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This is to certify that the
dissertation entitled

X-RAY IMAGING OF SUPERCONDUCTING RADIO
FREQUENCY CAVITIES

presented by

SUSAN ELIZABETH MUSSER

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Physics

 

 

 

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/ M‘ajor Professor’s Signature

MSU is an Affirmative Action/EquaI Opportunity Institution

 

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X-RAY IMAGING OF SUPERCONDUCTING RADIO FREQUENCY CAVITIES

By

Susan Elizabeth i\=‘lusser

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

2006

ABSTRACT
X—RAY IMAGING OF SUPERCONDUCTING RADIO FREQUENCY CAVITIES
By

Susan Elizabeth Musser

The goal of this research was to develop an improved diagnostic technique to
identify the location of defects that limit superconducting radio frequency (SRF)
cavity performance during cavity testing or in existing accelerators.

SRF cavities are primarily constructed of niobium. Electrons within the metal of
a cavity under high electric field gradient have a probability of tunneling through the
potential barrier, zle. leave the surface or are field emitted in regions where defects
are encountered. Field emitted electrons are accelerated in the electric fields within
the cavity. The electrons can have complicated trajectories and strike the cavity walls
thus producing x—rays via Coulomb interactions and/or bremsstrahlung radiation.
The endpoint energy of an x-ray spectrum predicts the electron maximum final kinetic
energy within the cavity. Field emission simulations can then predict the source of
the field—emitted electrons and the defect(s).

In a multicell cavity the cells are coupled together and act as a set of coupled
oscillators. There are multiple passbands of excitation for a multicell structure 0p-
erating in a particular mode. For different passbands of operation the direction and
amplitude of the fields within a cavity change from that of the normal accelerating
mode. Field emitted electrons have different trajectories depending on the mode and
thus produce x—rays in different locations.

Using a collimated sodium iodide detector and subjecting a cavity to multiple
passband modes at high electric field gradient the source of a cavity’s x-rays can be
determined. Knowing the location of the x—rays and the maximum electron kinetic

energy, field emission simulations for different passband modes can be used to deter-

mine and verify the source of the field emitted electrons from mode to mode. Once
identified, the defect(s) can be repaired or modifications made to the n'ianufacturing

process.

Copyright by
SUSAN ELIZABETH MUSSER
2006

iv

W

To Alex, De Anna, and Dustin Clark for encouraging me to follow my dreams too.

M

I thank god every day that all of my prayers have not been answered.

 

ACKNOWLEDGMENTS

First, I thank my advisor Terry Grimm for his perseverance as I navigated the
Ph.D. program. I thank him for taking a chance in me, a non—traditional student.
When I joined the accelerator group, my goal was to gain experience that I could
apply to the field of medical physics. The inspiration for this study came from him,
and the work in x—ray imaging perpetuated my interest in medical physics.

Secondly, I thank Walter Hartung for his inexhaustible patience and support. He
generously gave his time to discuss tOpics with me, most often multiple times. With
his unselfish guidance, particularly in computer program revisions, I am completing
this degree. It has been an honor to work with him. He is a true gentleman and
scholar.

Next, the members of my dissertation committee, S. D. Mahanti, Felix Marti,
Stan Schriber, and Kirsten Tollefson, I thank for patiently awaiting my completion.

Only I have read this dissertation more times than Renate Snider. I thank her over
and over for the numerous critiques of my writings. She not only made the writing
group an exhaustive punctuation and grammar lesson, but fun. I also thank her for
the incorporeal box of hyphens and commas for my birthday.

I thank John Bierwagen for his input in the design of the x-ray imaging device.
His initiation in constructing the device gave me the boost of confidence I needed to
complete it. I also thank James Colthorp and Ahmad Aizaz for their help with the
data acquisition system.

I could not have made it through this program without the support of friends
Elaine Kwan and Chandana Sumithrarachchi. I am thankful for our many insightful
conversations in the use of ORIGIN and Latex, as well as their tolerance of my
eccentricities. They are true friends. I also thank friends Mark Wallace, for guiding
me to the accelerator group, and Carla Gates, for encouraging me to attend Michigan

State Universary.

vi

Debbie Simmons was the physics undergraduate secretary when I was an under-
graduate student. Around the time that she became the physics graduate secretary,
I became a graduate student. I hope she never has to tolerate another student as
long she has tolerated me. I thank her for the semesterly enrollment reminders, the
course overrides, the required signatures, the answers to numerous questions about
everything.

I thank Doug Harris, Allyn McCartney, Kurt Niemeyer, Dan Scott, Mark Stod—
dard, Shelly Villarreal, and Jim Vincent for teaching me their interpretation of Hoyle’s
rules of euchre. These lessons and memories I will cherish forever.

A special thank you goes to the Michigan State University Intercollegiate Figure
Skating Team for accepting me, the oldest figure skater in the country, as a teammate
and friend. The comradery and joy of skating I experienced with this group was not
only instrumental in retaing my sanity but made my collegiate experience complete.
For those that were of legal drinking age, thanks for including me.

Finally, I thank Alex, De Anna, and Dustin Clark, my children. They cannot
remember a time when I was not engaged in some type of coursework. They have
witnessed the anxiety, apprehension, and frustration that I experienced. The greatest
pleasure of my life is supporting and encouraging my children as they pursue their
dreams. I have recently come to the realization, that at the same time, they have

supported and encouraged me to pursue mine. For this I am forever grateful.

vii

Contents

1 Introduction 1
2 Background 5
2.1 SRF Cavity Operation .......................... 7
2.1.1 Electric Fields ........................... 7

2.1.2 Field Emission .......................... 18

2.1.3 X-rays ............................... 26

2.1.4 Photon Interactions with lV’Iatter ................ 33

2.2 X-ray Detection .............................. 46
2.2.1 Detector .............................. 46

2.3 SRF Cavity Testing ............................ 53
2.3.1 Modulated Wave ......................... 54

2.3.2 Continuous Wave ......................... 56

3 Design 58
3.1 Experimental Setup ............................ 58
3.1.1 Cavity and Cryostat ....................... 58

3.1.2 Detector and Collimator ..................... 62

4 Experimental Results 68
4.1 Detector Calibration ........................... 68
4.1.1 Energy ............................... 68

4.1.2 Spatial Resolution ........................ 69

4.1.3 Spectrum ............................. 71

4.1.4 Detector and Collimator Performance .............. 73

4.2 Cavity Testing ............................... 77
4.2.1 Cavity #2 Performance ..................... 77

4.2.2 Cavity #2 7r Mode ........................ 79

4.2.3 Cavity #2 %Z£ Mode ....................... 86

4.2.4 Cavity #2 %71 Mode ....................... 90

4.2.5 Cavity #1 Performance ..................... 93

4.2.6 Cavity #1 7r Mode ........................ 95

4.2.7 Cavity #1 %’l Mode ....................... 97

viii

Simulations

5.1 SUPERLAN S ........................
5.2 Multipacting/ Field Emission Simulatirm .........
5.2.1 Program Operation .................
5.2.2 Input Parameters ..................
5.2.3 Code Output ....................
Comparison Between Data and Simulations
6.1 Cavity 2 ..........................
6.1.1 6: Mode .......................
6.1.2 7r Mode .......................
6.1.3 #3: Mode .......................
6.2 Additional Field Emitters Cavity #2 ...........
6.3 Cavity #1 ..........................
6.3.1 7r Mode .......................
6.3.2 56?! Mode .......................
Discussion and Conclusion
7.1 Summary ..........................
7.2 Previous Work .......................
7.3 Future Work .........................
Electromagnetic Fields
A.1 Fields Within a Rectangular Waveguide .........
A.2 Fields Within 3. Circular Waveguide ............
A.3 Fields Within a Cavity ...................
Field Emission
8.1 Transmission Coefficient D(W) ..............
B.2 Number of Electrons P(W) ................
8.3 Current Density j ......................
34 RF Current (I) .......................

Photon Interactions

C .1 Compton Effect .......................

Definitions

Floquet Theorem

E.1 Derivation ..........................

Data and Endpoint Energy

F .1 Cavity #2 7r Mode .....................
F.2 Cavity #2 %71 Mode ....................

F.3 Cavity #2 4671 Mode ....................
F.4 Cavity #1 7r Mode .....................
F.5 Cavity #1 fig: Mode ....................

ix

nnnnnnn

0000000

.......

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0000000

.......

101
101
104
107
107
115

128
128
128
131
133
136
140
140
142

146
146
148
149

150
150
155
158

161
161
166
168
170

172
172

174

177
177

183
183

....... 184
....... 186

188

....... 189

Bibliography 192

List of Figures

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21

Rectangular Waveguide .......................... 10
Circular Waveguide ............................ 11
Electric and Magnetic Fields in a Cylindrical Cavity. ......... 15
Single-Cell Cavity ............................. 16
Six-Cell Cavity .............................. 17
Surface Electric and Magnetic Fields .................. 18
Q and X-rays ............................... 19
Discharge Tube .............................. 19
Potential Barrier ............................. 21
Image Force ................................ 22
Bertha Roentgen’s Hand ......................... 27
Bremsstrahlung Radiation ........................ 30
Kramers Approximation ......................... 31
X-ray Angular Distribution ....................... 32
Compton Scattering ........................... 34
Compton Cross Section .......................... 39
Photoelectric Process ........................... 40
Photoelectric Cross Section ....................... 41
Pair Production .............................. 42
Pair Production Cross Section ...................... 43
Photon Interaction ............................ 44

xi

2.22
2.23
2.24
2.25
2.26
2.27
2.28

3.1
3.2
3.3
3.4
3.5
3.6
3.7

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13

Mass Attenuation ............................. 45

Energy Bands ............................... 47
Scintillator Detector ........................... 48
Collimator ................................. 51
Slit Spatial Resolution .......................... 52
Gamma Camera .............................. 53
Quality Factor Calculation ........................ 55
Cryostat End View ............................ 60
Cryostat Side View ............................ 61
Device Overhead View ......................... . . 63
Device Front, Back Views ........................ 64
Device Photograph ............................ 65
Wiring Diagram .............................. 66
Collimator ................................. 67
Detector Calibration ........................... 69
Spatial Resolution ............................. 70
X-ray Energy Spectrum ......................... 72
Test Set-up ................................ 74
Diagram of a Cryostat Scan ....................... 76
Cavity #2, 7r Mode, Q and X-rays versus Electric Field ........ 78

Cavity #2, 7r mode, Fowler-Nordheim Field Enhancement Factor . . . 78

Cavity #2, 7r Mode, Data ........................ 80
Cavity #2, 7r Mode, Energy Binning .................. 81
High Energy Binning ........................... 82

Cavity #2, 7r Mode, Endpoint Energy Linear Projection During Scan 84
Cavity #2, 7r Mode, Endpoint Energy Linear Projection Stationary . 85
Cavity #2, %75 Mode, Data ........................ 87

xii

4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
4.23
4.24
4.25

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9

5.11
5.12
5.13
5.14
5.15

Cavity #23— 5” Mode Energy Binning .................. 89
Cavity #2 Spec tra Comparison of 7r and' —6— \Iodes .......... 89
Cavity #2,36— 7er',Iode Data ........................ 91
Cavity #136— 7rIVIode, Energy Binning .................. 92
Cavity #2 Spectra Comparison of 7r, 56—, and4 —6- Modes ....... 92
Cavity #1, 7r Mode, Q and X-rays versus Electric Field ........ 94
Cavity #1, 7r Mode, Fowler—Nordheim Field Enhancement Factor . . 94
Cavity #1,7r Mode, Data ......................... 96
Cavity #1, 7r Mode, Energy Binning .................. 97
Cavity #1,%— 7rIVIode Data ........................ 99
Cavity #1,'—6- 7TMode, Energy Binning .................. 100
Cavity #1 Spectra Comparison of 7r and‘) :6. Modes .......... 100
Cavity Mesh ................................ 102
SUPERLANS Output Symmetric and Antisymmetric Wave Functions 102

Electric Fields On Axis Six-Cell Cavity ................. 104
Surface Electric Fields .......................... 105
Different Emitter Locations ....................... 109
Different Peak Surface Electric Fields .................. 111
1r Mode Different Cell Same S0, and :tSO ................ 112
—6- Mode, Different Cell Same S0, and :tSO ............... 113
Different Mode Same SO ......................... 114
Electron Trajectories and Power Spectrum ............... 116
Simulated Power Spectra, Different Number of Field Emitted Electrons 117
Simulated Power Spectra for Different Emitter Locations ....... 118
Simulated Power Spectra for Doubling Ep ............... 120
7r Mode Simulated Power Spectra for Different Cells and iSO . . . . 121
—6— Mode Simulated Power Spectra for Different Cells and iSO.122

xiii

5.16
5.17

5.19
5.20

6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9

A.1
A.2

El

E1
E2
E3

E1
F2
F3
F.4
F.5

7r, i6“ a11d4—6— Modes Simulated Power Spect1a Same S0 ....... 123
Power Spectrum with Increased 6 F N .................. 124
Similar Simulated Power Spectra .................... 125
Binned Simulated Power Spectra .................... 126
X-ray to Power Spectrum ........................ 127
Cavity #2,%— ”Mode, Data and Simulated Power Spectrum ..... 130
Cavity #2, 7r Mode, Data and Simulated Power Spectrum ...... 132
Cavity #2 A8 ............................... 133
Cavity #2,—6— 7rMode, Data and Simulated Power Spectrum ...... 135
Cavity #2, 3 Modes, SO 2 13.266 cm .................. 137
Cavity #2, 3 Modes, $0 = 13.5 cm ................... 138
Cavity #1, 7r Mode, Data and Simulated Power Spectrum ...... 141
Cavity #1 Ag ............................... 142
Cavity #1,§6— 7rIVIode Data and Simulated Power Spectrum ...... 144
Rectangular Waveguide .......................... 154
Fields in a Cylindrical Cavity ...................... 160
Potential Barrier ............................. 164
Periodic Structure ............................. 179
SUPERLANS Output .......................... 181
Symmetric and Antisymmetric Wave Output .............. 182
Cavity #2 7r Mode Data ......................... 184
Cavity #2§6_ 7TMode Data ........................ 185
Cavity #2516— 7rMode Data ........................ 187
Cavity #1 7r Mode Data ......................... 189
Cavity #186—7TIVIode Data ........................ 191

xiv

List of Tables

3.1

4.1
4.2
4.3

6.1
6.2
6.3
6.4
6.5
6.6

E1
F2
F3
F4

Absorber Materials ..........

Detector Calibration .........
Absorber Materials ..........

Characteristic X-rays .........

Cavity #2, 5675 Mode .........
Cavity #2, 7r Mode ..........
Cavity #2, #14 .............
Cavity #2, E675, SO 2 13.266 cm . . .
Cavity #1, 7r Mode ..........
Cavity #1, %75 Mode .........

Cavity #2, 7r Mode Endpoint Energy
Cavity #2, 56: Mode Endpoint Energy
Cavity #2, é67E Mode Endpoint Energy
Cavity #1, 7r Mode Endpoint Energy

Cavity #1, 56L Mode Endpoint Energy

XV

.................. 73

.................. 131
.................. 133
.................. 136
.................. 139
.................. 142
.................. 145

Chapter 1

Introduction

The goal of this research was to develop an improved x-ray imaging technique for
locating field emission sources in superconducting radio frequency (SRF) cavities.
Unlike other techniques, the present technique is unique in that it can be used after
the cavities are installed in an operating accelerator. Once defects are identified,
cavities can be subjected to high power pulsed processing or helium processing to
suppress field emission from an emitter source. For cavities being manufactured, x-
ray imaging could be used to locate defects resulting from the manufacturing process
such as improper clean room procedures or die defects. ApprOpriate steps could then
be taken to remedy the cause of the defects [1].

Cavities are tested and operated at high electric field gradients. In the region
of an inner surface defect the electric field is enhanced, causing electrons to leave
the surface of the metal. These field-emitted electrons are accelerated in the cavity
electric field, strike the cavity walls, dissipate power and create heat, and generate
bremsstrahlung x—rays. The maximum energy of the x—rays is equal to the maximum
energy of the impacting electrons (Duane and Hunt’s Law) [2], which is needed for
the particle tracking simulation to determine the l<;)cation of defects.

The x-rays travel through the cryostat materials such as magnetic shielding, ther-

mal shielding, 2 cm thick steel, and through detector shielding of 6-10 cm thick lead

and collimator, before entering the scintillator material of the detector. For these
materials and observed x-ray energies, Compton scattering is the most probable in-
teraction. During a Compton event, an x-ray interacts with an electron and both
the electron and x-ray are scattered, the x-ray to a lower energy and the electron
only traveling a short distance. The probability of a Compton event depends upon
the x-ray energy and the Z of the material through which it is traveling. There is
a small probability that an x-ray will have no interactions prior to entering the de—
tector [2]. Sodium iodide was chosen for the detector scintillator since it provides
adequate energy resolution and good efficiency [1].

A particle tracking code was used to infer the locations of the electron emitters
and the emitter parameters. The code provides information about the electron impact
site such as electron final kinetic energy, current, and power, which were fit to the
data.

Quality factor (Q) is a measure of cavity performance and is inversely proportional
to dissipated power. As cavities are pushed to high electric field gradients they expe-
rience a decline in Q. High electric field power loss mechanisms include field emission,
thermal breakdown, and exceeding the critical magnetic field. All radio frequency (rf)
cavities exhibit a surface resistance due to the nature of superconductivity. This small
resistance causes a cavity to heat at high electric field without the x-rays produced
during field emission being observed. All can lead to a quench of the cavity. Low field
power loss mechanisms include flux trapping, adsorbed gases, and resistive particles.
Of these, field emission is the chief degrader of cavity performance [3].

If the decline of Q is due to field emission, the general area radiation monitoring
equipment registers an x—ray exposure rate of the bremsstrahlung-generated x-rays.
X-ray energies cannot be determined based on the exposure rate and since the x—rays
enter the detector from all directions, the source location, and the possibility of mul-
tiple sources, cannot be determined. Although a good indicator of field emission, this

equipment cannot be used in conjunction with a particle tracking code to determine

defect sites.

A second technique for locating x—ray sources uses resistors arranged 011 the outer
cavity wall to locate an increase in temperature where electrons impact. With the
impact locations known, electron energies are calculated based on temperature, and
particle-tracking codes determine the source of the electrons, i.e., the locations of
defects. Verification of the defect and impact sites requires a cavity to be disassembled
and inspected under special conditions to prevent contamination [4]. This method
has proven very successful for cavity testing but is capital and labor-intensive, and
impractical for operating accelerators where cavities, housed in cryostats, are not
accessible.

The technique developed here identifies x—ray sources using a slit collimated sodium
iodide detector moving continuously along the cavity, outside the cryostat, while cav-
ities are tested. The collimator and detector housing are made of lead to attenuate
x-rays other than those entering the detector through the collimator. The sodium
iodide detector measures the energy spectrum of x-rays. Slit collimation makes it
possible to determine x-ray sources to within the slit resolution.

Particles are accelerated in cavities operated in the 7r mode, where successive cells
are 7r radians out of phase with each other. Operating a multicell cavity in a different
mode of excitation, such as the 577 mode for a six cell cavity, changes the electric fields
within the cavity. A particle emitted from a single defect has a different trajectory
and impact site from that of the 7r mode [4]. Thus x-ray imaging and the resulting
particle tracking of electrons from a cavity operated in another mode is used as a
verification of a defect site without having to cannibalize the cavity.

Operating accelerator cavities can develop defects from contaminants entering the
cavity through vacuum leaks or venting for maintenance. Once defects are identified,
cavities can be subjected to high power pulsed processing or helium processing to
suppress field emission from an emitter source. High power pulsed processing is ac-

complished by raising the electric field as high as possible for a very short duration.

The high current destroys the emitter in an explosive event. Helium processing in-
troduces helium gas into the cavity, where the field-emitted electrons ionize the gas.
Since they are of opposite charge, ions bombard the emitter at the same time electrons
are being emitted. Pressure builds in the area of the defect until it explodes [4].

During cavity manufacturing, if a defect or dimple develops in the die used to
form the cavities, x—ray imaging provides a simple technique for locating the defect
so necessary repairs can be made. X-ray imaging can also be used to locate defects
resulting from the manufacturing process such as improper clean room procedures
that lead to cavity contamination. High power pulsed processing or helium processing
can then be used to eliminate the defect. This method of x—ray imaging is the first
diagnostic technique for locating cavity defects from outside of a cryostat.

This work can be expanded upon with the accumulation of data in two dimen-
sions with a device similar to the one used in this study or with a gamma camera.
Multiple images acquired from multiple angles by using a gamma camera could be
reconstructed to yield a three dimensional image similar to PET or SPECT imaging.

Chapter 2 covers the mathematical background of cavity fields, field emission,
x-rays, and briefly discusses detector operation and the function of a collimator. In
Chapter 3 the experimental designs of the cavity, cryostat, detector, and collimator
are presented. Chapter 4 offers the experimental results of the data accumulated
during testing of two cavities in a cryostat. Chapter 5 covers the particle tracking
code used for this study. Chapter 6 compares the data and the simulated results.

Finally, Chapter 7 gives a summary of the study.

Chapter 2

Background

The attractiveness of superconducting over normal conducting cavities is that much
less power is dissipated in the cavity walls. The surface resistance of a superconductor
is on the order of 105 times lower than that of a normal conductor. This equates to
a quality factor (Q) on the order of 105-106 times higher (Equation 2.1).

Current flows in the cavity walls at the surface thus supporting the electromag—
netic fields in the cavity. As current flows through a defect, Joule heating heats the
surrounding area. If the temperature rises above a critical temperature (Tc) for the
material, the area becomes normal conducting, followed by a quench. Quench is the
sudden drop in a cavities Q due to the rapid increase in the power dissipation at the

defect:

wU

c2 = E’ (2.1)

where w is the angular frequency, U the stored energy, and Pd the dissipated power [4].

The DC resistance of a superconductor is zero below a critical temperature (Tc)
for the material. The rf resistance though, is on the scale of 119 for high purity
superconductors. According to a theory developed by Bardeen, Cooper, and Schrieffer
(BCS) in 1957, electrons in superconductors form Cooper pairs. Electrons in the pair
are of opposite momentum and spin. At T = 0 K all electrons are paired. At higher

temperatures there exists a finite probability of pair splitting due to thermal excitation

(two fluid model). The Cooper pairs and free electrons can collide with lattice defects,
impurities, or phonons [4].

In the DC case, collisions are of insufficient energy to scatter Cooper pairs out
of the ground state, so the pairs flow frictionlessly, i.e. without resistance. The free
electrons can be scattered and dissipate energy but the Cooper pairs shield the applied
field from the free electrons. The result is that electrical resistance vanishes [4].

In the rf case, Cooper pairs are again not scattered but do possess inertia and do
not shield the applied field completely. The free electrons of the fluid are continually
accelerated and decelerated leading to power dissipation, i.e. surface resistance. The
residual resistance of a superconductor depends in part upon impurities and micro—
scopic particles that scatter the free electrons, trapped flux localized at impurities,
called pinning centers, making the superconductor unable to expel the magnetic field,
and adsorbed gases such as N2 and 02 that possess dielectric properties that increase

the residual resistance. The rf surface resistance is:

2 _
R3 = A (if) e151) (~%D) + R0, (2.2)

where A is a material dependent constant, f is the frequency, k is the Boltzmann
constant, T is the temperature, 2A is the energy needed to break the Cooper pairing,
and R0 is the residual resistance [4]. Due to this surface resistance, all cavitites exhibit
power losses that limits cavity performance. The losses are present throughout the
entire cavity. With this as the only power loss mechanism present, the Q drops at
high electric fields (Equation 2.1) , but no x—rays associated with field emission are
detected.

The reduced resistance of SRF cavities, and thus power consumption, translates
to reduced operating costs even when the cost of refrigeration is considered. Higher
electric field gradients are attained due to lower power consumption. Lower power

consumption also means fewer klystrons are needed to power a bank of cavities.

Most importantly, higher currents can be accelerated due to the larger apertures that

minimize intrusive interactions of the cavity with the beam [3].

2.1 SRF Cavity Operation

2. 1. 1 Electric Fields

For this study the electric fields within a cavity need to be known in order to track the
trajectories of field emitted electrons from their impact sites back to their origin. To
explain the fields within a cavity one first begins with the fields within a waveguide. A
waveguide is a structure that guides a wave in a given direction in planes perpendicular
to that direction. In accelerator applications, rectangular and circular waveguides are
commonly used for high power transmissions between power sources and accelerating
structures. A circular waveguide structure with conducting end surfaces is called a
pill-box cavity [5]. Appropriate boundary conditions are then applied to the circular
waveguide to obtain the fields within a cylindrical or pill box cavity [5]. To allow a
beam through the cavity, beam pipes are added to both ends and cavity shapes are
generally elliptical due to multipacting issues. The addition of beam pipes increases
the peak surface electric field and peak surface magnetic field, thus increasing the
probability of field emission. Multipacting occurs when electrons that leave the cavity
wall impact and produce secondary electrons until an avalanche of electrons exist. Due
to the absorbtion of If power, cavity fields cannot be raised. This phenomenon can
take place in regions along the cavity wall where the surface magnetic field is nearly
constant. When cavity walls are rounded, the surface magnetic field varies along the
entire wall, thus minimizing multipacting.

The addition of these features makes it very difficult to calculate the fields ana-
lytically. The computer code SUPERLANS was used for numerical analyses in this
study [4].

Waveguide

The following derivation for the fields within a cylindrial cavity begins with the fields
within a rectangular waveguide, whose solutions are transformed to cylindrical coor-
dinates for the fields within a circular waveguide.

The derivation of fields within a rectangular waveguide begins with lV'IaxweIl’s

equations:
(i)v.E=§ (Ii)VxE=—%Iti _.
(iii)V-B=O (iv)vXB=IIJ+Zlg%§.

where p is electric charge density, 6 the permitivity, and ,u the permeability. In a

region free of charge, or free of current, Maxwell’s equations become:

(i)V-E=0 (ii)VxE=—%1ti
(iii) V-B :0 (iv) V x B 232%?

The fields within a rectangular waveguide are given dependent upon E2 and Bz

(Appendix A.1):

 

 

 

Ey = [22: (aha—8% — 1.100%) , (2.3)
Bx _ k—‘g (41.3%?— Z —:’2%i) (2.4)
Ex 2 If? (ikziiz +50%), (2.5)
By = [22 (ikrzgt)? — 632%) . (2.6)
(501; + :7: + £23 — 13) E3 = 0, (2.7)
(52+5‘Z/2) E..- 1§Ez_v2TEz, (28)

82 82 2
(— 2 +_ + 1— k2) Bz— —- 0, (2'9)

811: ()y2 c2
a2 a?
(07+ 0y, —) Bz = 423,. = v3.32, (2.10)

where kz is the wave number in thez direction (kg = g;), u) the angular frequency

(w = kc), kc the cutoff wave number (kc = V132 — kg), and VT: 52—? + {—9632 where
T implies the tranverse plane. For k > k2 propagation occurs, but for k < kz the
modes are called cutoff, or evanescent and decay exponentially along the waveguide.
The wave solutions are classified into the following types:
1) Transverse electromagnetic waves (TEM) if both E z and B z are zero. That is to say
the waves contain neither electric nor magnetic fields in the direction of propagation.
2) Transverse magnetic waves (TM) if Bz is zero. That is to say waves where the
magnetic field lies entirely in transverse planes and the electric field is oriented in the
direction of prOpagation.
3) Transverse electric waves (TE) if E z is zero. That is to say waves where the electric
field lies entirely in transverse planes and the magnetic field is in the direction of
propagation.
Of interest here is the derivation of the TM mode since it is the accelerating mode,
but a similar derivation could be carried out for the TE mode.
Boundary conditions dictate that Ez(.r = 0, a) = 0, Ez(y = 0, b) = 0, and

Ez(;r = %,y = g) 2 E0 (Figure 2.1):

m” sin— "7‘” 8i“ 2 (2.11)

E2 2 E0 sin b

 

For the TM mode, Equation 2.11 is substituted into Equations 2.3 through 2.6 to

solve for:

 

Ex = 3? [:tszOm cos (7mm) sin (_n7ry)] eiikzz,
kc a b

 

 

 

 

Xe

Figure 2.1: Rectangular Waveguide where lengths as = a and y = b.

 

_ _'i_ - n_7r ‘, mar ‘ w :t'ikzz
Ey — k2 [221,on b sm( a )cos( )] e ,

C

 

2' w 7277 _ 7mm: mry ilk z
B = — ——E —sm( ) cos (—) e Z ,
:1: kg I 02 0 b a b

 

73 (1) 77m 7mm: , mry iik z
B. =_ __E_...( )....(__) e z.
y kg I C2 0 a a b

A similar set of equations could be derived for the TE mode [6].

For waveguides of circular cross section, Equations 2.3 to 2.6 are transformed to

cylindrical coordinates (Appendix A.2):

 

 

 

 

 

_ kz 5E3 , (982
E¢_k2(i7~ 0¢ wa.,.)’

7733" 32—3:

43

10

(2.12)

E3 = AJm(.r) cos(mo)eiikzz, (2.16)

where A is a constant and Jm is a Bessel function of the first kind [7]:

 

a: m 00 —1 j .r 2j
“(1’): (2) gj!F(j(+ 371+ 1) (2) '

Bessel functions have an infinite number of roots with zeros occuring at integer values
of 72: 33ml: :rmg, 32.",3 (See Appendix A2 for more details). For integer values of
hi, the first few roots are
2:01 2 2.405, 2:02 2 5.520, 1:03 2 8.654,
$11 = 3.832, 11312 = 7.016, 113 = 10.173,
2:21 = 5.136, 31:22 2 8.417, 1'23 2 11.620, [8].

Since the electric field is perpendicular to a conducting surface and the magnetic

field parallel, the boundary conditions for the circular waveguide are (Figure 2.2)

Ez(r = R) = 0, E¢(r = R) = 0, and By-(r = R) = 0.

E2 : AJ771,(;13) COS(7an))€iikzz,

 

 

iii.» " '
Er : TiAJT’nQr) cos(m(b)eilkzza
1c
R
l/ \1
m l "

 

 

Figure 2.2: Circular Waveguide where lengths r=R and 2:1.

11

$ik3nl

 

E95 z 1637‘ AJmI’l )sin(m(_.j))e i’kzz

iwm . ; f iikzz
Br :k—_2C 2?” A1777.('I )SIII(NLQ)€ ’
B¢Z ICC-62 —w_2AJ;n(’I )COS(mgb)e i’kz 3

83:0.

Phase velocity is the velocity that any frequency component of a wave propagates.

