APPROVED: Ducan L. Weathers, Major Professor Carlos A. Ordonez, Co-Major Professor Floyd D. McDaniel, Committee Member Tilo Reinert, Committee Member Chris Littler, Chair of the Department

of Physics Mark Wardell, Dean of the Toulouse

Graduate School

AN ELECTRO- MAGNETO-STATIC FIELD FOR CONFINEMENT OF

CHARGED PARTICLE BEAMS AND PLASMAS

Josè L. Pacheco

Dissertation Prepared for the Degree of

DOCTOR OF PHILOSOPHY

UNIVERSITY OF NORTH TEXAS

May 2014

Pacheco, Josè L. An Electro- Magneto-Static Field for Confinement of Charged

Particle Beams and Plasmas. Doctor of Philosophy (Physics), May 2014, 101 pp., 34

figures, 40 numbered references.

A system is presented that is capable of confining an ion beam or plasma within

a region that is essentially free of applied fields. An Artificially Structured Boundary

(ASB) produces a spatially periodic set of magnetic field cusps that provides charged

particle confinement. Electrostatic plugging of the magnetic field cusps enhances

confinement. An ASB that has a small spatial period, compared to the dimensions of a

confined plasma, generates electro- magneto-static fields with a short range. An ASB-

lined volume thus constructed creates an effectively field free region near its center. It is

assumed that a non-neutral plasma confined within such a volume relaxes to a Maxwell-

Boltzmann distribution. Space charge based confinement of a second species of charged

particles is envisioned, where the second species is confined by the space charge of the

first non-neutral plasma species. An electron plasma confined within an ASB-lined

volume can potentially provide confinement of a positive ion beam or positive ion

plasma.

Experimental as well as computational results are presented in which a plasma or

charged particle beam interact with the electro- magneto-static fields generated by an

ASB. A theoretical model is analyzed and solved via self-consistent computational

methods to determine the behavior and equilibrium conditions of a relaxed plasma. The

equilibrium conditions of a relaxed two species plasma are also computed. In such a

scenario, space charge based electrostatic confinement is predicted to occur where a

second plasma species is confined by the space charge of the first plasma species. An

experimental apparatus with cylindrical symmetry that has its interior surface lined

with an ASB is presented. This system was developed by using a simulation of the

electro- magneto-static fields present within the trap to guide mechanical design. The

construction of the full experimental apparatus is discussed. Experimental results that

show the characteristics of electron beam transmission through the experimental

apparatus are presented. A description of the experimental hardware and software used

for trapping a charged particle beam or plasma is also presented.

ii

Copyright 2014

by

Josè L. Pacheco

iii

ACKNOWLEDGEMENTS

This work is dedicated to Nataly for her unconditional love, support, and patience! and

to my Mother and Father.†

I would like to thank Naresh T. Deoli and Allen S. Kiester for their help,

suggestions, and friendship; Kurt Weihe, Paul Jones, and Gary Karnes for assisting with

technical support; the Ion Beam Modification and Analysis Lab (IBMAL) for supplying

the necessary experimental equipment; and UNT's High Performance Computing

Initiative for providing computational resources.

The material presented is based upon work supported by the Department of

Energy under Grant No. DE-FG02-06ER54883 and by the National Science Foundation

under Grant No. PHY-1202428. The research that appears in this dissertation is, in

part, a compilation of published work:

• “Plasma Interaction With a Static Spatially Periodic Electromagnetic Field,” J. L. Pacheco, C. A. Ordonez, and D. L. Weathers. IEEE Transactions on Plasma Science, Vol. 39, no. 11, pp. 2424-2425, Nov. 2011 doi: 10.1109/TPS.2011.2158669.

• “Spatially Periodic Electromagnetic Force Field For Plasma Confinement and Control,” C. A. Ordonez, J. L. Pacheco, and D. L. Weathers. The Open Plasma Physics Journal, 5(2012). pp. 1-10. doi: 10.2174/1876534301205010001. (Section VI)

• “Artificially Structured Boundary for a High Purity Ion Trap or Ion Source,” J. L. Pacheco, C. A. Ordonez, and D. L. Weathers. Nucl. Instr. and Meth. in Phys. Res. B. Conf. Proc. 21st International Conference on Ion Beam Analysis. Seattle, WA. 2013.

• “Space-charge-based electrostatic plasma confinement involving relaxed plasma species,” J. L. Pacheco, C. A. Ordonez, and D. L. Weathers. Physics of Plasmas, 19, 102510 (2012), DOI: http://dx.doi.org/10.1063/1.4764076.

• “Electrostatic Storage Ring With Focusing Provided By the Space Charge of an Electron Plasma,” J. L. Pacheco, C. A. Ordonez, and D. L. Weathers. Application of Accelerators in Research and Industry, AIP Conference Proceedings 1525 (2013) 88-93.

iv

TABLE OF CONTENTS

Page ACKNOWLEDGEMENTS ............................................................................................ iii LIST OF FIGURES ....................................................................................................... vi CHAPTER 1. INTRODUCTION .................................................................................... 1 CHAPTER 2. PLASMA INTERACTION WITH A STATIC SPATIALLY PERIODIC ELECTROMAGNETIC FIELD ...................................................................................... 4

2.1. Introduction ............................................................................................... 4 2.2. Experiment: A Proof of Concept ................................................................ 6 2.3. Results ....................................................................................................... 9

CHAPTER 3. ARTIFICIALLY STRUCTURED BOUNDARY FOR A HIGH PURITY ION TRAP OR ION SOURCE ...................................................................................... 11

3.1. Introduction .............................................................................................. 11 3.2. Theory ...................................................................................................... 12 3.3. Results ...................................................................................................... 17 3.4. Conclusion ................................................................................................. 22

CHAPTER 4. SPACE-CHARGE-BASED ELECTROSTATIC PLASMA CONFINEMENT INVOLVING RELAXED PLASMA SPECIES ................................. 23

4.1. Introduction .............................................................................................. 23 4.2. Single-Species Non-Neutral Plasma ........................................................... 25 4.3. Two-Species Plasma .................................................................................. 31 4.4. Space-Charge-Based Electrostatic Confinement Conditions ...................... 35 4.5. Conclusion ................................................................................................. 38

CHAPTER 5. ELECTROSTATIC STORAGE RING WITH FOCUSING PROVIDED BY THE SPACE CHARGE OF AN ELECTRON PLASMA ........................................ 42

5.1. Introduction .............................................................................................. 42 5.2. Theory ...................................................................................................... 43 5.3. Results ...................................................................................................... 46 5.4. Space-Charge-Based Electrostatic Focusing .............................................. 51 5.5. Conclusion ................................................................................................. 53

v

CHAPTER 6. ELECTRON BEAM TRANSMISSION THROUGH A CYLINDRICALLY SYMMETRIC ARTIFICIALLY STRUCTURED BOUNDARY .... 55

6.1. Introduction .............................................................................................. 55 6.2. Apparatus ................................................................................................. 55 6.3. Electron Beam .......................................................................................... 56 6.4. Experimental Artificially Structured Boundary ........................................ 58 6.5. UHV Conditions During Experimentation ................................................ 61 6.6. Electron Detection System ........................................................................ 62 6.7. Accepance and Transmission Without Electrostatic Plugging .................. 63 6.8. Summary and Conclusion ......................................................................... 69

CHAPTER 7. CONCLUSION ........................................................................................ 70 APPENDIX A. NORMALIZATION OF MAXWELLIAN DISTRIBUTION ................ 72 APPENDIX B. PRODUCT LOGARITHM ................................................................... 75 APPENDIX C. TRAPPING FIELDS, PARTICLE BEHAVIOR, AND PLASMA BEHAVIOR ................................................................................................................... 82 APPENDIX D. CHARGED PARTICLE TRAPPING .................................................. 88 BIBLIOGRAPHY ........................................................................................................... 97

LIST OF FIGURES

2.1 Nested Penning trap. Rectangular segments are the locations of posi-

tively (red) and negatively (blue) biased electrodes. The center electrode

is typically grounded. A positively charged particle (red oval region) is

confined between the positively biased electrodes. A negatively charged

particle (blue oval region) is confined between the outer-most set of elec-

trodes. The magnetic field keeps particles with either sign of charge

radially confined.......................................................................................... 5

2.2 A section of a planar ASB: Four permanent magnets with electrostatic

plugging applied using copper electrodes (left). Corresponding simula-

tion of the magnetic field lines for magnets with a maximum magnetic

field magnitude Bmax = 1 T, and like poles facing each other (right). The

fields of interest lie in the two quadrants on the right. A typical magnetic

field cusp is present near the center of the figure, at the intersection of

the axes. ...................................................................................................... 6

2.3 Conceptual experimental setup as used to observe plasma interaction

with an ASB. See text for description of experiment.................................. 7

2.4 Left: Argon ions incident on magnet structure with electrostatic plugging

turned off (electrodes, magnets, and supporting structure are at ground

potential). Right: Electrostatic plugging turned on (reflection electrodes

at 30 V). Positively charged particles enter magnetic cusps in both left

and right panels. In the right panel, particles that enter a cusp experience

an E ×B drift that guides them into, or out of, the plane of the page,

thereby extending the plasma perpendicular to the plane of the page.

The brightness is enhanced where the E ×B drifts occur. ...................... 8

vi

2.5 Electrons incident on magnet structure. Left: Electrostatic plugging

turned off. Right: Electrostatic plugging turned on (−200 V). Right:

The E × B drift caused the plasma to reach and pass in front of the

ends of the magnets closest to the camera. ................................................. 9

3.1 Simulation environment representing two periods of a planar ASB. Ions

are confined to the region below the ASB (yn < 0). The lower edge of

the ASB is located at yn = 0. The dots mark the positions of the current

carrying wires, with current that alternates in sign from one column

of wires to the next, ±I. Magnetic field cusps are produced with the

direction of the magnetic field labeled by βt. The electrodes are marked

by lines, which represent their lengths and locations in the simulation

environment. The current carrying wires and the electrodes are infinite

in extent in the z dimension. The electrostatic potential energy barrier

is located in the region 0.5 ≤ yn ≤ 0.75, at the location of V1. V0 and V2

are at ground potential. φ0 is the electric potential at the center of the

anode gap, where the magnetic field has a magnitude B0. See Eq. (2)

for details regarding ηi, and ∆yn. ............................................................... 14

3.2 Simulation that represents a two period segment of an ASB. The differ-

ent shades show trajectories with φn0 = 1 and δ = 1000 (black), δ =

100 (dark gray), and δ = 20 (light gray). The trajectory calculation is

terminated when a particle reaches yn = 0.75............................................. 18

3.3 Simulation that represents a two period segment of an ASB. The different

shades show trajectories with δ = 20, and φn0 = 0.5 (light gray) and

φn0 = 5 (black). The trajectory calculation is terminated when a particle

reaches yn = 0.75......................................................................................... 19

vii

3.4 Profile of the spatial distribution of charged particles that reached yn ≥0.75 after entering a cusp and overcoming the electrostatic potential bar-

rier. The distribution of particles at yn ≈ 0.75 is for φn0 = 1 and δ =

10, 20, and 40. The data series are labeled according to the parame-

ter varied, and the corresponding percentages of particles that reached

yn ≈ 0.75 are indicated. The total number of trajectories simulated for

each of these plots was 100,000. .................................................................. 20

3.5 Profile of the spatial distribution of charged particles that reached yn ≥0.75 after entering a cusp and overcoming the electrostatic potential bar-

rier. The distribution is for δ = 20 and φn0 = 0.5, 1, and 2. The data

series are labeled according to the parameter varied, and the correspond-

ing percentages of particles that reached yn ≈ 0.75 are indicated. The

total number of trajectories simulated for each of these plots was 100,000. 21

4.1 Conceptual model of a plasma trapping volume with a field free region

at its center. A plasma is envisioned to relax within the volume and be

“edge-confined” by a reflecting surface such as an ASB. ............................ 24

4.2 Typical radial profile of the normalized electrostatic potential. The plots

are for rn,max = 100 and α = 1 (solid), 2 (long dash), 3 (short dash).

The normalized electrostatic potential difference between the center and

the boundary is 7.6 for α = 1, 7.2 for α = 2, and 6.8 for α = 3.................. 29

4.3 Typical normalized density profiles. The plots are for α = 3 and rn,max =

5 (dot-dashed), 10 (short dash), 20 (long dash), 30 (solid). Similar pro-

files occur for other values of α. ................................................................ 29

4.4 Normalized electrostatic potential difference (between plasma center and

edge) for the three geometries. The solid lines are Eq. (26). ...................... 30

viii

4.5 Normalized electrostatic potential of a two-species plasma (top). Self-

consistent distributions of the two plasma species (bottom) in logarithmic

scale. The plots are for, Tn = 5, rn,max = 30, and Nn = 0.004 (dashed),

0.04 (dot-dashed), 0.4 (solid). The arrows indicate the trend that the

system follows as Nn is increased. In the lower panel of this figure and

Figs. 4.6-4.8, the normalized distributions are n−(rn)/n0− , which are la-

beled by minus signs (–), and Zn+(rn)/n0− , which are labeled by plus

signs (+). Thus, each matching pair of plots are the normalized distri-

butions for the negative and positive plasma species. ................................. 33

4.6 Normalized electrostatic potential of a two-species plasma (top). Self-

consistent distributions of the two plasma species (bottom). The plots

are for Nn = 0.02, rn,max = 30, and Tn = 1 (solid), 15 (dot-dashed), and

30 (dashed). The ± labels are defined in Fig. 4.5....................................... 34

4.7 Two plasma species with equal temperatures and charge states. The

plots are for Tn = 1, rn,max = 30, and Nn = 0.1 (dot-dashed), 0.01

(dashed), and 0.001 (solid). The ± labels are defined in Fig. 4.5. .............. 36

4.8 Two plasma species with approximately equal charge densities at the

center of the plasma system. The plots are for rn,max = 30 and (Tn, Nn) =

(1, 0.05)[solid], (10, 0.0225)[dot-dashed], (25, 0.0152)[long dash], and (40,

0.0145)[short dash]. The normalized electron temperatures, Tn, were

chosen and the normalized positive plasma charge densities, Nn, were

then adjusted to the lowest value at which the two distributions have

approximately the same value at the center of the system. The ± labels

are defined in Fig. 4.5. ................................................................................ 37

ix

4.9 Normalized electrostatic potential energy well depth for space-charge-

based electrostatic plasma confinement as a function of normalized sys-

tem size. The dotted line in the top panel is for Nn = 0 and Tn = 1.

Top (bottom) panel is for Tn = 1 (Tn = 10), and Nn = 0.1 (solid), 0.01

(long dash), 0.001 (dash), 0.0001 (dot-dashed). .......................................... 39

4.10 Normalized electrostatic potential difference for increasing normalized

charge density of the positive species. There are plot points for rn,max =

300 and Tn = 1, 4, and 10, for each value of Nn, but the plot points are

indistinguishable. The solid line is Eq. (34)................................................ 40

5.1 Cross-sections of a segment of a cylindrical beam line. The electron

plasma is confined by an artificially structured boundary. The space

charge of the electron plasma creates an electrostatic potential that fo-

cuses a positive-ion beam or drifting plasma. ............................................. 44

5.2 Normalized electrostatic potential of a two-species system (top). Self-

consistent distributions of the two species (bottom) in logarithmic scale.

The plots are for, Tn = 5, rn,max = 30, and Nn = 0.004 (dashed), 0.04

(dot-dashed), 0.4 (solid). The arrows indicate the trend that the system

follows as Nn is increased. In the lower panel of this figure and Figs. 5.3-

5.5, the normalized distributions are n−(rn)/n0− , which are labeled by

minus signs (–), and Zn+(rn)/n0− , which are labeled by plus signs (+).

Thus, each matching pair of plots are the normalized distributions for

the negative and positive plasma species. ................................................... 47

5.3 Normalized electrostatic potential of a two-species plasma (top). Self-

consistent distributions of the two plasma species (bottom). The plots

are for Nn = 0.02, rn,max = 30, and Tn = 1 (solid), 15 (dot-dashed), and

30 (dashed). The ± labels are defined in Fig. 5.2....................................... 48

x

5.4 Two plasma species with equal temperatures and charge states. The

plots are for Tn = 1, rn,max = 30, and Nn = 0.1 (dot-dashed), 0.01

(dashed), and 0.001 (solid). The ± labels are defined in Fig. 5.2. .............. 49

5.5 Two plasma species with approximately equal charge densities at the cen-

ter of the plasma system. The plots are for rn,max = 30 and (Tn, Nn) =

(1, 0.05)[solid], (10, 0.0225)[dot-dashed], (25, 0.0152)[long dash], and

(40, 0.0145)[short dash]. The normalized electron temperatures Tn were

chosen and the normalized positive plasma charge densities Nn were then

adjusted to the lowest value at which the two distributions have approx-

imately the same value at the center of the system. The ± labels are

defined in Fig. 5.2. ...................................................................................... 50

5.6 Normalized electrostatic potential energy well depth for space-charge-

based electrostatic focusing as a function of normalized system size. The

plots are for Tn = 1, and Nn = 0 (dotted), 0.0001 (dot-dashed), 0.001

(dash), 0.01 (long dash), and 0.1 (solid). .................................................... 51

5.7 Normalized electrostatic potential difference for increasing normalized

charge density of the positive species. Points are plotted for rn,max =

300 and Tn = 1, 4, and 10, for each value of Nn, but these plot are

indistinguishable for the different values of Tn. The solid line drawn

through the points is a fit given by Eq. (47)............................................... 52

6.1 Schematic view of experimental apparatus. ................................................ 56

xi

6.2 Relative number of particles incident at the entrance of the trap as a

function of the magnitude of einzel lens focusing voltage. The einzel

lens was biased to provide focusing in decelerating mode. 5,000 electron

trajectories were simulated per data point marked by a cross. The maxi-

mum number of electron trajectories that collapsed onto the Faraday cup

electrode was 3472, occurring at an einzel bias voltage of −24 V. Data

points marked by dots are the electron currents observed at the Faraday

cup in the experimental setup, normalized to the maximum current of

−63.5 nA observed for an einzel lens bias of −24 V. ................................... 58

6.3 A length-wise cross-sectional view of the cylindrically symmetric ASB

and the fields produced within its interior. The rectangular features on

the top and bottom figures are the magnets and electrodes that create

the ASB. The lines in the top figure show contours of equal electric

potential. The lines on the bottom figure show the magnetic field. See

text for further details................................................................................. 59

6.4 Photographs of the experimental system. Panel A shows the alternating

sequence of copper ring electrodes and permanent ring magnets. Panel B

shows the trapping volume as viewed upstream from the exit side. Panel

C shows the phosphor screen that, along with the micro-channel plates

(not shown), constitute the electron detection system................................ 60

6.5 Phosphor screen as imaged by SBIG ST-7XMEI SBIG CCD camera

(left panel). An electron beam exiting the trap and incident on the

MCP/Phosphor assembly creates the time integrated fluorescence recorded

by the CCD camera (right panel). For reference, the phosphor screen

(major circular feature on left panel) is 1.9 cm in diameter (or≈ 500 pixels;

1 pixel unit (pu) = 38µm). The same scale applies to right panel............. 62

xii

6.6 Electron beam acceptance into the trap as a function of einzel lens volt-

age. Data points marked by dots are the normalized charge collected

on the unbiased plugging electrodes. Data points marked by crosses

are the time and space integrated relative beam intensities obtained by

processing the images recorded with the CCD camera. See text for details. 64

6.7 Spatial electron beam profile distribution as a function of focusing at the

entrance to the trap. The three dimensional shape that protrudes from

the x-y plane in the z direction is a plot of intensity I(in arbitrary units

(au)) as a function of position. The bands represent equal fractional

intervals of the peak intensity in each of the panels. A contour plot is

also shown at the top of each figure to illustrate the 2D beam profile. The

data processed for the plots shown are the pixel values that represent the

images of the beam as captured from the phosphor screen by the CCD

camera. An example of such an image is shown in the right panel of

Fig. 6.5. The x and y coordinates are in pixel size units (pu). ................... 65

xiii

CHAPTER 1

INTRODUCTION

The current project emerged from a quest for alternative ways to simultaneously

confine and control charged particles of either sign of charge. Confinement of both signs

of charge in overlapping volumes presents an ideal environment for experimentation with

non-neutral plasmas, partially neutralized plasmas, charged particles, and charged-particle

beams. Applications of such a system include the confinement of a two-species plasma for

recombination studies, lining of plasma facing components to minimize unwanted erosion,

guiding of neutral and partially neutralized ion beams, confinement of non-neutral or par-

tially neutralized non-drifting plasmas, ion accumulators, high purity ion sources, and for

experiments in atomic and molecular physics.

