

σIeff (u)2

+ ω2R/

2eume

E1R∂f(u)

∂u. (2.30)

26

Secondly lets re-write the anisotropic, Fourier order 0, equation of f10 (u),

f10 (u)σIeff (u) = −αRf(u)− E0R

∂f(u)

∂u

⇔ f10 (u) = − αR

σIeff (u)f(u)− E0R

σIeff (u)

∂f(u)

∂u. (2.31)

We now use these results in equation 2.27 to obtain the equation for the EEDF,

−u CI√2eume

f(u) +u3

α2R

σeff (u)f(u) +

u3

αRE0R

σeff (u)

∂f(u)

∂u+

+∂

∂u

[u

3

E0R

σIeff (u)

(αRf(u) + E0R

∂f(u)

∂u

)]+∂

∂u

(u

3E2

1R

σIeff (u)/2

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂u

)+

+∂

∂u

[2me

Mu2 σc(u)

(f(u) +

kB Tge

∂f(u)

∂u

)]+J0(u)

γ+JI(u)

γ+J0rot(u)

γ= 0 (2.32)

Usually, LoKI will be working with either the exponential temporal growth or the exponential spatial

growth of the electron density, and also with either a DC or a HF electric field. So four equations can

be obtained from 2.32. However, a spatial electron density growth is not realistic in a HF field since the

mean electron drift velocity is zero, meaning that an exponential spatial electron density profile, created

by electron-impact ionizations, is not plausible. 3

2.4.3 Temporal growth with DC electric field

−u CI√2eume

f(u) +∂

∂u

(u

3

E20R

σIeff (u)

∂f(u)

∂u

)+∂

∂u

[2me

Mu2 σc(u)

(f(u) +

kB Tge

∂f(u)

∂u

)]+

+J0(u)

γ+JI(u)

γ+J0rot(u)

γ= 0

2.4.4 Temporal growth with HF electric field

−u CI√2eume

f(u) +∂

∂u

(u

3

E21R

2

σIeff (u)

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂u

)+∂

∂u

[2me

Mu2 σc(u)

(f(u) +

kB Tge

∂f(u)

∂u

)]+

+J0(u)

γ+JI(u)

γ+J0rot(u)

γ= 0

2.4.5 Spatial growth with DC electric field

αR3

[u

σeff

(αRf(u)+E0R

∂f(u)

∂u

)+∂

∂u

(uE0R

σefff(u)

)]+∂

∂u

(u

3

E20R

σeff (u)

∂f(u)

∂u

)+

+∂

∂u

[2me

Mu2 σc(u)

(f(u) +

kB Tge

∂f(u)

∂u

)]+J0(u)

γ+JI(u)

γ+J0rot(u)

γ= 0

3 It is important to note that in LoKI, the electric field value is the measured one, not the field’s amplitude, this corresponds to

the effective RMS electric field Eeff1R = E1R√

2. However, during this formulation, the value kept is the field’s amplitude.

27

2.5 Conservation and transport equations

2.5.1 Drift-diffusion equation

The electron drift-diffusion equation is a particle transport equation that describes the various phenom-

ena responsible for electron motion. In gas discharges, usually this equation has a drift term, due to the

effect of the electric field, and a diffusion term, due to the effect of an electron density gradient,

~Γ = −∇r(Dne)− µ0ne ~E,

with the electron flux Γ is defined as,

~Γ = ne~vdrift. (2.33)

With the two-term Boltzmann equation, the electron flux can be calculated from the anisotropic part of

the distribution function

~Γ = ne~vdrift =

∫ ∞0

ne~v~F 1(v) cos θd~v

= ne

∫ ∞0

∫ π

0

∫ 2π

0

v3F 1(v) cos2 θ sin θdθdφdv ~ez

=4πne

3

∫ ∞0

v3F 1(v)dv ~ez

=ne3

√2e

me

∫ ∞0

uf1(u)du ~ez.

Since the electron distribution function was expanded in Fourier series, the electron flux will also be

expanded,

~Γ = ~Γ0 + ~Γ1 ej ω t.

Temporal or spatial growth of the electron density with DC electric field

Since we are in the DC case, we will use f10 (u). Although we will use either the spatial growth, or

temporal growth of the electron density, we will calculate the electron flux for both models simultaneously.

Using equation 2.31, with αR = 1neN

∂ne

∂z ,

ne3

√2e

me

∫ ∞0

uf10 (u)du = −ne

3

√2e

me

∫ ∞0

u

(1

neN

∂ne∂z

1

σIeff (u)f(u) +

E0R

σIeff (u)

∂f(u)

∂u

)du

= − 1

3N

√2e

me

∫ ∞0

u1

σIeff (u)f(u)du

∂ne∂z− ne

1

3N

√2e

me

∫ ∞0

u1

σIeff (u)

∂f(u)

∂uduE0

= −D∇rne + neµ0E0

Thus we have obtained the drift-diffusion equation,

Γ0 = −D∇rne + neµ0E, (2.34)

28

with the following expressions for the electron diffusion coefficient and mobility

D =1

3N

√2e

me

∫ ∞0

u1

σIeff (u)f(u)du (2.35)

µ0 = − 1

3N

√2e

me

∫ ∞0

u1

σIeff (u)

∂f(u)

∂udu. (2.36)

In the case of the exponential spatial growth, the diffusion and mobility coefficients have the same

form, with the quantity σIeff defined as,

σIeff (u) = σeff (u).

In the case of the exponential temporal growth, the drift-diffusion equation is simply

Γ0 = neµ0E,

since there is no electron density gradient. The effect of electron ionization collisions is then included in

the quantity σIeff defined as,

σIeff (u) = σeff (u) + CI

√me

2eu. 4

Temporal growth of the electron density with HF electric field

In the HF case the electric field oscillates with a high frequency ω. This means that the drift velocity will

now have an oscillatory term. In the HF case there is no DC field and as a result the DC component of

the electron drift velocity is zero,

Γ0 = 0.

The time-dependent term for electron flux is,

Γ1 =ne3

√2e

me

∫ ∞0

uf11 (u)du.

Using equation 2.30,

ne3

√2e

me

∫ ∞0

uf11 (u) = −ne

3

√2e

me

∫ ∞0

uσIeff (u)− jωR

√me

2eu

σIeff (u)2

+ ω2R/

2eume

E1R∂f(u)

∂u

= −ne1

3N

√2e

me

∫ ∞0

uσIeff (u)− jωR

√me

2eu

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂uE1

= neµ1E1,

and the drift-diffusion equation is

Γ1 = neµ1E1,

4A usual doubt arises from the fact that σeff already accounts for the ionization cross section and it seems that there is no need

to add the CI term. However, the CI coefficient comes from the electron density growth term, ∂ne∂t

and not from the collisional

operator.

29

with the HF mobility given by

µ1 = µr + j µi,

µr = − 1

3N

√2e

me

∫ ∞0

uσIeff (u)

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂u

µi =1

3N

√2e

me

∫ ∞0

uωR√

me

2eu

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂u.

Again, the effect of the temporal growth due to ionization collisions appears on the σIeff quantity.

2.5.2 Particle balance equation

In this subsection the calculation of particle balance will be made by calculating the zeroth velocity

moment of the EBE.

Looking back to equation 2.32 we will integrate it, term by term, in reading order.

Time variation term

−∫ ∞

0

uCI

√me

2 e uf(u)du = −CI

√me

2 e

∫ ∞0

f(u)√udu = −CI

√me

2 e,

in which we have used the normalization condition 2.2.1.

Space variation term ∫ ∞0

αR3

[u

σeff

(αRf(u) + E0R

∂f(u)

∂u

)]du

The first part of this integral gives,∫ ∞0

αR3

u

σeffαRf(u)du =

αRne

1

3N

∫ ∞0

u

σeff

∂ne∂z

f(u)du =

=αRne

√me

2 eD∇rne,

on which we have used 2.35. The second part of the integral gives,∫ ∞0

αR3

u

σeffE0R

∂f(u)

∂udu = αR

1

3N

∫ ∞0

u

σeff

∂f(u)

∂uduE0 =

= −αRne

√me

2 eneµ0E0,

on which we have used 2.36. So the integration of the spatial variation terms is

αRne

√me

2 eD∇rne −

αRne

√me

2 eneµ0E0 =

√me

2 e

αRne

(−nevdrift) = −αRvdrift√me

2 e.

Electric field term

.

Electron density exponential temporal growth with DC field∫ ∞0

∂

∂u

(u

3

E20R

σIeff (u)

∂f(u)

∂u

)du =

[u

3

E20R

σIeff (u)

∂f(u)

∂u

]∞0

= 0

30

Electron density exponential spatial growth with DC field

∫ ∞0

∂

∂u

[uE0R

3σeff

(αRf(u) + E0R

∂f(u)

∂u

)]du =

[uE0R

3σeff

(αRf(u) + E0R

∂f(u)

∂u

)]∞0

= 0

Electron density exponential temporal growth with HF field∫ ∞0

∂

∂u

(u

3E2R

σIeff (u)/2

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂u

)du =

[u

3E2R

σIeff (u)/2

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂u

]∞0

= 0


∫ ∞0

∂

∂u

[2me

Mu2 σc(u)

(f(u) +

kB Tge

∂f(u)

∂u

)]du =

[2me

Mu2 σc(u)

(f(u) +

kB Tge

∂f(u)

∂u

)]∞0

= 0


Using the operator before applying the Klein-Rosseland relation, that is, with explicit superelastic cross

sections,

∫ ∞0

J0

γdu =

∫ ∞0

∑i,j

1st term︷︸︸︷

δi(u+ uij

)σij(u+ uij) f(u+ uij)−

2ndterm︷︸︸︷δiu σij(u) f(u)

du+

+

∫ ∞0

∑i,j

3rdterm︷︸︸︷

δj(u− uij

)σji(u− uij) f(u− uij)−

4thterm︷︸︸︷δju σji(u) f(u)

du .

First term ∫ ∞0

δi(u+ uij

)σij(u+ uij) f(u+ uij) du, ε = u+ uij and dε = du

=

∫ ∞uij

δiε σij(ε) f(ε) dε =

∫ ∞0

δiε σij(ε) f(ε) dε−∫ uij

0

δiε*0

σij(ε) f(ε) dε︸︷︷︸σij(u)=0, if u<uij

=

=δi

√me

2 e

〈νij(u)〉N

= δi

√me

2 eCij

Second term

−∫ ∞

0

δiu σij(u) f(u)du = −δi√me

2 e

〈νij(u)〉N

= −δi√me

2 eCij

Third term∫ ∞0

δj(u− uij

)σji(u− uii) f(u− uij) du, ε = u− uij and dε = du

=

∫ ∞−uij

δjε σji(ε) f(ε) dε =

∫ ∞0

δjε σji(ε) f(ε) dε+

∫ 0

−uij

δjε*0

σji(ε) f(ε) dε︸︷︷︸σij(u)=0, if u<uij

=

=δj

√me

2 e

〈νji(u)〉N

= δj

√me

2 eCji

31

Fourth term

−∫ ∞

0

δju σji(u) f(u)du = −δj√me

2 e

〈νji(u)〉N

= −δj√me

2 eCji

In the end we have, ∫ ∞0

J0

γ(u)du = 0

Ionization collision term∫ ∞0

JIγdu =

∫ ∞0

∫ ∞2u+VI

εqisec(ε, u)f(ε)dε+

∫ 2u+VI

u+VI

εqisca(ε, u)f(ε)dε− uσI(u)f(u)

du

First term∫ ∞0

∫ ∞2u+VI

εqisec(ε, u)f(ε)dεdu =

∫ ∞0

∫ (ε−VI)/2

0

εqisec(ε, u)f(ε)dudε =

∫ ∞0

εσI(ε)f(ε)dε =

√me

2 e

〈νI〉N

,

in which we have used relation 2.11. Second term∫ ∞0

∫ 2u+vI

u+VI

εqisca(ε, u)f(ε)dεdu =

∫ ∞0

∫ ε−VI

(ε−VI)/2

εqisca(ε, u)f(ε)dudε, u′ = ε− VI − u

=

∫ ∞0

∫ 0

(ε−VI)/2

εqisca(ε, ε− VI − u′)f(ε)(−du′)dε =

∫ ∞0

∫ (ε−VI)/2

0

εqisec(ε, u′)f(ε)du′dε =

=

∫ ∞0

εσI(ε)f(ε)dε =

√me

2 e

〈νI〉N

,

in which we have used relation 2.11. Third term∫ ∞0

−uσI(u)f(u)du = −√me

2e

〈νI〉N

.