Phase velocity is determined by setting [9]:

ID = wt — kg: = constant

w

’Uph = Z = E (2.17)
= —-4‘i’-—— (2.18)

(4)2 _ k2

52' c
= _—€—— (2-19)

Inez

1 “' "a?
> c. (2.20)

A propagating wave has the form

EU", :3, t) = Ed(r, Z)€i(wt—kzz)’ (2.21)

where Ed is some periodic function with period d, such as sin(z + nd) for n = 1,2, . . ..
The phase velocity exceeds the speed of light and the velocity that a particle can

reach. This is not useful for particle acceleration because the. phase will roll over

12

the particles making net acceleration zero. To obtain a v < c, structures with

ph
recurring geometry or periodic geometries are used. Group velocity is the velocity of

the electromagnetic energy along a waveguide [9]:

 

 

dw
'U = _
9 dkz
(4)2 2 (k3 + kg) 02
w 2 VIC? + kgc
dw k 2k 2
21?: CZ =6 Z=£—<c. (2.22)
2 mg + 1:; w ”1’"
A cutoff frequency is defined as % > kc:
2.40:"
we 2 else 2 c R L), (2.23)

where R is the diameter of the waveguide.

Cavity

A cavity is formed when conductive ends are placed on the cylindrical waveguide at
z = 0 and at z = l. Now the cavity is filled with incident and reflected waves with
additional boundary conditions Er(z = 0,1) = 0, E¢(z = 0,1) = 0, and Bz(z =
0,1) = O.

The electric fields become (Appendix A.3):

 

E2 = E0J771($) cos<m¢> (”I”) ,

:pvr
E z
’" [kc

 

 

EOan (3:) cos(mgb) sin (p73) ,

13

 

 

 

_ :l:mp7r , p7rz

Ed) — leg E0J7n(.1.) sm(mq§) sin (——) ,
and magnetic fields:
82 = 07

iwm _ p7rz

B=—.—E.] ( ),
r kgcz'r 0 m(r)sm(mq§)cos 1

iw . p7rz

13¢ = k—chEoJ,’n(:-c>cos(m¢>cos( , ),

2
w
where kg. = —2— — k3 =>w = Cng +13,
c

 

w c 2
an“: a; : 27((k0)2+ ([315)-

In the Til-1010 mode the zero order Bessel function provides an accelerating field in
the longitudinal direction (direction of particle travel) and a magnetic field in the ()5
direction.

Charged particles must enter the cavity at the right time in order to receive the
maximum acceleration by the electric field in the direction of travel; they must also
be traveling at a velocity such that they traverse the length of the cavity in roughly
half an rf period. Electric and magnetic field lines are one direction (Figure 2.3) for
half of the rf period and are opposite in direction in the second half of the rf period.

Changing the shape of a cavity, from cylindrical to elliptical (Figure 2.4), or adding
cells to construct a multicell cavity (Figure 2.5), makes it very difficult to calculate the
fields. Codes such as Superfish, SUPERLANS and MAFIA, to name a few, numerically
solve the wave equation constructed from Maxwell’s equations.

A multicell cavity provides an accelerating electric field in successive cells by

14

 

 

 

Figure 2.3: Electric and magnetic fields in a cylindrical cavity. The maximum electric
field occurs on axis to accelerate the beam and falls to zero at the cavity wall (R).
The magnetic field is zero on axis and has a maximum at 0.77 R.

making the fields in neighboring cells 7r radians out of phase with each other (the
7r mode of operation) requiring the particle to cross a cell in half an rf period. As a
charged particle travels through each cell, an accelerating electric field points in its
direction of travel, providing continuous acceleration through the cavity.

The cells in a multicell cavity are coupled together and act as a set of coupled
oscillators. There are six modes of excitation for the full six-cell structure operating
in the TMOIO mode of operation, the 7r, 53675, Q67:, 367:, %73, and % modes. Electric fields
in each cell have different amplitude and direction compared to the accelerating or

7r mode. For different modes of excitation one expects different field emitted electron

trajectories and impact locations for the same emitter.

Cavity Geometry

A charged particle enters a cavity on axis at time t = 0 and exits at time t = 71,-,
where l is the length of the cell and v the charged particle velocity. The length of the

cell is determined by the charged particle velocity and the rf frequency:

_ U _BgC__,BgA
-2f_2f_ 2, (2.24)

15

 

(b) “

 

 

 

\ Equator

Figure 2.4: Panel (a): Photograph of a single-cell elliptical cavity. The cavity’s inner
diameter at the equator is 32.9 cm, the inner diameter at the iris is 7.72 cm, and the
active length is 8.78 cm. Panel (b): Electric and magnetic fields are shown in the right
panel.

where [39 is the cavity geometric 13 and /\ the rf wavelength. Since it is desireable for

the particle to traverse the cell in half an rf period:
I
t = —- = — (2.25)

where f is the rf frequency [4]
The accelerating voltage (Vacc) of a cavity is given by integrating the electric field,

in a particles direction of travel, over the length of the cell:
l ‘k
Vacc = / E;(’r = O,z)el ‘32 dz .
0
For a pillbox cavity operating in the TM010 mode:

. i
S n
Vacc = E01 1k?- = IEOT,
‘2'

 

16

     
 
 

Magnetic Field Iris

Equator Electric Field

Figure 2.5: Panel (a): Photograph of a six-cell elliptical cavity in a vertical test stand.
The cavity’s inner diameter at the equator is 32.9 cm, the inner diameter at the iris
is 7.72 cm, and active length is 52.7 cm. Panel (b): Electric and magnetic fields are
shown in the right panel.
where T is referred to as the transit factor. For a pill-box cavity operating in the
TM010 mode, T = % if Equation 2.25 is satisfied [4].

Most commonly referred to is the average accelerating electric field (Ea) that a
particle experiences as it transits a cell:

Ea = VOiCC

 

The accelerating fields in SRF cavities are limited by the peak surface electric field
(Ep) and the peak surface magnetic field (Hp) (Figure 2.6) [4].

It is desirable to maximize the accelerating electric field so the ratios % and
g5 are minimized. Adding beam tubes to a cavity cell increases these ratios. An
SRF cavity will quench if Hp exceeds a critical magnetic field (Hc) [4] and a cavity’s
performance will be degraded if electrons are field emitted in regions of high Ep.

During cavity testing, as Ea, and thus Ep, is increased, the cavity Q remains
constant until an additional loss mechanism is triggered (Figure 2.7 (a)). If the loss is

due to field emission, x-rays are detected on the area radiation monitoring equipment

17

805 MHz 6 Cell Frequency (MHz) 804.98102

 

H (A/m) E (MV/m)
/V\ .
r /1
Surface , \ Surface
9937'7 Magnetic ( . Electric 6'7995
Field Field
33 12.6 2.2665

  

Equator a uator

I .lJlHllfiillm '- -' 1 " . =3, '.-.. 1:” ' 5"":
= 3: '. l " l

‘._. I ‘ i
I v. Em" lllllll ' ' m .. -'--*~:"—.~' 1m net: ..

Iris /
-9938 v

. i l \
-l6563 \A/ w W‘/ -11.33

0 19.384 38.768 58.152 77.536 L(cm)

       
   
 

-3313 -2.2266

   

-6.799

 

 

 

 

Figure 2.6: Surface electric and surface magnetic fields as a function distance along
the axis of 3 cells of a 6 cell cavity with beam tube, in the 7r mode.

(Figure 2.7 (b)).

2. 1.2 Field Emission

The advantage of superconducting over normal conducting cavities is the much lower
power dissipation in the cavity walls. Cavities are designed to have as high an ac-
celerating electric field as possible, although the surface electric field must be kept
as low as possible to prevent field emission [4]. Field emission is the emission of an
electron from the surface of a metal in the presence of a high electric field gradi-
ent. “The first one to describe this phenomenon in detail was R. W. Wood, in 1897,

who reported enthusiastically what one would call today a kind of fire-works in his

18

1.0E+11 — (a)

01.0E+1O wu” ° ’°.

 

 

 

\
1.0E+09 i l
0 20 40
E 15000 — (b)
§ 10000 — .o
5 5000 . °
>2 0 «were» 0 o c 9 .
O 20 4O

Ep [MV/m]

Figure 2.7: Panel (a): The cavity Q remains constant as Ep is increased until the

triggering of field emitted electrons. Panel (b): X-rays are detected when the loss
mechanism is due to field emission.

discharge tube” [10] (Figure 2.8).

Sources of field emission in cavities include surface roughness, metallic dust, micro-
particles, grain boundaries, and impurities in the material. F ield-emitted electrons are
accelerated in the cavity electric field and have complex trajectories. When they im-

pact the cavity walls they dissipate power, creating heat, and generate bremsstrahlung

l C A
— +
To Vacuum
Pump

——'. High Voltage .—‘

Source

x-rays [4] .

 

 

 

 

Figure 2.8: Discharge tube where electrons are field emitted from the cathode, travel
to, and impact the anode.

19

For this study the average current, from which the average power is calculated, is
needed to determine the degree of electric field enhancement, Fowler-Nordheim (F lV)
field enhancement factor (F EF), from the data. This FEF is used by the particle
tracking simulation to determine the current and power of an impacting electron.
The simulated power is fitted to the x-ray power data until a likely electron source is

found.

Calculation of Current

The abundance of electrons inside a metal is almost inexhaustible. Within a metal an
electron has a potential energy of some constant value, Wa, relative to zero when the
electron and metal are separated. The Fermi energy (C) is the highest energy level
filled with electrons at 0 K. The energy required to take an electron from the Fermi
level within a metal and remove it to infinity, assuming a particle at infinity has zero
energy, is the work function qb of the metal.

Under normal circumstances the energy of an electron, confined inside a metal by a
potential well, has insufficient energy to escape from the metal (Figure 2.9 (a)) [4]. To
escape from the surface an electron must receive extra energy. Applying an electric
field to a metal, E in $, an electron encounters a potential of V(;L) = —-eE-.1:.
Quantum mechanically, the electron wave function attenuates outside the surface
potential barrier. If the potential barrier is lowered, the attenuation is not complete
and there is a finite probability that some electrons will tunnel through the barrier,
i.e. leave the surface (Figure 2.9 (b)) [4].

Once an electron has left the surface it also experiences an attractive force due
to the presence of the conducting surface. This force is represented as an image force

2
and lowers the height of the potential barrier by %E (Figure 2.9 (c)). The image force

is shown in Figure 2.10 [10].

20

The two contributions affecting the potential [11]:

V(:r) = —l’Va for a: < 0 (2.26)
2

: __e__ -— eErL‘O for :1; > 0. (2.27)
4330

1/2
By maximizing V(:1:), 330 = (2&3) , the maximum decrease of the potential barrier

is found to occur at go 2 63/2E1/2. The final potential barrier is in Figure 2.9 (d) [4]
Charge density, at the surface of a metal, is not identical to that within a metal.

Work is required to move an electron through the surface layer, W3. So the work

Lowered

(a) (b) Potential

Potential

Barrier
x=O x a/ x=O x —>/
V=0 V=0

 

 

 

 

 

 

 

 

 

    
   
  

 

   

 

 

 

 

44. ‘14/ -V1 2. ¢ N -V1
x I J» x I I \ \ }
\ / “’8 I x / I “’8
Wa V Wa \/ I
Electron Wave Electron Wave
Function Function
Lowered Final
(c) Potential (d) Potential
Barrier Barrier
x=0 x —>
V=O V=O
fi/I'L d5; /, -Vt 4 4’ -Vt
/ \\ /[ )- W8 W3
wa ”V Wa
Electron Wave Electron Wave
Function Function

Figure 2.9: Panel (a): An electron wave function at a potential barrier where (B is
zero at the metal surface, V the potential height, 45 the work function, C the Fermi
Energy, Wa the electron potential energy, and W3 the surface potential energy. Panel
(b): Effect of an electric field lowers the potential barrier to a triangular shape so
electrons have a higher probability of tunneling from the metal into the vacuum.
Panel (c): Effect of an image force in lowering the potential barrier. Panel ((1): The
potential barrier is lowered due to the applied electric field and the electron image
force.

21

J

X

///7//

 

Figure 2.10: An electron that has left the metal surface also encounters an image force

2
that lowers the height of the potential barrier by a)?

function is lower than the Fermi energy, (I) = —C + W3.

Fowler and Nordheim assumed that the conduction electrons in the metal form
a gas of free particles, which obey Fermi-Dirac statistics. They then calculated the
number, (In, of electrons within a volume, v, and with momenta within the range

dpxdpydpz arriving at the surface of the metal [10]:

2vdpxdpy dpz

[ca—masons ”'28)

 

(1n =

where h is Plank’s constant. k is Boltzmann’s constant, C the Fermi energy, E the
, )“+')‘;
total energy, i.e. W + [Jr/2%, and T the temperature. The supply of electrons, or

supply function N (W), is found by integrating Equation 2.28 over all py and pz [10]:

 

WW) 2 l/+°°/+°O ‘17)“!!de
(L3 -00 ~00 exp (TTQE_ )+1-

47rka ,. —W + C
= [1,3 in (1 + (31]) [7]) . (2°29)

 

Once at the potential barrier or surface, the probability of an electron tunneling

through or leaving the surface, is given by the transmission coefficient DOV) [10]

2,/2m|W'|3
D(W)=€:L‘P - 'v(y) , (230)

3eEh

(Appendix B.1):

 

22

 

where v = fiaflEUs) —— b2K(k)], K03) and E(k) are elliptical integrals of the first
and second kind, a, b, and k are defined in Appendix 8.1.
The number of electrons within energy dVV emerging from the surface per unit

area [10] (Appendix 82):

P(W)dW = N(W)D(W)dW. (2.31)

—4‘/2m|W|3

3eEh

 

 

4 AT — ’
P(W)dW _—. 7rm 172 [1+ erzrp< W + 4)] ea: v(y) ,

h3 kT

 

 

47rka —VV + C W — C
P(W) = [13 Zn [1+ exp (T):| erp (—c+ d ) ,

where

_ 4,/2m¢3v

C — BeEh

eEfi.

d: —.
2 2mqbt

In the low temperature limit [10]:

 

 

—W
len 1+e;rp ———+——C- =0 whenW>C
kT
=C—W whenW<(,
thus P(W) = 0 when W > C
4 _
= 2173716171) (-c+ C) (C -— W) when W < C.

The current density is found by integrating P(W) over all energies, i.e. from -VVa

to 00. Good and Miiller [10] assumed —Wa is far below the Fermi energy C, so the

23

lower limit is —00 (Appendix 83):

C
j = c/ P(l-i’)dH«’
—OO

 

 

3
—6 2
1.54 x 10 E 9gb?
: (M2 (21]) -fi.83 x 10 EU , (2.32)
where h. = Planck’s constant, 62 = 197 Meg/gnaw), and m = 9.109 X 10‘13 Kg.

The elliptic function is defined as [11]:

5(5) = 0.956 - 1.06232

1/2
.3 = 3.79 x 10—5ET

Both Good and Miiller [10] and Wang and Loew [11] set t = 1 in Equation 2.32 since

 

 

 

lgtgin:
1.54 10—5132 3/2 E
j = x » €171) 0 (—6.529 x 109 +10.42—2)
(D E ()5
_1
1.54 10*6 10452)“? 2E2 3/2
= X X9,) 6131) -—6.529 x 109% (2.33)

 

No metal surface is perfectly flat and clean therefore large variations in the micro-
scopic surface field lead to enhanced field emission. A few examples include roughness,
scratches, dust or micro particles, grain boundaries, and absorbed gas [11].

In a superconducting cavity, the current collected by a pickup probe and the x-ray
intensity outside the cavity follow the functional dependence of the F owler-Nordheim
law. But, just as Schottky observed in 1923, the function has to be modified by a
factor, E F N, referred to as the Fowler-Nordheim field enhancement factor [10]. Thus

E is replaced with [3 F NE in the equations for current density and current. The current

24

 

 

is I = jAe, where A8 is the effective emitting area. W’ithout the field enhancement
factor, current and x—ray intensity are many orders of magnitude lower than observed.

Therefore, current becomes [11]:

 

 

 

 

 

 

 

 

1 1.54 x 10*6 x 104-52A 532 ,I3/2
_§ 2 , PA FN e.1:p(g9_1/2)c.rp —6.529 x 109 ,Q ,
E <2 WW

1 4,132 3 2
171—2 :17. 1.54 x 10—6LF—1X + 10425“2 — 6.529 x 109 ff / ,
E (P 13FNE
1
d (“1752) _ —6.529 x 10933/2
44>
(1009104! '3/2
—————E—)— = (log/106’.) —6.529x 109 GD
43
—2.84 109 ‘3/2
z X f” . (2.34)

’[3F IV E

For DC electric fields the numerical value of 1’3 F N for a particular surface can be
obtained from I, E, and <15 and by plotting [0910 (E17) versus 71-}. The B F N value is
the slope of the line.

Electron emission for RF fields is modified from the DC version by assuming
that the microscopic electric field on a metal surface is of the form E 2 E0 sin wt.
The average field emission current is calculated by a time averaging substitution into

Equation 2.33 (Appendix 8.4):

1 T
1 = —— 1 t dt
< > T [0 o
5.7 x 10-12 x 104.52Aefi%§’vE(2)'56;rp(¢1‘5) (43.529 x 109,515)
— er ,

 

 

— (21-75 .BFN ED

25

 

 

 

(I) 5.7 x 10‘12Ae1'312;~'}5v _0_5 6.529 x 109(51-5
In. 2 5 = in. .1 75 + (10.413 ) — , ,
E . (p - (3FNE0

 

 

 

(I)
61(10910—2715 .3) = — (logloe)6.529 x 109315
d(%) HFN
——2.84 x 109315 (2 3,)
: . . 0
51w

Similar to the DC case, but with a plot of loglo (i525) versus 11;}, one can obtain

5 F N from the slepe of the line and Ae from the intercept [11]. For this study, a plot of
F

10910 ($532512) 2 10910 (#5) versus 11;}, where P is power, was used to determine

,6 F N since power was measured. The physical significance of 5 F N and A8 is still a

matter of debate [4].

2.1.3 X—rays

In 1895, the German physicist Wilhelm Conrad Roentgen was attempting to produce
radiation which would traverse matter opaque to ordinary light. He covered a dis-
charge tube (Figure 2.8) with ordinary paper and found that a screen covered with
crystals of platinum barium cyanide fluoresced. He found that the unknown radiation
could penetrate a 100 page book, but heavier objects stopped the fluorescence. He
also found that the radiation could expose a photographic plate. One of his earliest
photographic plates shows an image of his wife’s hand with a ring (Figure 2.11). He
named this radiation “x-rays”, indicating their mysterious nature [12]. Roentgen was
awarded the first Nobel Prize in 1901 for his achievements in x—ray studies [12].

As charged particles travel through matter they lose energy in Coulomb interac-
tions with electrons and nuclei, by emission of electromagnetic radiation (bremsstrah-
lung or breaking radiation), in nuclear interactions, and by emission of Cerenkov radi-

ation. The charged particle of interest in this study is the field emitted electron with

26

 

Figure 2.11: One of earliest photographic plates showing an x—ray image of Bertha
Roentgen’s hand with a ring produced on December 22, 1895 [13].

a maximum energy around 1 MeV, therefore, discussions of Coulomb interactions
and bremsstrahlung radiation follow. Cavities developed for the International Linear

Collider (ILC) could produce field emitted electrons with energies over 10 MeV.

Coulomb Interactions

As an electron (or any charged particle) travels through a material it may interact with
atomic electrons or the nucleus of the atom. The radius of the nucleus is approximately

10'14 m and the radius of the atom approximately 10‘10 m. It is expected that:

Number of interactions with electrons (R2)atom 00—10)2 108

Number of interactions with nuclei _ (R2)nucleus — (10*14)2

collisions are with atomic electrons and are more important than collisions with nuclei
[14].

Inelastic collisions with bound atomic electrons result in either one or more atomic
electrons transitioning to an excited state (excitation) or to an unbound state (ioniza—
tion). Elastic collisions are significant in cases of very-low—energy (< 100 eV) electrons
and therefore are omitted from this discussion [2]

Excitation occurs when a bound electron of the material acquires enough energy,

27

from electrons traversing the material, to transition to an empty state in another
orbit of higher energy. Transitions are typically between the K, L, M, shells in
atoms. In a time of 10"10 to 10‘8 seconds the electron will transition to a lower
energy state, provided one is empty. The excess energy is emitted in the form of
x-rays characteristic of the material [14].

Ionization occurs when an electron in the material acquires enough energy, from
electrons traversing the material, to leave the atom and become a free particle. The

kinetic energy of the freed electron is:
(K E )e 2 (energy given by electron) — (ionization potential).

The freed electron may cause ionization of another atom if its kinetic energy is high
enough. Fast freed electrons produced in ionizing collisions are called 6 (delta) rays
which produce secondary interactions while being brought to rest [14] [2].

An electron traveling through a material exerts Coulomb forces on many atoms
Simultaneously. Since every atom has many electrons, the incident electron interacts
with millions of atomic electrons, each occurance producing an energy loss. It is
impossible to calculate the energy loss from individual collisions; therefore, an average
energy loss is calculated per unit distance traveled. The energy loss or stopping power
per unit path length due to ionization—excitation for electrons is given by [14]:

T ,2 ,/-_—
(L) = 47W?) mOc NZ [In (5’7 7 1m062)]

 

 

2 2
, 1 —1
+ [rm-gingh, [(7 8 ) +1—(’)’2+27—1)ln2] (2.36)

28

 

 

 

in (lelv), where
6’2 15
To = 2 = 2.818 x 10_ m : classical electron radius
"’00
1
’7 = —f
1 — 1’3
11
fl = —
c
N A Avogadro’s Number
N = —— = densit
p( A ) y Atomic Weight
Z 2 atomic number of material
I = mean excitation potential of material

~ (9.76 + 58.8Z‘1'19)Z in CV.

A complete derivation can be found in [2].
In this study, the maximum electron kinetic energy was measured to be around 1

MeV, implying an electron velocity of 0.941 c. The energy loss per unit path length

in niobium was calculated to be approximately 7.6 cm .

Bremsstrahlung Radiation

Whenever a charged particle accelerates, part of its energy is converted to electromag-
netic radiation in the form of bremsstrahlung x-rays (Figure 2.12). In this study, the
charged particles of interest are the field emitted electrons impacting the walls within
a niobium cavity. For electrons with kinetic energy T (MeV) traversing a material
with atomic number Z, the energy loss or stopping power per unit path length due to

bremsstrahlung x-rays, is given in terms of the ionization and excitation energy loss

(a (9
‘15 brem 750 “is ion.

Bremsstrahlung emission increases with electron energy and atomic number of the

(Equation 2.36) [14]:

material. A full derivation is given in [2].

29

 

Incident
electron O
Bremsstrahlung
x-ray
emission

Figure 2.12: An electron accelerating near a nucleus and emitting bremsstrahlung
radiation.

In this study ((511) (Equation 2.36) was calculated to be 7. 64 Lgfilv, Z for
2071.

niobium is 41, and the maximum electron kinetic energy, T in MeV, 1 MeV.

dT dT
<—> we >
d3 brem db 3071

which means 5.5% of the 1 MeV x-rays are due to bremsstrahlung radiation.

The energy of x-ray photons can range from very small values up to the energy
of the incident electron, which is known as Duane and Hunt’s Law [2]. In 1922,
Kuhlenkampff [15] reported that the intensity of the continuous x-ray spectrum could

be represented experimentally by:

1 1 1 2_1_2
I =CZ——- ——— BZ 2.38
.x ,2 (,O ,) ,2 < >

81' '8
where I /\ is the x—ray intensity in units of 35135, B and C are constants, Z the atomic
number of the material, A0 the wavelength of the incident electron, and A the wave-
length of the x—ray photons.

H. A. Kramers furthered Kuhlenkampff"s work by theoretically deriving (to first

30

order) the constant C [16]:

8a 82h 1 1 1
I = s—-———Z— — — —
A 3MB? m6 A2 (A0 A)
_ ,r- —30. 1 1 1
— 4OX10 5Z:\§('X6-X)
= 2.35 x 10‘232E2(E0 — E), (2.39)

where s is the number of electrons entering the material, and E and E0 are in keV.
Equation 2.39 is known as Kramers approximation. Figure 2.13 shows a plot of the
continuous bremsstrahlung x-ray spectrum for 1 MeV and 0.5 MeV electrons in nio-
bium. In this study electrons of varying energies impact the cavity walls, thus the
x-ray spectrum is a sum of all the individual electron spectra.

Koch and Motz [17] describe the angular distribution of bremsstrahlung x—rays by

the fraction of the total incident electron kinetic energy that is radiated per steradian

1.6E+08 —
1.4E+08 —
1.2E+08 1
1.0E+08 ]
8.0E+07 ~
6.0E+07 «
4.0E+07 a
2.0E+07 a
0.0E+OO

  
  
   
  
  
 
  

$

 

A

Y I Y 1

X-ray Intensity [ergs/cmA2/sec]

 

f

0 200 400 600 800 1 000 1 200
Energy [keV]

Figure 2.13: X-ray spectrum for a 1 Mev (dotted line) and a 0.5 MeV (solid line)
electron traveling through niobium as predicted by Kramers approximation (Equa-
tion 2.39). The maximum x-ray energy ranges from very small values up to the energy
of the incident electron according to Duane and Hunt’s Law.

31

at the angle a as:

 

 

Kég —£80.2 —5302

R = -E- +— + E - , 2.40

“ 176077 [ l ( 1760t I 7.15 ( )

where K depends 011 the initial electron kinetic energy and atomic number of the
2

material, £0 is the incident electron energy in units of 7110c , a is the angle of the
bremsstrahlung x-rays in radians, t is the length given in units of radiation length,

E, is the exponential integral:

 

 

oo e—z
—Ei(—y)=/ dz, (2-41)
1/ z
and
R K681 246i 242
(1:0 — 1760 71, ,. ( . )

Figure 2.14 is a plot of the bren'isstrahlung angular distribution of x-rays for an
electron with energy 1 MeV and 300 keV incident on niobium. Incident electrons
with an energy of 1 MeV, typical of this study, show an anisotropic tendency. The

number of x-rays entering the detector through a collimator will be lower at angles

1 a

0.9 ~

0.8 a

0.7 4

Ra 0.6 .
_ 0.5 1
0.4 -
0.3 1
0.2 J
0.1 1
O 'I T—_' —_"_—“T'M' l

0 50 100 150
Angle [Degrees]

 

 

 

 

Figure 2.14: Angular distribution of bremsstrahlung x-rays. Electrons with incident
energy of 1 MeV are shown as a solid line and 300 keV as a dotted line.

32

not parallel with the electron forward motion.

2.1.4 Photon Interactions with Matter

Photons are classified according to their mode of origin, not their energy. Thus, '7 rays
are the electromagnetic radiations accompanying nuclear transitions. Bremsstrahlung
or continuous x—rays are the result of the acceleration/ deceleration of charged parti—
cles. Characteristic x-rays are emitted in atomic transitions of bound electrons be-
tween the K, L, M, shells in atoms. Annihilation radiation is emitted when a
positron and electron combine. Interactions of photons with matter are independent
of the mode of origin of the photon and dependent only upon the photon quantum
energy [2]

Photons may be treated as electromagnetic waves or as particles. An electromag-
netic wave is characterized by its wavelength (A) or frequency (u), 1/ = fi. As a
particle, a photon has zero charge and zero rest mass and travels at the speed of

light. The quantum energy of photons is expressed as:

E = hi/ 2 7,
where h is Planck’s constant 2 4.136 X 10—15 eVS [2] [14], l/ the photon frequency,
and A the photon wavelength.

A number of processes can cause photons to be scattered or absorbed. The energy
domains met in this study, 0.01 to 2 MeV, are explained in terms of photoelectric
effect, Compton effect and pair production. Other processes such as Rayleigh scat-
tering, Thomson scattering by the nucleus, and Delbruck scattering are explained in

Appendix 1.

33

Compton Effect

The Compton effect or Compton scattering is a collision between a photon and an
unbound or free electron (rest mass = mOc2 = 0.511 MeV) considered at rest. An
electron is considered unbound when the electron binding energy is much less than
the energy of the incident photon. The collision results in a scattered photon and a
scattered electron (Figure 2.15) [2]. The scattered electron loses energy by ionization
and excitation (Coulomb interactions) or by emission of bremsstrahlung radiation.
The scattered photon loses energy by Compton scattering, the photoelectric effect,
or at very high energies, pair production.

The incident photon momentum (1??) must be conserved between the scattered

photon and the struck electron. The relations for the conservation of momentum in

the direction of the incident photon for the collision of Figure 2.15 is expressed as:

h h, ’
—V—0 = —V cos6 + pcos (a, (2.43)
c c

where hI/ is the incident photon enerO' r, hu’ the scattered hoton ener , 6 the
0 o} P 83’

scattered photon angle, p the scattered electron momentum, and (0 the scattered

Compton
scattered
electron
Unbound O
electron
Incident
photon

 

Compton
scattered
photon

Figure 2.15: Compton scattering, a collision between a photon and an unbound elec-
tron. The photon is scattered at an angle 0 and the electron at an angle 1p [13].

34

electron angle [2].
Conservation of “10111811111111 normal to the direction of the incident photon is

expressed as:

h 1/ , '
0 = —— 8111 9 + psm (p. (2.44)
c
By the conservation of energy
hV0 = ’w' + T. (2.45)

where T is the scattered electron kinetic energy, and by using the relativistic rela-

tionship

 

[)C = \/T(T + 2moc2) (2.46)

any two parameters can be eliminated from the conservation equations 2.43, 2.44,
2.45 and relativistic relationship 2.46 [2].

Energy of the scattered photon (Appendix C.1):

 

 

Ill/I :- 771062 (2 47)
1 — cos 6 + l, i
(I
I
1
V (2.48)

% :1+a(1—cos6)’

hu ‘0“ . . _ o by _ ‘
where a — gag. For very large 1nc1dent photon energy "—109? — a >> 1 , the energy
of the backseattered photon, 6 = 1800 approaches hu’ ~ moc2 = 0.511 MeV. Energy

of the struck electron, or shift in the photon energy, is found from Equation 2.45,and

substituting Equation 2.47 for hu’:

Til-0C2

1 — cos6 +21;
III/000 — cos 6)

: _ 2.49
1+ 01(1— cos 6) ( )

 

T = hI/O—

 

Maximum energy transfer to the electron, Tm”, occurs at 6 2 180°:

b. 1/0

Timur 2 1+ 1 7
23
and minimum at 6 = 0° in which case Tm,” = 0 [2].
During Compton scattering, it is impossible for all the energy of the incident
photon to be given to the electron. The energy given to the electron will be dissipated
in the material within a distance equal to the range of the electron [14]. The kinetic
energy of the Compton scattered electron can be found dependent upon the electrons

scattering angle (cp) [2]:

2111/00 cos2 (,0

— (1+ 01)2 - 02008299.

