cal tech thesis physics

The BetaCage Prototype:Pulse Simulation and Noise Analysis

Charles Blakemore

Advisor: Sunil Golwala

May 15, 2015

Thesis submitted to

California Institute of Technology

in partial fulfillment for the award of the degree of

Bachelor of Science

Physics

Contents

1 Introduction 6

1.1 CDMS and the Origin of the BetaCage . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.1 WIMPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.2 Detecting WIMPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.3 The Necessity for Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Gaseous Time-Projection Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 The Need for a Pulse Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Status of the BetaCage 12

2.1 MWPCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 The Bulk MWPC and TPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Operating the BetaCage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 ROACH Components and Basic Structure . . . . . . . . . . . . . . . . . . . . 16

2.3.2 Communicating With the ROACH . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.3 Programming the FPGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Firmware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Pulse Theory 19

2

CONTENTS CONTENTS

3.1 Ionization and Track Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 The Electron Avalanche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 The Diethorn Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Current Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Ramos Theorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.2 Applying Ramos Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Electron Trajectories 28

4.1 MWPC Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 COMSOL Field Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Arrival Time Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.1 MAGBOLTZ and Electron Drift . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.2 Computing the Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Signal Propagation 37

5.1 The HV Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.1.1 Voltage Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2 The Readout Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Implementation 44

6.1 Pulses in the ProtoCage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.2 The Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.2.1 Arrival Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.2.2 Avalanche Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2.3 Computing the Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . 51

6.3 The Final Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3

CONTENTS CONTENTS

6.3.1 Captured vs. Simulated Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.4 Viability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.4.1 Signal Shaping Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.4.2 Confirmation Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.4.3 Further Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7 Noise 60

7.1 The HV Network: Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.2 Amplifier Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.3 Measured Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Appendices 68

A COMSOL Simulations 70

A.1 Purpose and Necessity of COMSOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A.2 An MWPC Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.2.1 COMSOL Physics and Model Specifications . . . . . . . . . . . . . . . . . . . 71

A.2.2 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.2.3 Boundary Conditions and the Solution . . . . . . . . . . . . . . . . . . . . . . 73

A.3 Ramo Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A.3.1 COMSOL Physics and Model Specifications . . . . . . . . . . . . . . . . . . . 76

A.3.2 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

A.3.3 Boundary Conditions and the Solution . . . . . . . . . . . . . . . . . . . . . . 78

B Assumption Check 81

B.1 Avalanche Center of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4

CONTENTS CONTENTS

B.2 Ramo Field Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

C Transmission Lines 85

C.1 Transmission Lines and Two-Ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

C.2 A Loaded Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

C.3 HV Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

D Pulse Program 93

5

Chapter 1

Introduction

A number of modern physics experiments are located underground in an attempt to shield detec-

tors from background solar and cosmic radiation and obtain sensitivity to low-energy interactions.

Unfortunately there is a trade-off, as increased radon in underground settings leads to radio con-

tamination from daughter isotopes like Lead-210, which is known for its low-energy beta particle

emission.

Low-energy beta and alpha particles are expected to be the dominant background for many

underground experiments. Examples of such include SuperCDMS [1], XMASS [2], LUX [3] and

newer experiments like LZ [4] and EXO-200 [5]. These experiments, and dark matter searches in

general, are a challenging prospect just from their very nature due to infrequency (small interac-

tion cross-sections) and low-energy signals dominated by background events. Many of the existing

background screening technologies are insufficiently sensitive in the desired energy range [6, 7].

1.1 CDMS and the Origin of the BetaCage

The Cryogenic Dark Matter Search (CDMS) is a collaboration currently working on their third dark

matter detector, SuperCDMS-SNOLAB. Briefly, this experiment primarily aims to detect nuclear

6

CHAPTER 1. INTRODUCTION 1.1. CDMS AND THE ORIGIN OF THE BETACAGE

recoil from WIMP interactions with a detector substrate. Weakly Interacting Massive Particles

(WIMPs) are a candidate dark matter particle, appealing for a multitude of reasons, which we will

briefly discuss in their relation to the BetaCage.

1.1.1 WIMPs

WIMPs nteract with standard model particles only through the weak force and gravity. They are

thermally produced in the early history of the universe when the temperature is sufficiently high.

When the temperature drops and thermal production of WIMPs ceases, self-annihilation continues

until its rate falls below the rate of expansion and WIMPs freeze out. The relic abundance of such

a particle with MWIMP 10 GeV is consistent with a self-annihilation cross-section on the order of

the weak force coupling, motivating a search for WIMPs [8].

Indeed, a particle with the properties described above is predicted by Supersymmetry as well.

Generally, Supersymmetry is the idea that the full Lagrangian of the universe is invariant under

transformations of fermions into bosons and bosons into fermions. However, mismatched quantum

numbers and currently accepted symmetries prevent the known particles from being partnered with

each other and thus new particles are necessary [9, 10].

Grazing over much of the complications inherent to SUSY, we can constrain a naive supersym-

metric model with the experimentally verified lifetime of the proton, > 1031 yrs, to yield a stable

supersymmetric mass eigenstate referred to as the Lightest Supersymmetric Particle (LSP) which

is a superb candidate WIMP [11].

1.1.2 Detecting WIMPs

Although the exact nature of any interaction is model dependent, it is believed that WIMPs are not

entirely non-interacting with the standard model, just that cross-sections are exceptionally small.

7

1.1. CDMS AND THE ORIGIN OF THE BETACAGE CHAPTER 1. INTRODUCTION

This is often called the crossing argument, an entirely qualitative claim that states WIMPs may

have interactions with the standard model on the same order as their self-annihilation cross-section

[9]. One model suggests WIMPs can coherently scatter off of atomic nuclei via a Z-boson exchange

[12]. It is exactly this interaction that CDMS, and other experiments, hope to observe.

To do this, CDMS makes use of Z-independent Ionization and Phonon (ZIP) detectors which are

semiconductor detectors that rely on ionization of substrate electrons and production of phonos to

determine the energy of some particle interaction. Ionizing radiation enters the detector substrate,

imparting energy into the electron system via various scattering processes. An electric field applied

over the substrate drifts the ionized electrons to a collection plate, inducing a charge on the electrode

proportional to the total charge of the ionized electrons [8]. Since the energy required to create an

electron-hole pair in millikelvin germanium or silicon is well known, we have a measure of the energy

deposited via ionization.

ZIP detectors are also sensitive to energy deposited in the phonon system of the detector sub-

strate, such as via a WIMP nuclear recoil. Phonon sensors are photolithographically etched onto the

germanium crystal and consist of an aluminum absorber in which phonons from the Ge or Si crystal

enter and break cooper pairs into quasiparticles which then diffuse into a tungsten filament whose

resistance is exceedingly sensitive to temperature. The energy from the quasiparticles diffusing into

the W filament increase the filimants temperature and thus the resistance, which is the quantity

CDMS measures.

1.1.3 The Necessity for Screening

Now that we understand how WIMPs are detected, we can briefly discuss why screening technol-

ogy is so crucial. Following the analysis in Zeeshan Ahmeds thesis [8], the CDMS-II experiment

characterized a wide variety of events by the energy deposited via ionization, the location of the

8

CHAPTER 1. INTRODUCTION 1.1. CDMS AND THE ORIGIN OF THE BETACAGE

interaction (which is derived from the shape of the phonon signal), the energy deposited in the

phonon system and the relative timing between the charge deposition and the phonon deposition.

In theory, the timing measurement allows one to discriminate WIMP recoils, which likely happen

deep in the substrate, from surface events which are due to ionizing radiation from radioactive

contaminants. WIMP events should exhibit a much longer phonon rise time as the phonons need

to propagate through the detector substrate, whereas a surface event would not.

An x-ray, -ray - or -particle from a radiocontaminant both ionize electrons as well as impart

energy in the phonon system via electron recoils. If the radiation is of sufficiently low energy, both

the charge and phonon energy deposited can be indistinguishable from those due to nuclear recoils.

Additionally, various sources of noise limit how accurately one can determine the phonon rise time,

which was the primary method of discriminating surface events from nuclear recoils.

In an effort to determine the source of surface events, the ZIP detectors were flooded with x-

rays, gamma rays and photo-ionized electrons, and the rate of flooding was compared to the rate of

corresponding electron recoils. Using this estimate as well as the low-background ambient photon

rate measurable via other means, it was found that the photon-induced surface-event rate accounted

for about half of the surface-event rate measured in CDMS II ZIP detectors. Thus it was concluded

that low-energy beta emitters have an almost equal contribution toward the background event rate.

