Study of High Voltage Charge Pumping with a Resonant Piezoelectric Device a Dissertation Presented to the Faculty of the Graduat

Study of high voltage charge pumping with a resonant piezoelectric device

A Dissertation presented to

the Faculty of the Graduate School at the University of Missouri

In Partial Fulﬁllment

of the Requirements for the Degree of Doctor of Philosophy

Brady B. Gall

Dr. Scott D. Kovaleski, Thesis Supervisor

May 2016 The undersigned, appointed by the Dean of the Graduate School, have examined the thesis entitled:

Study of high voltage charge pumping with a resonant piezoelectric device

presented by Brady B. Gall a candidate for the degree of Doctor of Philosophy and hereby certify that in their opinion it is worthy of acceptance.

Dr. Scott D. Kovaleski

Dr. Jae Wan Kwon

Dr. John M. Gahl

Dr. Robert A. Winholtz Acknowledgements

First, I would like to thank Dr. Kovaleski for your academic and professional guidance throughout my graduate education. Thank you for providing me with the time, energy, and resources necessary to cultivate my scientiﬁc curiosity. Your instruction has shaped my growth as a student and as a researcher. Thank you to my fellow students and colleagues for your camaraderie and advice. Thank you Brian Hutsel for your continuing friendship and encouragement. Thank you James VanGordon and Peter Norgard for always pushing me to do my best. Thanks to Abraham Dirnberger, Emily Baxter, and Erik Becker for your friendship both in lab and the real world.

Thank you to my mother, Jan Gall, for raising me with the values and work ethic necessary to succeed in both science and in life. You have loved me and believed in me since I was born and I could not be more grateful of your countless sacriﬁces. Finally, thank you to my beautiful wife and best friend, Jenny Gall. You are unwavering in your love and support for me, especially throughout the many challenges we have faced in preparing this dissertation. You are and always will be the joy of my life.

ii Contents

Acknowledgements ii

List of Tables vi

List of Figures vii

1 Introduction 1 1.1 Background ...... 1

1.2 Approach ...... 3 1.3 Thesis Overview ...... 6

2 Experimental Setup 7 2.1 PT Systems Overview ...... 7 2.2 Laser optical diagnostic ...... 10

2.3 Electron Gun Design & Calibration ...... 15 2.3.1 Filament ...... 17 2.3.2 Control Grid ...... 20

2.3.3 Optics ...... 22 2.4 Wire-loaded HVPT Conﬁgurations ...... 28

3 Theory & Analysis 32 3.1 Electrostatic Optics ...... 32

iii 3.1.1 Design ...... 32

3.1.2 Additional Comsol Modeling ...... 36 3.1.3 Beam Spot Imaging ...... 40 3.2 Laser diagnostic ...... 41

4 Experimental Results 45 4.1 HVPT with Electron Beam Optics ...... 45 4.1.1 X-ray Measurements ...... 45

4.1.2 Laser and X-ray Diagnostic Comparison ...... 48 4.1.3 Mode 1 vs. Mode 2 ...... 50 4.1.4 Current Response ...... 54

4.1.5 Long term operation ...... 55 4.1.6 Aggregate trends ...... 56 4.2 Wire-Loaded HVPT ...... 64

5 Conclusions & Future Work 73 5.1 Conclusions ...... 73

5.2 Future Work ...... 79

Appendix A Electron Source Calibration Data 82

Appendix B Vacuum Tube Theory 87 B.1 Richardson’s Law ...... 87 B.2 Temperature Determination ...... 88

B.3 Child’s Law ...... 89 B.4 Space Charge Eﬀects ...... 92

Appendix C Laser Diagnostic Data Processing Method 97 C.1 Data acquisition ...... 97

iv C.2 Data processing ...... 99 C.3 Data analysis ...... 103

Appendix D PTPS Neutron Generator Work 109 D.1 PTPS ...... 109

D.2 Coupled HVPT-PTPS Neutron Conﬁguration ...... 110 D.3 Diagnostics ...... 112 D.3.1 Neutron Counting Algorithm ...... 114

D.4 Modeling ...... 118 D.4.1 Excitation Function ...... 119

D.4.2 Trim Ion Beam Modeling ...... 120 D.4.3 Time-harmonic Analysis ...... 122 D.4.4 Calculated Neutron Yield ...... 124 D.5 Results ...... 126

Appendix E Laser Diagnostic Periscope Analysis 128

Bibliography 132

Vita 137

v List of Tables

Table page

3.1 Design parameters for electron beam optic systems ...... 36

3.2 Parameter descriptions and values for the laser optical diagnostic equations. Material constants assume a 45-degree rotated lithium niobate slab...... 44

B.1 Work functions and Richardson constants for common thermionic

emitter materials...... 88

D.1 Constants for calculating yield of piezoelectric neutron sources. . . . . 124

D.2 Excitation function integral solutions for the steady-state (DC) and time-variable (AC) conditions...... 125

vi List of Figures

Figure page

2.1 The High Voltage Piezoelectric Transformer (HVPT) with electrical

and mechanical profiles for mode 1 and mode 2 operation...... 8 2.2 3D graphic of the HVPT with mounting hardware for mode 2 operation. 10 2.3 General configuration of laser diagnostic for probing internal fields of

resonant PT. Top diagram is a view coplanar to the x0−z0 axes. Bottom diagram is a view coplanar to the y0 − z0 axes...... 10 2.4 Comparison between an optically clear polished HVPT (1) and a non

polished HVPT (2)...... 11 2.5 Laser diagnostic in internal reﬂection conﬁguration ...... 12 2.6 Experimental setup to measure PT operation with x-ray and optical

diagnostics. View emphasizes the x-ray diagnostic by orienting coplanar to the y0 − z0 axis of the PT coordinate system. Inset shows detail view for the electron beam, PT, and detector conﬁguration. . . 14

2.7 Photograph of the test stand for simultaneous x-ray and optical diagnostics for the HVPT inside the vacuum chamber...... 14 2.8 Diagram of laser-detector assembly for alignment of optical diagnostic. 15

2.9 Diagram of the electron gun...... 16 2.10 Electron gun current calibration circuit...... 17

vii 2.11 Derivation of beam current calibration equation using source transformation...... 18

2.12 Measured anode current response for varied ﬁlament currents. . . . . 19 2.13 Measured electron beam current vs. anode voltage for three grid voltages and three ﬁlament currents...... 21

2.14 Measured electron beam current vs. grid voltage with fixed anode voltage (6 kV) and three filament currents...... 22 2.15 Faraday collector configuration and diagnostic circuit for electron beam

optics output current measurement...... 23 2.16 Diagnostic circuit for measuring supplied thermionic emission current. 24 2.17 Electron gun (with focusing optics) calibration results...... 25

2.18 Current calibration results for a variety of configurations for the electron gun...... 27 2.19 Electron gun in x-ray reflection mode configuration...... 29

2.20 Electron gun in x-ray transmission mode conﬁguration...... 29 2.21 Circuit diagram for HVPT solid state charge pump (diodes conﬁgured for negative dc output)...... 30

3.1 Design for electron beam cylindrical optics...... 32

3.2 P−Q curves for a two-cylinder lens with equal diameter cylinders and

a gap of 0.1 D. Example shows method to determine V2/V1 for a magniﬁcation of 0.5...... 34

3.3 Beam spot intensity as a function of V2/V1 for optics with magniﬁca- tion of 0.5, 0.33, and 0.2...... 35 3.4 Comparison of beam focusing and intensity of the M=0.2 optic with

three diﬀerent voltage ratios...... 36

viii 3.5 Computed electron beam intensity and cross-section at the object (pupil) and image plane for each magniﬁcation power...... 37

3.6 Charged particle tracing for the M=0.2 optic coupled to the HVPT target for a range of HVPT output voltages...... 38 3.7 Electron beam intensity incident on HVPT output for varied HVPT

voltage and V2/V1 ratios...... 39 3.8 Photograph of electron beam spot on P22 screen. Pupil (object) size was 5 mm and spot (image) size was 2 mm, resulting in a measured

magniﬁcation of 0.4...... 40 3.9 Left axis: waveform example of the optical response of a resonant PT measured with a photodetector (PD). Right axis: waveform of applied

voltage signal to drive PT into resonance...... 42

4.1 Time-resolved x-ray count rate for three applied beam currents. . . . 46 4.2 Average x-ray count rate for varied electron beam currents...... 47 4.3 Time-resolved maximum x-ray energy for three electron beam currents. 47

4.4 Average maximum endpoint x-ray energy for varied electron beam currents...... 48 4.5 X-ray spectrum recorded simultaneously with laser optical measure-

ment from figure 4.6...... 49 4.6 Voltage profile of PT collected using the optical diagnostic simultaneously with x-ray data from figure 4.5...... 50

4.7 Measured voltage, electric ﬁeld, and stress proﬁle measurement comparison between PT operating at mode 1 and mode 2. Distance is measured from the center of the HVPT (0 mm) to the output extremity

(50 mm.) ...... 51

ix 4.8 Modeled voltage, electric ﬁeld, and stress proﬁle as determined using

Comsol ﬁnite element modeling for a mode 1 and mode 2 HVPT. The input, center, and output of the of the HVPT to -50 mm, 0 mm, and 50 mm, respectively...... 53 4.9 Box and whisker plot showing maximum voltage measured for a PT

operated at 7−8Vrms input for a range of applied beam current. . . . 55 4.10 Time-resolved voltage measurements for PTs operating for 1-hour. . . 56 4.11 Aggregated optical data of electrical and mechanical output for varied

input voltage and HVPT sample...... 57 4.12 Output voltage vs. input current...... 58 4.13 Output voltage vs. input impedance...... 59

4.14 Output voltage vs. input power...... 60 4.15 Eﬃciency vs. input power...... 61 4.16 Output voltage vs. maximum internal stress...... 63

4.17 X-ray Spectra for non-wired HVPT...... 66 4.18 X-ray spectra produced by transmission mode x-ray source energized with Emco DC high voltage supply...... 68

4.19 Time-resolved count rate for three applied beam currents for the DC transmission x-ray source...... 69 4.20 Charge pump voltage...... 70

4.21 Charge pump power...... 70 4.22 Charge pump eﬃciency...... 71 4.23 Charge pump capacitance...... 72

5.1 Proposed setup for improved laser diagnostic test bed...... 81

A.1 Filament=2.05 A, Optics=5 kV...... 82

x A.2 Filament=2.05 A, Optics=8 kV...... 83 A.3 Filament=2.05 A, Optics=10 kV...... 83

A.4 Filament=2.05 A, Optics=12 kV...... 83 A.5 Filament=2.25 A, Optics=5 kV...... 84 A.6 Filament=2.25 A, Optics=8 kV...... 84

A.7 Filament=2.25 A, Optics=10 kV...... 84 A.8 Filament=2.25 A, Optics=12 kV...... 85 A.9 Filament=2.35 A, Optics=5 kV...... 85

A.10 Filament=2.35 A, Optics=8 kV...... 85 A.11 Filament=2.35 A, Optics=10 kV...... 86 A.12 Filament=2.35 A, Optics=12 kV...... 86

B.1 Temperature vs applied current for a 140µm-diameter tungsten ﬁlament. 90

B.2 Left: thermionic current produced by a 140 µm-diameter tungsten ﬁlament over a range of applied currents. Right: Space charge limited current (blue) for a 60 mm-long vacuum tube. Horizontal lines (red)

show the thermionic limited current plateaus for a range of applied ﬁlament currents...... 91 B.3 Voltage potential as a function of position throughout an anode

cathode region when charge is present in the system...... 93 B.4 Position and voltage magnitude of the virtual cathode as ﬁlament current is increased...... 94

B.5 Space charge limited current as determined with Child’s law and the modiﬁed virtual cathode due to increased ﬁlament current...... 95

C.1 Process for optical data acquisition...... 98 C.2 Comparison of 4 lowess spans for data smoothing...... 101

xi C.3 Process for optical data acquisition...... 102 C.4 Functional block diagram for optical data processing method. . . . . 104

C.5 Process for optical data analysis...... 105 C.6 Example of measured data showing the change in phase transitions for an HVPT using the optical diagnostic and the corresponding electrical,

and mechanical responses...... 107

D.1 The Piezoelectric Transformer Plasma Source (PTPS)...... 110 D.2 Coupled HVPT-PTPS setup for neutron production...... 111 D.3 Systems diagram of the coupled HVPT-PTPS and support hardware. 112

D.4 Photograph of one RDT solid state neutron detector...... 113 D.5 Detection eﬃciency over a range of positions within an 8 lb stack of paraﬃn moderator...... 113

D.6 Photograph of the RDT detector array situated in the optimal position within the paraﬃn stack...... 114 D.7 Natural variation in neutron background count rate over one hour. . . 115

D.8 Distribution of measured background compared to the Poisson distribution (µ = 75 counts/10 min)...... 117 D.9 Excitation function for the 2H(d, n)3He reaction...... 120

D.10 Ion transmission ratio into a TiD2 target...... 121

D.11 Ion beam stopping power of a TiD2 target...... 122 D.12 Calculated yield for the HVPT-driven neutron source compared to a

DC neutron source...... 125 D.13 Box and whisker plot for 19, 1-hour-long experiments with HVPT- PTPS neutron source...... 126

D.14 X-ray spectra comparison of the HVPT-PTPS with and without target. 127

xii E.1 Laser diagnostic in periscope conﬁguration...... 129 E.2 Comparison of phase transitions measured using the periscope and

internal reﬂection beam paths...... 130

xiii Chapter 1

Introduction

1.1 Background

In 2013, the global demand for piezoelectric devices was valued at $18.65 billion, a ﬁgure which has experienced consistent growth in the last twenty years [1].

The application space for piezoelectric devices is vast, including industrial and manufacturing, automotive, medical instrumentation, and telecommunication. The piezoelectric transformer (PT) is a type of piezoelectric device which uses acoustic coupling to increase or decrease the amplitude of an applied voltage waveform, similar to a conventional ferromagnetic transformer [2]. However, in certain applications, the PT has many advantages over its magnetic counterpart [3]. Perhaps most striking of these advantages is its size. The PT is a compact, lightweight, and low power device and can weight as little as one-quarter of an ounce (7 grams) and requires only about 1 watt of input power. Due to these advantageous characteristics, the PT category makes up a substantial portion of the broader market of piezoelectric devices, with 25-30 million PTs sold annually and a valuation at the billion dollar level [4]. When the piezoelectric transformer was invented by Charles Rosen in 1954, the envisioned application space included band-pass ﬁlters and high voltage supplies. [5, 6]. This has remained fundamentally unchanged, however the scope of the device

1 has broadened substantially. In the 1970’s, development of the PT as an energy converter sparked the ﬁrst serious consideration of the device as a possible replacement for magnetic transformers. During this time, several manufacturers investigated the PT to generate high voltage for black and white CRT televisions [7, 8]. However, issues with hardware stability combined with an overall lack of commercial interest prevented the necessary development to launch the PT into mainstream production. In the 1980’s, with the LCD screen on the horizon in consumer technology, a demand arose for a compact, low power, high voltage supply to drive the display’s ﬂuorescent lamp backlight [9]. This demand catered precisely to the PT’s strengths and the device saw dramatic innovation in material science, manufacturing processes, and housing enclosures [4]. With these improvements, the PT became a robust platform for supplying high voltage, leading to its current market position as a integral component to many commercial and industrial systems. Clearly, the small size and light weight of the PT has been its key selling points, motivating development its for conventional uses. Alternatively, a broad range of more exotic applications have a need for compact high voltage solutions for which the PT is a compelling candidate. As a result, the PT as a compact high voltage supply enjoys a dual role as both mature commercial product and a thriving subject for research and development. Some researchers are pushing the PT to its extremes, using it to achieve voltages between 5 and 120 kV, which is largely uncharted territory [10–

14]. At these high voltages, the compactness of the PT enables innovation in ﬁelds otherwise untouched by piezoelectric transformers. Of particular interest is the use of the PT as a direct accelerator for charged particles, thus eliminating the need for bulky step-up transformers or voltage multiplier circuits. Throughout the last decade, researchers at the University of Missouri have been investigating the governing physics of lithium niobate PTs and characterizing

2 their performance as a charged particle accelerator. Lithium niobate is a favorable material choice for this application for many reasons, including high piezoelectric coupling coeﬃcients, high yield strength, a high curie temperature, and a high Q factor [15, 16]. By leveraging its small size and high voltage capabilities, the PT has been implemented as a compact plasma source and an x-ray generator. This has motivated the investigation of the viability of PTs in a wide range of application spaces which are in need of compact particle accelerator solutions, including micro- spacecraft propulsion, compact neutron sources for active interrogation and oil well logging, and hand-held x-ray sources for medical, industrial, and military radiography systems [17–21].

1.2 Approach

While the advantages of the PT as a particle accelerator are promising, measuring the device while operating in the high voltage regime has several non-obvious challenges which have motivated the development of unconventional diagnostic techniques. The PT operates at high AC voltage, so it is sensitive to the nature of its surroundings. Nearby grounded metal surfaces will act as capacitive loads, shunting the AC current of the PT. The device must also be operated in vacuum because discharges in air act as resistive loads which also reduce its voltage. Furthermore, the device is a mechanical resonator and any physical object which impedes its ability to freely oscillate will inhibit its performance. Because of its sensitivity to these loss mechanisms, conventional diagnostics such as resistive voltage dividers are incompatible with the PT [22]. Therefore, specialized diagnostic techniques have been developed to characterize its behavior.

The electron beam diode is one approach to diagnose the PT with minimal inﬂuence on its performance. Since the electron beam has extremely high resistance

3 and low capacitance, it mitigates the loss mechanisms associated with conventional voltage diagnostics. Since the PT is operated in vacuum, it is easily adapted to an electron diode configuration. One implementation of this diode is to adhere sub- micron scale field emission tips directly to the PT output. When operating at high voltage, field enhancement at the tips extracts electrons and accelerates them to a grounded surface, producing bremsstrahlung x-ray spectra [12, 13, 23]. These spectra can then be used to indirectly infer the accelerating voltage of the PT. An alternative approach to incorporate an electron diode is the hot cathode. In this approach, a tungsten filament is placed in the vicinity of the PT output and thermionic electrons emitted from the filament are accelerated from ground to the PT [24]. Both acceleration modalities (PT-to-ground and ground-to-PT) are possible due to the bipolar AC output of the PT. In practice, the thermionic mode has proven to be more reliable and convenient for characterizing the PT and has largely replaced the field emission mode.

However, while the the electron beam diode has been used to great effect to understand the PT and the underlying physics responsible for its behavior, the diagnostic is still not truly ‘invisible’ to the transformer. For example, the diode either extracts electrons (field enhancement mode) or deposits electrons (thermionic mode) at the transformer output. Due to the PT’s very small output capacitance ( 10−13 F [11]), a small change in net charge can dramatically influence the voltage of the transformer, effectively negating the piezoelectrically induced high output voltage. Additional charging at the surface also alters the electrical boundary conditions of the PT and introduces another source of variability. Furthermore, since the PT also responds to the pyroelectric effect, local heating due to electron beam impingement on the surface adds to the complexity of accurately diagnosing the PT [25]. Clearly, while the electron beam diode is a substantial improvement over the conventional

4 voltage divider as a diagnostic tool, it does not completely isolate the response of an unperturbed operating high voltage PT.

