UCLA Electronic Theses and Dissertations

UCLA UCLA Electronic Theses and Dissertations

Title Computational Modeling of Plasma-induced Secondary Electron Emission from Micro- architected Surfaces

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Author Chang, Hsing-Yin

Publication Date 2019

Peer reviewed|Thesis/dissertation

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Computational Modeling of Plasma-induced Secondary Electron Emission from Micro-architected Surfaces

A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Materials Science and Engineering

Hsing-Yin Chang

2019 c Copyright by

Hsing-Yin Chang 2019 ABSTRACT OF THE DISSERTATION

Computational Modeling of Plasma-induced Secondary Electron Emission from Micro-architected Surfaces

Hsing-Yin Chang Doctor of Philosophy in Materials Science and Engineering University of California, Los Angeles, 2019 Professor Jaime Marian, Chair

Advances in electrode, chamber, and structural materials will enable breakthroughs in future generations of electric propulsion and pulsed power (EP & PP) technologies. Although wide ranges of EP & PP technologies have witnessed rapid advances during the past few decades, much of the progress was based on empirical development of materials through experimentation and trial-and- error approaches. To enable future technologies and to furnish the foundations for quantum leaps in performance metrics of these systems, a science-based materials development effort is required. The present study aims to develop computational models to simulate, analyze, and predict the secondary electron emission of plasma devices in order to aid the design of materials architectures for EP & PP technologies through an integrated research approach that combines multi-scale modeling of plasma-material interactions, experimental validation, and material characterization. The range of materials of interest in EP & PP technologies include refractory metals, such as tungsten (W) and its alloys (W-Re) and molybdenum (Mo), ceramic composites, such as boron nitride (BN) and

alumina (Al2O3), high-strength copper alloys, and carbon-carbon composites. These classes of materials serve various design functions; primarily in cathode and anode applications, in accelerator grids, and in beam dumps of high power ( a few GW) microwave (HPM) sources. We ﬁrst give ∼ an overview of our fundamental understanding for the limits of using these materials in EP & PP technologies, and the opportunity to design material architectures that may dramatically improve

ii their performance. Next, we introduce the computational framework to model secondary electron emission from micro-architected surfaces. A detailed description of the underlying physics, computational models and methods are then provided, followed by simulation results, respectively. Finally, discussions and conclusions of our major ﬁndings as well as suggested future work are given.

iii The dissertation of Hsing-Yin Chang is approved.

Jenn-Ming Yang

Nasr M. Ghoniem

Richard E. Wirz

Jaime Marian, Committee Chair

University of California, Los Angeles

2019

iv To my parents and family, for their love, endless support and encouragement

v TABLE OF CONTENTS

List of Figures ...... x

List of Tables ...... xiv

Acknowledgments ...... xv

Vita ...... xvi

1 Introduction ...... 4

1.1 Motivation ...... 4

1.1.1 Electric Propulsion ...... 4

1.1.2 Magnetic Conﬁnement Fusion ...... 7

1.1.3 Other Applications ...... 9

1.2 Problem Statement ...... 9

1.2.1 Secondary Electron Emission ...... 9

1.2.2 Concept of Potential Solutions ...... 11

1.3 Thesis Outline ...... 12

2 Computational Approach ...... 14

2.1 Computational Framework ...... 14

2.2 Molecular Dynamics Simulation ...... 16

2.2.1 Ab Initio Molecular Dynamics Simulation ...... 16

2.2.2 Classical Molecular Dynamics Coupled With Electronic Subsystem Dy- namics Simulation ...... 17

2.3 Monte Carlo Simulation ...... 20

vi 3 Calculation of Secondary Electron Emission Yields from Low-energy Electron Depo- sition in Tungsten Surfaces ...... 24

3.1 Introduction ...... 24

3.2 Theory and Methods ...... 26

3.2.1 Electron Scattering Theory ...... 26

3.2.2 Elastic Scattering ...... 26

3.2.3 Inelastic Scattering ...... 28

3.3 Monte Carlo Calculations ...... 33

3.4 Results ...... 35

3.5 Discussion and Conclusions ...... 39

4 Monte Carlo Raytracing Method for Calculating Secondary Electron Emission from Micro-architected Surfaces ...... 43

4.1 Introduction ...... 43

4.2 Computational Model ...... 45

4.2.1 Intersection Detection Algorithm ...... 45

4.2.2 Generation of Secondary Rays ...... 47

4.2.3 Finite Element Model and Surface Geometry Development ...... 50

4.3 Results and Discussion ...... 51

4.3.1 Veriﬁcation ...... 51

4.3.2 Micro-architected Foam Structures ...... 53

4.4 Discussion and Conclusions ...... 58

5 Monte Carlo Modeling of Low Electron Energy Induced Secondary Electron Emis- sion Yields in Micro-architectured h-BN Surfaces ...... 61

5.1 Introduction ...... 61

vii 5.2 Theory and Methods ...... 62

5.2.1 Electron-Insulator Interaction Model ...... 62

5.2.2 Elastic Scattering ...... 63

5.2.3 Inelastic Scattering ...... 66

5.2.4 Phonon Excitation ...... 69

5.2.5 Polaronic Effects ...... 70

5.3 Monte Carlo Calculation ...... 71

5.4 Results ...... 73

5.4.1 Flat Surfaces ...... 73

5.4.2 Micro-Architectured Foam Structures ...... 74

5.5 Discussion and Conclusions ...... 75

5.6 Veriﬁcation ...... 78

6 Conclusions ...... 81

6.1 Discussion of Results ...... 81

6.2 Future Work ...... 83

6.2.1 Charging Effect on Secondary Electron Emission ...... 83

A List of Symbols ...... 84

B Constants & Kinematical Quantities ...... 87

C Deﬁnition of Coordinate System ...... 88

D Classical Scattering ...... 90

D.1 Deﬁnition of Cross Section ...... 90

D.2 Rutherford Scattering ...... 92

viii E Partial Wave Analysis ...... 94

E.1 Preliminary Scattering Theory ...... 94

E.2 Partial Wave Expansion ...... 95

References ...... 98

ix LIST OF FIGURES

1.1 Thrust and speciﬁc impulse ranges for various forms of propulsion...... 8

1.2 Schematic of a tokamak chamber and magnetic proﬁle. Figure reproduced from [1]. . .9

1.3 Plasma sheath proﬁle (a) in the absence of secondary electrons (b) in presence of secondary electrons. Figure reproduced from [2]...... 10

1.4 Candidates geometries: (a) micro-spears (b) micro-nodules (c) micro-velvets (d) micro- pillars (e) micro-foams (f) self-similar surface structures...... 12

2.1 Temporal snapshots of a collision cascade created by a 5 keV Ag atom impinging onto an Ag (111) surface. The excitation energy density is shown as a continuous color map in addition to atoms displaced during the cascade. Figure reproduced from [3]. . . . . 21

2.2 Flowchart of the combined Molecular Dynamics approach. Figure reproduced from [4]. 23

3.1 Schematic diagram of the discrete collision model of electron scattering simulated using Monte Carlo...... 33

3.2 Normalized distributions for 100-eV primary electrons incident at 0◦: (a) Energy distribution of secondary electrons; (b) Angular distribution of secondary electrons. . . . 36

3.3 (a) Depth distribution of both emitted and thermalized (captured) electrons for a pri-

mary electron energy and incident angle of 100 eV and 0◦. (b) Relative occurrence of the main scattering mechanisms as a function of primary energy for normal incidence. . 37

x 3.4 (a) Total SEE yield from an ideally-ﬂat W surface as a function of primary electron energy for electrons incident at 0 . = this work; black dashed line = Ahearn (1931) ◦ • [5]; cyan dashed line = Coomes (1939) [6]; = Bronshtein and Fraiman (1969) [7]; N = Ding et al. (2001) using the method where the SE is assumed to come from a distribution of electron energies in the valence band; H = Ding et al. (2001) using the method where the SEs are assumed to originate from the Fermi level [8, 9]; = Patino et al. (2016) [10]. (b) Total SEE yield from smooth W as a function of primary electron energy, for electrons incident at 45 . = this work; = Patino et al. (2016) ◦ • [10]...... 38

3.5 Total SEE yield from a ﬂat W surface as a function of primary electron energy and incidence angles...... 39

3.6 (a) Total SEE yield from an ideally-ﬂat W as a function of primary electron energy,

for electrons incident at 0◦, 30◦, 45◦, 60◦, 75◦ and 89◦. (b) Surface plot of the total SEE yield from an ideally-ﬂat W as a function of primary electron energy and angle of incidence...... 40

3.7 Flow chart of the Monte Carlo program ...... 42

4.1 Flowchart of the raytracing Monte Carlo code...... 49

4.2 (a) Finite element model of a real micro-architected foam structure rendered from X- ray tomography images. (b) Histogram of surface element normals...... 51

4.3 SEE yield as a function of primary energy for normal incidence on ideally ﬂat W surfaces obtained using (i) scattering Monte Carlo (raw data from ref. [11]), (ii) sampling functions given in Equation (4.6), and (iii) using the raytracing model described here. . 52

4.4 (a) Image of a cubic open cell foam structure with the cage size l and ligament size t indicated. (b) Secondary electron yield versus electron beam energy at 0 degree incidence from the cubic cage...... 53

xi 4.5 SEE yield versus electron beam energy initially projected from 0 degree incidence to the foam at varying volume-fraction percentages. The inset shows (in increasing order) the dependence of the yield with volume fraction for primary energies equal to 50, 100, 200, 300, 400, 500, and 600 eV...... 54

4.6 SEE yield versus electron beam energy for normal and random incidence primary electrons. Experimental results by Patino et al. [10] are shown for comparison. . . . . 55

4.7 (a) Penetration depth distribution of thermalized electrons in bulk W with primary electron energies of 100 and 1000 eV at normal incidence. (b) Depiction of a possible ray trajectory through a ligament with d as the traversed distance inside it. (c) Distance d distribution of rays in a 4%-volume fraction foam surface...... 56

4.8 Penetration depth of electrons in a 4% volume fraction foam with an electron beam energy of 100 eV at normal and random primary electron incidence. This distribution corresponds to locations at which electrons ‘thermalize’ and become absorbed into the material. Typical foam thicknesses are approximately 3 mm [12]...... 57

4.9 Computational cost (measured as CPU time) per primary ray as a function of primary energy. The CPU overhead loosely correlates with the number of daughter/granddaughter rays generated by each primary ray (in red)...... 60

5.1 Values of the DESCS of 50, 100, and 500 1000 eV electrons scattered by h-BN as a function of the scattering angle...... 65

5.2 (a) Energy loss function of h-BN in the optical limit. (b) Inelastic mean free path of h-BN as a function of primary electron energy...... 68

5.3 Plot of the various scattering cross sections of the processes considered here as a function of primary electron energy...... 71

xii 5.4 (a) Total SEE yield from smooth hBN surface as a function of primary electron energy for electrons incident at 0 . = this work; = Dawson (1966) [13]; = ONERA ◦ • H (1995) [14]; = PPPL (2002) [15]; N = Christensen (2016) [16]. (b) Total SEE yield from an ideally-ﬂat hBN as a function of primary electron energy, for electrons

incident at 0◦, 15◦, 30◦, 45◦, 60◦, 75◦ and 89◦...... 75

5.5 (a) Surface plot of the total SEE yield from an ideally-ﬂat hBN as a function of primary electron energy and angle of incidence. (b) Surface plot of the SEE energy distributions from an ideally-ﬂat hBN as a function of primary electron energy and angle of incidence...... 76

5.6 (a) SEE yield versus electron beam energy for normal incidence to the foam at varying volume-fraction percentages. The inset shows (in increasing order) the dependence of the yield with volume fraction for primary energies equal to 50, 100, 200, 300, 400, 500, and 600 eV. (b) SEE yield versus electron beam energy for random incidence to the foam at varying volume-fraction percentages...... 77

5.7 (a) Total SEE yield from smooth SiO2 surface as a function of primary electron energy for electrons incident at 0 . = this work; = Dionne (1975) [10]; = Barnard (1977) ◦ • [7]; N = Yong (1998). (b) Total SEE yield from an ideally-ﬂat SiO2 as a function of

primary electron energy, for electrons incident at 0◦, 15◦, 30◦, 45◦, 60◦, 75◦ and 89◦.. 79

5.8 (a) Surface plot of the total SEE yield from an ideally-ﬂat SiO2 as a function of primary electron energy and angle of incidence. (b) Surface plot of the SEE energy distribu-

tions from an ideally-ﬂat SiO2 as a function of primary electron energy and angle of incidence...... 80

C.1 Deﬁnition of coordinate system used in the Monte Carlo simulation program...... 89

D.1 Classical scattering. (r,φ) are the polar coordinates of the projectile, θ is the scattering angle, φ the azimuthal angle, b the impact parameter, d the distance of closest

approach, and d0 its minimum for a central collision...... 92

xiii LIST OF TABLES

1.1 Typical operating parameters for thrusters...... 7

5.1 Parameters used to model the energy loss function of h-BN...... 69

5.2 Parameters for h-BN in present calculation. Among these, αc, Wph, C and η are free parameters...... 72

xiv ACKNOWLEDGMENTS

First and foremost I would like to thank my advisor, Professor Jaime Marian, for his guidance, mentorship, and help over the past few years. The work presented in this dissertation would not be possible without his continuous support.

Furthermore, I’d like to express my gratitude to Professors Jenn-Ming Yang, Nasr Ghoniem, Richard Wirz, and Ya-Hong Xie for serving as my committee members for my qualifying exam and dissertation defense. Their guidance and suggestions have been invaluable for the completion of this dissertation.

Other current and past members of the Marian group, the Ghoniem group and the Wirz group have also played an instrumental role in my research. I would like to speciﬁcally thank Andrew Alvarado, Dylan Dickstein, Warren Nadvornick, Gary Li, Cesar Huerta, Dr. Chen-Hsi Huang Dr. David Cereceda and Dr. Nikhil Chandra Admal for valuable discussion and collaboration these past years.

To other friends – Z. Y.; K. T.; T. C.; – I’ve known all of you forever – thank you for your encouragement and friendship not only during my time at UCLA, but over the last decade or more.

And ﬁnally to my family – my parents T. C.; and S. Y.; and my sisters A. C. and N. C.; – thank you for your unconditional support and love.

xv VITA

2009–2013 B.S., Materials Science and Engineering, National Taiwan University (NTU), Taipei, Taiwan

2014 Research Assistant, Advanced Material Laboratory, Center for Condensed Mat- ter Sciences, NTU, Taipei, Taiwan

Winter 2016 Teaching Assistant, MAT SCI 143A Mechanical Behavior of Materials, Uni- versity of California Los Angeles (UCLA), Los Angeles, CA

Spring 2016 Teaching Assistant, MAT SCI 104 Science of Engineering Materials, UCLA, Los Angeles, CA

2016–2017 Simulation and Modeling Intern, Panel Process and Optics, Apple Inc., Cuper- tino, CA

Summer 2018 Computational Chemistry and Materials Science Graduate Intern, Quantum Simulations Group, Lawrence Livermore National Laboratory, Livermore, CA

Fall 2018 Teaching Associate, MAT SCI 243C Dislocations and Strengthening Mecha- nisms in Solids, UCLA, Los Angeles, CA

2014–Present Graduate Student Researcher, Department of Materials Science and Engineer- ing, UCLA, Los Angeles, CA

2016–Present Early Career Researcher, Institute for Digital Research and Education, UCLA, Los Angeles, CA

xvi PUBLICATIONS

H. Y. Chang, A. Alvarado & J. Marian, “Calculation of Secondary Electron Emission Yields from Low-energy Electron Deposition in Tungsten Surfaces” Applied Surface Science (2018)

A. Alvarado, H. Y. Chang, W. Nadvornick, N. Ghoniem & J. Marian, “Monte Carlo Raytracing Method for Calculating Secondary Electron Emission from Micro-architected Surfaces,” Applied Surface Science (2019)

H. Y. Chang, A. Alvarado, T. Weber & J. Marian, “Monte Carlo Modeling of Low-energy Electron- induced Secondary Electron Emission Yields in Micro-architected Boron Nitride Surfaces,” Nu- clear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, submitted for publication (2019)

D. Dickstein, H. Y. Chang, N. Ghoniem, J. Marian, M. Feldman, A. Hubble & R. Spektor, “Secondary Electron Emission of Copper Foam: An Empirical and Computational Comparison,” AIAA Journal, submitted for publication (2019)

H. Y. Chang, A. Alvarado & J. Marian, “Computational Modeling of Plasma-Induced Sputtering & Redeposition from Micro-Architected Surfaces,” Journal of Computational Physics, submitted for publication (2019)

H. Y. Chang, A. A. Correa, “Dynamics of Ejected Electrons from Laser Irradiation,” in prepara- tion

xvii PREFACE

The ability to predict materials behaviors with confidence and to further influence materials syn- thesis and design, is paramount to a number of industrial sectors, including aerospace, automotive, pharmaceutical, electronics, and energy. These sectors are among those that will greatly bene- fit from innovations that lead to the accelerated development and deployment of new materials. From the dawn of human existence, materials have been the cornerstone of the evolution of civi- lization. For thousands of years, materials science was solely empirical, which here corresponds to historical epochs by the materials used by the different civilizations such as the Stone, Cop- per, Bronze and Iron ages. Then came the paradigm of theoretical models a few centuries ago, characterized by the formulation of laws in the form of mathematical equations; in materials science, the three laws of thermodynamics are a good example. However, for progressively complex systems, the theoretical models sometimes became coupled that there exists no analytic solution. With the advent of computers a few decades ago, computational materials science became very popular, enabling simulation and modeling of complex real-world phenomena based on the theoretical models; in materials science, density functional theory (DFT) and molecular dynamics (MD) simulations are excellent examples. Traditionally materials modeling problems have been approached more along the lines of sequentially coupled length or time scales. Then the paradigm shifts toward the greater use of concurrent multiscale methods, which is crucial from both the application and the computer science perspectives, and maps well to the increasingly hierarchical and heterogeneous nature of computer architectures. Computational materials models, however, do not always perform as desired in the sense of being able to explain experimental data under different circumstances. Discrepancies between observations and predictions may be generally explained by modeling uncertainties, errors in approximations of the electronic structure calculations, the usage of surrogate models (e.g. cluster expansions in ab-initio simulations), inaccuracy of material-specific interatomic potentials in MD simulations, and nonlinear coupling across different time and length scales, and may more. As well, the mathematical model itself may be uncertain or incompletely known. Additional challenges in predictive multiscale materials models include the inability to model high dimensionality problems, attributed to information loss when

1 performing coarse-graining, rare events modeling, or a lack of scalable uncertainty quantiﬁcation techniques, and other issues. The revolution taken place over recent years in data science, along with the rapid advances in high performance computing and in the development of experimental means to collect structure and property data at different resolutions and scales, provide a means for a data-driven predictive modeling and design of material systems. To date, materials scientists have used machine learning techniques including partial least squares (PLS) regression, least absolute shrinkage and selection operator (LASSO) regression, ridge regression, decision trees and neural networks to build predictive models for diverse applications. For example, there are now models to predict the melting temperatures of binary inorganic compounds, the formation enthalpy crystalline compounds, the band gap energies of certain classes of crystals and the mechanical properties of metal alloys. Although these models demonstrate the promise of machine learning, they only cover a small fraction of the properties used in materials design and the data sets available for creating such models are limited. Generally speaking, if a physics-based theoretical model is capable of describing a system, it is often adopted. Nonetheless, this does not mean that a machine learning approach would not work. The ability of machine learning models to learn from experimental data also means they can learn from physics. Given enough examples of how a physical system behaves, machine learning models can learn the materials behavior and make accurate predictions. In addition, the computational cost often scales with the complexity of physics-based models. Even though a physics-based model is capable of describing the system in great detail, solving this model could be complicated and time-consuming. Therefore, a physics-based approach might break down for making real-time predictions on live streaming data. In this case, a simpler machine learning based model could be a good alternative. While the training phase of a machine learning model is often of high computational complexity, once the model has ﬁnished training, making predictions on new data is straightforward. This is where the hybrid approach of combining machine learning and physics-based modeling comes into play.