In the trapping system presented here, the reflection of charged particle trajectories

occurs near the confining boundary, where the confining fields have a relatively high strength.

Away from the boundary, an essentially field free region exists, where confined particles are

expected to reside. A field-free confinement region is highly desired as a prospecting tool

for experiments with particle trapping, particle-particle interaction, particle-external field

interaction, and self-consistent relaxation of plasmas. It is envisioned that the concept

described herein could be employed in conjunction with, or as an alternative to, systems

that currently exist for charged particle, or plasma, confinement and control.

In Chapter 2, initial research on an electro- magneto-static field for reflection and

confinement of charged particles is presented. The structure consists of an artificially struc-

tured boundary (ASB) with electrostatic plugging applied. An ASB generates a spatially

periodic sequence of magnetic field cusps. Electrostatic plugging occurs with applied poten-

tial variations along the magnetic field cusps that are similar to those applied to one side of

a nested Penning trap. A field thus created allows for the simultaneous reflection of charged

particles of either sign of charge that are incident randomly. Experimental results that show

the behavior of an argon plasma and an electron plasma are reported.

1

In Chapter 3, a plasma enclosed by an artificially structured boundary (ASB) is

proposed as an alternative to existing ion source assemblies. In accelerator applications,

many ion sources can have a limited lifetime or frequent service intervals due to sputtering

and eventual degradation of the ion source assembly. Ions are accelerated towards the exit

canal of positive ion sources, whereas, due to the biasing scheme, electrons or negative ions are

accelerated towards the back of the ion source assembly. This can either adversely affect the

experiment in progress due to sputtered contamination or compromise the integrity of the ion

source assembly. Charged particles in the proximity of an ASB experience electromagnetic

fields that are designed to hinder ion-surface interactions. Away from the ASB there is an

essentially field free region. The field produced by an ASB is considered to consist of a

periodic sequence of electrostatically plugged magnetic field cusps. A classical trajectory

Monte Carlo simulation is extended to include electrostatic plugging of magnetic field. The

conditions necessary for charged particles to be reflected by the ASB are presented and

quantified in terms of normalized parameters.

In Chapter 4, a volume that has its interior surface lined with an ASB, that is plugged

electrostatically, provides a field free region at the center of the configuration where a plasma

can self-consistently relax. A numerical study is reported on the equilibrium properties of

a surface-emitted or edge-confined non-drifting plasma. A self-consistent finite-differences

evaluation of the electrostatic potential is carried out for a non-neutral plasma, which fol-

lows a Boltzmann density distribution. The non-neutral plasma generates an electrostatic

potential that has an extremum at the geometric center. Poisson’s equation is solved for

different ratios of the non-neutral plasma size to the edge Debye length. The profiles of

the electrostatic potential and the plasma density are presented for different values of that

ratio. A second plasma species is then introduced for two-plasma-species confinement stud-

ies, with one species confined by the space charge of the other, while each species follows a

Boltzmann density distribution. An equilibrium is found in which a neutral region forms.

An equilibrium is also found in which the two species have equal temperatures and charge

states.

2

Space-charge-based focusing in electrostatic storage rings is presented in Chaper 5.

Electrostatic storage rings are used for a variety of atomic physics studies. An advantage

of electrostatic storage rings is that heavy ions can be confined. An electrostatic storage

ring that employs the space charge of an electron plasma for focusing is described. An

additional advantage of the present concept is that slow ions, or even a stationary ion plasma,

can be confined. The concept employs an artificially structured boundary that produces a

spatially periodic static field such that the spatial period and range of the field are much

smaller than the dimensions of a plasma or charged-particle beam that is confined by the

field. An artificially structured boundary is used to confine a non-neutral electron plasma

along the storage ring. The electron plasma would be effectively unmagnetized, except

near an outer boundary where the confining electromagnetic field would reside. The electron

plasma produces a radially inward electric field, which focuses the ion beam. Self-consistently

computed radial beam profiles are reported.

Experimental research on charged particle transmission through an electro- magneto-

static field configuration created by a cylindrically symmetric artificially structured boundary

(ASB) is presented. The ASB produces a periodic set of magnetic field cusps that are plugged

electrostatically. In the system presented, the reflection or modification of charged particle

trajectories occurs near the material wall boundary, where the confining fields have a rela-

tively high strength. Away from the boundary, an essentially field free region exists, where

confined particles are expected to reside. Such a system is expected to have applications as

a charged particle or plasma trap and as a beam guide. An overview of the experimental

system is given. Results that pertain to electron beam transmission through the system are

presented in Chapter 6 .

3

CHAPTER 2

PLASMA INTERACTION WITH A STATIC SPATIALLY PERIODIC

ELECTROMAGNETIC FIELD

2.1. Introduction

A concept referred to as an artificially structured boundary (ASB) has been predicted

to reflect charged particles of either sign of charge at grazing angles of incidence [1, 2, 3]. One

type of ASB produces a periodic sequence of field cusps [1, 2]. Such an ASB can be created

by an infinite array of current carrying wires with neighbouring wires carrying currents in

opposite directions, or an array of permanent magnets with like poles facing each other, to

create the cusping magnetic fields. By electrostatically plugging the magnetic field cusps, it is

also possible to reflect charged particles that are incident normally. A comprehensive review

of research related to electrostatic plugging of magnetic cusps is found in [4]. Electrostatic

plugging nominally consists of applying an electric field that stops charged particles from

passing through a magnetic cusp.

A variation of the Penning trap, the nested Penning trap, is designed to confine op-

positely charged particles by applying an electrostatic potential variation along a magnetic

field [5]. The nested Penning trap, see Fig. 2.1, has been successfully employed for antihy-

drogen production by the ATHENA [6] and ATRAP [7] collaborations and for antihydrogen

trapping by the ALPHA collaboration [8]. In order to achieve recombination of positrons

and antiprotons to produce and trap antihydrogen atoms in substantial numbers, many con-

flicting issues arise [5]. It is envisioned that the concept described herein could be employed

in conjunction with, or as an alternative to, the trapping environments already in place for

antihydrogen production and trapping.

The electro- magneto-static field considered here consists of a sequence of electrostat-

ically plugged magnetic cusps. The field envisioned here inherits desirable characteristics

produced by the ASB: (1) The field is short in range in comparison to the size of a nearby

source of charged particles; and (2) the field can reflect charged particles of either sign of

4

Nested Penning Trap.

B

B

V- V-V+ V+

Figure 2.1. Nested Penning trap. Rectangular segments are the locations of

positively (red) and negatively (blue) biased electrodes. The center electrode is

typically grounded. A positively charged particle (red oval region) is confined

between the positively biased electrodes. A negatively charged particle (blue

oval region) is confined between the outer-most set of electrodes. The magnetic

field keeps particles with either sign of charge radially confined.

charge. Electrostatically plugging magnetic cusps using the same applied potential variations

that are found in nested Penning traps brings forth the possibility of confining particles of

either signs of charge that are incident from random directions. As a consequence, a volume

with the inner surface lined by the electro- magneto-static field proposed here could be em-

ployed to trap oppositely signed charged particles. It is interesting to note that the interior

of such a trapping volume achieved in such a manner would be essentially field free, with

particle trajectories being affected only in close proximity to the boundary. Minimum-B

configurations can also be envisioned.

The possible applications of such a field configuration are numerous. This field could

be used to line plasma-facing components that would otherwise suffer (unwanted) erosion due

5

Figure 2.2. A section of a planar ASB: Four permanent magnets with elec-

trostatic plugging applied using copper electrodes (left). Corresponding simu-

lation of the magnetic field lines for magnets with a maximum magnetic field

magnitude Bmax = 1 T, and like poles facing each other (right). The fields of

interest lie in the two quadrants on the right. A typical magnetic field cusp is

present near the center of the figure, at the intersection of the axes.

to interaction with a plasma. Neutralized and partially neutralized charged particle beams

could be transported and guided with the fields of an ASB that is plugged electrostatically.

Confinement of quasi-neutral or partially neutralized nondrifting plasmas may be possible

with such a field. In Sec. 2.2 the initial experiment to observe plasma interaction with an

electro- magneto-static field is presented. Results are presented in Sec. 2.3.

2.2. Experiment: A Proof of Concept

A planar segment of the proposed electro- magneto-static field structure has been

produced experimentally and is shown in Fig. 2.2. Four neodymium-iron-boron permanent

magnets (dimensions: 5.08 cm×5.08 cm×0.635 cm and having a maximum field of 1 T) were

clamped with like poles facing each other and with a separation of 0.64 cm between them. To

achieve electrostatic plugging, copper electrodes were attached to, and electrically isolated

from, the faces of the magnets. When biased, these electrodes set up electrostatic potential

barriers to repel plasma particles that enter the magnetic field cusps.

6

Magnet assembly

Discharge region

Extraction and Focusing

CCD

Figure 2.3. Conceptual experimental setup as used to observe plasma inter-

action with an ASB. See text for description of experiment.

To perform the initial experimental testing, the magnet structure was placed in a

vacuum chamber that was evacuated to ≈ 1 mTorr and back-filled with Argon gas to 0.2 Torr

and throttled until the desired plasma was observed. See Fig. 2.3: An Argon plasma was

generated using a DC glow discharge plasma source. The plasma source consisted of a

straight tungsten wire encircled by a tungsten wire loop (wire thickness = 0.25 mm), with

a potential difference applied between these two wires to create a discharge. The plasma

was ignited by applying a potential difference of 250 V to 350 V between these two tungsten

wires; the potential difference necessary for plasma ignition varied depending on pressure

inside the chamber.

To obtain charged particles from the discharge region, an einzel-lens-like configura-

tion of three electrodes was employed, with grounded first and last electrodes and a biased

middle electrode. These extraction electrodes produced an electric field that penetrated the

discharge region and extracted a species of a particular sign of charge. Several conditions

(background pressure, discharge voltage, source bias with respect to ground, extraction elec-

trode voltage) were optimized empirically to obtain a visible plasma. It was found that

for the system described, a background pressure of 120 mTorr and a discharge voltage of ≈

7

Figure 2.4. Left: Argon ions incident on magnet structure with electro-

static plugging turned off (electrodes, magnets, and supporting structure are

at ground potential). Right: Electrostatic plugging turned on (reflection elec-

trodes at 30 V). Positively charged particles enter magnetic cusps in both left

and right panels. In the right panel, particles that enter a cusp experience an

E ×B drift that guides them into, or out of, the plane of the page, thereby

extending the plasma perpendicular to the plane of the page. The brightness

is enhanced where the E ×B drifts occur.

200 V were sufficient to sustain a plasma that could be imaged. Extraction was achieved

with the additional condition that the tungsten wires were biased to ≈ 80 V with respect to

the established ground, and the extraction electrode voltage was held at −660 V. Extraction

of electrons from the discharge region was achieved by literally reversing all the voltages

but increasing the discharge voltage to ≈ 300 V to keep the discharge stable. Ar ions (or

electrons) were extracted from the discharge region and directed onto the magnet assembly.

These Ar ions (or electrons) diffused through a region where the background pressure was

8

Figure 2.5. Electrons incident on magnet structure. Left: Electrostatic plug-

ging turned off. Right: Electrostatic plugging turned on (−200 V). Right: The

E ×B drift caused the plasma to reach and pass in front of the ends of the

magnets closest to the camera.

relatively high (120 mTorr), with a mean free path of ≈ 0.05 cm. The Ar ions (or electrons)

collisionally excited residual gas atoms, which primarily consisted of Ar atoms.

An ST-7XMEI SBIG CCD camera was employed to record light emitted by de-

excitations, using typical integration times of 2 to 3 minutes. The plasma was imaged with

electrostatic plugging either turned off or turned on. Images that represent the behavior

observed are shown in in Fig. 2.4 for Ar ion extraction and Fig. 2.5 for electron extraction

from the discharge region.

2.3. Results

For the conditions of the experiment presented here, the charged particle trajectories

are observed to follow magnetic field lines and to be confined to regions of low magnetic

field strength. This behavior is observed by inspecting Figs. 2.2, 2.4, and 2.5. Charged

9

particles that are incident on the magnet assembly near the middle of the magnet edges,

away from their corners, and have trajectories that are nearly parallel to the planes of the

magnets enter the regions of cusping magnetic fields. Experimental results associated with

electrostatic plugging of the magnetic field cusps are shown in the right panels of Fig. 2.4

and Fig. 2.5. Electrostatic plugging of the magnetic field cusps further modifies charged

particle trajectories and is observed to cause an E ×B drift that guides charged particles

into or out of the plane of the page.

10

CHAPTER 3

ARTIFICIALLY STRUCTURED BOUNDARY FOR A HIGH PURITY ION TRAP OR

ION SOURCE

3.1. Introduction

In the application of ion sources for accelerator physics, plasma physics, or plasma

processing purposes, a clean source of ions is desirable when unwanted sputtered contam-

ination is detrimental for the experiment at hand or for the ion source assembly itself. In

addition, in the accumulation of rare species of ions or antimatter particles, good confine-

ment is particularly important due to the limited availability of the particles being trapped.

Ions within an ion source that impinge on a surrounding material surface can alter the phys-

ical properties of the surface. Furthermore, ion sources that employ reactive metals can

require frequent service intervals. The study presented here proposes a configuration that

can minimize the interaction of a plasma with material surfaces.

An artificially structured boundary (ASB) is described here as a material boundary

that produces electrostatic and magnetostatic fields for the purpose of modifying charged

particle trajectories when charged particles approach the boundary. Such an ASB is consid-

ered here to form a periodic set of cusping magnetic fields with electrostatic potential barriers

at the location of the magnetic field cusps. An ASB that produces purely magnetic fields,

without the electrostatic barriers, is described in [2]. Two properties of such an arrangement

are notable: (1) The ASB is capable of simultaneously reflecting charged particles of either

sign of charge, but only when the particles are incident at shallow angles. (2) A nearly

field free region occurs away from the ASB so that the field only modifies charged particle

trajectories close to the material boundary. Charged particle trajectories that are normal

to the ASB can escape through magnetic field cusps if no electrostatic plugging is present.

Preliminary experimental research has been reported previously in which a plasma interacts

with a segment of an ASB with electrostatic plugging [9]. Also, theoretical research has been

reported on possible applications of an ASB for lining an electrostatic storage ring [10] and

11

for bounding a confined plasma [11]. The current study assesses the effect that incorporating

electrostatic plugging of the magnetic field cusps can have on the confinement of charged

particles of a single sign of charge. The configuration may serve to confine a two-species

plasma, with the first species confined by the ASB and the second, oppositely signed species,

confined by the space charge of the first species [11].

In Sec. 3.2, the fields employed for confinement are described. A normalization scheme

is developed and normalized equations of motion are derived. The method of solution is also

presented in Sec. 3.2. Results are presented in Sec. ??. Concluding remarks are presented

in Sec. 3.4.

3.2. Theory

The current study considers the interaction of a single charged particle with an ASB.

The effects due to the collective nature of plasmas are not taken into account here. Charged

particle trajectories near an ASB are determined by solving Newton’s second law. Figure 3.1

depicts the characteristics of the simulation environment.

The magnetic field developed in [12] is used here, except that (1) the magnetic field

dependence on the coordinates is changed, and (2) the strength of the field is defined by the

conditions necessary for magnetic confinement. Such a field has the form [12]

(1) B(x, y, z) = B0βt

(xS,y

S,z

S

),

with

(2) βt(xn, yn, zn) =N∑i=1

ηiβ(xn, yn − (i− 1)∆yn, 0),

and

(3) β(xn, yn, zn) =− cos(2πxn) sinh(2πyn)

cos(4πxn)− cosh(4πyn)x+

sin(2πxn) cosh(2πyn)

cos(4πxn)− cosh(4πyn)y.