To address this problem, a new diagnostic method was invented by Norgard and Kovaleski [26] to probe the PT without influencing its behavior. The method uses a laser to interrogate the internal electric fields of the lithium niobate PT. By analyzing the shift in polarization of the out-going laser and using the known electro-optic properties of lithium niobate, the internal electric fields, and thus, the voltage, of the device could be inferred. This method was further developed by VanGordon [11] to account for mechanical stress fields by incorporating the photo-elastic effect. This allowed for the rapid characterization of the PT under a range of operating conditions such as vacuum pressure and applied thermionic filament current. The piezoelectric effect is blind to the laser diagnostic since the probing beam is charge neutral, composed of photons rather than electrons. Also, by using a low power laser and a polished, optically clear finish on the PT, thermal effects caused by the laser are negligible. Thus, the laser diagnostic is the ideal technique for investigating the PT without influencing its behavior. The primary applications of interest for this work involve using the PT as a direct accelerator for charged particles. Therefore, the underlying motivation is to simultaneously develop favorable accelerator geometries and accurately diagnose the PT while in operation. Several experiments are presented using a test bed which combines two diagnostics: the laser optical diagnostic and an x-ray diagnostic. The laser diagnostic builds on the work of Norgard and VanGordon, while the x- ray diagnostic is an extension of work originally implemented by Kovaleski and

Benwell [12, 27]. By combining accurate diagnostic capability with the high energy particle accelerator, this test bed could represent a foundational design for a commercial prototype compact x-ray source.

5 1.3 Thesis Overview

Chapter 2: Experimental Setup The hardware and physical layouts for the various experiments are presented. The chapter begins with a brief introduction of piezoelectric systems. It continues with a diagram and discussion of the test bed for the laser-optical diagnostic. An electron gun is then presented, including its design and calibration results. The chapter concludes with a set of two experiments which investigate the eﬀect of adding a wire to the output of the high voltage PT.

Chapter 3: Theory & Analysis This chapter is divided into two sections. The

ﬁrst section details the theory and modeling-driven design of electrostatic optics for focusing the electron gun output into a small spot size. The second section describes the necessary mathematics and analytical technique to convert the measured signal from the laser optical diagnostic into real values of voltage, electric ﬁeld, and mechanical stress.

Chapter 4: Experimental Results This chapter begins with a detailed analysis of the laser-optical results, including a comparison to the x-ray diagnostic, a resonant mode study, the response to applied beam current, temporal eﬀects, and aggregate trends over the entire sample range. The chapter concludes with the results of the wire-loaded tests, which demonstrated that the addition of a wire substantially reduced the output voltage of the transformer.

Chapter 5: Conclusions & Future Work The thesis ends with a summary of the work completed and discussion of the results. Future work is suggested which explores possible experiments to build on research presented in this document.

6 Chapter 2

Experimental Setup

In this chapter, experiments are presented which explore applications and diagnostics for high voltage piezoelectric transformers. First, a brief description of PT technology is presented. Next, a laser optical diagnostic method is described which provides a zero-loss method of interrogating the internal electrical and stress ﬁelds of an operating high voltage piezoelectric transformer. The chapter continues with the design and calibration of an electron gun which can provide a controlled amount of current to a small location at the high voltage terminal of the PT. Finally, an alternative test bed is presented which uses a wire to couple the high voltage output of the PT to an external anode and charge pump circuit.

2.1 PT Systems Overview

The primary PT system used throughout this work was the high voltage piezoelectric transformer (HVPT), which was made from congruent slabs of lithium

◦ niobate (LiNbO3) with a crystallographic rotation of β = −45 (Euler notation) [18, 28]. Lithium niobate is a common piezoelectric material which has many favorable characteristics such as electro- and acousto-optic properties, high curie temperature, high yield strength, and strong piezoelectric coupling coeﬃcients [15, 16, 29]. A second geometry, the piezoelectric transformer plasma source (PTPS) was used for a

7 neutron generator experiment, and while that work is relevant to the thrust of this document, it is largely tangential and is therefore included in Appendix D.

Previous work has demonstrated that the HVPT can reach output voltages greater than 100 kV with as little as 10 V input drive voltage [13, 28]. Due to these results, the HVPT is investigated as an accelerator component for x-ray and neutron generator

systems. The design of the HVPT is shown in Figure 2.1. The transformer was made from a 100 × 10 × 1.5 mm lithium niobate bar. Input electrodes, shown as gray regions on the top and bottom (not visible) surfaces on the left portion of the bar were used to deliver electrical power to the device. The electrodes were applied manually using silver paint with a measured layer thickness of approximately 50 µm. For convenience, the x, y, and z axis orientations shown in ﬁgure 2.1 will be referenced

throughout this document where appropriate.

Figure 2.1: The High Voltage Piezoelectric Transformer (HVPT) with electrical and mechanical proﬁles for mode 1 and mode 2 operation.

A ﬂat bar of lithium niobate is shown in ﬁgure 2.1. A detailed description of the

material properties of lithium niobate can be found in a number of sources [16, 29, 30].

8 The primary geometric axes of the bar in ﬁgure 2.1 are x0, y0, and z0 and the secondary axis z is rotated by an angle θ about primary axis x0 [15]. This rotation indicates the crystallographic polarization direction of the lithium niobate. Electric ﬁelds in the z0 direction couple into mechanical displacements in the y0 direction as a result of the rotated polarization and an output voltage is produced at the extremity of the bar.

This is known as the length extensional mode [31]. The voltage gain of the transformer is nearly maximized by selecting a crystallographic rotation angle θ equal to 45 degrees [15]. Additionally, the PT should be

driven with an AC frequency at or near an integer multiple of its natural mechanical resonance, which depends on the length of the bar and the sound speed of the material [12]. Experiments in this study use the fundandental (mode 1) and second

harmonic (mode 2), which occur at a frequency of approximately 30.5 kHz and 61 kHz respectively. The stress, electric ﬁeld and voltage magnitude proﬁles for both

modes 1 and 2 are shown in ﬁgure 2.1 as determined using Comsol, a ﬁnite element

modeling environment. The details of modeling an HVPT using Comsol have been discussed at length in a number of other sources and are therefore not included in this document [12, 19, 28]. To mount the system with low mechanical loss, clamps

must be placed at the displacement nulls (co-located with stress maxima). Since mode 2 produced two nulls at the L/4 positions, it provided a more stable mounting conﬁguration than mode 1, which only had a single null in the center. As a result,

mode 2 was preferred in practice.

9 Figure 2.2: 3D graphic of the HVPT with mounting hardware for mode 2 operation.

2.2 Laser optical diagnostic

An optical method was implemented to diagnose the internal fields of a resonant PT without influencing its operation. The mathematical premise for the diagnostic is developed in complete detail in [26] and [11], and is summarized later in this document in Section 3.2. The purpose of this section is to introduce the experimental configuration used to support the laser optical diagnostic method. The basic premise of the method is shown graphically in Figure 2.3.

Figure 2.3: General conﬁguration of laser diagnostic for probing internal ﬁelds of resonant PT. Top diagram is a view coplanar to the x0 − z0 axes. Bottom diagram is a view coplanar to the y0 − z0 axes.

10 A laser beam is passed through the HVPT in the x0 direction and scanned along its length (y0 axis) from the center to the high voltage output. The energized

HVPT distorts the time-domain intensity of the laser beam via the photoelastic and electrooptic effects. Due to these effects, the laser beam carries mechanical and electrical information of the resonant HVPT after it exits the bar. A detector assembly captures the encoded laser light and sends the signal to an oscilloscope for recording. For the beam to enter and exit the bar effectively, the incident surface of the HVPT must be polished to a high degree of optical clarity. Polishing was performed by the supplier to a Scratch-dig clarity specification of 80-50 (standard quality) and was satisfactory for this work. [32]. Figure 2.4 shows a comparison between a polished and a non-polished HVPT. It is plausible that the Q factor differs slightly between the polished and non-polished due to the difference in surface quality.

Figure 2.4: Comparison between an optically clear polished HVPT (1) and a non polished HVPT (2).

The beam path shown in figure 2.3 shows the general configuration for the laser optical diagnostic, however different beam path solutions can be implemented if the vacuum chamber prohibits a 1-dimensional path geometry. An alternative path is presented in figure 2.5 which allow for the laser optical diagnostic using a single viewport into the vacuum chamber.

11 Figure 2.5: Laser diagnostic in internal reﬂection conﬁguration

As shown in figure 2.5, after traveling 10-mm into the PT, the beam encounters the boundary of the lithium niobate and undergoes a second surface internal reflection within the PT. The beam is then partially reflected back through the PT and exits the chamber to the detector assembly. To achieve this geometry, both the detector and laser must be at an angle of about 3-degrees from the normal of the incident surface. A second beam path mode was also investigated which used a periscope to return the beam to the detector. This method and a comparison to the internal reflection mode are presented in Appendix E. The laser used was a Research Electro-Optics, Inc 1.5 mW, 534 nm linearly polarized laser and the detector assembly consisted of three components fixed within a one-inch optical tube. The quarter-wave plate (QWP) was used as a Sénarmont compensator which, while not strictly necessary, improved the symmetry of the optical response, assisting with the data post-processing. The analyzing polarizer (AP) was used to filter out the ordinary wave component of the optical signal. The photodetector (PD) was a ThorLabs DET110, whose output was captured using a

Tektronix 4054B mixed signal oscilloscope with 2.5 GS/s sampling rate. The incident laser polarization, quarter wave plate fast axis, and analyzing polarizer orientation were all co-aligned along the z0 axis of the PT.

12 A test bed was developed which allowed for the simultaneous operation of both the x-ray and laser optical diagnostics. Since the vacuum chamber only had a single

viewport for transporting the laser beam to the HVPT, the test stand was configured for the internal reflection beam path. Figure 2.6 is a diagram of the setup with a view oriented coplanar to the y0 − z0 surface of the HVPT. The figure shows that the

test stand could be moved to either one of two positions within the vacuum chamber. Position A was in the center of the chamber and provided both an optical line-of-sight to the PT via the viewport and an x-ray line-of-sight via a 50 µm-thick Al foil window

located 23 cm away. Position B was inside of a 6 inch vacuum flange and located 7 cm away from a 650 µm-thick Al sheet window. Position A was required for the laser optical diagnostic but sacrificed x-ray signal due the 1/r2 losses associated with the longer source-to-detector separation. Position B reduced this separation for higher fidelity x-ray measurements but blocked laser access to the PT. A photograph of the test bed is shown in figure 2.7.

13 Figure 2.6: Experimental setup to measure PT operation with x-ray and optical diagnostics. View emphasizes the x-ray diagnostic by orienting coplanar to the y0 −z0 axis of the PT coordinate system. Inset shows detail view for the electron beam, PT, and detector conﬁguration.

Figure 2.7: Photograph of the test stand for simultaneous x-ray and optical diagnostics for the HVPT inside the vacuum chamber.

A critical aspect of the laser diagnostic is that alignment must be constantly maintained among the laser, the HVPT, and the detector. To facilitate this alignment

14 requirement, the laser and detector were coupled into a single assembly, which is diagrammed in ﬁgure 2.8.

Figure 2.8: Diagram of laser-detector assembly for alignment of optical diagnostic.

The laser assembly had three points of actuation, providing a robust platform for positioning the laser on the HVPT while maintaining alignment. Actuation point A was a y0−z0 linear stage which moved both the laser and detector. This actuation point was used to move the laser interrogation point across the longitudinal length of the HVPT while maintaining the relative positioning between the laser and the detector. For ﬁne adjustments, actuation point B provided tip-tilt and y0 − z0 positioning for only the detector and point C provided tip-tilt actuation for only the laser. Using this conﬁguration, three-body alignment was easily maintained throughout testing.

2.3 Electron Gun Design & Calibration

The electron gun was a 60 mm-long, 19 mm-diameter hollow cylindrical enclosure consisting of a cathode and an anode with a drift region between the two. The enclosure body was manufactured with a Formiga P100 selective laser sintering (SLS)

3D-printer using nylon polyamide powder. This material was chosen because it can

15 withstand the high temperatures generated by the thermionic emitter. The cathode was a stainless steel grounded plate with a slot to accept an SPI tungsten wire thermionic emitter. A drive current of between 2.05−2.35 A DC was used to energize the thermionic emitter. The anode was located 60 mm from the cathode grid and was energized to up to 2.1 kV. A stainless steel control grid with a 30% open area fraction was installed 20 mm from the cathode. An applied grid voltage of 0−48 V DC was used to gate the current to the anode to between 0−41 µA. A pupil with either a 1.2 mm- or 5 mm-diameter was used to transport the beam from the A-K gap to the focusing optics. The optics consisted of two coaxial cylindrical tubes with a 6.22 mm inner diameter and a total length of approximately 50 mm, aligned with the electron gun centerline. Electrons passed through the optics and exited the system with an energy V2 electron-volts. Figure 2.9 shows a diagram of the electron gun.

Figure 2.9: Diagram of the electron gun.

As shown in ﬁgure 2.9 there are three primary components that comprise the electron gun: the cathode ﬁlament, the control grid, and the focusing optics. The following sections describe the calibration methods and results of each of these components. All calibration experiments were conducted at vacuum chamber pressures between 1−5×10−6 Torr, achieved using a turbomolecular vacuum pump.

16 2.3.1 Filament

To isolate the response of the filament, the control grid and electron beam optics were removed from the gun. The circuit shown in figure 2.10 was then used to calibrate the thermionic filament source.

Figure 2.10: Electron gun current calibration circuit.

An Emco high voltage DC power supply energized the calibration circuit with a voltage VE between 1−12 kV. A network of resistors RS and RM in parallel with VE caused currents iE and iM to ﬂow through the circuit. The anode was connected at the node joining RS and RM, forming an essential node in the circuit. Whenever an

− electron beam e2 was collected by the anode, a current iB, equal to the electron beam current, would ﬂow from the essential node. Kirchoﬀ’s Current Law (KCL) analysis was then used to determine the electron beam current. Figure 2.11 shows the circuit derivation of this diagnostic to determine the electron beam current.

17 Figure 2.11: Derivation of beam current calibration equation using source transformation.

Figure 2.11(a) shows an equivalent representation of the calibration circuit in ﬁgure 2.10, where the anode and electron beam have been replaced by a variable resistor RB. Using a source transformation, the voltage source VE in series with RS was replaced with a current source IE in parallel with RS, shown in ﬁgure 2.11(b).

Finally, the two physical resistors RS and RM were combined in parallel, forming RP, shown in ﬁgure 2.11(c). Since the current source IE is ﬁxed by the drive level of the

Emco supply and the value of the source resistor RS, the beam current IB can be determined by measuring voltage VM and performing KCL. Equation 2.1 shows the KCL equation for measuring the output current produced by the electron tube.

VE VM ib = − (2.1) RS RP

18 The ﬁrst term on the right side of equation 2.1 is the Ohm’s Law equivalent of the source current IE. The second term on the right of the equation is the Ohm’s

Law equivalent of the current IP, or the current ﬂowing through the parallel resistor combination, RP. KCL requires that the sum of all currents entering the top node must equal zero, therefore IE is the algebraic sum of IB and IP, which is rewritten in equation 2.1 to solve for the electron beam current. Using this approach, the beam current can be solved directly by simply measuring the voltage VM, since all other variables are known. The calibration results for the vacuum diode are shown in ﬁgure 2.12.

100

2.1A A) 80 µ

Network resistance 2.08A 60

2.06A 40 2.04A

2.02A 20 2.0A

Electron Beam Current ( 1.98A 1.96A 0 0 1000 2000 3000 4000 5000 6000 Anode Voltage (V)

Figure 2.12: Measured anode current response for varied ﬁlament currents.

The family of curves shown in figure 2.12 correspond to different drive currents used to energize the thermionic emitter, ranging from 1.96 A to 2.10 A in 0.02 A increments. A prominent feature is that the measured electron beam plateaued at higher current levels as the filament current increased due to Richardson’s law. At the highest filament currents, electron current became space charge limited and increased with anode voltage. Furthermore, the space charge limited current increased

19 as the ﬁlament current increased, due to virtual cathode voltage eﬀects discussed in Appendix B.4.

2.3.2 Control Grid

With the ﬁlament current response known, the control grid was added with the

intent of increasing the stability of the electron gun response over a wider range of drive currents. The calibration results for the ﬁlament + control grid are shown in ﬁgure 2.13 for three grid voltages over a range of anode voltages.

In ﬁgure 2.13, three measurements were taken at each data point and the error bars correspond to one standard deviation of the data. The horizontal error bars arose because anode voltage could not be directly controlled for. Rather, a known

Emco output voltage was applied to the circuit and variability in the beam caused ﬂuctuations in the resultant anode voltage. The control grid response was repeatable and provided more control over the amount of current delivered compared to the bare

filament. For instance, for the 2.25 A filament case, a 9, 11, and 13 volt grid produce a 3.5, 5.5, and 8.5 µA beam current, respectively. Another key benefit of the control grid was that the anode current was not influenced by the anode voltage. This was a critical feature of the current source because it allowed for consistent comparison between the DC low voltage case of the calibration circuit and the AC high voltage case of the HVPT-driven accelerator. The grid decoupled the anode voltage from the anode current, shielding the cathode from the electric fields generated by the anode. This resulted in a flat current response across the entire range of anode voltages. To characterize the control grid in more detail, the anode voltage was fixed to 6 kV and the grid voltage was finely swept from 0 to 30 V. The results of this study are shown in figure 2.14. This test showed that higher filament currents produced more anode current for a given grid voltage

20 Figure 2.13: Measured electron beam current vs. anode voltage for three grid voltages and three ﬁlament currents.

21 which was an experimental observation of the virtual cathode phenomenon described in appendix B.4. The I-V curve exhibited a linear response from 2 µA until it began to saturate at approximately 10 µA, demonstrating predictable performance in this range.

12 2.25 A

A) 10 µ 2.05 A

8 1.85 A 6

2 Measured Anode Current (

0 0 5 10 15 20 25 30 Grid Voltage (V)

Figure 2.14: Measured electron beam current vs. grid voltage with ﬁxed anode voltage (6 kV) and three ﬁlament currents.

The improved level of control provided by the control grid came at the expense of the overall current output. With the ﬁlament alone, it was common to observe electron currents up to 100 µA, whereas the control grid typically only reached approximately 12 µA. However, this was a preferable trade-oﬀ considering that accurate knowledge of beam current was the primary goal of the electron gun.

2.3.3 Optics

A Faraday collector method was used to calibrate the complete electron gun assembly (ﬁlament, control grid, and electron beam optics), replacing the KCL circuit calibration method presented in Section 2.3.1. The Faraday collector was a concave stainless steel enclosure and was ﬁxed to the output of the electron gun with insulating

22 hardware. An 80% open area fraction grid was installed at the opening of the Faraday collector and three series 9V batteries energized the grid to -26.4 V relative to the collector to suppress secondary electrons. Two Emco E120 proportional high voltage supplies were used to energize the electron optics and Faraday collector. ‘Emco 1’ delivered between 5−12 kV to the optics output tube, a 1:5.7 voltage divider, and the negative terminal of ‘Emco 2’. The output voltage of ‘Emco 2’ was ﬁxed at 2 kV and was used maintain a constant voltage diﬀerential between the electron gun output and the Faraday collector. A digital multimeter (DMM) was used to determine the electron beam current by measuring the voltage across a current viewing resistor connected between the Faraday collector and the positive terminal of ‘Emco 2’.