Beta emitter contamination is primarily caused by three unstable isotopes. The trace 210Pb on

CDMS detectors is a result of plate-out from radioactive decay of 222Rn, a common radioisotope

found in air. In fact, 210Pb emits a number of x-rays and particles along its decay to a stable

isotope. Trace 40K and 14C are most likely the result of the processing of detector materials as well as

direct human contact. With reliable screening technology developed, material choice, manufacturing

methods, and a number of other experimental parameters can be modified to first identify sources

of contamination and eventually reduce their abundance.

9

1.2. GASEOUS TIME-PROJECTION CHAMBERS CHAPTER 1. INTRODUCTION

Toward this effort, members of the CDMS collaboration at Caltech, The University of Alberta

and South Dakota School of Mines (formerly Syracuse University) have directed efforts toward

developing a dedicated screening technology, called the BetaCage, which we will discuss herein.

1.2 Gaseous Time-Projection Chambers

Since we would like to reliably screen materials for both the identity of the contaminant, parame-

terized by energy of the ionizing radiation, as well as its location, we need sensitivity to low-energy

s (near the expected energy of a WIMP recoil) as well as a way to reconstruct the path of the

ionizing radiation so that we might determine a source [8]. To accomplish these goals, we will make

use of a gaseous time-projection chamber (TPC), together with a multi-wire proportional counter

(MWPC). We choose a gaseous detector as this medium prevents significant backscatter of ionized

electrons, which can affect a determination of the event energy.

Gaseous TPC detectors consist of a volume of detector gas, such as Argon or Neon, usually

with a uniform electric field established over the volume of the detector. Radiation that passes

through the detector gas, such as a high-energy -ray, will collide with gas molecules and ionize

electrons. The number of electrons produced is proportional to the energy deposited by the incident

radiation, assuming photoelectric absorption of the photon and/or successive electron collisions until

the incident has nearly the same energy as the ionized electrons.

The electrons are immediately drifted along the direction of the electric field until they are

deposited onto some electrode, usually one helping to source the E-field. The electrons deposit

a charge proportional to their own total charge, thus giving us a way to determine event energy,

assuming we collect all of the electrons produced by a particular ionization event. In this manner,

we have attained one design goal: energy reconstruction of ionization events.

10

CHAPTER 1. INTRODUCTION 1.3. THE NEED FOR A PULSE SIMULATION

1.3 Reconstructing Tracks and the Need for a Pulse Simulation

Toward the second design goal of reconstructing tracks, we make use of an MWPC which consists

of orthogonal arrays of high-voltage wires which are used for charge collection as well as signal

amplification. MWPCs are discussed in detail in a later chapter, but essentially they determine

the xy-location of a particular charge deposition (where the z-axis is assumed to along the axis of

the field established by the TPC) by comparing signals from wires along the x-axis to simultaneous

signals from wires along y-axis, choosing those that received the greatest charge deposition. In this

way, we can pinpoint the center of an event.

Consider a particle track under the assumption that electrons are uniformly distributed along the

path length. Clumps of electrons that are laterally spaced by the MWPC wire spacing will deposit

onto different wires. If the ionizing radiation that produced the track also had some z-component

of motion, the depositions on neighboring wires will be spaced in time due to the uniform drift

over the TPC. Thus, with the xy-projection of ionized electrons in a particle track given by the

MWPC, and the z-projection given by the spacing in time, we could reconstruct the path taken by

the ionizing radiation and thus determine its source.

In order to accurately construct the energy of the whole event, we need to determine the energy

in each track bin, corresponding to the energy deposited on a single sensing wire. To that end,

this work aims to determine the nature of the electronic signals produced by a cloud of electrons

depositing their charge on a single sensing wire. With a template of the voltage pulse we would

receive from just such a charge deposition, as well as a measurement of the observed background

noise, we can employ techniques of optimal filtering to accurately determine the pulse height, and

thus energy, of a particular signal [8, 13]. Not only that, but the exact shape of the signal may

be dependent on where a deposition happens within the detector, as the impedance of the readout

electronics network explicitly depends on the location of the charge deposition along a signal wire.

11

Chapter 2

Status of the BetaCage

The BetaCage project was originally proposed in the early 2000s, and since then has seen a myriad

of different students contribute to its development. We detail some of the major advances in the

BetaCage development as well as solidify the understanding necessary to create a pulse simulation.

2.1 MWPCs

Much work has been done developing the MWPCs for eventual use in a prototype BetaCage [16].

A prototype MWPC has been constructed that consists of two parallel layers of cathode wires

sandwiching a perpendicular layer of anode wires. Each layer consists of 79 wires with both a pitch

and inter-layer spacing of 5 mm. These wire arrays are housed in a gas such as P10 (90% Ar and

10% Methane) suitable for ionization based measurements.

An ionizing particle incident on our detector would pass through the P10 gas in the drift chamber

(discussed below) creating electrons and cation partners. The electrons produced via ionization drift

until they reach the cathode wires which are fixed at 100 V as compared to a ground plane located

some distance away. The space between the 100 V cathode plane and the ground plane is the

TPC discussed in the previous chapter and has copper spacers of linearly decreasing voltage to help

12

CHAPTER 2. STATUS OF THE BETACAGE 2.1. MWPCS

maintain field uniformity.

After drifting through the TPC, ionized electrons then proceed to avalanche onto the anode wire

array, which is held at 2100 V, thus amplifying the incident energy of the particle (the physics of

the electron avalanche is discussed in detail in Chapter 3). Following gaseous amplification during

collection on the anode wire arrays, also called the anode plane, the pulse is also electronically

amplified and finally sent to an ADC for digitization. One of the two cathode planes will eventually

be connected to readout electronics so that we might obtain the xy-projection of the path of an

incident particle.

In its final construction, the BetaCage will have the cathode plane fixed around 1000 V and

the anode plane fixed around 3000 V. Higher voltage is necessary to drift tracks through the full

chamber, as tests of the prototype only involved drifting tracks through a short section of the drift

chamber in order to demonstrate functionality.

In its prototype construction, the frame supporting these wire arrays is made of a plastic Noryl

which has been shown to have acceptable levels of bulk contamination for 238U, 232Th and 40K.

Using radiopure materials for construction is essential for the BetaCage to function as an effective

screener for surface contamination. To test functionality of the wire arrays, an individual MWPC

was housed in a P10 gas with a small electric field established between the cathode wire arrays

and a grounded boundary (in this case, a G10 sheet) in order to induce drift of an ionizing particle

[16]. Incident particles, in this case, x-rays from an 55Fe source, were collimated into this region to

induce ionization of the gas.

We obtain information about the energy of our incident particle first from the pulse heights and

also by fully integrating the collected charge that was produced in an interaction with the detector.

Spatial resolution (in terms of the two coordinates x and y) is obtained in the following manner. In

an ideal case, we would examine the voltage on each and every one of our wires in both the anode

13

2.2. THE BULK MWPC AND TPC CHAPTER 2. STATUS OF THE BETACAGE

array and one of the perpendicular cathode arrays. By comparing which wires exhibit a maximum

voltage or integrated charge, we can locate a particular charge deposition as an x-y ordered pair.

In practice, reading-out hundreds of channels is both costly and cumbersome to manage, so wires

are ganged together and read out using a single amplifier. With an appropriate ganging of distant

and dispersed wires, we can still obtain spatial resolution.

2.2 The Bulk MWPC and TPC

In its final design, the BetaCage will make use of three MWPCs designated the Veto, Trigger and

Bulk MWPC. The configuration of these MWPCs is shown in Figure 2.1. A particle, emitted in

a sufficiently normal direction from the plane of the MWPCs, with energy < 200eV will create a

ionization track fully within our chamber [14, 15, 17]. This track drifts up to the bulk MWPC for

imaging in the x-y spatial dimensions, defined on the plane of our MWPCs, as described above.

Directly above the sample, the trigger MWPC provides information about the initial time of a

particle interaction as well as whether or not our particle originated in the fiducial region of the

detector (A fiducial region, interior to the actual physical boundaries of the detector, is defined to

avoid edge effects on the drift field or detection electronics).

The timescale of the interaction is sufficiently shorter than the timescale of ionized electrons

moving across the drift region, so that we can examine the arrival times of our electrons on the

MWPC to obtain the tracks z-projection while the xy-projection is obtained from the perpendicular

wire arrays, as described above. This particular setup allows for full track reconstruction of ionizing

particles.