While it is not uncommon to have the juxtaposition of theoretical methods with data-driven methods, the real challenge is the integration, capitalizing on the strength and information present in each of them. Examples are in using data driven techniques to validate or calibrate simulation

2 models with test data or to reduce the vast amount of simulation model data to formats feasible for integration in system of system models. Alternatively, simulation models can be used to design optimal test sets or in the training phase of neural networks for quality assurance. Common to these applications is that physics based models and data driven approaches are blended and explored without prejudice.

As Charles Darwin wrote ”Ignorance more frequently begets confidence than does knowledge: it is those who know little, and not those who know much, who so positively assert that this or that problem will never be solved by science.” This is particularly true in materials science. By necessity we often rely on experimental data of questionable quality relevance and completeness to answer all manner of questions. Moreover the data often come from multiple sources of varying quality and reliability. Simulation and modeling can be an elegant and efficient way to get around the problem of not having the right data. With the advancement of large-scale parallel computing power and methodologies, the capability of simulating large-scale complex system becomes practical and achievable. Moreover, it has transformed simulation and modeling into a new paradigm where simulation results must be mined to extract useful data that can be compared to experiments on a meaningful basis. Simulated experimentation effectively accelerates and replaces effectively the ”wait and see” anxieties in discovering new insight and explanations of future behavior of the real system. Advanced materials are essential to human well-being and are the foundation for emerging industries. Yet, experimental testing for incorporating advanced materials into applications is prohibitively expensive and time-consuming. Predictive materials modeling therefore suggests itself as an ideal method to tackle this problem. The information-theoretic approach to materials predictive modeling and design has proven to have great potential for significant impact and relevance to technologically important industries and will transform life on the planet in the foreseeable future.

3 CHAPTER 1

Introduction

1.1 Motivation

The goal of the proposed research effort is to understand, predict, and ultimately control the plasma-material interactions of many applications, including electric propulsion devices, divertors and limiters of magnetic fusion devices, and plasma processing systems. To aid in early stage design and later qualiﬁcation of these plasma devices, development of a computational life-prediction tool is needed since experimental lifetime testing is prohibitively expensive and time-consuming, requiring up to tens of thousands of hours of testing and costing hundreds of thousands of dollars for propellant supply and use of facilities. Computational tools that can provide quick and accurate simulation results are therefore a valuable asset for the device design and planning processes. Such insights may lead to improvements in the performance and lifetime of plasma devices, thereby enabling space propulsion technologies that extend the capabilities for solar-system exploration and the production of nuclear energy.

1.1.1 Electric Propulsion

Electric propulsion (EP) is a class of space propulsion that utilizes electrical power to accelerate a propellant by different possible electrical and magnetic means. The use of electrical power enhances the propulsive performances of the EP thrusters by achieving low thrust-to-mass ratio

F/M, coupled with a high speciﬁc impulse Isp, resulting in a reduction in the amount of propellant required for a given space mission or application compared with conventional chemical thrusters. Reduced propellant mass can signiﬁcantly decrease the launch mass of a spacecraft or satellite,

4 leading to lower costs from the use of smaller launch vehicles to deliver a desired mass into a given orbit or to a deep-space target.

Electric thrusters propel the spacecraft using the same basic principle as chemical rockets’ accelerating mass and ejecting it from the vehicle. Starting with Newton’s second law of motion,

F = ma (1.1)

The force in this case is the thrust produced by the rocket

F = T = mv˙ exit (1.2)

wherem ˙ is the propellant mass ﬂow rate and vexit is the propellant exhaust velocity.

Rewriting Equation (1.1) and (1.2) to

∂m ∂v vexit = m (1.3) ∂t ∂t and integrating over time, while assuming that vexit is constant, produces the famed rocket equation,

m0 ∆v = vexit ln (1.4) m f which can be rearranged to be m f ∆v/v = e− exit (1.5) m0 Then for missions with a speciﬁed ∆v requirement, a higher propellant exit velocity allows for

the final mass m f to be a greater proportion of the total initial mass m0. Since the final mass is the mass of the spacecraft minus the propellant mass, maximizing the ratio of the final to initial mass increases the amount of mass that can be allocated for scientific equipment or other valuable payload. For every space mission, there is an optimal exit velocity, or in a more commonly used

form, the specific impulse. The specific impulse, Isp, is the ratio of the thrust to the propellant weight flow rate, mv˙ exit vexit Isp = = (1.6) mg˙ 0 g0 2 where g0 is the sea-level gravitational acceleration, 9.81 m/s .

5 Alternatively, EP enables missions with larger mission ∆v, leading to extended mission lifes- pans (e.g., for commercial satellites that provide cell-phone communication, broadband internet, and GPS, and for satellites studying the Earth’s climate, agricultural resources, and renewable resources) or the ability to reach previously inaccessible targets. The thrust produced from EP devices is smaller than that produced by chemical systems, but is more accurate, a requirement for high-precision formation ﬂying missions.

EP thrusters are generally categorized in terms of the mechanism through which the electricity is used to accelerate the thrust. These technologies group broadly into three divisions: electrothermal, electrostatic, and electromagnetic. Electrothermal devices (e.g. resistojet, arcjet) use electric power to heat the propellant, and as the propellant gains energy, it is then expelled thermodynam- ically through a nozzle. Electrostatic devices (e.g. ion thruster, field emission electric propulsion thruster) use an electric field to accelerate the ionized propellant. Electromagnetic devices (e.g. Hall thruster, pulse plasma thruster/PPT, magnetoplasmadynamic thruster/MPD) use combined electric and magnetic fields to accelerate the propellant.

Chemical rockets typically have specific impulses in the hundreds of seconds. Electrothermal thrusters have specific impulses typically ranging from the low hundreds to around one thousand seconds. Electromagnetic thrusters offer specific impulses in the thousands of seconds. Electro- static thrusters are able to achieve specific impulses in the thousands of seconds and possibly even tens of thousands of seconds. Typical operating ranges of thrusters can be found in Table 1.1 and Figure 1.1.

Two EP devices which have been employed for satellites and interplanetary missions include the ion thruster and Hall thruster. Since Hall thruster erosion issue has been resolved via magnetic conﬁnement, Hall thrusters are generally not a relevant motivation for the present research focused on erosion mechanisms from plasmas. However, Hall thrusters smaller than Magnetically Shielded Miniature (MaSMi) Hall Thruster [17] still suffers from erosion due to their small size and are unable to provide enough magnetic conﬁnement.

Researches on performance and lifetime assessment of ion thrusters have been conducted in several research groups [18], including the 3 cm Miniature Xenon Ion (MiXI) thruster developed

6 by Wirz, which is the ﬁrst miniature ion thruster to demonstrate stable operation and noteworthy total efﬁciency at these scales; the 25-cm Xenon Ion Propulsion System (XIPS-25) manufactured by L-3 Communications [19, 20], which serves both orbit-raising and station-keeping roles on the Boeing 702 communication satellite; and the NASA Evolutionary Xenon Thruster (NEXT) at Glenn Research Center [21, 22], an ion thruster about three times as powerful as the NASA Solar Technology Application Readiness (NSTAR) used on Dawn and Deep Space 1 spacecraft.

Thruster Specific Input Efficiency Propellant Impulse Power Range [s] [kW] [%] Cold Gas 50 75 Various − − − Chemical 150 225 N H − − − 2 4 (monopropellant) H2O2 Chemical 300 450 Various − − − (bipropellant) Resistojet 300 450 N H monoprop − − − 2 4 Arcjet 300 450 N H monoprop − − − 2 4 Ion Thruster 300 450 Xenon − − − Hall Thruster 1500 2000 1.5 4.5 35 60 Xenon − − − PPT 850 1200 < 0.2 7 13 Teflon − −

Table 1.1: Typical operating parameters for thrusters.

1.1.2 Magnetic Conﬁnement Fusion

Magnetic confinement fusion is an approach to generate thermonuclear fusion power that uses magnetic fields to confine the hot fusion fuel in the form of a plasma. A common reaction utilizing deuterium-tritium plasmas to produce helium and energetic neutrons is

D(2.014102 [amu])+T(3.016050 [amu]) 4 He (4.002603 [amu])+1 n (1.008665 [amu])+17.58 [MeV]. → 2 0 (1.7)

7 Figure 1.1: Thrust and speciﬁc impulse ranges for various forms of propulsion.

The energy released per a kilogram of deuterium is calculated as:

17.58 106 [eV/atom] 6.022 1023 [atoms/mole] 1.602 10 19 [J/eV] E = × × × × × − 2.014 10 3 [kg/mole] × − (1.8) = 8.42 1014 [J/kg] × Most magnetic fusion devices today make use of a toroidally shaped magnetic field to confine the plasma, as shown in Figure 1.2. Such a toroidal device with magnetic field coils is called a tokamak, after a Russian acronym. Plasma is surrounded by the first wall, which shields external components from high energy neutrons and harvests their energy. The divertor is located at the bottom of the tokamak and collects high particle and heat fluxes from energetic He and other impurities. As with the plasma-facing walls of electric propulsion devices, the divertor in a tokamak may also emit secondary electrons due to impact by energetic plasma electrons (electron energies are typically 1-100 eV in the divertor). These secondary electrons emitted from plasma-facing materials gyrate in a sheath, which is in the interface between an edge plasma and the plasma-facing material surface. As a result, some of them return to the surface within their first gyration, which results in low effective electron yield of the plasma-facing materials. Due to larger Lorentz force, the higher the energies of the secondary electrons, the more the secondary electrons return to the surface, causing a low-energy shift of the energy distribution of electrons entering into the plasma.

8 Figure 1.2: Schematic of a tokamak chamber and magnetic proﬁle. Figure reproduced from [1].

1.1.3 Other Applications

In particle accelerators (e.g. with positron beams), particle impact and cyclotron radiation incident on the walls can produce electrons that themselves can interact with the wall and lead to SEE. This multipacting effect can create a cloud of electrons that may lead to instabilities in the particle beams and overheating of facility components.

1.2 Problem Statement

Degradation of plasma-facing materials is a signiﬁcant problem in plasma devices for many applications, including fusion, high-energy-density plasma, electric propulsion and multi-pactor. Two interactions of particular importance causing the degradation of plasma-facing materials are (1) secondary electron emission from plasma-facing materials and (2) ion-atom collisions in the plasma bulk. In the present work, we focus on the effects of secondary electron emission.

1.2.1 Secondary Electron Emission

In plasma devices, a signiﬁcant number of secondary electrons is produced under plasma exposure, which reduces the sheath potential at the wall and increases the power loading. To satisfy the wall boundary condition, the requirement of local net current equal to zero and particle balance for the

9 three species gives I = I γI = I (1 γ) (1.9) iw ew − ew ew − where γ is the SEE yield under electron bombardment.

1 Equation (1.9) can be solved for the sheath potential φs, including the effect of SEE: " r # kTe 2M φs = ln (1 γ) (1.10) e − πm

The presence of secondary electrons tends to lower the sheath potential, making it more positive with respect to the bulk plasma, as shown in Figure 1.3. Under strong emissions, the electron population are reﬂected back to the bulk plasma and lead to formation of virtual cathodes.

Figure 1.3: Plasma sheath proﬁle (a) in the absence of secondary electrons (b) in presence of secondary electrons. Figure reproduced from [2].

The reduction of sheath potential due to SEE ﬁnd important applications in sputtering of channel walls in electric propulsion devices and divertors in fusion devices, as well as in deposition processes of thin ﬁlms. Yao [23] studied the effects of SEE on sheath potential in an ECR plasma source using RPA and found that the effects of SEE are important. Pandey and Roy [24] derived the sheath potential in the presence of SEE and sputtering yield following the works of Hobbs and Wesson [25]. They showed that the combined impact of secondary electrons and sputtering yield can be given by " p # kTe 1 γ Mi/2πme φs = ln − (1.11) Y E 1/2 e 1 Y 1 − 0 − 1 Y φs − −

1 1/2 This expression is slightly different than that found in some literature because we have approximated e− = 0.61 0.5 for the coefﬁcient in the expression for the Bohm current. ≈ 10 In the limit where sputter yield is zero, the wall potential is identical to the one by Hobbs and Wesson [25] and is given by Equation (1.10).

It is numerically shown that with increasing secondary emission from a wall, the velocity distribution of the primary electrons is depleted and the tail shifts [26]. This depletion is due to the reduction in sheath potential allowing energetic electrons to be lost to the wall. So the influence of SEE on plasma electron energy distribution function (EEDF) can be seen by a depletion of the tail. Another region where SEE may play a role would be modification of the bulk of the plasma, due to the cooling or heating of the plasma as a result of thermalization with the bulk population which would be reflected in the bulk population of the EEDF.

1.2.2 Concept of Potential Solutions

Recently, a new wall concept based micro-architected surfaces has been proposed to mitigate SEE and ion-induced sputtering [27, 28, 29, 30] issues. Demonstration designs based on high-Z refractory materials have been developed, including architectures based on metal micro-spears, micro- nodules, micro-velvets, micro-brushes and micro-foams [31, 32, 33, 34, 10] as shown in Figure 1.4. The idea behind these designs is to take advantage of very-high surface-to-volume ratios to reduce SEE and ion erosion by internal trapping and redeposition. Preliminary designs are based on W, W/Mo, and W/Re structures, known to have intrinsically low SEE and sputtering yields propensity. A principal signature of electron discharges in plasma thrusters is the low primary electron energies expected in the outer sheath, on the order of 100 eV, and only occasionally in the several hundred eV regime. Accurate experimental measurements are exceedingly difﬁcult in this energy range due to the limited thickness of the sheath layer, which is often outside the resolution of experimental probes [35, 36, 37]. Modeling then suggests itself as a complementary tool to experiments to increase our qualitative and quantitative understanding of erosion processes.

11 (a) (b)

(e) (f)

Figure 1.4: Candidates geometries: (a) micro-spears (b) micro-nodules (c) micro-velvets (d) micro-pillars (e) micro-foams (f) self-similar surface structures.

1.3 Thesis Outline

The main topic of the present work is the development of computational models to simulate, analyze, and predict the erosion proﬁle of the plasma devices in order to aid the design of materials architectures for electric propulsion and pulsed power (EP & PP) applications. Chapter 1 introduces the background of the present work, summarizes previous work on comprehending the erosion mechanisms and predicting the performance and lifetime of such plasma devices. Chapter 2 gives an overview of the computational approaches used to simulate particle-material interactions.

12 Chapter 3, 4 and 5 provide details of the secondary electron emission mechanisms and reviews current understanding of the phenomenon. Models and simulation results for different materials are then presented. A detailed description of how the models are implemented in the simulation as well as how the codes are tailored to forecast lifetime is also given. Finally, a summary of contributions and suggestions for future work are provided in Chapter 6.

13 CHAPTER 2

Computational Approach

2.1 Computational Framework

When an energetic particle impinges upon a surface, a sequence of collisions among near-surface atoms (atomic collision cascade) is initiated. In the course of the evolution of this atomic collision cascade in time and space, particles may be released from the surface into the gas phase. In general, the ﬂux of emitted particles not only consists of neutrals but also comprises excited atoms, ions, and electrons. This phenomenon clearly indicates that the development of the atomic collision cascade within the solid is accompanied by electronic substrate excitations. Although investigated for decades, the underlying mechanisms of this kinetic excitation of the bombarded solid and the effects associated therewith are still not yet completely understood. These so-called secondary electrons have a non-linear coupling effect with the bulk plasma and affect the performance of thrusters by changing the sheath potential as well as the electron energy distribution. This inﬂuence is not yet fully understood in the community and thus the computational models are based on assumptions that are not necessarily accurate.

The traditional approach to simulate SEE is to use particle-in-cell (PIC) codes that treat secondary electrons as a fluid particle flux in a highly ionized plasma and intense magnetic fields, and to estimate their (detrimental) effect on thruster plasma stability. However, it is not clear whether secondary electrons themselves result in wall material damage (sputtering, blistering, localized heating, etc) in addition to that due to plasma ion deposition.

Moreover, in analytical treatments of SEE in solids, three stages are considered, namely, the production of internal secondary electrons by collision in the material between fast primaries and

14 electrons, the cascade process by which these secondaries diffuse through the solid and reach the surface, and the transmission through the surface potential barrier. The collision cascade process has been primarily described using the Boltzmann equation [38]. Various assumptions are made in these models, concerning the rate of production of internal secondaries, the inelastic collision during diffusion toward the surface and the requisite simpliﬁcations for the resolution of integro- differential equations. However, none of these models based upon Boltzmann equation give a complete and satisfactory description of the SEE. The main criticism remaining is the very difﬁcult mathematical problem connected with the resolution of integro-differential equations. [39, 40, 41]

Other analytic models have also been developed to explain excitation and secondary electron formation but the shortcoming of practically all of these studies is that the particle kinetics have to be known as a prerequisite for the description of the resulting electronic excitation. All analytical formulations of the collision dynamics, however, must be regarded as rather rough approximations which are restricted to the prediction of average, macroscopic quantities. In particular, the local, microscopic character of the atomic motion within a collision cascade is neglected by such approximations.

Further, the Molecular Dynamics (MD) simulations can simulate particle dynamics by means of numerical integration of the corresponding coupled Newtonian equations of motion, which is in general more accurate as they consider many-body interactions and can provide more details about the cascade process. However, MD is very computationally intensive, which poses a limit for the time and length scales simulated. As well, it should be noted that the accuracy of MD relies on the accuracy of the interatomic potential used. Moreover, it has recently been shown that the electronic energy loss should be taken into account in MD simulations, while the majority of MD studies on radiation damage cascades ignore this effect [42, 43]. For nanostructured materials whose size is less than the mean-free-path of ions, it was recently shown that electronic stopping is the dominant energy loss mechanism for incident ions [44], indicating the necessity of considering electronic energy loss in MD simulations.

The Monte Carlo (MC) simulations, on the other hand, usually only consider two-body interactions and use universal potentials in many cases, gives us a more efﬁcient way to evaluate the many

15 physical quantities necessary to the study of the interactions of particle-beams with solid targets. Letting the particles carry out an artiﬁcial random walk and taking into account the effect of the single collisions, it is possible to accurately evaluate the diffusion process, and therefore suggests itself as an ideal method to approach this problem.