Figure 2.15: Faraday collector conﬁguration and diagnostic circuit for electron beam optics output current measurement.

In addition to the Faraday collector, an input current diagnostic circuit was implemented to further characterize the electron gun. This circuit is shown in

Figure 2.16.

23 Figure 2.16: Diagnostic circuit for measuring supplied thermionic emission current.

The calibration circuit consisted of two current meshes, as shown in Figure 2.16.

Mesh 1 contained the energizing voltage source, the thermionic emitter, and a high power current limiting resistor. Notably, Mesh 1 was ﬂoated from ground. Mesh 2

contained the thermionic emitter, a current viewing resistor CVR2, and a ground loop.

− Any electrons e1 which left the thermionic emitter to ground produced a current iB

which entered Mesh 1. Since Mesh 1 ﬂoated, the only path for the current iB to return to ground was through the current viewing resistor. By measuring the voltage

at CVR2, the amount of current supplied by the thermionic emitter was directly determined. A sample of the results of this calibration are shown in ﬁgure 2.17, which were

taken with a 1.2 mm pupil, an anode voltage (V1) of 2.1 kV, and an outlet voltage (V2) of 12 kV. Three measurements were made at each data point and error bars correspond to one standard deviation from the mean. Appendix A contains the complete set of results for calibrating this system. The ﬁrst plot shown in ﬁgure 2.17a is the measured thermionic beam input current to the vacuum tube. This measurement was taken using the input current diagnostic

24 10

2.35A 8 A) µ 6 2.25A

2.05A

Input Current ( 2

0 5 10 15 20 25 30 35 40 45 50 Gate Voltage (V) (a) Input Current Calibration

0.5

0.45 2.35A 0.4 A)

µ 0.35 2.25A 0.3

0.25 2.05A

0.2

Output Current0.15 (

0.1

0 5 10 15 20 25 30 35 40 45 50 Gate Voltage (V) (b) Output Current Calibration

0.5

0.45

0.4 2.35A A)

µ 0.35 2.05A 2.25A 0.3

0.25

0.2

Output Current0.15 (

0.1

0.05 -2 0 2 4 6 8 10 Input Current (µA) (c) Output vs Input cross-calibration

Figure 2.17: Electron gun (with focusing optics) calibration results.

25 circuit shown in figure 2.16. The gate voltage was swept from 0 to 48 V and three filament currents were investigated. As the gate voltage increased, the cathode delivered more electron current, in accordance with Child’s law. Furthermore, the beam current also increased with filament current, which was due to the virtual cathode effect seen previously in the no-optics results from figure 2.14. The diagnostic circuit was unable to distinguish the input current from background for gate voltages below 25 V. The maximum current emitted from the filament was 9 µA for a 48 V grid and a 2.35 A filament current.

The second plot in figure 2.17b shows the current measured by the Faraday collector from the outlet of the optical system. The results are again plotted against gate voltage for three filament currents. As anticipated, the outlet current increased with gate voltage and filament current. Again, measurements below 25 V gate voltage were indistinguishable from background. The highest outlet current for this configuration was 0.45 µA.

Finally, figure 2.17c plots the output current shown in figure 2.17b vs. the input current from figure 2.17a. The result is a nearly linear relationship between the current injected into the vacuum tube and the current measured at the outlet. The slope of the response is 0.05 Ain/Aout over the range of tested values, demonstrating the reliability of the electron source. Additional testing of the vacuum tube was conducted to further ensure reliable operation for varied physical configurations, including a sample-to-sample comparison between thermionic filaments and the effect of an increased pupil diameter. These results are shown in a summarized form in figure 2.18, however the contributing data set is included in Appendix A. Figure 2.18 shows the measured beam current for three configurations of the vacuum triode with optics using six drive conditions. The configurations included

26 Beam Voltage (kV) 1800 5 8 12 Emitter 1 (1.2 mm pupil) Emitter 2 (1.2 mm pupil) 1500 Emitter 2 (5 mm pupil)

1200

900

600 Beam Current (nA) 300

0 30 48 30 48 30 48 Gate Voltage (V) Figure 2.18: Current calibration results for a variety of conﬁgurations for the electron gun. two pupil diameters (1.2 mm and 5 mm) and two diﬀerent thermionic emitter samples. Two gate voltages of 30 V and 48 V and three optics voltages of 5, 8, and 12 kV are shown. Three measurements were made at each treatment level for error analysis.

The results in ﬁgure 2.18 showed little variation between emitter samples 1 and 2, with output current from each emitter typically staying within one standard deviation of one another across the range of tested conditions. The 5 mm pupil typically delivered about three times higher current than the 1.2 mm pupil, despite having a 17 times larger open area. Beam current increased with gate voltage and beam voltage, which was consistent with predictions from theory and previous experimental results.

The largest current observed in any condition was 1.5 µA, with the 5 mm pupil, 12 kV beam voltage, and 48 V gate voltage.

27 2.4 Wire-loaded HVPT Conﬁgurations

The primary focus of this document is the analysis of a freely oscillating HVPT for the direct acceleration of charge to the surface of the transformer. However it is of interest to investigate whether a wire can be used to electrically couple the HVPT output voltage to another separate physical anode. This would allow for the accelerator to accommodate a wider range of target geometries and materials. For example, earlier work conducted using deuterated titanium targets adhered on the transformer output (presented in Appendix D) showed that the additional mass of the target prevented the transformer from reaching high voltage. If the target was mechanically separated from the HVPT and instead connected with a wire, this problem might be avoided. This section presents two test configurations which examine the effect of adding a wire to the HVPT output. The first test configuration used the electron gun (minus the electron focusing optics) to deliver electron current to the HVPT target with no wire attached.

This provided the experimental control to compare against the wired HVPT. This conﬁguration is shown graphically in ﬁgure 2.19.

28 Figure 2.19: Electron gun in x-ray reﬂection mode conﬁguration.

With the no-wire case, the transformer was tilted to an angle of 45◦ to the beam, resulting in an oblique angle of incidence for the electron beam on the HVPT output, providing a path for x-rays to leave the system for external detection. Next, the wired conﬁguration is shown in ﬁgure 2.20.

Figure 2.20: Electron gun in x-ray transmission mode conﬁguration.

With the wired conﬁguration, the anode was a 50 µm-thin foil of aluminum, which due to its small thickness and low mass attenuation, permitted x-rays generated at the beam-incident surface to transmit through the anode for external detection. The

anode was energized via a wire connection to a high voltage source, including both

29 an HVPT and a commercial DC high voltage source for comparison. The second conﬁguration for the wired HVPT was a solid state charge pump circuit, which was built with conventional diodes, capacitors, and resistors. Figure 2.21 shows the circuit diagram for the solid state charge pump.

Figure 2.21: Circuit diagram for HVPT solid state charge pump (diodes conﬁgured for negative dc output).

The HVPT was driven as described in Section 2.1 and was coupled to the charge pump circuit using 18 AWG non stranded wire and adhered to the output electrode with silver paint. The charge pump circuit could be configured for either positive or negative rectification, depending on the bias direction of the diodes, essentially forming the first stage of a Cockroft-Walton generator. The diodes were VMI M50FF5 5 kV axial lead high voltage diodes, permitting a maximum DC output of 10 kV across

30 the capacitor. Several high voltage capacitors were used, ranging from 90 pF to 260 nF. A 1100:1 resistive voltage divider was used to measure the voltage at the output of the capacitor.

31 Chapter 3

Theory & Analysis

Included in this chapter is a discussion of the electron beam focusing optics and the laser diagnostic. First, theory and modeling is presented which details the design of the electron beam optics. Then, a summary of the mathematical algorithm for the laser optical diagnostic method is discussed.

3.1 Electrostatic Optics

In section 2.3, the cylindrical focusing optics were introduced which delivered a known amount of current to a small spot size. In this section, the design of that electrostatic optical system is presented. Two approaches are considered, including a numerical method and a ﬁnite element method.

3.1.1 Design

Figure 3.1: Design for electron beam cylindrical optics.

32 The cylindrical lens, shown in ﬁgure 3.1 is the most widely used system for focusing charged particles from a few eV up to several keV [33]. This optical system

consists of two energized coaxial segments of conducting material, separated by a dielectric for voltage standoﬀ. When a voltage diﬀerential is imparted between the two cylinders, the separation forms an electrostatic lens, bending a charged

particle beam much like a glass lens bends visible light. This analogy between optical and electrostatic phenomena gives rise to the terms ‘object’ and ‘image’, which correspond to the electron beam cross-section at the inlet and outlet of the system,

respectively. This design method is detailed in [33] and [34] and involves the selection of certain parameters to achieve a desired magniﬁcation for the optical system. These parameters include P , the object-to-lens distance, Q, the lens-to-image distance, `,

the sum of P and Q, D, the diameter of the optical cylinder, HO and HI , the height

of the object and image, respectively, and V1 and V2, the applied voltage to tube segment 1 and 2, respectively. The magniﬁcation M is deﬁned as the ratio of the size of the image to the size of the object, or HI /HO.

Numerical method In [34], authors E. Harting et. al. present a method for determining the ratio between V2 and V1, the energizing voltages of tubes 2 and 1, respectively. This method is determined using numerical solutions to the equation of motion of an electron in a lens and has an expected accuracy of between 1−3% [33]. Equations 3.1 and 3.2 and ﬁgure 3.2 provide a method to design a focusing system for a wide range of desired operating parameters.

Q M ≈ −0.8 (3.1) P

P + Q = ` (3.2)

33 Figure 3.2: P−Q curves for a two-cylinder lens with equal diameter cylinders and a gap of 0.1 D. Example shows method to determine V2/V1 for a magniﬁcation of 0.5.

Depending on the constraints of the system, a designer may choose to fix certain variables and solve for others. In the case of the work presented in this document, the limiting constant was `, the overall length of the optical tube, which had a maximum length of 61.3 mm (any longer and the optical cylinder would make contact with the HVPT). Once this parameter was fixed, the magnification power of the optical system could be selected. In practice, it is desirable to have the smallest spot size feasible. Therefore, M should have a value between 0 and 1, with smaller values of

M preferred. With this in mind, three optical systems with values of M equal to 0.5, 0.33, and 0.2 were designed. By ﬁxing ` and M, the system of equations 3.1 and 3.2 had solutions for P and Q, which are shown in table 3.1. The voltage ratio between tubes 2 and 1 is then determined using the plot shown in ﬁgure 3.2

34 Figure 3.3: Beam spot intensity as a function of V2/V1 for optics with magniﬁcation of 0.5, 0.33, and 0.2.

Finite element method The numerical method was used to establish the rough dimensions and voltages of the optical system, then finite element physics modeling software (Comsol) was used to refine the design. All Comsol modeling was performed in full 3D using the charged particle tracing and electrostatic physics available in the AC/DC module. The results are shown in figure 3.3. The y-axis value in figure 3.3 is normalized maximum beam spot intensity, which was used as an indicator of how well focused the beam was for a given ratio of

V2/V1. To collect this data, an accumulator boundary was placed over the image plane and V2/V1 was swept over a range of values. As V2/V1 changed, the beam went into and out of focus, corresponding to an increase and decrease in the intensity of incident electrons (#/cm2), respectively. A boundary probe was used to determine the maximum intensity over the surface, which was recorded for each V2/V1. Finally, the plots for each M were normalized to 1 so that they could be compared on the same scale. Figure 3.4 shows a qualitative representation of the focusing power of the beam for on- and oﬀ-focus V2/V1 ratios.

35 Figure 3.4: Comparison of beam focusing and intensity of the M=0.2 optic with three diﬀerent voltage ratios.

Table 3.1: Design parameters for electron beam optic systems

M Parameter 0.5 0.33 0.2 P [mm] 38.56 43.27 49.06 Q [mm] 22.76 18.05 12.26 V2/V1 (numerical) 5 6 8 V2/V1 (ﬁnite element) 3.1 3.6 5.3

The optic shown in figure 3.4 is the M=0.2 configuration. Three voltage ratios are shown, 3.3, 5.3, and 7.3. When V2/V1 is 3.3, the beam clearly does not converge to a tight focus at the image plane. Alternatively, at V2/V1 equal to 7.3, the beam converges before it reaches the image plane. Only when the voltage ratio was equal to 5.3 was the best focus was achieved. Table 3.1 shows a summary of the design parameters chosen for the three magnification powers of the cylindrical electrostatic optic systems.

3.1.2 Additional Comsol Modeling

In the previous section, the design methods and results for the electrostatic charged particle optics were presented. This section discusses additional modeling

36 Figure 3.5: Computed electron beam intensity and cross-section at the object (pupil) and image plane for each magnification power. results obtained using Comsol to test the optical systems. Figure 3.5 shows the computed effective magnification powers of each of the designs. The panels in figure 3.5 show the accumulated charge at the investigated surface for each case. The pupil was a 1.5 mm diameter inlet for charge, and was held constant for each of the three optical designs, serving as the object for the lens systems (it is shown here to demonstrate the relative size differences between object and image).

The color map in figure 3.5 indicates charge density, with blue corresponding to low density and red to high density. The designed magnification is shown on the top of the figure for each panel and the computed focusing is shown on the bottom. As shown, the magnification results computed using Comsol are close to the rough values determined using the Harting method. The analysis so far has only considered the electron optics as an isolated system with no external fields. However, in practice, the HVPT was used as a target for the electron beam, introducing additional asymmetric electrical forces. To verify that the

37 optical system maintained focusing when coupled to the HVPT, the Comsol model was modified to include an energized HVPT as a target. The HVPT was placed at a 45◦ oblique angle with the incident electron beam and given a voltage profile approximating that shown in figure 2.1. The distance between the HVPT output and electron optics outlet was fixed at 6.2 mm, which was equal to the outer diameter of the copper tubing used to fabricate the real-world optical system. Figure 3.6 shows the beam path for an optic configured for M=0.2, with an outlet tube voltage (V2) of 5 kV.

Figure 3.6: Charged particle tracing for the M=0.2 optic coupled to the HVPT target for a range of HVPT output voltages.

38 ×108

2.5 V /V =5.3 ) 2 1 2 − 2

1.5 V2/V1=2.3 1

0.5 V /V =8.8 0 2 1 Maximum Beam Spot Intensity (cm -0.5 0 10 20 30 40 50 60 70 80 PT Output Voltage (kV)

Figure 3.7: Electron beam intensity incident on HVPT output for varied HVPT voltage and V2/V1 ratios.

The graphics in ﬁgure 3.6 show the beam path for increasing values of voltage at the HVPT output, ranging from 0 up to 30 kV. At 0 V, the beam is completely deﬂected from the HVPT. This occurred because the electron beam exited the optical system with a voltage of 5 kV, thus the HVPT target was energetically unfavorable.

As the HVPT voltage increased above 5 kV, the beam was drawn to the target surface. Figure 3.7 shows a plot with additional data demonstrating this trend at higher HVPT output voltages and for multiple V2/V1 ratios. The plot in ﬁgure 3.7 shows that the beam intensity is constant for HVPT voltages approximately 4 times greater than the electron beam optics outlet voltage. The error bars are the standard deviation of 17 voltage ratio groups between 2.3 and 8.8. The best focus case of 5.3 stays within one standard deviation of the mean and transitions almost immediately from no beam collected to maximum intensity between 5 and 15 kV. Overall, these results show that with the correct voltage ratio, the optical

39 Figure 3.8: Photograph of electron beam spot on P22 screen. Pupil (object) size was 5 mm and spot (image) size was 2 mm, resulting in a measured magniﬁcation of 0.4.

system was capable of delivering electron current in a small focal spot on the HVPT throughout its range of AC output voltages.

3.1.3 Beam Spot Imaging

This section describes the experimental methods used to verify the beam spot

modeling results presented in section 3.1.2. A Kimball Physics 0.75-inch phosphor screen with 50 µW/cm2 sensitivity and 450 nm emitted wavelength was used to view the image of the electron beam spot. A bias voltage of 10 kV was applied to both the electron optics output tube as well as the stainless steel backing plate of the phosphor screen to provide a field free drift region for the electron beam. The screen was placed at the same distance from the optics as the PT output. A Sony A350 DSLR camera was used to capture photographs of the screen and was set to f/36 with a 1.6 second shutter to maximize the depth of field and provide sufficient exposure for the image.

40 Figure 3.8 shows a photograph of the beam spot on the phosphorous screen with the electron optics configured for M = 0.2 using the 5 mm pupil at 1.5 µA output current. The beam spot has a intense central region measuring 2 mm in diameter. A diffuse halo of scintillation is also visible around the central image, which is not focused and likely due to secondary electrons generated within the interior walls of the copper tubing. Considering only the focused central beam spot, the measured magnification power of the optics is the ratio of the pupil to the spot, or M = 0.4.

3.2 Laser diagnostic

In section 2.2, the experimental setup for the laser optical diagnostic was introduced. This included a discussion of the general configuration necessary to extract information from a resonant HVPT and variants of this configuration to operate using a vacuum chamber with a single viewport. Furthermore, an integrated laser-detector assembly design was introduced which allowed for rapid scanning of the HVPT and continued alignment of the optical system. In this section, a brief overview of the theory of operation for the laser diagnostic is presented. Appendix C contains a complete discussion of the methods used to collect the optical data, including data acquisition, post-processing, and analysis techniques. This intended to provide a clear set of procedures for a researcher seeking to recreate this experiment. Both the photoelastic and electrooptic effects govern the optical response of a material when subjected to a mechanical or electrical field, respectively. In both cases, the response is that the index of refraction of the material in the extraordinary axis changes, thereby changing the speed that light can move through the material along that axis. The index of refraction for the ordinary axis, however, remains constant, regardless of the stress or electric field. The variation in index of refraction between the ordinary and extraordinary components results in phase transitions, which appear

41 6 applied voltage 0.95 3 0.9

0.85 0

0.8 PD Voltage (mV) -3 Applied Voltage (V) 0.75 optical response

-6 0 5 10 15 20 Time (µs) Figure 3.9: Left axis: waveform example of the optical response of a resonant PT measured with a photodetector (PD). Right axis: waveform of applied voltage signal to drive PT into resonance. as oscillations in beam intensity when passed through an analyzing polarizer. By measuring the time-domain signal of the beam with a photodetector, the number of oscillations can be counted for a given time interval and used to calculate the stress and electric field of the PT. Figure 3.9 shows an example waveform of the optical response measured by a photodetector alongside the applied input voltage to the resonant PT. The optical signal in Fig. 3.9 is shown as a solid line and the voltage applied to the PT is a dotted line. The beam was directed along the x0 axis of the PT and the longitudinal position of the beam was 20 mm from the center of the PT in the positive y0 direction. Higher stress and electric fields result in more densely packed oscillations in optical signal. A Matlab script was written to count the number of oscillations that occurred over the time duration of the waveform, which are shown visually as filled circles at the maxima and minima along the optical signal trace. Equations 3.3−3.7

42 were used to compute the internal stress, electric ﬁeld, and voltage using the number of oscillations of the optical signal [11].

