Discovery of the Two-Neutrino Double-Beta Decay of Xenon-136 with Exo-200

DISCOVERY OF THE TWO-NEUTRINO DOUBLE-BETA DECAY OF XENON-136 WITH EXO-200

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF PHYSICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Russell Neilson November 2011

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/bg856gy1187

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Giorgio Gratta, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Blas Cabrera

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Michael Peskin

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

Recent neutrino oscillation measurements provide definitive evidence for non-zero neutrino masses. Oscillation measurements, however, only measure mass differences, not the absolute mass scale. Neutrinoless double-beta decay, a hypothetical nuclear transition, can probe the absolute neutrino mass scale. This process, if observed, would also imply that neutrinos are their own anti-particles and that lepton number is not a conserved quantity. The 200 kg Enriched Xenon Observatory (EXO-200) detector is currently taking data to search for the neutrinoless double-beta decay of 136Xe, with an expected half-life in excess of 1025 years. Located at the Waste Isolation Pilot Plant (WIPP) near Carlsbad, New Mexico, the detector is filled with 200 kg of isotopically enriched liquid xenon. The analysis presented here describes the recent observation with EXO-200 of the two-neutrino double beta-decay of 136Xe. This is the rarest process ever directly observed with a half-life of 2.11 ± 0.21 × 1021 years and provides important input to the measurement of the neutrinoless decay.

iv Summary

This dissertation describes initial results from the first phase of the EXO (Enriched Xenon Observatory) program of experiments. This first phase, known as EXO-200, is a detector filled with 200 kg of isotopically enriched xenon to search for a hypothetical decay process of the xenon, called neutrinoless double-beta decay. The xenon is cooled to a temperature of -104◦C, so that it liquefies in the detector, and is located deep underground in a New Mexico salt mine to evade cosmic rays. Neutrinoless double-beta decay is the subject of a major worldwide search—not just with xenon, but with a number of other elements as well. The reason for the intense interest in this process lies in what it may reveal about the nature of neutrinos, particles that interact so infrequently with other matter that their properties are very challenging to measure. Neutrinoless double-beta decay can be used to measure the neutrino mass—until recently believed to be zero, and only in the last two decades proven to merely be extremely small. It can also answer the question of if neutrinos and antineutrinos are the same particle, a result that has implications for the mystery of why the universe is filled with much more matter than anti-matter. The result reported here is not an observation of neutrinoless double-beta decay, but rather it is the discovery of two-neutrino double-beta decay in xenon-136—a related process that is an important first step in the search for neutrinoless double- beta decay. 136Xe is not the first nucleus for which this related decay has been seen, but there have been several previous unsuccessful attempts to observe the process in 136Xe. This decay has eluded experimenters until now because of its extremely long half-life, 2.11 × 1021 years according to our measurement. This is longer than any of the other two-neutrino double-beta decay half-lives that have been observed

v in the laboratory, and in fact is longer than any other decay process of any kind ever observed, with a half-life more than 100 billion times longer than the age of the universe.

vi Acknowledgments

EXO-200 is supported by the Department of Energy and National Science Foundation in the United States, the National Sciences and Engineering Research Council in Canada, the Swiss National Science Foundation, and the Russian Foundation for Basic Research. The experiment would not have been possible without the hospitality of the WIPP personnel. Part of my funding was provided by a grant from the Stanford Graduate Fellowship. This work is the result of many years of effort by the entire EXO collaboration, many of whom I have been fortunate to work closely with. Much of my education on liquid xenon detectors came from Thomas Koffas, Carter Hall, Al Odian and Ralph DeVoe, all of whom taught me a lot. I spent many days working with Francisco LePort, whose enthusiasm was always infectious. Thank you to everyone who made my time in the lab, in the cleanroom, and underground enjoyable, especially Andrea Pocar, Michelle Dolinski, Liang Yang, Bob Conley and Knut Skarpaas. I especially want to thank Phil Barbeau, Kevin O’Sullivan, Jesse Wodin and the rest of the analysis teams for the many conversations on EXO-200 data analysis. I would like to thank my thesis advisor, Giorgio Gratta, from whom I have learnt a lot that will help me in my future career. Many thanks to Matt Green, Mar´ıaMotero D´ıez,Kazumi Tolich, Nikolai Tolich, Peter Fierlinger, Karl Twelker, Axel Mueller, the rest of the Gratta research group, and the SLAC EXO group for making my time at Stanford enjoyable. And a final thanks to my wife, Naoko, for all of her support and encouragement.