The sum of all terms is then, ∫ ∞0

JI(u)

γdu =

√me

2e

〈νI〉N

Particle balance equation

The particle balance equation for an exponential temporal growth of the electron density is

CI + 0 =〈νI〉N

⇔ ∂ne∂t

+∂Γ

∂z= S,

in which it is possible to see that since the time variation of the electron density is equal to the ionization

rate (source term).

The particle balance equation for an exponential spatial growth of the electron density is

0 + αRvdrift =〈νI〉N

⇔ ∂ne∂t

+∂Γ

∂z= S.

Here the spatial variation of the electron flux is equal to the ionization source term. The number of

particles is also conserved, since the reduced Townsend ionization coefficient is defined as

α =〈νI〉vdrift

.

32

2.5.3 Energy balance equation

In this subsection the calculation of the energy balance equation will be made by calculating the second

velocity moment of the EBE, i.e. by multiplying the EBE by u and integrating in all energies.

This calculation is helpful in understanding the different sources of energy sharing. In LoKI, the

power balance equation is used to check the quality of the numerical calculations.

The following will be obtained in terms of the energy density and the energy flux,

nE = ne

∫ ∞0

uf(u)√udu = ne 〈u〉

ΓE =

√2 e

me

ne3

∫ ∞0

u2f10 (u)du,

writing the energy flux in terms of its diffusion and mobility components

ΓE = −DE∇rnE + nEµEE0 (2.37)

DE =

√2 e

me

1

3N

1

〈u〉

∫ ∞0

u2

σefff(u)du

µE = −√

2 e

me

1

3N

1

〈u〉

∫ ∞0

u2

σeff

∂

∂uf(u)du

This representation of the energy flux is somewhat unusual. Some authors prefer this formulation

due to its consistency with the two-term EBE and the fact that 2.37 uses transport parameters that can

be calculated from integration over the EEDF [16].

Time variation term

ne

∫ ∞0

u−CI√

2eume

f(u)u du = −ne CI√me

2 e

∫ ∞0

u f(u)√udu = −CI

√me

2 enE

Space variation term

ne

∫ ∞0

αR3

[u2

σeff

(αRf(u) + E0R

∂f(u)

∂u

)]du =

=αR3N

(∫ ∞0

u2

σefff(u)du

)∂ne∂z

+αR3N

ne

(∫ ∞0

u2

σeff

∂f(u)

∂udu

)E0

=αR

√me

2 e(DE∇rne 〈u〉 − ne 〈u〉µEE0)

=αR

√me

2 e(DE∇rnE − nEµEE0)

=αR

√me

2 e(−ΓE)

33

Electric field term

Electron density exponential temporal growth with DC field

ne

∫ ∞0

u∂

∂u

(u

3

E20R

σIeff (u)

∂f(u)

∂u

)du = − ne

3N

∫ ∞0

u

σIeff

∂f(u)

∂uduE0E0R

= +

√me

2 eneµ0E0E0R

=

√me

2 e(Γ)E0R

=

√me

2 ene vdriftE0R

Electron density exponential temporal growth with HF field,

ne

∫ ∞0

u∂

∂u

(u

3E2

1R

σIeff (u)/2

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂u

)du =− ne

3N

∫ ∞0

uσIeff (u)/2

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂uduE1

E1R

2

=

√me

2 eneµrE1

E1R

2

=

√me

2 e< (Γ1)

E1R

2

=

√me

2 ene<(vdrift)

E1R

2.

Electron density exponential spatial growth with DC field

ne

∫ ∞0

u∂

∂u

[uE0R

3σeff

(αRf(u) + E0R

∂f(u)

∂u

)]du =

=−1

3N

[∫ ∞0

u

σefff(u)du

∂ne∂z

+ ne

∫ ∞0

u

σeff

∂f(u)

∂uduE0

]E0R

=−√me

2 e(D∇rne − neµ0E0)E0R

=

√me

2 enevdriftE0R

In all of these cases, the power density (in eV cm−3s−1), gained from the electric field, can be written,

apart from a constant, as

PE = ~J · ~E = −~Γ · ~E = ΓE.

which corresponds to a Joule heating term.


ne

∫ ∞0

u∂

∂u

[2me

Mu2σc(u)

(f(u) +

kBTge

∂f(u)

∂u

)]du = −ne

2me

M

∫ ∞0

u2σc(u)

(f(u) +

kBTge

∂f(u)

∂u

)du

A suggestion to have a more intuitive insight of this power is to assume a Maxwellian distribution at

the electron temperature Te. Then the Chapmann and Cowling term is of the order −Tg/Te, and the

power lost in elastic collisions can be written approximately as [29],

−ne2me

M

√me

2 e〈u〉Cc

(1− Tg

Te

),

34

with Cc the rate coefficient of elastic collisions. As expected, the power lost to elastic collisions is zero if

Te = Tg.


The inelastic/superelastic collision term is composed by 4 terms and the the power is written for each

term separately.

First term∫ ∞0

δiu(u+ uij

)σij(u+ uij)f(u+ uij)du = , using ε = u+ uij

= δi

∫ ∞uij

(ε− uij

)εσij(ε)f(ε) dε = δi

∫ ∞0

(ε− uij

)εσij(ε)f(ε) dε− δi

∫ uij

0

(ε− uij

)ε

*0σij(ε)f(ε) dε︸︷︷︸

σij(u)=0, if u<uij

=

= δi

∫ ∞0

ε2σij(ε)f(u) dε− δi∫ ∞

0

uijεσij(ε)f(ε) dε =

= δi

√me

2 e

∫ ∞0

εCij(ε)f(ε)√ε dε− δi

√me

2 euij

∫ ∞0

Cij(ε)f(ε)√ε dε =

= δi

√me

2 e

[〈εCij(ε)〉 − uijCij

].

Second term

−∫ ∞

0

δiu2σij(u)f(u)du = −δi

√me

2 e〈uCij(u)〉 .

Analogous calculations could be made for the third and fourth term. In the end, summing all four

terms we have,

ne

∫ ∞0

uJ0(u)

γdu = ne

√me

2 e

∑ij

uij ( δjCji − δiCij )

This can be seen as the power gained and the power lost due to superelastic and inelastic collisions,

respectively.

Ionization collision term

ne

∫ ∞0

JI(u)

γdu = ne

∫ ∞0

u

∫ ∞2u+VI

εqisec(ε, u)f(ε)dε+

∫ 2u+VI

u+VI

εqisca(ε, u)f(ε)dε− uσI(u)f(u)

du

Starting with the second term,∫ ∞0

∫ 2u+vI

u+VI

uεqisca(ε, u)f(ε)dεdu =

∫ ∞0

∫ ε−VI

(ε−VI)/2

u ε qisca(ε, u)f(ε)dudε, u′ = ε− VI − u∫ ∞0

∫ 0

(ε−VI)/2

(ε− VI − u′)εqisca(ε, ε− VI − u′)f(ε)(−du′)dε =

∫ ∞0

∫ (ε−VI)/2

0

(ε− VI − u′)εqisec(ε, u′)f(ε)du′dε =

=

∫ ∞0

∫ ∞2u′+VI

(ε− VI − u′)εqisec(ε, u′)f(ε)dεdu′.

35

Now summing the first two terms we have,∫ ∞0

u

∫ ∞2u+VI

εqisec(ε, u)f(ε)dεdu+

∫ ∞0

u

∫ 2u+VI

u+VI

εqisca(ε, u)f(ε)dεdu =

=

∫ ∞0

u

∫ ∞2u+VI

εqisec(ε, u)f(ε)dεdu+

∫ ∞0

∫ ∞2u′+VI

(ε− VI − u′)εqisec(ε, u′)f(ε)dεdu′ =

=

∫ ∞0

∫ ∞2u′+VI

(ε− VI)εqisec(ε, u′)f(ε)dεdu′ =

∫ ∞0

∫ (ε−VI)/2

0

(ε− VI)εqisec(ε, u′)f(ε)du′dε

=

∫ ∞0

(ε− VI) ε σI(ε)f(ε)dε,

in which we have used relation 2.11. Finally, adding the third term,

ne

∫ ∞0

uJI(u)

γdu = ne

∫ ∞0

(ε− VI) ε σI(ε)f(ε)dε− ne∫ ∞

0

u2σI(u)f(u)du =

= −neVI∫ ∞

0

ε σI(ε)f(ε)dε

= −ne VI√me

2 eCI

Energy balance equation

For the exponential temporal growth model with DC field, the energy balance equation is,

CInE + 0− ne vdriftE0R = −ne2me

M

∫ ∞0

u2σc(u)

(f(u) +

kBTge

∂f(u)

∂u

)du+

+ne∑ij

uij (njCji − niCij )− ne VICI ,

which is similar to an energy transport equation

∂nE∂t

+∂ΓE∂z︸︷︷︸

here is zero

−ΓE0 = −Pel + Psup − Pinel − PI

with Γ the electron particle flux, ΓE the electron energy flux, Pel the power lost in elastic collisions, Psup

and Pinel the power gained and lost in superelastic/inelastic collisions, respectively, and PI the power

lost in electron-impact ionizations.

The energy balance equation for the case of the exponential temporal growth model with a HF

field is very similar, except that here we take into account the real part of the electron drift velocity

<(vdrift) = <(>

0v0 + v1e

jωt

)= v1 cos(ωt).

For the exponential spatial growth model the energy balance equation is,

0 + αRΓE − ne vdriftE0R = −ne2me

M

∫ ∞0

u2σc(u)

(f(u) +

kBTge

∂f(u)

∂u

)du+

+ne∑ij

uij (njCji − niCij )− ne VICI ,

which is similar to an energy transport equation,

∂nE∂t︸︷︷︸

here is zero

+∂ΓE∂z− ΓE0 = −Pel + Psup − Pinel − PI

36

Chapter 3

Computational Approach

3.1 Solving the Boltzmann equation

In LoKI the purpose is to solve equation 2.32, in order to obtain the EEDF, and then to calculate the

transport parameters and the rate coefficients needed on the Chemistry module.

Equation 2.32 can be written, detailing its functional dependence on the kinetic energy u, through f

and its derivatives

B

(u, f(u),

df(u)

du,d2f(u)

du2

)= 0.

In low-temperature plasmas, the probability of having electrons with energy much higher than the mean

electron energy is low. In consequence, the value of the EEDF at these energies will also be very low,

and we can calculate the electron coefficients using EEDF values between a certain range of electron

energies, with negligible errors.

Within this range it is then possible to define a discrete 1D energy-grid, with say N small energy

intervals, where the discretized equations is converted in a set of N equations

B1

(f1,

df1

du,d2f1

du2

)= 0

B2

(f2,

df2

du,d2f2

du2

)= 0

(...)

Bk

(fk,

dfkdu

,d2fkdu2

)= 0

(...)

BN−1

(fN−1,

dfN−1

du,d2fN−1

du2

)= 0

BN

(fN ,

dfNdu

,d2fNdu2

)= 0.

However there are still 3N unknown variables (f , dfdu and d2fdu2 ) on these N equations. In order to have

a solvable system, it is necessary to linearise the EEDF derivatives, for example by approximating them

37

with finite differences. This will lead to a system of equations Bk equations that are a function of the

EEDF at the different sites of the grid

Bk [(...), fk−2, fk−1, fk, fk+1, fk=2, (...)] = 0.

The result is a homogeneous system of N equations with N unknown variables, for which only the

solution that follows the normalization condition, equation 2.8, is the physical one. If the Bk terms are

linear in f , then the full system to be numerically solved is,

∑m

Bk,mfm = 0

∑k

fk√uk = 1.

The matrix B will be called EBE matrix throughout this work. Solving this system is the discrete

equivalent of solving equation 2.32.

If all EBE terms are linear on the EEDF, it is possible to add the normalization condition to a B matrix

line, rendering the system well defined and solvable. In LoKI, the normalization condition is added to the

first line of the matrix, and the system to solve becomes

B1,1 +√u1 B1,2 +

√u2 B1,3 +

√u3 · · · B1,N +

√uN

B2,1 B2,2 B2,3 · · · B2,N

B3,1 B3,2 B3,3 · · · B2,N

......