In the present work, we study plasma-induced sputtering and SEE from micro-architected surfaces using a combination of ray tracing and the method for particle emission yields calculation from ideally-ﬂat surface. The incident energy spectrum and angle of incidence for each type of impinging particle will be an input provided to the simulations, while, in exchange, our approach will furnish both ion sputtering as well as SEE yields and energy spectra from micro-architected surfaces. These yield and energy distribution functions can in turn be used as source terms to link with the plasma sheath region, to assess the associated impact on local plasma kinetics and stability in a self-consistent fashion.

For completeness, we provide the details of both the modiﬁed MD model and MC model to demonstrate the logical progression of thoughts, although our computational approach is based on the MC model as we have learned more things about the physics of the problem and have implemented things computationally.

2.2 Molecular Dynamics Simulation

2.2.1 Ab Initio Molecular Dynamics Simulation

From a fundamental point of view, a rigorous description of the particle dynamics together with the kinetic excitation requires one to numerically solve the corresponding many-body Schrodinger¨ equation for the entire particle system, where ab initio methods are used to generate the forces needed for Molecular Dynamics (MD) simulations. This way, the electronic degrees of freedom would inherently be incorporated into the computation and directly be coupled to the nuclear motion. In the Ab initio Molecular Dynamics (AIMD), instead of using a prescribed potential, one solves the interatomic forces at a given time instant as follows. From a quantum-mechanical per- spective, the system at a ﬁxed time can be parametrized in terms of the coordinates of the nuclei

16 and the relevant electrons. By invoking the Born-Oppenheimer approximation, one can regard the nuclei fixed at the instantaneous positions of the atoms. This way, one can write a time-independent Schrodinger¨ equation for the many-body wave function of the electrons. This Schrodinger¨ equation is then solved using time-independent density functional theory (DFT) to obtain the energy. The energy is then considered to be a function of the nuclear coordinates that were fixed earlier, and it can thus act as the interatomic potential that is needed to compute the forces in Newton’s equation of motion for the nuclei. Thus, by computing the gradients of the DFT energy at this fixed point with respect to the nuclear coordinates, the forces are obtained and the nuclei are moved accordingly to get to the next time step. The DFT process is then repeated with these new nuclear coordinates. Such ab initio calculations, however, are far too complex to date to be able to treat a particle ensemble large enough to embed an atomic collision cascade.

2.2.2 Classical Molecular Dynamics Coupled With Electronic Subsystem Dynamics Simu- lation

We then applied an MD-based computer simulation model capable of describing the generation of electronic excitation energy within atomic collision cascades in metals as well as the transport of excitation energy away from the spot of generation. The model is based on earlier work by Duvenbeck et al. [4, 45, 46, 3] to predict kinetic electron emission in unison with sputtering in a uniﬁed atomistic framework.

In this modiﬁed version of MD, the coupled Newtonian equations of motion are numerically integrated for the projectile and all target atoms in order to obtain the time evolution of the entire particle system given by

2 d ri(t) drri(t) M + K = ∇r V(r (t),r (t),...,r (t)). (2.1) dt2 dt − ri 1 2 N

The inter-atomic forces are modeled via a many-body potential ﬁtted to the properties of solid W (including recent W-Ar potentials). The incorporation of the electronic system of the metal is realized in terms of a quasi-free Fermi gas picture, i.e., the atomic collision cascade is assumed to be embedded into a free electron gas characterized by its Fermi energy EF and density ne.

17 The kinetic excitation via direct atom-electron collisions is described within the framework of the Lindhard theory of electronic stopping, yielding a velocity-proportional friction force acting on every moving cascade atom. Consequently, the electronic energy loss dE within the time interval th i dt for the i particle is proportional to its kinetic energy Ek. This energy loss is treated as a source of excitation energy fed into the electronic system. Thus, the atomic collision cascade generates a space- and time-dependent source of excitation energy given by

N dE i (r,t) = A∑Ekδ(r(t) ri(t)) (2.2) dt f ric i − where ri(t) is the position of particle i at time t, N is the total number of atoms, and A is an adjustable friction parameter.

The quasi-molecular orbitals (MOs) are assumed to be generated due to the mixing of atomic wave functions during a close encounter of two colliding atoms within the solid. The eigenenergies of these MOs, EMO, strongly depend on the interatomic distance and can be pre-calculated for metals such as W or Re. During the collision, some selected orbitals may be promoted to higher energy as to exceed the Fermi level of the metal and therefore transitioning from the quasi-MO into free conduction band states. The transition rate corresponds to a ﬁnite level width ∆ of the promoted MO which can inferred from experimental photoemission spectroscopy data and that here we assume to be constant. Within the MD simulation, the probabilistic nature of the electronic transition from the MO into a free conduction band state is implemented using a Monte Carlo scheme as follows:

At each MD time step i of length dt, the pair correlation function of the entire system is • calculated. In the case that there exists a pair of atoms with r < rc, an electronic transition is assumed to occur if ξ < ∆dt/h¯, where ξ is a random number in the interval [0, 1].

Once the transition has occurred, an additional autoionization process of the originally dou- • bly occupied MO is prohibited. The collisional excitation energy is given by

Eexc = E (r(t)) E (2.3) MO − F

where EF is the Fermi energy.

18 In order to enforce energy conservation, this energy is subtracted from the potential energy • such that the potential energy deﬁcit equals the excitation energy.

The transport of excitation energy generated by electronic friction or autoionization from its point of generation is treated in terms of the nonlinear diffusion equation:

exc ∂E(r,t) dE Ek (k) ∆(D(r,t)∆E(r,t)) = (r,t) + ∑ δ(t tk)δ(r r ) (2.4) ∂t − dt f ric k ∆t − − The l.h.s. of Equation (2.4) constitutes the transport term, whereas the r.h.s. contains all (time- and space-dependent) sources of excitation energy. The second sum on the r.h.s. loops over all close collisions k that lead to an autoionizing transition during the evolution of the cascade. The transport dynamics are governed by the excitation energy diffusivity D, which in general may depend

on the lattice temperature Tl, the local electron temperature Te, and a lattice order parameter. Equa- tion (2.4) is numerically solved by means of a ﬁnite differences scheme using Neumann boundary conditions at the surface in combination with pseudoinﬁnite boundary conditions at all other system boundaries. An image obtained with a method similar to the one proposed here is given in Figure 2.1, where a cascade of atoms produced by a particle impingement at the surface can be tracked as a function of time. The associated time and space evolution of the excitation energy density is shown on the right.

For each trajectory, the numerical solution of Equation (2.4) yields a space- and time-dependent excitation energy density proﬁle E(r,t) within the cascade volume that may be transformed into an

electron temperature distribution Te(r,t). This electron temperature Te should be primarily taken as an effective parameter to characterize the local excitation energy density rather than a ”real” electron temperature. In order to obtain the kinetic electron emission (KEE) yield γ, we assume a simple thermal emission model as proposed by Sroubek. Following these workers, the electron

current density je at the surface is given by:

emk2 T 2(r,t) ∞ ( 1)n 1 nφ(r,t) j (r,t) = B e − − exp (2.5) e 2 3 ∑ 2 2π h¯ n n − kBTe(r,t) where e is the elementary charge, m the (effective) electron mass, and φ = EF ?µ being the excess potential. The chemical potential µ can be taken from, e.g., the Sommerfeld model. The kinetic

19 electron emission yield γ is obtained by numerically integrating Equation (2.5) over surface area A and the duration time tc of the cascade, i.e.,

1 Z tc Z γ = je(r,t)dAdt (2.6) e 0 A It should be noted that in addition to the above treatment, electron-phonon coupling effects can also be included via a suitable model, such as, e.g., the Two-Temperature Model. However, for epithermal (<100 eV) incident energies, recent calculations have shown that this coupling is weak and the main contribution to the effective Te comes from KEE.

2.3 Monte Carlo Simulation

The investigation of the processes of electron-matter interaction requires the use of quantum mechanics-based techniques. And since, typically, the number of particles involved in these processes is huge, it is crucial to use statistical approaches, such as those represented by the Monte Carlo method. The Monte Carlo method was first developed by von Neumann, Ulam, and Metropo- lis at the end of the World War II to study the diffusion of neutrons in fissionable material. The name ’Monte Carlo’, coined by Metropolis in 1947 and used in the title of a paper describing the early work at Los Alamos (Metropolis and Ulam, 1949), derives from the extensive use of random numbers in the approach. The method is based on the idea that a determinate mathematical problem can be treated by finding a probabilistic analogue which is then solved by a stochastic sampling experiment (von Neumann and Ulam, 1945). The method is used, in particular, for evaluating the many physical quantities necessary to the study of the interactions of particle-beams with solid targets. By letting the particles carry out an artificial, random walk –taking into account the effect of the single collisions– it is possible to accurately evaluate the diffusion process.

The results are then used as sampling functions in ray-tracing Monte Carlo simulations of arbitrary geometry surfaces. Primary rays will be generated above the surface and the intersection points of these rays with surface elements will be determined.

From these intersection points, secondary rays will be generated according to the yield and emission energy distributions. These rays are tracked themselves, new intersection points are

20 Figure 2.1: Temporal snapshots of a collision cascade created by a 5 keV Ag atom impinging onto an Ag (111) surface. The excitation energy density is shown as a continuous color map in addition to atoms displaced during the cascade. Figure reproduced from [3]. determined, a new generated of secondary rays is generated. This process is repeated until a given ray either escapes the surface –in which case the event is tallied as a successful secondary electron emission (SEE) event– or dies by having a sampled energy less than the threshold energy (Fermi level plus workfunction). After tens of thousands of these events, the true SEE yield can be accurately calculated. We are currently working on such an implementation. The power of this approach resides in the fact that it can simulate arbitrarily complex surfaces, and is computationally

21 trivially parallel, so we can apply it to several surface designs to assess which one is superiori from a SEE point of view. As well, we study ion redeposition in the micro-architected surfaces by using a combination of binary collision approximation simulations and molecular dynamics. In a similar fashion as for SEE, we calculate erosion rates as a function of incident particle energy and angle. Then, using a modiﬁed ray tracing Monte Carlo (RTMC) approach again, we ’deposit’ atoms on the surface elements of the microstructure by calculating deposition rates consistent with sputtering yields. This gives rise to exposure time-evolving surfaces for which SEE yields have to be updated. In such a fashion, this becomes a coupled problem with coevolving dynamics.

22 Figure 2.2: Flowchart of the combined Molecular Dynamics approach. Figure reproduced from [4].

23 CHAPTER 3

Calculation of Secondary Electron Emission Yields from Low-energy Electron Deposition in Tungsten Surfaces

3.1 Introduction

Secondary electron emission (SEE) is the emission of free electrons from a solid surface, which occurs when these surfaces are irradiated with external (also known as primary) electrons. SEE is an important process in surface physics with applications in numerous ﬁelds, such as electric propulsion [25, 47, 48, 49, 50], particle accelerators [51], plasma-walls in fusion reactors [52, 53, 54, 55, 56], electron microscopy and spectroscopy [57, 58], radio frequency devices [59, 60, 61], etc. In Hall thrusters for electric propulsion, a key component is the channel wall lining protecting the magnetic circuits from the discharge plasma. These channel walls are a signiﬁcant factor in Hall thruster performance and lifetime through its interactions with the discharge plasma. These interactions are governed by the sheath formed along the walls, and so the properties of the sheath determine the amount of electron energy absorbed by the wall, which in turn affects the electron dynamics within the bulk discharge [25, 15, 62, 63]. Furthermore, the energy imparted by the sheath to the ions within the discharge determines the impact energy and incident angle of ions upon the surface, thus affecting the amount of material sputtered and consequently the wall erosion rate [64, 65]. Thus, understanding how SEE affects sheath stability is crucial to make predictions of channel wall lifetime.

Recently, a new wall concept based nano-architected surfaces has been proposed to mitigate surface erosion and SEE [27, 28, 29, 30]. Demonstration designs based on high-Z refractory materials have been developed, including architectures based on metal nanowires and nanofoams

24 [31, 32, 33, 34, 10]. The idea behind these designs is to take advantage of very-high surface-to- volume ratios to reduce SEE and ion erosion by internal trapping and redeposition. Preliminary designs are based on W, W/Mo, and W/Re structures, known to have intrinsically low sputtering yields secondary electron emission propensity. A principal signature of electron discharges in plasma thrusters is the low primary electron energies expected in the outer sheath, on the order of 100 eV, and only occasionally in the several hundred eV regime. Accurate experimental measurements are exceedingly difﬁcult in this energy range due to the limited thickness of the sheath layer, which is often outside the resolution of experimental probes [35, 36, 37]. Modeling then suggests itself as a complementary tool to experiments to increase our qualitative and quantitative understanding of SEE processes.

To quantify the net SEE yield from these surfaces, models must account for the explicit geometry of these structures, which requires high spatial resolution and the capacity to handle large numbers of degrees of freedom. However a precursor step to the development of these descriptions is the characterization of the SEE yield functions as a function of incident electron energy and angle of incidence in flat surfaces. Once defined, these functions can then be implemented at the level of each surface element to create a spatially-dependent emission picture of the SEE process. This is the subject of the present paper: to calculate SEE yield functions from flat W surfaces in terms of primary electron energy and incidence angle. To this end, we carry out Monte Carlo calculations of electron scattering processes in pure W using a series of scattering models specifically tailored to high-Z metals.

The chapter is organized as follows. First we discuss the theoretical models employed to study electron scattering in W. This is followed by a discussion of the implementation of these models under the umbrella of a Monte Carlo framework. Our results follow, with emphasis on emission yield and energy functions. We ﬁnalize with the conclusions and the acknowledgments.

25 3.2 Theory and Methods

3.2.1 Electron Scattering Theory

The present model assumes that electrons travel in an isotropic homogeneous medium undergoing collisions with bulk electrons. Each collision results in a trajectory change with an associated energy loss, which depend on the nature of the electron-electron interaction. As well, collisions may result in secondary electron production. We classify interactions into two broad categories: elastic and inelastic, each characterized by the corresponding collision mean free path and an angular scattering function. These processes are then simulated using a Monte Carlo approach, where collisions are treated stochastically and trajectories are tracked as a sequence of scattering events until the resulting secondary electrons are either thermalized or emitted back from the surface.

Scattering theory provides formulas for the total and the differential scattering cross sections, from which the mean free path and polar scattering angle can be obtained, respectively. Next, we provide a brief description of the essential theory behind each of the distinct collision processes considered here. Our implementation accounts for the particularities of low-energy electron scattering in high-Z materials. The validity range of the present approach in Z, which is for atomic numbers up to 92, and in primary electron energy, from 100 eV to 30 keV.

3.2.2 Elastic Scattering

Elastic scattering takes place between electrons and atomic nuclei, which –due to the large mass difference– results in no net energy loss for the electron, only directional changes [66]. A widely used electron-atom elastic scattering cross section is the screened Rutherford scattering cross section [67, 68], which provides a simple analytical form and is straightforward to implement into a Monte Carlo calculation. However, the screened Rutherford scattering is generally not suitable for low-energy electron irradiation of high-Z metals.

In this work, we use an empirical total elastic scattering cross section proposed by Browning et al. (1994), which is obtained via ﬁtting to trends in tabulated Mott scattering cross section data set

26 described by Czyzewski˙ et al. [69] using the relativistic Hartree-Fock potential. This is amenable to fast Monte Carlo computations at a high degree of accuracy. The equation for the total elastic scattering cross section is [70, 71]: 3.0 10 18Z1.7 σ = × − [cm2], (3.1) el (E + 0.005Z1.7E0.5 + 0.0007Z2/E0.5) which is valid for atomic numbers up to 92 and for energies from 100 eV to 30 keV. From this, the elastic mean free path can be derived: 1 AW λel = = [cm] (3.2) Nσel Naρσel where N is the number of atoms per cm3. For its part, the polar scattering angle can be obtained by a random number R uniformly distributed between 0 and 1: R θ dσR 0 dΩ dΩ R = (3.3) R π dσR 0 dΩ dΩ where dΩ = 2π sinθdθ is the inﬁnitesimal solid angle.

Solving the above equation for the Mott cross section requires numerical integration, as there is no simple analytical form for the polar scattering angle θ. Drouin [72] et al. (1994) gives a parameterized form of the function as 2αR cos(θ β ) = 1 ∗ (3.4) i − 1 + α R − ∗ where θi is given in degrees. Then ﬁrst parameter, α, as a function of the energy is obtained with

2 d log10(α) = a + blog10(E) + clog10(E) + (3.5) elog10(E) where E is the energy in keV, a,b,c and d are constants that have been calculated using the least- square method, and e = 2.7813. A tabulation form of a,b,c and d for the first 94 elements of the periodic table is found in Table 2 in reference [72]. For tungsten (Z = 74), a = 2.0205,b = − 1.2589,c = 0.271737,d = 0.695477. − − The second parameter, β, is calculated using the following equations: cln(E) d β ∗ = a + b√E ln(E) + + E E  1 if β ∗ > 1 (3.6) β = β ∗ if β ∗ 1 ≤ 27 where E is the energy in keV, a,b,c and d are constants that have been obtained using the least- squares fitting. A tabulation form of a,b,c and d for the first 94 elements of the periodic table is found in Table 3 in reference [72]. For tungsten (Z = 74), a = 0.71392,b = 0.00197916,c = 0.0172852,d = 0.0570799. − −

The third parameter, R∗ is obtained as:

R∗ = R R (3.7) × max

where R is a random number uniformly distributed between 0 and 1 and Rmax is the value of R∗ obtained when θi is set to 180◦ in eq. (3.4), i.e.: cos(180β ) + α cos(180β ) 1 α Rmax = − − (3.8) cos(180β ) 1 2α − − The azimuthal angle φ can take any value in the range 0 to 2π as determined by a random number R uniformly distributed in that range.

φ = 2πR (3.9)

3.2.3 Inelastic Scattering

In contrast to elastic scattering, inelastic scattering implies collisional energy loss. There are several distinct inelastic interaction processes to be considered, including phonon excitation, secondary electron excitation, Bremsstrahlung or continuum X-ray generation, and ionization of inner electron shells. Each mechanism is described by a model that provides expressions for the scattering cross section, scattering angle, and mean free path. The physics behind some of these processes is complex, and detailed expressions for the associated cross sections are often unavail- able [73, 74].

In conventional Monte Carlo approaches, Bethe’s theory of stopping power based on a continuous slowing-down approximation (CSDA) [75, 67, 76] is used to describe the average energy dissipation rate of a penetrating electron along its path, in which the contribution of all possible excitation processes to the energy loss has been represented by a factor called the mean ionization energy, J. However, this formula is not valid in the low energy regime (0.1-30 keV) or for high

28 atomic number elements (Z > 30). To resolve this, much effort has been devoted to modifying the Bethe formula, from which systematization of tabulated electron stopping powers for various elements and attempts to simplify the calculations have emerged. [77, 78, 79, 80, 81] In general, the use of these formulas for elements or compounds with ﬁtting parameters requires a detailed and accurate supply of experimental data on which to base its physics and against which to test its predictions. [82] Nevertheless, the CSDA strategy may still become obsolete when an electron occasionally loses a large fraction of its energy in a single collision as well as when secondary electron emission distribution spectra are required. To develop a more comprehensive Monte Carlo approach, incorporating differential cross sections for each of the inelastic events seems necessary [83, 84, 85, 86].