Although full track reconstruction has yet to be demonstrated, a single prototype MWPC has

been shown to respond to ionization by x-rays from an 55Fe source [16]. Additionally, using two

MWPCs housed in a P10 gas chamber, drift of ionizing particles has been shown in a very simple

14

CHAPTER 2. STATUS OF THE BETACAGE 2.3. OPERATING THE BETACAGE

Figure 2.1: A schematic side view of the BetaCage showing the configurationof the MWPCs and the drift chamber. Samples we wish to screen for radiocon-taminants would be placed between the veto and trigger MWPC. Courtesy ofBunker, Ahmed et. al., in preparation.

manner. Without any clever wire ganging, a initial particle signal from a trigger MWPC has been

shown to be followed by a signal on the bulk MWPC, where both arrays have all their wires ganged

to a single amplifier

2.3 Operating the BetaCage Detector

The detector itself responds to ionizing radiation as soon as the constituent wire arrays are biased

with the high-voltage supply, the only difficulty is collecting meaningful data. The wire arrays

are connected to various channels of an amplifier with large coupling capacitors to eliminate the

DC component from the HV supply. The signal is then amplified and sent through an ADC to a

ROACH board discussed below.

If the detector remains biased and the amplifier is running, we can access voltage traces from the

ADC at will, using the ROACH which is a type of hardware produced by the CASPER collaboration

for use in radioastronomy [18].

15

2.3. OPERATING THE BETACAGE CHAPTER 2. STATUS OF THE BETACAGE

2.3.1 ROACH Components and Basic Structure

The ROACH boards that will be used to control and capture data from the BetaCage consist of

a Xlinx Virtex5 FPGA, a PowerPC CPU (running Linux) and various RAM banks/data ports for

storing and transferring data. The native Linux OS is used to program the FPGA as well as read

data stored by the FPGA in shared memory banks.

Voltage traces from various amplifier channels are fed into an ADC which samples at 800 MHz.

However, the FPGA has a maximum clocking speed (maximum for reliable behavior) of 200 MHz.

Thus, to handle data with full temporal resolution, four consecutive ADC output data points are

split into four distinct 12-bit signals (the ADC has 4096 voltage bins) that are fed to the FPGA.

Memory banks on the FPGA are 64-bits wide which allows us to use a 16-bit signal (4 12+16 = 64)

from an external clock, such as a crystal oscillator, so pulse data from different amplifier channels

(different wires of the MWPC arrays) can be synced correctly.

On each clock of the FPGA, one 64-bit data word, which consists of four ADC data points and

a clock signal, is written into shared memory. At 800 MHz, these memory banks can hold 42 s

of continuous data (). The onboard CPU can then be used to read out this data and transfer it

to an external machine connected via the internet. A set of telnet commands for interacting with

the ROACH and its CPU, collectively referred to as the Karoo Array Telescope Control Protocol

or KATCP, have been developed by CASPER [19].

2.3.2 Communicating With the ROACH

Using compiled bitmaps (explained below) we can use a KATCP command to program the FPGA

with the on-board CPU,

?progdev

Using more basic telnet commands, we can read and write data off of the ROACH,

16

CHAPTER 2. STATUS OF THE BETACAGE 2.4. FIRMWARE

?(read/write)

We use the write command to control various software registers that can trigger data collection,

memory writing or control various parameters of the FPGA such as an automatic trigger threshold.

The read command is used almost exclusively for reading data traces off of shared memory banks.

A full list of commands exists on the CASPER wiki [19].

2.3.3 Programming the FPGA

Using MATLABs Simulink and a library of tools both from the FPGA manufacturer (ISE-14 from

Xlinx) and the CASPER collaboration, it is possible to create a simulink model of our desired FPGA

and then compile said design into verilog and finally a bitmap to program our specific FPGA.

These bitmaps (literally a map of bit values 0 or 1) specify the connection topology inside the

FPGA to obtain our desired circuit. Bitmaps output from CASPERs compiler can be programmed

directly onto the FPGA using the ROACHs CPU, as discussed above.

2.4 Firmware: Current State and Ideal Endpoint

To control a ROACH, we define a Python object that consists almost entirely of a TCP socket

together with a library of functions which read and write to that TCP socket. The choice of Python

as the front-end was due to the relative simplicity of interacting with TCP sockets via Python

modules. Currently, there exists a version of this software that can communicate with and control

a single ROACH board and thus 6 ADC channels.

Firmware has been developed that allows us to capture manually triggered data traces as well

as data traces which exhibit a falling edge that drops below a user-defined trigger threshold. For

the falling-edge trigger, the FPGA buffers the data as its looking for the falling edge so that once

an edge is seen, it can retrieve some amount of noise prior to the pulse itself. Simultaneous data

17

2.4. FIRMWARE CHAPTER 2. STATUS OF THE BETACAGE

traces can then be collected from each of the 6 ADC channels on a single ROACH board. If we

make use of the falling-edge trigger, we can only trigger off of one channels voltage, although data

traces from every other channel will be synced in time.

Eventually, we would like to construct a Python object for controlling an array of ROACH

boards, not just a single one. Such a task would involve temporal syncing between the boards so

that we could examine simultaneous data traces from each amplifier input and correctly space the

different charge depositions in time to find the z-projection of a particle track.

18

Chapter 3

Theory of Pulse Generation

As we mentioned previously, an interaction with the BetaCage detector is characterized by pro-

duction of electrons via ionization of gas molecules. These electrons are subsequently collected on

high-voltage wires, and as moving charges in an electric field, they create a current on the elec-

trode(s) producing the field. A deposition of electrons looks like a pulse in the voltage on the

detector wires. If these wires are connected to readout electronics, as described previously, we can

see these pulses.

By simulating a particle interaction with the detector, we would like to determine if the particular

shape of voltage pulses can be used to refine the energy resolution, or if the shape provides any

information about where on a wire an energy deposition occurred. This would allow for more

robust track reconstruction and identification of radio contaminants, two of the BetaCages primary

objectives as a screening technology.

3.1 Ionization and Track Production

Consider some ionizing radiation from a radio contaminant incident on the TPC volume of the

BetaCage. This radiation can be in the form of particles, -particles, x-rays and -rays, but

19

3.2. THE ELECTRON AVALANCHE CHAPTER 3. PULSE THEORY

its the energy deposition mechanism that matters. Incident photons are likely to photoelectrically

absorb within the gas, kicking out an electron. Thus, we really need only consider how incident

electrons behave in a gas (ignoring particles for this work), which is convenient since we are

concerned with sensitivity to s.

A charged particle traversing a drift chamber eventually collides with a gas molecule and scatters

in some direction, meanwhile imparting energy to the gas molecule and ionizing one of its electrons.

This happens repeatedly until the initial charged particle no longer has sufficient energy to ionize

more electrons. The result is a track of ionized electrons along the flight path of the initial decay

product. In truth, the picture is slightly more complicated and is treated in great detail in Particle

Detection with Drift Chambers [20].

A photoelectric electron or an incident deposits a relatively large amount of energy in a single

ionization event. The electron product of this ionization often produces secondary electrons in its

vicinity so the track is more like a string of clusters rather than a continuous line. Indeed, the

cluster size, ionized electron density and separation between clusters are all functions of the type

of detector gas, temperature, pressure as well as the momentum of the incident particle. The exact

nature of the tracks is not explicitly necessary to accomplish the goals set out in this thesis work,

so we suggest interested readers refer to the above reference.

3.2 The Electron Avalanche

Low-energy radiation in the form of -particles or x-rays produce relatively few electrons in typical

detector gases at atmospheric pressure. As an example, a 5.9 keV x-ray from 55Fe ionizes only

227 electrons in a mixture of 90% Argon and 10% Methane, which is a well studied drift chamber

gas called P10. In fact, this is the gas used in the prototype BetaCage and provides a relatively

well-understood platform on which to develop our detector technology.

20

CHAPTER 3. PULSE THEORY 3.2. THE ELECTRON AVALANCHE

With such a small number of electrons, however, their total charge would be minuscule. Recall

that a single Coulomb of charge consists of 1019 electrons so the initial 227 need to be signifi-

cantly magnified in order to be detectable. This is accomplished in part by making use of electron

avalanching. If an ionized electron in a gaseous chamber is under the influence of an electric field,

then it can be accelerated between collisions with gas molecules. If the field is of sufficient magni-

tude then the electron can be accelerated so much so that upon a collision with a gas molecule, it

ionizes another electron. With an appropriate field gradient, this doubling procedure can happen

indefinitely until the electrons collect on the electrode producing the high-field.

This is exactly how MWPCs operate. A large voltage, on the order of a few kV, is applied to

an array of very thin wires no more than 100 m in radius. Each wire generates a radial field that

increases significantly closer to the wire, enough to induce an avalanche within atmospheric pressure

P10 gas. A simulated electron avalanche onto a 25 m wire is shown in Figure 3.1, computed via

a Monte-Carlo technique and ignoring the effects of photons produced in any de-excitation of gas

molecules following a collision.

Figure 3.1: Electron density of an avalanche caused by a single electron ac-celerating toward a 25m wire. Developed using a Monte-Carlo integrationtechnique. Courtesy of Blum, Riegler and Rolandi.