π ∆φ = m (3.3) 2

λ(∆φ) E (y0 = 50) = (3.4) 2 πLa

0 0 0 0 D2 = ε22E2(y ) + d22T2(y ) (3.5)

0 0 πD2La − ε22λ(∆φ) T2(y ) = 0 0 (3.6) πL(d22a − ε22b)

Z k 0 0 V (k) = − E2(y )dy (3.7) 0

The value ∆φ is the change in phase diﬀerence between the ordinary and extraordinary component of the laser beam per quarter period of the applied voltage signal. This value is computed in equation 3.3 directly from m, the number of

oscillations in optical intensity per applied voltage period. The values E2, D2, and T2 are the electric field, electric flux density, and mechanical stress in the y0 direction, respectively. Table 3.2 contains a summary of the constants used to determine the electric and stress fields of a PT assuming an HVPT with a 45-degree rotated

crystallographic axis. This method is discussed in detail in Appendix C, but is summarized here for context and brevity. To solve equations 3.3−3.7, the photoeleastic and electrooptic eﬀects must be

decoupled. This is accomplished in two steps. First, the stress at the end of the

43 Table 3.2: Parameter descriptions and values for the laser optical diagnostic equations. Material constants assume a 45-degree rotated lithium niobate slab.

Term Description Value Unit L laser path length 0.01 or 0.02 m λ laser wavelength 5.34× 10−7 m a electrooptic term 2.916×10−10 m/V b stress-optical term 4.575×10−11 m2/N 0 −10 ε22 permittivity 5.041×10 F/m 0 −11 d22 piezoelectric strain 3.308×10 C/N

PT must be equal to zero since its motion at that location is not constrained. This

eliminates the rightmost term in in equation 3.5 such that D2 only depends on the

electric ﬁeld E2. When considering only the extremity, E2 is readily determined from

∆φ in equation 3.4. Next, the electric ﬂux density D2 must be constant throughout

the material, therefore the value for D2 determined at the extremity is valid for all

locations on the bar. With D2 fixed as a constant, the expression for T2 in equation 3.6 becomes purely a function of ∆φ and can be determined for all positions along the bar. With T2 known, the electric field can be determined for all positions by solving equation 3.5 for E2. The voltage at the output is determined by solving the line integral of electric field along the length of the bar, as shown in equation 3.7.

44 Chapter 4

Experimental Results

This chapter presents the results for the piezoelectric systems described in Chapter 2. First, the results for the HVPT with focusing optics and the laser optical diagnostic are presented. In the laser-diagnosed test bed, the HVPT produced x-rays up to 20 keV while simultaneously delivering optical voltage measurements. Several studies are discussed, including a presentation of the matched x-ray and optical data, a harmonic mode analysis, electron beam current response, the eﬀects of long- term operation, and aggregate trends throughout the entire data set. The chapter concludes with the wire-loaded HVPT experiments, which demonstrated that the addition of the wire to the HVPT output reduced the output voltage from as high as 30 kV in the unloaded case to 6 kV with the wire load.

4.1 HVPT with Electron Beam Optics

4.1.1 X-ray Measurements

The test stand for delivering a focused electron beam to the HVPT output, described in section 2.2, was used to determine the x-ray response of the transformer for a range of applied beam currents. The voltages applied to the two copper tube segments V1 and V2 of the cylindrical optics were ﬁxed at 0.88 and 5 kV respectively. Figure 4.1 shows the time-resolved count rate using this test stand for a selection

45 of three tested electron beam currents ranging from 62−190 nA. In these tests, the

HVPT was driven with an input voltage of 5.3 Vrms in continuous wave (CW).

250

200 Beam AVG Current (nA) CPS 150 190 128 100

Count Rate (CPS) 134 50 55.8 62 13.6 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time (s)

Figure 4.1: Time-resolved x-ray count rate for three applied beam currents.

In ﬁgure 4.1, the measured count rate per 300-second interval is shown with solid lines and the average count rate for the entire interval are horizontal dashed lines. The individual measurements have error bars corresponding to one standard deviation from the mean at each point. Figure 4.1 also shows that the average count rate increased with beam current. The variation in count rate also increased with the beam current. In the 62 nA case, little deviation from the average line was observed. In the 190 nA case, the count rate dropped as low as 76 CPM at 900 seconds and rose to a maximum of 212 CPM at the 300 second mark, or a range of 136 CPM. The count rate did change over time, but no consistent trends were observed. Figure 4.2 shows that the trend of increasing count rate with applied beam current was consistently observed throughout all nine measurements.

The HVPT output voltage was also measured to determine if it changed as a result of increased beam current. The output voltage was determined by examining the maximum endpoint energy of the x-ray spectra collected every 300 seconds. This

46 200

150

100

50 Count Rate (CPS)

0 20 40 60 80 100 120 140 160 180 200 220 Beam Current (nA) Figure 4.2: Average x-ray count rate for varied electron beam currents.

allowed for a temporal eﬀect study of the output voltage as well. Figure 4.3 shows the output voltage of the HVPT determined with the x-ray diagnostic for a selection of 3 beam currents of 38, 158, and 261 nA.

24 Beam Current (nA) 22

20 Energy (keV) 158 18 261 38 16 0 500 1000 1500 2000 2500 3000 3500 4000 Time (s)

Figure 4.3: Time-resolved maximum x-ray energy for three electron beam currents.

As shown in ﬁgure 4.3, the maximum measured voltage was 26 kV, which occurred

in the first 300 seconds of the 261 nA measurement. Many other current levels were tested, however the differences between each tested current were indiscernible therefore only three are shown in figure 4.3 for clarity. The temporal effect was

consistently observed, showing approximately a 5 kV drop in output voltage in the ﬁrst

47 1000 seconds, and remaining constant for the remainder of the test. Figure 4.4 shows the maximum HVPT output voltage averaged over the duration of the experiment for each tested current.

23 0

Energy (keV) 18

16 0 50 90 130 170 210 250 290 Beam Current (nA)

Figure 4.4: Average maximum endpoint x-ray energy for varied electron beam currents.

Figure 4.4 also shows that no signiﬁcant change in HVPT output voltage was observed as the electron current was increased, with values typically varying by less than one standard deviation from 19 kV. The results of this experiment series showed that the electron beam optics coupled to the HVPT were a reliable, well-controlled test platform. The x-ray count rate produced by the HVPT scaled linearly with the electron current supplied by the optics. Furthermore, the HVPT output voltage eﬀectively remained constant for each of the tested beam currents.

4.1.2 Laser and X-ray Diagnostic Comparison

As was shown in ﬁgure 2.6, the use of the optical diagnostic required the test stand to be moved to a less favorable position for x-ray detection (pos. A). Since it was more diﬃcult to collect x-ray data in the new position, the x-ray diagnostic was not used for a majority of the optical testing. However, to verify that the two diagnostics

48 were in agreement, one simultaneous measurement was made of both x-ray spectrum and optical response. Figures 4.5 and 4.6 show the results of that measurement. In the subsequent tests, the HVPT was driven in burst mode to reduce the likelihood of a fracture. The burst rep rate was 2 Hz with 3000 cycles per burst. Assuming a 61 kHz resonant frequency, this corresponds to a 9.8 % duty factor. However since electrons were only accelerated to the HVPT during the positive half cycle, x-rays were only produced with at a 4.9 % duty factor.

106 Accelerated by PT 105

104

103 X-ray Counts 102

101

100 0 5 10 15 20 25 Energy (keV)

Figure 4.5: X-ray spectrum recorded simultaneously with laser optical measurement from ﬁgure 4.6.

Figure 4.5, shows the x-ray spectrum from the simultaneous measurement. X-rays were generated in the same manner as the electron beam focusing tests in section 4.1, with a maximum endpoint energy equal to the maximum accelerating potential of the system. The spectrum was collected over the period of one hour and reached a maximum endpoint energy of 20 keV, which indicated that the HVPT output voltage was 20 kV. As a visual aid, the spectrum is divided into two portions by a line at

49 12 keV. This is to distinguish between x-rays that were generated at the PT (on the right) and those which may have been produced either by the HVPT or the electron beam optic itself, which was energized to 12 kV at the outlet.

10 Voltage (kV) 5

0 0 10 20 30 40 50 Longitudinal Position (mm)

Figure 4.6: Voltage proﬁle of PT collected using the optical diagnostic simultaneously with x-ray data from ﬁgure 4.5.

Using the optical diagnostic, the voltage proﬁle of the PT was measured and is shown in ﬁgure 4.6. Measurements were made at twelve longitudinal positions across the output portion of the transformer, with 0 mm corresponding to the center of the bar and 50 mm to the output extremity. Error bars along the curve represent one standard deviation of four measurements made at each position over the 1 hour-long test. The highest voltage occurred at the 50 mm position and was equal to 20 kV, matching the x-ray spectrum and verifying that both diagnostics were in agreement.

4.1.3 Mode 1 vs. Mode 2

The optical diagnostic was used to determine if mode 1 and mode 2 had similar performance. Mode 1 was tested at 30.5 kHz with 7.4 and 8.7 Vrms input signals.

These results are compared against a mode 2 PT operating at 61 kHz 7.3 Vrms in figure 4.7. 50 Figure 4.7: Measured voltage, electric field, and stress profile measurement comparison between PT operating at mode 1 and mode 2. Distance is measured from the center of the HVPT (0 mm) to the output extremity (50 mm.)

The top plot in ﬁgure 4.7 shows the measured voltage proﬁles for each drive condition measured from the center of the bar to the output extremity. The

7.3 Vrms mode 1 and 7.4 Vrms mode 2 cases both reach a similar output voltage

51 of approximately 20 kV, however the curvatures of each are distinct, with mode 1 increasing in voltage more rapidly and plateauing towards the end. This is caused

by the shape of the electric field profile, shown in the middle plot. For mode 1, since the electric field magnitude peaks closer to the center of the bar, the voltage profile is steeper towards the center. The shape of the stress profiles for modes 1 and 2 shown

at the bottom of figure 4.7 closely match the corresponding electric field profiles,

which was expected from the ﬁnite element modeling results. The 8.7 Vrms mode 1 measurement shows an increase in voltage at the output, to a value of 22 kV. For comparison, results from Comsol are shown in ﬁgure 4.8, which are plotted along the entire length of the HVPT.

52 Figure 4.8: Modeled voltage, electric field, and stress profile as determined using Comsol finite element modeling for a mode 1 and mode 2 HVPT. The input, center, and output of the of the HVPT to -50 mm, 0 mm, and 50 mm, respectively.

The voltage, electric field, and stress profiles from modeling closely match with the experimentally determined profiles. In mode 1 the electric field and stress are both maximized at the center of the bar, whereas in mode 2, the maximum occurs at the L/4 position. The rapid increase in voltage in mode 1 compared to mode 2 is

53 also apparent in the modeled results. Both modeling and experimental measurements show that while the internal stress and electric field profiles were different between modes 1 and 2, the output voltage response was effectively identical. Since voltage was the primary performance metric of interest for accelerator purposes, there was no advantage choosing one mode over the other. However, mode 2 had two mechanical clamping locations and mode 1 only had a single point in the center, therefore mode 2 was a more practical choice because it provided a more stable mounting configuration.

4.1.4 Current Response

Figure 4.9 shows a box and whisker plot for the measured output voltage of the PT grouped over a range of applied beam currents. The data set controls for input voltage by only including tests conducted with a PT input voltage between 7−8 Vrms. The results from this test showed that there was no signiﬁcant change in measured output voltage when output current of the PT was increased from 0 to 1.5 µA. The power supplied to the PT in each case was 530 mW. These results are also in agreement with the x-ray diagnostic results in ﬁgure 4.4, which showed little to no variation in HVPT output voltage as the beam current was increased to 290 nA.

54 25

n=12 n=5 n=8 n=6

20 PT Voltage (kV)

15 0 0.16-0.24 0.35-0.57 1.3-1.5 Beam Current (µA)

Figure 4.9: Box and whisker plot showing maximum voltage measured for a PT operated at 7−8Vrms input for a range of applied beam current.

4.1.5 Long term operation

To examine the temporal effects in each case, figure 4.10 shows a time resolved plot of PT output voltage integrated over all current ranges. In the figure, the optical measurements are shown alongside a summary of the x-ray measurements previously presented in figure 4.3, with the ‘×’ and ‘◦’ markers referring to the x-ray and optical measurements, respectively. The x-ray data had a mild effect on the PT output

voltage over time, with an average value of approximately 22.5 kV in the first 300 seconds and settling on a final value of approximately 18 kV. The optical diagnostic had good agreement with the x-ray data, showing a flat response between 700 and

3600 seconds. The optical diagnostic was unable to capture the early time-dependent drop in output voltage because it took between 10−12 minutes to make a complete scan of the output portion of the PT with the laser assembly.

55 26 X-ray 24 Optical

20 Voltage (kV) 18

16 0 1000 2000 3000 4000 Time (sec) Figure 4.10: Time-resolved voltage measurements for PTs operating for 1-hour.

4.1.6 Aggregate trends

Throughout the experimental series, 109 tests were conducted using the optical diagnostic with varied input voltages, electron beam currents, HVPT samples, and harmonic drive modes. Figure 4.11 shows a scatter plot of all tests to examine the relationship between input voltage and the electrical and mechanical response of the HVPT. Figure 4.11a shows scatter plot of all applied voltages against the measured output voltage, demonstrating that the output voltage scaled proportionally with the input voltage. The voltage gain of the HVPT was computed by examining the slope of the linear fit, which is equal to 3.5 kV/V (shown as the coefficient in the linear fit equation on the plot). The data in figure 4.11 are also discriminated by sample and drive mode. Two samples, PT-203 (blue ‘×’) and PT-204 (orange ‘◦’) were tested. Testing started at the low drive voltages with PT-203. As the applied voltage to that transformer was increased, the output voltage also increased until PT-203 unexpectedly fractured near 7 Vrms. Input voltages of up to approximately 8 Vrms were applied to PT-204 without any fracturing. 56 30 y = 3.5*x - 6.6 25 R2 = 0.83586 20

10 PT-203 (M2) Output Voltage (kV) 5 PT-204 (M2) PT-204 (M1) Linear Fit 0 23456789 Input Voltage (Vrms) (a) Output Voltage

8 y = 1.1*x - 1.9 R2 = 0.84234 6

PT-203 (M2) 2 PT-204 (M2) PT-204 (M1)

Max Electric Field (kV/cm) Linear Fit 0 23456789 Input Voltage (Vrms) (b) Max Electric Field

16 y = 1.9*x - 2.6 14 R2 = 0.87395 12

6 PT-203 (M2)

Max Stress (MPa) PT-204 (M2) 4 PT-204 (M1) Linear Fit 2 23456789 Input Voltage (Vrms) (c) Max Stress

Figure 4.11: Aggregated optical data of electrical and mechanical output for varied input voltage and HVPT sample.

57 While the vast majority of tests were conducted with a mode 2 transformer, two measurements were made using PT-204 in mode 1 (violet ‘+’). The ﬁrst mode 1

test was performed at a known safe input voltage of approximately 7 Vrms. Next, the

input voltage was increased to 8.5 Vrms, which also proved to be a safe drive level. The voltage gain of the mode 1 transformer appeared to be consistent with mode 2, although the sample size was too small to definitively demonstrate that trend. Figures 4.11b and 4.11c show the maximum electric field and stress of the HVPT for the same tests. As expected, both of these parameters increased in value with applied voltage. The data from PT-203 and 204 remain grouped since they were driven with different applied voltages and and the mode 1 data appear to not deviate substantially from the mode 2. It is likely that the strength of the correlation was reduced since the data set does not control for other variables such as applied beam current or temporal effects.

30 y = 0.28*x - 1.8 25 R2 = 0.94112 20

10 PT-203 (M2) PT-204 (M2) Output Voltage (kV) 5 PT-204 (M1) Linear Fit 0 20 30 40 50 60 70 80 Input Current (mArms) Figure 4.12: Output voltage vs. input current.

Figure 4.12 continues the aggregate investigation of the optical diagnostic, examining the relationship between input current and output voltage. As shown, the input current and output voltage were linearly correlated with an r-squared of 0.94, which was stronger than the input voltage trend in ﬁgure 4.11a. This suggests

58 that input current, rather than input voltage, is a better indicator of output voltage achieved by the HVPT. The current-resolved data in ﬁgure 4.12 also shows that

mode 1 results deviated substantially from mode 2. In both tests with mode 1, the input current necessary to reach 20 kV was half that for mode 2. This suggests that mode 1 more more eﬀectively uses input energy to generate high voltage. Figure 4.13

supports this claim, showing the output voltage of the data set plotted against input impedance.

PT-203 (M2) Output Voltage (kV) 5 PT-204 (M2) PT-204 (M1) 0 80 100 120 140 160 180 200 Input Impedance (Ω)

Figure 4.13: Output voltage vs. input impedance.

The impedance to the HVPT was determined by dividing the measured input voltage by the input current. The physical location of the voltage probe for this

measurement was located at the interconnection between the power supply and an RG-58 coaxial cable which tied the supply to the HVPT input. As a result, this value was actually the input impedance of the transmission line. Equation 4.1 was therefore

used to extract the HVPT input impedance (i.e. the load impedance ZL) from the measured input impedance of the cable [35]. The transmission line length l was 3.66 m and the phase constant β of the driving signal in RG-58 was 1.94×10−3 rad/m

(ν = 61 kHz, up = 0.659c). As a result, tan βl was very close to zero, causing the measured transmission line input impedance Zin to be nearly equal to the calculated

59 HVPT input impedance ZL. Due to the low discrepancy between the two, the values for HVPT input impedance reported in ﬁgure 4.13 are the measurements made directly at the transmission line input.

ZL + jZ0 tan βl Zin = Z0 (4.1) Z0 + jZL tan βl As shown, the three data groups are distinctly separated on the plot. PT-203, which fractured early in the experimental series, had a higher, but highly variable impedance. Values as low as 91Ω and as high as 148 Ω were observed, with an average of 126 Ω. PT-204, however was more tightly bound between 88 and 120 Ω with an average of 94 Ω. This suggests that input impedance may be a good indicator for mechanical durability of a particular HVPT sample, with higher impedances implying increased susceptibility to fracturing.

y = - 18*x2 + 42*x + 1.8 y = -18*x2 + 42*x + 1.8 25 R 2 = 0.92515 20

10 PT-203 (M2) PT-204 (M2) Output Voltage (kV) 5 PT-204 (M1) Quadratic Fit 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Input Power (W)

Figure 4.14: Output voltage vs. input power.

Additional analysis of the energy conversion eﬃciency of the HVPT is shown in

ﬁgure 4.14, which shows the output voltage measured against input power. The input power ranged from 0 to 600 mW, with higher power corresponding to higher voltage.

The mode 1 results are outliers from the bulk mode 2 data set. As was shown in the

60 current-dependent plot in ﬁgure 4.12, the mode 1 measurement reached 20 kV with half of the power necessary for the mode 2 cases.