Ritchie et al. (1969) have demonstrated that the stopping power described by Bethe’s formula is obtained by the summation of theoretical stopping powers for conduction electron, plasmon and L-shell electron excitations for aluminum. [85] Fitting (1974) [87] has also shown that this stopping power derived by Ritchie et al. is in very good agreement with experimental investigation even in the energy range between 0.8 and 4 keV. Accordingly, the model of inelastic scatterings considered in the present approach are electron-conduction electron scattering, electron-plasmon scattering and electron-shell electron scattering as shown in Figure 3.1.

3.2.3.1 Inner Shell Electron Ionization

The classical formalism of Gryzinski´ (1965) [88, 89, 90, 91] has been adopted to describe inner- shell electron ionization. The differential cross section can be written as: 4 dσ (∆E) πe E E 3/2 ∆E EB/(EB+∆E) s = B 1 d∆E (∆E)3 E E + E − E B (3.10) ∆E E 4 E ∆E 1/2 1 B + ln 2.7 + − × EB − E 3 EB where ∆E, E and EB are the energy loss, the primary electron energy, and the mean electron binding energy, respectively.

At each inelastic scattering event, the energy loss of the primary electron resulting from an inelastic scattering with the shell is determined using a uniform random number R and by ﬁnding

29 a value of ∆E which satisﬁes the relation

Z ∆E dσ (∆E ) d∆E R = s 0 0 (3.11) EB d∆E0 σs

The integral is given by the approximate expression [92]

Z ∆E dσs(∆E0) d∆E0 EB d∆E0 4 πn e E 3/2 ∆E 1+(EB/(EB+∆E)) = s 1 EE E + E − E B B (3.12) ∆E 2 ∆E E ∆E 1/2 + 1 ln 2.7 + − × EB 3 − E EB E2 B (∆E E ) × ∆E2 ≥ B

where ns is the occupation number of electrons in the shell.

The total cross section of the inner electron excitation is obtained by integrating over all possible values of ∆E Z ∆E max dσs(∆E0) σs(E) = d∆E0 EB d∆E0 3/2 14 ns EB E EB = 6.5141 10− 2 − (3.13) × EB E E + EB 2 E E 1/2 1 + 1 B ln 2.7 + 1 [cm2] × 3 − 2E EB − where the maximum amount of energy that can be lost ∆Emax is equal to E.

When the random number selection gives an energy loss less than the binding energy EB, the actual energy loss is set to be zero. The scattering angle for an inelastic electron-electron event is calculated according to the binary collision approximation (BCA) as

∆E 1/2 sinθ = (3.14) E

In tungsten, for primary energies E 1 keV, inner shell electron ionization can be safely neglected, ≤ as the energy is insufﬁcient to knock out inner shell electrons.

30 3.2.3.2 Conduction Electron Excitation

For metals bombarded by electrons, Streitwolf (1959) [93] has given the differential cross section for conduction electron excitation by using perturbation theory as

dσ (E ) e4N π c SE = a (3.15) dE E(E E )2 SE SE − F

The total energy loss cross section σc(E) can be obtained by integrating the above expression

between the lower energy limit EF + Φ and the upper energy limit E:

e4N π E E Φ σ (E ) = a − F − (3.16) c SE E Φ(E E ) − F The obtained relation samples the energy of the secondary electron with the random number R:

h i E (R) = RE A(E + Φ) /(R A) (3.17) SE F − F −

where Φ is the workfunction and A = (E E )/(E E Φ). Once the energy of the secondary − F − F − electron is known (equal to the energy lost by the primary electron), the next question is how the two electrons are oriented in space. More accurate results can be obtained if the classical BCA is used, which results from conservation of energy and momentum. The azimuthal angle is again assumed to be isotropic. For the incident electron, we then have: r ∆E sinθ = (3.18) E

φ = 2πR

where ∆E is the energy lost by the incident electron. For the secondary electrons, scattering angles can be calculated as follow: sinϑ = cosθ (3.19)

ϕ = π + φ (3.20)

The above expression is applied to the inner shell electron as well as conduction electron excitations.

31 3.2.3.3 Plasmon Excitation

The Coulomb ﬁeld of the primary electron can perturb electrons of the solid at relatively long range as it passes through the target. The primary electron can excite oscillations (known as plasmons) in the conduction electron gas that exists in a metallic sample with loosely bound outer shell electrons. The differential cross section for plasmon excitation is given by Ferrel (1956) [94, 95, 96], per conduction-band electron per unit volume

dσp(E,θ) 1 θp = 2 2 (3.21) dΩ 2πa0 θ + θp

∆E }ωp θ = = (3.22) p 2E 2E where a is Bohr radius (5.29 10 9 [cm]). In plasmon scattering, primary electron energy loss is 0 × − quantized and ranges from 3 to 30 eV depending on the target species, which is detected as strong features in electron energy-loss spectra (EELS). Plasmon scattering is so sharply peaked forward that the total plasmon cross section, σ , can be found by setting dΩ = 2π sinθdθ 2πθdθ: p ≈ θ Z θp Z 1 2πθdθ σp = dσp(θ) = 2 2 (3.23) 2πa0 0 θ + θp By assuming the upper integration limit as θ = 0.175 rad, where θ sinθ, and incorporating the 1 ≈ 2 factor (ncAW/Naρ) to put the cross section on a per-atom/cm basis gives the total cross section of the plasmon excitation as

ncAWθp h 2 2 2 i 2 σp = ln(θp + 0.175 ) ln(θp ) [cm ] (3.24) 2Naρa0 −

where nc is the number of conduction-band electrons per atom. Essentially, the scattering of

primary electrons due to plasmon excitations is restricted with θ < θmax, kc being the cut-off

wavenumber. Since θmax is so small, about 10 mrad in the energy range discussed here, the angular deﬂection due to plasmon excitation is neglected in this approach.

Again, the azimuthal angle φ can take on any value in the range 0 to 2π selected by a random number R uniformly distributed in that range.

φ = 2πR

32 E0 Vacuum α Primary electron

Sample ∆s0 E0 θ1 Elastic scattering (θ,φ) φ1 ∆s1 Secondary electron E ∆E 0 − θ2 Shell/conduction-electron Es φ 2 excitation (θ,φ,∆E) ∆s2

E0 ∆E ∆E0 − Plasmon− excitation (θ,φ,∆E0)

Figure 3.1: Schematic diagram of the discrete collision model of electron scattering simulated using Monte Carlo.

3.3 Monte Carlo Calculations

As indicated above, electron trajectories are simulated by generating a spatial sequence of collisions by randomly sampling from among all possible scattering events. The distance traveled by electrons in between collisions, ∆s, is assumed to follow a Poisson distribution deﬁned by the total mean free path λT [67] ∆s = λ logR (3.25) − T where 1 1 1 1 1 = + + + = N(σel + σp + σc + σs), (3.26) λT λel λp λc λs

33 N is the number of atoms per cm3 and R is a random number uniformly distributed in the interval (0,1]. From this, we deﬁne the following probabilities:

Pel = λT /λel : the probability that the next collision will be elastic

Pp = λT /λp : the probability that the next collision will cause a plasmon excitation

Pc = λT /λc : the probability that the next collision will cause a conduction electron excitation

Ps = λT /λs : the probability that the next collision will cause an inner shell electron excitation (3.27) The type of collision is then chosen based on the following partition of the value of R:.

0 < R P = elastic scattering ≤ el ⇒

Pel < R Pel + Pp = plasmon excitation ≤ ⇒ (3.28) P + P < R P + P + P = conduction electron excitation el p ≤ el p c ⇒ P + P + P < R 1 P + P + P + P = inner shell electron excitation el p c ≤ ≡ el p c s ⇒ The ﬂow diagram corresponding to the implementation of the model just described is provided in 3.7. Following this approach, electron trajectories1 are tracked in the energy-position space until a scattered electron either thermalizes, i.e. its energy follows below the surface escape threshold (Fermi level plus workfunction) within the material, or reaches the surface with a velocity having a component pointing along the surface normal with an energy larger than the escape threshold. In the latter case, the electron is tallied as a secondary electron and its energy and exit angle are recorded.

Next we analyze the Monte Carlo calculations performed following this method and present results of secondary electron yield and emission energies as a function of primary electron energy and angle of incidence.

1Trajectories are generated by stitching together each sequence of discrete collision events and referring each collision point to a global laboratory frame of reference. The coordinate transformation employed here can be found in ref. [97] and is provided in Figure C.1

34 3.4 Results

The total secondary electron yield for perfectly-ﬂat tungsten surfaces is calculated for primary

incident angles of 0◦, 30◦, 45◦, 60◦, 75◦ and 89◦ measured off the surface normal, and incident energies in the range 100-1000 eV. In this work, the typical number of primary particles simulated ranges between 104 and 105, which generally results in statistical errors around 3%. Our ﬁrst set of results includes the energy and angular distributions of emitted secondary electrons for normal incidence and 100 eV and a primary electron energy of 100 eV. The normalized distributions are given in Figures 3.2a and 3.2b, where the characteristic energy decay of 1/E and cosine angular distribution of collisional processes can be appreciated in each case. One of the advantages of using a discrete event method for simulating electron scattering processes is that useful information of discrete nature can be extracted from the data. For example, in Figure 3.3a we show the depths from which secondary electrons are emitted (last scattering collision inside the material) as well as the depth distribution of thermalized (non-emitted) electrons, i.e. the depth at which electrons attain an energy less than the threshold. Both cases are for normal incidence and E = 100 eV. In Figure 3.3b we break the total number of collisions down into the main scattering modes as a function of primary energy and normal incidence. It can be seen that scattering with conduction electrons is always the dominant mechanism, although its relative importance reduces with E.

Next, we plot the SEE yields as a function of primary electron energy for angles of incidence of 0 and 45◦ to facilitate comparison with existing experimental data and other published Monte Carlo simulation results. The results are shown in Figures 3.4a and 3.4b, respectively, with error bars provided in each case. In general, the simulation results are found to agree reasonably well with experimental data. The agreement is slightly worse for 45◦ than for 0◦ incidence, which we rationalize in terms of the higher incidence direction. It is well-known that the fraction of reﬂected particles increases with the angle of incidence [98]. In addition, the roughness of ‘real’ experimental surfaces compared to the ideally-smooth ones in the model likely plays a signiﬁcant role in the comparison. SEE yields as a function of E for all angles of incidence considered here are given in Figure 3.5.

Surface plots of both SEE energy distributions and the yields are given in Figures 3.6a and

35 0.6

0.1

0.4

0.2 Normalized Frequency Normalized Frequency

0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 90 45 0 45 90 − − ESE [eV] θ [deg]

(a) (b)

Figure 3.2: Normalized distributions for 100-eV primary electrons incident at 0◦: (a) Energy distribution of secondary electrons; (b) Angular distribution of secondary electrons.

3.6b As mentioned earlier, these data will be used in ray-tracing Monte Carlo simulations of SEE in arbitrary surface geometries. In these simulations, primary rays are generated above the material surface with the corresponding incident energy E. Intersections of these primary rays with surface elements determines the corresponding angle of incidence α. E and α are then used to sample from the data shown in, e.g., Figure 3.2, after which secondary rays with appropriate energies

ESE and exit angles (sampled from a cosine distribution) are generated. These ‘daughter’ rays are themselves tracked in their interactions with other surface elements, after which the sequence is repeated and subsequent generations of rays are produced. This process goes on until rays either escape the surface with an upward velocity –in which case the event is tallied as a successful SEE event– or until their energy is below the threshold escape energy (Fermi level plus workfunction). However, directly interpolating from our data tables potentially hundreds of thousands of times can slow down the simulations considerably. To avoid that, it is more efﬁcient to ﬁt the data to suitable analytical expressions that can be evaluated very fast on demand.

To this end, we ﬁt our raw data to bivariate mathematical functions obtained using symbolic regression (SR), which is a type of genetic evolutionary algorithm for machine learning which uti-

36 0.4

1.0 0.3 0.8

0.2 0.6

0.4 0.1 Normalized Frequency

Normalized Frequency 0.2 0.0 0 5 10 15 20 0.0 d [A˚] 100 200 300 400 500 600 700 800 900 ,000 1 E [eV] Emitted Thermalized Elastic Conduction Plasmon

(a) (b)

Figure 3.3: (a) Depth distribution of both emitted and thermalized (captured) electrons for a pri-

mary electron energy and incident angle of 100 eV and 0◦. (b) Relative occurrence of the main scattering mechanisms as a function of primary energy for normal incidence.

lizes evolutionary searches to determine both the parameters and the form of the fitting expressions simultaneously, to speed up our calculation. We have used the trial version of Eureqa [99, 100], a commercial SR engine, to generate a catalog of potential candidate expressions for mathematical fitting. We then select the final expression by capping the fitting errors to be no higher than the intrinsic statistical errors of the Monte Carlo calculations ( 3%). This ensures that the fitted ≈ functions always provide solutions to within the natural variability of the fitted data. Beyond that, we generally try to use functions that are well behaved numerically in as wide an energy range as possible, e.g. discarding those with logarithmic terms, etc. The final expressions for the total SEE yield and energy distributions are

5 2 7 2 γ(E,α) = 3.05288 + 1.7949 10− α + 6.15912 10− E × × 2 2 1 + (3.71317 10 exp(4.8259 10− α))/( 8.760195 10− E) (3.29) × − × − × − 4 1 1 1.974 10− E 1.0971 10− cos(1.20187 10− + 2.469421α) − × − × ×

37 2.5 2.5

2.0 2.0

1.5 1.5 γ γ

1.0 1.0

0.5 0.5

α = 0 ◦ α = 45 ◦ 0.0 0.0 0 100 200 300 400 500 600 700 800 9001000 0 100 200 300 400 500 600 700 800 9001000 E [eV] E [eV]

(a) (b)

Figure 3.4: (a) Total SEE yield from an ideally-ﬂat W surface as a function of primary electron energy for electrons incident at 0 . = this work; black dashed line = Ahearn (1931) [5]; cyan ◦ • dashed line = Coomes (1939) [6]; = Bronshtein and Fraiman (1969) [7]; N = Ding et al. (2001) using the method where the SE is assumed to come from a distribution of electron energies in the

valence band; H = Ding et al. (2001) using the method where the SEs are assumed to originate from the Fermi level [8, 9]; = Patino et al. (2016) [10]. (b) Total SEE yield from smooth W as a function of primary electron energy, for electrons incident at 45 . = this work; = Patino et al. ◦ • (2016) [10].

1 4 2 1 ESE(E,α) = 1.95 10− E + 1.69 10− E + 1.48 10− α sin(E) × × × (3.30) 15 7 + 3.44 10− Eα 6.54 × − We note that these expressions do not necessarily reflect the physics behind SEE and are just intended for efficient numerical evaluations strictly in the ranges shown in the figures.

38 2.5

2.0

1.5 γ

1.0

0 ◦ 30 ◦ 0.5 45 ◦ 60 ◦ 75 ◦ 89 ◦ 0.0 0 100 200 300 400 500 600 700 800 9001000 E [eV]

Figure 3.5: Total SEE yield from a ﬂat W surface as a function of primary electron energy and incidence angles.

3.5 Discussion and Conclusions

Electron-matter interactions are complex processes. To make the theoretical treatment of electron scattering a tractable analytical problem, it is assumed that elastic scattering occurs through the interatomic potential, while inelastic scattering only through electron-electron interactions. Ev- idently, the accuracy of the Monte Carlo simulations depends directly on how precisely the approximations introduced in the model are described. Most models treat elastic interactions within Mott’s formalism [66] (or adaptations thereof). For their part, inelastic scattering processes in this work are considered individually, each one characterized by its own differential cross sections, corresponding to valence, inner shell, conduction, and plasmon electron excitation. In contrast, Ding et al. [8] use Penn’s dielectric function [101] for electron inelastic scattering obtained from a modiﬁcation of the statistical approximation. Many other models for metals account for valence interactions only [102, 103, 67]. Here, we improve in these models, although we do not capture the generation of SE from plasmon decay, backscattered electrons, reﬂected electrons, and transmitted electrons (coming out from the back side of the sample). This must be kept in mind when

39 2.0

500 1.5 400

γ 300 [eV] 200 SE

1.0 E 100 90 0 75 90 0 60 0 75 45 60 200 45 200 400 30 400 30 600 15 600 15 800 0 [deg] 800 [deg] 1000 α 1000 0 α E [eV] E [eV]

(a) (b)

Figure 3.6: (a) Total SEE yield from an ideally-ﬂat W as a function of primary electron energy, for

electrons incident at 0◦, 30◦, 45◦, 60◦, 75◦ and 89◦. (b) Surface plot of the total SEE yield from an ideally-ﬂat W as a function of primary electron energy and angle of incidence. comparing the simulation results with experimental data (cf. Figure 3.4). In this sense, it can be said that our results provide a ﬁrst-order check of the importance of internal scattering processes, which helps us understand the governing physics behind SEE.

In any case, discrete event simulations –e.g. as the Monte Carlo model implemented in this paper– present the advantage that they provide a measure of the statistical errors associated with a given formulation. This is not just a numerical matter because experimental measurements themselves correspond to averages of a given realization of the scattering process. In the discrete approach the energy loss of electrons traveling through a solid is determined by considering different inelastic scattering processes –including conduction electron excitation, plasmon decay, and inner shell electron ionization– are considered individually, whereas within the so-called continuous slowing down approximation (CSDA), the overall inelastic scattering mechanisms are averaged out by using the total stopping power. From this point of view, the CSDA and the discrete-event simulation method would formally converge in the limit of an inﬁnite number of events. Discrete simulations also allow a better physical and spatial dissection of electron scattering processes,

40 providing spatial distributions and breakdowns among the different scattering mechanisms. This information is important to ascertain what scattering events dominate the secondary electron emission process in each material. This is what is shown in Figure 3.3b, where the partition of scattering mechanisms for normal incidence and 100 eV primary energy is given. A disadvantage of discrete vs continuous simulations is, however, the longer computational cost required to obtain acceptable statistics. Modelers, therefore, must weigh in each of these factors (better spatial resolution and statistical information vs worse computational efﬁciency) and decide what approach to use. To summarize, in this work we have carried out Monte Carlo calculations of low energy electron induced SE emission from ﬂat tungsten surfaces. Our model includes multiple elastic and inelastic scattering processes, implemented via a discrete energy loss approach. We compare predictions of our model with other Monte Carlo techniques as well as experimental data, with generally good agreement found. We have calculated the total SEE yield and secondary electron energy spectrum for primary electron beams at incident angles of 0◦, 30◦, 45◦, 60◦, 75◦ and 89◦, in the range 100-1000 eV. We have used SR to obtain analytical expressions that represent the numerical data. These functions are currently being used in ray-tracing Monte Carlo simulations of SEE in arbitrary surface geometries.