21

3.2. THE ELECTRON AVALANCHE CHAPTER 3. PULSE THEORY

3.2.1 The Diethorn Formula

It would be useful at this juncture to quantify the amplification inherent to the electron avalanche.

This is an exceptionally complicated subject in and of itself so interested readers are again directed

to Particle Detection with Drift Chambers [20] for a more detailed treatment.

Lets begin with our assumptions. The large fields induced by the sensing wires allow us to

assume the electric field produced by an avalanching electron has negligible effect on the field

experienced by other avalanching electrons. This is part of the proportionality of an MWPC since

electron-electron or ion-ion interactions could increase the net field strength experienced and thus

alter the signal. Essentially what this boils down to is that each of the 227 electrons from an 55Fe

x-ray can be considered independently, and their resulting signals can be added to yield to total.

The amplification of an avalanche is described by a differential increase in electrons per path

length ds and is given by dN = N ds where 1 is the called the first Townsend coefficient. The

coefficient is determined by the excitation and ionization cross-sections of the gas and depends

on the momentum/energy transfer mechanisms that were brushed over in Section 3.1. In fact, no

analytic expression exists for , rather, it is experimentally determined for various gas mixtures.

Integrating the above equation from some point smin, where the field is sufficient to start an

avalanche, to the wire radius a, we have that

Ntot/N0 = exp asmin

(s)ds = exp E(a)Emin

(E)

dE/dsdE (3.1)

So the natural quantization of amplification is given by the gain G Ntot/N0. Diethorn de-

veloped an expression for the gas gain by assuming that = E with some constant coefficient.

However, to fully develop the expression for the gain, we need to consider the field under which the

avalanche happens. The electric field of the n-th anode wire in an N wire grid of the MWPC, is

22

CHAPTER 3. PULSE THEORY 3.2. THE ELECTRON AVALANCHE

given by the expression,

~En(~x) =n

20rr =

1

20r

(Nm=1

cmnVm

)r (3.2)

where the cmn are elements of the many-port capacitance matrix between the anode and cathode

arrays and Um is the voltage on wire m. As such,N

m=1 cmnVm = (ca + caa)Va + cacVc where ca is

the anode self-capacitance, caa is the anode to all-other-anodes capacitance, cac is the array cross-

capacitance and Va and Vc are the anode and cathode voltages, respectively. In practice, the charge

on a sensing wire can be derived directly from a finite-element simulator like COMSOL, but it is

useful to understand the origin of the result.

We assume that if an electron is avalanching onto the n-th wire, then the field experienced by

the electrons and their cation pairs is entirely determined by ~En since rmin b by design. As

is demonstrated in Appendix A, the minimum electric field necessary to start an avalanche in the

BetaCage prototype is approximately 38m above the anode surface, while the wires are spaced by

5 mm! Thus, we are justified in assuming the field experienced by an electron is described solely

by the wire onto which it is avalanching. Indeed this correspondence holds to much larger radii as

well, which is convenient in developing the signal.

Substituting Equation 3.2 into 3.1 and assuming = E, we find (note that ds = dr)

Ntot/N0 = exp E(a)Emin

E

(/20r(E)2)dE = exp

E(a)Emin

20EdE

= exp

(

20ln

20Emin

)(3.3)

So all that remains is to find which is necessarily related to the average kinetic energy required

by avalanching electrons to ionize another electron eV , where V is an experimentally determined

quantity specific to the gas mixture and referred to as one of the Diethorn parameters. Consider

23

3.3. CURRENT INDUCTION CHAPTER 3. PULSE THEORY

the potential difference between the origin of the avalanche and the wire surface.

(a) (smin) = asmin

E(r)dr =

20lnsmina

=

20ln

20Emin(3.4)

If we assume that the number of electrons doubles over the potential difference V , then there

are ngen = ((a) (smin))/V generations of doubling, leading to a total of 2ngen electrons per

primary electron. Thus,

G = 2ngen = 2

20Vln

20Emin (3.5)

= lnG = ln 2V

20ln

20aEmin(3.6)

Where the final result is often referred to as the Diethorn Formula and n =N

m=1 cmnVm for a

particular wire n, just as before. We also include a table of the MWPC capacitance matrix values,

computed via COMSOL, for completeness.

Anode Self Anode to Other Anodes Anode to Cathode Array

8.75 pF/m -6.7 pF/m -2.3 pF/m

Table 3.1: Table of MWPC capacitance values computed via COMSOL. Notthe sign is simply a convention of the matrix, not the actual capacitance.

3.3 Current Induction On An Electrode via Ramos Theorem

With an understanding of the electron avalanche, one of the most important topics to discuss is how

the actual electrical signal is produced in the BetaCage electronics. This is generally accomplished

through electromagnetic induction on the electrodes due to changing field lines from moving charged

particles. In fact, the avalanching electrons are not responsible for the majority of the signal; its

24

CHAPTER 3. PULSE THEORY 3.3. CURRENT INDUCTION

their cation partners moving away from high-voltage wire that create the characteristic pulses in

charge collection detections, a topic we discuss below and in Appendix B.

3.3.1 Ramos Theorm

The authors of Particle Detection with Drift Chambers [20], discuss this principle in great depth as

well as provide quantitative justification for the simplifying assumptions made. We begin with an

introduction of the Shockley-Ramo Theorem. [The theorem] states: The charge Q and current i

induced by a moving point charge q are given by,

Q = q 0(~x)

I = q ~v ~E0 (3.7)

where ~v is the instantaneous velocity of charge q. The quantities 0(~x) and ~E0(~x) are the

electric potential and field that would exist at qs instantaneous position ~x under the following

circumstances: the selected electrode at unit potential, all other electrodes at zero potential and all

charges removed [22].

Consider our MWPC. Very near to a sensing wire, we can assume that the field is entirely

determined by the wire in question, with negligible effects from other wires in the array. Thus, the

weighting field ~E0(~x) discussed above, is simply that of an infinitely long wire at a fixed potential.

This assumption has been verified in Appendix A by computing the Ramo weighting field with

COMSOL Multiphysics and plotting it against the ideal field of a wire held at fixed potential.

This assumption is quite convenient since regardless of MWPC and chamber geometry, the signal

produced by any electron avalanching onto a high-voltage sensing wire is the same. Not only that,

but the signal is the same regardless of the electrons direction of approach, allowing us to more

accurately model the shape of the signal and use alternate, more robust methods to determine the

origin of the avalanche.

25

3.3. CURRENT INDUCTION CHAPTER 3. PULSE THEORY

3.3.2 Applying Ramos Theorem

Working from the above stated assumptions, we begin with the electric field of the n-th wire in an

N wire grid, but now with only the sensing wire at a non-zero voltage. Thus the weighting field is

given by,

~E0(~x) =0

20rr =

ca20r

r (3.8)

Where ca is the anode self-capacitance and it has been assumed that Va = 1. We also know

that positive ions have a drift velocity given by v = E where is the ion mobility and E is the

strength of the electric field determining their trajectory [20, 25]. Thus, if the sensing wires have

radius a, a positive ion starting at the surface of the wire will have the following trajectory.

dr

dt= E =

20r= r(t) = a

1 +

t

t0(3.9)

where1

t0=

a20

Nm=1

cmnUm (3.10)

Where we have made use of Equation 3.2 for the field which determines the electron trajectory.

A reasonable timescale for the cation to deposit on the cathode wires is given by td b2/U

where b is the array spacing and U is the sensing potential, or U = VaVc. Our implementation of a

pulse simulation will involve artificially forcing the signal to 0 after this timescale, which effectively

zero-pads our pulse for eventual fourier techniques discussed in Chapter 5.

With r(t) calculated and the weighting field known, the current on an anode sensing wire,

following Ramos theorem, is given by

Ia = Ntot e(

ca20r

) d

dt

(a

1 +

t

t0

),

= Ntot e ca20r

a2t0

(1 +

t

t0

)1/2,

26

CHAPTER 3. PULSE THEORY 3.3. CURRENT INDUCTION

= Ia = Ntot e

40

cat0 + t

(3.11)

Where ca is the anode self-capacitance (which will be determined via COMSOL), a is the anode

wire radius, e is the elementary charge, 0 is the vacuum permittivity, t0 given by Equation 3.10

and Ntot is the total number of cations moving away from the wire surface.

Now, we have implicitly made a few assumptions to obtain such a nice analytic result. These are

verbally addressed here, but treated in great detail in Appendix B. The most glaring assumption

is that the electrons dont produce a comparable signal compared to the ions, but to address this,

we need to understand our other major assumption. The above formula assumes that all the ions

formed in an avalanche originate at the wire surface and move outward from there. Due to the

exponential behavior of the electron avalanche, most of the electrons are actually very near to the

surface of the wire where the last round of ionization occurs before electrons deposit on the anode,

but not quite close enough to ignore completely.