Here rn = xnx + yny + znz = rS

; β(xn, yn, zn) describes the direction of the magnetic field

created by a planar array of current carrying wires that has a spatial period S, that is infinite

in z dimension and coincides with the y = 0 plane; and ηi assigns relative current factors for

each of the N(=10) planar arrays that are stacked ∆yn apart. η1 = η10 = 1.27 and ηi = 1 for

12

2 ≤ i ≤ 9. B0 is approximately equal to the magnitude of the magnetic field at the center

of the anode gap. An expression for B0 will be developed in a later section.

The electric field used for electrostatic plugging of the magnetic field cusps is obtained

by numerically computing the electrostatic potential φ(x, y, z) and then using E(x, y, z) =

−∇φ(x, y, z). The numerical computation of φ is described below.

Normalization

Consider a collisionless, non-drifting, unmagnetized plasma that follows a Maxwellian

velocity distribution. From this point onward, an ensemble of charged particles is loosely

referred to as a plasma. T is the temperature, in units of energy, associated with the

Maxwellian distribution and m is the mass of a plasma particle. Assume that the plasma is

composed of a single species of charged particles, each of which has a positive charge q (e.g.,

q = 2e for a doubly ionized positive species, and e is the electronic charge). In what follows,

the quantities m, q, S, and 3T/2 are employed to carry out a normalization procedure.

S is the spatial period of the magnetic field, and 3T/2 is the average kinetic energy per

particle in a Maxwellian source of particles. The normalized parameters are tn = tS

√3T2m

,

rn = rS

, vn = v√

2m3T

, an = a2mS3T

, Bn = BSq√

23mT

, En = E 2Sq3T

, and φn = 2qφ3T

, which are

the dimensionless normalized time, position, velocity, acceleration, magnetic field, electric

field, and electric potential, respectively. Newton’s second law for a charged particle that

experiences a Lorentz force is

(4) an = En + (vn ×Bn),

when written in terms of the normalized parameters. The normalized electric field is

(5) En = −∇φn(xn, yn, zn).

Consider a positive charged particle in a region with electrostatic potential φ, which is

positive or zero everywhere accessible to the particle. In particular, consider the electrostatic

potential present in the ASB described in Fig. 3.1, where φ = φ0 at the center of the anode

gap between electrodes labeled by V1. The electrostatic potential energy barrier, U0 = qφ0,

13

Electrodes

V0

V2

V1

S

Δyn

Current-carrying

wires

α

vx

vo

vy

+I +I -I -I

φ0 B0

βt

η1

η2

ηi

Figure 3.1. Simulation environment representing two periods of a planar

ASB. Ions are confined to the region below the ASB (yn < 0). The lower

edge of the ASB is located at yn = 0. The dots mark the positions of the

current carrying wires, with current that alternates in sign from one column

of wires to the next, ±I. Magnetic field cusps are produced with the direc-

tion of the magnetic field labeled by βt. The electrodes are marked by lines,

which represent their lengths and locations in the simulation environment.

The current carrying wires and the electrodes are infinite in extent in the z

dimension. The electrostatic potential energy barrier is located in the region

0.5 ≤ yn ≤ 0.75, at the location of V1. V0 and V2 are at ground potential. φ0 is

the electric potential at the center of the anode gap, where the magnetic field

has a magnitude B0. See Eq. (2) for details regarding ηi, and ∆yn.

14

reflects charged particles that start at zero potential with less kinetic energy than is required

to overcome the potential energy barrier. Define the ratio of the electrostatic potential energy

barrier, at the location of φ0, to the average kinetic energy of a plasma particle to be the

normalized potential barrier,

(6) φn0 =2qφ0

3T.

The Larmor radius, RL, is used to specify a condition for magnetic confinement.

At the center of the anode gap, the magnetic field has a magnitude specified by B0, so

that RL = mvpqB0

. Here vp is the magnitude of the velocity component perpendicular to the

direction of the magnetic field at the location of B0. In order for a charged particle to

experience magnetic confinement in the anode gap, its Larmor radius must be much smaller

than the space between two adjacent columns of wires, i.e. RL S4. Let the average thermal

energy be available to a plasma particle’s motion perpendicular to the magnetic field. In

such case 32T = m

2v2

p, which leads to

(7) B0 =

√3mT

qRL

.

With the magnetic field given by Eq. (1), and defining an inverse normalized Larmor radius

δ = SRL

, the normalized magnetic field becomes

(8) Bn =√

2δβt(xn, yn, zn),

where δ 4 is considered necessary for magnetic confinement.

Equations of Motion

The equations of motion are obtained from Eqs. (4), (5), and (8). For the planar

system considered in the current study, the fields are completely independent of the z-

coordinate. Therefore, the equations of motion become

(9) x′′n(tn) = Enx −√

2δ (z′n(tn)βty) ,

(10) y′′n(tn) = Eny +√

2δ (z′n(tn)βtx) ,

15

and

(11) z′′n(tn) =√

2δ (x′n(tn)βty − y′n(tn)βtx) .

Here, βtx, Enx and βty, Eny are the x and y components of βt and En, respectively, and the

notation x′n(tn) is the derivative of the normalized position with respect to normalized time.

The equations of motion (Eqs. (9)-(11)) were solved simultaneously to obtain parametric

trajectories in three dimensions.

Method of Solution

The initial conditions for the simulation of trajectories were obtained in the following

manner. Assume that the plasma has a temperature T associated with a Maxwellian velocity

distribution. Taking the present normalization into account, the initial components of the

velocity vector are obtained via

(12) vn0,i =

√2

3(−2 ln[R1i])

12 cos(2πR2i)

with i = x, y, or z. Equation (12) represents random components of the initial velocity

vector sampled from a Maxwellian distribution [13, 14], where Rji are all independent ran-

dom numbers with a uniform distribution between zero and one. The initial conditions

are obtained from the velocity vector vn0 = vn0,xx + vn0,yy + vn0,zz and position vector

rn0 = R[−1, 1]x− 3y + 0z, where x, y, and z are Cartesian unit vectors and R[x1, x2] is a

random number with a uniform distribution between x1 and x2. Note that the motion in the

y dimension takes charged particles towards [away from] the ASB when the velocity compo-

nent in the y dimension is positive [negative]; see Fig. 3.1. Consequently, the initial velocity

component in the y direction is calculated as prescribed by Eq. (12), and its absolute value

is used so that all charged particles initially travel toward the ASB. The initial x coordinate

is sampled over two full periods of the simulation environment, which directly corresponds

to the simulation region presented in Fig. 3.1. Additionally, two periods were chosen for the

simulation region in order to allow for a large sampling of the phase-space but not so large

that trajectories with glancing angles dominate the statistics obtained.

16

Taking advantage of the periodicity of the system, the electrostatic potential was

computed for a region that is 0.5S wide in the x dimension and 5.0S long in the y dimension

and then the solution was repeated in the x dimension to complete the simulation region.

The top of the simulation boundary corresponds to yn = 1 and the bottom to yn = −4.

The electrode labeled V0 starts at yn = 0 and ends at yn = 0.5, the electrode labeled V1

starts at yn = 0.5 and ends at yn = 0.75, and the electrode labeled V2 starts at yn = 0.75

and ends at yn = 1.0, with one grid unit between adjacent electrodes. The electrostatic

potential was computed using a finite differences sequential over-relaxation method [15].

In the calculation of the electrostatic potential, there are 40 grid units per period-lengths.

Values were specified for applied normalized potentials Vn0, Vn1, and Vn2 to establish the

boundary conditions at the electrode locations. Vn0 and Vn2 were set to ground potential,

Vn0 = Vn2 = 0, whereas Vn1 was biased to a positive value that was iteratively increased

until a chosen value for φn0 was reached at the center of the anode gap. The electrostatic

potential was obtained by first assigning values to the boundary regions where electrodes

are located, then applying the finite difference sequential over-relaxation algorithm to the

internal points, and assigning values to the remaining boundary points by requiring that the

derivative normal to the boundary be zero. Such a procedure was carried out a sufficient

number of times so that the difference in calculated normalized potential values from one

iteration to the next was less than 1× 10−5 for all internal points.

The equations of motion were solved simultaneously via a “leap-frog” numerical ap-

proach (see, for example [16]). The time-step size was adjusted until the energy throughout

the simulation was conserved to within 1% of the initial energy for all trajectories, during

code development on a desktop computer. The same code was submitted for batch process-

ing on a supercomputer. Some of the trajectories obtained from the supercomputer were

also chosen and checked for energy conservation to within 1%.

3.3. Results

A parameter study for a single-species plasma can be performed in terms of the

normalized parameters φn0 and δ. Figure 3.2 was obtained by solving Eqs. (9)-(11) with

17

Figure 3.2. Simulation that represents a two period segment of an ASB. The

different shades show trajectories with φn0 = 1 and δ = 1000 (black), δ = 100

(dark gray), and δ = 20 (light gray). The trajectory calculation is terminated

when a particle reaches yn = 0.75.

φn0 = 1 and with δ =1000 (1000 trajectories), δ =100 (2000 trajectories) or δ =20 (3000

trajectories) and by plotting the x and y components of the position vector in a parametric

form. A different number of trajectories was chosen for the purpose of achieving the contrast

18

Figure 3.3. Simulation that represents a two period segment of an ASB.

The different shades show trajectories with δ = 20, and φn0 = 0.5 (light gray)

and φn0 = 5 (black). The trajectory calculation is terminated when a particle

reaches yn = 0.75.

in the figure. Figure 3.2 shows the general behavior of charged particle trajectories near

an ASB as the inverse normalized Larmor radius δ is varied. Charged particle trajectories

were also calculated in a similar manner but keeping the inverse normalized Larmor radius

19

••••••••••••

•••••

••••

•

••••

•

•

•

••

•

••

•

••

•

•

•

•

•••

•

•

•

•••

•

•

•

•

•

•

••

•

•

••

•••

••••

•

•

••

••

•

••••

•

•

•

•••

•

••••••••

•••••ääääääääääääääääääääääääääääääääää

äääää

ää

ä

ä

ä

ä

ää

ä

ä

ä

ä

ä

ä

ä

ä

ää

ää

ä

ä

ää

äääää

ääääääääääääääääääääääääääääääää

∆ = 10 H•L∆ = 20 H L∆ = 40 H L

H•L 2.00 %

H L 1.02 %

H L 0.57 %

-0.2 -0.1 0.0 0.1 0.20

10

20

30

40

50

xn

Num

bero

fC

ount

s

Figure 3.4. Profile of the spatial distribution of charged particles that

reached yn ≥ 0.75 after entering a cusp and overcoming the electrostatic po-

tential barrier. The distribution of particles at yn ≈ 0.75 is for φn0 = 1 and

δ = 10, 20, and 40. The data series are labeled according to the parameter

varied, and the corresponding percentages of particles that reached yn ≈ 0.75

are indicated. The total number of trajectories simulated for each of these

plots was 100,000.

constant and varying the normalized electrostatic potential barrier φn0. The results are

shown in Fig. 3.3.

The trends observed are (1) increasing the magnetic field sufficiently can effectively

reflect charged particles away from most of the solid material and (2) increasing the electro-

static plugging sufficiently can reflect charged particles that would otherwise escape through

the magnetic field cusps. The latter trend is, of course, directly affected by the kinetic energy

20

äääääääääääääääää

ääääää

ä

ä

ä

ä

ä

äää

ä

ä

ä

ä

ä

ä

ä

ä

ä

ä

ä

ä

ä

ää

ää

ä

ä

ä

ää

ä

ä

ä

ää

ä

ä

ää

äää

äää

ääää

ää

ä

ääääääää

äää

ääääääääääääää•••••••••••••••••••••••••••••

••••••••••

•••••••••

•

••••••

•••••••••

••••••••••••••••••••••••••••••••••••

Φn0 = 0.5 H LΦn0 = 1.0 H LΦn0 = 2.0 H•L

H L 2.23 %

H L 1.02 %

H•L 0.25 %

-0.2 -0.1 0.0 0.1 0.20

20

40

60

80

100

xn

Num

bero

fC

ount

s

Figure 3.5. Profile of the spatial distribution of charged particles that

reached yn ≥ 0.75 after entering a cusp and overcoming the electrostatic po-

tential barrier. The distribution is for δ = 20 and φn0 = 0.5, 1, and 2. The

data series are labeled according to the parameter varied, and the correspond-

ing percentages of particles that reached yn ≈ 0.75 are indicated. The total

number of trajectories simulated for each of these plots was 100,000.

of the charged particles, and those particles that escape lie in the high energy tail of the

speed distribution. In the simulations, the trajectories were terminated when the particles

reached yn ≥ 0.75 (past the electrostatic potential barrier), or yn ≤ −3, or |xn| ≥ 1, or

|zn| ≥ 5. However, Figs 3.2 and 3.3 only show a region defined by yn ≤ 1, yn ≥ −1.5, and

|xn| ≤ 1 primarily to observe the modification of charged particle trajectories when charged

particles approach the ASB. When a particle reached yn ≥ 0.75, its position vector was

recorded. The x component of the position vector for each of these particles was evaluated

21

with respect to the center of the particular cusp that each of these entered and was assigned

to a bin. There were 100 total bins for half of a spatial period in the simulation. Figures 3.4

and 3.5 show the data from 100,000 trajectories obtained by such a procedure. Figure 3.4

presents the spatial profile of the particles that reached yn ≈ 0.75 for different values of the

magnetic field/Larmor radius. Figure 3.5 is identical except that the magnetic field is fixed

and the electrostatic potential barrier is varied. The solid curves in Figs. 3.4 and 3.5 are

there to help distinguish the general trend that each of the data sets follow. The data series

are labeled according to the parameter varied, and the corresponding percentage of particles

that escaped confinement is shown. The plots show the most probable location through

which particles can escape. Good confinement is defined here as the set of conditions that

minimize the interaction of charged particles with the solid material. In the present study,

magnetic confinement becomes apparent when the Larmor radius is smaller than 0.2 times

the spatial period. The effect of electrostatic plugging is observed in Fig. 3.5. When the

average thermal energy is less than the height of the electrostatic potential barrier, bet-

ter confinement is achieved. For φn0 > 5 and δ > 20 the number of particles that escape

confinement becomes negligible for the number of trajectories simulated in the present work.

3.4. Conclusion

An artificially structured boundary that produces electrostatically plugged magnetic

field cusps has been presented as an alternative way to confine charged particles or plasma

for ion source applications. Accumulation and confinement of highly pure or rare ions could

benefit by decreased particle loss due to particle-solid material interaction. An ASB produces

fields near the solid material boundary, and a nearly field free region exists away from the

ASB. Charged particles can be confined by suitable adjustment of the applied electric and

magnetic fields.

22

CHAPTER 4

SPACE-CHARGE-BASED ELECTROSTATIC PLASMA CONFINEMENT INVOLVING

RELAXED PLASMA SPECIES

4.1. Introduction

Suppose that a hollow and evacuated sphere, which is made from a refractory metal

such as tungsten or tantalum, is heated to a temperature sufficient for thermionic electron

emission to occur from the interior surface. A non-drifting non-neutral electron plasma would

be produced within the interior of such a sphere. Under certain conditions, the space charge

of that electron plasma can be used to confine a positive-ion plasma or a positron plasma.

In the work presented here, the electrostatic potential and the density profile of

a surface-emitted or edge-confined non-drifting non-neutral single-species plasma are self-

consistently evaluated assuming a relaxed plasma. Next, the equilibrium of a two-species

plasma, with one plasma species confined by the space charge of the other, is self-consistently

evaluated. Each species is assumed to be relaxed to a Boltzmann density distribution. An

edge-confined plasma would be effectively unmagnetized, except near an outer boundary

where a confining electromagnetic field would reside [9]. One possibility is for the confining

electromagnetic field to consist of a spatially periodic sequence of magnetic cusps that are

plugged electrostatically. This is a case where a magnetic multipole would be superimposed

on an electric multipole of higher order. The spatial period and range of the field would be

much smaller than the dimensions of the plasma.

A motivation for the work presented here is the prospect of testing fundamental

symmetries between the properties of matter and antimatter such as the gravitational ac-

celeration symmetry [12, 17]. However, plasma drifts within nested Penning traps represent

a formidable problem for producing antihydrogen with sufficiently low kinetic energy to be

trapped in useful quantities for experimentation [3, 18, 19]. An antihydrogen atom is born

with the kinetic energy of its antiproton, and plasma drifts can increase the kinetic energy

of antiprotons. Therefore, an ideal plasma confinement approach for antihydrogen studies

23

Trapping Volume: Conceptual Model.

N

N

N

N

S

S

S

S

Trapping VolumeASBField

Region

+ I

– I

+ I

– I

ASB with Current-Carrying Wires ASB with Permanent Magnets

Magnetic Field

Magnetic Field

Figure 4.1. Conceptual model of a plasma trapping volume with a field free

region at its center. A plasma is envisioned to relax within the volume and be

“edge-confined” by a reflecting surface such as an ASB.

would avoid plasma drifts and be capable of providing long confinement times for a cold,

dense, non-drifting (e.g., non-rotating) plasma of any desired size.

A trapping volume that is lined with an ASB could confine a non-neutral plasma of

either sign of charge, see Fig. 4.1. The center of the trapping volume is essentially field free

and presents an ideal scenario for prolonged electrostatic trapping of an oppositely charged

species that is free of plasma drifts. Three-body recombination rates within a multiple

species plasma can be about a factor of 10 larger within an unmagnetized plasma relative to

a magnetized plasma, all other parameters being equal [20]. The trapping volume envisioned

here is potentially suitable for recombination experiments without plasma drifts.

In Sec. 4.2, a self-consistent computation of the electrostatic potential that occurs

within an unmagnetized non-neutral plasma under equilibrium conditions is developed. In

Sec. 4.3, a second plasma species is introduced and the resulting equilibrium of a two-species

plasma is evaluated. In Sec. 4.4, the conditions necessary for achieving space-charge-based

electrostatic confinement are discussed. Concluding remarks are found in Sec. 4.5.

24

4.2. Single-Species Non-Neutral Plasma

A region of space that contains electric field sources must satisfy Poisson’s equation.