vii Contents

Abstract iv

Summary v

Acknowledgments vii

1 Neutrinos and double-beta decay 1 1.1 Neutrinos masses ...... 1 1.2 Neutrino-less double-beta decay ...... 3 1.3 Motivation for 2νββ measurements ...... 7 1.3.1 2νββ measurements and NMEs ...... 7 1.3.2 Theoretical motivation for 2νββ measurements ...... 10 1.3.3 Experimental implications of the 2νββ rate for 0νββ searches 11 1.4 Previous 136Xe experiments ...... 11 1.5 EXO ...... 14

2 Liquid xenon as a detection medium 15 2.1 Ionization in liquid xenon ...... 16 2.1.1 Charge collection and xenon purity ...... 17 2.2 Scintillation in liquid xenon ...... 19 2.3 Energy resolution and correlated ﬂuctuations ...... 20

3 The EXO-200 liquid xenon TPC 24 3.1 Ionization detection ...... 24

viii 3.2 Scintillation detection ...... 26 3.2.1 LAAPDs ...... 27 3.2.2 VUV reﬂection ...... 29 3.3 Drift ﬁeld ...... 30 3.3.1 Field uniformity ...... 30 3.3.2 Induction grid transparency ...... 30 3.4 TPC vessel ...... 32 3.5 Signal cables ...... 34 3.6 Detector component preparation ...... 35

4 LAAPD characterization 36 4.1 LAAPD handling and initial qualiﬁcation ...... 37 4.2 Multiple-LAAPD test setup ...... 37 4.2.1 Data acquisition ...... 41 4.2.2 Test procedure ...... 42 4.3 Noise ...... 43 4.4 Energy resolution ...... 45 4.5 Gain ...... 49 4.6 VUV response ...... 51 4.7 Stability ...... 52

5 External detector components 54 5.1 The cryostat and HFE system...... 54 5.2 Xenon handling system ...... 56 5.3 Underground installation, shielding and muon veto ...... 59

6 Data acquisition and calibration system 60 6.1 Electronics ...... 60 6.2 Calibration system ...... 62

7 Reconstruction and detector calibration 65 7.1 Event Reconstruction ...... 65

ix 7.1.1 u-wire reconstruction ...... 65 7.1.2 Clustering ...... 66 7.1.3 v-position assignment ...... 66 7.1.4 LAAPD reconstruction ...... 67 7.1.5 3D event location ...... 67 7.2 Detector Calibration ...... 67 7.2.1 LAAPD gain calibration ...... 67 7.2.2 Charge injection calibration ...... 68 7.2.3 Drift velocity measurements ...... 68 7.2.4 Purity measurements ...... 71 7.2.5 Energy scale calibration ...... 75 7.2.6 Energy resolution ...... 81 7.3 Detector threshold ...... 84 7.4 Comparison of calibration data to Monte Carlo simulations ...... 85 7.4.1 Clustering time ...... 87 7.4.2 Detector dead volume ...... 88 7.4.3 Charge detection threshold ...... 90 7.4.4 Spectral comparison ...... 90

8 Fits to low-background data 92 8.1 Data selection ...... 92 8.2 Background model ...... 93 8.2.1 Radioactive impurities in the TPC vessel and components . . 94 8.2.2 Cosmogenic activation of the TPC copper vessel ...... 94 8.2.3 Radiation sources external to the cryostat ...... 94 8.2.4 Dissolved 222Rn and 220Rn ...... 95 8.2.5 85Kr ...... 96 8.2.6 Cosmogenic activation of the Xe ...... 96 8.2.7 List of background PDFs ...... 97 8.3 Event selection ...... 98 8.3.1 Fiducial cut ...... 98

x 8.3.2 Scintillation coincidence ...... 98 8.3.3 Alpha cut ...... 99 8.3.4 Muon veto ...... 99 8.4 Fit results ...... 100 8.4.1 Time variation ...... 102 8.4.2 Fit results versus ﬁducial cut ...... 102 8.5 2νββ detection eﬃciency ...... 103 8.6 2νββ half-life result ...... 104 8.7 Systematic uncertainties ...... 105 8.7.1 Energy scale ...... 105 8.7.2 Multiplicity assignment ...... 107 8.7.3 Fiducial volume ...... 107 8.7.4 Background model ...... 107 8.8 Alternative hypotheses ...... 108

9 Conclusions 110

A LXe recombination model 111 A.1 Previous work ...... 111 A.2 A recombination model for EXO-200 ...... 114

B A magnetically coupled piston xenon pump 117 B.1 Design and construction ...... 118 B.2 Performance characterization ...... 125