In Particle Detection with Drift Chambers [20], they compute the center of gravity (center of

mass) of a cloud of avalanching electrons and find it to be just 1-2 m above the wire surface for a

25 m wire at 2 kV, like that in the BetaCage prototype. The expression for the induced current

can be appropriately modified to account for this center of gravity by taking a a(1 + ) with

no more than 0.1 in practice. This has no direct effect on Equation 3.11, only on the value of t0.

Now we come back to considering the electrons and their negligible influence on the signal.

Quite simply, electrons deposit on the anode wires in under a nanosecond after the beginning of the

avalanche, having traversed only 20 m, whereas the ions take tens of nanoseconds to produce their

signal over a much longer path length (at least an order of magnitude). The induced charge due

to the avalanching electrons was found to be just 1-2% or the total induced charge from the cation

partners. Again, this is addressed in great detail in Appendix B, which draws heavily from [20].

27

Chapter 4

Avalanche Electron Trajectories and

Spreading the Pulse in Time

With an understanding of single-electron avalanching onto high-voltage wires, we consider how a

multitude of electrons produced by a single particle of ionizing radiation might drift through an

MWPC and avalanche together. This implicitly involves simulating the electric field in the biased

BetaCage prototype and considering the various trajectories of charged particles in a gas under the

influence of that field. We begin by examining the MWPCs and simulating the electric field therein.

4.1 MWPC Symmetry

Each MWPC consists of three planes of wires: two parallel cathode planes and an orthogonal anode

plane where the avalanche takes places. A representation of this geometry is shown in Figure 4.1 as

well as an example of defining a fiducial region where we would can ignore edge effects of the wire

array. Thus, we can consider the MWPC in terms of a unit cell with a single anode wire and two

cathode wires, one from both the upper and lower cathode planes.

28

CHAPTER 4. ELECTRON TRAJECTORIES 4.1. MWPC SYMMETRY

Figure 4.1: Schematic of an MWPC wire array. The red wires constitute theanode plane and the black wires make up both cathode planes. The green boxdefines a typical fiducial region where we can ignore edge effects of the array.

The geometry of unit cell is shown in Figure 4.2. Tesselations of this cell in the xy-plane would

yield the full MWPC. Defining a unit cell like this is useful for field calculations as symmetry

demands an electric field perpendicular to the boundaries in the x and y direction.

Figure 4.2: Geometry of the unit cell. Consists of a single 25 m diameteranode wire parallel to the y-axis, sandwiched by two orthogonal 125 m diametercathode wires with a pitch of 5 mm. The cell is 5x5x20 mm, with 5 mm of freespace above and below the cathode wires to ensure the field approaches theuniformity we assume in the TPC.

29

4.2. COMSOL FIELD CALCULATIONS CHAPTER 4. ELECTRON TRAJECTORIES

4.2 COMSOL Field Calculations

Defining the above geometry of wires in COMSOL Multiphysics, we fill the intermediate space with

an arbitrary material that has permittivity equal to the vacuum permittivity, = 0 = 8.854

1012 F/m. This is the only relevant property to the electric potential/field (assuming no magnetic

fields), and is sufficiently close to the permittivity in P10 gas, as Ar = 1.000127.

Considering the electrostatics of our prototype BetaCage, we set the anode wire to 2100 V and

the cathode wires to 100 V. The top and bottom of the cell, 5mm away from the cathode wire planes,

are defined with a uniform potential equal to 1 5mm20cm = 97.5% of the 100 V cathode potential,

as the TPC is assumed to be a region of uniform field and linearly changing potential between

the cathode plane and the grounded end of the drift chamber. This is the setup of the prototype

BetaCage as it stands at Caltech, although these values can easily be changed to simulate the field

inside the final version of the BetaCage.

Finally, we assign boundary conditions of zero-charge accumulation to both the boundaries in

the x-direction and the y-direction (equivalent to a perpendicular field and mirror symmetry). Doing

so, we find the electric potential shown in Figure 4.3 which is displayed as a series of isopotential

surfaces at every 100 V. COMSOL outputs the computed potential as a list of points in three-space

each with an associated potential, as well as an electric field since COMSOL employs gradient

optimization and can readily output this information. With the potential and field inside a unit

cell solved, we can determine the particle trajectories through the cell.

A more detailed explanation of the unit cell electric potential and field calculations done in

COMSOL is included in Appendix A. This includes a brief introduction of how COMSOL works

followed by a detailed explanation of the geometry, electrostatics and even the finite element meshing

parameters used.

30

CHAPTER 4. ELECTRON TRAJECTORIES 4.3. ARRIVAL TIME MAP

Figure 4.3: Potential inside an MWPC unit cell. Surfaces drawn are isopoten-tials. We can see that in the majority of the unit cell, the potential is changinglinearly with respect to z.

4.3 Arrival Time Map

The formulas computed in Chapter 3 for the current signal on a wire assume the cations produced

in the avalanche are accelerated at time t = 0. The starting time is arbitrary since the events are

stochastic, but primary electrons from different parts of the initial ionization track may arrive at

the avalanching region at different times. Thus, we need an understanding of how electron starting

position is mapped to arrival time.

Since each primary electron produces its own independent avalanche, by the argument of pro-

portionality presented in the previous chapter, if we have N primary electrons with a normalized

arrival time probability distribution p(t) and current signal Ia(t) from a single electron avalanche,

31

4.3. ARRIVAL TIME MAP CHAPTER 4. ELECTRON TRAJECTORIES

then the signal induced on the avalanching wire would simply be

Itot = N (Ia p) (t) (4.1)

where (Ia p) (t) =

Ia()p(t )

Since we have Ia(t) from Chapter 3, all that remains is to find p(t).

As was mentioned briefly in Chapter 3, the exact shape and distribution of electrons in a

BetaCage track depends on a wide variety of factors. Indeed, the track shape is outside the scope of

this work, but we still wish to understand how different physical distributions of electrons correspond

to arrival time distributions. To achieve this, we develop an arrival time map which takes an electron

starting position external to an MWPC and yields the time it takes that electron to reach the

minimum field necessary to start an avalanche.

We assume that charged particles in an electric field, i.e. the avalanche electrons, will follow the

field lines computed via COMSOL. However, unlike positive ions, the speed at which the electrons

move in the gas is not simply proportional to the field for the field strengths we are concerned with,

as the scattering processes with gas molecules are governed by the negatively charged electrons. So

to accurately determine the arrival time map and spread the pulse in time, we need to understand

how electrons drift at various speeds in P10 gas.

4.3.1 MAGBOLTZ and Electron Drift

A relatively easy-to-use platform exists for computing electron drift properties in various detector

gases. MAGBOLTZ is a standalone component of the GARFIELD suite which integrates the

Boltzmann transport equation for electrons in nearly any imaginable gas mixture. The program

computes an electron drift velocity under the influence of an electric field, accounting for forces

involved in diffusion and collision with gas molecules. MAGBOLTZ also yields a diffusion coefficient

32


and thus a way to determine the lateral extent of the electron cloud prior to avalanche.

Using a wide range of input electric fields from 0.1 kV/cm up to 48 kV/cm, the minimum field

necessary to induce an electron avalanche in atmospheric pressure P10 gas, we found the relation

shown in Figure 4.4 for the electron drift velocity as a function of applied electric field.

Figure 4.4: Electron velocity vs. applied electric field in 90% Ar + 10%methane, i.e. P10. The relation is approximately linear above 15 kV/cm, butfor the low field strength present in the drift region of the MWPC, which reachesa minimum of 1.7 kV/cm, it is clear we cannot make this assumption

4.3.2 Computing the Map

Together with the electric field inside the MWPC and the relation between electron drift velocity

and applied electric field, we have all of the necessary tools to construct an arrival time map. This

was accomplished via a relatively short Python script.

Electrons were considered as test particles and assigned a location in 3-space as well as a velocity

vector. The initial position was chosen to be 3.0 mm above the cathode wire array and the initial

velocity vector was defined in the direction of the electric field at that location, with a magnitude

given by the relation between drift velocity and electric field computed via MAGBOLTZ. The

33


strength and direction of the field were sampled directly from the output of our COMSOL model,

which was imported into Python as a NumPy dataset and linearly interpolated in 3-dimensions over

an irregular grid with SciPys interpolation package.

Following an electrons instantiation, the test particle was advanced along its own velocity vector

for a small time-step, chosen to be 1000x smaller than the expected time scale of the pulse, simply

to ensure that we have full resolution of the paths taken by electrons especially near the avalanche

region where the field changes rapidly. The velocity vector was then updated as a function of the

new position and thus new electric field. This process was repeated until the electric field reached

the minimum value necessary to induce an avalanche.