 

The maximum electron kinetic energy occurs at (p 2 0° and the minimum at «p = 90°

[2]-
Another interesting effect is the Compton shift in wavelength of the incident pho—

ton to the scattered photon [2]:

 

 

c :
A, — A0 = '7 — i
V V0
hC C , , / .
: —, — —, substitutmg hu from Equation 2.47,
hi/ hz/O
hem c2
_ hc—hccos6+ 13%— “ ff:
771062 (“/0
he 1 — cos6
moc

The Compton shift in wavelength is independent of the energy of the incident
photon (hi/0) where as the Compton shift in energy, hug — hu’, is strongly dependent
upon hug. Thus low energy photons are scattered with moderate energy change but
high-energy photons experience a very large change in energy. For example, at 6 = 90°

if hl/O = 10 keV then hu’ 2 9.8 keV, a 2% change; but if hz/O = 10 MeV then

36

hu’ 2 0.49 MeV, a 95% change [2].
The Compton wavelength is defined as the wavelength of a photon whose energy

is equal to the rest. mass of the electron [2]:

c he he 11
/\C : —— z — : —2 :
l/C Iii/C mOC moc

4.136 x 10—21Mevs
0.511Mevc
——2——

C

= 2.424 x 10“12 m. (2.52)

 

 

The probablility that Compton scattering will occur is called the Compton cross
section [14]. To derive the Compton cross section from the initial state 0(h—IC/Q, p = 0)

to the final state F (I—lé—Cp), consideration must be given to the intermediate states.
The following two intermediate states are possible [18] [2]:
I) The incident photon is absorbed by the electron, which then has momentum Eta/Q,
and no photon is present. As the electron transitions to the final state of momentum
p, a photon of energy lw’ is emitted by the electron.
II) The electron first emits a photon of energy lw' , leaving the electron with momentun
iii—15:. Two photons are present until the electron transitions to the final state when
the incident photon is absorbed.

This representation of the scattering n'iechanism leads to the Klein-Nishina for-
mula of differential collision cross section:

7.2 VI 2 1/ VI
daz—OdQ(—) (—9+——2+4cos28),
4 V0 1/ V0

2
where r0 = —§—2 = 2.818 x 10‘15 n1 is the classical electron radius, d9 = sin6d6 dd)
moc
is the element of solid angle through which the scattered photon emerges after the
collision, and O is the angle between the electric vectors of the incident and scattered

radiation.

37

The total collision cross section , a, is the probability of photon removal from a

beam while passing through an absorber containing one electron per cm2. To obtain
I

the expressron for a, substltutions for (10 = 277 sin6d6 and 750 from Equation 2.48

are made and integration taken for all values of 6:

 

71'
21+a 20-1-07) 1 1 , 1+3oz
= d =2 ——l1 2 —l'1 2 ———,
0 /0 0 7W0] 02 [1+2o an( + (1) +207 ”( + a) (1+2a)2

C1112 [ ]

111 units Of m 2

As photons travel through a material they encounter NZ electrons per cm3

,where
N = pI—Yf, p the density, NA = Avagadro’s Number 2 6.022 x 1023%Ifi1§ and A the
atomic weight, and Z the atomic number of the material. Let n be the number of
incident photons and dn the number of photons removed from the number incident
per second:

(171

——n_ 2 (NZ d.r)0 = 01,17,611?-

Olin is defined as the Compton total linear attenuation coefficient [2]:
01m = NZa [cm—1] . (2.53)

Linear coefficients for various incident photon energies and materials can be found
in numerous reference books and on the web. Figure 2.16 shows how Olin changes as
a function of incident photon energy hug and 721 [14]. When a significant number of

photon interactions are due to Compton scattering, the fractional transmission —"—

1 710 7
of unmodified photons through a material of thickness .2: is:
1”— : e—Ulzt-nl‘. (2.54)
71.0

38

(a) (b)

o-lin U-Iin

 

 

 

 

hVO Z/A

Figure 2.16: Dependence of the Compton cross section 01,-” on (a) the incident photon
energy huO (ln vs 1n scale) and (b) the atomic number of the material (linear vs linear
scale).

Photoelectric Effect

An incident photon cannot be totally absorbed by a free electron as evidenced from
the momentum relationships in Compton collisions. The photoelectric effect takes
place when an incident photon is totally absorbed by an electron initially bound
in an atom. The photon disappears and one of the atomic electrons is ejected as
a free electron, creating an ion. The photoelectron loses energy by ionization and
excitation (Coulomb interactions) or by emission of bremsstrahlung radiation. The
electron vacancy can be filled through the capture of a free electron and/or by the
rearrangement of electrons from other shells of the atom, thus generating one or
more characteristic x-ray photons [19] (Figure 2.17). If the emitted x-ray photon has
sufficient energy and is absorbed by a second electron within the atom, it is emitted
as an Auger electron. The atom becomes doubly ionized [2] [19].

In a photoelectric interaction momentum is conserved by the recoil of the atom.
The most tightly bound electrons, K shell, have the highest probability, 80%, of

absorption when the incident photon energy exceeds the K-shell binding energy. The

39

Photoelectron Characteristic

x-ray
Electron photon
fills
vacancy vacancy
Incident
’ o

photon

Figure 2.17: Diagram of the photoelectric process. The incident photon is absorbed
and a photoelectron is ejected with energy T = hug — Be. When the vacancy is
filled the excess energy is carried off by a characteristic x-ray photon. Momentum is
conserved by the recoil of the atom.

kinetic energy of the free electron is:
T = hI/O -— Be, (2.55)

where Be is the binding energy of the electron [2] [14].

The probability of the photoelectric effect occuring is called the photoelectric
cross section. Unlike the Compton effect, exact solutions are difficult and tedious. In
the energy region of interest in this study (0.35 to 2 MeV), reference [20] provides
complete solutions. For a crude approximation of the total photoelectric cross section:

2 2 2% (2.56)
where a is a constant and 71 increases from 4—4.6 for fir/0 increasing from 0.1 to 3.0
MeV [2].

Similar to the Compton cross section, the linear photoelectric cross section is
defined as [2]:

Tlm : TN. (2.57)

Figure 2.18 shows how 71m changes as a function of incident photon energy and Z [14].

40

(a) (b)

Tlin Tlin

 

 

 

 

hVO Z/A

Figure 2.18: Dependence of the photoelectric cross section 01,17, on (a) the incident
photon energy 11110 (In vs ln scale) and (b) the atomic number of the material (linear
113 linear scale).

Pair Production

Pair production is an interaction between an incident photon with an energy above
1.022 MeV and a nucleus within a material. The photon is completely absorbed and in
its place a positron-electron pair appears. The nucleus does not undergo any change.

By conservation of energy:
_ 2 ,2
[1.1/0 — (T_ + 7710c ) + (T+ + mOc ),

where T_ and T+ are the kinetic energy of the electron and positron respectively [2]
[14].

The positron is not a stable particle and travels only a short distance before
combining with an electron or annihilating, creating two photons separated by 180°
and each with energy of m0c2 (0.511 MeV). The electron travels a few millimeters at
most before losing energy by ionization and excitation (Coulomb interactions) or by
emission of bremsstrahlung radiation (Figure 2.19) [2] [19].

The probability of the pair production occuring is called the pair production cross

section. Pair production was not a concern for the photon energies encountered in

41

electron

Incident /

positron

0.511 MeV

Figure 2.19: Pair production. The incident photon disappears and a positron-electron
pair is created, followed by two 0.511 MeV photons produced when the positron
annihilates.

this study. Unlike the Compton effect, the total pair production cross section is a
complicated function of Iii/0 and a complete derivation can be found in [2]. It may be

written in the form [2]:

 

2 2h 21
Ii. 2 OZ2 (—8l71 V0 — 8) .
9 moc2 27

Similar to the Compton cross section, the linear pair production cross section is
defined as [2]:
Run 2 Ii“.N. (2.58)

Figure 2.20 shows how 16”,, changes as a function of incident photon energy and

z [14].

Attenuation of Electromagnetic Radiation

In the previous three subsections, photon interactions with matter were discussed.
The following is a discussion of the over-all effects of these processes on photons as
they pass through a material.

The probability of a photon traversing a thickness t of absorbing material with-

t

out a Compton collision is e‘al’int, photoelectric interaction is e_Tl'i-'n , and a pair

42

(a) 0))

Klin Klin

 

 

 

 

1.022 MeV hvo Z/ A

Figure 2.20: Dependence of the pair production cross section film on (a) the incident
photon energy hz/O (ln vs 1n scale) and (b) the atomic number of the material (linear
vs linear scale).

production collision is e_""lint. Let [0 be the inital intensity of photons traversing a

material of thickness t. From Equation 2.54 the number of unaffected photons, I f, is:

[f = 102—Uzinf-T1mt—Hzmt

: [05"(011'11+Tliii.+Kli-71)t

= IOeWI'N-t, (2.59)

where 1112-7, is the total linear attenuation coefficient. A graphical relationship between
the three types of interactions is shown in Figure 2.21. The photoelectric interaction
is important at low photon energies and high Z materials, pair production at high
photon energies and high Z materials, and Compton effect at intermediate energies
and intermediate Z materials.

The total linear attenuation coefficient is distinguished from the total mass ab-

sorption coefficient, which is a smaller quantity and measures the energy absorbed by

43

 

 

 

 

   
 

  
   

 

 

i I JITTfiT—H 1171111 T ii‘TT'i'Tl T 1117
120 -~
100 ~— Photoelectric Effect Pair Production -
e Highest Probability \ Highest Probability
80 r- —
Atomic Number Z 7 '1
60 .. .1
_ 0'11n = Tim 0'11n = Klin T
40: Compton Effect
Hi best Probabilit
20]— g Y
i;__1__1_1.111111 - ”1.1 1-..] Ll 1.11;--. 1.. -1. !__L1Jll_l . __.1_.1 -1-
0.01 0.05 0.1 0.5 1 5 10 50 100
Photon Energy hvo [MeV]

Figure 2.21: Relative importance of the three main types of photon interaction as they
traverse a material. The lines show the values of hu and Z for which the probability
of the effects are equal.

the material. Often it is useful to know the mass attenuation coefficient:

 

 

It [c1112] _ 11“,,[cm—1]
ma _‘ " '1
9 p 7.33.

 

 

= Uma + 7"ma. + fima.
Olin + Tlin + I{11171

p

 

The thickness of the material is now measured in units of pm [3:37]:
I

[_f ___ e_i%u(p$) : e—amapm.
I0
The advantage of measuring material thickness in units of (3%? is that the mass
attenuation coefficient is nearly independent of the nature of the material since 7Z{ is
approximately constant for all elements.
There are many reference books and websites with figures and tables [21] that give

11),” and ”me for nearly all the elements and energies. Figure 2.22 shows the mass

44

 

1000

 

 

 

 

 

 

 

 

 

L1
100 L2H , 2 1
:L3 2
10 “ \
E “\
1 ~ \
cmzlg Compton
01 VEJ‘I‘CCI
: Photoelectric
0.01 ’ Effect x
°'°°‘ n)... \
: Production
0.0001 ‘ 1‘“fifli 1 1111..., 1 11111..
0.01 0.1 1 10

Energy [MeV]

Figure 2.22: Mass attenuation coefficient versus energy for photons in lead. This plot
was constructed from data retrieved from the NIST program XCOM [21].

attenuation coefficients for photons of different energies in lead. The plot shows the
photoelectric effect as the primary interaction at low energies. The discontinuities are
due to electrons being ejected from their shells and the vacancies filled. At interme-
diate energies the Compton effect is the primary interaction. Pair production comes
into play at 1.022 MeV, which is the minimum energy needed to produce an electron

and a positron.

For mixtures of elements, an over-all mass attenuation coefficient is given by:

 

1. - 1 - L -.
Ilm : Ilmlw1+ I 11712102 + . .. , (2.60)

P P1 P2
where w}, 1112, are the fractions by weight of the elements that make up the

absorber.
As a photon traverses a material with a total linear attenuation coefficient of 11),”,
the probability that it can travel a distance :5 without experiencing an interaction is

e—“linm. The mean free path length is the mean distance a photon will travel before

45

its first interaction:

1

l . , = —.
mfp Win

(2.61)

Photons with an energy of 1 MeV traversing lead have a mean free path of 1.25 cm.

2.2 X-ray Detection

When photons enter the scintillator material of a detector, sparks or scintillations
of light are produced. The amount of light produced is very small; to be detectable
it must be amplified with a device known as the photomultiplier tube (PMT). The
resulting voltage is proportional to the energy of the incident x-ray photon and is

displayed on a multichannel analyzer (MCA) [14].

2.2.1 Detector

Unlike the discrete electronic energy states of an atom (Subsection 2.1.4), the allowed
energy states in a crystal widen into bands. The luminescence of inorganic scintillators,
such as the one used in this study, is explained in terms of the valence, conductance,
and forbidden bands of a crystal. The valence band is completely filled with electrons
bound at lattice sites. The conduction band is void of electrons while in the ground
state. At energies between the valence and conduction bands lies the forbidden band,
in which electrons are never found in a pure crystal. Electrons that have obtained
sufficient energy from incident photons move from the valence band to the conduction
band and are free to migrate throughout the crystal. The displaced electron leaves a
hole in the valence band, which also moves freely. At energies insufficient to produce
complete ionization the electron stays bound to the hole and is called an exciton
(Figure 2.23) [14] [19].

In a pure crystal, when the electron de-excites and returns to the valence band,

a photon is emitted in the ultraviolet range. To enhance the probability of visible

46

 

. 4/ Electrons \ Conduction Band

// / ///(O}f/ ExcitonBand

Excited States

 

 

 

 

 

 

 

[ of Activator
Energy Exciton [ Forbidden Band
Ground State
of Activator
\\X,‘ \\\
‘— Holes Valence Band

\ \\\ \\\

 

Figure 2.23: Valence, conductance, forbidden, and excited states of activator for a
typical scintillator material. The energy between the valence and conductance bands
is around 8 eV.

photon emission, impurities, called activators, are added to the scintillator, creating
energy states within the forbidden band through which the electron can de-excite back
to the valence band. Electron transitions produce visible photons because the energy
is less than the full forbidden band. The light yield is proportional to the deposited
radiation energy. To prevent the entry of stray light and the escape of light from the
surface, both of which can influence the results, the scintillator is surrounded by a
reflector on all surfaces except where the PMT is mounted [19].

Photons emitted from the scintillator strike the photo cathode of a PMT and
transfer energy to electrons in the photo cathode. The electrons migrate to the surface
and escape (photoemission) [19].

Once free, an electron gains energy as it travels, guided by an electric field, to the
first dynode. The creation of an excited electron in the dynode requires energy on the
order of 2 to 3 eV. An electron traversing a 100 V gap may create 30 excited electrons.

Many of these electrons will not reach the surface due to energy loss and'their random

47

motion. Others may have insufficient energy to overcome the work function once they
reach the surface. The secondary emission ratio is around 10% for 1 keV electrons
entering cesiated antimony, such as the dynodes in the PMT used in this study [19].

Secondary electrons, from the dynode surface, travel to the second dynode where
they multiply again. Voltage between successive dynodes (up to 15) range from 80
to 150 V, thus amplifying the photon energy received from the scintillator material
(Figure 2.24) [19] [14]. Magnetic fields can misdirect some electrons away from the
next dynode, so that the photomultiplier must be covered with a thin sheet of mu

metal (magnetic shield comprised of nickel and iron) [14].

Source

3

 

 

 

Reflector
Scintillator
Photocathode
Photomultiplier
Photoelectron
Dynode

 

AnOde m

Figure 2.24: Scintillator detector. A photon travels through the scintillator produc-
ing visible photons that strike the photo cathode producing free electrons. The free
electrons are multiplied by the dynode system ending at the anode where an MCA
displays a pulse proportional to the original incident photo energy.

An anode is located at the end of the series of dynodes and collects electrons.
A corresponding voltage pulse develops across a load resistor. An MCA, operated in
the pulse height analysis (PHA) mode, sorts out the incoming pulses according to
their height and stores the number of pulses of a particular height in a corresponding
channel. The MCA channels are calibrated by recording radioactive spectra of sources
with known discrete energies [19] [14].

The energy resolution of a detector is a measure of its ability to distinguish photon

peaks that are close in energy:

_ CR — CL _ FWHM

R .
Cpeak Cpeak

(2.62)

where C R and C L are the channel numbers 011 either side of the peak at half of its
maximum, Cpeak the channel number of the peak, and FWHM the full width of the
peak at half the maximum value [19] [14].

When a photon enters the scintillor material of a detector it may not be counted.
The photon may travel through the detector without having an interaction, the inten-
sity of visible light produced may be too low to be recorded, or it may be scattered out
of the detector by the detector shielding. The absolute detector counting efficiency is

defined as [14]:

number of pulses recorded

 

5 Z . . . 3
abs number of radlation quanta ermtted by source

In practice, the detector counting efficiency depends on:
1. the fraction of space that a detector subtends:

G __ area of detector face
477R2 ’

 

where R is the source detector distance;

49

2. the fraction of photons absorbed by the intervening material:

1 : e—I‘linldl x e-Mm2d2 x ...,
where 11 is the linear attenuation coeflicient of the intervening material and d
it’s thickness;

3. the fraction of photons absorbed by the detector scintillator material (M):

M’ = 1 — e—ud

7

where 11 is the linear attenuation coefficient of the scintillator and d the scintil-

lator height.

The calculated detector efficiency becomes [22]:
D = GIM. (2.63)

The measured detector counting efficiency is determined by:

P
DE 2 — 2.64
RN. < >

where P is a ratio of the number of photons in the energy peaks to the total photon
count, R the number of counts in the photopeak, and N the number of photons

emitted by the source. N is calculated as:
N : TB x T x A,

where TB is the total branching ratio, T the total counting time interval in seconds,

and A the activity of the source in disintegrations per second. The activity of the

50

source is calculated as:

t
A: .4063,

where A0 is the activity of the source when it was calibrated, t the time interval since
the source was calibrated, and r the half—life of the source in the same units as the

time interval.

Collimator

X—ray and gamma ray imaging utilize collimators since such short wavelengths cannot
be focused with lenses as are optical or near optical wavelengths. A collimator is a
narrow tube, or slit in this case, through a strong radiation absorbing material. X-
rays traveling nearly parallel to the collimator axis traverse the entire length to the
detector. Other rays are absorbed by striking the front surface or side of the aperture
(Figure 2.25). Collimators are used to improve image resolution, but they also reduce

the intensity of the signal.

 

Shielding Dem“ hielding

 

 

 

 

 

 

 

 

 

 

/

Figure 2.25: A collimator showing only x-rays traveling nearly parallel to the collima-
tor axis reach the detector.

51

The slit spatial resolution is determined using the geometry configuration of Fig-
ure 2.26 and the following calculation:
_ 261.13 — tl‘
_ t ,

The slit was traveling horizontally at an average velocity ((0)) and data accumulated

in time intervals, so the total spatial resolution becomes:

_ 2d2: — ta:

t + ('0) (time). (2.65)

Since their discovery, x-rays have been the most important medical imaging diag-
nostic tool. A gamma camera consists of one or more PMT’s mounted to a scintillator
to map radiation entering through a, usually lead, collimator speckled with thousands
of holes (Figure 2.27). N 0 material totally attenuates photons so the crosstalk between
holes produces a blurred image. Gamma cameras are used to image patient organs,
track blood flow, scan shipping containers for radiation, and scan nuclear warheads
for fissile materials.

Nuclear medical imaging utilizes gamma cameras in the Single Photon Emission

 

A
L< -.
V

 

 

 

6

 

 

 

 

 

 

._x_.. 1

 

 

Figure 2.26: Geometric configuration of slit spatial resolution. The slit spatial resolu—

tion is y, the lead brick shielding thickness is t, and the detector to source distance
is d.

52

.u .I"II.I'IIJ{':'I.$£$‘:{{':'3.1-'3'2-n":":.':'$ Lead Shield
lectronics
Photomultiplier Tubes
Light Guide

3"” ‘3’3‘3‘3‘3‘

i1
. .2 .
o °.:.:.~.~.‘ 2.3 ~ 02352:? . 2 22

’5' .
.. . . ..
2... ..............2..
.' O. . 5: . . . .,.,...,.....,.,.,:,. . ..
..m....:. mu:
25:232.... m
. . .. N. . .‘.~.’.*.~»2. 3 22° 02* .,

Scintillator

Multihole
Collimator

 

Figure 2.27: Basic diagram of a gamma camera.

Computed Tomography (SPECT) scanning device. Patients are injected with a short-
lived radioactive tracer isotope mixed with a molecular probe. The isotope emits
gamma rays as it decays. The gamma camera is rotated around a patient. Multiple
images at multiple angles are obtained. A tomographic reconstruction algorithm yields
a 3 dimensional image.

In optics, a collimating lens focuses light to form a parallel beam. Medical colli-
mators are not only used for detection of radiation but for beam delivery as in proton

radiotherapy.

2.3 SRF Cavity Testing

The purpose of testing a cavity is to determine its response to rf fields. The unloaded
quality factor or quality factor of the cavity itself (Q0) as a function of the peak
electric field (Ep) is considered the most important cavity evaluation.

Q0 of a cavity can be determined either in a modulated, or pulsed, wave mode or
a continuous wave mode. Constants, determined by the modulated wave mode, are

needed for the calculation of Q0 in the continuous wave mode. Thus, the calculation

53

of Q0 for the modulated wave mode is presented first.

2.3. 1 Modulated Wave

During cavity testing the frequency, forward power (P f), reflected power (Pr), trans-
mitted power (Pt), standing wave transmitted power (13,;(5' l'V)) are recorded. Power

measurements are made in dBm and converted to Watts:

P(dBm)+AgdBm)
10 10

1000 ’

 

 

.muvz use

where A f, Ar, At, and At(S W) are the coupling factors for each of the measured
quantities.

The calculation of the electric field on axis (E0) of the cavity, operated in the
modulated wave mode, begins with the calculation of the loaded Q (Q L) or the
quality factor of the total system, circuitry plus cavity. When power is removed from
the cavity, the time taken for the cavity’s stored energy to dissipate (TL), or ”ring-

down”, is measured and Q L is calculated:

QL=qu wen

where w is the angular frequency.
Next, the coupling strength (’3) of the cavity is calculated from the measurements

of Pf and Pr (Figure 2.28):

tw—
: yfifisflg
Q0

2 ___, eta
Qeth

 

where the upper sign is for an overcoupled cavity ([3 > 1) and the lower sign is for an

54

Transmitted
Power
Meter

Output
Coupler

 

Reflected
Power
Meter
\l\
Il/
Amplifier
Forward
Power
Meter

Figure 2.28: Diagram of measured quantities used to calculate Q0.

undercoupled cavity (,6 < 1), Q0 the quality factor of the cavity itself, and QextJ a

quality factor of the input coupler.

The quality factor due to each loss mechanism:

Ptot _ Pd + P6 + Pt
wU _ wU ’

 

 

(2.69)

where P8 is power emitted back out the input. coupler when Pf = 0 (switched off),

and U is the stored energy of the cavity. Pd is the power dissipated in just the cavity

Pd 2 Pf — Pr — Pt — Pd(SW), where Pd(SW) is calculated from PASVV). The ratio

55

of Pd(Sl/V) to PdSl/V) was found to be. 19.98. Equation 2.69 becomes:

 

 

1 __ 1 + 1 + 1
QL Q0 Qexm Qt
~ 1 1
Q0 Qeztd
— 1 + ’8 (2 70)
Q0 Q0’ '
where Q1? is negligible. Q0 is calculated:
Qo = QL(1+ fl) (2.71)
The stored energy of the cavity is calculated:
P
U = —Q0 d. (2.72)
w

2.3.2 Continuous Wave

Knowing U, Ea and Q0 can be calculated for the continuous wave mode. First a

quality factor for the transmitted power is assumed:

wU

Qt = 7.5;, (2.73)

where U was calculated from data acquired in the modulated wave mode and Pt

measured. A constant, k1, is defined as:

SO, «(7 = klfpg.

Since U o< E3, another constant, k2, is defined by the output from SUPERLAN S:

The calculation of Ea is now possible:

Q0 is calculated:

Ea

Q0

 

Ep 2 3.34530.

_Pd_

57

wish/Pt
Pd '

(2.75)

(2.76)

(2.77)

(2.78)

Chapter 3

Design

Field emission (FE) in superconducting radio frequency (SRF) cavities is a primary
factor limiting the surface electric field. The field-emitted electrons are accelerated by
the rf field and strike the cavity wall, generating bremsstrahlung x-rays of a continuous
spectrum (Section 2.1.3). The purpose of this study was to locate the source, intensity,
and energy of the X—rays then, with a particle tracking code, Show that the origin of

the field-emitted electrons could be found.

3. 1 Experimental Setup

3.1.1 Cavity and Cryostat

A geometric beta [3920.47 805 MHz six-cell cavity was designed and fabricated
by a collaboration between Thomas Jefferson National Accelerator Facility, Istituto
Nazionale di Fisica Nucleare Milano, and the NSCL for the Rare Isotope Accelerator
(RIA) (Figure 2.5).

The cavities in this study were constructed of 4 mm thick sheet niobium with a
nominal residual resistivity ratio (RRR) of 250. RRR is the ratio between the electrical
resistivity at 273 K and the resistance in the normal conducting state extrapolated

to 4.2 K and zero magnetic field [23]. The Holder Corporation of Lansing, Michigan

58

water-jet cut disks from sheet Nb. The (lies used to form the cavity half cells were
machined at the NSCL. The following procedures were performed at J LAB by NSCL
faculty and staff. The disks were standard deep drawn into half cells and the equator
and iris edges machined prior to electron beam welding.

After electron beam welding, the cavities were etched with a 1:1:1 buffered chemi-
cal polishing solution of phosphoric acid, hydrofluoric acid, and nitric acid [24]. Etch-
ing removes a potentially damaged layer as well as any evaporated niobium that
may be deposited on the surface during welding; both are potential sources of field
emission [4].

The concentration of H within a niobium cavity can increase during etching. Q
disease, or residual losses from hydrides, is the accumulation of hydrides at the rf sur-
face as a cavity is cooled for testing. This increases the surface resistance and, in some
cases severely, lowers the cavity Q. Between 150 and 60 K the rate of accumulation is
greatest. Therefore, a cavity is cooled as quickly as possible between this temperature
range. The purpose of heat treatment of the cavity is to degas the H [4]. After heat
treatment the cavities were etched once more then rinsed with ultra—pure water [24].

The cavities were then tuned for field flatness in a jig, designed and custom-built
for the 09:0.47 cavity by N SCL staff. It is desirable to have equal accelerating fields
in all cells of a cavity. This is accomplished by securing one end iris (Figure 2.5) and
squeezing or stretching the cavity. The field flatness, A1;qu was brought to near 5%
for each cavity [24].

The cavities were tested in a vertical cryostat to verify their rf performance. The
accelerating electric field (Ea) was limited by the rf power available from the equip—

ment. One cavity only reached a Q of 8 x 109 so it was reetched and retested. During
M

m and a Q = 1010 were achieved. The other cavity had

the second test an Ea = 8
similar results [24]. Each cavity was housed in a titanium helium vessel with a wall
thickness of 0.48 cm and shipped to the NSCL, under partial vacuum, by truck [25].

At the NSCL, the helium vessels were surrounded by two n-metal shields that

 

Helium Dewar

 

 

 

 

Support Link

H

J/ Thermal Intercept Shield

m . . 9 w/ Inner Magnetic Shield

I Outer Ma netic Shield
r// g

 

 

 

 

 

 

 

 

 

 

Helium Vessel

 

\
\

T Vacuum Vessel

 

 

 

 

 

 

 

12 incnes

30 cm

Figure 3.1: Each cavity is housed in a titanium helium vessel, surrounded by two
p-metal shields and a copper thermal shield, inside a rectangular vacuum vessel.

reduce stray magnetic fields to 0.5-1 MT. The u-metal shields also serve as passive
thermal shields. A 0.16 cm thick copper sheet between the ,u-metal shields, served as
the thermal shield. Liquid nitrogen was used in the thermal shield. The outer rectan-
gular vacuum vessel was made of 1.91 cm thick low-carbon steel plates (Figure 3.1,
3.2). The cryostst absorber materials encountered by photons in this study are given

in Table 3.1 [21].

60

LNZ Supply/Return

    

Sensor & Instrumentation Port

\

  
 
 
  
  
 
  
 

Helium Supply

Helium Vessel

Vacuum Vessel

Alignment
Viewport

Titanium

Alignment
Rails

Beta = 0.47

 

 

Superconducting
Cavity
\
I 12 inches Fundamental Power Coupler 8:
Moveable Transmission Line
30 cm Transformer

Figure 3.2: Each cavity is housed in a titanium helium vessel, surrounded by two
u-metal shields and a copper thermal shield, inside a rectangular vacuum vessel.

Prior to testing, the cavitites were rapidly cooled to 20 K to avoid Q disease.
The cavities became superconducting (9.25:1:002 K) after approximately 5 hours.
Cooling proceeded to 4.3 K then to 2 K by vacuum pumping on the helium resevoir.
Vacuum pumping lowers the pressure on the liquid helium, causing it to boil at a
lower temperature. Heat is removed as the liquid reverts to a gaseous state.

Radio frequency power was supplied to each cavity via a coaxial cable in a trans-
verse electromagnetic (TEM) mode (Section 2.1.1). The center conductor couples
power to the input coupler, which protrudes into the beam tube, and excites a stand-

ing wave in the coupler (Figure 3.2). The outer conductor is connected through the

61

Table 3.1: Absorber materials.

 

 

 

 

 

 

 

 

 

Material t [cm] p [571—3] ,uma 9132— film [elm]
Nb 0.4 8.57 0.0587 0.503
He(liquid 2.2 K) 2.54 0.147 0.0636(gas) 0.009
Ti 0.48 4.54 0.0589 0.267
mu(80% Ni, 20% Fe) 0.10 8.70 0.0613 0.533
Ni - 8.90 0.0616 0.548
Fe - 7.87 0.0600 0.472
Cu 0.16 8.96 0.0590 0.529
Steel(75% Fe, 17% Cr, 8% Ni) 1.91 7.84 0.0600 0.470
Cr — 7.18 0.0593 0.426

 

 

power coupler to the cavity. A fixed output coupler, or transmitted power probe, picks
up power transmitted through the cavity. A sliding short provided input coupling ad-
justment. Forward power (Pf), reflected power (Pr), transmitted power (Pt), and the

standing wave transmitted power (Pt(SI'V)) were recorded during testing.