C Signal extraction with a matched ﬁlter 129

D In-situ LAAPD gain calibration 131 D.1 Laser pulser system ...... 131 D.2 Gain measurements ...... 132 D.3 Corrections ...... 133

xi List of Tables

1.1 A list of recent 0νββ experiments and their 90% conﬁdence-level (ex- 0ν cept as noted) limits on T1/2...... 6 1.2 Experimental half-lives and NME values for 2νββ...... 9

6.1 γ calibration sources...... 63 6.2 The ﬁve standard calibration source locations. Coordinates are given as (x, y, z) where the origin is the center of the TPC, x is away from the cryostat hatches, y is up and z is along the detector axis...... 63

7.1 Percentage diﬀerence in the weak source activities as measured by the TPC and from the source certiﬁcates for three cathode source positions. 90

8.1 Results of the maximum likelihood ﬁt. The amplitudes reported are the total amplitude in both single and multiple-cluster spectra in the simultaneous ﬁt...... 100 8.2 Systematic uncertainties in the 2νββ half-life...... 105 8.3 β-decays that could potentially mimic the observed 2νββ signal. The listed half-lives are either for the isotope or the parent with longer half-life if applicable...... 109

A.1 Recombination model parameters given by Dahl...... 114

xii List of Figures

1.1 Diagrams for two-neutrino double-beta decay (2νββ), a standard model process, and neutrinoless double-beta decay (0νββ)...... 4 1.2 Atomic masses of A = 136 isotopes...... 4

1.3 Spectra of the sum electron energies, Ke, from 2νββ and 0νββ, normalized to the decay Q-value...... 5

1.4 Allowed range of hmββi as a function of the minimum neutrino mass. 8 1.5 Majorana propagator resulting from 0νββ...... 9 1.6 The spectrum of events observed by Bernabei et al...... 13

2.1 Electron drift velocity in LXe (and solid Xe) as a function of electric ﬁeld...... 18 2.2 Field dependence of ionization and scintillation yields in LXe, relative to the yield at no ﬁeld...... 21

2.3 Resolutions for the ionization channel only, σe, and for the combination

of both ionization and scintillation channels, σmin...... 23

3.1 Photograph on one half of the TPC. The central cathode is not shown. 25 3.2 Simulation of the drift and collection of an electron cloud...... 26 3.3 Advanced Photonix LAAPD...... 28 3.4 A partially ﬁlled LAAPD platter...... 29 3.5 A section of the resistive divider that divides the potential to each of the ten ﬁeld-shaping rings on each half-TPC...... 31 3.6 The fully welded EXO-200 TPC being installed into the copper cryostat. 33

xiii 3.7 One of the signal cable bundles. The bundle is made from alternating signal ﬂex-cables and Teﬂon insulators...... 35

4.1 Test chamber design...... 39 4.2 The vacuum chamber of the multiple-LAAPD test setup...... 40 4.3 Electrical schematic of the test system up to the preamplifiers on the FEC...... 41 4.4 Dark noise as a function of temperature for fixed gains from 50 to 250 for one specific LAAPD...... 43 4.5 Leakage current as a function of the temperature for a typical device at fixed gain of 100...... 44 4.6 Noise of the EXO-200 production LAAPDs measured at 169 K and a gain of 100. A total of 667 devices are represented in the histogram. . 45 4.7 Pulse height spectrum of the 55Fe source for a typical device at 207 K. 46 4.8 Energy resolution of the 5.90 keV X-rays from the 55Fe source at 293 K and 197 K for a typical device. The best resolution is seen at gain of ∼100 for both temperatures...... 47 4.9 Energy resolution of the 5.90 keV X-ray line for 667 EXO-200 production LAAPDs...... 48 4.10 Energy resolution of the XSLS events for 667 EXO-200 production LAAPDs...... 49 4.11 Gain versus bias voltage near room temperature (data supplied by the manufacturer) and at 169 K (EXO data) for a typical device...... 50 4.12 Gain versus temperature at a fixed bias voltage (1439 V) for a typical device...... 50 4.13 Bias voltage required to operate at a gain of 100 for 667 EXO-200 production LAAPDs...... 51 4.14 Relative QE for 667 EXO-200 production LAAPDs...... 52 4.15 Variation of reference LAAPD gain at a bias of 1420 V and at 169 K. 53

5.1 The copper cryostat. A temporary steel hatch, later replaced with a copper one, is shown here...... 55

xiv 5.2 A set of cable feedthroughs installed on the inner vessel hatch. . . . . 57 5.3 Simpliﬁed schematic of the Xe handing system...... 58