Figure 4.15 shows the electrical power eﬃciency of the HVPT as a function of input power for each of the 48 electron beam-loaded measurements. Since the electron beam current was variable, the eﬃciency depended on the current supplied to the HVPT.

A maximum electrical eﬃciency of 2.35 % was determined, which occurred at the maximum output current of 1.5 µA. This value is perhaps counterintuitive considering that the Q of lithium niobate is nominally very high. However the low eﬃciency was

not a result of material losses, but rather due to the fact that the supply of charge at the output was limited to 1.5 µA. As a result, excess energy at the output was likely dissipated in the material as heat. Furthermore, a higher capacity electron gun would likely cause a linear increase in electrical eﬃciency of the system.

2.5

1.5

ciency (%) 1 ﬃ E 0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 Input Power (W)

Figure 4.15: Eﬃciency vs. input power.

Output power for each measurement was determined from the product of the RMS

current and voltage of the HVPT. Since the current source was DC, it’s RMS value was known directly, however current was only absorbed when the output voltage of the

61 HVPT exceed the electron beam voltage (as shown with modeling in section 3.1.2.) A truncated sinusoid waveform approach was used to determine the output rms voltage

of the HVPT, which is shown in equations 4.2 and 4.3.

v u n uX rms = t Dkuk (4.2) k=1

1 2 sin(θ2 − θ1) cos(θ2 + θ1) uk = Ipk 1 − (4.3) 2 (θ2 + θ1)

In equation 4.2, Dk is the duty cycle of the region of interest k, which for a partial sinusoid is either zero (no beam on target), or a sine wave with amplitude

and frequency matching the HVPT output (beam on target). The expression uk is the rms value of the region of interest k, which depends on the two phase angles θ1 and θ2. These angles deﬁne the two boundaries of the truncated sinusoid and were determined by solving the inverse sine of the ratio of the DC voltage of the beam to the peak voltage of the HVPT. The expression Ipk is the peak value of the sine wave and was set to equal the output voltage of the HVPT as determined using the laser optical diagnostic.

62 Figure 4.16: Output voltage vs. maximum internal stress.

Figure 4.16 shows the stress measured using the optical diagnostic of all tests plotted against PT output voltage. In practice, operating the PT in the ‘safe region’, as shown in ﬁgure 4.16 did not result in fracturing (except for PT-203). However, previous testing with non-polished crystals showed that as the input voltage exceeded

9 Vrms fracturing became a risk. Since only a small supply of polished crystals was available, higher input voltages were avoided and tests were restricted to the safe region. The optical diagnostic showed that 15 MPa defined the upper bound of safe operation. This experimentally obtained value differs quite substantially from values found in literature, quoting the yield strength for lithium niobate to be at least 30 MPa and even as high as 120 MPa [29]. Linear extrapolation of the trend shown in figure 4.16 suggests a PT output voltage of 54 kV for the maximum yield condition of 30 MPa. While it is possible individual samples may have been able to sustain higher stress and thus reach higher voltages, the purpose of the investigation was to determine trends in HVPT performance, not destructively test the device to identify limits of operation.

63 Overall, the optical diagnostic revealed several interesting trends of the HVPT.

Harmonics Mode 1 and Mode 2 were shown to reach similar output voltages for a given input voltage, however the impedance associated with these modes were diﬀerent by a factor of 2. Mode 1 therefore is the better choice for an energy-

constrained application such as a battery powered source. If there is no power constraint, mode 2 is a better selection because it is much easier to mount in a stable conﬁguration.

Input Voltage The transformer action of the HVPT resulted in a proportional

relationship between input and output voltage. Interestingly, input current also demonstrated this eﬀect, however with increased correlation strength. In the range of tested conditions, the input parameters were the dominant variable

in determining the output voltage of the system.

Output current The optical diagnostic agreed with the x-ray diagnostic, showing that for currents between 0−1.5 µA, no signiﬁcant change in output voltage was observed.

Temporal eﬀects Both the optical and x-ray diagnostics also agreed that the output

voltage of the HVPT did not change for time scales on the order of one hour. The x-ray measurement did show variation in the ﬁrst 10 minutes, however the optical diagnostic was not fast enough to measure such small time scales.

4.2 Wire-Loaded HVPT

In the previous section, the output of the HVPT was not physically connected to any structure other than a 10 mm-long silver electrode painted directly on the surface. This section investigates the eﬀect of adding a wire to the output of the

64 HVPT. All tests in this section were conducted with non-polished crystals since the optical diagnostic was not used. The ﬁrst experiment with a wired HVPT output was an x-ray test conducted with the electron gun as a charge source. For this test, the electron beam optics were not used on the electron gun. As a control, the HVPT was ﬁrst tested without a wire, positioned at a 45◦ angle to the electron beam direction

(shown in ﬁgure 2.19.) The electron gun was conﬁgured for the highest electron beam current of 16.1 µA to increase the likelihood of detecting x-ray output. The PT was driven with three input voltages of 5.3, 6.2, and 6.8 Vrms and x-rays were observed for each case.

65 PT reﬂection, electrode only, 7.5 Vin 102

Reg. 1 (0

101

Counts Reg. 3 (7200

100 0 5 10 15 20 25 30 Energy (keV)

(a) 5.3 Vrms PT reﬂection, electrode only, 8.8 Vin 103

Reg. 3 (7200

100 0 5 10 15 20 25 30 35 Energy (keV)

(b) 6.2 Vrms PT reﬂection, electrode only, 9.6 Vin 102

101 Counts

100 0 5 10 15 20 25 30 35 40 Energy (keV)

66 Figure 4.17 shows the measured x-ray spectra for each of the three input voltages measured at the beginning and end of a 2-hour test (except for ﬁgure 4.17c which fractured early into the test). The maximum endpoint energy increased with input voltage, with a value of 26, 28, and 30 keV, respectively. In the 5.3 Vrms test, the maximum endpoint energy dropped to 24 keV after 2 hours, however in the 6.2 Vrms case it remained constant at 28 keV. The count rate for each of these tests varied over time but constantly produced an average of between 10−40 CPS. These results establish a set of control data which demonstrate that the the

HVPT was capable of generating x-rays with no wire attached to the output in the electron gun configuration. Next, the system was reconfigured to use an anode which was physically separated from the HVPT output and electrically connected with a wire (shown in figure 2.20). The anode was made from 50 µm-thin aluminum foil, permitting the x-rays to be transmitted through the anode. However, with the addition of the wire and the use of the transmission anode geometry, no x-rays were observed under any circumstances. To verify that the loss of count rate was due to the addition of the wire rather than the change to a transmission mode geometry for the x-ray source, the HVPT was replaced with a commercial DC high voltage supply, energized to 12 kV. Three beam currents were supplied to the anode using appropriate settings determined from the calibration methods from section 2.3.2. Figure 4.18 shows the x-ray spectra produced by this test.

67 100 16.1 µA

-1 10 8.7 µA

10-2 CPS 3.8 µA

10-3

10-4 0 2 4 6 8 10 12 14 Energy (keV)

Figure 4.18: X-ray spectra produced by transmission mode x-ray source energized with Emco DC high voltage supply.

With the DC supply, a maximum endpoint energy of 11.6 keV was measured, which closely matched the applied DC voltage. Figure 4.19 shows the x-ray count rate integrated over the energy range of the spectra for each applied current. The count rate for each case was eﬀectively constant for durations up to 1 hour and the average count rate increased with applied current, with a measured count rate of 2.5, 18.3, 36.9 counts per second for the or the 3.8, 8.7, and 16.1 µA cases, respectively.

68 50 16.1 µA 40

30 8.7 µA CPS 20

10 3.8 µA 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time [s] Figure 4.19: Time-resolved count rate for three applied beam currents for the DC transmission x-ray source.

The results with the DC supply demonstrated that the geometry of the transmission mode x-ray source was viable for x-ray production. Therefore, the null result of the HVPT-driven transmission mode source was not a result of the design of the source, but rather the presence of the wire at the HVPT output. Certainly, with the wire attached, the HVPT was no longer capable of reaching ‘interesting’ voltages from a particle accelerator design standpoint. However, there are some applications, such as ozone generators [10], for which voltages in the 5−10 kV range are of interest. To investigate this range, a charge pump circuit was coupled to the HVPT output via a wire. The charge pump circuit design (shown in ﬁgure 2.21) allowed the transformer to deposit DC charge on a capacitor. Figure 4.20 shows the test results of this circuit measured with a 100 MΩ voltage divider with a 1100 to 1 division ratio.

69 6

5 Vacuum 4

3 Air 2

Output Voltage (kV) 1

0 1 2 3 4 5 6 7 8 9 10 Input Voltage (Vrms)

Figure 4.20: Charge pump voltage.

The HVPT was driven with between 1−5 Vrms in air and 1−10 Vrms in vacuum. A lower maximum drive voltage was used in air because electrical discharges were observed on exposed leads of the charge pump circuit which became increasingly prominent as the drive voltage increased. The two curves shown in ﬁgure 4.20 show a linear increase in DC output voltage as the rms voltage was increased. The gain for the vacuum test was similar to that for air. In vacuum, the maximum DC voltage achieved was 5.5 kV. Figure 4.21 shows the input and output power for this system.

0.8

Power IN 0.6

0.4 Power (W) 0.2 Power OUT

0 1 2 3 4 5 6 7 8 9 10 Input Voltage (Vrms)

Figure 4.21: Charge pump power.

Four curves are shown in ﬁgure 4.21, corresponding to power in and out for both air and vacuum. In general, the power measured for the vacuum and air tests were similar,

70 with higher power measurements associated with the vacuum case. Comparing at the

5.4 Vrms mark, the HVPT accepted 232 mW (vacuum) and 220 mW (air) at the input and delivered 63 mW (vacuum) and 49 mW (air) to the output. This corresponds to a 5 % increase in input power, but a 28 % increase in output power in vacuum as compared to air. Figure 4.22 shows the power efficiency (Pin/Pout, derived from figure 4.21) for these two cases. As shown in figure 4.22, the HVPT was consistently 33 % more efficient in vacuum than in air for every tested input voltage.

0.3

0.28 Vacuum

0.26 Air 0.24

Efficiency 0.22

0.2

0.18 1 2 3 4 5 6 7 8 9 10 Input Voltage (Vrms)

Figure 4.22: Charge pump eﬃciency.

The ﬁnal test conducted with the solid state charge pump was an investigation on the eﬀect of capacitances at the HVPT output between ∼10−11 to 10−7 at the charge pump output. Figure 4.23 shows the results of this study.

71 5 90 pF 900 pF

(kV) 4 10 nF 260 nF 3

2 Output Voltage 1

0 2.75 5.30 7.92 Input Voltage (Vrms) Figure 4.23: Charge pump capacitance.

Before conducting the capacitance test, the diagnostic circuit was refurbished by smoothing all of the unintended sharp features of the circuit connections. This

allowed the HVPT to be operated in air at higher drive levels with much less observed discharging. Using the refurbished circuit in air, four capacitors were tested with the charge pump circuit at three HVPT input voltages.

As shown in ﬁgure 4.23, the output voltage achieved for each capacitance remained relatively constant between ∼10−11 and 10−7 F. The practical implication of these results is interesting because it shows that the HVPT is insensitive to typical real world capacitances when conﬁgured as the driver for a charge pump, suggesting that the device could be used as an intermediate-value (∼5 kV) driver for a capacitive storage device.

72 Chapter 5

Conclusions & Future Work

5.1 Conclusions

The goal of this work was to generate high voltage using a piezoelectric transformer and accurately diagnose its output to better understand its capabilities as a compact charged particle accelerator. A system was implemented which permitted the simultaneous measurement of the operating high voltage piezoelectric transformer using both a laser-optical and an x-ray diagnostic. A well characterized electron source delivered controlled amounts of electron current to the HVPT output, allowing for a detailed study of the effect of electrical loading of the HVPT for beam currents in the hundreds of nano ampere range. A specialized laser probe apparatus was developed which allowed for the rapid and convenient diagnosis of the HVPT. The tests showed that an HVPT could operate continuously at 20−25 kV output voltage for hours without significant loss of voltage or x-ray output. These results suggest that a compact x-ray source based on the piezoelectric effect is a feasible technology. In Chapter 2, the various experimental configurations necessary to operate an HVPT were presented. After a brief introduction to the fundamental design of the

HVPT, the experimental configurations were discussed. The first configuration was the laser diagnostic system. The basic premise of the diagnostic was introduced, then

73 the physical test bed used to collect data was presented. Next, the electron gun was presented, including a summary of the design as well as calibration results for each component of the gun. Finally, two experiments were presented which investigated the eﬀect of adding a wired load to the output of the HVPT. The ﬁrst experiment explored a wire-coupled x-ray generator and the second was an external solid state charge pump circuit Chapter 3 contained the theory and analysis necessary to support the experimental work. In section 3.1, the electrostatic optics were presented. The optics were designed using a combination of two methods, which allowed for a rough approximation

(numerical method) followed by a fine optimization using Comsol. The electrostatic optics were also modeled with an HVPT as the anode to verify the electron beam would maintain a tight focus even with the asymmetry added by the transformer output. Section 3.2 contained a discussion of the mathematical model used to convert the time-domain optical signal of the laser diagnostic into the stress, electric field, and voltage of the operating HVPT. Section 4 contained the results measured using the laser diagnostic and the electron gun. X-ray spectra obtained simultaneously with laser-optical data showed agreement that the HVPT was operating at 20 kV output voltage. A comparison between the electrical and mechanical profiles of the two harmonic modes was then presented. This showed that both modes achieved the same output voltage for a given input, however the distribution of electric and stress fields were different. The current response of the HVPT was then investigated, which showed that no significant change in HVPT output voltage was observed for currents ranging from zero to 1.5 µA. Furthermore, the optical diagnostic showed that the output voltage remained relatively stable over 1-hour time intervals. An aggregate trend study showed that output voltage, electric field, and mechanical stress all increased with applied voltage to the HVPT input.

74 Since both the output voltage and current were known, the electrical eﬃciency of the beam-loaded HVPT was determined, with a maximum value of 2.35 %. The laser diagnostic was also used to calculate the stress of the transformer and results suggested that a ‘safe’ region extending to 15 MPa was present. This value was lower that expected, as reported yield strength from literature is 30−120 MPa for lithium niobate [29]. The chapter concluded with the results of the wire-loaded HVPT tests, which demonstrated that adhering a wire to the output of the HVPT dramatically reduced its voltage. Without a wire, a voltage of up to 30 kV was measured, however with a wire attached, the transformer could only reach 6 kV. As was mentioned in Section 1.1, the study of piezoelectric transformers as charged particle accelerators has been an ongoing topic of research at the University of

Missouri for almost a decade. In this time, several advances have been made in this area, such as, the x-ray and laser optical diagnostic techniques. One major change to the standard operating procedures for the HVPT during this time was the transition from mode 1 to mode 2. Mode 2 had a practical advantage over mode 1 and modeling suggested that the two should reach similar output voltages for a given input. The modal analysis presented in this document shows that for a given input, mode 1 and mode 2 reach the same output voltage. One unexpected observation was the impedance and power draw of mode 1 and mode 2 were different. The impedance of mode 1 was 180 Ω compared to 90 Ω for mode 2. Alternatively, mode 1 and mode 2 required 300 mW and 600 mW respectively to reach 20 kV. One real-world implication of this finding is that mode 1 would be a more favorable implementation when power draw is a consideration, such as for a mobile battery powered system. No significant variation in mechanical stress was observed between the two modes so they are likely to have the same fracturing limits.

75 Another signiﬁcant advancement was work completed in 2012 which showed that piezoelectric transformers were capable of reaching output voltages in the

120 kV range [28]. These results suggested that a neutron generator powered by a piezoelectric transformer was feasible, as the cross-section for D-D fusion is favorable at that voltage. In light of these results, development began on a PT-driven neutron generator, which is documented in Appendix D. After many months developing this system, it was determined that mass loading due to the presence of the titanium deuteride target at the HVPT output dramatically impacted its performance and prevented the system from generating neutrons. With no target, the HVPT reached 40 kV, but with the target (mass=170 mg), the voltage was reduced to less than 10 kV. As a result of this ﬁnding, a pivot in focus was made to accurately characterize the

HVPT and isolate potential loss mechanisms to maximize output voltage. The wire-loaded HVPT was intended to address the mass-loading problem by removing the target from the HVPT. Once separated from the crystal itself, it was thought that target mass would no longer inhibit performance the target since it was physically decoupled from the output. However, with a wire at the output, the HVPT was only able to reach 6 kV at the output. The added mass of the wire itself was actually quite small (only about 10 mg ≈ Ag output electrode mass), so it is unlikely that the mass of the wire was the cause of the reduced voltage. Instead, it is likely that energy was lost to other mechanisms such as the wire clamping the output or absorbing vibrational energy of the HVPT. The electron gun test showed that x- rays were observed only when no wire was present on the HVPT output. Ultimately, this test demonstrated that the a wire should not be used in designing an HVPT- driven charged particle accelerator. This is important knowledge moving forward with future designs because it emphasizes the importance of a direct charged particle accelerator geometry.

76 Another important loss mechanism which limited the 2012 piezoelectric x-ray system in [28] was its strong time-dependent decrease in count rate. With that system, electrons were produced using sharp field emitters at the HVPT output, but high voltage x-ray production typically ceased entirely within the first 10 seconds of operation, with few counts recorded even at lower energy afterwards. Two hypotheses were suggested to explain the phenomenon: physical degradation of the emitters and electrostatic charging. The degradation hypothesis was that the field emitters were damaged by surface chemistry effects and ion bombardment, both a result of an insufficient vacuum of 10−6 Torr [36]. An SEM was used to collect before-and-after images of the emitters and showed blunting in the ‘after’ case.