3.1.2 Detector and Collimator

The continous spectrum of x-ray photons of interest in this study were produced at
field-emitted electron impact sites. The field emitted electrons were emitted While
the cavities were rf tested. The cryostat materials and lead shielding surrounding the
detector imply a significant probability of an x-ray interacting before reaching the
detector (Subsection 2.1.4).

In this study the regions of x-ray emission were located by using a shielded, slit
collimated NaI detector placed outside the cryostat. The front of the detector assembly
was positioned 4 cm from the outer cryostat wall and moved perpendicular to the slit
orientation along the entire cavity length (Figure 3.3).

The single rail system was mounted on a 22.9 cmx 128.3 cmx 1.9 cm aluminum
platform. The 1.9 cm diameter shaft, Metric Shaft Support Blocks, and Super Ball
Bushing Bearing Pillow Blocks were purchased from Thomson Industries, Inc. (Port

Washington, NY). A 0.6 cm thick aluminum plate served as the detector-assembly

62

Sodium Iodide
Detector . .
Lead Shielding Slit Collimator

 

Slide Rail

6 Cell Prototype
Cryostat Nb Cavity Drive Motor

Figure 3.3: Schematic drawing showing an overhead View of the vertically oriented
slit collimator moving parallel to the cryostat wall.

table. The plate was mounted to the shaft by four bushing pillow blocks and supported
on the platform by three, 2.5 cm diameter roller bearings. The rail allowed 90.2 cm
travel of the table (Figure 3.4).

Table motion was accomplished by a 0.6 cm Berg drive chain (W. M. Berg, Inc.,
East Rockaway, NY) attached to the table and a 116 HP Dayton gear motor (Dayton
Electric Mfg. Co., Chicago, IL). The platform sat atop a Unistrut support system
assembled so the collimator slit was aligned with the cavity axis. Figure 3.5 shows a
photograph of the entire x—ray scanning device.

Motion control was achieved by a manual start and stop station attached to
the gear motor by 8 m of liquidtight flexible metallic conduit. Limit switches were
mounted near the rail supports to prevent table overtravel. The wiring diagram is
shown in Figure 3.6.

This study required the knowledge of x-ray photon energies and their number.

This eliminated the possibility of utilizing a Geiger-Muller counter since their signal

63

Slit
Collimator Drive Gear
Chain Motor

 

 

l"
[l]

  
  

   

Shielding

 
   
  

Slide
Rail Roller
(b) Bearings
Gear
Motor

Detector
Housing

   

Figure 3.4: a) Front view of the slit collimator slide table. b) Back view of the slit
collimator table.
is independent of particle energy. Proportional counters were not a possibility since
they detect charged particles. Ionization chambers are used mainly for detection of
alphas, protons, fission fragments, and other heavy ions. A scintillation detector, with
required hardware, was deemed most appropriate since both x-ray photon energies
and their number could be measured [14].

The choice of scintillator was restricted by the goals of the study, specifically scin-

tillator linearity over a wide range of x-ray energy, scintillator efficiency, x-ray count,

64

Cryostat Drive Motor Slide Rails

Lead Shielding

 

 

Figure 3.5: Photograph of the x—ray scanning device situated for a cavity scan.

and cost. Scintillator efficiency is the fraction of incident radiation that is converted
into visible light [19]. The inorganic scintillator, sodium iodide (Nal), was chosen over
plastic organic scintillators since it has better light output and linearity. High energy
resolution was not necessary for measurement of a continuous spectrum so Germa-
nium (Ge) was not chosen as the scintillator. Also, Nal does not require the constant
cooling by liquid nitrogen that is required of Ge [19]. A complete arrangement consist-
ing of an ORTEC 50 mm X 50 mm integral Nal crystal, PMT, and digital portable
MCA (digiDART) was selected. The detector and portable MCA are indicated in
Figure 3.5.

For this study the NaI(T1) detector was surrounded by lead shielding with an

isosceles trapezoid shaped slit open to the detector face (Figure 3.7). The lead shield-

65

 

 

 

 

 

 

 

 

 

     
 
 

 

 

 

 

 

 

 

 

Transformer
Forward
£9.11“)
‘57
Closed Forward
Forward
‘ .l/ 4 (LP 5
DPIDT BLACK
[______________l Closed
I Reverse
[ 8 I YELI;_<>W
l
I [\Y MOTOR
Clo‘sed'”"i{ ‘‘‘‘‘‘‘
Reverse everse
6 (.21.
DPDT Reverse
LED
0

Figure 3.6: Wiring diagram of the manually Operated gear motor driving the slide
table.

ing was thick enough that photons from a 20 u Cu 60Co radioactive source had a
very low probability of penetration.

Slit collimation of the x-rays was achieved by placing sheet lead pieces, shaped
to detect x-rays from full cavity diameter, between two lead blocks. The lead blocks
were water-jet cut at Holder Corporation of Lansing, Michigan. The smooth surfaces
provided a tight fit, and minimized x-ray entry. Blocks were also cut to house the de-
tector. A schematic diagram of the slit collimator and shielding is shown in Figure 3.7.
The slit was oriented vertically and moved on slide rails parallel to the cryostat wall
as shown schematically in Figures 3.3 and 3.4. A photograph of the detector assembly
is indicated in Figure 3.5.

The portable MCA was interfaced to a local computer with a USB cable. Con-
tinuous accumulation of x-rays in two-second time intervals was achieved through a

LABVIEW 7.0 program. LABVIEW labeled each two-second time interval file with

66

 

 

Front

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\
\ X\£).5
\ \
_ |<——— 30.5 ——>|<- 10.29] \\ 2
t..- \ \
20.3 813 _~_____:Ql 10-2 \\ 203
—""—"“‘m \\1‘22‘3 Slit \ i
K
\ / —)||<—0. 18
Sheet Lead - 20.3 '
0.18 / T
\ 1042
A Back
%
\{05
\
\\
\ \ Detector Opening
\\ \
\\ \ ' 20.3
\ l l l

 

 

 

 

[9162803334

Figure 3.7: Schematic diagram of slit collimation and lead shielding, dimensions in

cm.

the date and time and stored the files on the NSCL network.

67

Chapter 4

Experimental Results

The data for this study were collected at the National Superconducting Cyclotron
Laboratory (NSCL) situated on the campus of Michigan State University. At the
time of this study, the N SCL pursued a full range of Rare Isotope Accelerator (RIA)
related research, including the development of superconducting radio frequency (SRF)
accelerating structures. Superconducting (niobium) cavities were tested under contin-
uous wave (CW) conditions. To test a cavity, radio frequency (rf) power was supplied
to the structure while in vacuum and at superconducting temperature. When the
quality factor (Q) began to drop and x-rays were measured on the general radiation
monitoring equipment, field emission was present. The field emitted electrons, acceler-
ated in the cavity electric field, struck the cavity walls and generated bremsstrahlung

x-rays. These were the x-rays of interest in this study.

4.1 Detector Calibration

4.1.1 Energy

The energy per channel relationship of the NaI(Tl) detector was calibrated using
137Cs (662 keV), 60Co (1173 and 1332 keV), and 22Na (511 and 1280 keV) radioiso-

tOpe sources (Figure 4.1 (a)). The peak channel number of the radioisotope was found

68

(b)

 

 

 

 

E 1500 ~

3 S y = 7.6771x - 55.396

0 a)

'0 at. 1000 -

E a

m a 500 —

E c:

5 LU

z 1 1 r 1 1 0 l f l 1

0 50 100 150 0 50 100 150 200

Channel Number Channel Number

Figure 4.1: Panel (a): Calibration sources: circles are 22Na, squares are 137Cs, and
diamonds are 60C0. Panel (b): Plot of channel number versus energy of the peaks in

panel (a).

by applying a Gaussian fit to the peak with the scientific graphing and analysis soft-
ware, ORIGIN. Table 4.1 shows the channel number and the associated energy. A
linear trendline applied to the data produced the equation y = 7.67711: - 55.396,
which was used to calculate the energy per channel of the detector. Zero energy was

found at channel 7.22 (Figure 4.1 (b)).

Table 4.1: Detector Calibration

 

 

Channel Number Energy [keV]
73.4 511
94.2 662
158.5 1173
176.3 1280
179.5 1332

 

 

 

 

The energy resolution was calculated using Equation 2.62 and determined to be
7.3% for the 13705 at channel 94.2 (662 keV), with a FWHM of 6.9 channels. Both
the peak channel number and the F WH M were determined using ORIGIN.

4.1.2 Spatial Resolution

The thickness of the lead shielding (t) around the slit measured 10.2 i 0.2 cm and
the slit width (x) 0.16 :l: 0.02 cm (Figure 3.7). The most intense source of x-rays

was presumed near an iris, therefore the source-detector distance (d) varied from

69

900i

800; ji

1 /
7004 / \\\
600$ // f

. / \
500 -

. /i ~
/ \
300 d

2003

X-ray Count - Background

A
O
O
t
\lll

O
‘ I

 

 

761.5.77.017752307735290'79'.5'80.0'80.5'
Distance [cm]

Figure 4.2: High x-ray flux peak centered at 78.6 cm for the scan of cavity #2, 7r
mode. F WH M = 1.3 i 0.1 cm. Error bars are included in the figure (Equation 4.2).
39 i 0.5 cm to 47 :l: 0.5 cm. The collimator was also traveling at an average velocity
of 0.22 :l: 0.02 %. Data were acquired in two-second intervals, making the total
calculated spatial resolution (Equation 2.65) 1.5 i 0.2 cm to 1.8 i 0.2 cm.

Spatial resolution was measured using the x—ray peak centered around 79 cm of
one of the test cavities operated in the 7r mode (Figure 4.8 (c)). The data were plotted
in ORIGIN and, using a Gaussian fit, the centroid was found to be at 78.6 cm and
the F WH M was found to be 1.3 :l: 0.1 cm (Figure 4.2). The calculated and measured
F WH M values fell within the estimated error. The difference between the calculated
and measured F WH M is due to the uncertainty in the distance between the x-ray
source (electron impact site) and the detector. It has been assumed that electrons
impact at the iris. Electrons impacting off iris in the cell, and closer to the detector,

would yield a smaller calculated spatial resolution.

70

4.1.3 Spectrum

The cryostat absorber materials encountered by photons as they traveled to the de-
tector are given in Table 4.2. For an initial number of photons, 10, with energy 1 MeV:
[Final : [Ge—(‘the-“Hete_#T-ite_Mmute-flCute—#mute—Nsteclt

= 10(027).

Around one fourth of the 1 MeV photons traverse the cryostat materials without an
interaction. The number of interactions increases for lower energy photons and I Final
decreases.

A continuous x-ray energy spectrum of the high x-ray flux peak centered at 78.6 cm
(cavity #2, 7r mode, Figure 4.8 (0)) and scattered through the materials listed in
Table 4.2, is shown in Figure 4.3. X-ray energies are plotted along the abscissa and
x—ray count with background subtracted along the ordinate. The background count

of x-rays (y) was subtracted from the data (x):

 

 

 

 

'u : 1r — y. (4.1)
Table 4.2: Absorber materials.
2
Material t [cm] p [351-3] pma %— Win [513,-]
Nb 0.4 8.57 0.0587 0.503
He(liquid 2.2 K) 2.54 0.147 0.0636(gas) 0.009
Ti 0.48 4.54 0.0589 0.267
mu(80% Ni, 20% Fe) 0.10 8.70 0.0613 0.533
Ni - 8.90 0.0616 0.548
Fe - 7.87 0.0600 0.472
Cu 0.16 8.96 0.0590 0.529
Steel (75% Fe, 17% Cr, 8% Ni) 1.91 7.84 0.0600 0.470
Cr - 7.18 0.0593 0.426

 

 

 

 

 

 

71

 

 

 

 

 

 

 

 

 

 

 

 

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‘ [11“,].
40 ‘1IIIIII
.. I !l'l|||lillll]lzn'“
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8 10‘ Ill! "I” fill-InJ ll pil'llillllh
o I “'1'!“ ”If ..E :m. .
0- i Ink-‘1 l‘F:
-10 , e , - . .
0 200 400 600 800 W1000 1200 1400 1600 1800
Energy [keV]

Figure 4.3: Continuous x-ray energy spectrum of the high x-ray flux peak centered at
78.6 cm in cavity #2, 7r mode. Lead shielding in front of the slit collimator attenuates
40% of the 400 keV x-rays. Error bars are included in the figure (Equation 4.2).

Uncertainty was calculated from the error propagation formula [19]:

2 2 2
2 _ Bu 2 an 2 8a 2
an — (79$) 03; + (7%) 03/ + (792) oz + , (4.2)

where u = u(:r, y, z, . . .), and 03;, cry, 02, . .. are the errors of the variables 51:, y, z
A lead shield 0.34 cm thick placed in front of the collimator attenuates 40% of the
400 keV x-rays. Figure 4.3 shows a continuous x-ray spectrum for one high x-ray flux
peak.

The low energy of the characteristic x-rays (Sections 2.1.3 and 2.1.4) of Nb could
not be distinguished from the continuous x—ray spectrum. There is also a high prob—
ability of them having an interaction in the cryostat materials. The energy of char—

acteristic x-rays of Nb are shown in Table 4.3.

72

Table 4.3: Characteristic x—ray energies of Nb. * denotes a level which is unresolved
from the level above it [26].

 

 

 

Level Energy (eV)
K 18985.6
LI 2697.7

LII 2464.7

LII] 2370.5
1141 468.4

M11 378.4

M111 363.0

MIV 207.4

MV 204.3

NI 58.1

N11 33.9
N11] 339*
NIV,V 3.2

 

 

 

4.1.4 Detector and Collimator Performance

The performance of the detector and collimator was initially tested by scanning a
60C0 radioisotope source mounted on a 2.54 cm diameter disk. A diagram of the
arrangement, not to scale, is shown in Figure 4.4 (a). As the slit collimated Nal
detector moved past the radioisotope source, x-ray spectra (energy versus number of
x-rays at that energy) were collected in two-second intervals. Figure 4.4 (b) shows
the total number of x-rays entering the detector in each two-second interval. The
detector-collimator assembly moved 0.32 cm per two-second interval (0.16 9%). Error
bars are included in the figure (Equation 4.2). ORIGIN was used to measure the
F WH M weighted by the error. The F WH M was measured as 2.75 cm, a difference
of 8% from the calculated value.

A diagram of the slit collimated detector scanning the cryostat, not to scale,
is shown in Figure 4.5 (a) and (b). This simplified diagram shows two electrons
leaving one defect and impacting at two different sites. In reality, large numbers
of electrons are field emitted from defects, have complicated trajectories, and may

impact anywhere within the cavity or connecting structure.

73

 

 

[___
(a
. Radioisotope Source
Slit (disk)

 

 

I
l
l
l
Collimator :
l
l
n
l
I

——+———————————

 

 

 

 

” Direction
m of Travel

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/ —....r-——-——

  

S k Detector } Lead Drive Drive
proc et { Shielding Chain Motor
1 i
l s
l I
I I
(b) ‘ l {
14000 i :7 f
12000: i ._ l
E 10000- i 31,:
= * :2 ii
3 80001 H: i.
cu . -
“.- 4000 ’5‘
x 20001 "

 

 

 

o .--..---.....---.s
0 2 4 6 810121416
Distance[cm]

Figure 4.4: Panel (a): Diagram (not to scale) of a slit collimated detector scanning a
radioisotope source. The disk is 2.54 cm in diameter. Panel (b): Total number of x-
rays entering the detector per two-second interval. The detector-collimator assembly
moved 0.32 cm per two-second interval. The FWH M was measured as 2.75 cm. Error
bars are included in panel (b) (Equation 4.2).

74

The predicted performance of the slit collimated detector scanning the cryostat
is shown in Figure 4.5 (c). A silhouette of the cavity has been superimposed on the
mock data. \IV hen the slit is not opposite an x-ray source, x—rays that are primarily
Compton scattered in the cryostat materials can travel through the slit and enter the
detector. X-rays can also travel through the lead shielding and be Compton scattered
or experience no interaction. To compensate for the x—rays entering the detector from
the lead shielding, data acquired during a scan of the cryostat with the slit closed

(hereafter referred to as background) is subtracted from the scan with the slit Open.

75

(a) Defect

U
>
"A
WWW/WWW
(b)

 

 

 

a:

 

 

 

 

Direction
of Travel

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

I
I
l I
l l \ Collimator
I 1 h
\ ad Drive Drive
SPYOCRC‘ DC‘CC‘O‘ Shielding Chain Motor

A
O
v

X-rayCount
Amwammxiooco
OOOOOOOOOO
l.L I 1A1 I l‘

 

I v

I

I A O I...
0 10 20 30 40 50 60
Distance [cm]

Figure 4.5: Panels (a) and (b): Diagram (not to scale) of a slit collimated detector
scanning half of a cryostat containing two six-cell cavities. Panel (c): Predicted spec-
trum of the total number of x-rays entering the detector per two second interval. A
silhouette of the cavity has been superimposed on the mock data.

76

4.2 Cavity Testing

4.2.1 Cavity #2 Performance

The detector assembly and unistrut stand were positioned so that the front of the
collimator was 4 cm from the cryostat wall opposite cavity #2. Both cavities were
cooled to 1.99 K. Forward power to cavity #2 (7r mode) was raised until x-ray detec-
tion on the area radiation monitoring equipment measured an x-ray exposure rate of
25 yfiTR, signifying the drop in Q to be due to field emission. Forward power was held
nearly constant to maintain that level of x—ray exposure rate.

During the test, the forward, reflected, transmitted, and standing wave transmit-
ted power, bath pressure, frequency, liquid helium level, and cavity temperature were
recorded. From this data (7r mode), the cavity Q was measured (Equation 2.78))
as 1.25 x 1010 while Ea was held nearly constant at 7.6 kill—1V- (Equation 2.76), or
Ep 2 25.5 319—] (Equation 2.77) . The Q for cavity #2, for this study (7r mode), was
similar to the Q obtained from the previous cavity test (7r mode) (Figure 4.6 (a)). The
x-ray exposure rate for this study (7r mode) and the previous cavity test (7r mode)
are also shown to be similar (Figure 4.6 (b)). The x-ray exposure rate was difficult to
measure since the needle on the meter was in constant motion. (The x-ray exposure
rate appears to level off at 10. 000 11%, which is the scale maximum of the meter.)
The two plots demonstrate that no major changes in cavity performance occurred
between tests.

For this study the electric field was held nearly constant, thus a direct calcula-
tion of the Fowler-Nordheim field enhancement factor (L3 F N), needed for the particle
tracking code, could not be made. (3F N was inferred‘from 7r mode data acquired
during two previous tests and compared with data (7r mode) for this study (F ig—

ure 4.7)(Equation 2.35). ,B F N for cavity #2 operated in the 7r mode was in the range

120 S B F N g 220. This range was also used for the 5567: and 437: modes.

77

1.E+11 -

O1.E+10 0”... ° .’9

 

 

1.E+09 E I T r 1
0 10 20 30 40
Epeak[MV/m]
10000 — (b) .m-
0
1000 a
g 0
E, 100 a .-
E 10 -
>'< 1 7i "“" I I I I

 

 

0 1 0 20 30 40
Epeak [MV/m]

Figure 4.6: Cavity #2 performance, 7r mode. In both panels, diamonds indicate the
previous cavity test. The square indicates the measured quantity for this study. Panel
(a): Peak electric field versus Q. Panel (b): X-ray exposure rate detected on the area
radiation monitoring equipment when the observed drop in Q was due to field emis-

sion. The area radiation monitoring equipment has a maximum scale of 10,000 thB_
The two plots demonstrate that no major changes in cavity performance occurred

between tests.

1E-24
1E-25 ~
1E-26 -
1E-27 q
1E-28 -
V 1E-29 . I x .

0.025 0.035 0.045 0.055

1/Ep [1/MV/m]

    

Pd-Po)/(d*Ep"3.5)

 

 

Figure 4.7: Cavity #2, 7r mode, Fowler—Nordheim Field Enhancement Factor. Dia-
monds and squares represent data from two previous cavity tests; the triangle repre-
sents data from this study (7r mode). 120 3 3FN g 220.

78

4.2.2 Cavity #2 7r Mode

Two scans of the cryostat were made with the collimator open to show that the
data were reproducible (Figure 4.8 (a)). One scan was made with the collimator
closed (background) (Figure 4.8 (b)). The background was subtracted from the data
acquired with the collimator open (Figure 4.8 (c)). High x-ray flux peaks appear at
three irises (28, 37, and 46 cm). Peaks also appear in each beam tube, indicating
electrons traveled out of the cavity and impacted in the beam tube.

The highest probability for x-rays to enter the detector without having had an
interaction, is when the open slit is opposite the x-ray source (electron impact site).
To determine if the highest flux peaks in the data are real x-ray sources, and not
just x-rays scattered into the detector, energy binning was employed. Energy binning
is the method of examining the number of x—rays in a particular energy range. Real

x-ray sources contain the highest number of high energy x-rays.

79

1000 — (a)
800 -
600 ~
400 ~

200 Ly)

(b)

 

 

X-ray Count

 

 

200

l

150 ~

100 4

X-ray Count

 

 

 

1000 (c)

__l

800

600 r
400 a

 

 

N
O
O

l
”\
\
""‘\
H\

 

X-ray Count - Bkg

w

I

0 20 40 60 80
Distance [cm]

_1
‘5
5..
V‘—

—4
r.
K,
V'"
‘..

Figure 4.8: Cavity #2 operated in the 7r mode. X-ray count acquired in two-second
intervals. Panel (a): Slit of the collimator open. Panel (b): Slit of the collimator closed
(background). Panel (c): X-ray count with the background subtracted. A silhouette
of the cavity has been superimposed on the data.

80

500
j
400 ~
350 J
300 j
250 —
200 J
150 —.
100 -
50 4 j 9“.

7‘, in
" ELM ‘55—“; :15.»- xx.
0 -, . “AWL-Fr “:9"’¢4Afi .wl 7 my; ”msa‘ré’f‘fi-‘q
+ - .. _ . .*M ‘71 __‘ . v . v , 7

0 20 40 60
Distance [cm]

 

 

X-ray Count - Bkg

 

 

Figure 4.9: Cavity #2, 7r mode, energy binning of data presented in Figure 4.8 (c).
The black line represents the number of x—rays at all energies, the light-gray line
represents the number of x—rays with a minimum energy of 400 keV, and the medium-
gray line represents the number of x-rays with a minimum energy of 600 keV. A
moving average was applied to the data to smooth inconsistencies between adjacent
two—second intervals.

Figure 4.9 shows the energy binning for the data presented in Figure 4.8 (c). As
higher energies are binned, the x-ray flux decreases. The x-ray flux peak around 18 cm
has dissappeared at 600 keV, indicating an endpoint energy between 400 and 600 keV.
X-ray flux peaks at 28, 37, 46, and 79 em all have endpoint energies over 600 keV.
Endpoint energy will be considered when compared with the results of the particle
tracking code (Chapter 6). To better pinpoint endpoint energy, minimum energies of
1000 keV, 1100 keV, and 1200 keV are plotted in Figure 4.10. It appears that the
endpoint energy for this peak is around 1100 keV since the peak is indistinguishable
at 1200 keV.

81

15 r (a) 1000 keV

—L
O
1

X-ray Count - Bkg
0 (J1

 

0 20 40 60 80

T (b) 1100 keV

X-ray Count - Bkg

 

-4 I T I
0 20 40 60 80

 

3 r (c) 1200 keV

X-ray Count - Bkg
O

 

 

'3 1t—’_‘ r I T
0 20 40 60 80
Distance [cm]

Figure 4.10: Energy binning. Panel (a): 1000 keV minimum x-ray energy. Panel (b):
1100 keV minimum x-ray energy. Panel (c): 1200 keV minimum x-ray energy. The
moving average of the data is shown. The endpoint energy for the peak at 78 cm is
around 1100 keV since the peak is indistinguishable at 1200 keV.

82

An alternative evaluation of endpoint energy was achieved by a linear projection
of the high energy range of the x—ray spectrum. Each high x-ray flux peak in Fig-
ure 4.8 (c) is a collection of two—second energy spectra. The sum of individual energy
spectra was taken for each high x—ray flux peak. The energy spectrum for the high
flux peak at 78.6 cm is shown in Figure 4.11 (a). The endpoint energy projection,
for the same peak, is shown in Figure 4.11 (b). The endpoint energy obtained using
this method is 1062 keV. For consistency, the energy range considered for the linear
fit was chosen at an energy greater than the peak of the spectrum to a value where
the “X-ray Count - Background” becomes negative (Figure 4.11 (a)). This method is
in good agreement with the energy binning method where the endpoint energy was
found to be near 1100 keV. The endpoint energy of each peak and peak location
(centroid) in cavity #2, 7r mode, can be found in Appendix F.1.

To check the linear projection method, the detector was positioned at the location
of the highest x—ray flux peak in the 7r mode (79 cm). Data were acquired for 30 second
intervals with the detector assembly stationary. The energy spectrum and endpoint
energy projection are shown in Figure 4.12 (a) and (b). The endpoint energy for the
30 second time interval was found to be 1084 keV. The good agreement between
the three methods demonstrates that linear projection is an acceptable method in

determining x-ray endpoint energy.

83

X-ray Count - Background

 

 

 

l ' I ' l
500 1000 1500
Energy [keV]

o-

40 — (b)

N N w
0 01 O
I . I . I

X-ray Count - Background
E5
1

ill 5 _,
l lll" Illiiii Iii h,
l I l:

0'!
#1

I
II
‘I

O

 

I
01

 

600 800 1000
Energy [keV]

Figure 4.11: Cavity #2, 7r mode. Panel (a): X—ray energy spectrum for high x—ray
flux peak at 78.6 cm. Data were acquired during a scan of the cavity. Individual two-
second spectra within the peak were summed. Panel (b): Endpoint energy determined
by the linear projection method was found to be 1062 keV. Error bars are included
in the figures (Equation 4.2).

84

200 -

150-

1004

50-

X-ray Count - Background

 

 

 

 

 

 

 

 

 

0 t 500 ' 1000 ' 1500
Energy [keV]

1803 (b)

1603

140; “F

1203

1005

323 -- .igitg

20$ " fl

0 _ -- "I 0 m

 

 

X-ray Count - Background
m
‘?
I-EH

 

 

 

 

 

 

 

 

 

 

 

600 800 1000 1200
Energy [keV]

Figure 4.12: Cavity #2, 7r mode. Panel (a): X-ray energy spectrum for high x—ray flux
peak at 78.6 cm. Data were acquired with the detector stationary at 78.6 cm. Panel
(b): Endpoint energy determined by the linear projection method was found to be
1084 keV. Error bars are included in the figures (Equation 4.2). Some error bars are
larger due to the increased background count.

85

The linear projection method accounts for pile-up or summation effects in the
detector. Electrons, from a single emitter, are field emitted at differing times during
half of an rf period. The electrons are accelerated by the electric fields and guided
by the magnetic fields within the cavity. Electron trajectories are complex and, as
a result, impact at different sites at different times and with different final kinetic
energies. The electrons travel through the material they impact and generate x—rays.
The x-rays travel through the cryostat materials having interactions described in Sub-
section 2.1.4. This leads to the arrival of x-rays at the detector that are randomly
spaced in time. X-rays arriving at the detector sufficiently close together in time are
treated as a single event by the analysis system, called peak pile-up or summation
effects. Not only are counts recorded at higher energies, but removed from lower en-
ergies. The method of linear projection to estimate the x-ray endpoint energy ignores
the highest energy events due to pile—up. The endpoint energy and location for each
high flux peak for both scans of cavity #2 Operated in the 7r mode can be found in

Appendix F.1.

4.2.3 Cavity #2 % Mode

After data collection for the 7r mode was completed, the frequency was adjusted for
the 567i mode. Again, forward power was raised until an x-ray exposure rate of 25 1%
was measured. Ep was held nearly constant at 32.0 %. Two scans of the cryostat
were made with the collimator open to show that the data were reproducible. One
scan was made with the collimator closed (background).

Figure 4.13 shows the x-ray count acquired in two-second intervals with the slit
of the collimator, panel (a) open, panel (b) closed (background), and panel (c) x-ray
count with the background subtracted. Similar to the 7r mode spectrum, the highest
x-ray flux regions appear in the beam tube, at four irises, and possibly at 46 and

54 cm. The x-ray flux between peaks at 12 and 23 cm, 23 and 29 cm, and 29 and 37

cm may be regions of high Compton scattered x-rays entering the detector.

86

2500 ‘ (a)
2000 j

X-ray Count

 

 

 

01

O

O
J

(b)

N (A) .b.
O o O
O O O
1 I I

X-ray Count

_a

O

O
1

 

O

2000

1500 a

 

X-ray Count - Bkg
3
O
O

 

 

 

Distance [cm]

Figure 4.13: Cavity #2 operated in the fig; mode. X-ray count acquired in two-second
intervals. Panel (a): Slit of the collimator open. Panel (b): Slit of the collimator closed
(background). Panel (c): X-ray count with the background subtracted. A silhouette
of the cavity has been superimposed on the data.

87

Energy binning was performed for the %73 mode (Figure 4.14). The highest x-

ray flux peak at 12 cm becomes the lowest when x—rays are observed at a minimum
energy of 600 keV. This could be due to the low final kinetic energy of the electrons,
or a region of many x-rays being Compton scattered into the detector, or both. The
highest energy peak is around 30 cm. The possible peaks at 46 and 54 cm appear
to have x-ray endpoint energies between 400 and 600 keV. The endpoint energy and
location for each high flux peak, both scans of cavity #2 operated in the Edi mode,
can be found in Appendix F.2.

A comparison of the x—ray spectra obtained for both the 7r and 57f modes clearly
identifies differing electron impact sites and thus differing electron trajectories (Fig—
ure 4.15). The higher x-ray flux of the é675 mode spectrum is addressed in Chapter 5.
Both modes indicate electrons leaving the cavity and impacting in a beam tube. Both

modes also indicate electron impacts near irises.

88

Count - Bkg

 

 

 

 

!\ i
3“; z" a
i 5 if“ it {1.
x,
0 l T .—_- A\
0 20 40 60 80

Distance [cm]

Figure 4.14: Cavity #2, ’56: mode, energy binning of data presented in Figure 4.13
(c). The black line represents the number of x—rays at all energies, the light—gray line
represents the number of x-rays with a minimum energy of 200 keV, the medium-
gray line represents the number of x-rays with a minimum energy of 400 keV, and
the darkest gray plot represents x—rays with a minimum energy of 600 keV. A moving
average was applied to the data to smooth inconsistencies in adjacent two—second
intervals.