The xz and yz projections of an electron path are shown in Figure 4.5. The electron is initially

repelled from the cathode, as we might expect given the shape of the field lines, and is subsequently

attracted to the anode wire, eventually reaching the avalanching region.

Integrating a test particles motion for a large grid of starting positions, we were able to create a

map from starting position to travel time, with the travel time given as the number of steps it took

to reach the avalanching region multiplied by the time-step itself. The result is shown in Figure 4.6.

Figure 4.5: Example path from an arbitrary initial position. Shown are theprojections of the path onto the xz-plane and yz-plane. Cathode/anode wiresparallel to x-axis/y-axis, respectively, and colored red/black.

34


Figure 4.6: Contour plot of the arrival time of electrons as a function of theirx-y starting position 3.0 mm above the cathode wire array

As a brief sanity check, we found that mean distance at which the field was strong enough

to induce an avalanche was approximately 39 0.5m above the anode surface, which matches

our expectation from Chapter 3. Additionally, the map seems to match our intuition as electrons

incident directly above the cathode wire (which is parallel to the x-axis) experience a much longer

travel time as they have to circumnavigate the cathode to deposit on the anode. Also, electrons

closer to the anode wire (closer to x = 2.5 mm in the above figure) have comparatively shorter

times to those farther away, just as we might expect.

Finally, we bring it all back to Equation 4.1 and the distribution p(t). The arrival time distribu-

tion of electrons over any particular unit cell has two major components which we distinguish: the

distribution due to differing z-projection as well as the distribution due to differing x-y position.

First, we take each electrons initial x-y position and compute an arrival time assuming it originated

3.0 mm away from the cathode array and using the map computed above. The distribution in the

z-direction is then accounted for by assuming its center is at the starting plane 3.0 mm above the

35


cathode array. With a constant drift velocity prior to reaching the cathode array, we add to or

subtract from the arrival time of each electron based on its position relative to the starting plane.

This is summarized as,

tarrival = f(~x) T (x, y) + (z zc) vz (4.2)

where T (x, y) is the map shown in Figure 4.6, ~x = (x, y, z) is the electrons starting position,

z0 is the z-coordinate of the starting plane and vz is the average z-drift velocity experienced by

electrons 3 mm away from the cathode array. Note that if we assume the electron is coming from

large z, i.e. z > z0 then we should be adding time, which agrees with Equation 4.2.

If we have some initial distribution of electrons with starting positions ~xi, then their arrival time

distribution is given directly as

p(t) = PDF (f(~xi)) (4.3)

In practice, the probability density function (PDF) is computed as a histogram of the arrival

times ti = f(~xi), normalized by the total number of primary electrons avalanching together. We

discuss the discreetization of p(t) and our signal current in Chapter 6.

36

Chapter 5

Signal Propagation

and Readout Electronics

Weve now reached a point where we have a firm grasp on the current signal produced on an anode

sensing wire by a cloud of avalanching electrons having originated exterior to a unit cell of the

MWPC (and thus the whole MWPC). What remains is a discussion of how this signal propagates

through the BetaCage readout system including electronic amplification. We break this down into

two components: the network of high-voltage (HV) wires and cables between an avalanching location

and a readout amplifier as well as the readout amplifier itself, both of which have a complicated

frequency response.

5.1 The HV Network

From any particular avalanche location along an anode wire, the signal propagates down the re-

mainder of the wire and is split between the ganged anode wires as well as a 1 m long SHV cable

from a port on the BetaCage vacuum system. The SHV cable feeds into a bias resistor and a passive

37

5.1. THE HV NETWORK CHAPTER 5. SIGNAL PROPAGATION

HV-filter which connect to the HV supply and provide the voltage present on the sensing wires.

These bias elements are in parallel with an HV-blocking capacitor and a 50 amplifier input which

is where we readout the signal. Pulses are then amplified by a series of two cascaded amplification

stages which we will discuss in Section 5.2.

5.1.1 Voltage Transfer Function

Figure 5.1: BetaCage high voltage network. The HV-filter, coupling capacitorand amplifier termination resistance are all bundled into Z0, also written as Zout

A schematic of the HV-network is shown in Figure 5.1 which includes the sensing wire itself, other

anode wires ganged in parallel, the SHV cable, and finally the output impedance, which includes

the biasing circuitry and the HV-blocking capacitor/readout. Various currents and impedances are

indicated on the diagram and we should note that Zw and Zc are 2x2 impedance matrices for the

sensing wires and the SHV cables, respectively, whereas Zout is just a complex impedance.

Our goal is to determine the current/voltage transfer function given by the ratio of i3 to is, the

current into Zout and the avalanche signal current respectively. To do this, we begin by evaluating

the voltages V1, V2 and V3 via Kirchoffs laws and making use of our understanding of two-port

networks. A analysis of transmission lines, two-ports as well as the specific electronic characteristic

of the sensing wires and cables is included in Appendix C. Many details of the following derivation

are also included there. Note that Zl and Zg are not impedance matrices because the end of the

38

CHAPTER 5. SIGNAL PROPAGATION 5.1. THE HV NETWORK

sensing wires are open circuits (only 1-port) once the cations deposit on the cathode wire, as the

cations carry the current that completes the circuit and allows a pulse to form.

Proceeding with our analysis, we can write

V1 = Zl (is i1) = ZW11 i1 ZW12 ik (5.1)

V2 = Zg(ik i2) = ZW21 i1 ZW22 ik = ZC11i2 ZC12i3 (5.2)

V3 = Zouti3 = ZC21i2 ZC22i3 (5.3)

Where the values of the above impedances and the sign of the matrix elements follow the

convention established in Appendix C. The three equations above contain 5 unknowns, thus we

can solve this system for a ratio of two unknowns in terms of quantities we know. Solving these

expressions, we can obtain the ratio i3/is

i3is

= ZC21ZW21ZlZg/{ZC12Z

C21

[(Zl + Z

W11

) (ZW22 + Zg

) ZW12ZW21

](ZC22 + Zout

) [ZC11

[(Zl + Z

W11

) (ZW22 + Zg

) ZW12ZW21

]+ Zg

[ZW22

(Zl + Z

W11

) ZW21ZW12

]]}(5.4)

This exceptionally grungy expression is the transfer function from the avalanche point to the

junction between the amplifier input and the biasing circuitry. Now consider the final part of the

HV network. The HV filter, biasing resistor and HV coupling capacitor are shown in Figure 5.2.

Figure 5.2: BetaCage biasing circuit and HV coupling. The parallel combina-tion of the HV-filter and the coupling capacitor make up Zout

39

5.1. THE HV NETWORK CHAPTER 5. SIGNAL PROPAGATION

We define an impedance Zbias for the two-stage, low-pass filter in series with Rf and Rbias, and

an impedance Zin for the coupling capacitor in series with the termination resistance,

Zbias = Rf1 + jRfCf

1 2R2fC2f + 3jRfCf+Rf +Rbias and Zin = Rin +

1

jCc(5.5)

Finally, with Zbias and Zin acting as a voltage divider, we have that the full network transfer

function to the amplifier input is given as

Hnet iinis

=i3is

iini3

=i3is

ZbiasZbias + Zin

(5.6)

Where i3/is is given by Equation 5.4 and Zbias and Zin are given as above. Using the expressions

and numerical values given in Appendix C, we compute the network transfer function for various

avalanche points along the wire, and plot the results in Figure 5.3. As we might expect, the transfer

function is approximately flat at unity gain for much of the necessary bandwidth, obtaining structure

at high-frequency due to reflections from impedance mismatches.

Figure 5.3: Network transfer function computed for various avalanching loca-tions along the anode wire. Since we plan to sample at 800 MHz, the roll-offhappens just before the Nyquist frequency

40

CHAPTER 5. SIGNAL PROPAGATION 5.2. THE READOUT AMPLIFIER

5.2 The Readout Amplifier

The amplifier is considerably more simple to understand and model, because we have a way to

measure its transfer function. With such data in hand, we can empirically fit a generic bandpass

function to the observed amplifier transfer function.

In the language of the previous section, the amplifier consists of an ideal amplification stage

which applies some frequency-dependent gain and complex phase to each fourier component. In

truth, an SHV cable from the BetaCage is attached to the outside of a monolithic amplifier box

(produced by collaborators at the University of Alberta) which includes the biasing circuitry, cou-

pling capacitor and two cascaded amplifier chips with termination resistances. The amplifier box

has an SMA output that is fed to the ADC on the ROACH board, as was discussed in Chapter 2.

Each of the integrated circuit amplifier chips used has a well defined passband with smooth

roll-ons and roll-offs, to which the designers of the amplifier box (which includes Zout) have added a

high-order anti-aliasing filter just below the expected Nyquist frequency. Regardless of the expected

result, a measurement provides a more realistic basis on which to build the pulse simulation.