.... . .

...

BN,1 BN,2 BN,2 · · · BN,N

·

f1

f2

f3

...

fN

=

1

0

0

...

0

.

However, if secondary electrons are included, someBkm terms are non-linear on the EEDF. This non-

linear system needs to be linearised and then iterated until convergence on the non-linear coefficients

and the EEDF.

3.2 Discretization of the electron Boltzmann equation

In order to numerically solve the EBE a discretization of its different terms needs to be made.

These terms can be classified as linear terms on the EEDF, terms with integrals on the EEDF, and

terms that involve EEDF derivatives. Linear terms can be discretized with a conversion rule that makes

the connection between an energy on the continuous plane and on the discrete grid. Terms that involve

integrals can be discretized with a quadrature rule, here the simple mid-point (or rectangle) rule will

be used. Finally, terms with EEDF derivatives will be discretized using a finite difference method, here

centred differences will be used.

38

This discretization will be made using an energy grid that ranges from zero to a maximum energy

umax, divided in N points. The energy step is defined as ∆u = umax/N , with the energy at each point k

defined as u+k = k∆u. Quantities that are a function of the electron energy are defined at the interval’s

limit, for example, the cross-section σ+effK

= σeff (u+k ). The EEDF will be defined at the middle of the

interval

fk = f(uk)

in which uk =(k − 1

2

)∆u. Thus, the quadrature rule and finite difference method are applied with the

goal of arriving at expressions with the EEDF defined at the middle of the energy interval.

The details of the calculations performed in this subsection are given on the Annex A.

3.2.1 Linear terms on the electron energy distribution function

Linear terms are discretized directly by writing them at the middle of the energy interval. These terms are

the electron-density-growth terms, both spatial and temporal, and the conservative inelastic operators.

Time variation term

The electron density exponential temporal growth term is, for an energy uk = (k − 1/2)∆u,u CI√2 e ume

f(u)

u=uk

≡√m

2eCI fk

√uk.

Space variation term

One of the three terms that constitute the electron density exponential spatial growth term is linear on

the EEDF. It can be discretized for an energy uk,[α2R

3

u

σeff (u)f(u)

]u=uk

≡ α2R

3

ukσeffK

fk

Conservative inelastic/superelastic collisions term

For the conservative inelastic collisional operator, the discretization is[J0(u)

γ

]u=uk

≡∑i,j

[δi uk+mij σ

ijk+mij fk+mij − δi uk σijk fk +

gigjδj uk σ

ijk fk−mij −

gigjδj uk+mij σ

ijk+mij fk

]

in which mij defined as mij = floor(uij

∆u

)the lowest closer number of energy intervals that cor-

respond to the energy level difference (uij) of the excitation/de-excitation of the inelastic/superelastic

collisions.

39

3.2.2 Terms with integrals on the electron energy distribution function

Using the mid-point quadrature rule the discrete ionization operator for a specific interval of energy

u = uk,[JI(u)

γ

]u=uk

≡n∑

j=2k+MI+1

ujqisec(uj , uk)fj∆u+

2k+MI∑j=k+MI+1

ujqisec(uj , uj−k−MI

)fj∆u−ukfk(k−MI)/2∑

j=1

qisec(uk, uj)∆u.

in which MI is defined as MI = round(VI

∆u

), the closest number of energy intervals corresponding to

the ionization energy.

3.2.3 Terms with derivatives on the electron energy distribution function

For these terms a centred finite differences method will be used. There are many variations of this

method, our set of rules being described in Annex A.

Electron density exponential spatial growth terms

There are two of the three electron density exponential spatial growth terms that include EEDF’s deriva-

tive. The discretization for an energy interval u = uk is,[αR3

u

σeffE0R

∂f(u)

∂u

]u=uk

=αR3

ukσeffK ∆u

E0R

(fk+1 − fk−1

2

),

[αR3

∂

∂u

(u

σeffE0R f(u)

)]u=uk

=αRE0R

6

[fk+1

k

σ+effK

+ fk

(k

σ+effK

− k − 1

σ+effK−1

)− fk−1

k − 1

σ+effK−1

].

Rotational collisions term - continuous approximation

We will treat separately the two terms that compose the rotational continuous approximation collision

operator, ∂

∂u[4Bσ0uf(u)]

u=uk

≡ 2Bσ0 (fk+1k + fk − fk−1(k − 1)) ,

∂

∂u

[4Bσ0u

kBTge

∂f(u)

∂u

]u=uk

≡ 4Bσ0kBTge∆u

[fk+1k − fk(2k − 1) + fk−1(k − 1)] .

Elastic collisions term

The elastic collision operator can also be divided in two terms, with first order and second-order deriva-

tives, respectively∂

∂u

[2me

Mu2σc(u)f(u)

]u=uk

≡ me

M

fk+1k

2σ+c k∆u+ fk

[k2σ+

c k − (k − 1)2σ+c k−1

]∆u− fk−1(k − 1)2σ+

c k−1∆u,

∂

∂u

[2me

Mu2σc(u)

kBTge

∂f(u)

∂u

]u=uk

≡ 2me

M

kBTge

[fk+1k

2σ+ck − fk

(k2σ+

ck + (k − 1)2σ+c k−1

)+ fk−1(k − 1)2σ+

c k−1

].

40

Electric field term

The electric field term can be discretized for an energy u = uk,

∂

∂u

((E

N

)2u

3g(u)

∂f

∂u

)=

(E

N

)21

3∆u

[fk+1 k g

+k − fk(k g+

k + (k − 1) g+k−1) + fk−1 (k − 1) g+

k−1

],

with g+k being

g+k =

1CI√me√

2ek∆u+ σ+

eff k

× 1

1 +ω2

Rme

(CI√me+

√2ek∆u σ+

effk)2

.

Matrix form for the terms with EEDF derivatives

We can rearrange the terms with EEDF derivatives, with the exception of the here presented first term

of the spatial variation, to get a set of expressions on fk, fk−1 and fk+1,

Ak−1 fk−1 − (Ak +Bk) fk +Bk+1 fk+1,

with

Ak =

(E

N

)2k

3∆ug+k +

me

Mk2σ+

c k

[2kBTge−∆u

]+ 2Bσ0

[2kBTge

k

∆u− k]

+αRE0R

6

k

σeff k

Bk =

(E

N

)2k − 1

3∆ug+k−1 +

me

M(k − 1)2σ+

c k−1

[2kBTge

+ ∆u

]+ 2Bσ0

[2kBTge

k − 1

∆u+ (k − 1)

]+αRE0R

6

k − 1

σeff (k−1)

3.3 Solving the non-linear electron Boltzmann equation

On the previous subsection we presented the discretization of the EBE, with some terms involving CI

and α. These terms are calculated using the EEDF, hence they are non-linear. This means that they

have to be explicitly calculated, to then be used when writing the EBE matrix. This needs to be done

iteratively until CI and α convergence is achieved. Each of the electron-density growth mechanisms is

linearized in a different way, although the iterative algorithms are similar.

3.3.1 Convergence over the ionization rate or first Townsend coefficients

Using the exponential temporal growth model as example, the non-linear EBE to be solved is the dis-

cretized version of,

−u CI√2eume

f(u) +∂

∂u

(u

3E2

1R

σIeff (u)/2

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂u

)+I0(u)

γ+J0(u)

γ+JI(u)

γ+J0rot(u)

γ= 0.

In this equation, the ionization rate coefficient appears both on the time derivative and on the electric

field term through σIeff . By knowing CI we can solve this equation as a linear system and obtain the

EEDF. However the ionization rate coefficient is calculated as,

CI =

√2 e

me

∫ ∞0

uσI(u)f(u)du,

41

meaning that it is necessary to know the EEDF to calculate this coefficient.

The system is then non-linear. In order to solve it, an iterative method is necessary, with CI playing

the role of convergence parameter, being calculated in successive iterations until convergence.

3.3.2 Iterative algorithm

Instead of using a numerical method for non-linear algebraic systems, a simple convergence cycle was

used. Initially the EEDF is calculated assuming no electron-density growth model. With this first estimate

of the EEDF, the first Townsend coefficient, or the ionization rate coefficient are calculated. A new EBE

matrix is then constructed with terms that depend on these coefficients. Solving the resulting equations

system as a linear system, a new estimate of the EEDF is obtained.

The cycle can be described by:

1. An EEDF is calculated without including secondary electrons;

2. Using the previous EEDF as an initial guess, the convergence parameter is calculated. The conver-

gence parameters is the ionization rate coefficient (temporal growth model) or the first Townsend

ionization coefficient (spatial growth model);

3. Using the previous value of the convergence parameters, the EBE matrix is constructed now in-

cluding the production of secondary electrons due to ionization events;

4. The ionization routine cycle commences with a new calculation of the EEDF;

5. The new EEDF is used to update the convergence parameters;

6. The convergence criteria are checked. If convergence is not achieved the cycle continues back to

step 3.

Once the convergence criteria are met, usually corresponding to relative differences smaller than

10−10 for the convergence parameters and the EEDF, the cycle is terminated. A flow chart of this

iterative algorithm can bee seen in Figure 3.1. This simple method is stable for most working conditions.

3.3.3 Coupling with electron-electron collisions

The electron density growth due to electron-impact ionizations is not the only non-linear mechanism

included on the EBE. In particular, LoKI-B has also a routine for electron-electron collisions, in which

some of the coefficients on the electron-electron collisional operator (Jee) contain integrals over the

EEDF. When the different non-linear mechanism are activated simultaneously, we need to find an ade-

quate numerical algorithm that ensures convergence.

In order to couple two non-linear routines it is important to take into account some aspects. First it is

important not to try to converge too many parameters at the same time, since their values might start tp

oscillate between iterations. Second, there must be some communication between the routines so that

they both converge to the same solution.

The coupling was made as follows:

42

1. The EEDF is calculated without accounting for secondary electrons;

2. The ionization routine is performed without accounting for electron-electron collisions;

3. After the calculation of the ionization convergence parameters, the electron-electron collisions

routine is performed;

• Inside this routine, the terms due to electron-impact ionization are included but they are not

updated between iterations;

• The ionization convergence parameter is calculated at the end of each cycle;

4. After convergence on the electron-electron collisions routine, the convergence criteria are applied

to the ionization convergence parameters;

5. If the previous criteria are not satisfied, the ionization routine is performed again now including, but

not updating, electron-electron collisions terms;

6. The global cycle is repeated continuing from step 3.

A graphical description of this iterative algorithm can be seen in Figure 3.2.

The e-e collisions routine converges if the ratio of electric field’s power to electron-electron collision’s

power is bigger than 107. This global convergence can be hard to achieve with:

• Very-high frequency values:

• Large numbers of energy grid-points (∼5000 more);

• Very high ionization degree;

• Spatial electron density growth model.

It is sometimes hard to predict if a certain set of conditions will allow the convergence.

Electron density temporal growth model

In the case of the electron density temporal growth model, the EBE to be solved is the discretized version

of

−u CI√2eume

f(u)+∂

∂u

(u

3

E21R

2

σIeff (u)

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂u

)+I0(u)

γ+Jrot(u)

γ+J0(u)

γ+JI(u)

γ+Jee(u)

γ= 0,

for the HF case, and an equivalent form for the DC case by setting ω = 0 and E1R/2 → E0R. In

solving this equation, the rate coefficients CI will be used as convergence parameter. The ionization

rate coefficient appears on the term due to the time dependence, and on the electric field term since

σIeff (u) = σeff (u) +CI√

2eume

,

and both terms need to be updated at each iteration, following changes in CI .

43

After the calculation of the initial EEDF estimate, in which secondary electrons are not accounted

for, the program proceeds to calculate the ionization rate coefficient. By definition the ionization rate

coefficient is,

CI =

√2 e

me

∫ ∞0

uσI(u)f(u)du.

Initially the calculation of the ionization rate coefficient was done with this expression. However, there

were some particle balance errors and significant power balance errors when a final solution was ob-

tained. At the time the power balance errors seemed to be proportional to the energy step, which

indicated that it may be due to some discretization error. Indeed an error introduced by the discretization

was identified on the ionization collisional operator when doing analytical verifications of the discretized

EBE. In order to obtain a better correspondence between CI and the ionization operator, the calculation

of CI was made by integrating the corresponding operator over the energy grid,

CI =

√2 e

me

n∑k=1

JI

∣∣∣k∆u. (3.1)

This procedure ensures a perfect verification of the numerical particle balance equation, and minimizes

the relative error (keeping it bellow 10−12) in the calculation of the numerical energy balance equations.