The charging hypothesis was based on the premise that the output of the HVPT was electrically isolated from ground and the field emitters caused the output to develop a net positive charge by extracting electrons from the material. Because of the low output capacitance of the HVPT, even a small amount of charge buildup resulted in a rapid increase in static voltage. This negated the voltage generated by the piezoelectric transformer action and shut off the beam current. A model was presented which supported this claim and matched nicely with the observed time- dependence, however no experimental evidence was collected which objectively proved the hypothesis. The results presented in the present body of work provide additional data with the thermionic diode, allowing for a comparison to the field emission diode. In field emission mode, the emitters starved the HVPT output of charge with no means of replacement. In contrast, the thermionic mode continuously supplied the HVPT with electron charge with no means of removal. From an electrostatic standpoint, these two scenarios are simply an inversion of one another and should therefore experience the same charging effects. However, as was shown experimentally,

77 the thermionic diode continued to operate whereas the field emission diode quickly shut off. There are two conclusions that could be made from this result. First, since the thermionic case had no field emitters to degrade and did not experience a drop in count rate, degradation seems a plausible explanation for the decrease in count rate. Alternatively, the discrepancy between thermionic and field emission modes could be primarily due to charging, given a mechanism for charge replacement exists for the thermionic diode, but not the field emission diode. One explanation for this charge replacement mechanism arises from the fundamental difference between the two diode geometries. With field emission, electrons were removed from the HVPT output in the negative half cycle and no charge was deposited or removed in positive half cycle. With thermionic emission, charge was collected by the HVPT in the positive half-cycle, and the beam was deflected from the HVPT in the negative half-cycle. However, it is plausible that during the negative half-cycle, the surplus electron charge could have been emitted from the transformer output via

field enhancement at the corners of the bar or triple point emission at the interface among the lithium niobate, silver electrode, and vacuum. Experimental evidence was collected in the 2012 series in [28] which showed even with no emitters present at the output, x-rays were still observed, suggesting these mechanisms for charge removal in the negative half-cycle are realistic. This indicates that the thermionic electron source provided a method to use the HVPT in a net-zero charge pump configuration which was electrically similar to the solid state charge pump demonstrated at the end of Chapter 4, but without the loss mechanisms associated with the wired load. Losses associated with diagnostics have also been a prevalent issue throughout the development of the HVPT, and one of the most challenging parameters to measure has been the output current of the device. When field emitters were used to accelerate charge, the target was either a small tungsten target or the stainless steel walls of

78 a vacuum chamber − neither of which provided feasible platforms for an accurate current measurement. When the thermionic emitter was used without electrostatic optics, ray-tracing models showed the beam path was unfocused and there was no way of knowing how much current actually arrived at the HVPT output [11, 24]. However with the addition of calibrated focusing optics, the output current of the HVPT was

ﬁnally measured. Results showed that an electron beam current between 0−1.5 µA were applied to the HVPT operating at 20 kV. Little to no change in performance was observed between the 0 and the 1.5 µA case, indicating that these output current

levels were within the acceptable limits of operation.

5.2 Future Work

One area of improvement for this work is the electron gun, which could only

deliver a maximum of 1.5 µA, limiting the range of currents that could be tested with the system. Ideally, a higher current gun is desirable because it would increase the x-ray count rate of the system and permit a more feasible path-to-market for a

commercial grade x-ray source. One option to move forward would be to improve the design of the present electron gun system to produce higher output currents. The MU rapid prototyping laboratory is a valuable asset in this regard, providing a fast,

cost eﬀective way of manufacturing and testing many electron gun designs. Another option would be to purchase a commercial grade electron gun, which has the beneﬁt of pre-engineered performance. However, these systems are typically quite expensive,

with lower end models costing approximately $ 15 k [37]. Another interesting research topic would be a study of the thermal eﬀects of a long- term operating transformer. As was shown in section 4.1.6, the electrical eﬃciency

of the HVPT with the electron gun at the output had a maximum value of 2.35%. Perhaps some of the non-utilized energy of the HVPT is dissipating as heat in the

79 material, which could result in changes in performance of the transformer via the pyroelectric effect. One option would be to use a FLIR thermal camera to image the transformer and provide real-time measurements of its spatial temperature evolution. This temperature information could then be correlated with laser-optical diagnostic data as well as changes in applied voltage to the transformer and applied electron beam current at the output. The optical diagnostic is another area that could benefit from additional engineering design. The method presented in this document had a minimum temporal resolution of approximately 15 minutes, which is an improvement over previous methods but lacks the ability to resolve fine detail in the earliest, perhaps more interesting, 3−5 minutes of on-time. Also, from an operator standpoint, scanning the

PT was rather tedious, requiring manual translation of the laser stand and saving screen capture data at each position. Lastly, The data had to be offloaded from the oscilloscope and processed, preventing real-time analysis of the transformer. While this system was satisfactory from an R&D standpoint, future iterations would benefit greatly from a more streamlined approach. One possible approach to improve the optical diagnostic would be to use an array of light sources and photodetectors along the length of the bar, rather than scanning the diagnostic unit. An attempt was made to this effect, with the intention of using high luminosity LEDs as a replacement for the laser to miniaturize the optical diagnostic.

Unfortunately, this method failed because the optical bandwidth of the LEDs was 18 nm, which was too broad and washed out the signal such that no phase transitions were detectable. The laser however had a line width of only 0.3 nm and thus produced easily distinguishable phase transitions. One possible move forward could be to use laser diodes, which would have the compactness of the LED and the narrower line width of the laser. The data acquisition speed could be greatly improved by moving

80 from the manual recording and processing method to an automated system. Figure 5.1 provides a diagram for the envisioned next-generation test bed.

Figure 5.1: Proposed setup for improved laser diagnostic test bed.

In figure 5.1, and array of laser diodes is positioned along the length of the PT to probe the transformer, using the internal reflection beam path to return the signal to a coupled photodetector. Each output from the photodetector is then collected on a single channel of a sampling oscilloscope which sends the data to a PC using a GPIB or ethernet connection. A LabView virtual instrument (VI) then processes the time-domain signals using the methods detailed in Appendix C and displays the voltage, electric field, and stress waveforms of the PT in real time, allowing for rapid testing and diagnosis. This method would alleviate the operator ‘hassle’ and improve temporal resolution of the measurements. It would also provide the building blocks for an eventual microprocessor-based implementation used in the compact commercial x-ray product.

81 Appendix A

Electron Source Calibration Data

This appendix contains the complete electron gun calibration data set for 12 tested settings. Each ﬁgure uses the following layout:

Left input current vs. applied grid voltage

Center output current vs. applied grid voltage

Right output current vs. input current

×10-5 ×10-7 ×10-7 4.7 2.5 2.5

4.6

4.5 2 2

4.4

4.3 1.5 1.5

4.2

4.1 1 1 Input Current (A) Output Current (A) Output Current (A)

3.9 0.5 0.5

3.8

3.7 0 0 -10 0 10 20 30 40 50 -10 0 10 20 30 40 50 3.6 3.8 4 4.2 4.4 4.6 Grid Voltage (V) Grid Voltage (V) Input Current (A) ×10-5 Figure A.1: Filament=2.05 A, Optics=5 kV.

82 ×10-5 ×10-7 ×10-7 6.7 2.5 2.5

6.6 2 2

6.5

1.5 1.5 6.4

6.3 1 1 Input Current (A) Output Current (A) Output Current (A)

6.2

0.5 0.5 6.1

6 0 0 -10 0 10 20 30 40 50 -10 0 10 20 30 40 50 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 Grid Voltage (V) Grid Voltage (V) Input Current (A) ×10-5

Figure A.2: Filament=2.05 A, Optics=8 kV.

×10-5 ×10-7 ×10-7 8.2 2.5 2.5

8.1

2 2 8

7.9 1.5 1.5

7.8

1 1 Input Current (A)

7.7 Output Current (A) Output Current (A)

7.6 0.5 0.5

7.5

7.4 0 0 -10 0 10 20 30 40 50 -10 0 10 20 30 40 50 7.5 7.6 7.7 7.8 7.9 8 8.1 8.2 Grid Voltage (V) Grid Voltage (V) Input Current (A) ×10-5

Figure A.3: Filament=2.05 A, Optics=10 kV.

×10-5 ×10-7 ×10-7 9.65 3.5 3.5

9.6

3 3 9.55

9.5 2.5 2.5

9.45

9.4 2 2

9.35 Input Current (A) Output Current (A) Output Current (A) 1.5 1.5 9.3

9.25 1 1

9.2

9.15 0.5 0.5 -10 0 10 20 30 40 50 -10 0 10 20 30 40 50 9.2 9.25 9.3 9.35 9.4 9.45 9.5 9.55 9.6 Grid Voltage (V) Grid Voltage (V) Input Current (A) ×10-5

Figure A.4: Filament=2.05 A, Optics=12 kV.

83 ×10-5 ×10-7 ×10-7 5 2.5 2.5

4.8 2 2

4.6

1.5 1.5 4.4

4.2 1 1 Input Current (A) Output Current (A) Output Current (A)

0.5 0.5 3.8

3.6 0 0 -10 0 10 20 30 40 50 -10 0 10 20 30 40 50 3.6 3.8 4 4.2 4.4 4.6 4.8 5 Grid Voltage (V) Grid Voltage (V) Input Current (A) ×10-5

Figure A.5: Filament=2.25 A, Optics=5 kV.

×10-5 ×10-7 ×10-7 7 3.5 3.5

6.9 3 3 6.8

6.7 2.5 2.5

6.6 2 2

6.5

1.5 1.5 6.4 Input Current (A) Output Current (A) Output Current (A)

6.3 1 1

6.2 0.5 0.5 6.1

6 0 0 -10 0 10 20 30 40 50 -10 0 10 20 30 40 50 6 6.2 6.4 6.6 6.8 7 Grid Voltage (V) Grid Voltage (V) Input Current (A) ×10-5

Figure A.6: Filament=2.25 A, Optics=8 kV.

×10-5 ×10-7 ×10-7 8.5 4.5 4.5

8.4 4 4

8.3 3.5 3.5

8.2 3 3

8.1 2.5 2.5 8 Input Current (A)

Output Current (A) 2 Output Current (A) 2 7.9

1.5 1.5 7.8

7.7 1 1

7.6 0.5 0.5 -10 0 10 20 30 40 50 -10 0 10 20 30 40 50 7.6 7.7 7.8 7.9 8 8.1 8.2 8.3 8.4 Grid Voltage (V) Grid Voltage (V) Input Current (A) ×10-5

Figure A.7: Filament=2.25 A, Optics=10 kV.

84 ×10-5 ×10-7 ×10-7 9.9 4 4

9.8 3.5 3.5

9.7 3 3

9.6 2.5 2.5

9.5 2 2 Input Current (A) Output Current (A) Output Current (A)

9.4 1.5 1.5

9.3 1 1

9.2 0.5 0.5 -10 0 10 20 30 40 50 -10 0 10 20 30 40 50 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 Grid Voltage (V) Grid Voltage (V) Input Current (A) ×10-5

Figure A.8: Filament=2.25 A, Optics=12 kV.

×10-5 ×10-7 ×10-7 5.2 3 3

5 2.5 2.5

4.8

2 2 4.6

4.4 1.5 1.5 Input Current (A)

4.2 Output Current (A) Output Current (A) 1 1

0.5 0.5 3.8

3.6 0 0 -10 0 10 20 30 40 50 -10 0 10 20 30 40 50 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 Grid Voltage (V) Grid Voltage (V) Input Current (A) ×10-5

Figure A.9: Filament=2.35 A, Optics=5 kV.

×10-5 ×10-7 ×10-7 7.2 5 5

4.5 4.5 7 4 4

6.8 3.5 3.5

3 3 6.6

2.5 2.5

6.4 2 2 Input Current (A) Output Current (A) Output Current (A)

6.2 1.5 1.5

1 1 6 0.5 0.5

5.8 0 0 -10 0 10 20 30 40 50 -10 0 10 20 30 40 50 6 6.2 6.4 6.6 6.8 7 7.2 Grid Voltage (V) Grid Voltage (V) Input Current (A) ×10-5

Figure A.10: Filament=2.35 A, Optics=8 kV.

85 ×10-5 ×10-7 ×10-7 8.6 6 6

8.4 5 5

8.2 4 4

8 3 3 Input Current (A) Output Current (A) Output Current (A) 7.8 2 2

7.6 1 1

7.4 0 0 -10 0 10 20 30 40 50 -10 0 10 20 30 40 50 7.5 8 8.5 Grid Voltage (V) Grid Voltage (V) Input Current (A) ×10-5 Figure A.11: Filament=2.35 A, Optics=10 kV.

×10-5 ×10-7 ×10-7 10.2 5 5

4.5 4.5 10 4 4

9.8 3.5 3.5

3 3 9.6 2.5 2.5 Input Current (A) Output Current (A) Output Current (A) 9.4 2 2

1.5 1.5 9.2 1 1

9 0.5 0.5 -10 0 10 20 30 40 50 -10 0 10 20 30 40 50 9.2 9.4 9.6 9.8 10 10.2 Grid Voltage (V) Grid Voltage (V) Input Current (A) ×10-5

Figure A.12: Filament=2.35 A, Optics=12 kV.

86 Appendix B

Vacuum Tube Theory

The physical design of a vacuum tube electron source is relatively simple, consisting of a cathode, an anode, and an evacuated region separating the two. While more complex designs exist, all vacuum tubes are based on this common geometry. However, while the structure of the vacuum tube is simple, the physics that govern its behavior are substantially more complex. A rigorous discussion and derivation of the following physical phenomena can be found in a number of sources [38–44], however, for brevity and focus, this section will discuss only the most relevant physics necessary to support the experimental results in section 2.3.

B.1 Richardson’s Law

Before a vacuum tube can begin transporting charge, electrons must first be injected into the system. This is typically achieved using thermionic emission from a metallic cathode. To exit the material, an electron must first gain sufficient energy to overcome a potential barrier within the metal. This barrier is the result of image charge which acts to offset and negate the presence of an electric field within the conductor. The height of the barrier is known as the work function and varies among materials, however it is typically on the order of a few electron-volts. Electrons may escape the barrier through a variety of mechanisms, including intense external electric

87 ﬁelds (ﬁeld emission), bombardment by high energy photons (photoelectric emission) or other elections (secondary electron emission), or exposure to high temperatures

(thermionic emission). Equation B.1, known as Richardson’s Law, describes the electron current ejected into vacuum by a material due to the presence of high temperature.

I = AAT 2e−W/kT [A] (B.1)

In equation B.1, I is the current which is ‘evaporated’ into the vacuum, A is the area of the emission source in m2, T is the temperature in Kelvin, W is the work function in eV, k is Boltzmann’s constant (8.617×10−5 eV·K−1), and A(script) is the Richardson constant in A·m−2·K−2, which varies depending on the material. Table B.1 shows the work function and Richardson constants for some common materials used in thermionic emitters [42].

Table B.1: Work functions and Richardson constants for common thermionic emitter materials.

Richardson’s Work Material Contant (×104) Function (eV) Cu 65 3.9 W 60 4.5 Ta 55 4.2 W(Th) 4 2.7 BaO 0.6−60 1.4 SrO 100 2.2

B.2 Temperature Determination

The hot ﬁlament is a common device used to reach the temperatures necessary for thermionic emission in vacuum tubes. However, it is often impractical to directly measure the temperature of the cathode because of its small size and frailty, as well as the extreme heat it produces. However, temperature is a critical parameter in

88 Richardson’s law and must be known to calculate the thermionic emission current. One method to determine the temperature of a tungsten cathode is presented in [44]

and is shown in equations B.2 and B.3.

T = 117 + 56a0.5 + 0.00036a1.8 [K] (B.2)

a = I/d1.5 [A/cm1.5] (B.3)

In these equations, T is the temperature of the filament in K, and a is a parameter equal to the filament current I divided by the filament diameter d raised to the 3/2 power. This relationship has been found to be valid over a range of temperatures between 400 and 3000 K. The diameter of the filament used in the experiments in this study was 140 µm and the filament current typically ranged between 1.8 to 2.4 A. Figure B.1 shows the calculated filament temperature as a function of filament current.

This relationship between ﬁlament current and temperature along with Richardson’s law, provides a method to determine the thermionic emission current for an applied ﬁlament current.

B.3 Child’s Law

Once an electron has been ejected from the cathode into the space separating it from the anode, the positive voltage gradient accelerates the electron across the gap and to the anode. When considering only a single electron, this behavior is fairly simple, governed exclusively by the Lorentz force F = qE. However for a device operating in the µA regime, the number of electrons emitted from the cathode is on the order of 1013 per second, producing a dense cloud of electrons located very close to the surface of the cathode. The electric ﬁelds generated by this cloud, or virtual

89 2500

2000

1500

1000

500 Filament Temperature (K)

0 0 0.5 1 1.5 2 2.5 Filament Current (A) Figure B.1: Temperature vs applied current for a 140µm-diameter tungsten filament. cathode, is sufficient to limit the current which can be transported from the cathode to the anode. Child’s Law, shown in equation B.4, provides a method for determining this space charge limited current for a given vacuum tube configuration.

V 3/2 I = AK a [A] (B.4) x2

4ε0 −6 −3/2 K = p m ≈ 2.331 × 10 [AV ] (B.5) 9 2e As shown in equation B.4, the space charge limited current I depends on the

2 cross-sectional area A of the region in m , the anode voltage Va in volts the anode- cathode separation distance x in meters, and a constant K, shown in equation B.5.

The space charge limited current increases with V to the 3/2 power, thus higher anode voltages are capable of permitting larger currents for a given separation distance. With Richardson’s law and Child’s law, the current generated by the thermionic emitter and the space charge limited current imposed by the vacuum tube geometry can both be determined. The interaction between these two phenomena is shown 90 graphically in ﬁgure B.2

Richardson Child 140 140

120 120 2.15

A) 100 100 A) µ µ

80 2.10 80

Richardson 60 2.05 60 plateau Filament 40 Currents (A) 2.00 40 Anode Current ( Anode Current ( 1.95 20 1.90 20 1.85 1.80 0 0 1.6 1.8 2 2.2 2.4 0123456 Filament Current (A) Anode Voltage (kV)

Figure B.2: Left: thermionic current produced by a 140 µm-diameter tungsten ﬁlament over a range of applied currents. Right: Space charge limited current (blue) for a 60 mm-long vacuum tube. Horizontal lines (red) show the thermionic limited current plateaus for a range of applied ﬁlament currents.

The left-hand graph in figure B.2 shows the Richardson law for a 140 µm diameter tungsten thermionic filament supplied with between 1.6−2.2A. Clearly, only a small increase in filament current results in a dramatic increase in thermionic beam current. The right-hand graph shows the space charge limited current as determined by Child’s law for a 60 mm long vacuum tube supplied with an anode voltage ranging from 0 to 6 kV. The horizontal lines spanning from the space charge curve across the graph correspond to plateaus in actual current which reaches the anode for a given filament current. The transition from the Child law curve to the plateaus corresponds to a phenomenological transition from a space charge limited condition to a thermionic limited condition. For example, consider a vacuum tube operating at low filament current of 1.90 A. If the anode voltage is less than 1.5 kV, then the system is space charge limited because the the thermionic cathode is producing more electron current

91 than the vacuum tube can process. As the voltage increases beyond 1.5 kV, the tube transitions to a thermionic limited state because the tube is able to process more

current than can be supplied by the cathode. As a result, the anode current is constant for all anode voltages greater than 1.5 kV and the only option to increase anode current is to increase the ﬁlament temperature, thus increasing the supply of

available electrons at the cathode. This interaction between thermionic and space charge limited current is a fundamental characteristic of vacuum tube operation.

B.4 Space Charge Eﬀects

As it has been presented to this point, the only parameters capable of increasing the space charge limited current for a given vacuum tube are the anode voltage and the anode-cathode separation distance. However, and perhaps counterintuitively, it

is possible for the temperature of the thermionic to aﬀect the space charge limited current as well. Consider the plot shown in ﬁgure B.3 [38]. Figure B.3 shows the voltage potential as a function of distance from the cathode

(X = 0) to the anode (X = xcp) of an operating vacuum tube. Just to the right of the cathode, the voltage proﬁle dips negatively before rebounding and increasing to the full anode voltage vp at the anode surface. This dip in voltage is due to the accumulation of charge, which forms a virtual cathode. The minimum voltage magnitude Vm and the position of this minimum xm is dependent on many factors. Equations B.6, B.7 and B.8 describe the interdependent relationship between these factors [38].