 

2000 l
1500 .
3’
m ‘ i.
g 1000 “ f g
500 {/r-\\ (If: 'IE/‘wfi‘\ /'-‘\\ IA /— \
/ ‘ f 3'53}; {4.3 p.312. 3 t / (
o . 9 ‘ x” .. .4 m

 

40
Distance [cm]

Figure 4.15: Cavity #2 spectra comparison of 7r (black line) and §67I (gray line) modes.
A silhouette of the cavity has been superimposed on the data.

89

° 471'
4.2.4 CaVIty #2 0 Mode

Data were also collected for the 4” mode. Again, forward power was raised until an x—

ray exposure rate of 25 9% was measured. E1) was held nearly constant at 31.6 £1912
Two scans of the cryostat were made with the collimator open to Show that the
data were reproducible. One scan was made with the collimator closed (background)
(Figure 4.16). Both scans, collimator open and closed, show an increase in x—ray flux
from the beginning to the end of the scan. Once the background is subtracted, high
x-ray flux regions appear at 21, 28, 37, 45, and 66 ot 76 cm.

High x-ray flux regions were difficult to identify so energy binning was also per-
formed (Figure 4.17). Energy binning shows the same peaks identified at 21, 28, 37,
45, and 66 to 76 cm. The x—ray count with background subtracted (Figure 4.17 (c))
is negative around the 50-65 cm range. High x—ray flux peaks cannot be determined
in this range. The endpoint energy and location for each high flux peak, both scans
of cavity #2 operated in the "467: mode, can be found in Appendix F.3.

A comparison of the x—ray spectra obtained for all three modes is shown in Fig-

ure 4.18. All three modes appear to have electrons impacting primarily on irises. The

x-ray flux varies greatly between modes and is addressed in Chapter 5.

90

800 —

600 -+

400 ~

200 —

X-ray Count - Bkg

 

(a)

 

800

600 ,

400 7

200

X-ray Count - Bkg

 

20 40 60 80

(b)

 

400 ~

200 _

X-ray Count — Bkg
O

 

-200

20 40 60 80

(C)

 

40
Distance cm

20

Figure 4.16: Cavity #2 operated in the 47f mode. X-ray count acquired in two-second
intervals. Panel (a): Slit of the collimator open. Panel (b): Slit of the collimator closed
(background). Panel (0): X-ray count with the background subtracted. A silhouette

of the cavity has been superimposed on the data.

91

 

X-ray Count - Bkg

 

 

 

0 20 40 60 80
Distance [cm]

Figure 4.17: Cavity #2, %75 mode, energy binning of data presented in Figure 4.16
(c). The black line represents the number of x-rays at all energies, the light-gray line
represents the number of x-rays with a minimum energy of 400 keV, and the medium-
gray line represents x—rays with a minimum energy of 600 keV. A moving average was
applied to the data to smooth inconsistencies in adjacent two-second intervals.

2000 .
'l

1500 —

 

1000 —

 

 

500 -

X-ray Count - Bkg

0 ~F~wi°h’v"\s‘v"” #

 

 

-500

 

.40
Distance [cm]

Figure 4.18: Cavity #2 spectra comparison of 7r (black line), '56?!" (medium—gray line),

and 467! (light-gray line) modes. A silhouette of the cavity has been superimposed on
the data. Each mode appears to have electrons impacting primarily on iris. The x-ray

flux varies greatly between modes.

92

4.2.5 Cavity #1 Performance

Once the cryostat scans were completed on cavity #2 the entire detector assembly
and unistrut stand were positioned to collect data for cavity #1. The front of the
collimator was positioned 4.5 cm from the cryostat wall. Forward power to cavity #1
(7r mode) was raised until x-ray detection on the area radiation monitoring equipment
measured an x—ray exposure rate of 25 $l—i—1, signifying the drop in Q to be due to field
emission. As with cavity #2, forward power was held nearly constant to maintain
that level of x-ray exposure rate.

The cavity Q was measured (Equation 2.78)) as 8.97x109 while Ea was held nearly
constant at 7.5 Mir—1! (Equation 2.76), or Ep 2 25.1 .1.)ng (Equation 2.77). The Q for
cavity #1, for this study (7r mode), was similar to the Q obtained from the previous
cavity test (7r mode) (Figure 4.19 (a)). The x-ray exposure rate for this study (7r mode)
and the previous cavity test (7r mode) are also shown to be similar (Figure 4.19 (b)).
The two plots demonstrate that no major changes in cavity performance occurred
between tests.

As with cavity #2, the electric field was held nearly constant, thus a direct calcula-
tion of the Fowler-Nordheim field enhancement factor (,8 F N), needed for the particle
tracking code, could not be made. 6 F N was inferred from 7r mode data acquired dur—

ing two previous tests and compared with data (7r mode) for this study (Figure 4.20)

(Equation 2.35). 6 F N for cavity #1 was found to be in the range 120 g 5 F N S 300.

93

1.E+11 ~ (a)

 

 

 

 

 

01.E+10 "... . . . . 0 Q . . o
0....
1.E+09 "I I r I *7
O 10 20 30 4O 50
10000 (b)
@1000 ~ g ’
m
_E_, 100 — .
> I
e 10 . .
X o o
1 I I I I
0 10 20 30 40 50
Ep [MV/m]

Figure 4.19: Cavity #1 performance, 7r mode. Panel (a): shows peak electric field
versus Q. Diamonds indicate values of Q for a previous cavity test and the square,
the value of Q during this study. Panel (b): shows the increase in x—ray exposure rate
detected on the area radiation monitoring equipment when the observed drop in Q
was due to field emission. The two plots demonstrate that no major changes in cavity

performance occurred between tests.

A 1E-24 ~

1E-25 —

 

(Pd-Po)/(d*Ep"3.5

1 E-27

1E-26 a °

 

0.020 0.040 0.060 0.080 0.100

1/Ep [i/MV/m]

Figure 4.20: Diamonds and triangles represent data from two previous cavity tests
and the square, data from this study. 120 S [3 F N S 300.

94

4.2.6 Cavity #1 7r Mode

Two scans of the cryostat were made with the collimator open to show that the data
were reproducible as Ep was held nearly constant at 25.1 3%. One scan was made
with the collimator closed (background). Figure 4.21 shows the x-ray count acquired
in two second intervals. The regions of high x—ray flux appear to be located on irises
and possibly in the beam tubes. To aid in the determination of “real” x-ray flux
peaks, energy binning was employed (Figure 4.22). The endpoint energy and location

for each high flux peak, both scans of cavity #1 operated in the 7r mode, can be found

in Appendix F4.

95

X-ray Count
00
O
O

 

 

200 - (b)

X-ray Count
3 Si
O O

I 4

01
O
I

 

 

500 . (c)
400 .
300 ~

 

200 ~

X-ray Count - Bkg

100 *

 

 

Distance [cm]

Figure 4.21: Cavity #1 Operated in the 7r mode. X-ray count acquired in two-second
intervals. Panel (a): Slit of the collimator open. Panel (b): Slit of the collimator closed.
Panel (c): X-ray count with the background subtracted. A silhouette of the cavity has
been superimposed on the data.

96

 

X-ray Count - Bkg

 

 

 

0 20 40 60 80
Distance [cm]

Figure 4.22: Cavity #1, 7r mode, energy binning of data presented in Figure 4.21
(c). The black line represents the number of x-rays at all energies, the light-gray
line represents the number of x-rays with a minimum energy of 400 keV, and the
medium-gray line represents the number of x-rays with a minimum of 600 keV. A
moving average was applied to the data to smooth inconsistencies in adjacent two—
second intervals.

° 571’
4.2.7 Cav1ty #1 3- Mode

After data collection for the 7r mode was completed, the frequency was adjusted for

the 5675 mode. Again, forward power was raised until an x—ray exposure rate of 25 {ETR

M

was measured. Ep was held nearly constant at 31.2 m

. Two scans of the cryostat
were made with the collimator open to show that the data were reproducible. One
scan was made with the collimator closed (background).

During the scans of cavity #1, 7r mode, the temperature rose to 2.08 K. The
temperature continued to rise slightly during the fig: mode scans. After these scans,
liquid He was not available, so a scan of cavity #1 Operated in the 11675 was not
performed.

Figure 4.23 shows the x-ray count acquired in two-second intervals. The high x-
ray flux peak at 60 cm is evident in the figure. In order to locate additional “real”

peaks, energy binning and data smoothing were employed (Figure 4.24). High x-ray

flux peaks now become apparent around 22, 30, 44, 52, 61, and 68 cm. The endpoint

97

energy and location for each high flux peak, both scans of cavity #1 operated in the
5671 mode, can be found in Appendix F.5.
A comparison of the x-ray spectra of cavity #1 operated in both the 7t and %’5

modes is shown in Figure 4.25. The 7r mode spectrum has a higher flux, an explanation

follows in Chapter 5.

98

300 (a)

X-ray Count

 

 

150 (b)

100 ..

X-ray Count
01
O

 

O
l

20 40 60 80

O

200 (c)

100i

X-ray Count - Bkg

 

 

O
.,

l
—1

I

0 20 40 60 80
Distance [cm]

Figure 4.23: Cavity #1 Operated in the §67£ mode. X-ray count acquired in two—second
intervals. Panel (a): Slit of the collimator open. Panel (b): Slit of the collimator closed
(background). Panel (c): X—ray count with the background subtracted. A silhouette
of the cavity has been superimposed on the data.

99

200 -

 

 

C)
x
m
E100
3
o
O
>
9
>'<
O

 

 

Distance [cm]

Figure 4.24: Cavity #1, §67£ mode, energy binning of data presented in Figure 4.23
(c). The black line represents the number of x-rays at all energies, the light-gray
line represents the number of x-rays with a minimum energy of 400 keV, and the
medium-gray line represents the number of x-rays with a minimum energy of 600 keV.

A moving average was applied to the data to smooth inconsistencies in adjacent two—
second intervals.

500 -
400 a
300 r

200

i v ' I] ' \ i if ‘
100 — .. ' f "i ' ’ (d
I' ." : V f"! f ‘ f f ‘ ‘- I. ‘ I} l ,., ’i 5-21 _ ,1. 1 ,' .5 if , ' .5
' V 2‘ 2. 9 1! I 9 1‘1! i L ‘95. - . -. . . _ I f f ‘7' l : H
l J); I - -I a: I} , ( 1,: f . I a.) . = 1‘ 15w" _: -
' ' L f ' j k ' I ' '
o ___ I I . “I ' _ I

0 20 40 60 80
Distance [cm]

 

X-ray Count - Bkg

 

 

Figure 4.25: Cavity #1 spectra comparison of 7r (black line) and §67£ (gray line) modes.

Both modes indicate electron impacts near iris. A silhouette of the cavity has been
superimposed on the data.

100

Chapter 5

Simulations

5.1 SUPERLANS

SUPERLAN S is a computer code that calculates electromagnetic fields within a cav-
ity [27]. The code uses the finite element method with a quadrilateral mesh to cal-
culate the modes in axisymmetrical cavities and periodic structures, and calculates
the cut-off frequency in longitudinally-homogeneous waveguides [28]. The finite ele-
ment method is a numerical technique for solving problems that are represented by
partial differential equations. The range of interest is represented by a number of
finite elements, or the quadrilateral mesh [28]. A plot of the quadrilateral mesh for
half of a cell, required as input for a periodic structure, is shown in Figure 5.1. A
fine heterogeneous mesh with increased mesh density at the iris was chosen since the
fewest discontinuities were found in the surface electric field. The cavity has azimuthal
symmetry about the z axis.

Output from SUPERLAN S include symmetric and antisymmetric wave functions
(Figure 5.2 (a)). The wave functions are reflected about zero as either symmetric or
antisymmetric waves. The two wave functions for an entire six—cell cavity are shown

in Figure 5.2 panel (b).

101

RICH)

13.165

9.8736

6. 5824

3.2912

 

Figure 5.1: Quadrilateral mesh for a 6:047 805 MHz six-cell cavity. Half of a cell is
required as input for a periodic structure. The cavity has azimuthal symmetry and
the z axis is the contour in the axisymmetric problem.

 

 

 

31g: “0 .. E 207 (b)
S 0 em-fl"‘-—‘--' S 10 ] [
"2" '5 ‘ new”... %-1g+ W
m 212 “WWW”. ”J -20 i I i . . .
0 2 4 -10 0 10 20 30 40 50

2 [cm] 2 [cm]
Figure 5.2: Panel (a): Symmetric (black diamonds) and antisymmetric (gray squares)

wave function output from SUPERLANS for the 3671 mode. Panel (b): Symmetric
(black line) and antisymmetric (gray line) wave functions for an entire six-cell cavity.
The dashed lines in panel (b) identify the portion of the wave function shown in panel

(a)-

102

The two wave functions are combined in the particle tracking code, using the
F loquet theorem (Appendix E), to construct a standing wave with the desired phase
shift over a structure period. The electric field on axis of a six-cell cavity, constructed
using the Floquet theorem, for the 7r (180° phase shift), 5375 (150° phase shift), é675
(120° phase shift), 367: (90° phase shift), 2g- (600 phase shift), and g. (30° phase shift)
modes is shown in Figure 5.3. The 7r mode is used for particle acceleration since each
cell is 7r radians out of phase with the neighboring cell. The cavity magnetic fields are
generated in the same manner.

Electrons, field emitted from the same defect are accelerated by mode dependent
electric fields. The electron trajectories depend on which passband mode is excited,
thus their impact sites result in different x-ray spectra.

Output, also from SUPERLANS, includes electromagnetic fields along the cavity
surface. The absolute value of the surface electric fields, calculated for one Joule of
stored energy, for a six-cell [3 = 0.47 805 MHz cavity, 7r, E63, and 477 modes, are shown
in Figure 5.4. The field for the g675 was normalized and the other two modes scaled
accordingly. The surface electric field reverses direction at the equator. Field emission
current is dependent upon the surface electric field at the emitter (Equation 2.35).
The peak electric field on axis for the 7r mode is calculated using Equation 2.76. Data
were acquired for these three modes so only these three are shown.

The surface electric field is identical from cell to cell for the 7r mode and is maxi-
mum at each iris and zero at each equator. The surface electric field for the 53675 mode
is greater than the other two modes at the first and last irises (adjacent to the beam
tube) and between the first and second and fifth and sixth cells. Data acquired with
the highest x—ray flux for the %75 mode, has the highest probability of a field emitter
located along one of the irises where the surface electric field is greatest for the fig:

mode. The same is true for any mode with the highest x-ray flux.

103

151 (a) 15 (d)

 

 

 

 

 

 

 

 

 

10 10 a
E 5 r 5 a
E 0 0 ..
LU 5 1 5
10 ~ -10 ]
15 r f T I '15 I I I
5 5 15 25 35 45 -5 5 15 25 35 45
10 ~ (b) 15 (e)
5 _i 10
E o 4 5 i
> .
2 1 0
w '5 I -5
10 i -10
15 [ I I T I I 15 7'" I I I
5 5 15 25 35 45 -5 5 15 25 35 45
15 (C) (D
10 _ 15 r
"E 5 —
E o 10
L” '5 1 5 a
-10 1
-15 ‘ v“ I’" of“ I A" I ‘ I O I I 1 I
'5 5 15 25 35 45 -5 5 15 25 35 45
Length [cm] Length [cm]

Figure 5.3: Electric fields on axis for a six-cell cavity. Panel (a): 7r mode (180° phase
shift). Panel (b): '56: mode (150° phase shift). Panel (c): 4675 mode (120° phase shift).
Panel ((1): %75 mode (90° phase shift). Panel (e): 26?: mode (60° phase shift). Panel (f):
gr mode (30° phase shift). The electric fields were constructed from the symmetric
and antisymmetric wave functions using the F loquet theorem.

5.2 Multipacting/ Field Emission Simulation .

The code used for predicting the trajectories of field-emitted electrons was “Mul-
tip”, originally developed at Cornell University. The code predicts electron impact
locations, impact energies, and the power deposition along the inner surface of the
cavity [29]. The original code was modified to allow different passband modes with the

electromagnetic field distribution numerically calculated using the Floquet theorem.

104

Code Modifications

A significant part of this research, nearly one year, was devoted to code modifications

and verifications. The modifications and verifications are outline in this subsection.
The first step to running the Multipacting/ Field Emission Simulation is to produce

a field table using SUPERLANS. For periodic boundary conditions, which the original

code would not allow, SUPERLAN S produces two field tables: one for the symmetric

and one for the antisymmetric wave functions.

The symmetric and antisymmetric wave functions were first combined in EXCEL
using equations derived from the Floquet theorem (Appendix E). The construction
of the fields was verified with the electromagnetic field output from MAFIA [30].

Two sub-programs, one that reads the field tables, and another that calculates

Equator

Iris
1 - / \. [

MI
‘- .

    

m
. ~ "“"
I

—;

AM
A- . IW‘um n.

Es [MV/m]
O
01

 

 

 

 

 

 

 

 

 

 

 

 

 

2 [cm]

Figure 5.4: Normalized absolute value of the surface electric fields for a six-cell fl =

0.47 805 MHZ cavity for the 7r (black line), §67£ (dark-gray line), and £167: (light-gray

line) modes. First and second irises, surface electric field greatest for the é675 mode.

Third iris, surface electric field greatest for the 7r mode. Fourth iris, surface electric
field greatest for the 11673 mode. Fifth iris, surface electric field greatest for the 7r mode.

Sixth and seventh irises, surface electric field greatest for the §61£ mode.

105

the electromagnetic field components were initially modified. The initialization code
reads information from three text files and translates the information into a format
compatible with the remaining programs. This program was modified to read two
tables, symmetric and antisymmetric, and store the field tables as sets of ordered pairs.
The other program was modified to calculate the electromagnetic field components
using the Floquet theorem.

The initialization program also identifies the maximum surface electric field. The
program originally examined only one cell since the fields do not vary from cell to
cell for the 7r mode. The program was modified to examine an appropriate number
of cells since the fields vary from cell to cell for modes other than the 7r mode. The
appropriate number of cells depends on the phase advance per cell and number of
cells. Another text file identifies the cell number and the field emitter location and
stores that information.

Two programs are used to examine the tables graphically. One program draws the
cavity boundary. The same program can also plot contour maps of [E I, H051 or TH (15'
This code called the revised program to read the fields that were calculated using the
Floquet theorem. Output for the 7r mode were compared to the unmodified output.
It was assumed the other modes were functioning properly.

Another program has the capability of plotting the electric field magnitude, mag—
netic field magnitude, radial component of the electric field (Er), or axial component
of the electric field (Ez) as a function of R (radially), Z (horizontally), or S (along
the cavity surface). This code was modified to call the revised programs to read the
fields and calculate the field components. The fields for the 7r mode, periodic bound-
ary conditions, were compared to the fields output for the unmodified program. The
fields for the other modes were compared with the EXCEL output.

The next program for comparison calculates the energy stored in the magnetic
and electric fields. This program also called the revised programs to calculate the

fields. The correct output for all three programs verified that the Multipacting/ Field

106

 

 

Emission Simulation was calculating the fields correctly.

5.2. 1 Program Operation

The rf period is divided into a number of equal time intervals (number chosen by
the user). At the beginning Of each time interval an electron is emitted with a given
velocity from a point on the cavity surface for a given amplitude of the surface electro-
magnetic field. Electron trajectories are calculated by integrating the following vector

equations:

 

= —f[(t), (5-1)

 

Emn+lmnx3mn, am

where 7"(t) is the position vector of the electron at time t, E(F, t) and 13.0", t) are the
electric and magnetic field at position F and time t, q is the electron charge, m is the

electron mass, 7 = ——1-——§, I? is the electron velocity, and {17 is the proper velocity

3

with I7 E 75' [29].

The instantaneous field-emitted current is calculated from Equation 2.32, where
the electric field at the time of emission EU) is replaced with the enhanced electric
field 3150) and 12(5) and t(s) (Appendix B.1) are evaluated in terms of complete

elliptic integrals.

5.2.2 Input Parameters

Input parameters to the code include the cell number of the emitter, the location of
the emitter along the cell wall (SO), the peak surface electric field (Ep), the maxi-
mum number of rf periods that an electron trajectory is tracked, and the number of
electrons tracked in an rf period. The average cavity current due to field emission

depends on the Fowler-Nordheim field enhancement factor (I3 F N), the emitter area

107

 

(Ae(cn12)), the work function (,9(eV)) of the metal, and the electric field at the emit-
ter (Eem) (Equation 2.35). A range of 6 F N was determined from data for each cavity.

2 was used for A6. Since the average cavity

Initially the default value of 9 x 10‘11 cm
current is directly proportional to A8, the shape of the simulated power spectrum
(Subsection 5.2.3) is unchanged with changes in A8. An estimate of Ae can be found
in Chapter 6. The assumed value for (,0 was 4.3 eV; published values for Nb range
from 4.0 to 4.9 eV, depending on the crystal orientation [26]. Eem is calculated by
the code for the input value of Ep.

Output from the code includes the time of electron emission during the rf period,
the peak electric field at the emitter (Eem), the electric field at the emitter at the time
of emission E (t), the electron impact location along the cavity wall and with respect
to the z axis, the electron impact kinetic energy (K f), and the angle at which the
electron impacts with respect to the z axis. The number of electrons tracked during
27r radians is chosen by the user (€732). Electrons are only field emitted during half
the RF period.

Changing S0 changes the field emitted electron trajectories. S0 is measured in cm
from the cavity equator; negative S0 is to the left and positive SO is to the right. An
example of field emitted electron trajectories for two different emitters in cell #2 of
a six-cell cavity is shown in Figure 5.5. Eem varies with field emitter location and
the maximum K f also varies. The change in the electron trajectories, and thus their
impact sites, will produce different x-ray spectra. Electrons that appear to strike an
end plate of the cavity, would exit the cavity and travel down the beam tube. Only

50 trajectories are shown. Increasing the number of trajectories to 10,000 increases

the difficulty to visually trace individual electron trajectories, in most cases.

108

 

(a)
/\ / ’\
Field I \ / '
Emitter [

.o

 

 

 

”-—
~_-
.2...’

.2—
I -

 

 

Electron
Impacts

 

 

 

L
[1,,
CT

 

 

 

 

 

/\ K
[[1 [[1

. Electron
I I Impacts
’ I

 

 

 

 

f1 . f) '7
I, I
[ ‘1 I, ] [
I
f i , I I [
e) L ’ I ’ ‘
Figure 5.5: 7r mode field emitted electron trajectories differ for different emitter sites
with all other input parameters equal. Field emitter in cell #2, Ep 2 26.0 %.
Panel (a): so = -13.0 cm, Eem = 24.26 %, and Kf = 1.673 MeV. Panel (b):

SO = —13.6 cm, Eem = 24.41 M and Kf = 1.684 MeV.

III ’

 

109

Changing E1), the peak surface electric field, also changes the field emitted electron
trajectories. An example of field emitted electron trajectories for two different Ep
values are shown in Figure 5.6. All other input parameters are identical. Again, not
only the electron trajectories but the x-ray spectra would change.

For the 7r mode, simulating a field emitter in another cell with the same SO,
shifts the electron trajectories since the electric fields are symmetric from cell to cell
(Figure 5.7 (a) and (b)). Simulating a field emitter on the opposite side of a cell,
the same distance from the equator (iSO), reverses the electron trajectories since the
electric field is symmetric within the cell (Figure 5.7 (b) and (c)). This is an essential
aid in locating field emitters for the 7r mode.

The electric field for the other cavity modes, #5675, 3675, 3675, ZBE’ and gr, are not
symmetric from cell to cell, nor symmetric within a cell (Figure 5.3). Figure 5.8 (a)
and (b) shows the electron trajectories of a field emitter in a neighboring cell but
with the same SO for the 26: mode. The pattern of the electron trajectories does not
shift by a cell as in the 7r mode. Figure 5.8 (c) shows the electron trajectories of a
field emitter on the opposite side of a cell for the .4671 mode. Electron trajectories do
not simply reverse direction as they do in the 7r mode.

Figure 5.9 shows electron trajectories from a field emitter in 7r, %75, and E167: modes
with all input parameters, cell #, SO, and Ep, identical. Changing modes changes
the surface electric field distribution, which in turn changes Eem. The electrons are
accelerated by the electric field within the cavity and the trajectories directed by the
magnetic field. Changing modes changes these fields. Electrons impact with differ-
ing K f from mode to mode. The number and energy of the x-rays produced from

impacting electrons depends on Kf.

110

 

 

 

 

 

l l . l
1’ Field 1 f f' l l
f Emitter l l! l '
f , .
l \ \ 1i \ - v“- :- M’A‘r-‘L‘

 

  
  
 

 

(b) p
ng ( l
’ Emitter

 

 

 

Figure 5.6: Field emitted electron trajectories differ for different peak surface electric
fields with all other input parameters equal. Field emitter in cell #2, SO 2 —13.2 cm.
Panel (a): Ep = 13.0 me, Eem = 12.33 %, and K, = 861.8 keV. Panel (b):

E,, = 26.0 1%‘1, Ep = 24.67 Mg, and K, = 1.292 MeV.

111

 

 

 

 

(a) / \ Flfld / \x //..‘-\l /M\ /"\
f ( Enntter 3 ; l l l, i" '1
/ i' f ’l .‘l ‘I‘ \ .i .‘| l l.
f l l , ~ 1’ l l l 1' l
i R f l , i ' i l l i ll

‘C
C

 

 

 

 

 

x/

l

l’ l,
\J k} k/ \J
m fl /\ /\ Field (‘\.

 

(C)

 

 

 

 

    

’ l a l ' l ." l

’ , l. l l l
\J' P.
n, H

I l l

.. l , ;

-.': l l l .‘

‘ e. ' l f ( I

‘ ‘ l ( ;' l ,‘ l l
' . ‘ t ‘ I‘ i' ', l i f

\J k) \J \/ \\J k,»

Figure 5.7: 7r mode, 15,, = 26.0 Mg, Eem = 24.41 Mg, and Kf = 1.684 Mev.
Panel (a): Field emitter in cell #3, 30 = —13.3 cm. Panel (b): Field emitter in
cell #4, 80 = —13.3 cm. Panel (0): Field emitter in cell #4, SO 2 +13.3 cm. In the

7r mode, the electric fields are symmetric from cell to cell and within the cell.

112

(a) ,,

   
  
   
     
  
 
     

I 1""~ / 1"\ ’— _
‘\ ,/ N / \ ,/ \ ./ \
Field 3 ‘1, .‘ ‘~. ; , ,1 l, l '-.
Emitter 1' i i' l " l, ,' '1, f I.
l ' l ‘ , l l
i '1. 1‘ t : i . l ,1 'i I l
I ‘ 8 ,' ‘~, ‘1 ' l l' l

J
:D':

- ~-""'
_\
y...-

C
C

\

/}

\\\
//
\

J

(C‘)\.. j \\a / \\. _/;
/\, 1//‘\‘ (W
l ' - l
5 l. j, ‘. l l
l ‘1, l ‘l l l l
i‘ l l I l I l

l I .' . l l

J \J \

V7 : ' 1‘: I T - - - é

( p T‘. ‘ ‘ in, 1"

9 - r j . l‘ i '
i‘ f l l l l I. I i (I i 'i
l, r l ' l 1 l ’1 / ‘-. I l :
~ ’ . ' z 1 ‘ l .'

l l l f l l’ i ,1 ‘U l ,l
l . ,‘ ‘ ‘ I
\/ k/ L) v __/

Figure 5.8: 467E mode, Ep = 26.0 MTnY' Panel (a): Emitter in cell #2, 50 = —13.8 cm,
Eem = 4.014 % and Kf = 2.607 Mev. Panel (b): Emitter in cell #3, 30 =
—13.8 cm, Eem = 21.11 Mg, and Kf = 2.555 Mev. Panel (c): Emitter in cell #3,

so = +13.8 cm, Eem = 25.12 %, and Kf = 2.304 MeV. In all bllt the 7r mode, the
electric fields are not symmetric from cell to cell and not symmetric within the cell.

113

   

   

\-l

r

I

   

f

     

7.272 l‘fn—V
and Kf = 2.221 MeV.

. Panel (a): 71' mode,

M
m

22
, and Kf = 1.422 MeV. Panel (b): fig: mode, Eem

m
In

Figure 5.9: Field emitter in cell #3, SO 2 —13.8 cm, Ep

7

17.86 kg

and Kf = 1.039 MeV. Panel (c): 5167: mode, Em

114

Some symmetry of the field elllitted electron trajectories does exist for modes
other than the 7r mode. For exalllple, referring to Figure 5.3, electrons emitted in
cell #1 of the 3675 mode would have the same trajectories if field emitted at the same
SO but in cell #4, shifted by three cells. Also, electrons field emitted in cell #1 of
the fig mode would have their trajectories reversed relative to those of electrons field
emitted from ~80 in cell #6.

In the preceding examples of electron trajectories, the electrons travel in a plane
through the cavity. According to the Lorentz Force Law, there are no forces acting

on an electron in the (:5 direction:
F = q [E + (17X §)], (5.3)
where q is the charge of an electron, E : Err + E32, 17 2 127-7“ + 1232, and 3:15 = quffi.

5.2.3 Code Output

To compare the simulation output with the data, power was calculated for both. For
each field emitted electron, the field emission simulation provides the electric field at
the emitter Eem(t) (Lg/l), Kf(eV), and impact site both along the cavity wall and
with respect to the z axis. Since power is the product of current and voltage, the
power dissipated by each electron is calculated:

[1(-
P :il,

i

e (5.4)

where I,- is the instantaneous current (Equation 2.35) of the ith electron for a given
[1 F N and g0, and K f,- is the final kinetic energy of the ith electron. The power can
now be summed for a given distance along the cavity z axis. Figure 5.10 (a) shows

trajectories of electrons field emitted in the %75 mode from cell #1, and (b) shows

115

 

 

 

 

 

 

3 E-08
2.E-08 7
1.E-08 ~

526d

0. E+00

40 50 60

30
Length [cm]

20

10

 

 

 

 

 

 

 

 

 

[67808 2
E] 1211]
m

765323
Ella/.3455
Z

 

 

 

Figure 5.10: Electrons field emitted in the 4671 mode from cell #1, SO = 13.99 cm,

19.28 %. Panel (a): 500 electron trajectories. Panel (b):

Corresponding power spectrum with fl F N = 100 for the current calculation. The table

lists the electron maximum final kinetic energy for each peak.

and Eem

m
111’

116

the corresponding simulated power spectrum with 13 F N = 100. At 0 cm, electron Kf‘s
are found as if they strike an end plate on the cavity. In reality, those electrons travel
out of the cavity and can account for x—rays detected in the beam tube. Around 9 cm,
electrons impact an iris but their total power is very low (IO—11). The highest K f
electrons impact around 35 cm. Power is left unitless since the shape of the spectrum
will be compared with the data. A table with electron maximum final kinetic energy
is included in Figure 5.10.