Since the initial EEDF estimate is done without accounting for secondary electrons, there can be

large differences between the initial and the final values of CI . A large difference between consecutive

coefficients, may cause the EEDF or CI to overshoot (or oscillate), in a way that convergence is difficult

or impossible to obtain. In order to ensure that the convergence parameter converges slowly, a mixing

of solutions was used. Let C0I be the value used on the EBE matrix on the previous iteration, and CxI its

current value calculated with equation 3.1. The CI value to be used, when writing the next EBE matrix,

is

CI = x CxI + (1− x) C0I ,

with x ∈ [0; 1] being the mixing parameter. Usually, we have adopted x = 0.7 as the mixing parameter

for the temporal electron density growth.

Having a new CI estimate, the temporal growth term and the electric field term can be updated. A

new EEDF is calculated, a new value is assigned to C0I , a new calculation of CxI is performed, and the

next CI is calculated with mixing of solutions. The process goes on until convergence.

Convergence tests are performed after the calculation of the new CI value and before updating the

EBE matrix. Since CI works as a convergence parameter, the first test is on the relative difference

between the previous and the current ionization rate coefficient,

||CI − C0I ||

C0I

< 10−10.

If this criterion is met, a second test is performed upon the EEDF, calculating the maximum relative

difference between corresponding values of the consecutive EEDFs,

max

(||fnew − fold||

fold

)< 10−10.

44

Electron density spatial growth model

With exponential spatial growth model for the electron density, the EBE to be solved is the discretized

form of

αR3

[u

σeff

(αRf(u) + E0R

∂f(u)

∂u

)+

∂

∂u

(uE0R

σefff(u)

)]+

∂

∂u

(u

3

E20R

σeff (u)

∂f(u)

∂u

)+

+I0(u)

γ+Jrot(u)

γ+J0(u)

γ+JI(u)

γ+Jee(u)

γ= 0.

Here, the convergence parameter is the reduced first Townsend ionization coefficient, of which calcula-

tion is not as direct as for the ionization rate coefficient.

The first Townsend coefficient α was defined in this work, see sec 2.3.4, as

α =∇rnene

· ~ez.

corresponding to a spatial growth frequency. The reduced coefficient relates to the net electron produc-

tion as [16]

αR =CIvdrift

, (3.2)

in which the ionization rate coefficient is calculated with the same expression 3.1 as in the temporal

growth model case, and the electron drift velocity is calculated using equations 2.34 and 2.33. Writing

this drift-diffusion equation in respect to the drift velocity we have,

vdrift = −N DαR + µ0E0,

which leads to a quadratic equation on αR,

α2RND − αRµ0E0 + CI = 0,

with solution

αR =µ0E0 −

√(µ0E0)

2 − 4NDCI

2ND, (3.3)

in which µ0 and D are calculated using

D =

√2 e

me

1

3N

n∑k=1

ukσeff k

fk∆u;

µ0 = −√me

2 e

1

3N

n∑k=1

ukσeff k

fk+1 − fk−1

21.

On the numerical code the solution corresponding to the smaller value of αR (minus sign) is used.

The larger root was also tested but either the convergence wasn’t reached or the EEDF solution was not

physically relevant (yielding negative EEDF’s values).1 At initial stages of the numerical code implementation, the calculation of the electron mobility in Argon was sensible to

variations of the energy step . For some ∆u values, the mobility was underestimated, leading to negative or complex values of

α and preventing convergence. We think that this was caused by a wrong user defined energy step. The elastic (or momentum

transfer) cross section of noble gases has a minimum (between 0.1 eV and 1eV) due to the Ramsauer-Townsend effect. This

minimum affects the mobility calculation through the effective cross section σeff . If the energy step is not small enough to

describe this minimum, the calculation of the derivative with finite differences may not be good enough.

45

With the spatial growth model a mixing of solutions for the convergence parameter αR was used.

The new αxR is calculated with equation 3.3. Then this solution is mixed with the ionization coefficient,

α0R, obtained in the previous cycle according to

αR = x · αxR + (1− x) · α0R.

The new αR is used to update the EBE matrix terms, and a new EEDF is calculated, resulting in new

CI , D and µ0 estimates, which lead to the next calculation of αxR and αR. The cycle goes on until

convergence. The calculation of the ionization coefficient through equation 3.3 is delicate, since big

changes on the EEDF may lead to negative αxR values that jeopardize the convergence. In order to

ensure a slow convergence, a small mixing parameter should be used. Usually the mixing parameter for

this growth model is x = 0.5.

The convergence tests for the spatial growth model are very similar to the temporal growth’s. The

tests are performed after the calculation of the new αR and before updating the EBE matrix. Here αR is

the convergence parameter, hence the first convergence test is

||αR − α0R||

α0R

< 10−12.

A second test is performed upon the EEDF corresponding values of two consecutive EEDFs calculating

the maximum relative differences between

max

(||fnew − fold||

fold

)< 10−10.

3.4 Numerical verification of the conservation equations

The numerical code LoKI performs a numerical verification of the conservation equations, using the

discretized form of the EBE, to control the quality of the calculations. In particular,

• the calculation of the particle balance equation is used to identify possible discretization errors;

• the calculation of the power balance equation is used to monitor the error in solving the EBE and

obtaining the EEDF.

As mentioned before (see sec 2.5.2 and 2.5.3), the particle and the energy balance equations are

obtained by multiplying the discretized form of the isotropic EBE 2.32, by 1 and u, respectively, and by

integrating the resulting matrix equation over all energy intervals. Taking, as example, the EBE for the

HF temporal growth case, one has for the particle balance and the energy balance equations

∑k

γk

−u CI√2eume

f(u)+∂

∂u

(u

3

E21R

2

σIeff (u)

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂u

)+I0(u)

γ+Jrot(u)

γ+J0(u)

γ+JI(u)

γ

√uk∆u = 0

∑k

γk

−u CI√2eume

f(u)+∂

∂u

(u

3

E21R

2

σIeff (u)

σIeff (u)2

+ ω2R/

2eume

∂f(u)

∂u

)+I0(u)

γ+Jrot(u)

γ+J0(u)

γ+JI(u)

γ

(uk)3/2∆u = 0.

46

3.4.1 Particle balance

The calculations are detailed in Annex B. Here we will only show the results for each term of the discrete

electron Boltzmann equation.

After calculations we have the following contributions:

• The exponential temporal growth term =-√

m2eCI ;

• The exponential spacial growth terms =-√

m2eCI ;

• The continuous terms = 0;

• The conservative inelastic collision terms = 0;

• The ionization collisional operator = 〈νI(u)〉N

√m2e +O(∆u).

The ionization collisional operator introduces an error of the order of the energy step, given by

O(∆u) =

(n−MI)2∑k=1

u2k+MIqisec(u2k+MI

, uk)f2k+MI∆u2,

The particle balance equation can be written, for either growth model, as

−√m

2eCI +

〈νI(u)〉N

√m

2e+O(∆u) ≈ 0.

This equation confirms that the discretized EBE preserves particle conservation, if we ignore the term

of the order of the energy grid-step.

3.4.2 Energy balance

The calculations are detailed in Annex C. Here we will only show the results for each term of the discrete

EBE.

After calculations we have the following contributions:

• The exponential temporal growth term = −√

m2eCI 〈u〉 ;

• The exponential spacial growth term = −αR

ne

√me

2 e ΓE ;

• The continuous terms =∑∞k=1(Ak −Bk)fk∆u;

• The conservative inelastic/superelastic collisions terms =√

me

2 e

∑i,jmij

(δj〈νji〉N − δi 〈νij〉N

)∆u;

• The ionization collisional operator =(MI + 1

2

)∆u√

me

2 e〈νI〉N +O(∆u).

Again, the ionization collisional operator introduces an error of the order of ∆u,

O(∆u) =

(n−MI)2∑k=1

uku2k+MIqisec(u2k+MI

, uk)f2k+MI∆u2,

47

In principle, the power lost in ionizing collisions (PI ), should be given by an similar to those of the other

inelastic collisions. However, due to the discretization of the ionization collisional operator, the correct

expression for the power lost in ionizations is

PI =

(MI +

1

2

)∆u

√me

2 e

〈νI〉N

.

in which an extra shift of ∆u/2 is introduced.

The final expression for the case of exponential temporal growth of the electron density is,

CI 〈u〉+n∑k=1

(Ak −Bk) fk∆u2

√2e

m=∑i,j

(δj〈νji〉N− δi〈νij〉N

)mij∆u+

〈νI(u)〉N

(MI +

1

2

)∆u+O(∆u) = 0,

and for the case of the exponential spatial growth,

αRne

ΓE+

n∑k=1

(Ak −Bk) fk∆u2

√2e

m=∑i,j


)mij∆u+

〈νI(u)〉N

(MI +

1

2

)∆u+O(∆u) = 0.

48

writing of EBE matrix

w/o secondary electrons

start

calculation of initial ionization

rate coefficient (CI) Townsend coefficient (α)

update of EBE matrix with secondary electrons

calculation of EEDF

update CI or α

calculation ofconvergence

criteria

reading cross-section

data

calculation of the initial

EEDF

calculation ofpower balance

equations

output

end

end/continuecycle

choice of electron-

-density growthmodel

EBE solver

Ionization routine

Figure 3.1: Flowchart of the EBE solver in LoKI. In blue we present the original routine if secondary

electrons were not included. On the right hand side, we present all the various steps of the ionization

routine.

49

startcalculation of

calculation of EEDF with secondary

electrons and w/oe-e coll op

C I /α

test convergence

eedf and C I /α

calculation of EEDF w/o

non-linear terms

calculation of

C I /α

calculation of EEDF with secondaryelectrons and

e-e coll op

calculation of e-e coll op

test convergence

eedf

calculation of C I /α

calculation of

C I /α

test convergence of

C I /αend

test convergence

eedf and C I /α

No

No

No

No

Yes

Yes

Yes

Yes

calculation of EEDF with secondaryelectrons and

e-e coll op

Figure 3.2: Flowchart of the coupling between the ionization and the electron-electron collisions routine.

In red are the steps done within the ionization routine. In green are the steps done within the electron-

electron collisions routine.

50

Chapter 4

Results

The inclusion of secondary electrons caused a significant change on the EEDF shape for high E/N

fields. Also, the choice of the electron density growth model had a significant impact on the high-energy

part of the distribution, the so-called tail of the EEDF. Different energy sharing assumptions change the

amount of high and low energy electrons.

In order to validate the new ionization routine, it was necessary to perform benchmark tests with an

established electron Boltzmann equation solver, such as BOLSIG+ [16]. This computational tool has the

option to include secondary electron due to electron-impact ionizations with also two different electron

density growth models, exponential spatial or temporal electron density growth. However, it doesn’t

provide the option to use an energy sharing using a single differential ionization cross section (SDCS).

Consequently, these benchmarks were performed for the equal energy sharing and "one electron takes

all" cases, available in BOLSIG+.

Since the benchmark tests for a energy sharing using a SDCS could not be performed (against

another EBE solver), validation tests with experimental data were done in this case.

4.1 Comparison between energy sharing modes

Energy sharing modes refer to the way in which the scattered and secondary electrons share the energy

available after the ionization collision. Different energy sharing definitions will impact the EEDF shape in

different forms.

Two limiting cases would be the equal energy sharing, and "one electron takes all" sharing. With

equal sharing the secondary electron has always the maximum energy possible and the scattered elec-

tron the least energy possible. With the "one electron takes all" the secondary electron has always the

minimum possible energy and the scattered electron the maximum energy possible. In-between these

cases, there is the energy sharing using an SDCS.

The shape of this cross section depends on the energy of the primary electron and on the gas in

which the ionization occurs, however, there are some common features. Experimental data suggest that

there is a maximum near the zero energy of secondary electrons [32, 24], with the lowest probability

51

(eV)secu10210

/eV

2

cm

-20

10

× se

c

i q

10

210

310

2N = 100 eV∈ = 200 eV∈ = 300 eV∈

Figure 4.1: Experimental data on the differential ionization cross section on the secondary electron

energy for molecular nitrogen, and for three different primary electron energies [1].

values being for secondary electrons with energies near its maximum, that is usec ≈ (ε− VI) /2. An

example of SDCS is represented in Figure 4.1 for molecular nitrogen. According to this information, the

equal energy sharing scenario would be the least probable one, and the "one electron takes all" the

most probable one.