Using a vector network analyzer (VNA), we input a frequency-swept sinusoid into the SHV port,

i.e. where the pulse signal would go, and examined the output from the amplifiers SMA port. The

resulting transfer function, both magnitude and phase, are shown in Figure 5.4. Included is our

empirical fit for the amplifier transfer function.

Briefly, the data was fit with a function by constructing successive high and low-pass filters of

arbitrary order and with arbitrary corner frequencies until the constructed model matched the data.

Generic low- and high-pass filters of order n are given by,

HLPF (fc, n) =1

1 + j ( )nHHPF (fc, n) =

j ()n

1 + j ( )n(5.7)

Where = f/fc with fc the corner frequency.

41

5.2. THE READOUT AMPLIFIER CHAPTER 5. SIGNAL PROPAGATION

Figure 5.4: Measured transfer function plotted against our ideal estimate. Themeasured data is in black while the empirical fit is in red and the measured datais normalized by the VNA transfer function. The non-smooth behavior at high-frequency is believed to be an artifact of the VNA, despite the normalization,as is shown in Figure 5.5. Note that there is a small amount of phase lag at lowfrequency, most likely due to the coupling capacitor which is not included in theideal model. Additionally, the phase rolls off much more quickly than expectedat high frequency, but again, this could be an artifact of the VNA itself.

The ideal function overlaid on the measured data in Figure 5.4 is given by the following transfer

function, assembled from the simple filters above,

Hamp = G HHPF (45 kHz, 1) HHPF (150 kHz, 2)

HLPF (380 MHz, 3) HLPF (650 MHz, 8) (5.8)

with G 78.5 the gain in the pass-band. Unfortunately, the frequency range that is of concern

to us includes the upper-limit of the audio band as well as the lower limit of the RF band. Thus,

VNAs that can probe this frequency range are very hard to come by as they commonly specialize

for the audio or RF bands. Initially, the transfer function was cobbled together from two distinct

measurements, but the bandpass gain was not consistent.

42

CHAPTER 5. SIGNAL PROPAGATION 5.2. THE READOUT AMPLIFIER

At the very least, the high-pass corner frequencies from the above measurement were confirmed

by taking a series of data points by hand, using another old, but reliable and well-calibrated VNA.

Thus, our model of the amplifier transfer function was consistent with two separate measurements

(and instruments) for the majority of the pass-band and around and the high-pass corner frequencies.

Only near the Nyquist frequency are we unsure of consistency.

Figure 5.5: Measured transfer function of the VNA itself (VNA output VNA input). We can see there is some artifact at high frequency and phasethat continuously rolls off. Indeed, the rolloff becomes much more significantabove 107 Hz, which corresponds to the early roll-off seen in Figure 5.4.

43

Chapter 6

Implementation of a Pulse Generator in

Python

The previous three chapters have detailed the physics of pulse generation in the BetaCage allowing

for a clear and concise explanation of our actual simulation. Without further ado, we begin with

an outline of the steps involved in computing pulses so that we might reorient ourselves.

1. Derive current signal generated by single electron avalanching onto wire (Ramos Theorem)

2. Simulate electric potential/field of an MWPC unit cell with COMSOL

3. Propagate electrons through MWPC field to find arrival time map

4. Compute arrival time distribution from an initial physical distribution

5. Multiply signal by number of primary electrons, convolve with arrival time distribution

6. Propagate signal through readout electronics

In practice, the current signal for a single event is computed at one time, which implicitly

involves convolving the signal from multiple constituent primary electrons with their arrival time

distribution. We begin with a few constraints on what wed like to model.

44

CHAPTER 6. IMPLEMENTATION 6.1. PULSES IN THE PROTOCAGE

6.1 Pulses in the ProtoCage

The BetaCage prototype at Caltech has a test source positioned within the detector with which

we can obtain reliable pulses with a well-defined energy. The test source is a sample of 55Fe

which produces 5.9 keV x-rays through electron capture and subsequent relaxation of electrons to

unoccupied lower energy orbitals [27]. The source is positioned just a few centimeters exterior to

the cathode plane, with a hole punched through a grounded copper plate serving as a rudimentary

collimator as well as a field-shaper. The source is located at l = 0.5 lW .

As such, we limit ourselves to simulating the expected pulses from 55Fe x-rays avalanching at

this location. Due to the relatively low-energy x-ray, most of the primary electrons are concentrated

around the initial photoelectric absorption, so much so that we often consider the 227 electrons

produced in atmospheric pressure P10 to be located on a single point [20]. The cloud then gains

some lateral extent via diffusion as it drifts toward the MWPC, which we can model using our

MAGBOLTZ result from Chapter 4.

We also need to understand how the data is sampled. The ADC card which digitizes data before

sending it to the ROACH operates at 746 MHz and collects 215 data points, equivalent to traces

about 40s long. Thus we have upper and lower limits on the frequency content of our data,

following the Nyquist-Shannon Sampling Theorem [29].

6.2 The Program

The simulation consists of four main helper functions and one synthesis function that brings all the

results together. We begin by explaining the helper functions, starting with the most basic. The

program in its entirety is included in Appendix D.

45

6.2. THE PROGRAM CHAPTER 6. IMPLEMENTATION

6.2.1 Arrival Times

This function, arrival_times, serves to interface with a saved dataset that contains the map from

starting position to arrival time shown in Figure 4.6. This function has 6 arguments:

1. xloc: x-coordinate of electron cloud center, in meters

2. yloc: y-coordinate

3. spread: standard deviation of lateral extent, assumed gaussian due to diffusion

4. z_spread: standard deviation of longitudinal extent

5. delay: an artificial delay to place the pulse anywhere in a time stream

6. fil_name: the file containing the arrival time map, in HDF5 binary format

There is also a hardcoded constant, ZSPEED, which is the drift speed due to the uniform field,

5 mm above the cathode array. Refer to Appendix D for the exact value used.

Initially, the function loads a vector of x-coordinates, a vector of y-coordinates and a 2D grid of

travel times into NumPy arrays, where coordinate/time pairs come directly from the map presented

in Chapter 4. Both x and y range from 0 to 5 mm with 50 points each (chosen to limit computation

time). This data is used to construct an interpolating function which takes x-y pairs as input and

outputs the appropriate travel time.

Using the xloc, yloc and spread arguments, the program samples a bi-variate gaussian distri-

bution 10000 distinct times, where the distribution has center (xloc, yloc) and standard deviation

spread in both directions. Each coordinate sample is added to a list which is then transformed to

arrival times via the interpolating function for the arrival time map. We can label these times as

txy,i, for future reference, with i = 1, 2, . . ., 10000. Thus we have incorporated the lateral extent of

the avalanche in determining pulse shape.

The longitudinal extent of the primary electrons is accounted for in the following manner. Each

starting location is assigned a random z-projection which is sampled from a guassian distribution

46

CHAPTER 6. IMPLEMENTATION 6.2. THE PROGRAM

with a mean of 0 and a standard deviation of z_spread. The z-projection is transformed to an

arrival time via tz,i = zi/ZSPEED where zi is the z-projection of the i-th data point. Note that

because the zi are centered on 0, this can increase or decrease individual arrival times. This is just

a computational shortcut in computing Equation 4.2.

Bringing it all together, the final arrival times, and thus the output of the function, are given as

tarr,i = txy,i + tz,i + tdelay with i = 1, 2, . . ., 10000 (6.1)

This type of treatment is appropriate if the initial deposition of primary electrons is reasonably

point-like, as their lateral and longitudinal extent would be completely determined by diffusion.

Note that because of non-ideal edge effects in both the COMSOL simulation of the field and

the subsequent interpolation of the arrival time map, we exclude any starting points within 0.1 mm

of the boundary. Although this potentially limits the accuracy of the arrival time distributions, we

have not significantly explored methods to resolve edge-effects.

A distribution of arrival times is not particularly elucidating in and of itself, so we refrain from

reproducing any data here. We will eventually demonstrate how the distribution spreads a pulse.

6.2.2 Avalanche Current

It naturally follows to consider the avalanche current associated with some ionization event since this

would make use of the arrival times function from the previous section. The function avalanche_current

makes use of the previously listed arguments as well, only adding five new ones:

1. F_s: ADC sampling frequency

2. N: number of data points to sample in a single trace

3. event_energy: incident radiation energy in keV

4. gain: expected gas gain, determined externally from COMSOL

5. FFT: boolean variable. If TRUE, function outputs the FFT of the pulse as well

47


The arguments F_s and N are used to discretize a continuous signal while event_energy is the

fiducial free-parameter of this simulation.