As the number of field emitted electrons increases, the number of impact sites
increases, thus changing the simulated power spectrum. The power also increases due

to the increasing number of electrons. Figure 5.11 shows the simulated power spectra

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7'E'09 7 (a) 2 [cm] KE [keV]
6.E-09 — 1 953
‘- 5.E-09 “ 17 707
g) 45.09 — 34 2076
a? 3.E-09 a 43 1820
2.E'09 d 53 1081
1.5-09 . (
0.E+00 T T l A 1 A 14‘ l
0 10 20 30 40 50 60
Distance [cm] Z [cm] KE [keV]
8.E-08 _ (b) l 953
7.E-08 7 10 632
6.E-08 ‘ 17 708
5 5.E-08 ~ 26-28 1858
g 4.E-08 — 34 2097
0' 3.E-08 ‘ 43 1839
2.E-08 ‘ 52 1444
1.E-08 “
O.E+OO If i‘ I Mn A l A TA 1

 

0 1 O 20 30 40 50 60
Distance [cm]

Figure 5.11: Simulated power spectra of electrons field emitted in the %’5 mode from
cell #1, SD = 13.99 cm, Ep 2 25.0 M and Eem = 19.28 M—V. Panel (a): 100 electron

III ’ Ill
trajectories (g; = 0.005). Panel (b): 1250 electron trajectories (a? = 0.0004). The
tables list the electron maximum final kinetic energy for each peak.

117

for the electron trajectories in Figure 5.10 but with the number of field emitted
electrons increased from 100 to 1250 trajectories (523% = 0.005 to 0.0004). Peaks in
the simulated power spectrum emerge at 9, 26, and 28 cm. It was found that the
shape of the simulated power spectrum for 1000 impacting electrons did not differ
significantly from the power spectrum of 10,000 impacting electrons. Therefore, the
simulated power spectra for 1000 field emitted electrons were used for comparison
with the data power spectra.

Moving the field emitter location (SO) changed the electron trajectories (Fig-

ure 5.5), likewise, the simulated power spectrum changed. Figure 5.12 shows the

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.E-09 n (a) 2 [cm] KE [keV]

5.E-09 d 17 1056
‘- 4.E-09 4 27 1673
‘é’ l
8 3.E-09

2.E-09 —

1.E-09 ~

0.E+00 . i 1 ‘ l ' ‘ ‘

0 10 20 30 40 50 60
Distance [cm]

6.E-09 — (b) 2 [cm] KE [keV]

5.15-09 ~ 10 544
,_ 4.E-09 — 2313 1865854
a:
:3 3.E-091 45 891

2.E-09 4

1.E-09 4

0.E+OO 1 1 l ‘ i ’ ‘

 

 

 

0 10 20 30 40 50 60
Distance [cm]

Figure 5.12: 7r mode simulated power spectra for different emitter sites Ep = 26.0 .1%¥’
and 4%; = 0.01. Panel (a): Field emitter in cell #2 SO 2 —13.0 cm, and Eem =

24.26 %. Panel (b): Field emitter in cell #2 so = —13.6 cm, and Eem = 24.41 3,131.
The tables list the electron maximum final kinetic energy for each peak.

118

simulated power spectra generated when S0 was moved by 0.6 cm within the same
cell and with all other input parameters remaining the same. In some cases moving
SO by 0.001 cm significantly altered the simulated power spectrum.

Simulated power calculations depend upon current (Equation 5.4), current calcula-
tions depend upon Eem(t), and Eem(t) calculations depend upon Ep (Equation 2.35).
Figure 5.13 shows the change in a power spectum when Ep is doubled (Figure 5.6).
Doubling Ep increases the power by 10 orders of magnitude and greatly changes the
shape of the spectrum, in most cases.

Identical to the electron trajectory example mapped in Figure 5.7, moving the field
emitter to another cell in the 7r mode shifts the simulated power spectrum. Moving a
field emitter to the opposite side of the cell reverses the spectrum (Figure 5.14).

It was also shown that moving a field emitter to a neighboring cell, in a mode
other than the 7r mode (Figure 5.8) , completely changes the electron trajectories and
thus the simulated power spectrum. Figure 5.15 shows the simulated power spectrum
for the 47f mode when the field emitter site is moved to a neighboring cell. The low
Ecm in cell #2 (Figure 5.4) accounts for the simulated power spectrum of zero.

Changing modes, but keeping all input parameters identical, changed the electron
trajectories (Figure 5.9) and the simulated power spectrum. Figure 5.16 shows the
simulated power spectra for a field emitter in the same location for the 7r, égi, and %75
modes. When the emitter is near the tip of the iris (S0 = —14.49364 cm) in cell #3,
Eem is the greatest for the 7r mode, lower for the 11675 mode, and lowest for the 5675
mode. This is evidenced ill the magnitude of the simulated power spectra for the three
modes. Of course this would not be true for the simulated power spectra if the field
emitter were on the opposite side of the cell (Figure 5.4).

Changing {3 F N changes the instantaneous current and thus the power spectrum.
Figure 5.17 shows the power spectrum when ,8 F N was increased from 100 to 300.

Aside from the difference of 29 orders of magnitude between the two spectra, the

overall shape of the spectrum changed. The peak at 28 cm went from equaling the

119

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 [cm] KE [keV]
1.6E-14 . (a) 1 75'
1.4544 1 16 348
1.2E-14 ~ 20 543
., 3: 3::
g 8.0E-15 1 43 694
6.0E-15 1 50 419
4012-15 . 52 526
205-15 4 A 53 650
0.0E+00 \ . “ "r ‘ . . . ‘
o 10 20 30 4o 50 60
Distance [cm] 2 [cm] KE [keV]
1 1600
3 981
1.6E-03 ~ (b) 11 “00
1.4E-03 . 13 396
1.2E-03 i 17 1360
a.) 105-03 ~ l 27 1570
g 8.0E-O4 - 32 377
“- 6.0E-04 - 36 719
4.0E-O4 i :8 1915150
2.0E-04 — I” j 53 463
0.0E+00 “i“ . " 1‘ ‘ ‘ ’

 

 

 

0 1 0 20 30 40 50 60
Distance [cm]

Figure 5.13: 7r mode simulated power spectra for doubling Ep. The emitter is in

cell #2, so = —13.2 cm, and 49% = 0.0005. Panel (a): 15,, = 13.0 M and Eem =

Ill’

12.33 Mm—V. Panel (b): 13,, = 26.0 MV and Eem = 24.67 % The tables list the

III ’
electron maximum final kinetic energy for each peak.

peak at 18 cm to half its value. A peak emerged at 44 cm that was absent from
simulated power spectrum with ,8 F N = 100. The peak on the iris at 53 cm increased
in relative height by a factor of 10. In many instances, changing B F N by I allowed
a better or worse fit of the simulated power spectrum to that of the data power

spectrum.

120

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 [cm] KE[keV]
1.5E-05 — (a) 0 682
1 979
10 1770
6 1.0E-05 i 18 973
g 25 841
o. 28 791
5.0E-06 — 35 1100
A 43 1890
0.0E+00 . l . ~ , i . 53 841
0 20 40 60
Distance [cm] z [cm] KE [keV]
1.5E-05 1 (b) 1 54
8 735
10 979
B 1.0E-05 - 19 1770
g 27 973
o. 2 33 841
5.0E 06 37 791
A 44 863
0.0E+OO i . . 52 1890
o 20 40 60
Distance [cm] 2 [cm] KE [keV]
1.5E-05 — (Cl 2 841
10 1890
19 1100
a 1.0E-05 ~ 24 791
g 28 841
o. 35 967
5-05'05 " 43 1770
52 979
0.0E+OO . . . 53 682
0 20 40 60

 

 

 

Distance [cm]

Figure 5.14: 7r mode simulated power spectra for SO in a neighboring cell and on
the opposite cell wall, 13;, = 26.0 Mle Eem = 24.67 MnTY’ Kf = 1.841 MeV, and

$5? = 0.0005. Panel (a): Field emitter in cell #3, SO = —13.3 cm. Panel (b): Field
emitter in cell #4, SO 2 —13.3 cm. Panel (c): Field emitter in cell #4, SD = +13.3 cm.
The tables list the electron maximum final kinetic energy for each peak.

121

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.E+00 " (a)
E
o 5.E‘01 T
o.
O.E+OO 1 r 1
O 20 40 60
Distance [cm] 2 [cm] KE [keV]
1.E-06 - (b) 1 1800
10 611
8 6.E-O71 25 1020
g 27 1070
o. 4.E-07 ~ 35 326
43 930
2-5‘07 L II 1 52 2130
O.E+00 A -. ,. , , 53 2520
0 20 40 60
Distance [cm]
4.E-03 —~ (C)
2 [cm] KE [keV]
3.E-03 ~ 1 2280
a, 7 783
g 2.E—03 ~ 10 305
0- 19 1050
1.E-03 ~ 43 1580
53 2650
0.E+00 ~ Ar . A ’ i
0 20 4O 60

Distance [cm]

Figure 5.15: 7r mode simulated power spectra for SO in different cells, Ep = 26.0

and Ad) = 0.0005. Panel (a): Field emitter in cell #2, SO 2 -—13.8 cm, and Eem =

2?

4.014 2.4112 Panel (b): Field emitter in cell #3, SO = ~13.8 cm, and Eem = 21.11 1%}:
Panel (c): Field emitter in cell #3, SO 2 +138 cm, and Eem = 25.15 Mfily. The tables

list the electron maximum final kinetic energy for each peak.

122

 

 

4.E-04 — (a)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.E-O4 '1
a z[cm] KEIKCVI
g 2.E-04 4 43 1520
n. 52 1440
1.E-04 7
0.E+00 1 ‘ if T ‘
0 20 40 60
Distance [cm]
1.E-29 n (b)
as z[cm] KE[keV]
g 5.E-30 - 1 1180
n. 17 766
25 511
O.E+OO ‘ F i ‘ ”I
0 20 40 60
Distance [cm] 2 [cm] KE [keV]
1.E-08 — (c) 1 1450
10 477
19 231
a? . 27 868
35 211
43 802
\ LA J ( 52 1830
0.E+00 " ‘1 A ‘ ' 53 2200
0 20 40 60

Distance [cm]

Figure 5.16: Simulated power spectra for a field emitter ill cell #3, SO = —13.8 cm,
Ep = 26.0 MV and 9%!) = 0.0005. Panel (a): 77 mode, and Eem = 21.25 MEX.

ln’

Panel (b): fig: mode, and Eem = 7.272 h—ffiyn Panel (c): 4375 mode, and Eem =

17.86 l\_11fi\[. The tables list the electron maximum final kinetic energy for each peak.

123

 

 

 

 

 

 

 

 

 

 

4.E-36 7 (a)
E
0 2.5-361
D.
0.12.400i M L Al .4 i if n
O 10 20 30 40 50 60
Distance cm
2.E-07 - (b) I 1
E
o 1.E-07 .
CL
OE+OO AT L M J If fl
0 10 20 30 40 50 60

Figure 5.17: Field emitter in

Distance [cm]

mode in cell #3, Ep = 26.0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

z [cm] KE[keV]
1 1100
8 317
18 411
25 569
27 44
34 176
36 504
43 837
52 800
53 1240

2 [cm] KE [keV]
1 1100
8 317
18 411
25 569
27 44
34 176
36 504
43 837
52 800
53 1240
%, and Eem =

7.272 l‘f—n‘l. Panel (a): 4m = 100. Panel (b): 6“, = 300. The tables list the electron

maximum final kinetic energy for each peak.

It was also possible to generate similar shaped simulated power spectra for dif-

ferent emitters. Figure 5.18 shows the simulated power spectra for the 7r mode with

field emitters in different cells, located at a different S0, and with a different 6 F N-

Certainly the specta are not identical, but the shape of either simulated power spec-

tra would be a reasonable match to a data power spectrum. Thus when making a

comparison with the data, the electron Kf for each peak should also be compared

with the x-ray endpoint energy.

124

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a)
2 [cm] KE[keV]
ES 1 1010
E 10 830
0- 13 1380
16 814
19 1010
I -1 Mi l f / 1 27 1520
34 830
o 20 40 60 42 824
Distance [cm] 45 722
— x (A, 52 581
.3 . -, 1 1‘ 53 1210
2 [cm] KE[keV]
(b) 2 1600
9 577
a; 16 748
5 19 782
‘L 26 1050
34 1470
A M 43 881
, I i’\ . 5 ___, 45 616
0 20 4o 60
Distance [cm]

x’ \ x
. . .1 ‘ ~ 1
5 1 - l ‘.
1 s .1 .
O I u

I/.-. -11--

 

Figure 5.18: Two simulated power spectra for different emitters in the 7r mode. The
spectra have the same general shape. Panel a): Field emitter located in cell #1,
SO = —13.52 cm, fiFN = 100, and Ep 2 22.9 MITIY' Panel b): Field emitter located in

cell #3, SO 2 13.38 cm, BFN = 400, and Ep 2 23.1 -1\—f£1¥' The tables list the electron

maximum final kinetic energy for each peak.

125

Finally, the simulated power spectrum can also be binned by K f, similar to energy
binning described in Subsection 4.2.2. Since power is proportional to K f, the sum of
the power bin is limited by choosing an electron final kinetic energy range. Figure 5.19
(a) shows the simulated power spectrum of Figure 5.18 (a) binned to an energy range
of 500 keV to 2 MeV. The highest peak moved from 34 cm to 27 cm. Many of the
electrons impacting around 34 cm have final kinetic energies below 500 keV and those
impacting around 27 cm above 500 keV. Figure 5.19 (b) shows the impacting electron
final kinetic energy as a function of distance along the cavity z axis.

The acquired data x-ray spectra includes information about the energy and num-

ber of x—rays entering the detector in two second intervals. Since power is defined as

 

 

 

 

 

 

 

 

6.E-06 a (a)
5.E-06 i
,_ 4.E-06 7
3’
8 3.E-06 7
2.E-06 7
1.E-06 ~ I
0.E+00 . - i 1 l . i .
0 20 4O 60
Distance [cm]
1.8E+06 - (b)
5" 0 5
.9. 1.2E+06 -
§ 9
c o o o '
o 3
:3 6.0E+05 .0 O. . o
8 .0 ° 3
El . 8.. f o: 9
0.0E+00 ~ 1 I l
O 20 4O 60

Distance [cm]

Figure 5.19: Panel (a): Binned simulated power spectrum of Figure 5.18 (a). Panel
b): Impacting electron final kinetic energy.

126

energy per unit time, the x-ray spectra were converted to power spectra:

233 En,
P- = J— 5.5
3 72:1 2sec’ ( )

where Pj is the power at distance interval j, Ez- is the energy of the acquired x-rays (233
energy intervals (subsection 41.1)), and n,- is the number of x—rays with energy Ei-
The data x—ray spectrum becomes a data power spectrum for direct comparison with
the simulated power Spectrum. Figure 5.20 shows an x—ray spectrum acquired from
cavity #2 operated in the fig: mode (Figure 4.13 (c)) and the data power spectrum
of the same scan. This method can be used for comparison of the data power spectra

with the simulated power spectra.

2000 — (a)
1 500

1000 1

 

X-ray Count - Bkg

500 7

 

 

7.E+05 (b)
6.E+05 7
5.E+05 7
4.E+05 7
3.E+05 7
251-05 7
1.E+05 7
0.E+00 r r r ‘ 1‘7““

0 20 40 60 80
Distance [cm]

Power

 

 

Figure 5.20: Cavity #2, 577, acquired data. Panel (a): X-ray flux spectrum. Panel (b):
Power spectrum.

127

Chapter 6

Comparison Between Data and

Simulations

Using the Multipacting/ F ield Emission Simulation, the simulated power spectra for
field emitted electrons were examined until a reasonable match to the power spectra
for the data was found. The task of finding a match of the simulation to the data
meant defining field emitters along the cavity surface in 0.001 cm increments while
varying I} F N and Ep. The number of power bins along the z axis was chosen to reflect
the number of two—second time intervals (118) that the slit collimator was opposite
the cavity as data were acquired. Once a reasonable match was found, the same field
emitter parameters were applied as input to another mode to compare that simulated

power spectum with the data power spectrum.

6.1 Cavity #2

6.1.1 46’: Mode

The 5671 mode had the highest x-ray flux (Figure 4.18) of the three modes of which
data were acquired. Since a field emitter has the same S0, A8, and (.9 for various modes,

a field emitter in the 5371 mode is limited to regions where E1957r was greatest, that is
6

128

along the irises of cells #1 and #6. Initially, current, and subsequently power, were
calculated with nominal values of Ep%7[ (32 5%) and (31:1 N (150) (Section 4.2). When
a power spectrum resembling the data was found, closer evaluation of Ep57r :1: 5%

_6
(30.4 % g 123,057r s 33.6 W) and 13m, (120 g ,BFN g 220) (Section 4.2) were
7,7

m
pursued until the best fit to the power spectrum was achieved. ,8 F N was measured
for the 77 mode only, so a range was chosen between two previous cavity tests for use
with the 56L and fifii modes.

Three power spectra resembling the data power spectrum were identified. The
results of one emitter is presented in detail here, and the remaining two emitters are
presented in Section 6.2.

The data power spectrum, generated from the x-ray flux spectrum presented in
Figure 4.13 (c), is compared with the simulated power spectrum in Figure 6.1 (a). The
field emitter is located in cell #1, SO = 14.025 cm, lap-5675 :7- 32.5 9%, fiFN = 120,
and Ag 2 2.4 X 10"9 c1112. Other than the peak at 22 cm, the simulated power
spectrum is a reasonable match with the data. The high power, in the simulated
power spectrum, at 21 and 74 cm are those electrons impacting the cell end caps of
the simulated cavity. Figure 6.1 (b) shows 500 simulated electron trajectories and (c)
shows the kinetic energy of 1000 impacting electrons. The x-ray endpoint energy of

the acquired data and the kinetic energy of the impacting electrons are compared in

Table 6.1

129

Normalized Power

0.5 -

 

(a)

 

 

 

 

 

\ i 7 n AAA A AA]
I l'

 

Electron Kinetic Energy [eV

20 4O 60 80
Distance [cm]
(b) .7" ~\. z’f \\ /’ 7‘. ,x' ’\ /’ \

\ 1T\ /.\ {‘6 f
1 l 1 l‘. i l I ‘1 j!
i ‘ i" l i i, i
' 1 (G ' j .
1' ' it“ :1 " :f
\ / i / \ / x // \\J \\J'
(C)
2.E+06 1
O
O o
’ o
O O
1.E+06 ‘ ,
O
. .0

O
0.E+00 I T ‘ l . l

 

20 4O 60 80

Distance [cm]

Figure 6.1: Cavity #2, §67£ mode, power spectrum. Panel (a): The black line represents

the simulated power spectrum for a field emitter located in cell #1, SO = 14.025 cm,

E1957r = 32.5 %, EFN 7—7 120, and A3 = 2.4 x 10’9 cm2. The gray line represents

the data power spectrum. Panel (b): 500 field emitted electron trajectories. Panel (c):
Final kinetic energy of 1000 electrons.

130

Table 6.1: Cavity #2, 56: mode, emitter in cell #1, S0 7—7 14.025 cm, E1057r = 32.5 01—13;,
6
(BFN = 120, and A6 = 2.4 x 10’9 cm2.

 

 

z [cm] X—ray Endpoint Energy [keV] Electron KE [keV]
12.3 755 1210

22.8 800 922

29.4 923 1110

37.4 874 1630

45.7 790 1310

53.9 752 1710

61.9 940 1200

73 —— 1180

 

 

 

 

 

6.1.2 77 Mode

Using the same field emitter site and emitter parameters, the power spectum and
electron trajectories were generated for the 7r mode. The data power spectrum, from
the x—ray flux spectrum presented in Figure 4.8 (c), is compared with the simulated
power spectrum in Figure 6.2 (a). The 7r mode data power spectrum was normalized
to the same scale as the 56: mode data power spectrum. The 77 mode simulated power
spectrum was normalized to the g675 mode simulated power spectrum. Em was varied
until the simulated power spectrum peak at 28 cm was just greater than the data.
An Em :7 22.5 % was low compared to the measure range of 24.1 MTHY 3 Elm _<_
26.7 L121. The peaks at 37, 45, and 78 cm are absent from the simulated power
spectrum. Figure 6.2 (b) shows 500 simulated electron trajectories and (c) shows the
kinetic energy of 1000 impacting electrons. The x-ray endpoint energy of the acquired
data and the kinetic energy of the impacting electrons are compared in Table 6.2.
The 77 mode power spectrum could be reproduced for this field emitter but with
(3 F N = 30. Not only was this an unreasonable value for 6 F N, since it was determined

that 120 S 6 F N S 220, but the %75 mode power spectrum was no longer a match to

the data power spectrum.

131

Normalized Power

 

 

 

 

 

 

 

 

 

 

 

0.25 7 (a)
0.2 7
0.15 7
0.1 -
0.05 ~ (
( 1
l
0 r " T I 1
0 20 4O 60 80
Distance [cm]
(b) , A A\
in; \/ '
2/ 7
(C) 7
. .1.5E+06
i; .
3
a3 1.0E+06 .
C O .
”J t
.9 .
f‘g’ 5.013+05 t °
5:
c ‘Q 0.
.9. ,f . .
_33, 0.015400 . , ,
L“ 20 4O 60 80

Distance [cm]

Figure 6.2: Cavity #2, 7r mode, power spectrum. Panel (a): The black line represents
the simulated power spectrum for field a emitter located in cell #1, SO 2 14.025 cm,

MV
7—7 22.5 —

m 1
the data power spectrum. Panel (b): 500 field emitted electron trajectories. Panel (c):

Final kinetic energy of 1000 electrons.

BFN = 120, and Ag 7—7 2.4 x 10‘9 cm2. The gray line represents

132

Table 6.2: Cavity #2, 7r mode, emitter in cell #1, SD = 14.025 cm, Elm = 22.5 MIX

.BFN = 120, and A8 = 2.4 x 10‘9 01112.

11’

 

7. [cm]

X—ray Endpoint Energy [keV]

Electron KE [keV]

 

13.4
22
28.8
37.5
45.6
55.3
78.6

 

 

577
900
733
878

1073

 

840
819
726
770
1310
576
970

 

 

The emitter area, A6, was determined by matching simulated values of the cavity
dissipated power as a function of E37r with the data (Figure 6.3). An emitter area of

A6 = 2.4 x 10-9 cm was used for the simulations.

 

§§€K)1
5 o
0?? l! f“.
.0 40 a as O
-9 gi '
.3 6
fl)” figm . .
D O 3'1 5 it '" "'3 .E O ' l ’ ' 1
0 400 800 1200 1600
EpEp

Figure 6.3: Cavity #2 Black diamonds - data acquired for the test just prior to
this study. Medium-gray squares — data acquired for another cavity test. Light-gray
triangles - simulated results for an emitter in cell #1, SO 2 14.025 cm, ,8 F N = 120,

22.5 M and A6 = 2.4 x 10—9 cm.

E1?” = m ’

6;143 é; hdtnie

Again, using the same field emitter site and emitter parameters, the power spectum
and electron trajectories were generated for the 51675 mode. The power spectrum, from
the x-ray flux spectrum presented in Figure 4.16 (c), is compared with the simulated

power spectrum in Figure 6.4 (a). The data and simulated power spectra were nor-

133

malized by the same method used for the fig: and 7r modes. Figure 6.4 (b) shows the
simulated electron trajectories. Figure 6.4 (c) shows the kinetic energy of the impact-
ing electrons (1000). 1000 field emitted electrons were used to generate the simulated
power spectrum.

For this field emitter site an Epéér— = 31.6 1%}1 was needed to genterate a power

spectrum of a magnitude to match the data power Spectrum (30.0 % S Ep47r S
'6'

33.2 1%!) The x—ray endpoint energy of the acquired data and the kinetic energy of
the impacting electrons are compared in Table 6.3

Binning of the data and simulated power spectra for the same emitter found in the
ESE mode was also tried, but the 7r and é67E mode simulated power spectra still did not
match the data power spectra. Another approach would be to locate a field emitter
for the binned data power spectrum of the 567: mode and then make comparisons with
the other modes.

It is possible that multiple field emitters reside within the cavity. The other 57f

simulated power spectra, eliminated by comparison with the endpoint energy, would

require multiple emitters as well in order to reproduce the data power spectra.

134

 

 

 

 

 

 

 

 

 

l
l
0.2 1
5 l
3
O
a 0.15
u l
G.)
.E '
a |
O l
2 l
0.05 j A
1 ll
0 WWWM’S T. MUL
0 20 40 50 80 l
Distance [cm] i
(b) \ z’x /" ‘ \ / \ ‘ X
i 1'; , 3",: a. . 5 'X, (it.
gr”? -
E\ f“. {9"}! c “ A
(C) l“ 2". if -. , 'i ‘ ,
E 2.0E+06 “ V H J ‘ '
>
9
a) 1.5E+06
C
LlJ
g 1.0E+06 ' , .
.5
E 5.05.05 I
g 0.0E+00 ' H . .
20 40 60 80

Distance [cm]

Figure 6.4: Cavity #2, 3675 mode, power spectrum. Panel (a): The black line represents

the simulated power spectrum for field a emitter located in cell #1, SD = 14.025 cm,

Ep47r = 31.6 M fiFN = 120, and Ag 2 2.4x 10—9 cm2. The gray line represents the
—6_

111 ’

power spectrum for acquired data. Panel (b): 500 Field emitted electron trajectories.
Panel (c): Final kinetic energy of 1000 electrons.

135

Table 6.3: Cavity #2, 16,: mode, emitter in cell #1, SO : 14.025 cm, 13104,.r = 31.6 -I\l%/-,
' 6

zaFN = 120, and A6 = 2.4 x 10—9 cm2.

 

 

2 [cm] X—ray Endpoint Energy [keV] Electron KE [keV]
21.5 745 1220
28.5 670 1610
37.1 750 962
45.0 771 —-—
55 —— 910
64 —* 1990
70.1 669 1250

 

 

 

 

 

6.2 Additional Field Emitters Cavity #2

Two additional field emitters were likely candidates to match the data power spectrum
of the Q65 mode. Both were located near an iris in cell #6. A plot comparing the data
power spectrum with the simulated power spectrum for first field emitter, SO =
13.266 cm, for all three modes is shown in Figure 6.5 . The x-ray endpoint energies
and the electron kinetic energies are shown in Table 6.4. The emitter area was found
using the method described in Section 6.1.2. The results of the second field emitter
are shown in Figure 6.6 and Table 6.4. A decision as to which field emitter most

closely matches the data cannot be made since 6 F N is not known for each mode.

136

1- (a)

0.5 ~

Normalized Power

 

0 . 1J1 (l '1 4114 _

O 20 40 60 80
Distance [cm]

 

 

 

 

 

 

 

 

 

 

0.2 « (b)
E
g
0
0.
‘0
g 0.1 .
E
E
O
2 z i
0 7" . I I l 6'. ' A A, 1
0 20 40 60 80
Distance [cm]
0.1 , (c)
(T:
3
O
0.
"O l
g 0.05 -
E
E
O
z t
. . a 6 l . ' I l
0 :- l iv ‘ rLJfi! H fl
0 20 40 60 80
Distance [cm]

Figure 6.5: Cavity #2, cell #6, $0 = 13.266 cm, ,0“, = 200, and A6 = 4x 10-12 cm2.

Black line is the normalized simulated power spectrum and the gray line the normal-

ized data power spectrum. Panel (a): 5675 mode, Ep57r = 32.0 1%,‘1. Panel (b): 1r mode,
6

E1977 2 27.4 MrfiY_ Panel (c): 267': mode, E1346: 2 31.6 %.

137

1 —~ (a)

0.5 4

 

Normalized Power

 

 

 

0 I ( 1AA REA“ J T
0 20 40 60 80

Distance [cm]
0.2 (b)

0.1 W

Normalized Power

 

 

 

 

Distance [cm]

0.1 1 (C)

0.05 -.

l l l l

0 20 40 60 80
Distance [cm]

 

Normalized Power

 

 

 

Figure 6.6: Cavity #2, cell #6, SO 2 13.5 cm, fiFN = 120, and A6 = 2.4 x 10"9 cm2.

Black line is the normalized simulated power spectrum and the gray line the normal-

ized data power spectrum. Panel (a): 767 mode, E1957r = 33.0 9%. Panel (b): 7r mode,
‘6—

Eprr = 31.0 %. Panel (c): 4675 mode, Epégfl = 31.5 Mix:

138

Table 6.4: Cavity #2, ESE mode, emitters in cell #6.

 

 

 

 

 

 

 

 

 

 

i? Mode X-ray Endpoint Energy Electron KE [keV] Electron KE [keV]
2 [cm] [keV] S0 = 13.266 cm SO 2 13.5 cm
12.3 755 855 354
22.8 800 1300 1240
29.4 923 1030 1980
37.4 874 1710 1890
45.7 790 1550 1530
53.9 752 1860 1890
61.9 940 1120 1170
73 -- 1430 1050
1r Mode X-ray Endpoint Energy Electron KE [keV] Electron KE [keV]
2 [cm] [keV] S0 = 13.266 cm S0 = 13.5 cm
13.4 577 1750 1860
22 -~ 93 —-
28.8 900 1060 951
37.5 733 1110 1290
45.6 878 1690 2080
55.3 — 1140 1620
65.4 --~ 720 1450
78.6 1073 607 945
3671 Mode X-ray Endpoint Energy Electron KE [keV] Electron KE [keV]
2 [cm] [keV] SO = 13.266 cm SO 2 13.5 cm
21.5 745 3580 3580
28.5 670 -—- 1120
37.1 750 -— ——
45.0 771 —— 362
55.0 —— - 968
64.0 —— 493 1180
70.1 669 1410 1930

 

139

 

6.3 Cavity #1

For cavity #1, the x—ray flux was greater for the 7r mode than the 365 mode, indicating
a higher Eem. Eem is higher for the 7r mode than the %75 mode in cells #2, #3, #4,
and #5 (Figure 5.4). A reasonable match to the data, satisfying both the 7r and the
%75 modes, could not be found in those cells. A field emitter displaying reasonable
agreement with both modes, after binning both the data and the simulated power

spectrum, was found in cell #1.