These tests were performed for Argon, the general behaviour being similar for other gases. A com-

parison of the three energy sharing models was done for E/N = 1000 Td 1. The different energy sharing

effects were similar between growth models, so the comparisons were done for the exponential spatial

growth model only.

With wither spatial or temporal growth, and for both energy sharing modes, there are less high en-

ergy electrons when secondary electrons are included. This is caused mostly by the introduction of the

secondary electrons at low energies that, by imposing the normalization condition, force the EEDF to be

lower at high energy values. Physically speaking, the increase in the number of low energy electrons is

high enough so that the probability of having a high energy electron is reduced. This is well illustrated in

Figure 4.2 with the energy sharing mode "one electrons takes all". This mode is in part identical to the

treatment of electron-impact ionizations as conservative inelastic collisions (see 2.3.3), the description

previously implemented in LoKI, the difference being that here secondary electrons are introduced at

zero energy. As a result, the change seems to be caused by the introduction of secondary electrons,

and not by the energy partitioning between the electrons produced after the ionization.

1For E/N = 1000Td the near-isotropy assumption, necessary to justify the two-term approximation, is not plausible since

the energy that electrons gain by the electric field during the relaxation frequency of the first anisotropy starts to be comparable

to the electron thermal velocity. However, as will be showed in sec 4.4, LoKI’s predictions are still good when compared with

experimental data. Also, higher electric fields augment the differences between energy sharing modes and electron density

growth models, facilitating this analysis.

52

u (eV)0 100 200 300 400 500 600

)

-3

/2f (

ev

20−10

18−10

16−10

14−10

12−10

10−10

8−10

6−10

4−10

2−10 Ar E/N=1000Tdwithout secondary electronsequal energy sharingone electron takes allusing a SDCS

(a) Logarithm plot of the various EEDFs for different en-

ergy sharing models.

u (eV)0 5 10 15 20 25 30 35

)

-3

/2f (

ev

0

0.005

0.01

0.015

0.02

0.025

0.03 Ar E/N=1000Tdwithout secondary electronsequal energy sharingone electron takes allusing a SDCS

(b) Linear plot of the low-energy part of the various

EEDFs for different energy sharing modes.

Figure 4.2: Plot of EEDFs calculated in LoKI for Argon with DC E/N = 1000Td and the electron density

spatial growth model.

In Figure 4.2a we can better see the behaviour of the high-energy region of the EEDF, mainly corre-

lated with the most energetic of the product electrons, the scattered electron.

In the "one takes all" case, the energy of the scattered electrons after the collision has the maximum

value usca = ε − VI . In this case the tail of the EEDF is higher, because the scattered electrons loose

less energy. Most of the high energy electrons of the distribution function will remain within the high

energy region, even after an ionizing collision.

In contrast, on the equal energy sharing case, the energy of a scattered electron after the collision is

usca = (ε− VI) /2 which is the lowest possible value for a scattered electron. This shows why this case

will have the lowest probability of finding a high energy electron. For most of these electrons, the energy

lost is high enough, to remove them from the tail into the body of the EEDF.

The tail of the EEDF for the energy sharing using a differential ionization cross section is between

these two limiting cases.

Observing Figure 4.2b we can better see the behaviour of the low energy region of the EEDF, mainly

related with the energy of the least energetic of the product electrons, the secondary electron.

For the "one electron takes all" case, the probability is the highest. This is expected, since with this

type of sharing secondary electrons are introduced at zero energy usec = 0.

The EEDF of the equal energy sharing case has the lowest probability (apart from the case without

secondary electrons). This can be explained by the fact that with this type of energy sharing, secondary

electrons have the highest possible energy usec = (ε− VI) /2.

Again, the case in which the energy sharing is described with a differential ionization cross section

is between the previous limiting cases.

53

It is interesting to see that the EEDF for the equal energy sharing case has a probability higher than

the other distributions between 7eV and 20eV. This explains why it has the lowest probability both for

low and high energy regions, while complying with the EEDF normalization condition. An analogous

observation can be made for the "one electron takes all" energy sharing mode.

4.2 Comparison between electron density growth models

Electron-density growth models refers to the way in which the introduction of secondary electrons influ-

ence the electron density. In this work, two electron density growth models were adopted, assuming a

spatial growth and a temporal growth.

The results obtained using these models are shown in Figure 4.3, for a DC Argon plasma at E/N =

1000Td, where we have also included the case in which secondary electrons are not considered.

u (eV)0 100 200 300 400 500 600

)

-3

/2f (

ev

19−10

17−10

15−10

13−10

11−10

9−10

7−10

5−10

3−10

Ar E/N=1000Td without secondary electronsexponential temporal growthexponential spatial growth

(a) Logarithm plot of the various EEDFs for different elec-

tron density growth models.

u (eV)0 5 10 15 20 25 30 35

)

-3

/2f (

ev

0

0.005

0.01

0.015

0.02

0.025

0.03Ar E/N=1000Td

without secondary electronsexponential temporal growthexponential spatial growth

(b) Linear plot of the low-energy part of the various

EEDFs for different electron density growth models.

Figure 4.3: Plot of EEDFs calculated in LoKI for Argon with DC E/N = 1000Td and the energy sharing

using a SDCS.

A significant difference is observed between the two growth models. Spatial growth seems to pro-

duced lower tails than the temporal growth model, which also relates to a higher probability of finding

low energy electrons.

This result is coherent with lower drift velocities for the spatial growth model, where the electron flux

is composed by two opposite components, due to the drift in the electric field and the diffusion caused

by a pressure gradient.

electron density exponential spatial growth ~Γ = −∇r(Dne) + µ0ne ~E;

electron density exponential temporal growth ~Γ = µ0ne ~E.

54

4.3 Benchmarks against BOLSIG+

One of the most popular EBE solvers is BOLSIG+ [16]. This EBE solver has some of the electron density

growth models of LoKI. However, it can only use either the equal sharing or the "one electron takes all"

energy sharing mode. Thus, our benchmark tests against BOLSIG+ will not include the description of

ionization using a differential cross section.

The first tests were for the EEDF in Argon at E/N = 1000Td, with the temporal growth and the spatial

growth models. After, the first Townsend ionization coefficient, calculated with LoKI and BOLSIG+, was

compared for different E/N values in Argon and in molecular Nitrogen.

u (eV)0 100 200 300 400 500 600

)

-3

/2f (

ev

11−10

10−10

9−10

8−10

7−10

6−10

5−10

4−10

3−10

2−10BOLSIG equal energy sharing

LoKi equal energy sharing

BOLSIG one takes all energy sharing

LoKi one takes all energy sharing

Ar E/N=1000Td electron density exponential temporal growth

(a) Comparisons between LoKI and BOLSIG+ for differ-

ent energy sharing in ionization, using the exponential

temporal growth model for the electron density.

u (eV)0 100 200 300 400 500

)

-3

/2f (

ev

16−10

15−10

14−10

13−10

12−10

11−10

10−10

9−10

8−10

7−10

6−10

5−10

4−10

3−10

2−10 BOLSIG equal energy sharing

LoKi equal energy sharing

BOLSIG one takes all energy sharing

LoKi one takes all energy sharing

Ar E/N=1000Td electron density exponential spatial growth

(b) Comparisons between LoKI and BOLSIG+ for differ-

ent energy sharing in ionization, using the exponential

spatial growth model for the electron density

Figure 4.4: Comparisons of the EEDF, calculated with LoKI and BOLSIG+, for Argon with DC E/N =

1000Td.

First we will compare EEDFs calculated with BOLSIG+ and LoKI for equivalent electron-density

growth models and energy sharing modes. Contrary to LoKI, BOLSIG’s energy grid is not completely

defined by the user. BOLSIG+ discretizes the electron density growth and electric field terms using an

exponential scheme [33], which is accurate for some convection and diffusion conditions [16], meaning

that it automatically defines the number of energy grid-points and energy limit for a given set of dis-

charge conditions. Comparing two distribution functions with different initial and final energy grid points

is difficult, since different energy limits can change the EEDF in a significant way when enforcing the

normalization condition. So, whenever possible, we have tried to match LoKI’s user-defined energy grid

with BOLSIG’s.

For the exponential temporal growth model there is very good agreement between BOLSIG+ and

LoKI for all energy sharing cases.

The EEDFs for the exponential spatial growth model are similar in shape, and in the trend observed

55

for the different energy sharing modes, but they don’t match as well as in the temporal case. One of the

factors for this discrepancy was the difficulty in matching the energy grid-points. If the BOLSIG’s upper

limit for the energy grid is used in LoKI, large power-balance errors are present. However, even when

these limits are made very similar, there are still some differences between the EEDFs that might be

due to the distinct discretization procedures in the two EBE solvers.

E/N (Td)210

310

2

/N m

α

0

5

10

15

20

25

30

35

40

45

21−10×Ar DC discharge

BOLSIG+ equal energy sharing

LoKI equal energy sharing

BOLSIG+ one electron takes all sharing

LoKI one electron takes all sharing

(a) First Townsend ionization coefficient in Argon DC

plasmas for various E/N values.

E/N (Td)210

310

2

/N m

α

0

5

10

15

20

25

30

35

21−10× DC discharge2N

BOLSIG+ equal energy sharing

LoKI equal energy sharing

BOLSIG+ one electron takes all sharing

LoKI one electron takes all sharing

(b) First Townsend ionization coefficient in Nitrogen DC

plasmas for various E/N values.

Figure 4.5: Comparison between first Townsend ionization coefficient calculated with BOLSIG+ and

LoKI, using the exponential spatial growth model and adopting different energy sharing modes.

For the first Townsend ionization coefficient, the EBE solvers predict similar results. However, there

are some deviations for high E/N values in Nitrogen and even for low E/N values in Argon.

There is a strange behaviour, more evident in nitrogen, around E/N = 700Td, where BOLSIG’s

predictions deviate, when compared with LoKI’s prediction path. Between E/N = 1000Td and E/N =

2000Td, BOLSIG’s predictions for the "one electrons takes all" case approach LoKI’s corresponding

energy sharing mode prediction.

4.4 Validation of the first Townsend ionization coefficient against

experimental data for Ar and N2

In order to assess the improvements in the values of transport coefficients predicted by LoKI, a com-

parison of the first Townsend ionization coefficient with experimental data was made. The experiments

were made in a Steady State Townsend (SST) discharge, accordingly the spatial electron-density growth

model was adopted. Comparisons will use the energy sharing using an SDCS and the case in which

secondary electrons are not included.

In Argon the simulation predictions improved significantly but they are still above experimental data

56

E/N (Td)210

310

2

/N m

α

21−10

20−10

19−10

Argon SST discharge

Exp data

LoKI with secondary electronsLoKI without secondary electrons

(a) Comparison between LoKI’s calculated first

Townsend ionization coefficient and experimental data

for Argon SST discharges [34].

E/N (Td)210

310

2/N

mα

24−10

23−10

22−10

21−10

20−10

19−10 SST discharge2N

Exp dataLoKI with secondary electronsLoKI without secondary electrons

(b) Comparison between LoKI’s calculated first

Townsend ionization coefficient and experimental data

for Nitrogen SST discharges [2].

Figure 4.6: First Townsend ionization coefficient as a function of the reduced electric field. LoKI’s simula-

tions use the exponential spatial growth model or conservative ionization collisions (secondary electrons

not included).

uncertainty. In Nitrogen the inclusion of secondary electrons was enough to shift LoKI’s predictions into

experimental data uncertainty. These results show that LoKI can now operate at higher E/N values.

LoKI does yet not consider attachment or recombination non-conservative processes. Since attach-

ment and recombination have an effect on the number of electrons, it may have an influence on the

electron density growth terms of the EBE (in the case of Argon and molecular Nitrogen, attachment

would not improve prediction since these gases are not electronegative).

In a discharge there are other ionization processes that contribute to the spatial growth of the elec-

tron density, such as associative and Penning ionization, chemionization, photoionization, associative

detachment, and ionization by neutral particles and ions [35, Ch. 9.3]. Photoionization and heavy-

particle collisional ionizations are not included in this EBE, but they do influence the EEDF.