Therefore, the electrostatic potential resulting from a Boltzmann distribution of charged par-

ticles can be obtained by solving Poisson’s equation and imposing the appropriate boundary

conditions according to the geometry of the problem. Poisson’s equation (in SI units) reads

(13) ∇2φ(r) = −ρ(r)

ε0,

where φ(r) is the electrostatic potential, ρ(r) is the charge density, and ε0 the vacuum per-

mittivity. Assume that a single-species plasma is in a steady-state equilibrium. Furthermore,

assume that the plasma is relaxed, such that the Boltzmann density distribution represents

the charged particle distribution in space [21]. Also, let the electrostatic potential be repre-

sented by the average of the local electrostatic potential (averaging over the discreteness of

the plasma constituents). The Boltzmann relation for the plasma density is

(14) n(r) = nse−q[φ(r)−φ(rs)]/T .

Here, ns is a known plasma density at rs, q is the charge of a plasma particle (e.g., q = −e for

an electron plasma where e is the unit charge), and T is the plasma temperature in energy

units. The plasma temperature is assumed to be temporally constant and spatially uniform.

Equation (13), becomes

(15) ∇2φ(r) = −qnsεoe−q[φ(r)−φ(rs)]/T .

At the location where the plasma density is specified (i.e., at rs), the electrostatic potential is

defined to be zero: φ(rs) = 0. By solving Eq. (15) for the electrostatic potential, the plasma

equilibrium can be obtained. In order to generalize the study, the equation is normalized by

introducing a dimensionless potential ψ(r) = qφ(r)/T , and defining the Debye length at rs

as λD =√ε0T/(q2ns). With these modifications the governing equation simplifies to

(16) ∇2ψ(r) = −e−ψ(r)

λ2D

.

Equation (16) is solved for spherical, cylindrical, and planar geometries.

25

(1) For the spherical geometry, a system that has spherical symmetry is assumed. Let

r denote the radial coordinate of a spherical coordinate system. The governing

equation reads

(17)2

r

∂ψ

∂r+∂2ψ

∂r2= −e

−ψ

λ2D

.

(2) For the cylindrical geometry, a system is assumed that has infinite length in the

axial dimension and is cylindrically symmetric. Let r denote the radial coordinate

of the cylindrical system. The governing equation in cylindrical coordinates is

(18)1

r

∂ψ

∂r+∂2ψ

∂r2= −e

−ψ

λ2D

.

(3) For the planar geometry, assume that the system is contained between two infinite

planes. Let the variable r be defined as a Cartesian coordinate normal to the

planes. A system that has mirror symmetry about r = 0 is assumed. In this case,

the governing equation is

(19)∂2ψ

∂r2= −e

−ψ

λ2D

.

In the previous three equations, ψ is a function of the variable r, ψ = ψ(r). The notation

has been suppressed for brevity.

Equations (17), (18), and (19) are combined into a single equation. By introducing

a coefficient of the form (α − 1)/r in place of the term multiplying the first partial deriva-

tive, and changing variables to the spatial coordinate rn = r/λD, the following equation is

obtained:

(20)(α− 1)

rn

∂ψ(rn)

∂rn+∂2ψ(rn)

∂r2n

= −e−ψ(rn).

Here, α takes the value of 1, 2, or 3 for the planar, cylindrical, or spherical geometry,

respectively. Thus, Eqs. (17), (18), and (19) are simultaneously represented by Eq. (20) in

terms of the normalized coordinate rn.

26

Boundary Conditions

The symmetry of the charge distribution dictates a set of mixed boundary conditions.

(1) Neumann Boundary Condition

The electric field is zero at the origin:

(21)

[∂φ(r)

∂r

]r=0

=

[∂ψ(rn)

∂rn

]rn=0

= 0.

(2) Dirichlet Boundary Condition

The electrostatic potential is defined to be zero at the plasma edge:

(22) φ(rmax) = ψ(rn,max) = 0.

Here, the plasma edge is located at rmax and at rn,max = rmax/λD, where λD is the Debye

length at the plasma edge. The plasma diameter, or thickness in the planar geometry,

is 2rmax. The method by which the plasma is produced, sustained, and confined is not

considered here. The description is only applicable for the region 0 ≤ r ≤ rmax.

Finite Differences

A finite-differences computational approach has been used to predict plasma equilibria

in nested-well and single-well Malmberg-Penning traps [3, 22]. Equation (20) is solved using

a finite-differences approach. Written in terms of finite differences, the first partial derivative

of a general function, f(x, y, z), with respect to x (in symmetrical form) becomes:

(23)∂f(x, y, z)

∂x≈ f(x+ ∆x, y, z)− f(x−∆x, y, z)

2∆x.

The second partial derivative is

(24)∂2f(x, y, z)

∂x2≈ f(x+ ∆x, y, z) + f(x−∆x, y, z)

(∆x)2− 2f(x, y, z)

(∆x)2.

In principle, this recipe can be used to represent any second order partial differential

equation. Applying the finite-differences approach to Eq. (20) gives, after some algebraic

manipulations,

ψ(rn) =ω∆r2

n

2

[(α− 1)

rn

ψ(rn + ∆rn)− ψ(rn −∆rn)

2∆rn

27

+ψ(rn + ∆rn) + ψ(rn −∆rn)

∆r2n

+e−ψ(rn)]− (ω − 1)ψ(rn),(25)

where the left-hand side represents the new value each iteration, and the normalized grid

spacing is ∆rn = ∆r/λD. ω is introduced for the purpose of reducing computation time [15].

Equation (25) is implemented using a sequential over-relaxation method, with ω having a

value in the range 1 ≤ ω < 2 [22].

Self-Consistent Solution

A computer program was developed to solve for the normalized electrostatic potential

ψ using the finite-differences approach. The parameter ω and the number of iterations were

chosen to achieve the desired convergence. The self-consistent computation of the electro-

static potential was achieved by iteratively solving for the electrostatic potential according to

Eq. (25). All computations were run until the absolute difference between one iteration and

the next was less than 10−10 at every grid point. It has been reported that if the grid spacing

is on the order of, or smaller than, the Debye length, code instabilities are reduced, and the

convergence of a solution is more likely to be achieved [23]. The computation assigned at

least three grid points per Debye length for rmax λD and significantly more grid points

(≈ 50) per Debye length for values of rmax ≈ λD.

Figure 4.2 shows typical profiles of the electrostatic potential. The three plots cor-

respond to the planar, cylindrical, and spherical geometries (α = 1, 2, and 3, respectively).

The non-neutral plasma generates an electrostatic potential that has an extremum at the

center of each geometry (at rn = 0).

In Fig. 4.3, radial profiles of the normalized density function, e−ψ, are shown. The

profiles are for a spherical geometry, α = 3, with rn,max = 5, 10, 20, and 30. The non-

neutral plasma density has a minimum at the geometric center of the system. The behavior

of the plasma distribution in the vicinity of the boundary is observed to change in a more

pronounced manner as the value of rn,max increases, behavior that agrees qualitatively with

previous results for magnetized non-neutral plasmas [24].

28

0.0 0.2 0.4 0.6 0.8 1.0

rn

rn,max0

2

4

6

8

ΨElectrostatic Potential Profile

Figure 4.2. Typical radial profile of the normalized electrostatic potential.

The plots are for rn,max = 100 and α = 1 (solid), 2 (long dash), 3 (short dash).

The normalized electrostatic potential difference between the center and the

boundary is 7.6 for α = 1, 7.2 for α = 2, and 6.8 for α = 3.

0.0 0.2 0.4 0.6 0.8 1.0

rn

rn,max

0.2

0.4

0.6

0.8

1.0e-Ψ

Radial Plasma Distribution Profiles

Figure 4.3. Typical normalized density profiles. The plots are for α = 3 and

rn,max = 5 (dot-dashed), 10 (short dash), 20 (long dash), 30 (solid). Similar

profiles occur for other values of α.

29

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

ææ

æ

æ

æ

æ

à

à

à

à

à

à

à

à

à

à

àà

à

à

à

à

ì

ì

ì

ì

ì

ì

ì

ì

ì

ì

ìì

ì

ì

ì

ì

Α = 1 (è)

Α = 2 ()

Α = 3 (ì)

0 100 200 300 4000

2

4

6

8

10

rn,max

DΨ

0,Α

Normalized Electrostatic Potential Difference

Figure 4.4. Normalized electrostatic potential difference (between plasma

center and edge) for the three geometries. The solid lines are Eq. (26).

The behavior of the plasma has also been characterized by evaluating the normalized

potential at the geometric center of the volume in question. Recalling the boundary condition

ψ(rn,max) = 0, the normalized potential at rn = 0 is equal to the difference in normalized

potential between the geometric center and the boundary: ∆ψ0,α = ψ(0) − ψ(rn,max). The

value of the α subscript indicates the geometry being studied. Figure 4.4 shows how the

normalized potential difference, ∆ψ0,α, changes for increasing values of rn,max for planar,

cylindrical, and spherical geometries (α = 1, 2, 3). The value increases rapidly for small

values of rn,max and increases at a much slower rate for larger values of rn,max. Notice that

values exceeding 10 are predicted. Such large values indicate that it should be possible to

confine a second species of particles with opposite sign of charge within the electrostatic

potential well created by the first species, with both species having the same temperature

and charge state. Such a possibility is considered in Sec. 4.3.

The numerical results in Fig. 4.4 were fitted by an analytical expression. The approx-

imate analytical expression obtained for the normalized electrostatic potential difference

30

between the origin at rn = 0 and the boundary at rn,max is

(26) ∆ψ0,α = ln[(rn,max)0.217α0.139

]+W

(r2n,max

2α

).

Here, W (x) is the product logarithm function, which satisfies the equation x = W (x) exp [W (x)].

The last term on the right of Eq. (26) is the functional dependence that is expected in the

asymptotic limit of large rn,max. In such a limit, the density is spatially constant, except near

the plasma edge, where the variation is ignored, see Appendix B. Equation (26) agrees with

numerical results to within 1% for 30 ≤ rn,max ≤ 440 and to within 15% for 1 ≤ rn,max ≤ 30.

As a specific case, consider the electrostatic potential generated for rn,max = 100 and α = 3.

In such a case, ∆ψ0,α has a value of 6.85, and a 1 eV (temperature) electron plasma generates

an electric potential difference between the plasma edge and center of 6.85 V.

4.3. Two-Species Plasma

Consider a surface-emitted or edge-confined electron plasma that follows a Boltzmann

distribution as described in Sec. 4.2. An electrostatic potential energy well is created for

positive particles. Assume that a positive plasma species, such as a positive-ion or positron

plasma, is introduced near the center of the electron plasma. The charge density is ρ(r) =

−en−(r) + Zen+(r). Z represents the average charge state of the positive plasma particles.

The first term on the right-hand side refers to the electron plasma, which follows a Boltzmann

density profile of the form n−(r) = n0−exp(e[φ(r) − φ(rs)]/T−). The second term refers to

the positive plasma that is confined by the electrostatic potential well created by the space

charge of the electron plasma. The positive plasma is also assumed to follow a Boltzmann

distribution: n+(r) = n0+exp(−Ze[φ(r)− φ(0)]/T+). Poisson’s equation is

(27) ∇2φ(r) = − 1

ε0

[−en0−e

e[φ(r)−φ(rs)]T− + Zen0+e

−Ze[φ(r)−φ(0)]T+

],

where n−(rs) = n0− and n+(0) = n0+ are known densities of the respective plasma species.

Let Nn = Zn0+/n0− and Tn = ZT−/T+. Nn is the positive plasma charge density at the

center of the configuration normalized by the magnitude of the electron charge density at

the edge. Tn is the average charge state of the positive plasma particles multiplied by the

31

ratio of the electron temperature to the temperature of the positive species. Write

(28) ∇2φ(r) = −en0−

ε0

[−e

e[φ(r)−φ(rs)]T− +Nne

−eTn[φ(r)−φ(0)]T−

].

Introduce the normalized electrostatic potential, ψ(r) = eφ(r)/T−, with φ(rs) = 0, where rs

defines the boundary of the system:

(29) ∇2ψ(r) = −e2n0−

ε0T−

[−eψ(r) +Nne

−Tn[ψ(r)−ψ(0)]].

Define the Debye length at the boundary of the electron plasma as λ2D− = (ε0T−)/(e2n0−)

and scale the coordinates with respect to this quantity. Now,

(30) ∇2ψ(rn) = eψ(rn) −Nne−Tn[ψ(rn)−ψ(0)],

where rn = r/λD− is the normalized radial coordinate and ψ(0) is the value of the normalized

electrostatic potential at the center of the system. The Laplacian can be replaced by the

left-hand side of Eq. (20). The boundary conditions for the two-species plasma are identical

to those for the single-species plasma computations presented in Sec. 4.2, namely, Eqs. (21)

and (22).

A finite-differences approach was used to self-consistently evaluate the properties of

the plasma system. The shape of the electrostatic potential profile must now adjust to

include the effect of the positive species occupying the central region. Comparing Eq. (30)

to Eq. (20), one can see that Eq. (25) is applicable here, except that the density term becomes

(31) e−ψ(rn) → Nne−Tn[ψ(rn)−ψ(0)] − eψ(rn).

The computer program developed to solve for the normalized electrostatic potential

for a two-species plasma configuration is similar to the one used for Sec. 4.2, except that it

was adapted to include the density term for the second plasma species. The number of grid

points was determined by the smallest Debye length (smallest of the two species). At least

three grid points per such Debye length were used. The normalized electrostatic potential

was iteratively computed in a self-consistent manner until the absolute difference between

one iteration and the next was less than 10−10 at every grid point. Initial results indicate that

32

Increasing Positive Species Density

0.2 0.4 0.6 0.8 1.0

rn

rn,max

5

4

3

2

1

ΨNormalized Electrostatic Potential

+

+

+

_

_

_

Increasing Positive Species Density

0.0 0.2 0.4 0.6 0.8 1.0

rn

rn,max

0.0050.010

0.0500.100

0.5001.000

TwoSpecies Plasma Distribution

FIG. 5. Electrostatic potential of a two-species plasma (top). Self-consistent distributions of the

two plasma species (bottom) in logarithmic scale. The ratio of the temperatures is held constant

so that Tn = 5. The densities are varied: Nn = 0.004 (dashed), 0.04 (dot-dashed), 0.4 (solid).

The arrows indicate the trend that the system follows as the density of the positive plasma is

increased. The normalized distributions are n−(rn)/n0− , which are labeled by minus signs (–), and

n+(rn)/n0− , which are labeled by plus signs (+). The corresponding plot styles are for the two

distributions for a given value of Nn. rn,max is fixed at 30.

21

Figure 4.5. Normalized electrostatic potential of a two-species plasma (top).

Self-consistent distributions of the two plasma species (bottom) in logarithmic

scale. The plots are for, Tn = 5, rn,max = 30, and Nn = 0.004 (dashed),

0.04 (dot-dashed), 0.4 (solid). The arrows indicate the trend that the system

follows as Nn is increased. In the lower panel of this figure and Figs. 4.6-

4.8, the normalized distributions are n−(rn)/n0− , which are labeled by minus

signs (–), and Zn+(rn)/n0− , which are labeled by plus signs (+). Thus, each

matching pair of plots are the normalized distributions for the negative and

positive plasma species.

33

Increasing Positive Species Temperature

0.2 0.4 0.6 0.8 1.0

rn

rn,max

5

4

3

2

1

0Ψ

Normalized Electrostatic Potential

Increasing PositiveSpecies Temperature

_

+

0.0 0.2 0.4 0.6 0.8 1.0

rn

rn,max0.00

0.01

0.02

0.03

0.04

0.05TwoSpecies Plasma Distribution

FIG. 6. Electrostatic potential of a two-species plasma (top). Self-consistent distributions of the

two plasma species (bottom). The relative densities are held constant at Nn = 0.02 and the tem-

perature is varied: Tn = 1 (solid), 15 (dot-dashed), and 30 (dashed). The positive species density

is low enough that the negative species distribution and the normalized electrostatic potential

change only slightly. Notice the self-consistent equal-temperature equilibrium, Tn = 1, plot. The

normalized distributions are n−(rn)/n0− , which are labeled by a minus sign (–), and n+(rn)/n0− ,

which are labeled by a plus sign (+). rn,max is fixed at 30.

22

Figure 4.6. Normalized electrostatic potential of a two-species plasma (top).

Self-consistent distributions of the two plasma species (bottom). The plots

are for Nn = 0.02, rn,max = 30, and Tn = 1 (solid), 15 (dot-dashed), and 30

(dashed). The ± labels are defined in Fig. 4.5.

the electrostatic potential is similar for different geometries even when the positive plasma is

introduced. Only computations pertaining to a spherically symmetric system are presented

throughout the remainder of this chapter. The results obtained for cylindrically symmetric

system are presented in Chapter 5.

34

The depth of the electrostatic potential well tends to decrease as the positive plasma

charge density is increased. Such behavior is shown in Fig. 4.5. Also, for sufficiently high

charge densities of the positive plasma, both species approach the same charge density at the

center of the system. If the positive plasma charge density is sufficiently small, the electro-

static potential and density of the negative plasma are relatively unaffected by the presence

of the positive plasma. However, Fig. 4.6 shows that, for such a case, the volume occupied

by the positive plasma increases as the temperature of the positive plasma is increased, or

its average charge state is decreased.

The normalized electrostatic potential and distribution profiles for two plasma species

with equal temperatures and charge states under equilibrium conditions are shown in Fig. 4.7.

Notice that for sufficiently high charge densities of the positive plasma, the system reaches

neutrality near the center of the system. When the charge density of the positive plasma is

sufficiently low, there exists a state of partial neutralization. Figure 4.8 shows the minimum

positive species charge density at which the system achieves approximate neutrality at the

center of the system for different temperatures or charge states of the two plasma species.

The minimum neutral density decreases as the positive plasma temperature is lowered or its

average charge state is increased.

4.4. Space-Charge-Based Electrostatic Confinement Conditions

Good confinement of the positive plasma species can be expected to occur when the

electrostatic potential energy well that is self-consistently created by the plasma is much

deeper than the temperature (in energy units) of the positive plasma species [25]:

(32) T+ Ze∆φ0.