41 Start

i=0 Ns=0 Yes i+=1 No Read in data of SE i>N Yes Stop Ns=Ns-1, L=0 No Set initial condition L=0, Ns=0

Transmitted Yes L>Lm

New position

Store data of BSE Yes Z>0

Select collision type

Determine Store data of SE Yes Elastic scattering angle Ns+=1 No Yes Determine scattering SE angle and energy created No Store data of Compute a Yes E

Figure 3.7: Flow chart of the Monte Carlo program

42 CHAPTER 4

Monte Carlo Raytracing Method for Calculating Secondary Electron Emission from Micro-architected Surfaces

4.1 Introduction

The release of electrons from a material surface exposed to a primary electron beam, known as secondary electron emission (SEE), is an important phenomenon with applications in a wide variety of physical processes, such as in electron multiplication devices [104, 105], electron microscopes [57, 106], and plasma devices [15, 48, 107, 108, 109, 110, 53, 111], among others. While SEE can be induced to amplify electron currents, such as during photoemission spectroscopy [112], it can also be detrimental for performance, such as in the case of the multipactor effect in radio frequency devices [113].

In the case of Hall thrusters for electric propulsion [108], an electrostatic sheath forms between the plasma and the inner lining of the plasma-facing surface material. This sheath potential acts as a thermal insulator and as a deterrent of current ﬂow that protects the wall from particle discharges [25]. SEE weakens this sheath potential [25, 53, 111, 114], which has detrimental effects for the stability of the thruster, as is known to occur as well in magnetic fusion devices and radiofrequency plasma sources [109, 53]. Thus, as a crucial phenomenon affecting the efﬁciency of these devices, there is an increasing interest in mitigating –or at least controlling– secondary electron emission. A direct method to reduce the overall SEE yield is to engineer the structure of the material surface, leading to a class of surfaces known under the umbrella term of microarchitected surfaces.

Aside from early efforts in surface texture development to control SEE [115, 31], the use of advanced characterization and new processing techniques to develop microarchitected surfaces and

43 experimentally determine the reduction in SEE yield is relatively recent [10, 116, 117]. There are now numerous examples of successful designs that are seen to lower the secondary electron yield [118, 10, 119, 120]. These surfaces can be fabricated ex situ and deposited over existing chamber walls to achieve the desired level of functionalization.

The theory of secondary electron emission is generally well known and has been studied for decades [121, 122, 104]. However, studies of how the surface geometry and morphology affect the rate of electron emission are relatively limited. Modeling and simulation can play an important role in predicting the expected reduction rates of SEE before a costly effort of surface texture development, fabrication, and testing needs to be mounted. Simulations involving surfaces with grooves [28], ‘velvet’-like ﬁbers [116] and open-cell structures [117] have been recently carried out, showcasing the versatility of numerical simulation but also its relatively high computational cost. In this paper, we present a two-pronged simulation approach in which SEE yields and energy spectra are precomputed for ideally ﬂat surfaces, and are later used to describe the constitutive response at the local material point level of a discretized surface with arbitrary geometry. The connection between both descriptions is made via a ray-tracing Monte Carlo algorithm coupled to an intersection detection algorithm. The method simulates individual electron tracks, one at a time, and generates secondary tracks on the basis of incident energies and angles sampled from the precomputed physical relations. The number of tracks that escapes the surface is tallied and compared to the total number of simulated tracks to compute the effective secondary electron emission yield.

The paper is structured as follows: in Section 4.2.1 we provide a description of the intersection detection algorithm and the raytracing Monte Carlo method. In Section 4.2.3 we describe the discretized model of the microarchitected surface considered, while in Section 4.3 we demonstrate the validity of the method by reproducing results for ﬂat surfaces and open cell foam unit cells, followed by SEE yield calculations in in actual foam structures of various porosities. We conclude with a brief discussion section and the conclusions.

44 4.2 Computational Model

Our model is based on a raytracing model to track particle trajectories from a random point above the surface as they are directed towards the material. This primary ray is defined by its energy E and angle of incidence with respect to a laboratory (global) frame of reference. Once this primary ray is generated, the next step is to determine whether its trajectory intersects the material surface, discretized into a finite element mesh, i.e. whether the the ray crosses a surface element of the mesh. The algorithm used to detect such intersections is an extension of the Moller¨ and Trumbore algorithm [123], which we briefly review in the following.

4.2.1 Intersection Detection Algorithm

Next, we provide a brief overview of the Moller-Trumbore¨ (M-T) method [123]. The procedure is described for triangular surface elements, although it can be extended to other geometric shapes in a straightforward manner. A ray defined by an origin O~ and a direction ~D is defined by the equation: ~R(t) = O~ +t~D (4.1) where t is a scaling parameter that defines the length of the ray. If the vertices of the surface element triangle ~V0,~V1,~V2 are known, any point on the element can be defined by

~T(u,v) = (1 u v)~V + u~V + v~V . (4.2) − − 0 1 2

Where u and v are barycentric coordinates that deﬁne the plane of a triangle, and satisfy u 0, ≥ v 0, and u2 + v2 1. Note that the surface element normal can be determined as1: ≥ ≤ ~s ~n = , ~s V~ V~ V~ V~ ~s ≡ 1 − 0 × 2 − 0 k k As outlined by Moller¨ and Trumbore, the intersection point can be uniquely found by equating eqs. (4.1) and (4.2) above:

O~ +t~D = (1 u v)~V + u~V + v~V (4.3) − − 0 1 2

1Assuming that the vertices are given a counter-clockwise order as seen from a direction opposing the normal.

45 Rewriting and rearranging in terms of matrices gives   t h i  ~ ~ ~ ~ ~   ~ ~ D, V1 V0, V2 V0 u = O V0 (4.4) − − −   − v

Deﬁning~L =~V ~V ,~L =~V ~V , and ~T = O~ ~V , the solution can be obtained through Cramer’s 1 1 − 0 2 2 − 0 − 0 rule:     t ~T ~L1 ~L2    × ·    1   u =  ~D ~L2 ~T  (4.5)   ~ ~ ~  × ·    D L2 L2   v × · ~T ~L ~D × 1 · The algorithm must scan through all elements of the structure, following for each element the steps above in search for possible intersections. For large meshes, the computation can rapidly become prohibitive. For this reason, several checks are employed to quickly determine if an intersected point lies on a triangular element:

1. First, a back-face culling technique is applied so that if the normal of a triangular element is in the direction of an incoming ray, that element is ignored.

2. Second, if the ray is parallel to the plane of a triangle within an allowed tolerance, that element is ignored.

3. If the above two checks are satisﬁed, then a last check is made to determine the conditions for barycentric parameters u and v. If all conditions are satisﬁed, then the algorithm determines the intersection point within a triangular element.

4. Finally, a micro-architected surface may have many surface elements intersecting a given ray, but only the closest one is considered.

46 4.2.2 Generation of Secondary Rays

Once a collision is detected via the M-T procedure, the incidence angle of the primary ray on the selected surface element is determined as: ~ ! 1 D ~n α = cos− · ~D ~n k k k k Note that this angle of incidence is a local variable (given in a relative frame of reference). The pair (E,α) is then used to sample from bivariate relations giving, ﬁrst, the number of secondary rays per incident primary ray [11]:

5 2 7 2 371.32 exp(0.05α) γ(E,α) = 3.05 + 1.80 10− α + 6.15 10− E − + × × − 87.60 + E (4.6) 3 1.97 10− E 0.11cos(0.12 + 2.47α) − × − Specifically, this function gives the SEE yield from a flat tungsten surface for a primary electron beam of energy E at an incident angle α. So once a collision has been confirmed, we evaluate this function with the primary ray’s energy and angle of incidence and the resulting fractional yield is rounded up or down using a uniform random number generator2. This means that rays that produce yields γ < 1 may result in no secondary electron emission. If one secondary ray is emitted, the energy of the resulting electron is obtained by evaluating the accompanying function [11]:

4 2 15 7 E (E,α) = 0.19E + 1.60 10− E + 0.15α sin(E) + 3.44 10− Eα 6.54 [eV] (4.7) SEE × × −

Strictly speaking, expressions (4.6) and (4.7) are valid for E > 100 eV, but to capture rays with energies lower than 100 eV a smooth stitching of a polynomial function is performed. In the event that more than one secondary rays are produced, the energy of one of them is obtained as:

p E1 = ESEE ξ2. (4.8)

and emitted with an angle sampled from a cosine distribution:

1 1/2 β1 = sin− ξ2

2For example, if a yield γ = 1.2 is obtained, we sample uniformly in the interval (0,1] and if the random number ξ 0.2, then we round the yield down to γ = 1. Else, it is rounded up to 2. 1 ≥ 47 The second emitted ray then has an outgoing angle of π β = β 2 2 − 1 and energy 2 E2 = cos β1ESEE

If E1,E2 < Ec, the corresponding ray is terminated, where Ec is a threshold energy equal to Ec =

Φ + E f with Φ the material’s workfunction and E f the material’s Fermi energy, 4.55 and 5.58 eV, respectively. [124, 103]. Therefore, once non-primary rays reach an energy less than the threshold, they no longer have the ability to emit secondary electrons.

The rays are tracked one at a time, from intersection to intersection until they terminate. Each primary ray creates a tree. For example, a primary ray that generates two secondary rays will add two branches to the tree.

λ λ 0 + λ 00 p → d d

λpλd0 λd00 where the subindex d stands for ‘daughter’ ray. Once the calculations of a given branch are com- pleted the algorithm moves to the next branch. If a given branch spawns another branch –third ‘generation’ (or gd, ‘granddaughter’) branch–, then it is added to the end of the tree and the tree moves on to next branch.

λ 0 λ 0 d → gd

λpλd0 λd00λgd0 .

This sequence repeats until a tree has no further branches and encounters a null element. With this procedure, each ray’s energy, angle, generation, and starting point/termination is tracked. Periodic boundary conditions are used along the x and y directions, while a ﬂat boundary is used at z = zmin to mimic a solid substrate beneath the foam. A ray (of any generation) that is found to reach zmax with an polar angle between π/2 is considered and counted as a secondary electron emission ± event.

This raytracing approach is trivially parallelizable, in the sense that each primary ray is independent and the geometry of the foam remains unaltered for all rays. This allows the method to

48 run on multiple replicas and quickly generate large subsets of data, allowing the control of number of initial rays, energy, position and direction. The ﬂow diagram of the entire process is given in Figure 4.1.

Figure 4.1: Flowchart of the raytracing Monte Carlo code.

49 4.2.3 Finite Element Model and Surface Geometry Development

The foam model is computationally reconstructed from a series of grayscale X-ray tomography images of a real foam with 65 pores per inch (PPI) and approximately 4% volume fraction Vf [12]. Each image is filtered and stacked to create a three-dimensional array with elements assigned either a value of 0 (space) or 1 (material). This voxel representation of the foam can be manipulated to change the foam’s morphology, allowing for SEE yield comparisons to be drawn between foams of various porosities. The volume fraction of the foam is increased by adding layers of material voxels to the surface voxels, which are identified by their immediate proximity to voxels of value equal to zero. To prevent the growth of large, flat surfaces for the higher volume fraction foams which can adversely affect meshing quality, some randomness is included in the voxel layering process. This procedure is used to generate foams with Vf = 4%, 6%, 8% and 10%. Volume fractions are computationally determined by summing the material voxels and dividing by the total number of voxels in the domain. Although the procedure to generate these surfaces is general, this particular structure is based on foams with pore and ligament sizes of approximately 270 and 80 µm, respectively [12].

An iso-surface routine is run on the voxel model to create the finite element model used for the raytracing study. The number of triangular surface elements generated is typically of order 107 and is reduced using a mesh coarsening routine to order 105. This ensures reasonable simulation runtimes and introduces a wider variety of element angles in the finalized mesh. An example of a finite element foam model is shown in Figure 4.2a. An important aspect of the mesh is the distribution of surface element normals that will be encountered by the simulated rays. Figure ?? shows a histogram of surface element normals for the geometry shown in Figure 4.2b. As the figure reveals, the distribution of normals is not uniform, with a maximum observed for orientations near

90◦ (perpendicular to the z-axis). This is indicative of (nonuniform) pore shapes elongated along the vertical axis, a factor which will be invoked in Section 4.3.2 to explain some particular results.

50 0.05

0.04

0.03

0.02

0.01 Normalized Frequency 0.00 0 90 180 θ [deg]

(a) (b)

Figure 4.2: (a) Finite element model of a real micro-architected foam structure rendered from X-ray tomography images. (b) Histogram of surface element normals.

4.3 Results and Discussion

Next we show several key results of the method. First, we carry out a series of verification tests to confirm the correctness of the implementation. We then deploy the verified methodology to cases of practical interests such as the microfoam architected surface.

4.3.1 Veriﬁcation

The first test performed involves studying flat W surfaces by sampling from the SEE yield surface as a function of incident energy and angle calculated in our previous work [11] (Equation (4.6)). This simply verifies the implementation of the sampling procedure. The thickness of the sample is chosen so as to ensure that the no primary ray has enough energy to traverse it up to 1000 eV. 105 rays per energy point are simulated, distributed over 10 computational cores. As Figure 4.3 shows

51 the results using the raytracing method match exactly those given by the sampling function for normal incidence. The actual data from scattering Monte Carlo from which the sampling function is obtained is also shown for reference. 2.0

1.5

γ 1.0

0.5 Solid W (Chang et al.) Solid W (Raytrace) Source Yield Function 0.0 0 200 400 600 800 1000 E [eV]

Figure 4.3: SEE yield as a function of primary energy for normal incidence on ideally ﬂat W surfaces obtained using (i) scattering Monte Carlo (raw data from ref. [11]), (ii) sampling functions given in Equation (4.6), and (iii) using the raytracing model described here.

The second test is performed on an open cell (also known as ‘open cage’) structure, shown in Figure 4.4a, which is a simple way to represent foams with arbitrary porosity. It was proposed by Gibson and Ashby, who express the solid volume fraction of the structure as [125]:

ρ t 2 v = c C (4.9) f ρ ≈ l where ρc is the relative density, ρ is the density of the material of which the cell is made, t is the ligament thickness, l is the cell size, and C is a proportionality constant with a value around 28. These cells can be used as repeat units of periodic arrangements simulating foams of arbitrary size. Next we calculate the SEE yield for an open cage with volume fractions of 1.00, 2.25, 4.00, and 6.25%. A full-dense ﬂat surface is assumed to lie beneath the cage, as to simulate an underlying solid substrate. The results are shown in Figure 4.4b for normal primary electron incidence. Interestingly, the maximum yield is obtained for the lowest volume fraction. This is an

52 1.0

0.8

0.6 γ 0.4

1.00% 0.2 2.25% 4.00% 6.25% 0.0 0 200 400 600 800 1000 E [eV]

(a) (b)

Figure 4.4: (a) Image of a cubic open cell foam structure with the cage size l and ligament size t indicated. (b) Secondary electron yield versus electron beam energy at 0 degree incidence from the cubic cage.

artifact due to the simulation setup, as in that case the ligaments are too thin to absorb any primary electrons and most of the primary rays hit the bottom substrate and secondary electrons are able to escape unimpeded.

4.3.2 Micro-architected Foam Structures

Four foam structures as the one shown in Figure 4.2b with varying volume fractions are are studied. Figure 4.5 shows the secondary electron emission yield for solid volume fractions of 4, 6, 8, and 10% in the 50-to-1000-eV energy range using normal incidence. The inset to the ﬁgure shows the dependence of the yield with the material volume fraction at energies of 50, 100, 200, 300, 400,

500, and 600 eV. In the high porosity range explored here the dependence of the SEE yield on Vf is clearly linear in the high porosity range explored here. For the sake of comparison, the maximum

53 SEE yield for Vf = 4% (which occurs for E = 600 eV) is approximately 0.76, compared with a value of 1.49 for the flat surface (from Figure 4.3). This decrease in SEE yield by about a factor of two is indicative of the potential performance gains that micro-architected surfaces might offer relative to fully dense surfaces. It is also worth noting that the results in Figure reffig:figure2b for an open cell cage with 4% material volume fraction is approximately 0.71, suggesting that such a simple model could be an acceptable surrogate for more complex geometries.

1.0 4% 6% 8% 10%

0.8

1.0 0.6 γ 200 eV 0.4 γ 100 eV

50 eV 0.2 0.2 4 10 Vf % 0.0 0 200 400 600 800 1000 E [eV]

Figure 4.5: SEE yield versus electron beam energy initially projected from 0 degree incidence to the foam at varying volume-fraction percentages. The inset shows (in increasing order) the dependence of the yield with volume fraction for primary energies equal to 50, 100, 200, 300, 400, 500, and 600 eV.

One issue that must be kept in mind when using material structures with ﬁnite characteristic lengths, such as the columns in the open-cell structure or the ligaments in the foam, is that electrons may be capable of traversing them in their entirety. Because our model only considers surface elements, this possibility is not captures in our calculations. In our prior study [11], we calculated the absolute penetration depth of 100 eV electrons, which came out to be no more than 2 nm. We have now extended these calculation to 1000-eV electrons, to have an upper bound in the absolute penetration depth. The result is shown in Figure 4.7a, where it can be seen that the tail of the depth distribution stops at approximately 5 nm. While this suggests that the effect can be

54 1.0

0.8

0.6 γ 0.4

4%, 0 ◦ 0.2 4%, Random Patino et al., W Fuzz, 0 ◦

Patino et al., W Fuzz, 45 ◦ 0.0 0 200 400 600 800 1000 E [eV]

Figure 4.6: SEE yield versus electron beam energy for normal and random incidence primary electrons. Experimental results by Patino et al. [10] are shown for comparison. neglected in our foam, with characteristic ligament diameters of 80 µm, we have carried out a study to estimate the relative occurrence of these events. We have calculated the travel distance, d (shown schematically in 4.7b), of more than 200,000 random rays within the ligaments of the foam, and have integrated the relative frequency of having trajectories less than 5 nm. The results are shown in Figure 4.7c, with the inset showing an ampliﬁed view of the ﬁrst 20 nm of the histogram. Our results indicate that the probability of having rays that traverse ligaments over distances of 5 5 nm or less is approximately 10− . Since most of our simulations involve approximately 100,000 trajectories, this indicates that we are missing one ray in one hundred thousand, which can be considered negligible for all practical purposes.

Next we study the effect of the incident angle on the results by carrying out a study considering random primary incidence vs. just normal incidence. Given that the distribution of surface normals in the foam is not uniform (cf.— Figure 4.2b), we do expect some differences between both cases. As Figure 4.6 shows, these differences are more pronounced at high incident energies where primary rays with random incidence are able to impinge on high-angle surface elements at shallower depths and create more secondary electrons than normal rays, which are capable of penetrating deeper distances and become absorbed there. This is the reason behind the higher yields

55 0.1 0.15 100eV 8 ) 1000eV -5 6

4 0.10 2 Normalized Frequency (10 0 0 5 10 15 20 d [nm] 0.05 Normalized Frequency Normalized Frequency 0.0 0 5 10 15 20 25 30 35 40 45 50 0.00 d [A˚] 0 50 100 150 200 d [µm]

(a) (b) (c)

Figure 4.7: (a) Penetration depth distribution of thermalized electrons in bulk W with primary electron energies of 100 and 1000 eV at normal incidence. (b) Depiction of a possible ray trajectory through a ligament with d as the traversed distance inside it. (c) Distance d distribution of rays in a 4%-volume fraction foam surface.

for random incidence shown in the figure. To confirm this, one can compute the penetration profile of the electron beam in both cases, obtained by tallying the depths at which electrons –regardless of what ‘generation’ they belong to– thermalize and become absorbed by the material. The results are shown in Figure 4.8, where it can be seen that, overall, normal-incidence rays penetrate further than the random incidence counterparts. The normalized depth can be scaled to typical foam thicknesses of approximately three millimeters, as described by Gao et al. [12].