6.3.1 7r Mode

The nearest match of the simulated power spectrum to the data power spectrum,
was for a field emitter located in cell #1, S0 = —13.47 cm, and Elm = 23.1 Mill.
This choice for Epw is low, but provided the best match with the data (23.8 _h;1n_V _<_
Epw S 26.4 914%). Energy binning was also employed for electrons with a minimum of
K f = 500 keV. The simulated power spectrum was normalized to the peak at 42 cm.
The power spectrum, binned for electrons with a minimum K f = 500 keV, is shown
in Figure 6.7 (a), panel (b) shows all of the electron trajectories, and panel (c) the
electron final kinetic energies. Table 6.5 shows a comparinson of the x-ray endpoint

energy with the electron kinetic energy.

140

Normalized Power

1— (a)

0.5 J

(c)
1 .6E+06 5

eV

'—'1 .2E+06 -

l o

8.0E+05 9

Energy

IC

met

4.0E+05 ~

0.0E+00 v

 

 

 

3
3
o

O O '0.

O

O

9 o 6
’0»—

O
y .0
O

 

Electron K

10

“___,___1

30 50 70
Distance [cm]

Figure 6.7: Cavity #1. 7r mode, power spectrum. Panel (a): The black line represents
the simulated power spectrum for a field emitter located in cell #1. SO : —13.47 cm,

E?” = 23.1 % de = 120, and Ag 2 5.5 x 10’10 cm2. The gray line represents the

power spectrum for acquired data. Panel (b): 1000 field emitted electron trajectories.
Panel (0): Final kinetic energy of 1000 electrons.

141

Table 6.5: Cavity #1, 7r mode, emitter in cell #1, SO 2 —13.47 cm, Elm = 23.1 %,
,BFN = 120, and A6 = 5.5 x 10‘1” c1112.

 

 

2 [cm] KE [keV] Data KE [keV] Simulation
14.4 913 976
23.5 846 838
31.5 926 1350
39.9 768 1450
47.8 1065 818
51 —-- 842
56.1 845 522
61.9 776 749
72.3 878 1280

 

 

 

 

 

The same method to determine the emitter area for cavity #2 was used for cav-

ity #1 (Figure 6.8). An emitter area of A6 = 5.5 x 10_10 cm2 was used for the

 

 

simulations.
E 60 T .
t o
a) ..
g 40 « .
O.
D o
2 I -
(a 20 ~ ' °.
.5 5
8 . .
O fiWfi‘A‘wQfi—a- _..‘._ _-_0_T._ ___. T 1
0 500 1 000 1 500 2000
EpEp

Figure 6.8: Cavity #2 Black diamonds - data acquired for the test just prior to
this study. Medium-gray squares - data acquired for another cavity test. Light-gray
triangles - simulated results for an emitter in cell #1, 30 = -—13.47 cm, 6 F N = 120,

Elm = 23.1 M and A8 = 5.5 x 10‘10 cm.

111’

6.3.2 5?” Mode

The input parameters, used for the 7r mode simulated power spectrum, were used as

input for the _6_57r mode. Ep5,r was adjusted until the intcsity of the power spectrum
3—
lVIV

m ). This value for

peak at 60 cm was in range of that of the 7r mode (Eme = 23.5
_6_

142

Ep 5,r was low r than the measured range of 29. 6 L11}, < Epi- ,7, < 32. 8 L3,. Energy

‘6— (*6—
binning was also untilized; electrons with a minimum [8 f = 500 Ice V for the simulated

power spectrum and a minimum of 500 keV x—rays for the data power spectrum. The
simulated and data power spectra are shown in Figure 6.9 (a), panel (b) shows all of
the electron trajectories, and panel (c) the electron final kinetic energies. Table 6.6
shows a comparinscm of the x—ray endpoint energy with the electron kinetic energy.

A better fit of the simulation to the data may be to scale E19567E with the peak at 68
cm and include a second field emitter to fill in the remainder of the spectrum. Peaks
are emerging at 26 and 40 cm. It may be possible to adjust the input parameters
more closely to find a more suitable match to the data.

It is reasonable for some simulated power peaks to be higher than data power
peaks. The angle that electrons impact effects the data power spectrum since bremsstrahlung
x-rays are anisotropic for 1 MeV electrons (Subsection 2.1.3). Some data power peaks
could be lower by 30%. Calculation of the data power spectrum includes the lower
energy of Compton scattered x-rays and x-rays with higher energies due to summa-
tion events (Subsection 4.2.2). The method of power binning may best be applied to

individual peaks since the endpoint energy of x-rays varies for each peak.

143

 

0.5

 

Normalized Power

 

0 l . LM__.___,1 - \ 1U .

0 20 40 60 80

Distance [cm]
(b) t.” ' ' 3 7 ,' " ' (”\\2 .’ ‘ ‘\‘. If"

 

(c)

 

 

E1. .6E+06 '1 ‘
a , .
51.2E+06 ~ '
c
LU 0
ga. OE+05 - f I) 3
C o
">3 , t ’ f
g4. OE+05 a °
.. . . 1
8 0,. . I ‘
LTJ O. OE+00 I
10 30 50 70

Distance [cm]

Figure 6. 9: Cavity #1,53— ”mode, power spectrum. Panel (a): The black line represents

the simulated power spectrum for a field emitter located in cell #1, S O — -13. 47 cm,

Epgm = 23.5 MV ,=BFN 120, and Ae— — 5. 5 x 10 10 cm2. The gray line repre-
‘6'

sents the power spectrum for acquired data. Panel (b): 1000 field emitted electron
trajectories. Panel (c): Final kinetic energy of 1000 electrons.

144

Table 6.6: Cavity #1, ESE mode, emitter in cell #1, S0 = —13.47 cm, Ep5,r =
F

23.5 M fiFN = 120, and A6 = 5.5 x 10—10 cn12.

111 ’

 

 

2 [cm] KE [keV] Data KE [keV] Simulation
15.1 556 658
22 733 887
26 -— 1310

29.5 652 -—

33.5 736 —-

43.9 554 998

52.2 605 -—

61.1 859 1540

68.8 937 850

 

 

 

 

 

145

Chapter 7

Discussion and Conclusion

7. 1 Summary

This study examined a new method for locating defects in SRF cavities in operating
accelerators and during the manufacturing of cavities. Two 6 = 0.47 805 MHz six-cell
cavities, encapsulated in a cryostat, were tested. An x-ray image was acquired for
each mode tested for each cavity. A particle tracking code was used to determine a
defect site. Operation of the cavities in other modes verified the location of the defect.

111 an operating accelerator, contaminants could enter the cavity through vacuum
system leaks or while cavities are vented for maintainance. Cavity defects may oc-
cur during the manufacturing process, and may result from improper clean room
procedures, defective dies used to form the cavities, or impurities in the material.

Cavity defects enhance the surface electric field causing electrons to leave the
surface or are field emitted. Field emitted electrons are accelerated by electric fields
within the cavity and guided by the magnetic fields. The electrons strike the inner cav-
ity walls or adjoining structures and generate bremsstrahlung x-rays, dissipate power,
and create heat. The power dissipation degrades cavity performance by lowering the
cavity Q.

The x-rays travel through the surrounding cryostat materials with Compton scat-

146

tering as the most probable interaction. There is a 25% probability that 1 MeV x—rays
will have no interaction with the cryostat materials. A slit collimated Nal detector
shielded with lead scans along the side of the cryostat, recording the number and
energy of x-rays entering the detector in two second intervals. The flux of x-rays trav-
eling through the slit and entering the detector is highest when the slit is opposite
the x-ray source. There is a finite probability that x-rays may travel through the lead
shielding and enter the detector. Data were acquired with the slit of the collimator
open and closed so x-rays traveling through the lead shielding could be subtracted
from those entering through the slit. To reduce the number of Compton scattered
x-rays from the data, energy binning and subsequently power binning were employed.

The x-ray endpoint energy is deduced from the x-ray spectrum. The endpoint
energy infers the highest final kinetic energy of the impacting electrons. This energy
is used as a comparison with the particle tracking code.

Multiple cell cavities can be operated in different modes of excitation. The acceler-
ating electric field within a cavity, operated in a mode other than the accelerating or
7r mode, is not symmetric within a cell or from cell to cell. The field emitted electrons
impact at different sites and thus produce a different x-ray spectrum. In this study,
it was demonstrated that electron trajectories and thus x-ray spectra are dependent
upon the mode of operation.

The Multipacting/Field Emission Simulation was used as the particle tracking
code. The code was modified to perform for other modes of operation. Input param-
eters for the code were estimated from previous cavity tests. The code generated an
output that could be used to calculate power dissipated in the cavity walls. Since the
simulation does not account for Compton scattered x-rays, the power was binned. The
kinetic energy of the impacting electrons was also compared to the endpoint energy
of the x—ray data.

A reasonable single emitter was found for cavity #1 Operated in the 7r mode. This

was verified by the simulated power spectrum for the 5375 mode. A single emitter for

147

cavity #2 was not confirmed. Another approach would be to start with the binned
power spectrum when comparing the simulation with the data. The same binning
could then be used as input for the other modes. There exists the possibility that the
cavities have multiple emitters. Once either the defective cavity or specific defect site
is identified, the cavity can be subjected to high power pulsed processing or helium
processing to effectively eliminate or burn off the emitter.

For this study a range of 13 F N values were inferred from previous tests of the 7r
mode. The simulated power is proportional to current. The calculation of current is
dependent up the measured quantities of Ep and 6 F N- Ep values were determined
from the acquired data, but not B F N- A more accurate calculation of the simulated

power would require the measurement of 13 F N for each mode.

7 .2 Previous Work

Area radiation monitoring equipment is a good indicator of x-ray exposure ( 1&1) but

x-ray energy cannot be inferred from the measurement. X—ray energy is needed for
comparison with the particle tracking code results. Area radiation monitoring equip-
ment is also capable of distinguishing between areas of high and low x—ray exposure,
but is incapable of pinpointing an x-ray source. X-ray source location must be accu-
rate to some degree in order to determine the location of impacting electrons, also for
comparison with the particle tracking code results.

The method of locating cavity defects developed in this study has an advantage
over area radiation monitoring equipment because a NaI detector is used for measure—
ment of x-ray energy. Slit collimation of the detector provides the means of locating
x-ray sources to within the slit spatial resolution. Knowing the x-ray source location,
or electron impact location, a comparison is made with the results of the particle
tracking code to determine the electron source, or defect.

Extensive studies have been performed to locate cavity defects using thermometry.

148

Resistors are arranged 011 the entire outer cavity surface. A change in resistance
signifies a change in temperature, indicating a region of electron impacts. Energy of
the impacting electrons is inferred from the temperature change. A particle tracking
code is used to locate the defect. Defects are verified by disassembling the cavity and
viewing the region with a scanning electron microscope. This method is cost and labor
intensive but has been proven successful for locating cavity defects. This method is
not practical for operating accelerators where the cavities are sealed in cryostats and
not accessible.

The method of locating cavity defects developed in this study has an advantage
over thermometry in that it can be used on operating accelerators. Cavity defects
are verified by operating the cavity in different modes of excitation. This noninvasive

method for verifying cavity defects is simpler and less costly than thermometry.

7 .3 Future Work

This work can be expanded upon with development of a gamma camera to take
a snapshot of the x-ray image. Not only could data be acquired more quickly, but
the gamma camera could be positioned to acquire x-ray spectra in more than one
dimension. Other dimensions would provide azimuthal defect location. Tomographic
techniques could be used for image reconstruction. The method of locating cavity

defects presented here could also be used on a liner accelerator in operation.

149

Appendix A

Electromagnetic Fields

A.1 Fields Within a Rectangular Waveguide

The derivation of fields within a rectangular waveguide begins with Maxwell’s equa-
tions:

(i)V-E=g% (ii)VxE=—3§ _.
(111)V-—§=0 (iv)_v"x§=po7+1‘9E.
Z???
In a region free of charge, or free of current, Maxwell’s equations become:
. -—> —> _ .. —> ——> _ _03
(1)V-E—O (11)VXE— _87—
(iii) 6’ - f? = 0 7 %%

. ' —-> ___ 1
(1V)V X B 32‘
The wave equation, used for solving the fields, is derived by taking the curl of (ii):

——>

~-—-) 2-—)
——> —>_—>—>.——>-——>2—>_—+ _BB ___-24 —)—__!_BE
VX(VXE)—V(V E) VE—Vx( _8t)_ 8t(VXB)_ (9&2
sothat,
102 —>
172—___ 13:0. A.1
( c28t2) ( )

150

A similar derivation by taking the curl of (iv) would yield

2
2 1 a - _

Equations A1 and A2 have a solution for the z or longitudinal direction in Cartesian
coordinates:

E02121) = Etc, netic—M (A3)

and

B(:L‘,y, z,t) = B($,y)ei’ikzz—wt, (A.4)

where kz is the propagation constant or wave number in the z direction, kz = 2%,
w the angular frequency, w = 27w 2 kn, u the frequency, and v the velocity. The
sign of k depends on the wave direction. Substituting Equations A.3 and A4 into

Equations Al and A2 respectively:

1122 1122

v21: = —C—2E = 4.313 and v23 :2 -—C—ZB = 4&3.

From Maxwell’s equations and Equations A.3 and A4 respectively [31]:

9x3: —=—-f—E. (A.6)

The Laplacian operator,

 

151

can be separated into two parts

r2
2 _ 2 d E
V E — VTE '1' fl
2
where T implies the tranverse plane and 5% the derivatives in the longitudinal di-
rection:
2 2
2 _ 2 0 E _ w .2
r 2 2
2 _ 2 d B _ w .2
VTB—VB-bj——(—C—2-—kz)3. (A.8)

Let 7'22— — k3 2 kg where kc is the cutoff frequency, kc = V1.52 — kg. For k > kz
propagation of the mode occurs, but for k < leg the modes are called cutoff, or
evanescent and decay exponentially along the waveguide [32].

Two differential equations, A7 and A8, must be satisfied in dielectric regions or

regions free of charge, as in this case. First the components of the fields are found

using Maxwell’s equations, Equations A5 and A6, and Equations A.3 and A4:

0E3 813,, _ aEz

 

 

 

 

 

 

 

 

 

 

 

MB]; 2 0y — 8.: 0y $ 1711'; Ey. (A.9)
1.03,, = _8aiz + 8:3” = ~80? i viszx, (Ale)
1.53,. = 86:” — 803‘”, (A.11)

i—zEx = 80% —— 8:” = 85": a 11533)], (A.12)
3:53,, 2 —% + 8:? = —a£fi11:zBI. (A.13)

152

—“ E3 = r '—
‘2 d1: 83]

 

 

. (A.14)

c.
Combining Equations A9 and A13 . . . the fields within a rectangular waveguide are

given dependent upon E z and Bz:

 

 

 

 

’1: BBZ u.) 8E3
B — — A.1
“3 kg ( z 31 C2 By ) ( 6)
i 8E2 aBz
E. = — ik —— A.
I kg( Z81: +w0y)’ ( 17)
3 QB; LU 8E2
B, = — :tk — — — . A.1
y 1:? ( Z By 02 8:1: ) ( 8)

By taking Equations A.15 to A18 and substituting them into Equations All and

A14, E z and B 7, are obtained independent of one another [6]:

(g§+%+§ 43) E2 =0. (A-19)
(£72, + 5%) E. = -1313. = VQTE... (A20)
(5732—, + 5722 + “CL: -— k3,) B. = 0. (A21)
(81; + 55:13,) 13,... = 4&8... = V%Bz. (A22)

Of interest here is the derivation of the transverse magnetic mode (TM) mode

153

 

<——>

 

 

 

/

2!

X ——->
Figure A.1: Rectangular Waveguide where lengths x=a and y=b.

since it is the accelerating mode for particle propagation. A similar derivation could
be carried out for the transverse electric mode (TE) where the magnetic field is in

the direction of particle propagation. From Equation A.20, E z is found by separation
of variables: Let E; = X(.t-)Y(y)eiikzz then X”Y + xv” = —k2XY after factoring
out the eiilrzz term. Dividing through by XY leaves X? + 116;”— — kg. Since x and
y can be changed independently of each other bothX —XI- andY '17" must be different
constants. 7— : —k% and T: = —k.§. Solving X = Aeos(kg;:1:)+Bsin(kx:r) and Y =
C cos(kyy) + D sin(liryy), where A, B, C, and D are constants that will be determined.
So E3 = XYeiii-tzz = (Acosflcgflt) + Bsin(kx;r))(Ccos(kyy) + Dsin(kyy))eiikzz [6].
Boundary conditions dictate that Ez(2: = 0,0.) = O, Ez(y = 0,0) = O, and
Ez(.r= g, ,—y — 2)— — E0 (Figure A. 1): S0 A and C must both equal zero, and so
E3 = (B sin(k$.r))(Dsin(l~:yy))ei ikzz, which requires kxa = mrr, kyb = mr and
BD = E0. Altogether,
7mm: nvry eiikza

' A.2
sm— b ( 3)

 

Ez 2 E0 sin

For the TM mode, Equation A23 is substituted into Equations A15 to A18 to solve

for Ex, Ey, Bx, andBy [6]:

 

' m7r 1 ;. ' '~
EI 2 — [3:19; EO—cos (mar) sin (_niry)] ei'l‘zz,
1.3 a b

154

 

'i 717T 'rmrzr mr' . ‘ ~
Ey : —— [iszO—sin( a ) cos (779)] Biz/1.22,

 

 

'2' w mr , 7mm: mry :t-ik z
B =— ——.—E —s1n( )cos (————) e Z ,
1: kg l CZ 0 b a b
i w m7r 7mm: , mry iik~z
B. = — —-—.E —cos( )s1n (—) 6 ~ .
g kg [ (:2 0 a a b

A.2 Fields Within 3 Circular Waveguide

For waveguides of circular cross section, Equations A.15 to A.18 are transformed to

cylindrical coordinates:

 

 

Er = i? (fig-618E; + $88?) , (A24)
Br 2 kl? (twig—Z— — 57%;) , (A25)
15,, = é (i%%% —- magi), (A26)
8,, _—_ If? (fig-93% + 595%). (A27)

A similar set of equations could be derived for the TE mode [6].
Starting with the wave equation, Equations A.1 and A2, the Laplace operator
V2, in cylindrical coordinates, Operating on E 2 can also be separated into two parts,

transverse and longitudinal:

1 a ( 8E3) 1 82Ez 82Ez 1 82132 —0

7‘07‘ 07“ 7'2 at? 822 c2 8232

155

82EZ 18E~ 1 82E~ 2
— . ~ a y A: E : 7
87‘2 7' (97' + r2 0&2 + C Z 0

 

62E; 1&9EZ 182a,;

2
—— —— —-——— = —k, E.
872 + 7‘ 87“ + 7'2 [M2 C 2

VCQIVEz 2

This differential equation can also be solved by separation of variables. Let E z =

RFqfieiikt-Tz, where 12(7) and F((;'>):

RI

I; 1 ll _ ,2
R a, + 7F), + T—2RF _ —ACRF¢ (A28)

where the eiikzz term has been factored out. Dividing Equation A.28 by RF¢ and

multiplying by T2:
ll
Fat 2 R” R’

2 2
If both sides of Equation A.29 are equal they must be equal to a constant, say m2:
F/I
——‘*" = m2 (A.30)
F05
or r212” + TR, + (kgr2 — m2)R : 0. (A31)

Equations A.30 and A31 have solutions of the form F4, 2 Ccos(m¢) + Dsin(m(j))
and R = AJn-L(:L') + BNm(;r), where A, B, C, and D are constants, :1: = kcr, and

kc = 91}? making :rmn = 351%}: Since E2 = RFgeiikZz:

EZ = [A.Im(;1:) + BN,,,,(.1:))[C cos(mgb) + Dsin(m¢)]ei“~'22, (A.32)

156

where Jm is a Bessel function of the first kind [7]:

00 - j _ 2.
M = (9771233)“, (5) J,

 

and Nm a Bessel function of the second kind [7]:

J :1: cos mvr — J__- a:
N']n(lr) : 77l( ) ( ) "ll )

 

sin(m7r)

Nm(;r) approaches 00 at integer values of m (at the origin) so B:O in Equation A32.
The origin is chosen such that (b = 0 and only the cos(m¢) term varies there so D = 0.
Altogether,

Ez z AJm(;1:) cos(m¢)eiik32 (A.33)

where the new constant for AC = A [6] [7]. Bessel functions have an infinite number
of roots with zeros occuring at integer values of n: 517ml, 23mg, $m3 For integer
values of n1, the first few roots are

3:01 :2 2.405, 1602 = 5.520, 11703 = 8.654,

1:11 2 3.832,:1312 = 7.016, 2:13 = 10.173,

.2721 = 5.136, 2:22 2 8.417, 3:23 2 11.620, [8]

Since the electric field is perpendicular to a conducting surface and the magnetic
field parallel, the boundary conditions for the circular waveguide are (Figure 2.2)
Ez(r = R) = 0, E¢(7” = R) = 0, and B,»(r = R) = 0. These boundary conditions
imply

_ 71an _ k ‘
33mn - R —- (:7-

 

 

u, 7' . _
E2 = AJm ( "A" )cos(m¢)ei7’kzz,

_ ilsz

“mm

11 r - .
Er AJ,,’n (—le—) cos(772<z'))ei2k'zz,

R

157

I
“‘24:..-

_ :FI [£3771R2

 

 

Umn'l' . . '_ -~
Ed) AJm ( .sz'n(mc9)eifl‘°z,
“(HINT R
'Iime‘? unmr , :tik~ z
3,. : _Ajm s1n(m(b)e ‘~ ,
“712.710 T R
I’w’ R I Ulrnn 7" it k Z Z
' UnmC R
B2 2 O.

A.3 Fields Within a Cavity

A cavity is formed when conductive ends are placed on the cylindrical waveguide at
z = O and at z = I. Now the cavity is filled with incident and reflected waves with
additional boundary conditions Er(z = 0,1) : 0, Ed,(z : 0,1) = O, and Bz(z =

0,1) :2 0. Since both incident and reflected waves are present:

 

E2 = Aan (unglr) COS(m¢)eiikzZ

: AJm (WELT) cos(mq§)(B+e+ikzz + B’e_ikzz),

 

where B+ and B _ are the wave amplitudes in the forward and backward direction.

The condition Ez(r = 0, (,6 = O, z = 0,1) = E0 implies B+ 2 B" and 2AB+ = E0:

 

u- r
Ez = EOJm ( "A” )cos(mq’)) cos(kzz).

Again, since incident and reflected waves are present the radial electric field be-

COIIIGSZ

E
r kc

 

u r -. _ _-...
Aan (4’2" ) cos(m<f))(B+c+"kzz + B e “‘22).

158

The boundary conditions require that ET must vanish, so B+ 2 —B_ so:

 

 

 

iIkz I 'Unnfl' , 2IB+(€+iAzZ — e—IATgZ)
Err : TAJ,” ( R ) COS(IIIQ9) 21
Ili‘v U T
= i. ” EOJ'hi (J%—) cos(m<f)) sin(kzz).
C
E-,‘(z = l) = 0 implies k3! = [fit for some integer p:
I» '1 7‘ 7T2
Br 2 LEOJ'r/n (fl) cos(m¢>) sin (L) .

kc [13 1

Similar expressions can be obtained for E05, Br, and B4,. Altogether the electric fields

become:

 

Ez = EOJm (”1172117“) cos(m¢) cos (pH—l?) ,

 

 

E. _ WREOJ;,, (T) COW) sin (W) ,

l'lllrnn l

 

 

 

i772p7rR2 11.771717" p7rz
E = —————E Jm (——) sin mqb sin ( ),
45 7111.72,”, 0 R ( ) l
and magnetic fields:
Bz = 0,
'Iw'rrzrzp7nR2 Umvfl' . P7”5
Br 2 —————E0Jm ( ' ) s1n(mq§) cos ( ) ,
“mnc 7' R l
B — ___iwnmpRE J, (___—11.771717“) cos(m(b) cos (Iv—7m)
('15 “7717102 0 m R I L I l ’

 

.2 2 . 2
w u )71
Where kg = —2 — k? :> w = CV k3 + k2 = C\/(-%fl) + (LB) 1‘ WWI/77,1):
c

159

 

 

 

Figure A.2: Electric and magnetic fields in a cylindrical cavity. The maximum electric
field occurs on axis to accelerate the beam and falls to zero at the cavity wall (R).
The magnetic field is zero on axis and has a maximum at 0.77R.

 

__ wmnp C \/(Umn)2 (P77)2
( d - _ — = ._ ._ __ .
“1 fmnp 27r 27f R + I

In the TA’IOIO mode the zero order Bessel function provides an accelerating field in
the longitudinal direction (direction of particle travel) and a magnetic field in the (b

direction.

160

Appendix B

Field Emission

B.1 Transmission Coefficient D(W)

+00 (1 d
N(W 2—3/_OO [LOO (py_ W (13.1)
:h 00 exp( qut)+1

Equation BI is solved by using the following substitutions:

 

pg 2 pc036 => dpy = —psir16d6 + cosOdp,

pz 2 psi116 => dpz = pcos0d6 +sin6dp.

The only term that survives in the product is pdp dB:

271' d
N()W 253/0001) " ”d6 . (B2)
e$p(W

2

 

161

 

'_ '2
Substituting (u : WWQ + 77%) and (111 = 512% into Equation B2 [10]:

, 47msz 00 (in,

 

 

 

 

 

 

 

 

h‘3 ('u.) + 1
_ 47rka W -—( p3 W W —( p2
_ 12,3 ( kT + 2ka) " I” (up kT — 2ka + 1
4, ' kT —W
: 7:; In. (1 + carp [ kT+ C]) . (B3)
The time independent Schrodinger equation [10]:
h? d2
-———— 1: f —
2m (11:2 d; (W VW)’ (B4)

where 1,0 is a complex function that has both amplitude (A) and phase ((25) [31],

A) : AWM), (as)

 

% :: A($)tiei¢(x)gb’(;r) + A’(.r)ciq§(x) = (fl/(33) + iA($)¢’($))€i¢($)a
2), . , . . .
:1; = A,(:E)igbl(;13)el¢(“r) + A”(r)e'¢(‘”) + iA(a:)¢’(w)i¢’(-r)e’¢("’)

+ tA’(;t)¢’(.-t)ei¢(1’) +1A(.r)¢”ei¢(x)

= (A” + 2A’2Ttb' + Aid)” — A(¢’)2)e"¢. (B.6)
Substituting Equation B.5 into into Equation B4:

(A” + 2A’z'tb’ + Ate” — A(¢’)2)€i¢ = :7ng — V)Aei¢.

162

The real part [31]:

2 .
A” = (gig—RV — W) + (6)2) A,

is very difficult to solve, so the Wentzel—Kramers—Brillouin (WKB) approximation is

u
used. Assume A” varies slowly or €13]— is much less than 273-(1/ — W') and (qb')2. Then:

 

 

 

2m , 2 dab \/2m(l-’V — V)
— — f = , —— :
1220/ W) ((b ) i di- 12 ’
1
(b : ifl / \/2m(W — V) (1.1:. (B7)

The imaginary part [31]:

d
(A2621) = 0,

2A’t'o’ + A‘Ié” = 0 :> —'
(11'

A2975, 2 Constant = C2,

_ C
\/2m(l/V — V) .

 

 

Substituting Equations B7 and BS into Equation B5:

21:", 1:2
119(2) Ch )e;1:p (71: / \/2m(W — V)d:1:) .

 

 

 

: \/2m(l’V —— v 3:1

To the left of the barrier, 13 < 11:1, there is a wave component traveling to the right

163

X1 X”)
‘<—— b %

Incident \/\_Transmitted
Wave Wave

 

 

Barrier

Figure B.1: An electron has a probability of tunneling through the potential barrier
if the barrier is either too high or too wide.

and a reflected component traveling to the left:
$0.13) 2 Aeikx+Be—ik$,

where k :— ——.—V2;1”E. To the right of the barrier as > 1:2:

 

 

 

111(2) 2 Feikx.
W'ithin the barrier 1:1 < :r < $2:
' "l
Ch +z' f“?
at = 6.1: — 2mW—Vrcda:
1%) \/2m(w—v1 p 5 $1 \/ < ()1

 

 

_' 2:2 _
\/2m(Dl/l,: _ V)ea:p [ z \/2m(W — V(1‘))d;1: .