Here, ∆φ0 = φ(rmax)− φ(0) is the ordinary (unnormalized) electrostatic potential difference

between the edge and the center of the configuration. It may also be possible to sustain the

positive plasma species, even when Eq. (44) is not satisfied, provided that sufficient fueling

35

Increasing Positive Species Density

0.2 0.4 0.6 0.8 1.0

rn

rn,max

5

4

3

2

1

0Ψ

Normalized Electrostatic Potential

_

_

+

+

+

0.0 0.2 0.4 0.6 0.8 1.0

rn

rn,max

0.001

0.0050.010

0.0500.100

0.5001.000TwoSpecies Plasma Distribution: Equal Temperatures

FIG. 7. Two plasma species with disparate densities and equal temperatures (Tn = 1). The density

of the positive species is varied so that Nn = 0.1 (dot-dashed), 0.01 (dashed), and 0.001 (solid).

In the lower figure, the matching pairs of plots correspond to the negative and positive plasma

species. The density of the negative plasma species is normalized to unity at the boundary of the

system, rn = rn,max, and self consistently adjusts to accommodate the positive plasma species. The

normalized distributions are n−(rn)/n0− , which are labeled by minus signs (–), and n+(rn)/n0− ,

which are labeled by plus signs (+). rn,max is fixed at 30.

23

Figure 4.7. Two plasma species with equal temperatures and charge states.

The plots are for Tn = 1, rn,max = 30, and Nn = 0.1 (dot-dashed), 0.01

(dashed), and 0.001 (solid). The ± labels are defined in Fig. 4.5.

and heating are used. In terms of a normalized quantity, the condition is written as

(33) ∆ψ+0,α =Ze∆φ0

T+

1.

∆ψ+0,α is the normalized electrostatic potential energy well depth that confines the positive

plasma, and the value of the subscript α is used to indicate the geometry considered. ∆ψ+0,α

is evaluated in Fig. 4.9. A value for ∆ψ+0,α is first obtained by employing the approximate

36

Increasing Positive Species Density

0.2 0.4 0.6 0.8 1.0

rn

rn,max

5

4

3

2

1

0Ψ

Normalized Electrostatic Potential

_

_

_

+

++

0.0 0.2 0.4 0.6 0.8 1.0

rn

rn,max0.00

0.01

0.02

0.03

0.04

0.05

0.06

TwoSpecies Plasma Distribution: Equal Densities

FIG. 8. Two plasma species with disparate temperatures and equal densities at the center of the

plasma system. The normalized temperature and normalized density (Tn, Nn) are; (1, 0.05)[solid],

(10, 0.0225)[dot-dash], (25, 0.0152)[long dash], and (40, 0.0145)[short dash]. The normalized tem-

peratures were chosen and the normalized densities were then adjusted to the lowest value at which

the two distributions have the same value at the center of the system. The normalized distributions

are n−(rn)/n0− , which are labeled by minus signs (–), and n+(rn)/n0− , which are labeled by a

plus signs (+). rn,max is fixed at 30.

24

Figure 4.8. Two plasma species with approximately equal charge densi-

ties at the center of the plasma system. The plots are for rn,max = 30 and

(Tn, Nn) = (1, 0.05)[solid], (10, 0.0225)[dot-dashed], (25, 0.0152)[long dash],

and (40, 0.0145)[short dash]. The normalized electron temperatures, Tn, were

chosen and the normalized positive plasma charge densities, Nn, were then ad-

justed to the lowest value at which the two distributions have approximately

the same value at the center of the system. The ± labels are defined in Fig. 4.5.

37

expression developed in Sec. 4.2 [Eq. (26)], which represents the limit Nn → 0. The result

is shown in Fig. 4.9 (top panel, dotted line). To explore the effect that Tn and Nn have on

∆ψ+0,α, this quantity was evaluated as a function of system size for two different values of

Tn and varying Nn values. It is found that the normalized electrostatic potential energy well

depth tends to decrease as the normalized charge density of the positive plasma is increased,

and to increase with normalized system size. However, for a sufficiently large normalized

size, the normalized well depth saturates to a value nearly independent of the normalized

size of the system. The normalized well depth is also found to increase with Tn.

Figure 4.10 shows ∆ψ0,α = ∆ψ+0,α/Tn = e∆φ0/T−, the normalized electrostatic

potential difference, evaluated in the saturated regime for several values of Tn and Nn. An

expression for the normalized electrostatic potential difference is readily derived by assuming

that the plasma is neutral at the center. The expression is

(34)∆ψ+0,α

Tn= ∆ψ0,α = − ln(Nn).

Inspection of Figs. 4.8 and 4.9 indicates that, for the spherical geometry, Eq. (34) is applicable

for rn,max & 200, 10−4 . Nn < 1, and 1 ≤ Tn ≤ 10. The close agreement between Eq. (34)

and the numerical values suggests that the saturated regime is an equilibrium in which a

neutral region forms. It is conjectured that the electric potential well depth in the two-species

plasma is given by the smaller of Eq. (26) and Eq. (34). The predicted electric potential

well depth can be used together with collision-based theory, such as that in Refs. [25, 26] to

evaluate space-charge-based electrostatic plasma confinement time scales.

4.5. Conclusion

A self-consistent computation of the electrostatic potential generated by an edge-

confined or surface-emitted non-neutral plasma that follows a Boltzmann density distribution

with planar, cylindrical, or spherical symmetry has been carried out. The electrostatic

potential profile, plasma density distribution, and the electrostatic potential well depth for

different values of the normalized (to the edge Debye length) plasma size have been evaluated.

Relatively deep electrostatic potential wells are predicted by the computation when the

38

0 50 100 150 200 250 300 3500

2

4

6

8

rn,max

Ψ0,3;T n1

0 50 100 150 200 250 300 3500

20

40

60

80

rn,max

Ψ0,3;T n10

FIG. 9. Electrostatic potential difference between the edge and the center of the system composed

of two plasma species which is plotted vs. system size. Top (bottom) plot shows the electrostatic

potential difference that a positive plasma experiences when the temperatures are equal, Tn = 1

(disparate, Tn = 10). Notice the difference in magnitude of the vertical axis. The different plot

styles are for Nn = 0.1 (solid), 0.01 (long dash), 0.001 (dash), 0.0001 (dot-dash). The dotted line

in the top figure illustrates the trend as Nn → 0.

25

Figure 4.9. Normalized electrostatic potential energy well depth for space-

charge-based electrostatic plasma confinement as a function of normalized sys-

tem size. The dotted line in the top panel is for Nn = 0 and Tn = 1. Top

(bottom) panel is for Tn = 1 (Tn = 10), and Nn = 0.1 (solid), 0.01 (long dash),

0.001 (dash), 0.0001 (dot-dashed).

plasma size is much greater than the Debye length. An approximate expression has been

fitted for the electrostatic potential well depth as a function of the normalized plasma size.

39

10-4 0.001 0.01 0.1 10

2

4

6

8

10

Nn

DΨ0,3

Figure 4.10. Normalized electrostatic potential difference for increasing nor-

malized charge density of the positive species. There are plot points for

rn,max = 300 and Tn = 1, 4, and 10, for each value of Nn, but the plot points

are indistinguishable. The solid line is Eq. (34).

A self-consistent computation has been carried out for a positive plasma that is con-

fined by the space charge of an edge-confined or surface-emitted electron plasma, with both

species following Boltzmann density distributions. The results also apply to a negative

plasma that is confined by the space charge of an edge-confined positive plasma such as a

positron plasma. The results presented for the two-species system pertain to a spherically

symmetric system, but are expected to be qualitatively applicable to planar and cylindrical

geometries. The two-species system has been characterized by varying normalized param-

eters, which include (1) the ratio of the plasma radius to the Debye length at the plasma

edge, (2) the ratio of the positive plasma charge density at the center of the system to that

of the electron plasma at the edge, and (3) the ratio of the temperatures multiplied by the

average charge state. The electrostatic potential profile and the charge density distribution

have been computed, and the cases explored indicate the following: (1) Increasing the charge

density of the positive plasma species decreases the depth of the electrostatic potential well.

(2) Increasing the temperature or decreasing the average charge state of the positive plasma

40

causes the positive plasma to occupy a larger volume. (3) An equilibrium is possible in which

the two plasma species have equal temperatures and equal charge states. (4) Approximately

equal charge densities of the two plasma species at the center of the system occurs for a

sufficiently high charge density of the positive plasma species, even when the temperatures

and charge states are equal.

The electrostatic potential well depth has been evaluated for the two-species system.

It has been found that, once the positive plasma is introduced, the electrostatic potential

well depth reaches a saturation regime when the normalized plasma size is sufficiently large.

41

CHAPTER 5

ELECTROSTATIC STORAGE RING WITH FOCUSING PROVIDED BY THE SPACE

CHARGE OF AN ELECTRON PLASMA

5.1. Introduction

In what follows, a study similar to that found in Chapter 4 is carried out for a cylindri-

cally symmetric system. A cylindrically symmetric system that employs space-charge-based

confinement is presented. Based on the study, propositions and remarks for constructing a

cylindrically symmetric experimental apparatus are made.

Several purely electrostatic systems can be employed to confine ion beams or drift-

ing ion plasmas, such as electrostatic storage rings, electrostatic ion beam traps, Kingdon

traps, and electrostatic beam guides [27, 28, 29, 30, 31, 32, 33, 5]. Confinement of charged

particles by purely electrostatic means is important when time-varying electro magnetic or

magnetostatic confinement interfere with the experiment at hand. A system is proposed here

for investigation of atomic physics processes in plasmas (e.g., processes characterized by the

interaction of trapped heavy ions and plasma electrons). A system is envisioned where a

hollow and evacuated cylinder, which is made from a refractory metal such as tungsten or

tantalum, is heated to a temperature sufficient for thermionic electron emission to occur

from the interior surface. A non-drifting, non-neutral electron plasma would be produced

within the interior of such cylinder, and an electrostatic potential well for positive particles

would be created by the space charge of the electron plasma. Similarly, suppose a cylindrical

beam line has an interior surface lined with an artificially structured boundary [2] so that

it confines an electron plasma. Under certain conditions, the space charge of that electron

plasma can be used to focus a positive-ion beam or drifting plasma; see Fig. 5.2. Note

that focusing is loosely defined here as the nominal effect of ion bunching towards a desired

region, and its definition is not the typical ion optics one.

The equilibrium of a two-species system, with one species confined by the space

charge of the other, is self-consistently evaluated. Each species is assumed to be radially

42

relaxed to a Boltzmann density distribution. An edge-confined electron plasma would be

effectively unmagnetized except near an outer boundary where a confining electromagnetic

field would reside. One possibility is for the confining electromagnetic field to consist of a

spatially periodic sequence of magnetic cusps that are plugged electrostatically with a single

potential energy barrier for negative species (electrons) and the space charge of the electrons

to confine a positive species plasma. This is a case where a magnetic multipole would be

superimposed on an electric multipole of higher order. The advantages of multipole magnetic

fields on confined plasmas are envisioned to positively affect the system here proposed [34].

The spatial period and range of the field would be much smaller than the dimensions of the

plasma, such that the positively charged species (beam or plasma) would experience only an

electrostatic field that is produced by the space charge of the electron plasma.

5.2. Theory

Consider a surface-emitted or edge-confined electron plasma that follows a Boltzmann

distribution in the radial direction of an infinitely long cylindrically symmetric device. Such a

configuration serves as a model for a storage ring that has a radius that is much larger than

the characteristic electron plasma parameters. An electrostatic potential well is therefore

created at the radial center of such device for positive particles. Assume that a positively

charged species, such as a positive-ion or positron beam or plasma, is introduced near the

center of the electron plasma. The charge density is ρ(r) = −en−(r) + Zen+(r), where Z

represents the average charge state of the positive species. The first term on the right hand

side refers to the electron plasma, which follows a Boltzmann density profile of the form

n−(r) = n0−exp(e[φ(r) − φ(rs)]/T−), rs specifies the location where the plasma density is

known. The second term refers to the positive particle beam or plasma that is confined

by the electrostatic potential well created by the space charge of the electron plasma. The

positive plasma is also assumed to be of a Boltzmann type. Its density profile is n+(r) =

n0+exp(−Ze[φ(r)−φ(0)]/T+). From this point on, we refer to a positive ion beam or plasma

as positive species where the temperature, T+, is associated with the transverse energy of

43

rn

zn

n0+

n0_

n0_

Artificially StructuredBoundary

Positive Ion Beam or Plasma

Electron Plasma

Electron Plasma

rn,max

Figure 5.1. Cross-sections of a segment of a cylindrical beam line. The

electron plasma is confined by an artificially structured boundary. The space

charge of the electron plasma creates an electrostatic potential that focuses a

positive-ion beam or drifting plasma.

the positive ion beam, or T+ is the characteristic temperature of the positive species when

it is referred to as a plasma.

Poisson’s equation is

(35) ∇2φ(r) = − 1

ε0

(−en0−e

e[φ(r)−φ(rs)]T− + Zen0+e

−Ze[φ(r)−φ(0)]T+

),

where n−(rs) = n0− and n+(0) = n0+ are known densities of the respective species. Let

Nn = Zn0+/n0− and Tn = ZT−/T+. Nn is the positive species charge density at the center

of the configuration normalized by the magnitude of the electron charge density at the edge.

Tn is the average charge state of the positive species multiplied by the ratio of the electron

44

temperature to the temperature of the positive species. Equation (35) can be written as

(36) ∇2φ(r) = −en0−

ε0

(−e

e[φ(r)−φ(rs)]T− +Nne

−eTn[φ(r)−φ(0)]T−

).

Introducing the normalized electrostatic potential, ψ(r) = eφ(r)/T−, with φ(rs) = 0, where

rs is the boundary of the system, gives

(37) ∇2ψ(r) = −e2n0−

ε0T−

(−eψ(r) +Nne

−Tn[ψ(r)−ψ(0)]).

Then defining the Debye length at the boundary of the electron plasma as λ2D− = (ε0T−)/(e2n0−)

and scale the coordinates with respect to this quantity produces

(38) ∇2ψ(rn) = eψ(rn) −Nne−Tn[ψ(rn)−ψ(0)],

where rn = r/λD− is the normalized radial coordinate, and ψ(0) is the value of the normalized

electrostatic potential at the center of the system. For a storage ring, in the limit of a large

radius, a cylindrical geometry is assumed. The Laplacian operator in Eq. (38) takes its usual

cylindrical form.

(39)1

rn

∂ψ(rn)

∂rn+∂2ψ(rn)

∂r2n

= eψ(rn) −Nne−Tn[ψ(rn)−ψ(0)].

Note that the system has been scaled with respect to the electron plasma Debye length.

However, quantities pertaining to the positive species can be obtained via the following

relationships:

(40) λ2D− = TnNn

ε0T+

e2Z2n0+

= TnNnλ2D+

and

(41) ψ+(rn) = −Tnψ(rn).

The notation ψ(rn) = ψ−(rn) could be used in Eqs. (37), (38), and (41), and note that

electron and positive species Debye lengths are specified with respect to the densities at the

boundary and the center of the system, respectively.

The boundary conditions for the two-species plasma are:

45

(1) Neumann Boundary Condition

(42)

[∂φ(r)

∂r

]r=0

=

[∂ψ(rn)

∂rn

]rn=0

= 0.

(2) Dirichlet Boundary Condition

(43) φ(rmax) = ψ(rn,max) = 0,

where rmax = rs and rn,max = rsλD−

.

A finite-differences approach was used to self-consistently solve Eq. (18) to calculate

the properties of the system when the two species are present [15]. The number of grid

points was determined by the smallest Debye length (smallest of the two species). At least

three grid points per such Debye length were used. The electrostatic potential was iteratively

calculated in a self-consistent manner until the absolute difference between one iteration and

the next was less than 10−10 for all grid points.

5.3. Results

For sufficiently high densities of the positive species, both species approach the same

charge density at the center of the system and the electrostatic potential well depth decreases

as the charge density of the postive species is increased. Such behavior is shown in the

Fig. 5.2. If the positive species charge density is sufficiently small, the electrostatic potential

generated by, and density of, the electron plasma are unaffected by the presence of the

positive species. However, Fig. 5.3 shows that as the temperature of the positive species is

increased, its distribution occupies a larger volume.

The normalized electrostatic potential and distributions profiles for two species with

equal temperatures and charge states under equilibrium conditions are now examined. Fig-

ure 5.4 shows a two-species system in which these have equal temperatures and charge states.

Notice that for sufficiently high charge densities of the positive species, the system reaches

neutrality near the center. When the charge density of the positive species is lower, there

exists a state of partial neutralization.

46

Increasing Positive Species Density

0.2 0.4 0.6 0.8 1.0

rn

rn,max

-5

-4

-3

-2

-1

ΨNormalized Electrostatic Potential

Increasing Positive Species Density

_

_

_

+

+

+

0.0 0.2 0.4 0.6 0.8 1.0

rn

rn,max

0.005

0.010

0.050

0.100

0.500

1.000Radial Distribution Profiles

Figure 5.2. Normalized electrostatic potential of a two-species system (top).

Self-consistent distributions of the two species (bottom) in logarithmic scale.

The plots are for, Tn = 5, rn,max = 30, and Nn = 0.004 (dashed), 0.04 (dot-

dashed), 0.4 (solid). The arrows indicate the trend that the system follows

as Nn is increased. In the lower panel of this figure and Figs. 5.3-5.5, the

normalized distributions are n−(rn)/n0− , which are labeled by minus signs (–),

and Zn+(rn)/n0− , which are labeled by plus signs (+). Thus, each matching

pair of plots are the normalized distributions for the negative and positive

plasma species.

47

Increasing Positive Species Temperature

0.2 0.4 0.6 0.8 1.0

rn

rn,max

-5

-4

-3

-2

-1

ΨNormalized Electrostatic Potential

Increasing Positive

Species Temperature

_

+

0.0 0.2 0.4 0.6 0.8 1.0

rn

rn,max0.00

0.01

0.02

0.03

0.04

0.05Radial Distribution Profiles

Figure 5.3. Normalized electrostatic potential of a two-species plasma (top).

Self-consistent distributions of the two plasma species (bottom). The plots

are for Nn = 0.02, rn,max = 30, and Tn = 1 (solid), 15 (dot-dashed), and 30

(dashed). The ± labels are defined in Fig. 5.2.