In SST experiments, the ejection of electrons from the cathode due to ions or photons is significant.

These secondary processes can be difficult to separate from the current growth due to electron impact

ionization. To avoid this problem, some experiments are made in a time scale shorter than the ions

transient time [35, Ch. 9.3]. On the Argon SST experiment [34], the first Townsend coefficient was

measured through the spatial variation of light emitted by the de-excitation of metastable states. This

spatial measurement was done by moving the detector in a platform with a stepper motor controlled by

a computer, with the light ouptup being measured in 1mm steps until 25mm [34]. Although there is no

reference to the total time that these measurements took, it is safe to assume that these measurements

were not shorter than the ions transient time, or the metastables diffusion time, and so, electrons from

57

E/N (Td)3

10

)2/N

(m

α

20−10

SST discharge2NExp dataequal energy sharingone electron takes all

Figure 4.7: Comparison between LoKI’s calculated first Townsend ionization coefficient with equal en-

ergy sharing mode, "one electron takes all" mode, and experimental data for Nitrogen SST discharges

[2].

ion and photon cathode emissions probably played a role on the electron density spatial profile. In this

experiment, the first Townsend coefficient α was calculated by adjusting an exponential spatial growth

of the electron density, similar to 2.19, to the measured electron density profile. In sum, the measured

first Townsend ionization coefficient, was done by measuring an electron density spatial profile created

by electron-impact ionization and possibly by other processes that are not included in the EBE solver.

Some measurements of the secondary ionization coefficients were made in nitrogen [36], and sep-

arated in fast and slow coefficients, due to cathode emissions by ions and by photons or metastables,

respectively. With this discrimination of the secondary ionization coefficients, it may be possible to in-

clude these processes in the EBE.

4.4.1 The use of the equal energy sharing mode

The non-conservative ionization operator with predefined equal energy sharing is popular in the LTP

community. One of the reasons is that the numerical writing of this operator uses much less computa-

tional resources compared to the one with energy sharing using a SDCS. However, the equal energy

sharing mode continues to be preferred even when the "one electron takes all" mode is available, and

despite the fact that the latter consumes comparable resources. This may be because it leads to closer

estimates when compared to the experimental data of the first Townsend ionization coefficient (see

Figure 4.7).

Nonetheless, we have seen in section 4.1 that the equal energy sharing scenario is much less prob-

able than the "one electron takes all" scenario (when compared to a differential description of the energy

sharing), so the better estimates cannot be attributed to a more realistic ionization operator. Also in

section 4.1, we have seen that the EEDF can suffer significant changes between energy sharing modes.

58

One of the possibilities, is that the EEDF calculated with the equal energy sharing mode, somehow

underestimates the value of the first Townsend ionization coefficient, leading to closer estimates.

Although the equal energy sharing mode leads to better estimates of the first Townsend ionization

coefficient, the fact that it is the least probable scenario means that its use cannot be properly justified

with physical arguments. A better option to improve the calculation of the first Townsend ionization coef-

ficients would be to include the other non-conservative processes that were discussed in this section.

59

60

Chapter 5

Prospective

An ionization routine was fully integrated into LoKI-B. Both the ionization routine, and electron-electron

collisions coupling strategies are outlined in work-flows, aiding in future changes. This work also pro-

vides a general description of LoKI’s EBE solver operating procedure.

The basic functionalities of the ionization routine have been developed. It has opened some paths

to calculate new coefficients and include some more mechanisms. However, some numerical improve-

ments and tests can be made as well in order to help users, as well as other validations with experimental

data.

• Now that electron density growth is included on the EBE, the path to include other non-conservative

processes is simplified.

Attachment collisions can easily be included with a collisional operator. Recombinations can also

be included but they are more complex, due to it being a three body process.

For some gases, such as nitrogen, frequencies of electron cathode ejection for fast processes

(photoionization and ion impact ionization) and for slow processes (metastable impact ionization)

have been measured. This approach is not common, but may improve the similarities between the

measured and simulated first Townsend coefficient;

• The spatial growth model permits the calculation of new parameters, such as the longitudinal

diffusion coefficient and the bulk drift velocity [17]. It is also useful when doing electron density

gradient expansions. The implementation of these calculations would allow to take full part of the

spatial growth model;

• Inclusion of a nonlinear standard algorithm in the ionization routine, such as Newton-Raphson or

Broyden’s method.

In some situations, of very high/low electric-field, very large number of grid-points, and high ioniza-

tion degree, the convergence can be quite difficult. These methods could diminish the number of

iterations required for convergence, as well as improve the convergence pathway when including

electron-electron collisions;

61

• In many cases, LoKI performs sequential batch simulations, e.g. for increasing values of the elec-

tric field, in which case many operating conditions remain the same (e.g. the number of intervals

and the maximum value of the energy grid). In these cases, it is not necessary to re-write the

ionization collisional operator for each simulation, which could significantly reduce run times;

62

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65

http://dx.doi.org/doi.org/10.1088/0022-3727/10/7/011

http://dx.doi.org/doi.org/10.1088/0022-3727/10/7/011

http://dx.doi.org/10.1088/0022-3700/2/5/308

http://dx.doi.org/10.1088/0022-3700/2/5/308

http://dx.doi.org/10.1007/978-3-319-09253-9

http://dx.doi.org/10.1088/0963-0252/24/3/035002

http://dx.doi.org/10.1088/0963-0252/24/3/035002

http://dx.doi.org/10.1103/PhysRevA.6.977

http://dx.doi.org/10.1109/T-ED.1969.16566

http://dx.doi.org/10.1109/T-ED.1969.16566

http://dx.doi.org/10.1103/PhysRevE.53.4007

http://dx.doi.org/10.1103/PhysRevE.53.4007

http://dx.doi.org/10.1007/978-3-7091-4028-4

http://dx.doi.org/10.1088/0022-3727/9/3/018

http://dx.doi.org/10.2307/2325085

66

Appendix A

Discretization of the electron

Boltzmann equation

An energy grid will be used to discretize energy values, cross sections, the energy distribution function,

and other functions of the electron energy.

The energy grid will be from u = 0 to u = umax, being umax the maximum energy defined by the user.

The grid will be divided into N intervals, each with a ∆u = umax

N energy step. When applying the finite

difference method, EEDF values are defined at the middle of the energy interval, and so it was decided

that every term should be written in a way so that it is multiplied by the EEDF defined at the middle of

the energy interval.

A value defined at the upper energy interval limit, will be identified with an interval number in sub-

script, and a plus sign "+" in superscript. For example, the cross section define at the upper limit of the

interval k is identified as,

σ+k = σ(k ∆u).

Quantities defined at the lower energy interval limit will be identified with an interval number in sub-

script and a minus "-" sign at superscript. For example, the energy at the lower limit of the interval k

is,

u−k = (k − 1)∆u.

Quantities defined at the middle of the energy interval will be identified with an interval number in

subscript and no superscript. For example, the EEDF value at the middle of the interval k is,

fk = f (k∆u−∆u/2) .

These definitions are summarized in Figure A.1.

67

0

u_max

fK

u

Δu(k-1)Δu kΔu (k+1)Δu

fk fk+1fk-1

nΔu

uk+uk-1

+

uk-

uk+1+

uk+1

-uk-1

-

fk+ σ

k+σ

k-1+f

k-

u k=(k-

1/2)Δu

u k+1=(k+

1-1/2)Δu

Figure A.1: The energy grid and the various notations of quantities used on the discretization.

A.1 Linear terms on the electron energy distribution function

A.1.1 Time variation term

The electron density exponential temporal growth term is, for an energy uk = (k − 1/2)∆u,

−

u CI√2 e ume

f(u)

u=uk

≡ −√m

2eCI fk

√uk.

A.1.2 Space variation term

One of the three terms that constitute the electron density exponential spatial growth term is linear on

the EEDF. It can be discretized for an energy uk,[α2R

3

u

σeff (u)f(u)

]u=uk

≡ α2R

3

ukσeffK

fk

A.1.3 Inelastic/superelastic collisions term

The energy interval between levels uij , are usually not of the interval size ∆u. To take care of this we

will define

u+ij = mij∆u = floor

( uij∆u

)∆u

uk+mij= (k +mij − 1/2)∆u = uk +mij∆u,

. where floor(x) is a function that takes the integer part of x.

J0

γ

∣∣∣∣k

=∑i,j

[δi uk+mij σ

ijk+mij fk+mij − δi uk σijk fk +

gigjδj uk σ

ijk fk−mij −

gigjδj uk+mij σ

ijk+mij fk

]

68

n∑j=k

uj = uk + uk+1 + uk+2 + ...+ un−1 + un .... ...

uk uk+ 1 uk+ 2 un

(k+ 1)Δu (k+ 2)Δu nΔukΔu0

Summand index

Sum index

Figure A.2: Numerical and graphical representation of an energy sum. The summed energy is repre-

sented in light red.

A.2 Terms with integrals on the electron energy distribution func-

tion

A numerical integration is calculated with a sum. In order to be consistent with the grid notation (see

Figure A.1), the index in a sum, corresponds always to the upper limit of a given energy interval. A sum

can be viewed as represented in Figure A.2,

In terms of quadrature rules it will be used the simpler midpoint or rectangle rule. In this rule, the

integral is described by a discrete sum of the integrand function evaluated at the middle of the interval,

times the interval’s width. This can be thought as a sum of rectangles with height and width equal to the

integrands value and interval width respectively. A demonstration of this rule is in Figure A.3.

Midpoint integration algorithm

Let g be a function which is integrable

on the interval [a, b], with a = k∆u and

b = n∆u, n > k. In the midpoint rule the

integral of g is approximated by,∫ n∆u

k∆u

g(u)du ≈ Imid(∆u) = ∆u

n∑j=k+1

gj .

Where gj = g(uj) with uj being the en-

ergy at the middle of the interval j. uk+2

g (u)

uuk=kΔu+

uk+1

gk +1

gk +2

gk +3gk +4

gn

un=nΔuuk+2 uk+3 uk+4 un

+uk +1+ uk+3

++ uk +4+

u j=( j−12)Δ u

+u j= jΔu

g j=g (u j)

Figure A.3: Representation of the midpoint quadrature rule.

69

A.2.1 Ionization collisional operator

JI(u)

γ=

∞∫2u+VI

ε qisec(ε, u) f(ε) dε

︸︷︷︸J1I (u) secondary electrons

+

2u+VI∫u+VI

ε qisec (ε, ε− VI − u)f(ε) dε

︸︷︷︸J2I (u) scattered electrons

−u∫ (u−VI)/2

0

qisec(u, ε)dε f(u)︸︷︷︸J3I (u) primary electrons

For the ionization potential energy the interval number is MI and defined as,

MI = Round

(VI∆u

)with round(x) being the nearest integer function applied to x.

Using the mid-point quadrature rule the discrete ionization operator for a specific interval of energy k

is,

JIγ

∣∣∣k

=

n∑j=2k+MI+1

ujqisec(uj , uk)fj∆u+

2k+MI∑j=k+MI+1


)fj∆u−ukfk(k−MI)/2∑

j=1

qisec(uk, uj)∆u.

A.3 Terms with derivatives on the electron energy distribution func-

tion

Here a finite difference method will be used in terms that involve EEDF’s derivatives. For example, the

derivative of a function h(u) in respect to the electron energy will have N components of the type

∂h(u)

∂u

∣∣∣∣k

.

This finite difference method will be a centred one. It will approximate a derivative of a function with

respect to u to the difference of the function at half of an interval forward and backward, divided by the

interval’s width. For example,

∂h(u)

∂u

∣∣∣∣k

=h(u+

k )− h(u−k )

∆u=h+k − h

−k

∆u=h+k − h

+k−1

∆u.

In the case of a second order derivative the process is the same. However the terms should be

proportional to to the EEDF at the middle of the interval, so if a quantity is proportional to the EEDF at

an interval’s limit, the EEDF is re-written as an average between two values,

f+k =

fk + fk+1

2.

With other energy functions not the EEDF, this interpolation usually is not made. On the case of

cross sections and energies, the values remain defined at the energy interval limit.