Finally, we compare our results with experimental data. While no measurements have been made for W micro-foams, there are data available on He-plasma-exposed W surfaces [10], which are seen to develop a nano-tendril structure (commonly known as ‘fuzz’) at temperatures above approximately 900 [126]. These fuzz structures with characteristic ligament sizes of 10 20 nm ◦ ∼ resemble open foam surfaces with high porosity and can therefore be considered for comparison against our raytracing Monte Carlo calculations. The results are also shown in Figure 4.6, where

very good agreement is found between the calculations for the foam with Vf = 4% and the fuzz

surfaces used in the experiments. The good match between the experimental data for 0 and 45◦ primary incidence serves as indirect indication that the distribution of surface normals in the fuzz

56 0.3 Normal Random

0.2

0.1 Normalized Frequency 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Penetration Depth

Figure 4.8: Penetration depth of electrons in a 4% volume fraction foam with an electron beam energy of 100 eV at normal and random primary electron incidence. This distribution corresponds to locations at which electrons ‘thermalize’ and become absorbed into the material. Typical foam thicknesses are approximately 3 mm [12]. is closer to uniform.

Next, we brieﬂy turn back to the discussion in Section 4.3.2 on the effect of backscattered electrons (electrons that traverse an entire ligament) on the results of the model. In this case, the nanofuzz structure has comparable tendril sizes to that of the penetration depth of electrons. While discarding their contribution is not a concern for the foam (with ligament sizes of 80 microns), it could a priori result in a larger error in the calculations for the fuzz. However, there is another consideration to keep in mind here. Electrons become ‘thermalized’ within 5 nm of distance (in the most conservative scenario of 1000-eV primary energy). However, in the unlikely case that they traverse the entire tendril, their residual energy will be below the threshold to create further secondary electrons. Therefore, while we acknowledge that this is missing in our model, we are conﬁdent that it has little impact on the results shown here. Indeed, the fact that our model predictions produce such good agreement with the experimental measurements is an indirect indication of this.

57 4.4 Discussion and Conclusions

Secondary electron emission is an important process in materials exposed to charged particle distributions (cf. Section 4.1). Measurements of SEE yields are challenging, and modeling and simulation can play an instrumental part in predicting the response of complex surfaces to electron exposure. Models can also be used as a way to pre-assess the suitability of a specific surface morphology prior to developing costly fabrication techniques [127, 128, 129]. For this, computational methods must display sufficient efficiency to parse through the parametric space, which can be large if one takes into account the multiple length scales of the problem, such as pore size, ligament size, total thickness, total exposed area, etc. Here, we develop an experimentally-validated methodology that simulates electron irradiation on a surface using rays generated randomly with a given set of properties. This technique, known as ‘raytracing’ Monte Carlo, is routinely used in the visual graphics industry to create shades and lighting effects [130, 131, 132]. The interaction of each ray with the surface consists of a mathematical determination of the possible intersections with it, and a physical description of an incident ray impinging on a surface. Per se, our model is trivially-scalable and can run on multiple processors without any communication cost.

However, to be fully applicable, the methods proposed in his paper must satisfy three premises: (i) that a surface structure with arbitrary geometric complexity can be reduced by discretization to a piecewise collection of flat surface elements on which to apply the secondary electron physics of flat surfaces, (ii) that electron irradiation on these discretized geometries can be effectively simulated with individual rays representing electron trajectories, and (iii) that the typical ligament size always be much larger than the electron penetration in the material at all energies considered. In this sense, the raytracing approach proposed here is not conceptually too different from the original neutron transport Monte Carlo methods developed several decades ago to study neutronics in nuclear reactors [133, 134]. The M-T algorithm acts as the bridge between the discretized material surface and the raytracing approach. A feature not to be overlooked is the pre-computation of SEE yields and energy distributions for flat (ideal) surfaces. It is in these calculations where all the physics around electron-matter interactions is contained, such that the problem of SEE from complex surfaces can be easily separated into a ‘physics’ part (in idealized scenarios) and a

58 ‘geometry’ part (discretized to take advantage of the physical calculations). This division affords a great deal of versatility, so that as long as a sufficiently fine mesh can be generated one can conceivably study geometric features as fine as nanopillars, internal voids, surface islands, or even asperities associated with surface roughness.

This is the case in this work, where porous foams with volume fractions < 10% have been studied. Foams of this type, with thicknesses as thin as just a few microns lying on solid substrates are shown to reduce the SEE yield by over 50%. This is an encouraging ﬁnding to promote the use of these micro-architected structures in materials exposed to charged plasmas. Work to extend this methodology to dielectric materials of interest in plasma thrusters for electric propulsion (BN [135, 136, 137, 138]) is currently underway.

We note again that, while the present model ignores backscattered electrons, we have convincingly shown that their effect is negligible for the systems considered in this paper. There are computer codes that do capture this contribution, chief among which is Geant4 [139, 140], although they are somewhat less adaptable to complex geometries such as the one considered here. Another aspect worth mentioning is the fact that porous materials have traditionally been used to enhance SEE, not lessen it as is the case here. However, secondary electron multiplication is usually ac- complished under a high voltage condition, with energies much higher than those employed in this 4 work. As well, the typical Debye length (on the order of 10− m) of plasma devices of interest is larger than the surface architecture characteristic length scale (on the order of tens of microns), and it is therefore safe to neglect the electric and magnetic ﬁelds when tracking the trajectories of electrons. This is showcased in studies with textured surfaces, e.g. as in refs. [141, 31].

We ﬁnalize by showing the computational cost of the calculations in Figure 4.9. The ﬁgure shows the CPU time per primary ray monotonically increasing with primary energy for normal incidence in the foam with 4% porosity. The explanation for this increase lies in the number of branches (rays) created by each primary ray, which depends on the SEE yield at each energy. This is shown in red in the plot (vertical scale shown on the right). The curves suggest that a representative CPU cost per ray is approximately 0.035 seconds3.

3This is for Intel Xeon E5-2650v4 CPUs installed on the hoffman2 cluster at UCLA, running at 2.2 GHz, with 4

59 4

0.10

3 0.08

0.06 2

0.04

1 Rays/Primary Ray

CPU Time/Ray0 [s] .02

0.00 0 0 200 400 600 800 1000 E [eV]

Figure 4.9: Computational cost (measured as CPU time) per primary ray as a function of primary energy. The CPU overhead loosely correlates with the number of daughter/granddaughter rays generated by each primary ray (in red).

To conclude, we list the main ﬁndings of our work:

1. We have developed a raytracing Monte Carlo approach that generates random electron trajectories and determines their intersection with solid surfaces via the Moller-Trombore¨ algorithm.

2. Each intersection is characterized by an impinging angle and energy, from which partial electron emission trajectories can be generated from pre-calculated relations.

3. All trajectories are tracked until either an electron emission is recorded or until the energy of the ray falls below the threshold energy for escape.

4. We ﬁnd that microfoams with 96 to 90% porosity reduce the net SEE yield by approximately 50% in W surfaces.

5. We ﬁnd very good agreement between our full approach and measurements of SEE yields in W-fuzz surfaces across a wide energy range.

GB of memory.

60 CHAPTER 5

Monte Carlo Modeling of Low Electron Energy Induced Secondary Electron Emission Yields in Micro-architectured h-BN Surfaces

5.1 Introduction

Advances in electrode, chamber, and structural material technology will enable breakthroughs in future generations of electric propulsion and pulsed power (EP & PP) concepts [142, 49]. Although signiﬁcant advances have been achieved during the past few decades, much of the progress has re- lied on empirical development of materials through experimentation and trial-and-error approaches [143, 144]. Materials and channel wall designs are sought by optimizing performance against weight, power density, and cost. Under extreme operating environments, the discharge channels and cathode insulators used in Hall thrusters require a very demanding set of properties. Typically, the range of materials of interest in EP & PP include refractory metals, such as W, Mo, and their alloys, ceramic composites, such as BN and Al2O3, high-strength copper alloys, and carbon-carbon composites [138]. These classes of materials possess great mechanical strength, thermal shock re- sistance, refractoriness and machinability. Surface erosion and secondary electron emission (SEE) from the channel walls have been identiﬁed as among the most critical life-limiting factors for Hall thrusters [145]. To mitigate surface erosion and SEE, a promising route is to modify the surface roughness. Recently, demonstration designs have been developed, including surface architectures based on metal micro-spears, micro-nodules and micro-velvets [118, 119, 120, 146, 116]. See reviews on the topic for further information [48, 147, 148, 149, 50, 138].

The main objective of our work is to develop reliable physical models of SEE, to be applied to

61 the calculation of effective SEE yields in micro-architected surfaces for space propulsion thrusters. The methodology relies on two distinct but complementary elements. The ﬁrst is an experimentally validated theoretical model of electron scattering in solids. This model is built as a transport Monte Carlo simulator of individual electron trajectories in a solid, capturing the pertinent scattering mechanisms in terms of an interaction differential cross section that is integrated across the relevant energy and angular ranges. These cross sections reﬂect different elastic and inelastic (e.g. core electrons, valence electrons, polarons, plasmons, etc) scattering mechanisms in each material, and is formulated according to the best available physics. The second element of the methodology is a discretization procedure to represent arbitrary surface geometries in terms of discrete boundary elements. Both modules (physics and geometry) are coupled by way of a raytracing algorithm that captures the intersection of primary electron rays with different boundary elements. The material considered here is BN (used in current thrusters thanks to its low mass density, low thermal expansion, and highly dielectric properties). The procedure is described in two papers from our group focused on W [150, 151].

The structure of the paper is as follows. First, the theoretical models employed to study electron interaction with insulators is described. Subsequently, these models are validated experimentally for SiO2 and hexagonal BN. The resulting SEE yield and emitted energy distributions for BN are then used as response functions in ray-tracing Monte Carlo simulations of SEE in discretized complex foams. We ﬁnalize with a summary of the main ﬁndings and the conclusions and acknowledgments.

5.2 Theory and Methods

5.2.1 Electron-Insulator Interaction Model

The energy loss mechanisms for internal secondary electrons differ between metals and insulators. In metals, the internal secondary electrons lose energy through interactions with conduction electrons, lattice vibrations –known as polarons–, and defects. In order to escape, the kinetic energy of an internal secondary electron must be above the energy barrier –taken to be equal to the Fermi

62 level EF plus work function Φ (typically >10 eV)–when it reaches the surface. This large threshold escape energy, as well as the high collision probability due to the large number of conduction electrons, result in the low SEE yields (usually <1) found in metals. Conversely, due to the low number of conduction electrons in insulators, internal secondary electrons lose energy primarily through the excitation of valence electrons into the conduction band. This prevents secondary electrons with kinetic energies below the bandgap energy from participating in such electron-electron collisions, signiﬁcantly increasing their mean escape depths compared to that of metals [152]. The mean escape depth for insulators ranges from 10-50 nm, compared to 0.5-1.5 nm for conductors [57]. Consequently, SEE yields in insulators are typically substantially higher than in metals. In some reported cases, the SEE yields of certain insulators can exceed those of standard conductors by as much as a factor of 20 [57].

When the energy of the primary electron beam is considerably higher than the bandgap energy

Eg of an insulator material, the elementary scattering processes are essentially those encountered for metals. However, at low energies, several new aspects of the electron-material interaction processes become important. For example, at a low-energy electron within an insulator can locally distort the lattice and greatly reduce its mobility by forming a polaron. As well, defects and impurities can also act as traps. In any case, a distinctive feature of secondary electron emission in insulators is the buildup of charge, such that subsequent SEE must be considered in the context of the existence of an internal electric ﬁeld. In the present work, our calculations consider the following electron-material interaction processes: (i) Mott’s theory for electron-atom interactions, (ii) Ritchie’s theory for electron-electron interactions, (iii) Frohlich’s¨ perturbation theory for electron- phonon interactions, and (iv) Ganachaud and Mokrani’s semi-empirical model for electron-polaron interaction.

5.2.2 Elastic Scattering

Due to the large mass difference of electrons and atomic nuclei, electron-atom collisions can be approximated as being perfectly elastic. A commonly used elastic scattering cross-section is the screened Rutherford cross-section, which has a convenient analytical form and is straightforward

63 to implement in a Monte Carlo calculation. However, the screened Rutherford cross-section can be applied only to high-energy electrons and solids with a low atomic number. An alternative to the screened Rutherford cross-section is the relativistic partial wave expansion method of the Mott scattering cross-section. In Mott’s theory [153], the differential elastic scattering cross section (DESCS) with respect to solid angle Ω can be calculated as

dσ el = f (θ) 2 + g(θ) 2 (5.1) dΩ | | | |

where f (θ) and g(θ) are the direct and indirect scattering amplitudes, respectively, given by | | | | 1 ∞ f (θ) = (l + 1)[exp(2iδ +) 1] 2iK ∑{ l − l=0 (5.2)

+ l[exp(2iδ −) 1] P (cosθ) l − } l ∞ 1 + 1 g(θ) = ∑[exp(2iδl−) exp(2iδl )]Pl (cosθ). (5.3) 2iK l=1 − In these equations, K is the momentum of the electron, E the total energy, m the electron mass, 1 c the speed of light, Pl(cosθ) the Legendre polynomials, and Pl (cosθ) the ﬁrst-order associated Legendre polynomials: 1 dP (x) P1(x) = 1 x2 2 l . (5.4) l − dx

The phase shifts δl± can be computed by using the equation

K jl+1(Kr) jl(Kr)[ξ tanφl± + (1 + l + k±)/r] tanδl± = − (5.5) Kn (Kr) n (Kr)[ξ tanφ ± + (1 + l + k )/r] l+1 − l l ± where E + mc2 ξ = . (5.6) }c k+ = l 1, k = l, j are the regular spherical Bessel functions, n the irregular spherical Bessel − − − l l functions and

φ ± = lim φ ±(r) (5.7) l r ∞ l →

where φl± is the solution of the Dirac’s equation which can be reduced, as shown by Lin, Sherman, and Percus [154] and by Bunyan and Schonfelder [155], to the ﬁrst-order differential equation

2 dφl±(r) k± mc E V(r) = sin[2φ ±(r)] cos[2φ ±(r)] + − (5.8) dr r l − }c l }c 64 with V(r) being the electron-atom potential.

In the present calculation, the atomic differential cross-sections for elastic scattering are ex-

tracted from the NIST Electron Elastic-Scattering Cross-Section Database [156] ranging from 0◦

to 180◦ for 24 incident energies between 50 eV and 1 keV (in increments of 10 eV from 50 to 100 eV, and in increments of 50 eV from 100 eV to 1 keV) and put into a data ﬁle in tabulated form. The elastic differential cross-sections for energies and angles other than those in the table can be calculated by linear interpolation accurate to two decimal places.

For compounds, the DESCS can be approximated by using the additivity rule, as the sum of the atomic DESCS of all atoms in the molecule. With the DESCSs of the single elements B and N, the electron-molecule DESCS is obtained for hexagonal boron nitride (h-BN) through dσ (E,ϑ) dσ (E,ϑ) dσ (E,ϑ) el = el + el . (5.9) dΩ BN dΩ B dΩ N Figure 5.1 shows the DESCS of 50, 100, 500 and 1000-eV electrons scattered by h-BN as a function of the scattering angle.

E = 50 eV E = 100 eV 101 E = 500 eV E = 1000 eV

100 /sr] 2 ˚ A [ Ω 1 /d 10− σ d

2 10−

0 50 100 150 θ [deg]

Figure 5.1: Values of the DESCS of 50, 100, and 500 1000 eV electrons scattered by h-BN as a function of the scattering angle.

The total elastic scattering cross section for electron-molecule interaction can then be calcu-

65 lated as: 1 Z π λel− dσel(E,ϑ) σel(E) = = 2π sinϑdϑ (5.10) N 0 dΩ

where N is the number of molecules per unit volume in the target, λel is the elastic mean free path and ϑ is the polar scattering angle.

At low energies, the elastic mean free path calculated using the phase-shift method becomes very small –on the order of 0.1 nm–, which is much smaller than the interatomic distances and therefore unphysical. This is probably a consequence of using a rigid static potential, as well as neglecting dynamic effects such as that associated with the polarization of the electron cloud. In any case, electrons undergoing elastic collisions become conﬁned in a very small region of space, effectively becoming trapped. Consequently, these processes do not contribute appreciably to secondary electron emission, but may slow down calculations signiﬁcantly. To address this issue, in this work the elastic cross-sections calculated by the phase-shift method are multiplied by a cut-off function whose role is to gradually decrease the importance of the elastic effect at very low energies. This function must tend towards unity as the energy increases so that the behavior of the static potential used in the phase-shift method is recovered. The choice of function is of course not unique but an expression of the form

2 Rc(E) = tan[αc(E/Eg) ] (5.11)

is seen to behave well for this purpose (αc is a dimensionless parameter and Eg is bandgap).

The angle of electron scattering at a certain step can be expressed by a random number

Z θ 1 dσel(E,ϑ) R = Pel(E,θ) = sinϑdϑ. (5.12) σel 0 dΩ

5.2.3 Inelastic Scattering

Inelastic scattering is characterized by the energy loss function (ELF) Im 1 , where q − ε(q,∆E) is the momentum transfer and ∆E is the energy loss. The dielectric function ε(q,∆E) in the ELF reﬂects the response of a solid to an external electromagnetic perturbation. Due to the difﬁculties of determining the energy loss function experimentally, Ritchie and Howie suggested an approximate

66 function from the optical dielectric constants by fitting the measured optical data into a finite sum of Drude-Lindhard model functions in the optical limit (q = 0) [157]. In this fashion one can extend the explicit formula to the required Im 1 for finite q-values. − ε(0,∆E) The ELF of the material is parametrized in terms of an expansion of Drude-Lindhard-type oscillators at the optical limit with N-term analytic form, which are directly obtained from the features observed in the reflection electron energy loss spectroscopy (REELS) spectrum:

1 n A γ ∆E Im = i i (∆E E ), (5.13) ∑ 2 2 2 2 2 θ g − ε(q,∆E) ((}ω0iq) ∆E ) + γ ∆E × − i=1 − i where }2q2 }ω0iq = }ω0i + ζ (5.14) 2m

and Ai, γi, and }ω0iq are the oscillator strength, the damping coefficient, and the excitation energy of the ith oscillator, respectively. The step function θ(∆E E ) simulates a bandgap for the case of − g semiconductors and insulators so that θ(∆E E ) = 0 if ∆E < E and θ(∆E E ) = 1 if ∆E > E . − g g − g g Although the dependence of }ω0iq on q is generally unknown, eq. (5.14) is generally accepted using ζ as an adjustable parameter. The value of ζ is related to the effective mass of the electrons, so that for free electrons ζ = 1 and for insulators with flat bands ζ = 0. The calculated fitting parameters of energy loss function for some pure elements and oxides are stored in an open online database [158, 159]. The parameters used to model the energy loss function of h-BN are listed in Table 5.1.