3/2 1/2 (Vp − Vm) 0.0247T PJ = K 2 1 − 1/2 [A/m] (B.6) (xcp − xm) (Vp − Vm)

92 Figure B.3: Voltage potential as a function of position throughout an anode cathode region when charge is present in the system.

T 3/4 x = 0.0156(1000J)−1/4 [m] (B.7) m 1000

T 1 V = log [V] (B.8) m 5040 P Equations B.6, B.7 and B.8 form a system of equations that can be used to determine the position and magnitude of the virtual cathode, given that the

thermionic temperature T , thermionic emission current density J, anode voltage Vp, and anode-cathode separation xm are known. There are three unknown variables in this system, including Vm, xm, which have already been introduced, and P , which is the fraction of emitted current that actually transmits beyond the virtual cathode and is collected at the anode. In practice, the virtual cathode returns most (>99%) of the current that is emitted from the thermionic emitter back to the hot ﬁlament where

93 it is reabsorbed into the metal. This fraction is diﬃcult to know with certainty and therefore must be determined arithmetically by solving the system of three equations.

Solving this system manually is an arduous task, especially if one is sweeping one or more parameters. However, this task can be greatly simpliﬁed using analytical

software such as Mathematica. Using the FindRoot function, solutions to for the three unknown variables can be quickly determined. Figure B.4 shows the position

xm and voltage magnitude Vm for a 140 µm-diameter tungsten ﬁlament driven with between 1.4 and 2.4 A of current.

3 2

2 1 Vm Voltage (V) Distance (mm)

1 0 1.4 1.6 1.8 2 2.2 2.4 Filament Current (A) Figure B.4: Position and voltage magnitude of the virtual cathode as ﬁlament current is increased.

In ﬁgure B.4, the left and right y-axes correspond to distance and voltage,

respectively. As shown, both xm and Vm increase with filament current. At 2.4 A filament current, the virtual cathode has been pushed almost 3 mm from the cathode surface and has a voltage magnitude of 2 V, which is an appreciable figure. This action of moving the virtual cathode and increasing its negative potential has direct

94 implications when solving Child’s law, which is a function of both the anode-cathode distance and voltage potential. When taking into account the reduced distance x and increased voltage potential Va due to this effect, Child’s law can be solved over a range of filament currents, producing the family of curves shown in figure B.5.

Figure B.5: Space charge limited current as determined with Child’s law and the modiﬁed virtual cathode due to increased ﬁlament current.

The plot in ﬁgure B.5 was determined using the same tungsten ﬁlament used

throughout this analysis and assumed a 20 mm anode cathode separation distance and an anode voltage ranging from 0 to 30 V. As shown previously in ﬁgure B.2, as the anode voltage increases, so does the space charge limited current. However, in

figure B.5, higher filament current shifts the entire space charge limited current curve up, thereby increasing the overall current that the vacuum tube can process [43]. This affect was not observed in figure B.2, where the space charge limited current remained

95 constant, regardless of the filament current. This discrepancy is due to the fact that figure B.5 accounts for the modification of the virtual cathode as determined using equations B.6−B.8. Again, this discussion of Richardson’s law, Child’s law, and space charge effects on the virtual cathode was intended as a supplement to the calibration results of the electron gun in section 2.3. Many of the phenomena observed experimentally are only fully explained with an understanding of the fundamental physics in mind.

96 Appendix C

Laser Diagnostic Data Processing Method

The following three sections include the methods necessary to reproduce the laser diagnostic for measuring an operating HVPT in vacuum. This is primarily presented in the form of functional block diagrams with supporting text and ﬁgures where necessary. A complete derivation of the mathematical theory for this method can be found in [11].

C.1 Data acquisition

The first diagram is shown in figure C.1 and shows the procedure used for operating the diagnostic and collecting raw data. The procedure starts by configuring the system for desired operating conditions, such as input voltage, electron beam current, and chamber pressure, and energizing the HVPT. Often, to account for error, multiple redundant measurements are made. This number S should be determined prior to beginning. Typically, four sets of measurements are sufficient to capture an accurate assessment of the HVPT, however this number can be modified depending on time or error tolerance requirements. The diagnosis begins by energizing the laser, verifying alignment to the detector, and positing the laser spot at the extremity of the HVPT which is 50 mm away from

97 Figure C.1: Process for optical data acquisition.

98 the center of the bar. Phase transitions should be visible on the oscilloscope and recorded using the screen capture functionality. The position of the measurement

should also be recorded along with the screen capture ﬁle name for compilation in post processing. Once the data is recorded, the laser stage should be incremented by a distance ∆y0 toward the center of the bar. The value of ∆y0 may be as coarse

or ﬁne as necessary for the experiment, however 5 mm is typically satisfactory. This should be repeated until the laser has been swept across the output length of the bar. If redundant measurements are required for error analysis, the laser should be placed

back at the output of the PT and again swept down the length of the output. When suﬃcient data has been collected, it is ready for processing.

C.2 Data processing

Before the laser optical data can be converted into the desired field profiles, the number of phase transitions ∆φ must be extracted from the time-domain photodetector output. As was shown in figure 3.9 the phase transitions occur at the maxima and minima of the oscillations (or m, in equation 3.3) of the transmitted laser intensity waveform. One method to determine the phase transitions is to manually count the maxima and minima for a period and multiply by π/2. This method is fairly arduous and quickly becomes infeasible for large data sets. However, this procedure can be greatly simplified using a computing platform such as Matlab. Phase transitions occur when the signal reaches either a maximum or a minimum, or in other words, whenever the first derivative of the slope is equal to zero. However, since the data is discretely quantized, it is unlikely that the solution of the first derivative of the signal is exactly equal to zero at any point. The method must therefore search for zero-crossings rather than points with a zero-value. To accomplish this, the derivative is taken a second time and a subroutine analyzes the sign of the

99 resultant array. Whenever a sign inversion is detected, the subroutine increments the counter and a phase transition is recorded.

For signal with a high signal-to-noise ratio (SNR), the above method would be the end of the processing phase. However, the typical SNR of the photodetector output was fairly low, making zero-crossing discrimination diﬃcult to discern from background. For this reason, the signal had to be smoothed for the phase transitions to be accurately counted. Figure C.2 shows the results of the counting algorithm for a sample of data using diﬀerent smoothing spans.

Four plots are shown in figure C.2, all based on the same sample of raw photodector output, shown as the noisy grey signal. A lowess smoothing method was used to reduce the noise of the raw signal so that zero-crossings could be easily counted. The term ‘lowess’ is derived from the phrase ‘locally weighted scatter plot smooth’, which is natively available in Matlab and produced the best results of any of the smoothing methods provided in the software package. The lowess smooth is primarily defined by a user-specified span of data points, with shorter spans corresponding to a rougher output. The thin black line in each of the plots in figure C.2 is the resultant smoothed data for a selection of lowess spans (noted in the column labeled ‘L’). With low spans, the smoothed data very closely follows the raw data set. As the span increases, the smoothed data deviates farther from the raw trace. The second column labeled ‘Z’ is the number of detected zero-crossings of the signal for each of the lowess spans, which are shown graphically as filled circles on the plot. There are 11 transitions total for this sample of data, however the lowest and the highest span generate incorrect results. In the 75 sample span, the smoothing function has not completely filtered the noise from the signal, and therefore erroneously categorizes random fluctuations as a true count. Alternatively, when the sample span is set to 1000, the smoothed output

100 Figure C.2: Comparison of 4 lowess spans for data smoothing.

101 104

103

102 Counted Zero Crossings

101 0 200 400 600 800 1000 Lowess Span

Figure C.3: Process for optical data acquisition. deviates so substantially from the raw signal that the counting function completely disregards true zero-crossings. Using the middle sample spans, the counting function determines the correct number of zero crossings. In the 600 sample case, it is evident that the smoothed output deviates slightly from the raw data, however not so much as to miss a zero-crossing. Figure C.3 shows the number of detected zero-crossings for 40 µs of data for a range of lowess spans. Figure C.3 shows that for a lowess span between 200 and 700 samples, the detected zero crossing equals 33, which was the correct value. However as the lowess span decreases past 200, noise is falsely counted as true signal and the count dramatically increases to several thousand. On the right of the plot, the count decreases, due to the deviation from the true signal mean, as was shown in the bottom plot of ﬁgure C.2. This curve is typical of most, but not all optical measurements. Often, the long plateau of correct zero-crossing values is much shorter or the knee on the left side of the curve occurs at lower lowess spans. These variations make it diﬃcult to automate the smoothing algorithm without visually verifying that the smoothed

102 waveform matches the raw data signal. However this method is still dramatically faster than counting phase transitions manually. Figure C.4 shows the functional

block diagram for processing the optical data. The data consists of digital ﬁles containing the photodetector waveforms and a logbook of the corresponding measurement locations, which must be copied to a PC

for processing. Once loaded to the processing PC, the data is sorted by position, smoothed, and veriﬁed to ﬁt the raw data trace. If the counting algorithm produces false high counts or the smoothed data deviates from the raw signal as was shown

in the top and bottom plots in ﬁgure C.2, the lowess span should be adjusted. Once the smoothed data ﬁts the raw signal, the number of phase transitions is recorded for that position. This process is repeated for each position until the entire length of the

HVPT has a corresponding number of phase transitions, ∆φ(y0). The ﬁnal step is to analyze the phase transition data using equations 3.3−3.7.

C.3 Data analysis

The data processing method reduces the series of time-domain voltage waveforms from the photodetector and position information associated with each individual

waveform into a two column array of phase transitions vs. longitudinal position, or ∆φ(y0). In this form, the data is ready to be converted into stress, electric ﬁeld, and voltage information. Figure C.5 shows the block diagram for this data analysis. Equations 3.3−3.7 are repeated again here for convenience, corresponding to equations C.1−C.5, respectively.

π ∆φ = m (C.1) 2

λ(∆φ) E (y0 = 50) = (C.2) 2 πLa

103 Figure C.4: Functional block diagram for optical data processing method.

104 C.2

C.3

C.4

C.3

C.5

Figure C.5: Process for optical data analysis.

105 0 0 0 0 D2 = ε22E2(y ) + d22T2(y ) (C.3)

0 0 πD2La − ε22λ(∆φ) T2(y ) = 0 0 (C.4) πL(d22a − ε22b)

Z k 0 0 V (k) = − E2(y )dy (C.5) 0

It is important to note that the photoeleastic and electrooptic eﬀects are superimposed at every position throughout the material. It is possible to decouple the two eﬀects by measuring the optical response at the output of the transformer

(y0=50 mm), which has a stress equal to zero. Any optical response measured at the extremity must therefore be due exclusively to the electrooptic eﬀect, which is

shown in equation C.2. Then, using equation C.3, the electric displacement ﬁeld D2

can be computed at the extremity. However, D2 must be constant throughout the

0 material. With this in mind, equation C.4 can be used to determine the stress T2(y )

0 for any longitudinal position since it depends directly on D2. Once the stress T2(y )

0 is known, the electric ﬁeld for all positions E2(y ) can be determined by again using

0 equation C.3 with the calculated values of D2 and T2(y ). Finally, the voltage for all points V(y0) can be determined using equation C.5, where k is an indexing variable

which increments the upper limit of integration from 0 mm< k <50 mm. Figure C.6

0 0 0 shows an example of the stress T2(y ), electric ﬁeld E2(y ), and voltage V(y ) for a given change in phase transitions ∆φ(y0).

The top plot in ﬁgure C.6 are the measured changes in phase transitions for a mode 2 HVPT determined through the data analysis method described in

106 30

10 Phase Transitions 0 0 10 20(a) 30 40 50 Distance (mm) 15

5 Stress (MPa)

0 0 10 20(b) 30 40 50 Distance (mm) 6

Electric Field0 (kV/cm) 0 10 20(c) 30 40 50 Position (mm) 20

Voltage (kV) 5

0 0 10 20 30 40 50 Distance (mm) (d)

Figure C.6: Example of measured data showing the change in phase transitions for an HVPT using the optical diagnostic and the corresponding electrical, and mechanical responses. 107 appendix C.2. The circles correspond to the discrete measurement points and the connecting lines are drawn for visual aid. The number of counted transitions follows a cosine and reaches a maximum of 27 transitions per quarter-period at the 25 mm position. The stress and electric field profiles closely resemble the phase transition curve, with the notable difference that at the extremity of the bar (50 mm) the stress is zero, whereas the electric field is non-zero. Finally, the voltage profile is the integral of the electric field, starting at zero, increasing rapidly in the middle of the bar when the peak of the electric field is high, and leveling off when the electric field returns to zero. In summary, the laser optical diagnostic method is an extremely useful tool for probing the internal properties of an operating HVPT. First, unlike the voltage divider or other more conventional diagnostics, it does not influence the operation of the transformer, allowing for the most accurate assessment of the performance of the device. Second, this method provides a measure of the electrical and mechanical effects, allowing for a much deeper analysis of the transformer than can be provided by any other diagnostic technique.

108 Appendix D

PTPS Neutron Generator Work

D.1 PTPS

The PTPS is a compact RF-driven plasma source which uses an integrated piezoelectric transformer to achieve high voltage [19]. The PTPS was used as an electron and ion source in the piezoelectric x-ray and neutron generator system. The design of the PTPS transformer is shown in Fig. D.1a. The transformer is a disk with 5 mm radius and 2 mm thickness. Electrical energy was delivered through the thickness of the crystals via input electrodes on the top and bottom surfaces of the disk. The bottom electrode (not shown) spanned the entire area of the ﬂat surface of the disk and was energized with RF voltage. The top electrode was annular and

ﬁxed at ground potential. The high voltage developed in the center portion of the top of the crystal. The disk was situated within a diﬀerentially pumped chamber to comprise the PTPS system, shown in Fig. D.1b.

109 Ion Beam Output 5 mm Ground 1 mm Aperture Plate

Plasma D2 Flow 2 mm Ground Radial PT Electrode Input Energized Electrode Electrode (a) Radial LNO transformer dimensions (b) PTPS cross-sectional view and electrode pattern

Figure D.1: The Piezoelectric Transformer Plasma Source (PTPS).

To drive the transformer, an RF voltage was applied to the input electrodes in a similar manner to the HVPT. The frequency of the RF was matched to the fundamental mechanical frequency of the radial transformer, typically between 385–

415 kHz [19]. Deuterium was ﬂowed into the PTPS at rates between 5−10 sccm, producing an internal pressure of approximately 1 torr [20]. Ionization occurred within the chamber at the high voltage surface of the transformer and a 600 µm- diameter aperture was used for ion extraction.

D.2 Coupled HVPT-PTPS Neutron Conﬁguration

The HVPT and PTPS were coupled together to comprise the HVPT-PTPS neutron source, which is shown in Fig. D.2. All experiments were conducted in vacuum at pressures between 3 – 7×10−4 Torr. The gap distance between the PTPS and the HVPT was 25 mm and an electronic linear stage was used for beam-target alignment in the transverse direction. The target material was titanium deuteride

(TiD2) and was adhered to the output of the HVPT using silver paint. Due to the sinusoidal bipolar output of the HVPT, the composition of the beam was time-variable and phase-locked with the HVPT, with electrons extracted during the positive half- cycle and ions during the negative. 110 Vacuum chamber Al h window HVPT TiD2 xray det. n Linear stage e–, i+ beam PTPS neutron det.

D2 flow

Figure D.2: Coupled HVPT-PTPS setup for neutron production.

Electrons, while not directly useful in producing neutron radiation, did produce bremsstrahlung x-rays, a convenient by-product that allowed for real-time monitoring of the HVPT output voltage. To measure the x-rays, a CdTe x-ray detector was placed outside of the vacuum chamber at a distance of 10 cm from the source. A 50 µm-thick aluminum window provided low-attenuation line-of-sight from the source to the x-ray detector. Neutrons were produced during the negative half-cycle, when positive ions were accelerated to the HVPT to fuse with deuterium nuclei in the TiD2 target. For visual simplicity, a beam-like proﬁle for the neutron radiation is shown in Fig. D.2, however in practice, the proﬁle was assumed to be near-isotropic [45]. The neutron detector was located external to the vacuum chamber at a distance of 10 cm from the source. A system diagram of the coupled HVPT-PTPS neutron source is shown in

Fig. D.3. Two individual drive circuits were required, with each piezoelectric transformer having its own circuit. Agilent 33210A function generators with internal modulators were used to drive the PTs. On the HVPT line, the function generator produced 60 kHz waveforms which were ampliﬁed by an Ampliﬁer Research 25A250A

25 W, 43 dB RF power ampliﬁer to drive the HVPT at 8.5 Vrms. A Pearson 2877 current monitor with 1 V/A output sensitivity measured the input current to the

111 HVPT. On the PTPS line, another function generator produced 385 kHz waveforms which were amplified by another AR 25A250A RF amplifier. A 1:16 step-up matching transformer further amplified the voltage to drive the PTPS at 400 V. A Tektronix TDS 2024B oscilloscope was used to measure HVPT and PTPS input voltages and currents.

Function HVPT Line Gen. 1 Vacuum Chamber Carrier 1 Current Diagnostics RF Amp 1 Monitor 1 Modulator 1 X-ray HVPT DPP Detector Function Neutron Gen. 2 PTPS Line PTPS PC Carrier 2 Detector Current RF Amp 2 Monitor 2 Modulator 2 Step-up O-scope Xfrm

Figure D.3: Systems diagram of the coupled HVPT-PTPS and support hardware.

D.3 Diagnostics

Neutron counting was conducted with an RDT Domino, which uses the 6Li(n, t)4He reaction to capture and detect incident neutrons. The Domino has a 4 cm2 detection area, 20% thermal neutron detection eﬃciency, < 3 mW power consumption, and digital TTL output. A photograph of the detector is shown in Fig. D.4. In practice, the detector was comprised of 3 individual Domino sensors combined in series to increase the active area of the system. The detector was coupled to a PC running

PRA spectroscopy software for pulse counting.

112 Figure D.4: Photograph of one RDT solid state neutron detector.

Once assembled, the detector array was placed in an 8 lb stack of paraﬃn

moderator [46]. The detector was then experimentally calibrated using a sample of Californium-252 (252Cf), a material often used for calibrating neutron detectors [47]. Californium-252 has a broadband neutron spectrum with an average energy of

2.1 MeV, making it a suitable stand-in for the 2.45 MeV mono-energetic output of the 2H(d, n)3He reaction. The quantity of 252Cf present in the sample was 5.33×10−2 µg, which produced 1.2 × 105 n/s [48]. The known source activity was compared to the count rate measured by the detector, producing an eﬃciency curve shown in Fig. D.5.

2.0 Paraffin Stack 1.5

1.0 Neutron 0.5 Direction 0.0 Detection Efficiency (%) Detection Efficiency -2 0 2 4 6 8 10 12 14 16 18 Detector Position (cm)

Figure D.5: Detection eﬃciency over a range of positions within an 8 lb stack of paraﬃn moderator.

113 Figure D.6: Photograph of the RDT detector array situated in the optimal position within the paraﬃn stack.

Figure D.5 shows the position of the detector was positioned in 2-cm increments

throughout to the paraffin stack. The count rate at each position was measured and a peak detection efficiency of 1.6 × 10−2 was observed. The optimal position for the detector in the stack of paraffin was not in the center of the stack, but rather 2 cm offset from the center towards the source. A photograph of the detector configured for optimal detection efficiency is shown in Fig. D.6.