A.3.1 Space variation term

The space variation term with an EEDF derivative can be written as,

αR3

u

σeffE0R

∂f(u)

∂u=αR3E0R

ukσeff k

fk+1 − fk−1

2

70

[αR3

∂

∂u

(u

σeffE0R f(u)

)]u=uk

=αRE0R

6

[fk+1

k

σ+effK

+ fk

(k

σ+effK

− k − 1

σ+effK−1

)− fk−1

k − 1

σ+effK−1

].

A.3.2 Rotational collision term - continuous approximation

∂

∂u[4Bσ0uf(u)] = 4Bσ0

(f+k u

+k − f

−k u−k

∆u

)= 4Bσ0

[fk + fk+1

2

k∆u

∆u− fk + fk−1

2

(k − 1)∆u

∆u

]= 2Bσ0 (fk+1k + fk − fk−1(k − 1))

∂

∂u

[4Bσ0u

kBTge

∂f(u)

∂u

]= 4Bσ0

kBTge

∂f∂u

∣∣∣+ku+k −

∂f∂u

∣∣∣−ku−k

∆u

= 4Bσ0

kBTge

[fk+1 − fk

∆u2k∆u− fk − fk−1

∆u2(k − 1)∆u

]= 4Bσ0

kBTge∆u

[fk+1k − fk(2k − 1) + fk−1(k − 1)]

A.3.3 Elastic collisions terms

∂

∂u

(2me

Mu2σc(u)f(u)

)=

2me

M

[σc(u

+k )u+2

k f+k − σc(u

−k )u−2

k f−k∆u

]=

2me

M

[σ+c kk

2∆ufk+1 + fk

2− σ+

c k−1(k − 1)2∆ufk + fk−1

2

]=me

M

fk+1k

2σ+c k∆u+ fk

[k2σ+

c k − (k − 1)2σ+c k−1

]∆u− fk−1(k − 1)2σ+

c k−1∆u

∂

∂u

(2me

Mu2σc(u)

kBTge

∂f(u)

∂u

)=

2me

M

kBTge

u+2k σ+

c k∂f∂u

∣∣∣+k− u−2

k σ+c k−1

∂f∂u

∣∣∣−k

∆u

=

2me

M

kBTge

[k2σ+

c k (fk+1 − fk)− (k − 1)2σ+c k−1 (fk − fk−1)

]=

2me

M

kBTge

[fk+1k

2σ+ck − fk

(k2σ+

ck + (k − 1)2σ+c k−1

)+ fk−1(k − 1)2σ+

c k−1

]

A.3.4 Electric field Term

The Electric Field Term can be written as ∂∂u

((EN

)2 u3 g(u)∂f∂u

)where

g(u) = Re

(CIv

+ σeff +iωRv

)−1

=1

CI√me√

2eu+ σeff (u)

× 1

1 +ω2

Rme

(CI√me+

√2euσeff)

2

71

∂

∂u

((E

N

)2u

3g(u)

∂f

∂u

)=

(E

N

)21

3

u+k g

+k∂f∂u

∣∣∣+k− u−k g

−k∂f∂u

∣∣∣−k

∆u

=

(E

N

)21

3

[g+k

fk+1 − fk∆u

k − g+k−1

fk − fk−1

∆u(k − 1)

]=

(E

N

)21

3∆u

[fk+1 k g

+k − fk(k g+

k + (k − 1) g+k−1) + fk−1 (k − 1) g+

k−1

]in which g+

k :

g(u+k ) = g+

k =1

CI√me√

2ek∆u+ σ+

eff k

× 1

1 +ω2

Rme

(CI√me+

√2ek∆u σ+

effk)2

In the spatial growth model, this electric field term is analogous with CI = ω = 0. For the DC temporal

growth model, this electric field term is analogous with ω = 0.

72

Appendix B

Verification of the discretized form of

the particle balance equation

In order to obtain this equation we multiply the isotropic Boltzmann equation by "1" and integrate it in

energy.

B.1 Time variation term

−√m

2eCI

n∑k=1

fku1/2k ∆u = −

√m

2eCI

B.2 Space variation term

n∑k=1

α2R

3

ukσeff k

fk∆u = αRα

√me

2 e

1

3N

n∑k=1

2 e

me

ukσeff k

fk∆u

=α2

N

√me

2 eD

=αRne

√me

2 eD∂ne∂z

n∑k=1

αR3E0R

ukσeff k

fk+1 − fk−1

2=αRne

√me

2 e

ne3N

n∑k=1

E0uk

σeff k

fk+1 − fk−1

2

=αRne

√me

2 eneµ0E0

73

The sum of both terms is

n∑k=1

α2R

3

ukσeff k

fk∆u+

n∑k=1

αR3E0R

ukσeff k

fk+1 − fk−1

2=αRne

√me

2 e

(D∂ne∂z

+ neµ0E0

)= −αR

ne

√me

2 enevdrift

= −αR√me

2 evdrift

using equation 3.2

= −CI√me

2 e

B.3 Terms with derivatives on the electron energy distribution func-

tionn∑k=1

[Ak−1fk−1 − (Ak +Bk) fk +Bk+1fk+1] ∆u =

n−1∑k=0

Akfk∆u−n∑k=1

(Ak +Bk)fk∆u+

n+1∑k=2

Bkfk∆u =

=A00

f0∆u−>0

Anfn∆u+

n∑k=1

Akfk∆u−n∑k=1

(Ak +Bk)fk∆u−>0

B1f1∆u+Bn+1*0

fn+1∆u+

n∑k=1

Bkfk∆u =

=

n∑k=1

Akfk∆u−n∑k=1

(Ak +Bk)fk∆u+

n∑k=1

Bkfk∆u = 0

B.4 Inelastic/superelastic collision terms

n∑k=1

∑i,j

[δiuk+mijσ

ijk+mijfk+mij − δiukσijk fk +

gigjδjukσ

ijk fk−mij −

gigjδjuk+mijσ

ijk+mijfk

]∆u =

=∑i,j

n+mij∑k=1+mij

δiukσijk fk −

n∑k=1

δiukσijk fk + δj

gigj

n∑k=1

ukσijk fk−mij − δj

gigj

n+mij∑k=1+mij

ukσijk fk−mij

∆u =

=∑i,j

δi

n∑k=1

(ukσijk fk − ukσ

ijk fk)−

>

0mij∑k=1

ukσijk fk +

*

0n+mij∑k=n+1

ukσijk fk

∆u+

+∑i,j

gigjδj

n∑k=1

(ukσijk fk−mij − ukσ

ijk fk−mij)−

*0

mij∑k=1

ukσijk fk−mij +

*0

n+mij∑k=n+1

ukσijk fk

∆u = 0

σijk ≡ 0, k ≤ mij and fk ≡ 0, k > n

B.5 Ionization collisional operator

In this case the approach was to simplify these three terms by getting the differential cross section to be

dependent of the same variables. To be able to do this we’ll change variables and summation limits.

In order to do this clearly we have used Iverson’s brackets, or the Iverson convention. Square brack-

ets, with a condition inside, are 1 if the condition is true and 0 if it is not (for more information on the

74

Iverson convention the reader is requested to see [37]).

With this in mind

n∑k=1

JI

∣∣∣k

=

n∑k=1

n∑j=2k+MI+1

ujqisec(uj , uk)fj∆u

2 +

n∑k=1

2k+MI∑j=k+MI+1


)fj∆u2 −

n∑k=1

ukfk

(k−MI)/2∑j=1

qisec(uk, uj)∆u2

We can then re-write the ionization operator where on the third term we have exchanged index

names, between j and k,

n∑k=1

JI

∣∣∣k∆u =

(n−MI−1)/2∑k=1

n∑j=2k+MI+1


2+

+

n−MI−1∑k=1

2k+MI∑j=k+MI+1


)fj∆u2−

−n∑

j=MI+2

(j−MI)/2∑k=1


2.

Our objective will be to re-write the second and third terms in order to have similar summands as the

first term. This way we have a better possibility to add the three terms together.

Secondary electron term

When dealing with the first term we will separate the sum in two terms, in order to have them in a suitable

form to compare to the other terms.

n∑k=1

J1Ik

∆u =

(n−MI−1)/2∑k=1

n∑j=2k+MI+1


2 =

n∑k=1

n∑j=2k+MI+1


2.

We can then add and subtract a term in j = 2k +MI summed over k,

n∑k=1

J1Ik

∆u =

n∑k=1

n∑j=2k+MI+1


2+

+

n∑k=1

[j = 2k +MI ]ujqisec(uj , uk)fj∆u

2 −n∑k=1


2

And we can add them into,

n∑k=1

J1Ik

∆u =

n∑k=1

n∑j=2k+MI


2 −n∑k=1


2

⇔n∑k=1

J1Ik

∆u =

n∑k=1

n∑j=2k+MI


2 +O(∆u) (B.1)

Scattered electron term

Focusing know on J i2 we will use Iverson’s brackets to re-write these limited sums as infinite sums,

75

n∑k=1

J2Ik

∆u =

n−MI−1∑k=1

2k+MI∑j=k+MI+1


)fj∆u2 =

=∑j,k

[1 ≤ k ≤ n−MI − 1][k +MI + 1 ≤ j ≤ 2k +MI ] uj qisec(uj , uj−k−MI

)fj∆u2

=∑j,k

[(j −MI)/2 ≤ k ≤ j −MI − 1] [2 +MI ≤ j ≤ N ] uj qisec(uj , uj−k−MI

)fj∆u2

=∑j,k

[1 ≤ −k + j −MI ≤ (j −MI)/2] [2 +MI ≤ j ≤ N ] uj qisec(uj , uj−k−MI

)fj∆u2

making the variable change L = j − k −MI

=∑j,L

[1 ≤ L ≤ (j −MI)/2] [2 +MI ≤ j ≤ N ] uj qisec(uj , uL)fj∆u

2

=∑j,L

[1 ≤ L ≤ (n−MI)/2] [2L+MI ≤ j ≤ N ] uj qisec(uj , uL)fj∆u

2

=

(n−MI)/2∑k=1

n∑j=2k+MI


2

Where on the last step we renamed variable L to k. The final relation to be used in the followed

calculations is,

n∑k=1

J2Ik

∆u =

n−MI−1∑k=1

2k+MI∑j=k+MI+1


)fj∆u2 =

(n−MI)/2∑k=1

n∑j=2k+MI


2.

(B.3)

Primary electron term

Focusing now on the primary electron operator we have,

n∑k=1

J3Ik

∆u =

n∑j=MI+2

(j−MI)/2∑k=1


2 =

=∑j,k

[2 +MI ≤ j ≤ n][1 ≤ k ≤ (j −MI)/2]ujqisec(uj , uk)fj∆u

2

=∑j,k

[2k +MI ≤ j ≤ n][1 ≤ k ≤ (n−MI)/2]ujqisec(uj , uk)fj∆u

2

=

(n−MI)/2∑k=1

n∑j=2k+MI


2

The relation to be used in following calculations is,

n∑k=1

J3Ik

∆u =

n∑j=MI+2

(j−MI)/2∑k=1


2 =

(n−MI)/2∑k=1

n∑j=2k+MI


2 (B.4)

76

The sum of the three terms

In the end, and using B.4 the other way around on B.3 and on B.1, we get

n∑k=1

JI

∣∣∣k

=

=

J1I︷︸︸︷

(n−MI)/2∑k=1

n∑j=2k+MI


2 +O(∆u) +

J2I︷︸︸︷

n−MI−1∑k=1

2k+MI∑k+MI+1


2−

J3I︷︸︸︷

n∑j=MI+2

(j−MI)/2∑k=1


2 =

=

using B.4︷︸︸︷n∑

j=MI+2

(j−MI)/2∑k=1


2 +O(∆u) +

using B.3 and then B.4︷︸︸︷n∑

j=MI+2

(j−MI)/2∑k=1


2−n∑

j=MI+2

(j−MI)/2∑k=1


2 =

=

n∑j=MI+2

(j−MI)/2∑k=1


2 +O(∆u) =

=

n∑j=1

(j−MI)/2∑k=1


2 +O(∆u)

in which j can start at 1 since the terms from 1 to MI + 1 are null terms.

σI(uj) =

(j−MI)/2∑k=1

qisec(uj , uk)∆u

〈νI(u)〉 =

n∑j=1

N

√2e

mujσI(uj)fj∆u

We finally get that.

n∑k=1

JI

∣∣∣k

=n∑j=1

(j−MI)/2∑k=1


2 +O(∆u) =〈νI(u)〉N

√m

2e+O(∆u)

B.6 Particle balance equation

Summing the terms from subsections B.3 to B.5 we have

CI

√m

2e− 〈νI(u)〉

N

√m

2e+O(∆u) = O(∆u) ≈ 0

It is possible to see that there is an error O(∆u).