 

 

The tunneling probability (Figure B.1):

If the barrier is very high or very wide the probability of tunneling is small, and the

164

(+) exponent term must go to zero [31] [10]:

]——i||~ c1p[h/ \/2m( W—V) )1],

 

 

 

D(W) =t1p[:h— Q/T \/2m W V)d

where 11 and 12 are the zeros of the randicand. The potential of an applied electric
field is more complex:

8 (,2
37;]; +;eE1:’—W]=O,
, I

 

|W| :1: \/|W|2 — e35
26E

[WI 63E

—— 1:1; 1— —

2eE W2

 

Thus the transmission coefficient is written as:

 

12 ,2
—lnD(W) = f 8—7; (W — 5— - 6E1)d1, (8.9)
11 h 35

 

2 .
_ 2eE12eE e c __
LCt y — W (111(l€—— —]—_lV] 33’ (162 [___Wldl- Thus “lg“: ‘ZIWIL => 173 —-— T and

= £129] (1 :1: V1 — y?) = l—Cgé Substituting into Equation B.9:

 

 

 

 

 

,. IWI 1+ “1 y _IWK y2|W|
2W3/2 7n l—Zl:+\fl—y2( 212
= —— Y 2 — é — '—d£
26Eh 1- 1.3, i
V ”/3771 /1+\/1_y2 \/ £2 + 25 2 (16
eEh 1—-\/1—y§ x/E

165

Letn=fl=>dn=2dfiz

 

2VlV 3m
—lnD(lV)= —E_h ———E\{E\/-+—\/___——/ —‘I]4 + 202 — (1 +y/1- y2) (1— 1— y2)d7).
e

Leta=\/1+\/1—y§ and=\/1—\/1—y2

 

 

 

2“ 3m
—lnD(l/V) = TEhm [ba \/—774 +2772 —a2b2dr]

: —e_2Eh fl/ba\/( 22)—7)( )(7) 2—.b2)d7}

This is a standard form for an elliptical integral:

 

 

-—InD(W) = 4” mlWl3 [13(1) — b2K(k)] ,

3eEh

 

dd) . — g .2 - 2 ~2 _ a2_b2
:f073 \/1__ k25i112¢ E(k) —' f0 \/1—k sm z/idzj), and It _ T-

Let U = i \/a[E (k) — b2K(k)] then

\/2a
2(/2m W 3
D(W) = exp ———|——|——'v(y)) . (B.10)

 

where K(

 

— 3eEh

B.2 Number of Electrons P(W)

The number of electrons within energy dW emerging from the surface per unit area
[10]:
P(l/V)le = N(W)D(l/V)dW, (B.11)

166

where N(VV) is the number of supply electrons and D(W) the probability that they

will leave the surface. Substituting Equations B.3 and BIG into Equation B.11:

 

 

4 1:1 —W -4\/2m|W|3
P(l’ll”)a’W = mm In [1 + 61]) (45)] 61])

I13 kT 3eEh ”(3’) ’

Field emission electrons have energies ltV ~ C, so the exponent of the tunneling

probability is expanded in a power series at W = C:

f(-r) = Na) + f’(a)(r — a) + . .. ,

BeEh ” y N 3eEh 3eEh U
2 277ng
+ v<y>(-———'—')<W—o

—4(/2n W3 _. ' _ 7 "
l] l (l 4 2111a v(y)+( 4 27ng ’(y))(VV—C)

eEh

. . _- 3 _,/ 3 __ ,
Substitutin 'v’ = d” — d” I , y = e E => dy = e EdC, and dC = _daé into
g a? T T y

_@

—4]/2m|W’]3 w' _ C

3eEh

the above equation:

 

where

2 d1)
andt=v——' —

3 dy’

 

 

.. _ 47rka , ,. —W +C .. W _C .
thus P(IV) —- 113 In [1+ 61p( kT )] e11) ( c+ d )_

167

111 the low temperature limit [10]:

 

 

—W
It‘T In 1 + 61p ——J:—C = 0 when W > C
1th

= C — W when l’V < C

thus P(lV) 2 0 when W > C
4 ' W —
_ 7:18:17) (—c + d C) (C — W) when W < C.
1

B.3 Current Density j

The current density is found by integrating P(W) over all energies, i.e. from —Wa to
00. Good and Miiller [10] assumed —Wa is far below the Fermi energy C, so the lower

limit is —00:

C
j = e/ P(W)dW
—OO

 

 

C _
z e/ 4fgle1p(—C + W C) (c — W)dW.
l

-00 d

168

Let 1 = —C+ 1%; => (l1: %l—:

47r-me

j = fe.1:p(1) (—(I(.I: + 0)) (Ida:
I13

 

 

 

4 d2 . ,

2 W21; / [e‘l'(—d.r) — 617(dc)] d1
4 , . d2 ,.

= WIT; (ex(;r + 1) — etc)

 

 

 

 

_ 47rmed2 _ .+ W” —C _ +11” —C + _ + W —-C C
_ I13 erp c d c c (.611) c d _00

2
47rmed _C

 

 

 

 

I13
63E2 4 2771653
I 61‘? ———-—’U
87rhcbt2 p BeEh
3
1.54 10-6E2 2
2 X, 2 611) —6.83 X 10965—1) , (13.12)
at E
where h = Planck’s constant, e2 = 197A»Ieii?{;ri.(47reo)’ and m = 9.109 x 10‘13Kg.

The elliptic function is defined as [11]:

5(3) = 0.956 — 1.06232

E1/2
s = 3.79 x10-5—
(9
Both Good and lV’Iiiller [10] and Wang and Loew [11] set t = 1 in Equation B.12 since

IStS 1.11:

3/2 E
L (45529 x 10“’+10.42-¢—2)]

1.54 x 10—612
' E
9963/2
(2:17) —6.529 X 10 _E—- . (B.13)

(I)

 

61p

 

__1
1.54 x 10—6 x 10462W 2E2
d9

 

 

169

B.4 RF Current (1)

Electron emission for RF fields is modified from the DC version by assuming that
the microscopic electric: field on a metal surface is of the form E = E0 sin wt. The
average field emission current is calculated by a time averaging substitution into

Equation B.13:

 

 

 

 

 

 

1 T
I = — I t dt
< > T [0 <1
_ _ , 2 t
= 1.54 x10 6 x104’5261p(¢ 1/2)Ae,13%NE3 (if?) I
T/4 _ 1 9 3/2
/ (sin2 wt)e1p( 6629 X 0 Cb )di‘ .
0 BFNEosmwt a
_ 9 3/2
Let 11 = 652372123096 and sin(wt) : 31,— => tacos(wt)dt 2 —;%d1 => cos(wt) =
\/1 —sin2 t: ,/1— (:12) [11]:
2 1 1 —d:r
(I) = C— —c1p(—11'1) ————————-
_ C/OO €1p(—m:)d1
7T 1 $3VI2—1
_ 0536—“
Wm
5.7 x 10’12 x 104.52Aefl%§’vEg'5exp(¢l°5) —6.529 x 109¢1-5
: 175 6” ’
cf) - fiFNEo

 

 

 

ln( (1) ) _ m 5.7 x 111-1214121312143, + (10 412—05) _ 6.529 x 10941-5
' E25 $1.75 ' 517N130 ’

170

(D r

d (109105-25) _ — (10(1106‘) 6.529 x 10951-0
d(%) fiFN
—2.84 x 10951-5

: . B.14
AFN ( )

 

 

 

Similar to the DC case, but with a plot of [0910 (%€—>5) versus 11;, one can obtain

[31: N from the slope of the line and Ae from the intercept [11]. The Fowler-Nordheim
parameters, 1’3 F N and Ag, only express the dependence of the emitted current on the

field. The physical significance of B F N and A8 is still a matter of debate [4].

171

Appendix C

Photon Interactions

C. 1 Compton Effect

Energy of the scattered photon is found by combining equations 2.44 and 2.46:

 

 

 

 

hz/ \/T(T + 27n0c2)
——- sin 9 = sin (0
c c
, hu' sin 6
sm (,0 = .
\/T(T + 2m0c2)

Squaring each side and solving for cos «,2:

(hr/’)2 sin2 6
cos «p = 1 — .
T(T + 2711062)

 

 

Substituting cos 1p and Equation 2.46 for p into Equation 2.43:

 

hI/O — hr/Icosfl = \/(IlI/0)2 — 2hI/0Iw’ + 2moc2(hI/O — hu’) + (1111’)2 cos (9,

172

squaring both sides and and solving for hu’, the scattered photon energy is obtained

[21

I
I111

Leta: M7:

7710c.

 

 

 

 

_ hz/Omoc2
121/0 + mOc2 -— hI/O cos 6
__ mOc2
_ 2 .
ch _
1 + Ill/0 cos6
, c2
Ill/l 2 m0 1 a (0.1)
1 — cos9 + 5
I
1/ 1
-— (C2)

V0 :1+a(1—cos(})'

173

Appendix D

Definitions

Auger electron — A photon absorbed by an electron can excite the electron to a higher

 

energy level. When an electron from an outer atomic shell fills the vacant inner shell,
the excess energy may be either emitted as a characteristic X-ray or transferred to
another outer electron which is ejected from the atom thus becoming the Auger
electron. [2].

Bremsstrahlung radiation — As a charged particle travels through a medium it loses

 

kinetic energy when it is deflected by other charged paricles. The energy loss is emitted
as electromagnetic radiation and ranges in energy from zero up the the kinetic energy
of the charged particle [2].

Cerenkov radiation - Light emitted when the phase velocity of a charged particle
exceeds c as it travels through an optically transparent material, ,Bn > 1, where B is
the ratio of the particle velocity in the material to c and n is the index of refraction
of the material [19].

Compton scattering - The interaction between a photon and a free electron in which

 

both the photon and electron are scattered [14].
Coulomb interactions — As a charged particle travels through a material it interacts
with atomic electrons and the nucleus of atoms. Interactions with electrons include

ionization and excitation [14].

174

Delbruck scattering - Referred to as elastic nuclear potential scattering when a photon

 

is scattered by a nucleus and an electron pair is formed [2].

Delta ray - A free electron, freed by ionization, that has enough kinetic energy to
create additional ions as it comes to rest [2].

Duane and Hunts Law - The maximum photon energy produced by a charged particle
traveling through a material is equal to the energy of the incident particle [‘2].
Excitation - An atomic electron obtains sufficient energy to move to an unoccupied
state of higher energy [14].

Exciton - Bound state of an electron with an electron hole [14].

Floquet theorem — For a given mode the electric field, separated by one period, differs

 

by a constant factor (eiikzd) [33]:
5(1', :5 + d, t) = EU“, 2, t)eiik3d, (D.1)

where d is the period of the structure.

Geiger - Muller counter - Constructed similar to the ionization chamber but with a

 

higher voltage applied to the electrodes. The number of electrons produced during
ionizations does not depend on the nature of the initial radiation. The discharge
pulses have equal amplitude that are displayed on a meter or audio device. No energy
information is gathered from the incident radiation [19].

Group velocity - Velocity of electromagnetic energy [9].

 

Ionization - An atomic electron obtains sufficient energy to leave the atom [14].
Ionization chamber - Typically a metal chamber (cathode) surrounding a stretched
wire, rod, or disk (anode) and filled with a gas (air, argon, krypton, neon,...). The
electrodes are connected to a power supply. Radiation traveling through the gas cre-
ates ion pairs. As the ion pairs migrate to their respective electrodes additional ion
pairs are created (avalanche). A current is measured and interperted as close.

Pair production - Pair production is an interaction between an incident photon with

 

175

an energy above 1.022 MeV and a 1111010118 within a material. The photon is completely
absorbed and in its place a positron-electron pair appears. The positron is not a
stable particle and travels only a short distance before combining with an electron
or annihilating, creating two photons separated by 180° and each with energy of mg
(0.511 MeV). The electron travels a few millimeters at most before losing energy by
ionization and excitation (Coulomb interactions) or by emission of bremsstrahlung
radiation (Figure 2.19) [2] [19].

Phase velocity - Velocity that any frequency component of a wave propagates [9].

 

Phonon — A vibrational quantum of energy, a form of mechanical energy such as the
way a sound wave propagates through a medium [34].

Photoelectric effect — Total absorption of an incident photon by an atomic electron.

 

The photon disappears and the electron is ejected from the atom, creating an ion [14].

Proportional counter - Constructed similar to the ionization chamber. A low voltage

 

is applied to the electrodes such that the number of electrons produced is due only
to the initial ionizing event. The counter’s output is thus proportional to the number
of ions making it possible to distinguish between different types of radiation [19].

Q disease - Hydrogen, found in bulk Nb, precipates to the RF surface and dissipates
power, preventing an increase in cavity electric fields [4].

Rayleigh scattering - Scattering of photons by tightly bound atomic electrons. Scatter-

 

ing angles are small since the recoil imparted to the atom must not produce excitation
or ionization [2].

Standing wave - A disturbance that is confined to a given space in a medium.

 

SUPERLANS - A computer code that calculates electromagnetic fields within axisym-

 

metrical cavities, periodic structures, and cut—off frequencies in long homogeneous
waveguides [28].

Traveling wave - A disturbance that travels through a medium, transporting energy.

 

Work Function — Energy required to take an electron from the Fermi level within a

 

metal and remove it to infinity, assuming a particle at infinity has zero energy [19].

176

Appendix E

Floquet Theorem

E. 1 Derivation

The SUPERLANS code provides two real functions, the symmetric and antisymmetric
standing waves, that can be combined using Floquet’s theorem to construct another
standing wave with a phase shift over a structure period.

A standing wave is the sum of two traveling waves so the symmetric and antisym-

metric standing waves can be written as:

—¢

(E(r,z,t) + E(r, —z,t)) (E.1)

NIH

ESL/m0“, 3, t) =

—o

Ea.,,,,(r, z, t) = (EU, 2, t) — Ev, —z, 12)), (B.2)

where EU, 2, t) is a wave traveling in the +2 direction and 15(7) —z, t) a wave traveling

in the —z direction. Taking the sum and difference of the two equations yields:

5(7', Z, t) : ESE/771(T’ Z, t) + Ean‘t.i(r, Z, t) aIld (B.3)

177

—4

Eu, —.~, t) = E...,,,i,l,,(r, z. t) — Ea.,,t,-(7', z, t) (13.4)

For example. let Asin(wt — 153:) be a wave traveling in the +2 direction and

Asin(wt + k;:) be a wave traveling in the —z direction. Then

 

 

A
Esym = —2—(sin(u2t — A732) + sin(wt + k;z)) (E.5)
_ _A_ rather—L332) _ e—i(wt—kzz) _+_ ei(wt+kzz) _ e—i(wt+kzz) (E 6)
’ 2 2i '
: :1—4. 'C_7jk:z (e'iwt _ C—‘iwt) + eikzz (eiwt _ e-iwt)] (E7)
2 _
A - . ', -. _._
Z 4_ (e7wt __e—ztut) (elltzz +6 2A;z)] (E8)
'1. -
= Acos(kzz)sin(wt), (E9)

a standing wave. Similarly:

A
Emit,- = E (sin(wt — kzz) — sin(wt + 1:32)) (BID)

2 Acos(wt)sin(—kzz). (E11)

The F loquet theorem states that for a given mode the electric field, separated by

one period, differs by a constant factor (eiikzd) [33]:
EXT, 2 + d, t) = EXT, z, t)eiikzd, (13.12)

where d is the period of the structure (Figure E.1).
The standing wave of the periodic structure is the sum of two such waves traveling

in opposite directions:

(jikzd

E+Zd,;,.(r, z + d, t) = E+Zd,§,.(r, z, t) and

178

 

 

 

 

 

 

 

 

 

 

 

___—J ___——

 

""3?"-
a.

Figure E.1: Periodic structure with period (I.

E—zdir(7’~ z + d, t) = E—zdirm z, t)e—ikzd.

-O

E(r, z + d, t) = E+zdir(r, z + d, t) + E"_Zd,,.(r, z + d, z) (B.13)

" , 7k d " -—'k d
: E+zdi,,.(7,z,t)c? Z +E_Zdz-,.(r,z,t)e 2 Z .

The amplitude of the two traveling waves differ by a phase:

EU", 2 + d, t) 2 [3(1', 2. flew:d + EU, —z, t)e_ikzdew. (E.14)

where the subscripts have been omitted.

Substituting Equations E3 and E.4 into Equation E.14 yields:

—0

EU» 2+(1, t) Z ( fisy171(T‘,Z,t)+E.ay,—m'(r, 3~t))e?kzd+(E-:3ym(rvZ:t)—Ea'nti(rvzit))e_zkzd€up-
(13.15)

179

Recombining terms in Equation E15 yields:

5(7‘, :5 + d, f.) : 26H}. [E's-y‘nfir, :, f) cos(kzd — g) + iE‘muflr, :, 1‘) sin(kzd — §)] .
(13.16)
Continuing with the previous example where Esym = A(cos(kzz)sin(wt)) and

Eanti. = A(cos(wt) sin( —k?; 2)):

—o

E(r, Z + (1* t) : A(cos(kzz) 3111(wt)€’ikzd + Afcos(wt) Sirl(—kzz)eik32d
+ A(COS(kzZ) 5111(wt)e—ikzde’i99 - A(cos(wt) sin(—kzz)e—ikzzdei99

. —i- . i
= A(cos(kzz)sin(wt) (clad—12f + e_7'kzde_§2)
. ‘ —i . , i«
+ A(cos(wt)sin(—kzz) (elkzdeji — e_"]‘zde732)

= 262']; [Esymva 3: t) cosUrzd - g) + zlE—fa'ntifrv z, t) Sin(k3d _ §)] ’

which was found above in Equation E16.

Extending the Floquet theorem (Equation E12) to multiple cells results in:

E(r,z+2d,t) = E(r,z,t)ei2ik3d

EV, z + md, t) 2 EU, z, t)eimik3d.
Equation E.16 becomes:

EU“, 2+md, t) = 267:); [553177101Z,t)COS(‘nlkz(l — g) + iEantifl‘» z, t) sin(mkzd — §)] ,
(E.17)

where E (r, z + md, t) is the electric field along the z axis, m is the cell number, (1 is

180

Ez [MV/m]

 

 

2 [cm]

Figure B.2: Symmetric (diamonds) and antisymmetric (squares) wave output from
SUPERLANS for the half cell geometry and the £167: mode.

half a wavelength of the structure, go is the phase advance per cell, kz is the wave
number in the z direction, kzd = «p, Egymh, z,t) is the symmetric standing wave
output from SUPERLANS, and Ea,,,ti(r,z,t) is the antisymmetric standing wave
output from SUPERLAN S.

The symmetric and antisymmetric wave output from SUPERLANS, for the half
cell geometry and the é67I mode, is shown in Figure B.2. The symmetric and an-
tisymmetric wave output from SUPERLANS, for the six-cell structure is shown in
Figure E.3 panels (a), (b), and (c). The corresponding electric field on axis for the

full six-cell structure is shown in Figure E.3 panels ((1), (e), and (f).

181

15 - a) d)

I l

_‘L A
omomo
Lg; 1i
(F
_. .
m1 I:
IN)
(i
o.) I"
ma >
'i
g.
L;
I I
__8—3 _b—b
UNDU‘IDU‘IDU‘I
J l l l

 

 

 

 

51 A b) A 5‘ e)

 

 

 

 

 

 

 

 

-15 I r 15 1 .
-5 15 35 5 15 35
15 - C) 15 - f)
10 - 10 ~
5 4 f\ A /\ 5+
0 J ~.’ S +3 .— "I: ‘3 ’r‘ J," h if "£3. 0 "
-5 j *Lru V ”__ _5 d
-10 - ~10 -
-15 r 1 -15 I .
-5 15 35 -5 15 35
2 [cm]

Figure B.3: Panels (a), (b), and (c): The dark symbols represent the symmetric stand-
ing wave output from SUPERLAN S and the light symbols represent the antisymmet-
ric standing wave output from SUPERLAN S. Panels (d), (e), and (f): Electric fields on
axis. Symmetric and antisymmetric standing waves were combined using the Floquet
theorem. Modes observed in this study were 7r, gal, and 4375.

182

Appendix F

Data and Endpoint Energy

F.1 Cavity #2 7r Mode

Two scans were made of each cavity, and each mode, to show that the results were
reproducible. The x—ray energy spectra obtained from cavity # 2, 7r mode, are pre-
sented in Figure F1 Panels a) and b) show the x—ray count with the background
subtracted. Panels c) and (1) show the data with energy binning applied. The high in-
tensity peaks are numbered and questionable peaks labeled alphabetically. Table F.1
indicates the centroid and endpoint energy, obtained in ORIGIN, of the high intesity
x-ray peaks identified in the corresponding figures.

Table F .1: Cavity #2, 7r mode, high intensity peak location and endpoint energy.
Numbering corresponds to Figure F.1.

 

 

 

Cavity # Mode Scan # Peak ID Centroid [cm] Endpoint Energy [keV]
2 7r 1 A 12.4 :1: 0.1 540 :1: 5
” ” 1 1 28.9 i 0.1 928 :t 1
” ” 1 2 37.9 i 0.1 774 :1: 1
” ” 1 3 45.8 :f: 0.1 965 :1: 1
” ” 1 4 79.5 i 0.1 1066 :f: 1
2 7r 2 B 13.4 :1: 0.1 577 :1: 2
” ” 2 5 28.8 :1: 0.1 900 :t 2
” ” 2 6 37.5 i 0.1 733 :1: 2
” ” 2 7 45.6 i 0.1 878 i 1
” ” 2 8 78.6 :f: 0.1 1073 i 1

 

 

 

 

 

 

183

 

1000 —« (a) 1000 _, (b)
4 .

 

 

 

 

 

 

C)
85
I 8
E
:3
8 500 ~
>5
9
>'<
C)
x
m
E
8 500 ~
O
>
e
>'<
o 1

 

 

   

0 20 40 60 80
Distance [cm] Distance [cm]

Figure F1: Cavity #2 7r mode. Panels (a) and (b): Data accumulated for two scans
of the cryostat with the background subtracted. Panels (0) and (d): Energy binning
of data in panels (a) and (b). The black plot represents the number of x—rays at
all energies, the light-gray plot represents the number of x—rays with energies of 400
keV and above, and the medium—gray plot represents the number of x—rays with
energies of 600 keV and above. A moving average was applied to the data to smooth
inconsistencies in adjacent two-second intervals.

F.2 Cavity #2 £361 Mode

Two scans of cavity #2 were made in the 567: mode. The x—ray energy spectra obtained
from cavity # 2, Q5: mode, are presented in Figure F2 Panels a) and'b) show the
x-ray count with the background subtracted. Panels c) and d) show the data with
energy binning applied. The high intensity peaks are numbered and questionable
peaks labeled alphabetically. Table F.2 indicates the centroid and endpoint energy,
obtained in ORIGIN, of the high intesity x-ray peaks identified in the corresponding

figures.

184

2000 : (a)9
1500 : 10

1000 ~ 12

  
 
  
  
 
 
 

 

0'1

0

O
1
O
p—t
M

X-ray Count - Bkg

 

 

 

O
P
_I-

o 20 4o 60 80
1600 a. (c)
1400-] 10] 11
10

Bkg

1200 a
1000 ]
300 .
600 5
400 J
200 «

 

 

X-ray Count

  

Distance [cm] Distance [cm]

Figure F2: Cavity #2 §67£ mode. Panels (a) and (b): Data accumulated for two scans
of the cryostat with the background subtracted. Panels (c) and ((1): Energy binning
of data in panels (a) and (b). The black plot represents the number of x-rays at
all energies, the light-gray plot represents the number of x—rays with energies of 400
keV and above, and the medium-gray plot represents the number of x-rays with
energies of 600 keV and above. A moving average was applied to the data to smooth
inconsistencies in adjacent two—second intervals.

185

Table F.2z Cavity #2, 5675 mode, high intensity peak location and endpoint energy.

Numbering corresponds to Figure F2

 

 

 

 

 

 

 

 

Cavity # Mode Scan # Peak ID Centroid [cm] Endpoint Energy [keV]
2 8;: 1 9 12.9 :1 0.1 756 3: 1
1 10 22.0 i 0.1 857 d: 1
” ” 1 11 28.6 a: 0.1 979 i 1
’1 1 12 37.0 i 0.1 957 :1: 1
” ” 1 C 45.6 :1: 0.1 723 3: 3
” 1 D 53.7 :1: 0.1 728 j: 2
” 1 13 63.0 :t 0.1 902 :t 2
2 57;; 2 14 12.3 :i: 0.1 755 :i: 1
” ” 2 15 22.8 :1: 0.1 800 i 1

2 16 29.4 :1: 0.1 923 a: 1
” ” 2 17 37.4 i 0.1 874 3: 1
” ” 2 E 45.4 :i: 0.1 790 :1: 1
” 2 F 53.3 :1: 0.1 752 :i: 2
” ” 2 18 61.9 a: 0.1 940 :t 1

 

 

- 47r
F.3 Cav1ty #2 -6— Mode

Two scans of cavity #2 were made in the 467i mode. The x-ray energy spectra obtained
from cavity # 2, $67: mode, are presented in Figure F3. Panels (a) and (b) show the
x-ray count with the background subtracted. Panels c) and (1) show the data with
energy binning applied. The high intensity peaks are numbered and questionable
peaks labeled alphabetically. Table F .3 indicates the centroid and endpoint energy,
obtained in ORIGIN, of the high intesity x-ray peaks identified in the corresponding

figures.

186

400

 

 

 

 

 

 

 

 

 

 

300 “ (a) " (b)
C” 300 «
x 20 23
:5 2122 27
g 100 . G 100 . .
a 0 0*
,2 -100 4
400 . . . 1 -200 . .
o 20 4o 60 30 o 20 4o 60 80
200 . (c) 20 200-
150
27
a) 1 _4
5 100 - 0°
‘5 50
8 o M.
13 0] ‘
>'<
-50 -100 - e-eeee— .- ——
o 20 4o 60 80
Distance [cm] Distance [cm]

Figure F .3: Cavity #2 i4675 mode. Panels (a) and (b) Data accumulated for two scans
of the cryostat with the background subtracted. Panels (c) and ((1): Energy binning
of data in panels (a) and (b). The black plot represents the number of x—rays at
all energies, the light-gray plot represents the number of x-rays with energies of 400
keV and above, and the medium-gray plot represents the number of x—rays with
energies of 600 keV and above. A moving average was applied to the data to smooth
inconsistencies in adjacent two—second intervals.

187

Table F.3: Cavity #2, £1675 mode, high intensity peak location and endpoint energy.

Numbering corresponds to Figure F.3.

 

 

 

 

 

 

 

 

 

 

Cavity # Mode Scan # Peak ID Centroid [cm] Endpoint Energy [keV]
2 {7i 1 19 18.6 i: 0.1 678 i 2
” ” 1 20 26.3 i 0.1 724 :l: 2
’ ” 1 21 36.2 i 0.2 814 :f: 5
” ” 1 22 43.3 :1: 0.1 813 :1: 2
” ” 1 G
2 461 2 23 21.5 i 0.1 745 i 1
” ” 2 24 28.5 :1: 0.2 670 i 3
” ” 2 25 37.1 :1: 0.2 750 :f: 4
’ ” 2 26 45.0 :1: 0.1 771 :1: 2
” ” 2 27 70.1 :1: 0.2 669 :t 2

 

 

 

F.4 Cavity #1 7r Mode

The x-ray energy spectra obtained from cavity #1, 7r mode, are presented in F ig-
ure F.4. Panels (8) and (b) Show the x-ray count with the background subtracted.
Panels (c) and (d) show the data with energy binning applied. The high intensity
peaks are numbered and questionable peaks labeled alphabetically. Table F .4 indi-
cates the centroid and endpoint energy, obtained in ORIGIN, of the high intesity

x-ray peaks identified in the corresponding figures.

188

500 (a) 5 500 (b) 12
g 400 a 400 .
game 1234 300-« 910 ll 14
O
0 200 6 200 8 13
5‘ . A 7 B
>'< 100 — ~ ' , 100i

 

 

 

 

X-ray Count - Bkg

 

 

 

 

Distance [cm] Distance [cm]

Figure F4: Cavity #1 77 mode. Panels (a) and (b): Data accumulated for two scans
of the cryostat with the background subtracted. Panels (0) and ((1): Energy binning
of data in panels (a) and (b). The black plot represents the number of x—rays at
all energies, the light-gray plot represents the number of x-rays with energies of 400
keV and above, and the medium-gray plot represents the number of x-rays with
energies of 600 keV and above. A moving average was applied to the data to smooth
inconsistencies in adjacent two—second intervals.

F.5 Cavity #1 5’61 Mode

The x—ray energy spectra obtained from cavity #1, 53675 mode, are presented in F ig-
ure F .5. Panels (a) and (b) show the x-ray count with the background subtracted.
Panels (c) and ((1) show the data with energy binning applied. The high intensity
peaks are numbered and questionable peaks labeled alphabetically. Table F.5 indi-
cates the centroid and endpoint energy, obtained in ORIGIN, of the high intesity

x-ray peaks identified in the corresponding figures.

189

Table F .4: Cavity #1, 7r mode, high intensity peak location and endpoint energy.

Numbering corresponds to Figure F4.

 

 

 

 

 

 

 

 

Cavity # Mode Scan # Peak ID Centroid [cm] Endpoint Energy [keV]
1 7r 1 1 14.5 :1: 0.1 972 :1: 1
” ” 1 2 22.9 :1: 0.1 827 :1: 2
” ” 1 3 30.8 :1: 0.1 957 :1: l
” ” 1 4 40.0 :1: 0.1 797 :1: 3
” ” 1 5 47.9 :1: 0.1 972 :1: 1
” ” 1 6 56.5 :1: 0.1 821 i 3
” ” 1 A 64.5 :1: 0.1 737 i 1
” ” 1 7 70.9 :1: 0.1 806 i 4
1 7r 2 8 14.4:1: 0.3 913 :1: 1
” ” 2 9 23.5 :1: 0.1 846 i 1
” ” 2 10 31.5 :1: 0.1 926 i 2
” ” 2 11 39.9 :1: 0.1 768 :1: 2
” ” 2 12 47.8 i 0.1 1065 :1: 1
” ” 2 13 56.1 :t 0.1 845 :1: 2
” ” 2 B 61.9 :1: 0.1 776 :1: 1
” ” 2 14 72.3 i 0.1 878 i 1

 

Table F .5: Cavity #1, '55: mode, high intensity peak location and endpoint energy.
Numbering corresponds to Figure F.5.

 

 

 

 

 

 

 

 

Cavity # Mode Scan # Peak ID Centroid [cm] Endpoint Energy [keV]
1 1567! 1 15 15.1 _+_ 0.1 556 :t 10
” ” 1 16 22.0 :1: 0.2 733 a: 2
” ” 1 17 29.5 :t 0.1 652 :t 3
” ” 1 18 33.5 i: 0.1 736 :1: 2
” ” 1 19 43.9 :1: 0.1 554i 6
” ” 1 20 52.2 :1: 0.1 605 a; 2
” ” 1 21 61.1 a: 0.1 859 :1: 1
” ” 1 22 68.8 :1: 0.1 937 i 2
1 ‘56; 2 23 15.0 i 0.1 573 :1: 9
” ” 2 24 21.5 :1: 0.1 697 i 3
” ” 2 25 31.1 :t 0.1 678 :1: 5
” ” 2 26 33.6 i 0.2 961 i 4
” ” 2 27 47.1 a: 0.1 532 :1: 30
” ” 2 28 50.9 i 0.1 749 i 4
” 2 29 61.0 a: 0.1 876 i 2
” ” 2 30 68.8 d: 0.1 871 :1: 1

 

190

 

 

 

21

 

 

 

 

 

 

 

 

200 1 (a) 200 1 (b)

29
3’
m
*3 15 6 8 20
g 100 : 17 1 22 100 ‘ 23 24 2526 28 3O
8 7
>.
9
x O T I 1 O 7 1 1 1

0 20 40 60 80 0 20 40 60 80
200 I (C) (d)
21 150 :
a? 29
CD
E
E); 5 6 28 30
E .3", 27 fl
X fir1 4": ,. ’1 r ‘.
15"N U” thK/‘J’ ' \ W1 N;
0 20 40 60 80 0 20 40 60 80

Distance [cm]

Distance [cm]

Figure F .5: Cavity #1 §67£ mode. Panels (a) and (b): Data accumulated for two scans

of the cryostat with the background subtracted. Panels (0) and (d): Energy binning
of data in panels (a) and (b). The black plot represents the number of x-rays at

all energies, the light-gray plot represents the number of x-rays with energies of 400

keV and above, and the medium-gray plot represents the number of x—rays with
energies of 600 keV and above. A moving average was applied to the data to smooth

inconsistencies in adjacent two—second intervals.

191

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111]

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194

   

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