To obtain ELF(q,∆E) from ELF(0,∆E) we use Ashley’s model [160] by which the electron differential inelastic scattering cross section can be deﬁned as

dσ (E,∆E) me2 1 ∆E inel = Im S (5.15) d∆E 2π}2NE − ε(0,∆E) E

Here m is the electron mass, e the electron charge, N the number of molecules per unit volume in the target, E the electron energy, and ∆E the energy transfer. The function S(x) takes the form:

4 7 33 S(x) = (1 x)ln x + x3/2 x2 (5.16) − x − 4 − 32

so that the inelastic scattering cross section σinel(E) for the electron-electron interactions can be

67 2.0 30

1.5 20 ˚ A] [ 1.0 ELF inel λ 10 0.5

0.0 0 0 20 40 60 80 200 400 600 800 1,000 ∆E [eV] E [eV]

(a) (b)

Figure 5.2: (a) Energy loss function of h-BN in the optical limit. (b) Inelastic mean free path of h-BN as a function of primary electron energy.

written as Z W max dσinel(E,∆E) σinel(E) = d∆E Wmin d∆E 1 2 (5.17) λ − me Z Wmax 1 ∆E inel ∆ = = 2 Im S d E N 2π} NE Wmin − ε(0,∆E) E

Wmin is set to zero for conductors and to the bandgap energy for semiconductors and insulating

materials; Wmax = E/2 is the maximum energy transfer.

me2 Z Wmax 1 ∆E 1 ∆ λinel− (E) = 2 Im S d E (5.18) 2π} E Wmin − ε(0,∆E) E

Figure 5.2a is the ELF of h-BN in the optical limit; Figure 5.2b is the inelastic mean free path of h-BN as a function of primary electron energy.

For the stopping power SP = dE/ds , the following expression is used [76]: − dE me2 Z Wmax 1 ∆E ∆ ∆ = 2 Im G Ed E (5.19) − ds π} E 0 − ε(0,∆E) E where 1.166 3 x 4 1 x2 4 31 G(x) = ln x ln + x3/2 ln x2. (5.20) x − 4 − 4 x 2 − 16 x − 48

68 In order to ﬁnd the energy loss, W, of an inelastic collision of an incident electron with kinetic

energy E, it is necessary to calculate the function Pinel(W,E) providing the fraction of electrons losing energy less than or equal to W.

Z W 1 dσinel R = Pinel(W,E) = d∆E (5.21) σinel 0 d∆E where R is a random number uniformly distributed in the range (0,1].

Table 5.1: Parameters used to model the energy loss function of h-BN.

2 h-BN }ω0i [eV] A0i [eV ] γ0i [eV] (ζ = 0.05)

1 8.65 6.6 0.5 2 19.0 18.0 5.0 3 25.2 270.0 9.5 4 35.8 140.0 10.0 5 51.0 80.0 20.0 6 65.0 20.0 20.0

5.2.4 Phonon Excitation

At low energies, when E does not exceed two or three times the value of the bandgap Eg, an electron has a high likelihood of interacting with the lattice vibrations. The interaction of a quasi-free electron with the longitudinal optical (LO) phonons in a polar medium can be treated by Frohlich’s¨ perturbation theory [161]. The interaction with the lattice is accompanied by the creation or by the absorption of a phonon. For the optical branch, it is reasonable to ignore the dispersion relation

of the longitudinal phonon and to characterize it by the unique frequency ωLO. Then, an electron with energy E has a probability per unit of path length to create a phonon of frequency ω (thus losing an energy ∆E = }ω) given by p 1 1 n(T) + 1 ε(0) ε(∞) }ω [1 + 1 }ω/E] λph− = − ) ln p − (5.22) a0 2 ε(0)ε(∞) E [1 1 }ω/E] − − 69 where a0 is the Bohr radius, kB is the Boltzmann constant, }ω is the electron energy loss (on the order of 0.1 eV), ε(0) is the static dielectric constant, ε(∞) is the high frequency dielectric 1 constant and n(T) = }ω/k T 1 is the occupation number for the phonon level at temperature T, e B − taken here equal to 300 K. For the present calculation, we assume ε(∞) = 4.5 and ε(0) = 7.1 and only one LO phonon mode has been considered (with energy ∆E = }ωLO = 0.1 eV). Since the phonon generation probability is higher than the absorption probability by a factor of about 10, the annihilation of the LO phonons along the electron path is neglected.

The polar scattering angle is given according to Llacer et al. [162] by E + E cosθ = 0 (1 BR) + BR 2√EE − 0 (5.23) E + E + 2√EE B = 0 0 E + E 2√EE 0 − 0 where E and E0 are the electron energy before and after electron-phonon scattering, respectively. R is also a random number uniformly distributed in the range [0,1].

5.2.5 Polaronic Effects

A low-energy electron moving in an insulating material induces a polarization ﬁeld that has a stabilizing effect on the moving electron. This phenomenon can be described as the generation of a quasi-particle called polaron. The polaron has a relevant effective mass and mainly consists of an electron (or a hole created in the valence band) with its polarization cloud around it. The polaronic effect is important in the description of low-energy electron transport in insulators as it allows the description of the electron trapping and de-trapping necessary for the investigation of electric current inside insulators and charging-up phenomena. However, cross sections of these channels of electron interaction with matter are scarce in the literature. Here, a semi-empirical formula proposed by Ganachaud and Mokrani [163, 164] is adopted. They assume that the inverse inelastic mean free path that rules the phenomenon –and which is proportional to the probability for a low-energy electron to be trapped in the ionic lattice– is given by

1 ηE λpol− (E) = Ce− (5.24)

where C and η are constants depending on the dielectric material. [163, 164]

70 5.3 Monte Carlo Calculation

The probability for any given scattering to occur is in proportion to its cross section. Thus, spec- ifying the cross section for a given reaction is a proxy for stating the probability that a given scattering process will occur. Figure 5.3 shows various scattering cross sections as a function of primary electron energy.

13 10− Elastic Inelastic Phonon Polaron

15 10− ]

2 17 10− [cm σ 19 10−

21 10−

50 100 150 200 E [eV]

Figure 5.3: Plot of the various scattering cross sections of the processes considered here as a function of primary electron energy.

The stochastic process for multiple scattering is assumed to follow Poisson statistics. If R is a random number uniformly distributed in the interval (0,1], the step length ∆s is given by

∆s = λ lnR, (5.25) − T where λT is the electron mean free path, given by

1 1 1 1 1 = + + + . (5.26) λT λel λinel λph λpol

71 The procedure to sample different scattering events follows the sequence:

1/λ 0 < R el = elastic scattering ≤ 1/λT ⇒ 1/λ 1/λ + 1/λ el < R el inel = inelastic scattering 1/λT ≤ 1/λT ⇒ 1/λ + 1/λ 1/λ + 1/λ + 1/λ el inel < R el inel ph = phonon excitation (5.27) 1/λT ≤ 1/λT ⇒ 1/λ + 1/λ + 1/λ 1/λ + 1/λ + 1/λ + 1/λ el inel ph < R 1 el inel ph pol 1/λT ≤ ≡ 1/λT = polaron generation ⇒ In this fashion, events are sequentially sampled and executed, conforming effective trajectories that are tracked until an electron reaches the surface with an energy larger than the workfunction or until the electron is thermalized inside the material. Electrons that escape the surface are tallied and the net yield s computed as the ration of the number of escaped electrons relative to the total number of primary trajectories generated.

Our model accounts for the main physical processes on which the secondary electron emission of metal oxides depends. However, several parameters appear in the empirical laws we have proposed. The physical meaning of these parameters is quite clear but their values can be found only by comparing the simulation results to the experimental measurements. Preliminary calculations have allowed us to estimate what can be considered as a set of reference parameters. These values will be further varied to check the inﬂuence of their choice, particularly on the secondary electron emission yields. The parameters for h-BN in present calculation can be found in Table 5.2.

Table 5.2: Parameters for h-BN in present calculation. Among these, αc, Wph, C and η are free parameters.

Parameters

1 1 h-BN αc Eg [eV] ε(∞) ε(0) Wph [eV] C [nm− ] η [eV − ] χ [eV]

0.5 5.2 4.10 5.09 0.1 1.0 0.1 4.5

72 5.4 Results

5.4.1 Flat Surfaces

The total secondary electron yield for ideally-ﬂat h-BN surfaces is calculated for incident angles

of 0◦, 15◦, 30◦, 45◦, 60◦, 75◦ and 89◦ measured off the surface normal, and incident energies in the range 50-1000 eV. In this work, the typical number of primary particles simulated ranges between 104 and 105, which generally results in statistical errors around 3%.

Experimental data on h-BN secondary electron emission yield is scarce and subjected to high

uncertainty, so ﬁrst we test our models on a material system such as SiO2 for which such data

exist. This is a ﬁrst step aimed at validating our codes before the study of BN. The results for SiO2 are given in the Section 5.6, which convincingly demonstrates the validity of our models. On this basis, the simulations of 50 eV-1 keV electron irradiation on ﬂat h-BN surfaces are performed.

At low energy regime, the simulation results are found to agree reasonably well with experimental data, which corresponds exactly to normal operating conditions (<100eV) of Hall thrusters. The agreement is slightly worse in the intermediate energy regime, which we rationalize in terms of the charging effect of insulators under electron irradiation. Again, at high energy regime, the simulation results converge to the experimental data. Note that under steady state, the total SEE yield at high temperatures should tend to unity. This can be attributed to the onset of a charge gradient due to the existence of holes created by the departure of secondary electrons from the lattice. This charge gradient creates a ‘shielding’ electric ﬁeld that captures further SEE until charge neutrality is achieved again. Once the material is neutral, the process starts again, leading to an oscillatory steady state that keeps SEE balanced [165]. This picture can be altered by factors such as radiation-induced conductivity changes, sheath potential modiﬁcations, or slowly evolving internal charge distributions. In addition, the roughness of ’real’ experimental surfaces compared to the ideally-smooth ones in the model surely plays a role in the comparison. SEE yields as a function of E for all angles of incidence considered here are given in Figure 5.4b.

Surface plots of both the SEE energy distributions and the yields are given in Figures ?? and ??. As mentioned earlier, these data will be used in ray-tracing Monte Carlo simulations of SEE in

73 arbitrary surface geometries. The details of the function ﬁtting process can be found in our prior publications [150, 151]. The ﬁnal expressions for the total SEE yield and energy distributions are generated using machine learning software [99]:

 15 4 15 3 6 2 0.0185E + 1.53 10− E + 1.53 10− E + 1.2 10− Eα  × × ×   15 4 3 5 2 12 2 4  + 1.53 10− E α + 3.915 10− E 1.4 10− E α (0 E 200eV) γ(E,α) = × × − × ≤ ≤ . 3.82 + 1.54 10 6E2 cos(2.48α) 3.68 104/(2.63 10 1α  − −  × − × ×  2 2 4 3  + E Eα cos (2.48α)) 7.88 10− E 3.37 10− E cos(2.48α)(200 E 1000eV) − − × − × ≤ ≤ (5.28)

 6 2 7 3 2 8 2 0.73E + 3.88 10− αE + 6.60 10− Eα + E sin(5.31 10− E )  × × ×   3 2 5 2  5.12 10− E 4.22 10− Eα (0 E 200eV) ESE(E,α) = − × − × ≤ ≤ . 27.81 + 7.69 10 5E2 1.99 105/(E3 cos(E))+  −  × − ×  2 7 2 (12.16 1.94E)/(α 97.58) 3.97 10− E 4.08 10− αE (200 E 1000eV) − − − × − × ≤ ≤ (5.29)

5.4.2 Micro-Architectured Foam Structures

Next we calculate SEE from microfoam structures with various porosities. The details of these structures are given in our past studies [150, 151]. A ﬁnite element reconstruction of the material is used to extract surface elements that may be intersected by electron trajectories. Special algorithms are then used to identify intersections between primary and secondary rays. Daughter rays are generated from parent rates using correlations (5.28) and (5.29).

Figures 5.6a and 5.6b show the secondary electron emission yield for solid volume fractions,

Vf , of 4, 6, 8, and 10% in the 50-to-1000-eV energy range for normal and random incidence. The inset to Figure 5.6a shows the dependence of the yield with the material volume fraction at energies of 50, 100, 200, 300, 400, 500, and 600 eV for normal incidence. In the high porosity

range explored here the dependence of the SEE yield on Vf is clearly linear in the high porosity

74 2.5 4.5 4.0 2.0 3.5 3.0 1.5 2.5 γ γ 2.0 1.0 1.5 1.0 0.5 0 ◦ 15 ◦ 30 ◦ 45 ◦ 0.5 60 ◦ 75 ◦ 89 ◦ 0.0 0.0 0 200 400 600 800 1000 0 200 400 600 800 1000 E [eV] E [eV]

(a) (b)

Figure 5.4: (a) Total SEE yield from smooth hBN surface as a function of primary electron energy for electrons incident at 0 . = this work; = Dawson (1966) [13]; = ONERA (1995) [14]; ◦ • H = PPPL (2002) [15]; N = Christensen (2016) [16]. (b) Total SEE yield from an ideally-ﬂat hBN as a function of primary electron energy, for electrons incident at 0◦, 15◦, 30◦, 45◦, 60◦, 75◦ and 89◦.

range explored here. For the sake of comparison, the maximum SEE yield for Vf = 4% (which occurs for E = 700 eV) is approximately 1.3, compared with a value of 2.3 for the ﬂat surface (from Figure 5.4b). This decrease in SEE yield by about a factor of two is indicative of the potential performance gains that micro-architected surfaces might offer relative to fully dense surfaces.

5.5 Discussion and Conclusions

The main objective of our work is to develop reliable physical models of secondary electron emission (SEE), to be applied to the calculation of effective SEE yields in micro-architected surfaces for space propulsion thrusters. The methodology relies on two distinct but complementary elements. The ﬁrst is an experimentally validated theoretical model of electron scattering in solids. This model is built as a transport Monte Carlo simulator of individual electron trajectories in a solid, capturing the pertinent scattering mechanisms in terms of an interaction differential cross section

75 3.5 3.0 200 2.5 γ

2.0 [eV] 1.5 1.0 SE 100 0.5 E 0.0 0 90 90 0 75 200 75 0 60 60 45 400 45 200 30 600 30 400 600 15 800 15 800 0 [deg] 0 [deg] 1000 α E [eV] 1000 α E [eV]

(a) (b)

Figure 5.5: (a) Surface plot of the total SEE yield from an ideally-flat hBN as a function of primary electron energy and angle of incidence. (b) Surface plot of the SEE energy distributions from an ideally-flat hBN as a function of primary electron energy and angle of incidence. that is integrated across the relevant energy and angular ranges. These cross sections reflect different elastic and inelastic (e.g. core electrons, valence electrons, polarons, plasmons, etc) scattering mechanisms in each material, which are formulated according to the best available physics. The second element of the methodology is a discretization procedure to represent arbitrary surface geometries in terms of discrete boundary elements. Both modules (physics and geometry) are coupled by way of a raytracing algorithm that captures the intersection of primary electron rays with different boundary elements. The material considered here is BN (used in current thrusters thanks to its low mass density, low thermal expansion, and highly dielectric properties).

Our main findings are that, in the primary electron energy range of interest (<100 eV) microfoam architected surfaces are seen to decrease the SEE yield by about a factor of two. This is already a significant advantage over flat surfaces (or with as-fabricated surface roughness). As well, these micro-foams are seen to suppress the SEE yield peak typically observed at 400 eV of primary energy. Our results have been validated at two different levels. As mentioned above, the electron scattering model has been compared to experiments in smooth surfaces for both SiO2

76 1.8 1.8 4% 6% 8% 10% 1.6 1.6 1.4 1.4 1.2 1.6 1.2 1.0 1.0 γ 200 eV γ 0.8 γ 0.8 100 eV 0.6 0.6 50 eV 0.4 0.4 4% 0.2 6% 0.2 4 10 0.2 8% Vf % 10% 0.0 0.0 0 200 400 600 800 1000 0 200 400 600 800 1000 E [eV] E [eV]

(a) (b)

Figure 5.6: (a) SEE yield versus electron beam energy for normal incidence to the foam at varying volume-fraction percentages. The inset shows (in increasing order) the dependence of the yield with volume fraction for primary energies equal to 50, 100, 200, 300, 400, 500, and 600 eV. (b) SEE yield versus electron beam energy for random incidence to the foam at varying volume- fraction percentages. and BN. As this is the ‘physics’ building block of the methodology, great care has been placed on ensuring that the parameters of the model are consistent with available measurements in each case. Second, we have compared results for the W micro-foam to measurements in ‘fuzz’ W surfaces (smooth W surfaces pre-exposed to a He plasma), which have a similar degree of porosity and morphology as foams.

Our approach has several advantages. First, it allows us to study any arbitrary surface morphology thanks to the ﬁnite element discretization scheme and the raytracing Monte Carlo method to generate electron trajectories. This endows the methodology with an extraordinary versatility, as complex surface geometries such as foams, pillars, fuzz, cells, etc., of any size can be treated in a straightforward manner. As well, our physical scattering model being capable of describing both metals and ceramics, we can now study materials of high relevance for electric propulsion.

77 We believe that this approach will enable a rapid parsing of thruster lining material surface concepts prior to costly development and characterization, to narrow down the parametric space and accelerate materials development.

To summarize, our main ﬁndings are:

1. In the primary electron energy range of interest (<100 eV) BN microfoam architected surfaces are seen to decrease the SEE yield by about a factor compared to ideally ﬂat surfaces.

2. These micro-foams are seen to suppress the SEE yield peak typically observed at 400 eV of primary energy.

3. Our results have been validated by comparison to experiments in smooth surfaces for both

SiO2 and BN.

5.6 Veriﬁcation

78 4.5 2.5 4.0 3.5 2.0 3.0 1.5 2.5 γ γ 2.0 1.0 1.5 1.0 0.5 0 ◦ 15 ◦ 30 ◦ 45 ◦ 0.5 60 ◦ 75 ◦ 89 ◦ 0.0 0.0 0 200 400 600 800 1000 0 200 400 600 800 1000 E [eV] E [eV]

(a) (b)

Figure 5.7: (a) Total SEE yield from smooth SiO2 surface as a function of primary electron energy for electrons incident at 0 . = this work; = Dionne (1975) [10]; = Barnard (1977) [7]; ◦ • N = Yong (1998). (b) Total SEE yield from an ideally-ﬂat SiO2 as a function of primary electron energy, for electrons incident at 0◦, 15◦, 30◦, 45◦, 60◦, 75◦ and 89◦.