As shown in Fig. 5.4, relatively high (low) densities of the positive species can yield a

region of neutrality (partial neutralization) near the center of the system. Figure 5.5 shows

the minimum positive species charge density at which the system achieves approximate

charge neutrality at the center of the system for different temperatures or charge states of

48

Increasing Positive Species Density

0.2 0.4 0.6 0.8 1.0

rn

rn,max

-5

-4

-3

-2

-1

ΨNormalized Electrostatic Potential

+

+

+

_

_

0.0 0.2 0.4 0.6 0.8 1.0

rn

rn,max

0.001

0.0050.010

0.0500.100

0.5001.000

Radial Distribution Profiles: Equal Temperatures

Figure 5.4. Two plasma species with equal temperatures and charge states.

The plots are for Tn = 1, rn,max = 30, and Nn = 0.1 (dot-dashed), 0.01

(dashed), and 0.001 (solid). The ± labels are defined in Fig. 5.2.

the two species. The minimum neutral density decreases as the positive species temperature

is lowered or its average charge state is increased.

The space charge of the electron plasma becomes partially compensated as the charge

density of the positive species is increased at the center of the configuration. Consequently,

the electrons near the center of the configuration experience less repulsion from each other

(due to the presence of the positive species) and the system self-consistently adjusts to an

49

0.2 0.4 0.6 0.8 1.0

rn

rn,max

-5

-4

-3

-2

-1

0Ψ

Normalized Electrostatic Potential

_

_ +

+

0.0 0.2 0.4 0.6 0.8 1.0

rn

rn,max0.00

0.01

0.02

0.03

0.04

0.05

0.06

Radial Distributin Profiles: Equal Central Densities

Figure 5.5. Two plasma species with approximately equal charge densi-

ties at the center of the plasma system. The plots are for rn,max = 30 and

(Tn, Nn) = (1, 0.05)[solid], (10, 0.0225)[dot-dashed], (25, 0.0152)[long dash],

and (40, 0.0145)[short dash]. The normalized electron temperatures Tn were

chosen and the normalized positive plasma charge densities Nn were then ad-

justed to the lowest value at which the two distributions have approximately

the same value at the center of the system. The ± labels are defined in Fig. 5.2.

50

0 50 100 150 200 250 300 3500

2

4

6

8

10

rn,max

DΨ

0

Figure 5.6. Normalized electrostatic potential energy well depth for space-

charge-based electrostatic focusing as a function of normalized system size.

The plots are for Tn = 1, and Nn = 0 (dotted), 0.0001 (dot-dashed), 0.001

(dash), 0.01 (long dash), and 0.1 (solid).

equilibrium in which the central electron density is higher than would occur if the positive

species were absent.

5.4. Space-Charge-Based Electrostatic Focusing

Focusing of the positive species is expected to occur when the electrostatic potential

energy well created by the electron plasma is much deeper than the energy associated with

the transverse degree of freedom of the positive species [25];

(44) T+ Ze∆φ0.

Here, ∆φ0 = φ(rmax)− φ(0) is the ordinary (unnormalized) electrostatic potential difference

between the edge and the center of the configuration. In terms of a normalized quantity, the

condition is written as

(45) ∆ψ+0 =Ze∆φ0

T+

1.

51

10-4 0.001 0.01 0.1 10

2

4

6

8

10

Nn

DΨ0

Figure 5.7. Normalized electrostatic potential difference for increasing nor-

malized charge density of the positive species. Points are plotted for rn,max =

300 and Tn = 1, 4, and 10, for each value of Nn, but these plot are indis-

tinguishable for the different values of Tn. The solid line drawn through the

points is a fit given by Eq. (47).

∆ψ+0 is the normalized electrostatic potential energy well depth that confines the positive

species. Also, ∆ψ0 is the electrostatic potential energy well created by the electron plasma,

that is,

(46) ∆ψ0 =∆ψ+0

Tn

∆ψ+0 is evaluated in Fig. 5.5. To explore the effect that Tn and Nn have on ∆ψ+0, this

quantity was evaluated as a function of system size for two different values of Tn and varying

Nn values. It is found that the normalized electrostatic potential energy well depth tends

to decrease as the normalized charge density of the positive plasma is increased, and to

increase with normalized system size. However, for a sufficiently large normalized size, the

normalized well depth saturates to a value nearly independent of the normalized size of the

system. The normalized well depth is also found to increase with Tn. Figure 5.7 shows

∆ψ0 = ∆ψ+0/Tn = e∆φ0/T−, the normalized electrostatic potential difference, evaluated for

52

several values of Tn and Nn. Assuming that a state of charge neutrality exists at the center

of the system, the normalized electrostatic potential difference is given by the following

expression for rn,max & 200 and 10−4 . Nn ≤ 1:

(47)∆ψ+0

Tn= ∆ψ0 ≈ − ln(Nn)

This result can be used together with collision-based theory, such as that in Refs. [25] [26],

to evaluate space-charge-based electrostatic confinement time scales.

5.5. Conclusion

A self-consistent computation of the electrostatic potential generated by the space

charge of an edge-confined or surface-emitted electron plasma that follows a Boltzmann

density distribution in the radial direction of a cylindrically symmetric system has been

carried out. The elecrostatic potential energy well generated was employed to investigate

the possible focusing of a positive ion beam or drifting plasma. The results obtained are

applicable for a cylindrically-symmetric charged-particle focusing system such as a beam

guide or a storage ring in the limit of large ring radius. The two-species system has been

characterized by varying normalized parameters, which include (1) the ratio of the plasma

radius to the Debye length at the plasma edge, (2) the ratio of the positive species charge

density at the center of the system to that of the electron plasma at the edge, and (3) the

ratio of the temperatures multiplied by the average charge state.

The electrostatic potential profile and the charge density distribution have been com-

puted, and the cases explored indicate the following: (1) Increasing the charge density of

the positive species decreases the depth of the electrostatic potential well; (2) increasing the

temperature or decreasing the average charge state of the positive species causes the positive

species to occupy a larger volume; (3) an equilibrium is possible in which the two species have

equal temperatures and equal charges states; and (4) approximately equal charge densities

of the two species at the center of the system occur for a sufficiently high charge density of

the positive species, even when the temperatures and charge states are equal.

53

The electrostatic potential well depth has been evaluated for the two-species system.

It has been found that, once the positive species is introduced, the electrostatic potential

well depth reaches a saturation regime when the normalized size is sufficiently large. The

electrostatic potential well depth has been described analytically in this saturated regime.

the results obtained can be applied to evaluate space-charge-based electrostatic confinement

time scales. The concept could be of particular utility for investigation of atomic or molecular

processes in plasmas.

54

CHAPTER 6

ELECTRON BEAM TRANSMISSION THROUGH A CYLINDRICALLY SYMMETRIC

ARTIFICIALLY STRUCTURED BOUNDARY

6.1. Introduction

Experimental research on charged particle confinement and control by an electro-

magneto-static field configuration created by a cylindrically symmetric artificially structured

boundary (ASB) is presented. The ASB produces a periodic set of magnetic field cusps that

are plugged electrostatically. In the trapping system presented, the reflection or modification

of charged particle trajectories occurs near the confining boundary where the confining fields

have a relatively high strength. Away from the boundary, an essentially field free region

exists where confined particles are expected to reside. A field-free confinement region is

highly desired as a prospecting tool for experiments with particle trapping, particle-particle

interaction, particle-external field interaction, and self-consistent relaxation of plasmas.

The system presented here has several potential applications in electron or ion beam

physics and plasma physics such as in ion sources, ion or electron beam guides, and as

a charged particle trap. A cylindrically symmetric experimental apparatus has been con-

structed and its description is given. The characteristics of beam acceptance and transmis-

sion for the system operated as a beam guide are presented.

6.2. Apparatus

The components of the experimental apparatus are an electron source, a charged

particle trap or beam guide, and charged particle detection system. The electron source

consists of an electron gun and an electrostatic einzel lens that focuses the electron beam

at the trap entrance. The charged particle trap consists of a cylindrically symmetric ASB

configuration. This configuration produces spatially periodic electro- magneto-static fields

for the confinement of charged particle trajectories. Charged particles that leave the trap

through the exit side are detected with a position- and (relative) intensity-sensitive charged

particle detection system. An Ultra-High-Vacuum (UHV) chamber houses the electron gun,

55

Camera

Einzel LensElectron Gun

Entrance Electrode Exit Electrode

MCP

Phosphor ScreenVacuumPermanent Ring Magnets

OFHC Copper Electrodes

EntranceFaraday Cup

Exit Cup

Figure 6.1. Schematic view of experimental apparatus.

the charged particle trap, and the detection system. Figure 6.1 is a schematic representa-

tion of the experimental apparatus. The experimental hardware and software developed to

operate the apparatus as a charged particle trap is shown in Appendix D.

6.3. Electron Beam

The source of electrons is a Kimball Physics Incorporated (KPI) electron flood gun

(EFG-8) and electron gun power supply (EGPS-8) capable of supplying an electron beam

from a few eV to 1.5 keV with an energy spread of less than 0.4 eV and an emission current

of up to hundreds of µA, as per manufacturer specifications. The einzel lens assembly

was custom designed to focus charged particles on the center of the entrance to the trap.

Space limitations constrained the dimensions of the einzel lens but were optimized through

simulation. The einzel lens was simulated in SimIon [35] to determine the dimensions of

the lens elements which would allow focusing, in decelerating mode, of charged particles

before or after the center of the entrance electrode. The einzel lens elements are made of

oxygen-free high conductivity (OFHC) copper and, when assembled, are isolated from each

other by ceramic balls (D = 0.317 cm), which also serve to align the lens elements.

Refer to Fig. 6.1: For an electron beam with an energy of 30 eV, the voltage biasing

the center einzel lens element was varied to optimize empirically electron beam current at

the entrance of the trap, as measured by the entrance Faraday cup. This procedure was

carried out both with the experimental system and in the SimIon simulation environment.

56

The results are shown in Fig. 6.2. The procedure to obtain the plotted points from the

electron gun-einzel lens simulation is as follows. A simulation environment was created in

SimIon with the dimensions of the electron gun-einzel assembly. A classical trajectory Monte

Carlo simulation was carried out in SimIon where the electrons originated uniformly from a

1 mm disc into a velocity cone with half-angle α = tan−1(v⊥v‖

)= 15o. Here v‖ and v⊥ are the

components of the velocity vector parallel and perpendicular to the axis of the cylindrically

symmetric system simulated, respectively. 5,000 trajectories were simulated to obtain each

plotted point in Fig. 6.2. For each simulated charged particle trajectory, the position and

velocity vectors were recorded when the simulated trajectory terminated at the boundary

of the volume or at the surface of any of the electrodes in the simulation. The simulated

charged particle trajectories that terminated at an electrode that represented the entrance

Faraday cup were summed. The values thus obtained are plotted as Nn,i = NiNmax

in Fig. 6.2,

where Nmax = 3472 is the value with the einzel biased to −24 V (decelerating mode). The

plot shows the magnitude of the bias voltage as the abscissa. The SimIon simulation of

the electron gun-einzel lens-Faraday cup system can be used to determine the parameters

necessary to focus electrons or ions with different energies and other initial conditions.

An experiment was performed where the electron current on the entrance Faraday cup

was recorded as a function of einzel biasing voltage. Inside the gun, electron emission occurs

by heating of an yttria coated (Y2O3) disc of 1 mm in diameter. The gun construction limits

the emission to 11o , when ballistic trajectories are assumed, via a collimating aperture. The

entrance Faraday cup in the experimental setup was biased to 27 V to suppress secondary

electron current. The corresponding electrode in the simulation was also set to 27 V. The

plotted points in Fig. 6.2 which correspond to the recorded current on the Faraday cup were

normalized with respect to the maximum current observed; Imax = −63.5 nA, which occurred

at an einzel biasing voltage of −24 V. The plotted points are In,i = IiImax

, a quantity that is

proportional to the number of electrons incident on the Faraday cup per unit time. During

experimentation, the electron gun was set in electronic current control (ECC) mode, which

kept the current nearly constant over several hours after conditioning for 30 min. The current

57

• • • • • • • • ••

• • ••

•

••

•

´´

´´ ´ ´

´

´´´´

´´

´´

´

Experiment H •L = In,i

SimulationH ´ L = Nn,i

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

V

Num

bero

fPar

ticles

HNor

mali

zedL

Figure 6.2. Relative number of particles incident at the entrance of the trap

as a function of the magnitude of einzel lens focusing voltage. The einzel lens

was biased to provide focusing in decelerating mode. 5,000 electron trajectories

were simulated per data point marked by a cross. The maximum number of

electron trajectories that collapsed onto the Faraday cup electrode was 3472,

occurring at an einzel bias voltage of −24 V. Data points marked by dots are

the electron currents observed at the Faraday cup in the experimental setup,

normalized to the maximum current of −63.5 nA observed for an einzel lens

bias of −24 V.

was measured for less than 1 h to produce all In,i data points. In Fig. 6.2, the current at the

entrance cup shows a broad maximum for einzel lens voltage of −22 V to −27 V. Given the

close agreement between the simulated and experimental points plotted in Fig. 6.2, electron

beam characteristics such as beam intensity and phase-space properties can be inferred from

the simulation with confidence.

6.4. Experimental Artificially Structured Boundary

The charged particle trap presented here is defined as a cylindrically symmetric vol-

ume that has its interior boundary surface lined by an ASB with electrostatic plugging. The

system has been constructed to explore its possible application as an ion source, as an ion

58

beam guide, or as an ion trap. The electro- magneto-static fields have been designed for

trapping charged particles and plasmas or for guiding ion or electron beams. An ASB that

creates a periodic set of magnetic field cusps has been predicted to reflect charged particles

of either sign of charge simultaneously. Reflection of charged particles occurs most effec-

tively when their trajectories have grazing angles of incidence [2]. An ASB with electrostatic

plugging to has been predicted to confine effectively a single species of charge [36]. The

behavior of an electron plasma and an argon plasma in the vicinity of a planar segment of an

ASB has been observed to be strongly governed by the shape of the magnetic fields [12, 9].

These considerations have been taken into account to develop the electric and magnetic

fields within the volume lined by the ASB described here. Figure 6.3 shows the magnetic

Plugging Electrodes Magnets

Ring Cusp Point Cusp Point Cusp

Equipotential Contours

Figure 6.3. A length-wise cross-sectional view of the cylindrically symmetric

ASB and the fields produced within its interior. The rectangular features on

the top and bottom figures are the magnets and electrodes that create the

ASB. The lines in the top figure show contours of equal electric potential. The

lines on the bottom figure show the magnetic field. See text for further details.

59

field produced by a periodic set of ring magnets and a possible electrostatic biasing scheme.

The electrostatic field forms electrostatic mirrors axially at the entrance and exit of the trap.

The magnetic field forms magnetic mirrors. Charged particles can be confined in the axial

direction by either or both of these two fields. Charged particles are confined radially by the

magnetic fields near the magnet surfaces and by electrostatic barriers present at the location

of the magnetic field ring cusps.

A B

C

Figure 6.4. Photographs of the experimental system. Panel A shows the

alternating sequence of copper ring electrodes and permanent ring magnets.

Panel B shows the trapping volume as viewed upstream from the exit side.

Panel C shows the phosphor screen that, along with the micro-channel plates

(not shown), constitute the electron detection system.

Images of the actual system are shown in Fig. 6.4. The material structure consists of

a sequence of 12 neodymium-iron-boron high-strength permanent ring magnets (dimensions:

ID = 1.27 cm, OD = 2.54 cm, thickness = 0.254 cm) that alternate positions with 13 OFHC

copper ring electrodes (dimensions: ID = 1.78 cm, OD = 3.75 cm, thickness = 0.200 cm). The

60

magnets are arranged with like poles facing each other so that magnetic field ring cusps are

created between any two magnets. The magnets and electrodes are kept electrically isolated

from each other by mica washers (dimensions: ID = 1.90 cm, OD = 2.54 cm, thickness =

0.0254 cm). The magnet/electrode structure is electrically isolated from the entrance and

exit electrodes by ceramic balls (D = 0.317 cm). The spatial period of the magnetic field is

0.95 cm, with six full periods constituting the length of the trap. The distance from the center

of the entrance electrode to the center of the exit electrode is 7.62 cm, and from the center

of the exit electrode to the front face of the first MCP is 0.64 cm. The maximum magnetic

field at the surface of the magnets is specified by the manufacturer to be 0.3 T. A maximum

magnetic field of ≈ 0.2 T was measured before assembling the structure. A simulation of the

magnetic fields was developed in Vizimag [37] in order to observe the magnetic fields with

in the trap. In the simulation, a maximum magnetic field strength of 0.3 T was used along

with scaling set to represent the physical dimensions of the magnets and the assembly itself.

The resulting simulated magnetic field is shown in Fig. 6.3. The magnitude of the magnetic

field in the middle of the central ring cusp is approximately 0.08 T, and 0.017 T at the center

of the axial point cusps. Reflection or confinement of charged particles by the magnetic

field of an ASB has been predicted, see for example [36], to occur when the Larmor radius

is significantly smaller than the spatial period of the fields. The magnets are nickel plated

and are connected in three groups of four magnets each; one inner and two outer groups.

The electrodes are connected in three groups with three electrodes in the inner group and

five electrodes in each of the outer groups. In this manner, different biasing schemes can be

explored for diagnostic purposes.

6.5. UHV Conditions During Experimentation

The vacuum chamber is evacuated from atmospheric pressure to a base pressure of

5× 10−9 Torr in four stages: 1) A dual sorption pump stage achieves a vacuum of 100 mTorr

with the first sorption pump and ≈ 20 mTorr with the second one. 2) A turbomolecular

pump, backed by a rotary vane pump, takes over and evacuates the chamber until the ion

pump can be started, typically at 10−6 Torr. 3) Once the ion pump starts, the vacuum

61

Figure 6.5. Phosphor screen as imaged by SBIG ST-7XMEI SBIG CCD

camera (left panel). An electron beam exiting the trap and incident on the

MCP/Phosphor assembly creates the time integrated fluorescence recorded by

the CCD camera (right panel). For reference, the phosphor screen (major

circular feature on left panel) is 1.9 cm in diameter (or ≈ 500 pixels; 1 pixel

unit (pu) = 38µm). The same scale applies to right panel.

chamber is isolated from the previous two pumping stages by a gold seal valve. The ion

pump keeps the system at ≈ 5 × 10−9 Torr. 4) The last pumping stage is composed of a

titanium sublimation pump and a getter pump with a cryostat cooled by liquid nitrogen.