77

78

Appendix C

Verification of the discretized form of

the energy balance equation

C.1 Time variation term

−n∑k=1

CIuk√2euk

me

uk∆ufk = −√m

2eCI

n∑k=1

ukfk√uk∆u = −

√m

2eCI 〈u〉

C.2 Space variation term

n∑k=1

α2R

3

u2k

σeff kfk∆u =

αRne

me

2 eDE

∂nE∂z

n∑k=1

αR3E0R

u2k

σeff k

fk+1 − fk−1

2= −αR

ne

me

2 enEµEE0

The sum of both terms is,

n∑k=1

α2R

3

u2k

σeff kfk∆u+

n∑k=1

αR3E0R

u2k

σeff k

fk+1 − fk−1

2= −αR

ne

√me

2 eΓE

C.3 Terms with derivatives on the electron energy distribution func-

tion

Using the definitions of the energy intervals:

uk+1 = (k − (1/2) + 1)∆u = uk + ∆u

uk−1 = (k − (1/2)− 1)∆u = uk −∆u

79

n∑k=1

[Ak−1fk−1 − (Ak +Bk) fk +Bk+1fk+1]uk∆u =

=∑k

[0 ≤ k − 1 ≤ n− 1]Ak−1fk−1uk − [1 ≤ k ≤ n] (Ak +Bk)uk + [2 ≤ k + 1 ≤ n+ 1]Bk+1fk+1uk∆u =

=∑k

[1 ≤ k ≤ n]Akfkuk+1 − [1 ≤ k ≤ n] (Ak +Bk)uk + [1 ≤ k ≤ n]Bkfkuk−1∆u+

+A00

f0u1∆u−>0

Anfnun+1∆u−>0

B1f1u0∆u+Bn+1*

0fn+1un∆u =

=

n∑k=1

Akfk(uk + 1)−

(Ak +ZZBk

)uk +Bkfk(HHuk − 1)

∆u

=

n∑k=1

(Ak −Bk) fk∆u

C.4 Inelastic/superelastic collision term

Using the definitions in energy intervals:

uk+mij = (k − 1

2+mij)∆u = uk +mij∆u

uk−mij = (k − 1

2−mij)∆u = uk −mij∆u

n∑k=1

∑i,j

δiuk+mijσ

ijk+mijfk+mij − δiukσijk fk +

gigjδjukσ

ijk fk−mij −

gigjδjuk+mijσ

ijk+mijfk

uk∆u =

=∑k

∑i,j

[1 ≤ k ≤ n]

δiuk+mijukσ

ijk+mijfk+mij − δiukukσijk fk +

gigjδjukukσ

ijk fk−mij −

gigjδjuk+mijukσ

ijk+mijfk

∆u =

=∑k

∑i,j

δi

[1 +mij ≤ k ≤ n+mij ]ukuk−mijσ

ijk fk − [1 ≤ k ≤ n]ukukσ

ijk fk

∆u+

+∑k

∑i,j

gigjδj

[1−mij ≤ k ≤ n−mij ]uk+mijuk+mijσ

ijk+mijfk − [1 ≤ k ≤ n]uk+mijukσ

ijk+mijfk

∆u =

=∑k

∑i,j

δi

[1 ≤ k ≤ n]uk(uk −mij)σ

ijk fk − [1 ≤ k ≤ n]ukukσ

ijk fk

∆u+

+∑k

∑i,j

δi

[n+ 1 ≤ k ≤ n+mij ]ukuk−mij

0

σijk fk − [1 ≤ k ≤ mij ]ukuk−mij

0

σijk fk

∆u+

σijk≡ 0 if k > n ∧ k ≤mij

+∑k

∑i,j

gigjδj

[1 ≤ k ≤ n]uk+mij(HHuk +mij)σ

ijk+mijfk − [1 ≤ k ≤ n]uk+mijHHukσ

ijk+mijfk

∆u+

+∑k

∑i,j

gigjδj

[1−mij ≤ k ≤ 0]uk+mijuk+mij

*0

σijk+mijfk − [n−mij ≤ k ≤ n]uk+mijuk+mij*

0σijk+mijfk

∆u =

σijk≡ 0 if k > n ∧ k ≤mij

80

= −n∑k=1

∑i,j

δimij∆u2 ukσ

ijk fk +

n∑k=1

∑i,j

gigjδjmij∆u

2 uk+mijσijk+mijfk

= −n∑k=1

∑i,j

δimij∆u2 ukσ

ijk fk +

n∑k=1

∑i,j

δjmij∆u2 ukσ

jik fk

in which we have used the Klein-Rosseland relation

giσijk+mijuk+mij = gjσijkuk,

on the last step.

Knowing that the frequency is given by

〈νij(u)〉 =

√2e

mn

n∑k=1

σijkukfk∆u,

we get that energy balance of the inelastic/superelastic terms is

n∑k=1

∑i,j

δiuk+mijσijk+mij

fk+mij − δiukσijkfk +gigjδjukσijkfk−mij −

gigjδjuk+mijσijk+mij

fk

uk∆u =

=∑i,j

mij

√m

2e

(−δi〈νij〉n

+ δj〈νji〉n

)∆u

=∑i,j

mij

√m

2e(−δiCij + δjCji) ∆u

C.5 Ionization collisional operator

n∑k=1

JI

∣∣∣kuk∆u =

(n−MI−1)/2∑k=1

n∑j=2k+MI+1

ukujqisec(uj , uk)fj∆u

2+ (C.1)

+

n−MI−1∑k=1

2k+MI∑j=k+MI+1

ukujqisec(uj , uj−k−MI

)fj∆u2− (C.2)

−n∑

j=MI+2

(j−MI)/2∑k=1

ujujqisec(uj , uk)fj∆u

2. (C.3)

Our objective will be to re-write the second and third terms in order to have similar summands as the

first term. This way we have a better possibility to add the three terms together.

Secondary electron term

When dealing with the first term C.1 we will separate the sum in two terms, in order to have them in a

suitable form to compare to the other terms.

n∑k=1

J1Ikuk∆u =

(n−MI−1)/2∑k=1

n∑j=2k+MI+1


2 =

n∑k=1

n∑j=2k+MI+1


2.

We can then add and subtract a term in j = 2k +MI summed over k,

81

n∑k=1

J1Ikuk∆u =

n∑k=1

n∑j=2k+MI


2 −n∑k=1

[j = 2k +MI ]ukujqisec(uj , uk)fj∆u

2

⇔n∑k=1

J1Ikuk∆u =

n∑k=1

n∑j=2k+MI


2 +O(∆u) (C.4)

Scattered electron term

Focusing know on C.2 we will use Iverson’s brackets to re-write these limited sums as infinite sums,

n∑k=1

J2Ikuk∆u =

=

n−MI−1∑k=1

2k+MI∑j=k+MI+1


)fj∆u2 =

=∑j,k

[1 ≤ k ≤ n−MI − 1][k +MI + 1 ≤ j ≤ 2k +MI ]uk uj qisec(uj , uj−k−MI

)fj∆u2

=∑j,k

[(j −MI)/2 ≤ k ≤ j −MI − 1] [2 +MI ≤ j ≤ N ] ukuj qisec(uj , uj−k−MI

)fj∆u2

=∑j,k

[1 ≤ −k + j −MI ≤ (j −MI)/2] [2 +MI ≤ j ≤ N ] ukuj qisec(uj , uj−k−MI

)fj∆u2

making the variable change L = j − k −MI

=∑j,L

[1 ≤ L ≤ (j −MI)/2] [2 +MI ≤ j ≤ N ] uj−L−MIuj q

isec(uj , uL)fj∆u

2

=∑j,L

[1 ≤ L ≤ (n−MI)/2] [2L+MI ≤ j ≤ N ] uj−L−MIuj q

isec(uj , uL)fj∆u

2

=

(n−MI)/2∑k=1

n∑j=2k+MI

uj−k−MIujq

isec(uj , uk)fj∆u

2

Where on the last step we renamed variable L to k. The final relation to be used on the following

calculations is,

n∑k=1

J2Ik

∆u =

n−MI−1∑k=1

2k+MI∑j=k+MI+1


)fj∆u2 =

(n−MI)/2∑k=1

n∑j=2k+MI

uj−k−MIujq

isec(uj , uk)fj∆u

2.

(C.5)

82

Primary electron term

Focusing now on the primary electron part C.3 we have,

n∑k=1

J3Ikuj∆u =

n∑j=MI+2

(j−MI)/2∑k=1


2 =

=∑j,k

[1 ≤ k ≤ (j −MI)/2][MI + 2 ≤ j ≤ n] ujujqisec(uj , uk)fj∆u

2 =

=∑j,k

[1 ≤ k ≤ (n−MI)/2][2k +MI ≤ j ≤ n]ujujqisec(uj , uk)fj∆u

2 =

=

(n−MI)/2∑k=1

n∑j=2k+MI


2. (C.6)

But now it would be useful to have an energy term uk. We can use the energy interval definition to

re-write the uj term,

uj =

(j − 1

2

)∆u =

(k − 1

2+ j − k − MI

2+MI +

1

2

)∆u = uk−j−MI

+

(MI +

1

2

)∆u

Thus the relation to be used in following calculations is,

n∑k=1

J3Ikuk∆u =

n∑j=MI+2

(j−MI)/2∑k=1


2 =

(n−MI)/2∑k=1

n∑j=2k+MI


2 =

=

(n−MI)/2∑k=1

n∑j=2k+MI


2+

+

(n−MI)/2∑k=1

n∑j=2k+MI

uj−k−MIujq

isec(uj , uk)fj∆u

2+

+

(n−MI)/2∑k=1

n∑j=2k+MI

(MI +

1

2

)∆u ujq

isec(uj , uk)fj∆u

2

(C.7)

83

The sum of all three terms

In the end we have the following relations,n∑k=1

JIkuk∆u =

=

J1I︷︸︸︷

(n−MI)/2∑k=1

n∑j=2k+MI+1


2 +

J2I︷︸︸︷

n−MI−1∑k=1

2k+MI∑k+MI+1


2−

J3I︷︸︸︷

n∑j=MI+2

(j−MI)/2∑k=1


2 =

=

using C.4︷︸︸︷(n−MI)/2∑

k=1

n∑j=2k+MI


2 +O(∆u) +


k=1

n∑j=2k+MI

uj−k−MIujq

isec(uj , uk)fj∆u

2−

−


k=1

n∑j=2k+MI


2−(n−MI)/2∑

k=1

n∑j=2k+MI

uj−k−MIujq

isec(uj , uk)fj∆u

2 +

+


k=1

n∑j=2k+MI

(MI +

1

2

)∆u ujq

isec(uj , uk)fj∆u

2 =

=

(n−MI)/2∑k=1

n∑j=2k+MI

(MI +

1

2

)∆u ujq

isec(uj , uk)fj∆u

2 +O(∆u) =

=

using C.6︷︸︸︷(j−MI)/2∑

k=1

n∑j=MI

(MI +

1

2

)∆u ujq

isec(uj , uk)fj∆u

2 +O(∆u)

It is now possible to retrive some physical significance of this term using

σI(uj) =

(j−MI)/2∑k=1

qisec(uj , uk)∆u,

and

〈νI〉 =

n∑j=1

N

√2e

mujσIfj∆u,

we get a contribution to the power balance equation, from the discrete ionization operator, of

P dI =

(MI +

1

2

)∆u

√m

2e

〈νI〉N

.

C.6 Energy balance equation

In the case of electron density exponential temporal growth,

−n∑k=1

(Ak −Bk) fk∆u2

√2e

m−CI 〈u〉+

〈νI(u)〉N

(MI +

1

2

)∆u+O(∆u)+

∑i,j


)mij∆u = 0.

(C.9)

And on the case of electron density exponential spatial growth,

−n∑k=1

(Ak −Bk) fk∆u2

√2e

m−αRne

√me

2 eΓE+

〈νI(u)〉N

(MI +

1

2

)∆u+O(∆u)+

∑i,j


)mij∆u = 0.

(C.10)

84

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