79 2.0

1.5 400 300 γ [eV] 200

1.0 SE

E 100 90 0 75 90 0 60 0.50 75 45 60 200 45 200 400 30 400 30 600 15 600 15 800 0 [deg] 800 [deg] 1000 α 1000 0 α E [eV] E [eV]

(a) (b)

Figure 5.8: (a) Surface plot of the total SEE yield from an ideally-ﬂat SiO2 as a function of primary electron energy and angle of incidence. (b) Surface plot of the SEE energy distributions from an ideally-ﬂat SiO2 as a function of primary electron energy and angle of incidence.

80 CHAPTER 6

Conclusions

This chapter discusses the results, identiﬁes contributions made to the ﬁeld and concludes with recommendations for further work.

6.1 Discussion of Results

This work investigated the interaction of particles with the discharge chamber walls and the resulting secondary electron emission (SEE) and surface erosion. For many plasma devices, including EP and fusion energy devices, SEE and surface erosion can affect device performance and life. Thus, understanding the underlying mechanism is crucial to make predictions of channel wall lifetime. The main ﬁndings of our simulation results are given in the following.

1. We have developed a discrete MC model to simulate the secondary electron emission from ideally ﬂat metal and insulator surfaces for primary electrons in low energy ranges (100-1000 eV). In this discrete approach, the energy loss of electrons traveling through a solid is determined by considering different inelastic scattering processes –including conduction electron excitation, plasmon decay, and inner shell electron ionization– individually, whereas within the so-called continuous slowing down approximation (CSDA), the overall inelastic scattering mechanisms are averaged out by using the total stopping power (SP). From this point of view, the CSDA and the discrete-event simulation method would formally converge in the limit of an inﬁnite number of events. However, discrete simulations allow a better physical and spatial dissection of electron scattering processes, providing spatial distributions and breakdowns among the different scattering mechanisms. This information is important to

81 ascertain what scattering events dominate the secondary electron emission process in each material.

2. We have developed a raytracing Monte Carlo approach that generates random electron trajectories and determines their intersection with solid surfaces via the Moller-Trombore¨ algorithm. Each intersection is characterized by an impinging angle and energy, from which partial electron emission trajectories can be generated from pre-calculated relations. All trajectories are tracked until either an electron emission is recorded or until the energy of the ray falls below the threshold energy for escape. Our approach has several advantages. First, it allows us to study any arbitrary surface morphology thanks to the ﬁnite element discretization scheme and the raytracing Monte Carlo method to generate electron trajectories. This endows the methodology with an extraordinary versatility, as complex surface geometries such as foams, pillars, fuzz, cells, etc., of any size can be treated in a straightforward manner. As well, our physical scattering model being capable of describing both metals and ceramics, we can now study materials of high relevance for electric propulsion.

3. Our simulation results show that microfoams with 96 to 90% porosity reduce the net SEE yield by approximately 50% in W surfaces. As well, very good agreement between our full approach and measurements of SEE yields in W-fuzz surfaces across a wide energy range has been achieved. This is an encouraging ﬁnding to promote the use of these micro-architected structures in materials exposed to charged plasmas. Work to extend this methodology to dielectric materials (e.g. boron nitride) of interest in plasma thrusters for electric propulsion is currently underway.

4. Our simulation results show that in the primary electron energy range of interest (<100 eV) BN microfoam architected surfaces are seen to decrease the SEE yield by about a factor compared to ideally ﬂat surfaces. These micro-foams are seen to suppress the SEE yield peak typically observed at 400 eV of primary energy. Our predictions have been validated

by comparison to experiments in smooth surfaces for both SiO2 and BN.

82 6.2 Future Work

6.2.1 Charging Effect on Secondary Electron Emission

Dielectric materials usually suffer charging effects when irradiated by charged particles. When transporting in an insulating solid, electrons encounter elastic and inelastic scattering events where the Mott cross section and a Lorentz-type dielectric function are employed to describe such scatterings, respectively. As well, the band gap and the electron long optical phonon interaction are taken into account. The electronic excitation in inelastic scattering causes generation of electron?hole pairs. These negative and positive charges establish an inner electric field, which in turn induces the drift of charges to be trapped by impurities, defects, vacancies etc in the solid, where the distributions of trapping sites are assumed to have uniform density. Under charging conditions, the inner electric field distorts electron trajectories, and the surface electric potential dynamically alters SEE. In the present study, we have adopted a more empirical approach to account for this so-called ”polaron effect”. However, a more rigorous approach should be developed to simulate the building of the charge in the target and that of the electrostatic field have to be treated simultaneously and self-consistently.

83 APPENDIX A

List of Symbols

a0 Bohr radius AW atomic weight Z atomic number

Na Avogadro’s number ρ density of the target

N = ρNa/AW atomic number density ns number of electrons in shell or subshell nc number of conduction-band electrons per atom e electron charge kB Boltzmann constant ε permittivity of vacuum me mass of electron } reduced Planck constant

ωp plasma frequency E primary electron energy

ESE secondary electron energy

EF Fermi energy kF Fermi wave number

EB binding energy of the shell

Epl = h¯ωp plasmon energy

Eg energy gap ∆E energy loss of primary electron

84 q momentum transfer

} reduced Planck constant ω phonon frequency χ electron afﬁnity [eV] ε(0) static dielectric constant ε(∞) high frequency dielectric constant n refractive index k extinction coefﬁcient Φ work function J mean ionization potential α incident angle of primary electron [deg] γ secondary electron emission yield

Rc cut-off function

σel elastic scattering cross section

σp plasmon excitation cross section

σc conduction electron ionization cross section

σs inner shell electron ionization cross section

σinel inelastic scattering cross section

σph phonon excitation cross section

σpol polaron cross section dσ/dΩ differential scattering cross section with respect to direction dσ/dE differential scattering cross section with respect to energy

λel elastic mean free path

λp plasmon excitation mean free path

λc conduction electron excitation mean free path

λs inner shell electron excitation mean free path

λinel inelastic mean free path

λph phonon excitation mean free path

85 λpol polaron excitation mean free path

λT total mean free path θ polar scattering angle of the primary electron ϑ polar scattering angle of the secondary electron φ azimuthal scattering angle of the primary electron ϕ azimuthal scattering angle of the secondary electron

θp plasmon loss scattering angle d depth

86 APPENDIX B

Constants & Kinematical Quantities

N = 6.022 1023,Avogadro’s number a × 12 ε = 8.85 10− [F/m],permittivity of vacuum × 31 m = 9.1 10− [kg] e × 19 e = 1.6 10− [C],electroncharge × 16 h¯ = 6.58 10− [eV s/rad],reduced Planck constant × · 4 2 Eh = mee /h¯ = 2Ry = 27.2114[eV],Hartree energy

Ry = 13.6[eV],Rydberg energy

2 2 9 a = h¯ /(m e ) = 5.29177 10− [cm],Bohr radius 0 e × 4 2 14 2 2 πe = π(a E ) = 6.5141 10− [cm eV ] 0 h × 2 mec = 510.999[keV],rest energy of the electron The Fermi energy can be estimated using the number of electrons per unit volume as

15 2/3 2/3 E = 3.64645 10− n [eV] = 1.69253n [eV] F × 0 where n and n are in the units of [cm 3] and n = n 1022. The Fermi wave number is calculated 0 − 0 × as 7 1/3 1 k = 6.66511 10 n [cm− ]. F × 0 The Fermi velocity is calculated as

v = 7.71603 107n1/3[cm/s]. F × 0

87 APPENDIX C

Deﬁnition of Coordinate System

The basic geometry for the simulation assumes that the electron undergoes an elastic scattering

event at some point Pn, having traveled to Pn from a previous scattering event at Pn-1. To calculate

the position of the new scattering point Pn+1, we ﬁrst require to know the distance, or ”step,”

between Pn+1 and the preceding point Pn.

The path is described using direction cosines, ca, cb and cc. The coordinates at the end of the

steps at Pn+1, xn, yn and zn, are then related to the coordinates x, y, z at Pn by the formulas

x = x + step ca n · y = y + step cb n · z = z + step cc n · The direction cosines ca, cb, cc are found from the direction cosines cx, cy and cz with which the electron reached Pn. The result is

ca = (cx cosφ) + (V1 V3) + (cy V2 V4) · · · · cb = (cy cosφ) + (V4 (cz V1 cx V2)) · · · − · cc = (cz cosφ) + (V2 V3) (cy V1 V4) · · − · · where V1 = AN sinφ · V2 = AM AN sinφ · V3 = cosψ

V4 = sinψ

88 and cx AM = −cz 1 AN = √1 + AM AM ·

Figure C.1: Deﬁnition of coordinate system used in the Monte Carlo simulation program.

89 APPENDIX D

Classical Scattering

D.1 Deﬁnition of Cross Section

Consider a classical measurement where a single particle is scattered off a single stationary target particle. Conventionally, a spherical coordinate system is used, with the target placed at the origin and the z axis of this coordinate system aligned with the incident beam. The angle θ is the scattering angle, measured between the incident beam and the scattered beam, and the φ is the azimuthal angle. The impact parameter b is the perpendicular offset of the trajectory of the incoming particle, and the outgoing particle emerges at an angle θ.

For a given interaction (Coulombic, magnetic, gravitational, contact, etc.), the impact parameter and the scattering angle have a deﬁnite one-to-one functional dependence on each other. Gen- erally the impact parameter can neither be controlled nor measured from event to event and is assumed to take all possible values when averaging over many scattering events. The differential size of the cross section is the area element in the plane of the impact parameter, i.e. dσ = bdφdb. The differential angular range of the scattered particle at angle θ is the solid angle element dΩ = sinθdθdφ. The differential cross section is the quotient of these quantities and is given by dσ b db = , (D.1) dΩ sinθ dθ which is a function of the scattering angle (and therefore also the impact parameter), plus other observables such as the momentum of the incoming particle. The differential cross section is always taken to be positive, even though larger impact parameters generally produce less deﬂection. In cylindrically symmetric situations (about the beam axis), the azimuthal angle φ is not changed

90 by the scattering process, and the differential cross section can be written as dσ 1 Z 2π dσ = dφ (D.2) d(cosθ) 2π 0 dΩ

In situations where the scattering process is not azimuthally symmetric, such as when the beam or target particles possess magnetic moments oriented perpendicular to the beam axis, the differential cross section must also be expressed as a function of the azimuthal angle.

For scattering of particles of incident ﬂux Finc off a stationary target consisting of many parti- dσ cles, the differential cross section dΩ at an angle (θ,Ω) is related to the ﬂux of scattered particle

detection Fout(θ,φ) in particles per unit time by dσ 1 F (θ,φ) (θ,φ) = out (D.3) dΩ nt∆Ω Finc Here ∆Ω is the ﬁnite angular size of the detector (SI unit: sr), n is the number density of the target particles (SI units: m3), and t is the thickness of the stationary target (SI units: m). This formula assumes that the target is thin enough that each beam particle will interact with at most one target particle.

The differential cross section is extremely useful quantity in many ﬁelds of physics, as measur- ing it can reveal a great amount of information about the internal structure of the target particles. For example, the differential cross section of Rutherford scattering provided strong evidence for the existence of the atomic nucleus.

Instead of the solid angle, the momentum transfer may be used as the independent variable of differential cross sections.

Differential cross sections in inelastic scattering contain resonance peaks that indicate the creation of metastable states and contain information about their energy and lifetime.

The total cross section σ may be recovered by integrating the differential cross section dσ/dΩ over the full solid angle (4π steradians): I dσ Z 2π Z π dσ σ = = sinθdθdφ (D.4) 4π dΩ 0 0 dΩ It is common to omit the ”differential” qualiﬁer when the type of cross section can be inferred from context. In this case, may be referred to as the integral cross section or total cross section. The

91 latter term may be confusing in contexts where multiple events are involved, since total can also refer to the sum of cross sections over all events.

Figure D.1: Classical scattering. (r,φ) are the polar coordinates of the projectile, θ is the scattering angle, φ the azimuthal angle, b the impact parameter, d the distance of closest approach, and d0 its minimum for a central collision.

D.2 Rutherford Scattering

For the derivation of the Rutherford scattering cross section we assume:

1. The projectile and the scattering center (target) are point particles (with Gausss law it can be proved that this is also fulﬁlled for extended particles as long as the charge distribution is not touched upon).

2. The target nucleus is inﬁnitely heavy (i.e. the laboratory system coincides with the c.m. system).

3. The interaction is the purely electrostatic Coulomb force (more precisely: the monopole term of this force) 2 1 Z1Z2e C FC = 2 = 2 (D.5) ±4πε0 r r

92 with the Coulomb potential V = C/r. C ±

The deﬂection function is most simply determined by applying angular-momentum conservation and the equation of motion in one coordinate (y)(v∞, E∞, and p∞ are the asymptotic (i.e. valid or prepared at r ∞) quantities: projectile velocity, kinetic energy, and momentum). → ±

2 L = mv∞b = mr φ˙ = mvmind (D.6) and from this 2 dt = r dφ/v∞b Z (D.7) m∆vy = Fydt

C Z ∞ v∞ sinθ = φ˙ sinφdt mv∞b ∞ − C Z π θ = − sinφdφ (D.8) mv∞b 0 C = (1 + cosθ) mv∞b After transformation to half the scattering angle the deﬂection function is

2 cot(θ/2) = mv∞b/C = v∞L/C (D.9) and C θ b = cot( ) (D.10) 2E∞ 2 and db C 1 C 1 = 2 2 = 2 (D.11) dθ 2mv∞ sin (θ/2) 4E∞ sin (θ/2) and thus for the Rutherford cross section

2 2 db Z1Z2e 1 = 4 (D.12) dΩ 4E∞ sin (θ/2)

2 db Z1Z2 1 mb = 1.296 4 (D.13) dΩ E∞[MeV] sin (θ/2) sr

93 APPENDIX E

Partial Wave Analysis

E.1 Preliminary Scattering Theory

The following description follows the canonical way of introducing elementary scattering theory. A steady beam of particles scatters off a spherically symmetric potential V(r), which is short ranged so that for large distances r ∞, the particles behave like free particles. In principle, any → particle should be described by a wave packet but we describe the scattering of a plane wave traveling along the z-axis exp(ikz) instead, because wave packets are expanded in terms of plane waves and this is mathematically simpler. Because the beam is switched on for times long compared to the time of interaction of the particles with the scattering potential, a steady state is assumed. This means that the stationary Schrodinger¨ equation for the wave function Φ(r) representing the particle beam should be solved:

h¯ 2 ∇2 +V(r) Φ(r) = EΦ(r) (E.1) − 2m

We make the following ansatz:

Φ(r) = Φ0(r) + Φs(r) (E.2)

where Φ0(r) ∝ exp(ikz) is the incoming plane wave and Φs(r is a scattered part perturbing the

original wave function. It is the asymptotic form of Φs(r that is of interest, because observations near the scattering center (e.g. an atomic nucleus) are mostly not feasible and detection of particles takes place far away from the origin. At large distances, the particles should behave like free

particles and Φs(r) should therefore be a solution to the free Schrodinger¨ equation. This suggests that it should have a similar form to a plane wave, omitting any physically meaningless parts. We

94 therefore investigate the plane wave expansion:

∞ ikz l e = ∑(2l + 1)i jl(kr)Pl(cosθ). (E.3) l=0

The spherical Bessel function jl(kr) asymptotically behaves like

This corresponds to an outgoing and an incoming spherical wave. For the scattered wave function, only outgoing parts are expected. We therefore expect Φs(r) ∝ exp(ikr)/r at large distances and set the asymptotic form of the scattered wave to

exp(ikr) Φ (r) f (θ,k) (E.4) s → r

where f (θ,k) is the so-called scattering amplitude, which is in this case only dependent on the elevation angle θ and the energy. In conclusion, this gives the following asymptotic expression for the entire wave function:

exp(ikr) Φ(r) Φ+(r) = expikz + f (θ,k) (E.5) → r

E.2 Partial Wave Expansion

In case of a spherically symmetric potential V(r) = V(r), the scattering wave function may be expanded in spherical harmonics which reduce to Legendre polynomials because of azimuthal symmetry (no dependence on φ):

∞ ul(r) Φ(r) = ∑ Pl(cosθ). (E.6) l=0 r

In the standard scattering problem, the incoming beam is assumed to take the form of a plane wave of wave number k, which can be decomposed into partial waves using the plane wave expansion in terms of spherical Bessel functions and Legendre polynomials:

∞ ikz l φin(r) = e = ∑(2l + 1)i jl(kr)Pl(cosθ). (E.7) l=0

Here, we have assumed a spherical coordinate system in which the z-axis is aligned with the beam direction. The radial part of this wave function consists solely of the spherical Bessel func-

95 tion, which can be rewritten as a sum of two spherical Hankel functions:

1 j (kr) = (h(1)(kr) + h(2)(kr)) (E.8) l 2 l l

(2) (l+1) ikr This has physical signiﬁcance: hl asymptotically (i.e. for large r) behaves as i− e /(kr) (1) l+1 ikr and is thus an outgoing wave, whereas hl asymptotically behaves as i e /(kr) and is thus an incoming wave. The incoming wave is unaffected by the scattering, while the outgoing wave is modiﬁed by a factor known as the partial wave S-matrix element Sl:

l ul(r) r ∞ i k (2) (2) → h + Slh (kr) (E.9) r −−−→ √2π l l where ul(r)/r is the radial component of the actual wave function. The scattering phase shift δl is deﬁned as half of the phase of Sl:

2iδl Sl = e (E.10)

If ﬂux is not lost, then S = 1 and thus the phase shift is real. This is typically the case unless the | l| potential has an imaginary absorptive component, which is often used in phenomenological models to simulate loss due to other reaction channels. Therefore, the full wave function is, asymptotically,

∞ (1) (2) r ∞ l hl (kr) + Slhl (kr) φ(r) → ∑(2l + 1)i Pl(cosθ) (E.11) −−−→ l=0 2

Subtracting φin yields the asymptotic outgoing wave function:

∞ r ∞ Sl 1 (2) → l φout(r) ∑(2l + 1)i − hl (kr)Pl(cosθ) (E.12) −−−→ l=0 2 Making use of the asymptotic behavior of the spherical Hankel functions, one obtains:

ikr ∞ r ∞ e Sl 1 φout(r) → ∑(2l + 1) − Pl(cosθ) (E.13) −−−→ r l=0 2ik Since the scattering amplitude f (θ,ϕ) is deﬁned via:

ikr r ∞ e φ (r) → f (θ,φ) (E.14) out −−−→ r

It follows that

∞ ∞ iδ Sl 1 e l sinδl f (θ,k) = ∑(2l + 1) − Pl(cosθ) = ∑(2l + 1) Pl(cosθ) (E.15) l=0 2ik l=0 k

96 and thus the differential cross section is given by

∞ iδl 2 dσ 2 1 e sinδl = f (θ,k) = 2 ∑(2l + 1) Pl(cosθ) (E.16) dΩ | | k l=0 k This works for any short-ranged interaction. For long-ranged interactions (such as the Coulomb interaction), the summation over l may not converge. The general approach for such problems to treat the Coulomb interaction separately from the short-ranged interaction, as the Coulomb problem can be solved exactly in terms of Coulomb functions, which take on the role of the Hankel functions in this problem.

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