The last stage is typically turned on only during an experiment and nominally achieves

a vacuum better than 5 × 10−10 Torr. Particular care was taken to choose materials and

cleaning practices compatible with UHV conditions. The ring magnets that generate the

magnetic fields and the mica spacers are expected to be the main factors that limit the base

pressure of the vacuum system from achieving a lower value.

6.6. Electron Detection System

An electron detection system has been fitted to the exit side of the trap, and is

electrically isolated from the exit electrode (and to every other element on the system)

by ceramic balls that serve also as spacers. The electron detection system consists of a

Chevron pair of micro-channel plates (MCPs) that are biased so that the input side of the

62

first MCP is at +500 V with respect to ground and there is an 1800 V difference across the

MCP assembly. The electron cascades produced by the MCPs when electrons are incident

on the MCP assembly produce light when they impinge on a (P-22 Blue) phosphor screen

held in a metal cage structure biased at +2.8 kV with respect to ground. Activity occurring

on the phosphor screen is recorded using an ST-7XMEI SBIG CCD camera located outside

the vacuum chamber. Figure 6.5 (left panel) shows the phosphor screen when the vacuum

chamber is illuminated by an external light source and the image is recorded by the CCD

camera. Figure 6.5 (right panel) shows the emission from the phosphor screen as captured

by the CCD camera when the phosphor emission, due to the MCP electron cascade, is

the predominant source of light. The actual size of the phosphor screen is 1.91 cm, which

corresponds to the main circular shape in Fig. 6.5 (left panel) that covers ≈ 500 pu (pu =

pixel units).

6.7. Accepance and Transmission Without Electrostatic Plugging

The characteristics of electron beam transmission through the cylindrically symmetric

volume that has its interior lined with an ASB are now presented. The experimental results

obtained in this section show the dependence of a transmitted electron beam on input pa-

rameters. In this experiment, an electron beam was incident at the entrance to the trap,

and a portion of the beam was transmitted through the trap. The exiting electrons, and

their locations were detected and recorded by the MCP-phosphor screen and CCD camera

detection system. Relative beam intensity as a function of position could be deduced from

the magnitudes of the pixel values in the recorded images.

A 30 eV electron beam with a steady beam current of −63.5 nA was incident on the

entrance side of the structure. The electron beam focal point was varied by changing the

einzel lens voltage so that a diverging, focused, and over-focused beam was incident at the

entrance point cusp. Electron beam acceptance into the structure, as a function of einzel lens

voltage, is shown in Fig. 6.6. The ring electrodes were virtually grounded through a current

integrator and the ring magnets grounded through an ammeter. In Fig. 6.6, the normalized

integrated current (normalized with respect to the maximum value observed of 6 nC in 180 s

63

•••

•

•••

•

••

•••••••

••

´´

´

´

´´

´

´

´´´´´´´

´

IntegratedCurrent on Ring Electrodes H•LIntegratedTransmittedBeam Intensity H ´ L

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

1.2

V

Num

bero

fPar

ticles

HNorm

alize

dL

Charge on HUnbiasedL PluggingElectrodes

Figure 6.6. Electron beam acceptance into the trap as a function of einzel

lens voltage. Data points marked by dots are the normalized charge collected

on the unbiased plugging electrodes. Data points marked by crosses are the

time and space integrated relative beam intensities obtained by processing the

images recorded with the CCD camera. See text for details.

at −24 V einzel voltage) that reached the grounded ring electrodes is plotted as a function of

einzel lens voltage. The integrated current is directly proportional to the number of electrons

hitting the ring electrodes but is not equal to the total number of electrons collapsing onto

the electrode since secondary electron suppression was not employed. Secondary electron

emission can be suppressed by biasing the electrodes positive but electrons within the trap

would also be accelerated toward the ring electrodes. The secondary electron emission from

copper due to a 30 eV electron beam has been shown to be a constant fraction of the electron

current striking the copper surface. The effect that biasing the ring electrodes may have on

electron beam transmission is not considered here.

Also plotted in Fig. 6.6 is the time integrated beam intensity exiting the trap as

acquired by the CCD camera with an exposure time of 180 s. The image obtained from the

64

Figure 6.7. Spatial electron beam profile distribution as a function of focus-

ing at the entrance to the trap. The three dimensional shape that protrudes

from the x-y plane in the z direction is a plot of intensity I(in arbitrary units

(au)) as a function of position. The bands represent equal fractional intervals

of the peak intensity in each of the panels. A contour plot is also shown at the

top of each figure to illustrate the 2D beam profile. The data processed for

the plots shown are the pixel values that represent the images of the beam as

captured from the phosphor screen by the CCD camera. An example of such

an image is shown in the right panel of Fig. 6.5. The x and y coordinates are

in pixel size units (pu).

65

camera for each einzel setting was converted to a 2D array of values, where each value in the

array corresponds to a recorded value of a pixel in the image. The values of the array which

corresponded to a circular region containing the beam spot were summed to yield a single

value representing the integrated beam intensity. These values are normalized to the peak

value of the integrated intensity and are shown by a crosses in Fig. 6.6.

The number of electrons that reach the ring electrodes is observed to peak when the

einzel lens focusing voltage is −24 V. With this observation, and with the assumption that

the number of electrons reaching the ring electrodes is proportional to the number of electrons

that enter the trap, the maximum number of electrons that enter the trap occurs at an einzel

lens voltage of −24 V. The number of electrons incident on the electron detection system is

also observed to peak at the same einzel lens focusing voltage. However, notice that although

the plots in Fig. 6.2 and Fig. 6.6 peak at the same value of the einzel focusing voltage, Fig. 6.6

peaks more sharply. The decrease in intensity away from the maximum admitted number

of particles is attributed in part to a loss of unfocused electrons on apertures upstream of

the ring electrodes (same as the cause of the peak observed in 6.2), and additionally to a

portion of the electrons in the beam not meeting the phase space requirements to penetrate

the magnetic point cusp at the entrance to the trap (see Appendix C).

The results presented here pertain only to a 30 eV electron beam. There were two

reasons for choosing such an energy: (1) The electron gun can supply a steady current at this

energy; it is more stable when operating near the middle of its emission current and energy

range. (2) 30 eV electron trajectories within the trap have been predicted to be confined by

the magnetic field inside the trap [36]. For 30 eV electrons in this apparatus, the maximum

Larmor radius possible is 1 mm at the center of either point cusp and 250µm at the center

of a ring cusp, in the region between two magnets. An electron will be confined radially

within the trap if its Larmor radius at the center of the ring cusp is smaller than the spatial

period of the magnetic field by at least a factor of five, a condition that is clearly met by

30 eV electrons.

66

Information regarding electron beam characteristics upon exiting the trap was ob-

tained by processing the images acquired with the CCD camera; see an image example in

Fig.6.5 (right panel). The detection system is position sensitive to within a few MCP-channel

pores (diameter 25µm), and intensity sensitive to one part in 65536 (a digitized dynamic

range of 16 bits), where a pixel value of zero corresponds to a dark pixel and 65536 corre-

sponds to a bright saturated pixel. The pixel values are directly proportional to the number

of electrons incident on the MCP assembly at a given location. As a result, the transmitted

beam shape and intensity can be obtained as a function of electron beam parameters at the

entrance to the trap. The integrated beam intensity exiting the trap is shown in Fig. 6.6.

In Fig. 6.7, the relative spatial intensity distribution of the exiting beam is shown as

a function of einzel lens focusing voltage. The relative intensity is shown by the 3D shape

that protrudes from the x-y plane toward the positive z-axis. A cross-section of the beam

profile is shown by the contour plot included at the top of each figure panel. In each of these

figures, the intensity was normalized to the peak intensity of the profile measured for an einzel

voltage of −24 V, with electrostatic plugging turned off (ring electrodes grounded through

the current integrator, magnets grounded through ammeter, entrance and exit electrodes

directly connected to ground). To produce the plots shown in Fig. 6.7, the images obtained

from the camera were processed in the following manner: (1) The image data were changed

to a 2D array of values that corresponded to pixel values and then cropped to a square region

sufficiently large to contain the beam spot. (2) All pixel values were normalized by the peak

pixel value found for the image that was obtained when the einzel voltage was set to −24 V.

(3) A surface plot was generated for each of these 2D arrays. Figure 6.7 shows the electron

beam intensity profile, in arbitrary units (I (au)) as a function of position (in pixel units

(pu)) obtained by such a procedure. The four panels shown correspond to an over-focused

(einzel voltage = −26 V), optimally focused (einzel voltage = −24 V), and under-focused

(einzel voltage = −22 V and einzel voltage = −20 V) electron beam at the entrance to the

trap.

67

The information obtained from the integrated intensity is a proportional measure of

the current density of the electron beam exiting the trap because (1) The data for each of

the figures was obtained by setting an exposure time to 180 s and (2) the cross sectional

area of the transmitted beam is nearly constant and independent of focusing or defocusing

at the entrance to the trap. The maximum electron beam current density exiting the trap

occurs when the electron beam is optimally focused at the entrance to the trap. Focusing

or defocusing does not affect the size of the transmitted beam; only the transmitted beam

intensity is affected. To help show this, the full range of pixel values for each of the panels in

Fig. 6.7 was divided into ten intervals of equal magnitude, represented by alternating blue

and orange bands along the z direction in the figure. The highest point of each of these

shapes represents the most probable location where electrons exit the trap. The alternating

bands represent cylindrical bins, each of which has a 10% lower pixel value, as compared to

the maximum value, than the one above it. In the panel for einzel voltage of −24 V, the

value of the pixel at the top is unity (with all other values in Fig. 6.7 normalized with respect

to this value). The highest pixel value that corresponds to an einzel lens voltage of −22 V is

about half the pixel value that corresponds to the einzel lens voltage of −24 V. In this case,

the top most band covers a range of pixel values from 0.5 to 0.45, the next band from 0.45

to 0.40 and so on.

The contour plot at the top of each panel is a projection of the 3D figure onto the

top plane (top view) where the contours correspond to the inner and outer radii of the

corresponding bands in the 3D figure. These contour plots show that the beam size is

essentially constant with a beam diameter of ≈ 100 pu or 3.8 mm. The characteristics of

the beam incident at the entrance to the trap, as determined by the einzel setting, are not

present at the exit side. These observations lead to the conclusion that the beam profile is

governed by the shape of the fields present within the trap and by the shape of the magnetic

field at the exit point cusp and not by the ion-optical elements before the trap. The beam

waist size and cross-sectional profile are essentially constant, independent of focusing at

the entrance to the trap. However, the intensity of the electron beam observed exiting the

68

trap is dependent on the focusing at the entrance to the trap. The optimal electron beam

acceptance conditions manifest as the maximum electron beam current density at the exit

side of the trap. A similar behavior is expected for higher energy electrons, or ions, as long

as the fields are scaled accordingly.

6.8. Summary and Conclusion

A cylindrically symmetric system that employs an Artificially Structured Boundary

(ASB) to produce electro- magneto-static fields for charged particle confinement and control

has been presented. Electron beam transmission through the experimental apparatus con-

structed shows that electron trajectories through the trap are primarily controlled by the

magnetic field at the entrance to, inside of, and at the exit of the trap. The beam spot size

was observed to not change for a range of electron beam focusing conditions at the entrance

to the trap. For 30 eV electrons incident at the entrance to the trap, optimized transmission

manifests as a high intensity beam exiting the trap. A similar behavior is expected for other

ions or other energies as long as the fields are scaled appropriately. The results presented

show that this system could potentially be employed as a charged particle beam guide or ion

source.

69

CHAPTER 7

CONCLUSION

A system capable of modifying charged particle trajectories of either sign of charge

has been presented. Such a system consists of an ASB which generates periodic magnetic

field cusps that are plugged by electrostatic potential energy barriers. A segment of a planar

ASB was shown to reflect charged particles of either sign of charge when the appropriate

potential energy barrier was activated. A system that employs nested potential energy bar-

riers along the magnetic field cusp, where the magnetic field is nearly constant, has been

predicted to confine charged particles of either sign of charge simultaneously. A sequence of

current carrying wires, which alternate in the sign of current carried, could be employed to

generate the magnetic field cusps that could be electrostatically plugged by a nested config-

uration of potential energy barriers. The system experimentally studied employs permanent

magnets, which are stacked with like poles facing each other to generate a periodic sequence

of magnetic field cusps. Due to the shape and length of the magnetic field cusps generated

by such an arrangement of permanent magnets, only a single potential energy barrier can be

applied. Activation of the electrostatic potential energy barrier had a significant effect on

charged particle trajectories when these penetrated the ASB through a magnetic field cusp.

Neither the extent to which the activation of the potential energy barrier reflected charged

particle trajectories nor the actual behavior of charged particles in the region of where a

potential energy barrier overlaps with the magnetic field cusp could be determined from the

experimental system built.

A computer simulation was created to determine such behavior and to quantify the

capabilities of an ASB to reflect charged particle trajectories when a single potential energy

barrier is present along the magnetic field cusp. This simulation showed that a single species

of charge can be confined very effectively when the magnetic and electric fields generated have

moderate strengths as compared to the average energy of the charged particles in question.

Due to the short range of the fields generated by an ASB, it had been suggested that a

70

volume which has its interior surface composed of an ASB could provide a field-free region in

which charged particles or plasmas could be confined. Additionally, a single species plasma

would relax, due to self fields, within such volume. A study was carried out where the self-

consistent relaxation of an edge-confined plasma was computed assuming that such a plasma

follows a Boltzmann distribution. Deep potential energy wells are thereby predicted to occur

a the center of such an edge-confined and self-consistently relaxed plasma. An example of

an edge-confined plasma would be an electron plasma that relaxes within the a volume lined

by an ASB. Consequently, the potential energy well created by the space charge of the edge

confined plasma could be used to confine a plasma species which is oppositely charged. In

this manner, simultaneous confinement of charged particles of either sign of charge has been

predicted to occur.

An experimental system was built in which a cylindrically symmetric volume has

the interior surface lined by an ASB. Such a system has potential applications as an ion

source, ion beam guide, and as a charged particle trap. Results that pertain to electron

beam transmission were presented.

71

APPENDIX A

NORMALIZATION OF MAXWELLIAN DISTRIBUTION

72

Consider a Maxwellian source of charged particles with an associated temperature T ,

which has units of energy. The Maxwellian velocity distribution normalized to unity reads

(48) f(v) =( m

2πT

) 32e−

mv2

2T ,

where |v| = v, and v2 = v2x + v2

y + v2z . The Maxwellian distribution for each dimension is

separable and can be written as

(49) g(vi) =( m

2πT

) 12e−

mv2i2T ,

with i = x, y, z. The first moment (the average of a Cartesian velocity component) of the

Maxwellian distribution is of course zero, but the second moment is

(50) v2 =

∫v2f(v)d3v =

3

mT.

Equivalently,

(51)1

2mv2 =

3

2T,

which is the average kinetic energy of a particle in a system of particles that follows a

Maxwellian distribution. This is the amount of energy with respect to which the system is

normalized in Chapter 3.

Random variables with normal distributions can be used to construct an initial ran-

dom velocity vector from a Maxwellian distribution [13]. A distribution of the form

(52) f(Xi) =1√2πe−

X2i2

is adequately sampled by the random variable Xi

(53) Xi = (−2 log[R1i])1/2 cos(2πR2i),

73

where Rji are independent random numbers in the range (0, 1) sampled from a uniform

distribution [14]. From Eq. (49), define

(54) fn(vi) = g(vi)

√T

m=

1√2πe−

12 [√

mTvi]

2

so that, by inspection of Eqs. (53) and (52), the random components of velocity sampled

from a Maxwellian distribution are

(55) vi =

√T

m(−2 log[R1i])

1/2 cos(2πR2i).

Equation (55) is used to sample the initial conditions for the velocity vector in Sec. 3.2.

74

APPENDIX B

PRODUCT LOGARITHM

75

Here is given a derivation of the functional behavior of the electrostatic potential

well generated by an edge confined plasma. This approximation applies for systems that are

> 100 Debye lengths in size.

Spherical symmetry

Assume a plasma sphere with radius rp that is confined by a spherical boundary with

a radius rw. Next, the electric fields at rp and at rw are found for a spherically symmetric

system; see Fig. B.1. Use Gauss’s law:

(56)

∮E · dA =

Qencl

ε0,

where E is the electric field, A the surface area that encloses the region of interest and has

a normal vector pointing radially outward, ε0 the permittivity of free space, and Qencl the

enclosed charge. The total enclosed charge is

(57) Qencl =

∫ rp

0

qn(r)dΩr2dr = 4πq

∫ rp

0

n(r)r2dr.

Notice that for a system of 100 Debye lengths, the number density n(r) ≈ n0; see Sec. 4.2.

Making this approximation, the charge enclosed is approximately

(58) Qencl =4

3πqn0r

3p,

so that the electric field at the edge of the plasma becomes

(59) Ep4πr2p =

4

3ε0πqn0r

3p → Ep =

Ze

3ε0n0rp

where q = Ze has been substituted.

At the wall that confines the plasma, rw > rp, and the electric field there is obtained

via

(60)

∮Ew · dA =

Qencl

ε0,

76

(64) Qencl =

∫ rp

0

qn(r)dφdzrdr = 2πqL

∫ rp

0

n(r)rdr

where L is the length of the cylindrical boundary. For a system of 100 Debye lengths in size,

n(r) ≈ n0; therefore

(65) Qencl = Lπqn0r2p.

The electric field at the edge of the plasma becomes

(66) Ep =Ze

2ε0n0rp,

where q = Ze has been substituted.

At the wall where the plasma is confined, rw > rp, the electric field is obtained via

(67)

∮Ew · dA =

Qencl

ε0,

which yields

(68) Ew =Qencl

2πrwLε0=Zen0r

2p

2ε0rw

;

therefore,

(69) Ew = Ep

[rp

rw

]for cylindrical symmetry.