The gain used was computed from our COMSOL result. Assuming the relationG = 2((a)(smin))/V

and a value of V = 23.6 5.4 V for P10 gas [20], we found that G 3 104 by directly sampling

our from COMSOL. However, because V is in the exponent, G is very sensitive to changes

in V which, of course, has quite a large experimental error. Thus, we take G to be a malleable

quantity allowing for minor adjustment following comparison of simulated pulses to real pulses.

Regardless, with the gain fixed, the avalanche current function computes the charge on a biased

sensing wire, , by optimizing the Diethorn Formula for gas gain given by Equation 3.6. The

program uses least-squared optimization to compute charge, starting with a value of = Va ca.

Now, recall the signal current we computed in Chapter 3 which was given by Equation 3.11

Ia = Ntot e

40

cat0 + t

(3.11*)

where the definition of the characteristic time t0 is given by Equation 3.10, ca is the anode

self-capacitance, e is the elementary charge, 0 the vacuum permittivity and Ntot is calculated as,

Ntot = Floor(G Eevent

EP10

)

where G is the user-input gas gain, Eevent = event_energy and EP10 is the average energy

deposited by incident s as they ionize a single electron in P10 gas. EP10 is taken directly from

[20], while G was computed previously.

Briefly, we discuss how the signal is discretized. Since the sampling frequency Fs and the number

of samples N is controlled by the experiment, we can construct a discrete-time array t[n] as a space

of N linearly spaced points with separation dt = 1/Fs. This is easily accomplished with NumPys

linspace function, as is shown in Appendix D.

48


With this array t[n], we can directly compute a discretized current signal given by the continuous

signal sampled at the discrete times t[n],

Ia[n] Ia(t[n]), n = 0, . . . , N 1

where Ia[n] and t[n] are easily stored and manipulated in Python as NumPy arrays, allowing

for element-by-element operations or discrete-time convolution.

Finally, before we incorporate the arrival times and spread the pulse in time, we enforce that

the signal is vanishing prior to t = 0 to avoid numerical errors or artifacts as well as vanishing for

t > td, where the deposition time td is the time it takes cations to reach the cathode array defined

in Chapter 3. This is both a physical limit, as the cations eventually deposit somewhere, in addition

to a zero-padding measure as it yields a signal with almost half of the values exactly equal to 0.

I a[n] = Ia[n] (t[n]) (1(t[n] td)) (6.2)

where (t) is the heaviside step function. All that remains is to generate the actual signal

current by convolving with the arrival time distribution.

Here, the avalanche current function calls the arrival time function to obtain a set of 10,000

arrival times based on a user-defined location and extent of the initial electron cloud. Having

obtained these times, we compute an arrival time probability distribution as a histogram of the

arrival times in which the bin centers are equally spaced from t = 0 to the longest arrival time, with

a spacing of dt = 1/Fs. Lets label the value of the histogram in bin-m as H[m], which corresponds

to the number of arrival times between mdt and (m+ 1) dt with m = 0, 1, . . . , (M 1) and where

N > M as td is a assuredly shorter than the span of data traces, by construction.

The value H[m] of arrival times in each bin is then normalized by the total number of arrival

times so thatM1

m=1 Hnew[m] = 1, i.e. Hnew[m] represents a discrete probability density function.

As such, upon convolution with the signal, it wont alter the total integral, i.e. the charge deposited

49


from the avalanche, but rather just distribute the currents differently in time.

Since we have a discretely sampled signal for some number of primary electrons and a discrete

PDF of arrival times for a distribution of those electrons, we simply convolve the two to obtain

the discretely sampled signal with contributions from individual primary electrons having their

avalanche currents spread in time. For discretely sampled signals, the convolution is given as a

summation,

Isig[n] Ia[n] H[n] =k=0

Ia[k]H[n k] (6.3)

where the output of the discrete convolution is formally of length (N + M 1). This is a

consequence of the nature of the discrete convolution as an approximation to a continuous operation,

as detailed in [28] as well as the NumPy documentation for the functions used. We keep only the

first N elements (as indicated by the argument for Isig in Equation 6.3) to match the expected array

size of data from the BetaCage, noting that the large amount of zero-padding employed before keeps

the signal well away from the boundaries and thus we avoid removing any of the pulse itself.

And thus the avalanche function is complete, yielding two arrays of N = 215 data points each,

which are the sampling times t[n] and the discretely sampled current signal Isig[n], given in Equation

6.3, for an avalanche with a defined event energy.

We now consider the output of this function. Shown in Figure 6.1 is the current induced by

a simultaneous deposition of 227 electrons which is then spread in time according to the expected

distribution due to diffusion. The field strength just outside the cathode in the BetaCage prototype

is given by E 102 V/cm. With this value as input to MAGBOLTZ, we find 0.1 mm of lateral

diffusion and 0.05 mm of longitudinal diffusion over 2 cm of electron drift, where these distances

represent the value of for a gaussian distribution of possible displacements from the expected

location without diffusion.

50


(a) (b)

Figure 6.1: Spreading an 55Fe pulse in time (a) is the current from a simulta-neous deposition, while (b) is the appropriately smeared pulse. We can see thegeneral shape of the signal remains the same, while the width increases and theheight decreases.

6.2.3 Computing the Transfer Functions

Once a signal is produced on a sensing wire, it propagates through the impedance network of readout

cabling and electronics to eventually reach an amplification point. This propagation changes the

shape of the signal since the wires and cables act like transmission lines and thus have some non-

uniform frequency response as we demonstrated in Chapter 5. The amplifiers frequency response

also changes the shape of the signal, but most of the frequency content of a pulse is within the

pass-band of the amplifier and thus the effect is less significant.

Both of these functions HV_network_transfunc and amplifier_transfunc begin by construct-

ing an array of discrete frequencies f [n] = n f0 = {0, f0, . . . , Fs/2} where f0 = 1/T = 1/Ndt

is the lowest frequency we can obtain information about due to the finite length of the signal.

The frequency arrays are then manipulated into frequency dependent transfer functions according

to Equations 5.6 and 5.8 which yield some Hnet[n] and Hamp[n] respectively. Each function then

returns the array of frequencies f [n] and the array of frequency responses H[n].

We neglect to reproduce these equations here, or any corresponding plots, as they are reasonably

51

6.3. THE FINAL FUNCTION CHAPTER 6. IMPLEMENTATION

complicated and shown in the actual implementation in Appendix D. Note also that we only

consider frequencies up to Fs/2, often called the Nyquist frequency, since we can only know the

frequency content of a signal up to half of the sampling frequency due to aliasing. This concept, the

Nyquist-Shannon Sampling Theorem, is quite well understood and conventions for discrete Fourier

transforms are well established [28, 29].

6.3 The Final Function

The final step in the program is simply a synthesis function, which is what most end-users will

interact with. This function has all relevant inputs and parameters as user-defined arguments, such

as geometrical properties of the detector, the expected gas gain, ionization event energy, primary

electron cloud distribution as well as default values for a menagerie of other parameters we assume

to be relatively fixed, such as the electronic properties of the SHV cable and sensing wires.

Calling pulse_template, the function proceeds and calls the avalanche_current subroutine

which generates a current signal on the anode wire, making use of the arrival_times subrou-

tine to account for electron propagation. Following signal generation, the template function com-

putes the expected HV-network and amplifier transfer functions, via HV_network_transfunc and

amplifier_transfunc respectively.

However, to understand how the signal shaping was implemented with these transfer functions

we need a brief introduction to the Discrete Fourier Transform (DFT), although not one so detailed

as to warrant a dedicated appendix. In general, a Fourier transform decomposes an arbitrary signal

into its various frequency components each with some associated strength in the final signal, relying

on the completeness of an infinite set of single-frequency oscillations. A discretely sampled signal

has both an upper and lower limit on frequency content due to the sample spacing and finite signal

length, respectively [28, 29].

52

CHAPTER 6. IMPLEMENTATION 6.3. THE FINAL FUNCTION

Following generally accepted conventions of the definition of the DFT, which differ in normal-

ization and sign of the exponent, we have that

Isig[k] N1n=0

I[n] exp(j2nk

N

)k = 0, . . . , N 1 (6.4)

where k = 2k/N is the kth discrete frequency sampled to approximate the continuous variable

. This is also the definition of the DFT as used by NumPys FFT algorithms. This transform

convention has a corresponding inverse transform given by,

Isig[n] 1

n

N1k=0

Isig[k] exp(j2

nk

N

)n = 0, . . . , N 1 (6.5)

To compute the final signal, we inverse-Fourier transform the element-by-element product of

the HV transfer function and the amplifier transfer function (since theyre binned over the same

frequencies) into a convolution kernel using NumPys inverse Fast Fourier Transform for real inputs,

as is shown in Appendix D. This

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