D.3.1 Neutron Counting Algorithm

Unlike gamma radiation and counting, neutron background measurements do not follow traditional Poisson distributions [49]. The primary source of neutron background is the spallation of dense materials nearby the detector due to high energy cosmic radiation, often referred to as the ‘ship eﬀect’ [50, 51]. As a result, neutron background rates tend to vary based on a number of factors including time of day, atmospheric pressure, and detector proximity to dense objects.

114 Figure D.7 demonstrates the variability of the neutron background over one hour. The dotted line is the average background counts (10 min)−1 corrected to zero. The measured time-resolved count rate was digitally filtered using a four point running sum to smooth the data [49]. Figure D.7 shows that the background count rate can significantly deviate above and below its own average value by multiple Poisson standard deviations, indicated by the error bars on the figure. The numerous measurements significantly above the mean would typically suggest that a source was present, however the measurements were taken without any active source. Furthermore, the occurrence of measured count rates below background is non-physical and verifies that Poisson statistics are incompatible with neutron background characterization. For these reasons, special considerations must be made to avoid falsely identifying a high background fluctuation as a positive indicator of source activity during an experimental measurement.

-1 30 20 10 0 -10 Rate (10 min) Corrected Count Corrected Count -20 -30 0 10 20 30 40 50 60 Time (min)

Figure D.7: Natural variation in neutron background count rate over one hour.

An algorithm for analyzing neutron background and source count rates is detailed in [49]. The essence of the approach is to deﬁne a false alarm probability (FAP)

threshold, K. Typically, a false alarm probability of 5 % is considered a suﬃcient

115 indicator of a positive treatment eﬀect and is used throughout the analysis of the piezoelectric neutron source. The FAP cannot be directly selected due to the discrete

nature of the Poisson distribution. Rather, an iterative discrete summation, shown in Equation D.1, can be performed to determine a minimum threshold K that sets an upper bound on the FAP.

K−1 X µx 1 − e−µ < FAP (D.1) x! x=0 In Equation D.1, K is the threshold count rate, FAP is the desired false alarm probability, µ is the mean background count rate, and x is a loop counter which monotonically increases from zero as successive iterations of the summation are

performed. The loop is exits once K gets suﬃciently high to reduce the value on the left side of the inequality to less than the FAP. This method of determining K is valid for short-duration measurements where the background is not expected to vary by much. However, measurements of the piezoelectric neutron sources are made over the duration of an hour, which is long enough for a substantial shift in background to occur, as was shown in Figure D.7.

For this reason, the method shown in Equation D.1 must be modiﬁed slightly. The modiﬁcation involves replacing the discrete summation in Equation D.1 with

µx −µ an integral and replacing the Poisson summation term, x! e , with an interpolated analytic distribution function P (x) computed from a histogram of hundreds of hours of background counting. Equation D.2 shows this expression.

Z ∞ FAP(µ, K) = P (x)∂x (D.2) K While the methods for determining the threshold K from Equations D.1 and D.2 are functionally similar, the eﬀect of accounting for the measured background count rate probability distribution rather than a Poisson distribution has a strong eﬀect on

116 the threshold determination and is particularly important for low count rate systems. Figure D.8 shows how these two approaches can change the interpretation of measured data. Thresholds

K0 Km Measured Background n=2343 FAP = 18.5%

Expected Background

FAP

Rel. Probability (arb) FAP = 5% = 5%

0 50 100 150 Counts in 10 Minutes Figure D.8: Distribution of measured background compared to the Poisson distribution (µ = 75 counts/10 min).

Figure D.8 shows two probability distributions. The ﬁrst is the measured background, shown as a solid line. This background is the cumulative histogram of 2,343 10-minute-long counting periods. The second is a Poisson distribution with the same mean value as the measured background, shown as a dashed line. It is evident that the width of the background distribution is much greater than that of the Poisson. The values K0 and Km are the thresholds for a FAP = 0.05 for the

Poisson and background distributions, respectively. If the K0 threshold were to be used to analyze the background, 18.5% of all 10-minute measurements would indicate a count rate above the threshold, erroneously leading the analyst to conclude that a source was present in the system. To avoid this mistake, the threshold Km should be

117 used. As shown in Figure D.8, using a threshold Km = 108.3 counts/10 minutes, a background FAP of 5% is achieved.

In summary, the variability of the neutron background precludes the use of conventional gamma counting methods (i.e. 1.64 standard deviations above background 6= source present). Rather, an iterative algorithm based on an

interpolated long-term accumulation of background measurements was used to numerically arrive at a minimum count rate threshold K = 108.3 counts in a ten minute period. Any measurement above this value can be said to have at most a 5%

chance of being a false positive.

D.4 Modeling

A model was developed to estimate the total neutron yield from the HVPT-

PTPS neutron source. The model was based on the thick target neutron yield

equation [52], the 2H(d, n)3He (D–D) excitation function, Trim simulation results, and time-harmonic analysis. The neutron yield Y (n·s−1) of a thin target fusion system can be generally described as a three-variable function, shown in Equation D.3.

Y = nσφ (D.3)

The value n is the number of target nuclei (cm−2), σ is the reaction cross section

(cm2), and φ is the incident particle rate (s−1). For a target to be considered ‘thin’, its thickness must be small enough to not change the properties of the beam, such as by decreasing the energy or number of ions. Most real-world targets however are

‘thick’ because they absorb and scatter the beam, introducing variability in the x direction. This is variability is accounted for in diﬀerential form in Equation D.4.

dY = Nσφ dx (D.4)

118 The value N is the number density of the target (cm−3). Equation D.4 can be expressed in terms of energy and reduced by adding a dE term to the right side of

the equation, shown in Equation D.5.

dE −1 dY = Nσφ dE (D.5) dx

dE The term dx is the stopping power of the material and describes the energy loss of the incident particles as they travel through the target. Finally, the total neutron yield (s−1) for a thick target fusion system is found by integrating Equation D.5, shown in Equation D.6.

dE −1 Z Emax Y = Nφ σ(E)dE (D.6) dx 0 Equation D.6 shows that neutron yield for thick targets is dependent on the ion current (φ), the target doping concentration N, the stopping power of the material

dE dx , and the excitation function σ(E). The stopping power is taken as a constant (demonstrated in Section D.4.2) and is therefore evaluated outside of the integral.

D.4.1 Excitation Function

The excitation function σ(E) is the dependence of the cross section on the

incident deuteron beam. This function can be numerically approximated and is shown graphically in Fig. D.9 [53].

119 10-1 10-2 10-3 10-4

-5

Cross Section (barns) 10 20 30 40 60 100 Deuteron Energy (keV)

Figure D.9: Excitation function for the 2H(d, n)3He reaction.

Figure D.9 shows that the cross section for the D–D reaction has a strong, non- linear dependence on the energy of the incident deuteron. Using this numerical approximation of the excitation function, the R σ(E)dE term from Equation D.6 can be readily calculated with trapezoidal numerical integration using analytical software.

D.4.2 Trim Ion Beam Modeling

Trim (Transport of Ions in Matter) is a a Monte-Carlo simulation suite and has been used to compute the behavior of the ion beam as it passes through the target.

Two studies were conducted: ion beam transmission and stopping power.

2 + The model assumed a beam composed of diatomic deuterium ions (1H2 ), accelerated in discrete increments from 20 keV to 120 keV. The target was modeled as a three-layer system [54]. The ﬁrst layer was a contaminant layer, composed of hydrocarbons, carbon monoxide, and molecular nitrogen with a thickness of 13 nm.

The second layer was a titanium dioxide (TiO2) layer with a thickness of 37 nm. The last layer was the titanium deuteride (TiD2) bulk layer in which nuclear fusions can occur (deﬁned in SRIM as a 1:2 ratio of titanium to deuterium atoms). Figure D.10 shows the penetration of a deuterium ion beam into a three-layer titanium target for

120 a selection of six initial ion beam energies as computed by Trim modeling.

1.0 Accelerating Potential (kV) 120 0.8

80 0.6 100 60 0.4 40

0.2 20 Transmission Ratio Transmission

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Penetration Depth (µm)

Figure D.10: Ion transmission ratio into a TiD2 target.

Figure D.10 shows that the ions penetrated the contaminant layers with negligible losses. Once inside the bulk TiD2 layer, the transmission ratio for each accelerating potential is approximately 100% until it eventually drops sharply to zero. This behavior approximates a step function and the transmission ratio is therefore simply taken as unity. Next, the stopping power of the target was calculated, shown in Figure D.11.

121 120

Accelerating 100 Potentials

80 120 kV 100 kV 60 80 kV

40 60 kV 40 kV 20 20 kV

Average Ion Energy (keV) Ion Energy Average 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Penetration Depth (µm)

Figure D.11: Ion beam stopping power of a TiD2 target.

Figure D.11 shows that the ion energy E(x) has a nearly ﬁrst-order response

dE −1 to penetration depth. The slope dx is the stopping power, equal to 120 keV·µm . With the stopping power and excitation function computed, Equation D.6 is satisﬁed, however additional analysis is required to account for the AC voltage output of the HVPT accelerator.

D.4.3 Time-harmonic Analysis

The derivation of the thick-target model for neutron yield assumed a time- invariant acceleration potential, thus the excitation function was simply dependent on distance x. Since the HVPT produces AC output, this model must be adapted to include time-variability. This variation is known to be sinusoidal, so time-harmonic analysis is used and carried out over one negative half-cycle of HVPT operation. For convenience Equation D.3 is reintroduced as D.7, the general form of the thin target neutron yield.

Y = nσφ (D.7)

122 For thick targets and time-variable acceleration potentials, the problem becomes two-dimensional with one spatial dimension and one temporal. This is shown in

diﬀerential form in Equation D.8.

d2Y = Nσφ dx dt (D.8)

In a similar manner to the steady-state case derivation, Equation D.8 can be

reduced and put in terms of deuteron energy E and phase angle θ. This is shown in Equation D.9.

dE −1 dθ−1 d2Y = Nσφ dE dθ (D.9) dx dt

dθ rad The term dt is now introduced and is the angular frequency in s of the deuteron energy. Taking the double integral of both sides of Equation D.9 results in the expresssion for neutron yield for a thick target time-harmonic source, shown in

Equation D.10.

−1 −1 π E dE dθ Z 2 Z max Y = 2 Nφ σ(E, θ) dE dθ (D.10) dx dt 0 0 The double integral in Equation D.10 is again computed using analytical software.

π The factor of 2 is due the even function σ(θ) integrated from 0 to 2 . The angular dθ frequency dt is a constant and therefore evaluated outside of the integral. Unit analysis of Equation D.10 will show that the value Y is dimensionless, as compared to

Equation D.6 where its unit is s−1. This is due to the time-harmonic approach where only a single half-cycle of operation was considered. To express the time-harmonic Y

in s−1, it must be multiplied by the of cycles per second of the operating HVPT, which can vary among experiments. The remaining variables including angular frequency, stopping power, target doping concentration, and ion beam current are all known,

123 estimated, or computed and the neutron yield of the HVPT-PTPS neutron source may be calculated.

D.4.4 Calculated Neutron Yield

Using the presented model, the neutron yield for the piezoelectric neutron sources has been calculated. Table D.1 shows a list of the constants used in the calculation.

Table D.1: Constants for calculating yield of piezoelectric neutron sources.

Parameter Variable Value Unit Ion rate φ 6.25 E+10 s−1 Target doping N 1.12 E+23 cm−3 −1 Stopping power dE/dx 1.20 E+06 keV·cm −1 Angular freq. dθ/dt 3.83 E+05 rad·s Cycles/sec. f 6.10 E+04 s−1 Duty factor D 5.20 E+00 %

In Table D.1, ion rate φ was calculated from a conservative assumption of 0.01 µA of ion current from the PTPS. The target doping concentration N was determined from manufacturer speciﬁcations. The HVPT was assumed to be operating in FM mode with f=61,000 cycles per second. The duty factor D of the system was limited in both cases by the PTPS on-time, which was ﬁxed at 5.2 %. The solutions for the integrals in the right hand side of Equations D.6 and D.10 are shown in Table D.2 for a selection of relevant accelerating potentials. Figure D.12 combines the data from Tables D.1 and D.2 to show the calculated neutron yield for the HVPT-driven neutron source as well as the ideal case of a DC accelerator for comparison.

124 Table D.2: Excitation function integral solutions for the steady-state (DC) and time- variable (AC) conditions.

Accelerating Ex. Fun. Integral Solns. Potential [kV] DC [barn·keV] AC [barn·keV·rad] 20 4.72E-05 8.26E-05 40 3.80E-03 6.30E-03 60 2.60E-02 4.1E-02 80 8.12E-02 1.23E-01 100 1.75E-01 2.57E-01 120 3.08E-01 4.35E-01

100 HVPT 80 DC

Source Rate (n/s) 20

0 20 40 60 80 100 120 Accelerating Potential (kV) Figure D.12: Calculated yield for the HVPT-driven neutron source compared to a DC neutron source.

As shown in Fig. D.12, both the AC and DC case accelerators show a non-linear dependence on accelerating potential. The ideal case of a DC-driven source reaches a yield of 93 s−1 at 120 kV compared to 42 s−1 for the HVPT-driven source. This discrepancy is largely due to the fact that ions are only accelerated during the negative half-cycle of the HVPT on-time. The values in Fig. D.12 are linearly proportional to the constants in Table D.1, therefore scalar multiplication can be used to ﬁnd the yield of these systems with diﬀerent initial assumptions.

125 D.5 Results

Figure D.13 shows the measured neutron results from 19 individual 1-hour experiments with the HVPT-PTPS source.

140

120 5% FAP Theshold 100

80 Mean Background

60 Counts in 10 Minutes 40

20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Experiment Reference Number

Figure D.13: Box and whisker plot for 19, 1-hour-long experiments with HVPT-PTPS neutron source.

In figure D.13, the experiments are presented in a box and whisker format with each experiment corresponding to a single box plot. The line in the center of the boxes is the median count rate for that experiment and the values bounded by the box correspond to the 25–75 percentiles. The mean background of approximately 78 counts/10 minutes is shown as a solid line across the figure and the 5% FAP threshold is shown as a dashed line at 108 counts/10 minutes. Each box plot has a sample size n = 26 10-minute measurements accumulated using a 5-point running sum digital filter. For a particular experiment to demonstrate that counts above background were measured, the median line of the box plot should be at or above the FAP threshold. As figure D.13 shows, while some experiments do have data points above this threshold, they are in the minority, either in the upper quartile of the population or outliers, and do not reflect the presence of a statistically

126 signiﬁcant treatment eﬀect. An investigation of the HVPT operating voltage was conducted to address the null results of the HVPT-PTPS neutron source. It was found that the presence of the

171 mg TiD2 target aﬃxed to the end of the HVPT substantially reduced the output voltage of the HVPT. The spectra shown in ﬁgure D.14 show the x-ray output of the

HVPT-PTPS source both with and without the titanium target.

1 HVPT without target HVPT with target ray CPS

X 0.1

0.01 0 10 20 30 40 50 60 Xray Energy (keV)

Figure D.14: X-ray spectra comparison of the HVPT-PTPS with and without target.

The results in ﬁgure D.14 suggest that a mass-loading eﬀect occurred at the output that inhibited high voltage and is likely the primary reason why no neutron production from the HVPT-PTPS was observed. It is likely that to produce measurable counts above background, a system must be implemented which removes the mass from the HVPT output.

127 Appendix E

Laser Diagnostic Periscope Analysis

In addition to the internal reflection beam path discussed in section 2.2 an alternative path was also implemented and is shown in figure E.1. In this path mode, two mirrors were placed behind the HVPT at 45◦ angles to the incident beam vector. The bottom mirror transports the beam to the top mirror, which passes the beam out of the chamber via the viewport and to the detector assembly. The second beam path is the internal reflection mode, shown in figure 2.5. The periscope was made with four ThorLabs 1×1 inch Square Broadband Dielectric Mirrors with a range between

400−750 nm held in place with custom designed 3D-printed mounting platforms. Two mirrors were ﬁxed side-by-side in both the top and bottom mirror mounts to provide 2 inches (or roughly 50 mm) of scanning travel.

128 Figure E.1: Laser diagnostic in periscope conﬁguration.

As a result of these path geometries, the length L for the laser was 0.01 m for the periscope and 0.02 m for the internal reflection. The expected result of this discrepancy was that the number of changes in phase transitions ∆φ for the internal reflection path should be twice that of the periscope path. Figure E.2 shows the measured number of changes in phase transition for periscope and internal reflection for an HVPT with three applied voltages.

129 Figure E.2: Comparison of phase transitions measured using the periscope and internal reﬂection beam paths.

The HVPT was driven at 3.4, 4.6, and 5.7 Vrms input and ∆φ was measured across the length of the transformer using each beam path. In each case, ∆φ measured with the internal reﬂection path was approximately twice the magnitude of the corresponding periscope path measurement. For convenience, the periscope

measurements were artificially doubled in post processing to compare to the internal reflection. Both the 2× periscope and internal reflection plots are very closely overlapped, demonstrating that the ∆φ increased with increasing path length. In

130 practice, the internal reflection mode was substantially easier to operate because it did not rely on multiple independent reflection surfaces to be in alignment. In general, as long as the beam was incident on the HVPT surface, an internally reflected ray was always produced. For this reason, all analysis was conducted with internal reflection.

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136 Vita

Brady Gall was born in San Diego, California on April 20, 1986 and moved to Kansas City, Missouri in 1998. After graduating from Winnetonka H.S. in 2004, he received his B.S. and M.S. in Electrical Engineering from the University of Missouri in May 2009 and August 2012, respectively. His M.S. thesis was entitled “Investigation and Optimization of a High Voltage Piezoelectric Particle Accelerator” and was supported by Los Alamos National Laboratory and Qynergy, Inc. He will graduate with his Ph.D. in Electrical Engineering in May of 2016. Brady has presented his research in ﬁve international conferences and at the 2013 High Energy Density Physics summer school at Ohio State University. As of this writing, he has published two peer-reviewed journal papers, one conference paper, and is coauthor on several others. He worked as a student researcher at the Trident short pulse laser accelerator at Los Alamos National Laboratory for three summers between 2009 and 2011. Brady’s awards include First place presenter at the 31st annual Research and Creative Arts forum in 2014, Outstanding Ph.D. student in ECE in 2013, and a full scholarship to the 2013 HEDP summer school. Brady also received Coulter Translational Seed Funding in 2014 for development of medical x-ray sources driven by piezoelectric transformers. Brady played trombone in Marching Mizzou from 2004 to 2008, traveling to four NCAA football bowl games in this time. Brady’s teaching roles included physics tutor, teaching assistant for intermediate electromagnetics, and teaching fellow for circuit theory. On September 13, 2014 Brady married Jenny Leeper of St. Louis, Missouri. After graduation, Brady will be employed at National Security Technologies, LLC in North Las Vegas, Nevada as a Senior Engineer on the Dense Plasma Focus neutron source.

137