Planar symmetry

Assume a plasma planar sheet as described in Fig. B.1 (right panel). The plasma

sheet thickness is again assigned the varialbe rp and the location of the planar boundary is

at rw. Next, the electric fields at rp and at rw are found for such system:

(70)

∮E · dA =

Qencl

ε0, and

(71) Qencl =

∫ rp

0

qn(r)dr1dr2dr3 = qAn0rp

78

where A is the planar area considered. For a system of 100 Debye lengths n(r) ≈ n0, so that

the electric field at the edge of the plasma becomes

(72) Ep =Ze

ε0n0rp

where q = Ze has been substituted. At the wall where the plasma is confined, rw > rp, the

electric field there is

(73) Ew = Ep.

Electrostatic potential difference between the wall and the plasma center

Notice that the electric fields calculated for the three different geometries can be

written as

(74) Ew = Ep

[rp

rw

]α−1

where α = 1, 2, 3 for planar, cylindrical, spherical symmetry, respectively. Define ∆φ to be

the electrostatic potential difference between the wall and the center of the charge distribu-

tion. Using the electric field, the electrostatic potential can be calculated:

(75) ∆φ = −∫ rp

rw

Ewdr −∫ 0

rp

Epdr.

For the spherical geometry considered,

(76) ∆φ = Q3r2p

[3

2− 1

R

]where Q3 = Zen0

3ε0, and R = rp

rw. For the cylindrical geometry considered,

(77) ∆φ = Q2r2p

[1

2+ ln(R)

]where Q2 = Zen0

2ε0. For the planar system,

(78) ∆φ = Q1r2p

[R− 1

2

]where Q1 = Zen0

ε0. Defining βα as

79

βα =

R− 1

2: α = 1

ln(R) + 12

: α = 2

32− 1

R: α = 3

so that ∆φα applies to all three systems as

(79) ∆φα =Zen0

αε0r2

pβα.

Assume that n0 = nsexp[−ZeT

∆φ]. In such case, we can write

(80) ∆φα =Ze

αε0r2

pβαnse−ZeT

∆φα .

This equation can be normalized with Ze = q and q∆φT

= ∆ψ:

(81) ∆ψα =1

α

[q2nsTε0

]r2

pβαe−∆ψα .

The quantity in brackets is the squared inverse Debye length. Substituting the Debye length

and assuming that rp ≈ rw = rmax, which is the case when the system is ≥ 100 Debye

lengths, then βα ≈ 12

and

(82) ∆ψα =1

2α

[rmax

λD

]2

e−∆ψα ,

where the quantity in brackets is now the maximum, normalized, system size rn,max. Making

that substitution and re-arranging the equation, we obtain

(83) ∆ψαe−∆ψα =

r2n,max

2α.

This equation has the form of the product logarithm (Lambert) function: Ξ = χexp[χ] ↔χ = W(Ξ), so that

(84) ∆ψα = W

(r2n,max

2α

).

The normalized electrostatic potential well depth is a function of the system size and the

geometry in question. Equation (26) was obtained by subtracting the product logarithm

80

function W(Ξ) from the self-consistent evaluation of ∆ψα; the resulting values were fit-

ted with a non-linear regression algorithm which yielded the best fit to be of the form:

ln[(rn,max)0.217α0.139

].

81

APPENDIX C

TRAPPING FIELDS, PARTICLE BEHAVIOR, AND PLASMA BEHAVIOR

82

Trajectories that Escape Through a Cusp

Acceptance of charged particle trajectories is in part governed by direction or charged

particle trajectories relative to the direction the magnetic field, and by the relative magnetic

field strength between the point where the trajectory starts and the center of the magnetic

field cusp. T. J. Dolan [38] presented the following derivation as applied to charged particle

trajectories near the point cusps of a magnetic mirror configuration assuming that only

adiabatic processes take place. Derivation that is equally applicable to point, ring, and line

cusps.

Assuming that charged particles that are electro- magneto- statically trapped expe-

rience only adiabatic processes, then the energy conservation equation becomes:

(85) E = (1/2)mv2‖ + (1/2)mv2

⊥ + qφ = (1/2)mv2‖ + µB + qφ = constant

where q and m are the test particle’s charge and mass, respectively, µ = mv2⊥/2B is its

magnetic moment and φ is the electrostatic potential. The components of the velocity

vector, v‖ and v⊥, are defined with respect to the direction of the magnetic field line at the

instantaneous location of the particle. When the trajectory of the particle and the magnetic

field are co-linear, then:

(86) (1/2)mv2‖ = E − µB − qφ

which shows that the particle can be stopped either magnetically or electrically. In the

presence of a magnetic field only (i.e, φ = 0) the particle will momentarily stop if

(87) µB = E.

83

As initial conditions, suppose that the particle starts with a velocity v0 in a region of zero

electrostatic potential and magnetic field with magnitude B0; when that is the case, then

(88) E − (1/2)mv2‖0 = µB0 = (1/2)mv2

⊥0,

and

(89) µ = mv2⊥0/2B0

since

(90) E = (1/2)mv20.

Substituting Eq. (89) and Eq. (90) back into Eq. (87), and rearranging gives

(91) B/B0 = v20/v

2⊥0 = 1/sin2α0

where α0 is the included angle between the direction of the magnetic field and the velocity

vector v0 and defines the upper limit on the largest angle that will penetrate the magnetic

field cusp. This angle is typically called the pitch angle in the literature related to magnetic

mirrors. This pitch angle places a constraint on the charged particles that enter or exit the

charged particle trap through the point cusps located at the axial ends. The charged particle

trap is described in Chapter 6.

An additional force to consider is

(92) F = −µ∇B

which acts to accelerate particles, positive or negative, in the direction of decreasing B. This

force can be a significant contribution to the modification of charged particle trajectories

where the magnetic fields change abruptly. Systems in which the magnetic field lines are

everywhere convex toward the location of the confined plasma are known to have Magneto-

Hydro-Dynamic (MHD) stability [39].

84

Charged Particle Motion

Charged particle motion within the trap is expected to be complex, especially when

the trapped entity is described as a plasma [34]. A qualitative description of the motion and

the general behavior of single particles within the trap is now given; Fig. C.1 is included here

for reference. At the center of the trap, both electric and magnetic fields are essentially zero

and charged particle trajectories are thus unaffected; only cold particles can reside there.

Due to the electrostatic fields, the motion in the axial direction is very similar to the motion

of charged particles in an electrostatic bottle or a harmonic trap [40]. However, the confining

magnetic field is periodic and non-uniform. Charged particles that travel radially and are

incident on the on the surface of the magnet experience a v ×B force that guides charged

particles in or out of the plane of the page. Once such trajectories have a velocity in the

azimuthal direction, these experience a v × B force that guides charged particles to the

center of the trap. Additionally, a ∇B force (see Eq. (92)), especially near the boundary,

has the effect of confining particles to the center of the trap, or focusing them at the location

of the magnetic field cusps. This effect of the ∇B force is the same for either sign of

charge and has a direction that points away from increasing magnetic field magnitude. It

is expected that the majority of charged particles that do enter the cusps are reflected back

toward the center of the trap. In the cusp, near the plugging electrode, charged particles can

experience an E×B force. However, this effect is expected to be small since the electric and

magnetic fields at the center of the electrode have nearly identical directions (but not so at

the center of the magnet!). Such an E ×B drift can have many effects, some of which are:

(1) generation of a constantly rotating layer, (2) migration back into the confining volume,

(3) enough energy gain that causes the particle to escape confinement. The first two are

desired, especially for space-charge-based confinement. With respect to the third of these,

particles that enter the magnetic field cusps are nearly parallel to the cusp and would fall

in the high energy tail of a Maxwellian speed distribution (see Sec. 3.3), a condition that

presents itself as a possible diagnostic tool. The probability of a charged particle escaping

through any one cusp is relatively low and proportional to the ratio of the total area of the

85

Plugging Electrodes Magnets

Ring Cusp Point Cusp Point Cusp

Equipotential Contours

Figure C.1. Cross-sectional view of an charged particle trap consisting of

overlapping electric and magnetic multipole fields. Trapping expected to occur

when both fields are physically superimposed.

ring cusps to the total lateral surface area of the trap, correlated with the energy distribution

of the trapped constituents; see Chapter 3. The point cusp fields at either end of the trap

very closely resemble magnetic mirrors and charged particles are also confined axially by this

effect, especially, and more strongly, when the trap is biased in such a configuration as to

create electrostatic mirrors along the axis.

Plasma Behavior.

Notice that the description given thus far deals only with the single particle limit.

When the trapped bunch of particles is described as a plasma, first and foremost, trapping is

limited by the density and temperature of the plasma. This is predicted to be a dominating

factor in plasma confinement as presented in Chapter ?? for an edge confined plasma and

for space-charge-based confinement, and dictated by the Debye length. A plasma with low

temperature and high density meets the ideal qualities for edge confined plasma and space

charged based plasma confinement.

86

An upper limit on the temperature is dictated by the electrostatic-plugging-potential

energy barrier of the ASB; see Chapter 3. An upper limit on the density is given by the

Brillouin limit. Such temperature and density can be used to define an edge confined plasma

where the actual experiment is expected to have an order of magnitude less on either parame-

ter. The energetic tail of the charged particle distribution will most likely escape confinement,

and although not desired, such an effect provides a potential plasma diagnostic tool and self

temperature regulation via evaporation. An equilibrium is thus expected for a range of tem-

perature and density conditions once a suitable temperature is reached. Plasma heating is

expected at the edge of the plasma, in the region of strong magnetic fields. However, periodic

counter rotations, due to the periodicity of the magnetic field, may mitigate such effect.

87

APPENDIX D

CHARGED PARTICLE TRAPPING

88

The electronic circuitry developed to operate the cylindrically symmetric ASB as a

charged particle trap is presented. A typical trapping cycle consists of (1) a narrow pulse of

charged particles that is injected into the trap, upon injection, (2) the entrance and exit gate

electrodes are biased to reflect and confine the charged particles for a preset time period,

and (3) the exit electrode is grounded so that trapped charged particles exit the trap and

are detected. The details regarding the equipment employed, software developed, and fast

timing circuitry to drive charged particle trapping cycle is shown.

Equipment

The fast timing circuitry developed to drive a the laboratory equipment for trapping

cycle is shown in Fig. D.2. A virtual instrument developed in LabView assigns the gate/delay

time intervals by programming two LeCroy dual gate generators (DGG, model 2323A) so

that a total of four gates/delays can be created to generate the pulses necessary for trapping.

The DGGs can be programmed manually or via LabView-GPIB-LeCroy 1434 crate interface

with a timing resolution of tenths of µs to few s with the possibility of delaying the output

Camera

Einzel LensElectron Gun

Entrance Electrode Exit Electrode

MCP

Phosphor ScreenVacuumPermanent Ring Magnets

OFHC Copper Electrodes

EntranceFaraday Cup

Exit Cup

Figure D.1. Trapping system description

89

for 10, 30, 100, 300ns as compared to the triggering signal. The gate/delay signals generated

by the DGGs have rise and fall times of < 50nc for all outputs (with matched impedances).

Each of the pulses used to trigger the pulsed power supplies is now explained as each

one is traced through the diagram shown in Fig. D.2. The transistor-transistor logic (TTL)

output from the DGG that starts the cycle triggers the opto-coupling circuit (Vishay High

Speed Optocoupler P/N 6N137). This circuit is triggered when a TTL signal is present at the

input side. With the correct pulse shaping circuit, a TTL pulse is present at the output side.

The output side of this circuit is can be electrically floating and is powered via a 1:1 isolation

transformer. An HP3310B function generator is also powered by the isolation transformer

and floating at −24 V with respect to ground. The nuclear instrumentation module (NIM)

pulse generated by this same DGG module is used to synchronize the timing between the

pulsed electron beam entering the trap and the entrance and exit gate electrodes. The width

of W1 is programmable and adjustable to lead, coincide with, or lag the leading edge of W1..

When the second module of the first DGG is triggered, a TTL pulse passes through

pulse shaping amplifier to provide the correct input for a Cober high power pulse generator

(605P). The width of W4 is programmable; the height of W4 is adjusted via a front panel

knob. The Cober pulse generator can provide a variable pulse height from 50 V to 2.2 kV

positive or negative with typical fall and rise times of < 30ns. W3 is created by a TTL pulse

that triggers an Instrument Research Co. (model 80S) pulsed power supply. The width and

height of W3 are set via front panel knobs. The Ins. Res. Co. pulse generator can provide a

negative variable pulse from −150 V to −350 V with rise and fall times of < 20ns. Matching

loading impedances is very important so that the power supplies can provide pulses with the

characteristics given.

Einzel lens bias and pulsed electron beam

In this system, the einzel lens focuses charged particles at the entrance of the trap

and, by pulsing the biasing voltage for the middle einzel lens element, the electron beam

is also pulsed. The einzel lens drive system is now described; see Fig. D.3. An HP3310B

function generator is electrically floating and negatively biased to a voltage, Vf , by a Kikusui

90

Einzel lens

Exit electrode

Entrance electrode

Ins. Res. Co. 80S

S

NIM

TTL

S

TTL

LeCroy 1434Dual Gate Gen.

2323A (1)

S

DLY

S

TTL

Dual Gate Gen. 2323A (2)

Pulse Amplifier

CoberHPPG605P

W4

Opto-CouplingCircuit

start

PC LabView

Floating Func.Gen.

HP3310B

W2

W1 synchronization of pulsed electron beam and gating electrodes

W3

Figure D.2. Experimental equipment for trapping charged particles with the

trap presented.

Electronics Corp. power supply. Such voltage is the optimal voltage for focusing an electron

beam with the einzel lens when in decelerating mode. For example, a 30 eV electron beam is

optimally focused, in the current system, when the einzel middle element is biased to −24 V;

see Fig. D.3. The pulse width and height are adjustable via front panel knobs. The function

generator pulse height is added to the focusing voltage and has a value of 0 V when triggered

and −9 V otherwise. In this manner, the beam is switched from being optimally focused at

the entrance to the trap to being off. The rise and fall times of the pulse generated by the

HP3310B are < 30ns. The function generator requires a TTL pulse for triggering, a fact that

poses a problem. This issue was resolved by using an opto-coupling circuit that activates a

light emitting diode by the leading edge of a TTL pulse coming from a LeCroy delay and gate

generator (2323A). The floating optical-detector side provides the appropriately referenced

signal to trigger the function generator. The DC-offset in the HP3310B function generator

is set so that the baseline is at −9 V and 0 V when triggered. The einzel lens voltage is thus

−33 V at baseline and −24 V when triggered, thereby achieving a pulsed and focused 30 eV

91

Trap

Einzel Lens Bias for Beam Pulsing.

Func. Gen. HP.

Vcc

Trigger

Vf

(W,H)

W

H

TTL

Entrance

Faraday CupEinzel LensElectron Gun

Gate/DelayGen.

LeCroy2323A

Opto-CouplingCircuit

Figure D.3. Einzel lens focusing and pulsed electron beam.

Pulse Sequence for One Trapping Cycle.

W1

W2

W3

delay

Wait

Load Dump

delay

Trapping Cylcle

Ins. Res. Co.

Cober

Floating Func. Gen. HP.

LabViewStart

Electron BeamON/OFF

Exit Gate

Entrance Gate W4

Time

Figure D.4. Time line of events to drive the trapping cycle.

electron beam at the entrance to the trap. A similar scheme is expected for other charged

particles and other energies.

92

Fast timing circuitry

Figure D.4 represents the event time line during a typical trapping cycle. In this

diagram, spikes are trigger signals to start the consequent delay or pulse. The horizontal

dashed lines indicate variable time intervals. Vertical dashed lines indicate the time at

which other events are triggered or stages during the trapping cycle. A driver was coded in

LabView which provides the initial trigger and repetition rate for the number of trapping

cycles desired. The initial trigger starts the first pulse, W1. The leading edge of W1 triggers

the electron beam ON. The delay between the trigger signal and electron beam pulse is

inherent to the circuitry necessary to switch the beam ON and OFF; see Fig. D.4. The

trailing edge of W1 triggers the biasing of the exit electrode to repel electrons. The leading

edge of W2 and W3 can be made to temporally coincide by varying the width of W1. The

flexibility is such that the leading edge of W3 can come before or after the leading edge

of W2. The correct overlap of W2 and W3 is dictated by the absence of signal at the

MCP/Phosphor screen electron detector. The trailing edge of W1 also triggers the biasing

of the entrance electrode to repel electrons. The delay between the trigger signal and the

leading edge of W4 is necessary to allow electrons to enter the trap and essentially defines the

time during which the trap is loaded with electrons. The signal transit time which produces

W4 is significantly longer than the signal transit time to generate W3. For this reason, W3

is delayed so that it nearly coincides in time with W4. Once W4 starts, the entrance to the

trap is blocked and no electrons enter the trap. The temporal overlap beteween W3 and W4

define the time during which charged particle trapping occurs. W2 is turned off after the

loading occurs but sufficiently long before the trailing edge of W4. The trailing edge of W3

marks the end of the trapping time, at which point charged particles can exit the trap and

be incident on the MCP/Phosphor screen detector. W4 is made sufficiently long to ensure

trap evacuation and maximize the events at the MCPs. During the dump time interval,

electrons are repelled from the entrance electrode and attracted by the fields present at the

exit due to the biasing of the MCPs. It is necessary that W4 stays ON for the remainder

of the trapping cycle but the duty cycle of the pulsed power supply limit this possibility.

93

Figure D.5. Front-end panel of virtual instrument (VI).

Additionally, the cycle repetition rate cannot be faster than the luminescence decay time of

the phosphor screen (≈ 25µs). In this manner, it is ensured that the source of electrons is

not ion-optically visible to the detector at any time; only electrons that could have become

trapped can be incident on the detection system.

Virtual instrument system driver

The basic components of the virtual instrument (VI) developed for the trapping cycle

are the front panel (Fig. D.5), block diagram of cycle driver (Fig. D.6), and block diagram to

program anyone DGG (Fig. D.7). In the front panel, the values of the delays and gates can

be adjusted by the user to change the pulse width or delay for the trapping cycle. The values

shown are those that would allow 1000 repetitions of a load-wait-dump cycle as specified in

the previous sections of this appendix.

94

Figure D.6. VI code segment that generates the start signal, cycle repetition

rate, and number of cycles.

95

Figure D.7. Typical VI code to program the anyone of the two modules in

a LeCroy 2323A DGG.

96

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