6.1 Schematic of the EXO-200 electronics...... 61 6.2 The two copper E-boxes located in front of the cryostat (which is not visible)...... 61 6.3 EXO-200 signal read-out scheme...... 62 6.4 Drawing of the TPC vessel, welded to the inner vessel hatch, and calibration tube, in which calibration sources are inserted...... 64

7.1 A typical u-wire signal...... 66 7.2 Measurements of the wire gains in e/ADC...... 69 7.3 The distribution of drift times for 228Th calibration source clusters. . 70 7.4 Measurements of the drift velocity in the TPC for several electric ﬁelds. 71 7.5 Reconstructed energy and drift time of single-cluster events from one 228Th source calibration...... 72 7.6 Energy resolution of the 2614 keV 228Th γ line for a set of possible electron-lifetime corrections...... 73

7.7 Variation of the electron lifetime, τe, over time...... 74 7.8 Peak fits of the single-cluster 228Th (top) and 60Co (bottom) spectra. 77 7.9 Variances of the peak position measurements for the 228Th and 60Co sources at both anode and cathode positions...... 78 7.10 Energy calibration of single-cluster γ events...... 79 7.11 Comparison of the pair-production and multiple-cluster peaks to the calibration curve determined from single-cluster events. This suggests the need to include multiple calibrations as a function of that topology. 80 7.12 Ratio of reconstructed energy to true energy for a number of simulated signals...... 82 7.13 The three different energy scales: β-like events, single-cluster γ events and multiple-cluster γ events...... 83 7.14 Energy resolution for the calibration points with a parameterized fit. The three components of the fit are plotted individually...... 84

xv 7.15 Scintillation reconstruction eﬃciency...... 86 7.16 The amplitude ratio of signals on neighboring u-wires versus the time separation of the signals...... 88 7.17 Radial distributions of events for the weak 228Th source for data and true position from the simulation...... 89 7.18 Successful comparison of simulation (shaded blue) to data (points) for a weak 228Th source run...... 91

8.1 222Rn decay chain...... 96 8.2 Ratio of ionization to scintillation in arbitrary units for α and β-decays from 214Bi-214Po β-α coincidence events...... 99 8.3 Energy distributions of the low-background data...... 101 8.4 Residuals from the fit to the low background data...... 101 8.5 Fit results vs time of the 2νββ signal and the largest background component (40K)...... 102 8.6 Rate of 2νββ and 40K vs standoff distance...... 103 8.7 Distribution of the fit amplitude of the 2νββ PDF for a large number of randomly sampled energy calibrations...... 106

A.1 Comparison of the recombination model to data from the 1.5 L chamber with 500 V/cm and 1000 V/cm drift ﬁelds...... 116

B.1 Schematic diagram of the magnetically coupled piston pump...... 119 B.2 Top: Photograph of the assembled xenon pump. Bottom: Photograph of the piston removed from the pump...... 120 B.3 MAXWELL calculations of the restoring force acting on the piston magnet versus displacement...... 122 B.4 A view of the piston with sprung UHMWPE gaskets...... 124 B.5 The measured xenon ﬂow rate through the two pumps and four total sets of gaskets at inlet pressures of approximately 760 torr...... 126

xvi B.6 The measured ﬂow rate, produced by the production pump, through the xenon recirculation loop over more than 6 months of detector operations, including the period of low-background data taking described in Chapter 8...... 127

D.1 A Teﬂon diﬀuser being illuminated from behind by the laser pulser during a bench test...... 132 D.2 Simplied schematic of the LAAPD signal readout...... 134 D.3 Capacitance measurements of a single LAAPD...... 135

xvii Chapter 1

Neutrinos and double-beta decay

Neutrinoless double-beta decay (0νββ) is a hypothetical nuclear transition that if observed would have profound physics implications. The observation of 0νββ would reveal that neutrinos are Majorana particles—that is that neutrinos are their own antiparticles—as opposed to Dirac particles, and would provide a measurement of the absolute mass scale of the neutrino sector. 0νββ is also a total lepton number violating process, and is thus possibly linked to the matter-antimatter asymmetry in the universe. The implications of 0νββ, along with results from neutrino oscillation experiments suggesting that its observation may be possible in the near future, have triggered an intense world-wide campaign of experiments. Numerous reviews of the ﬁeld have been written [1–5].

1.1 Neutrinos masses

Recent observations of ﬂavor oscillation of solar, atmospheric, reactor and accelerator neutrinos [6–9] imply the existence of three massive neutrino eigenstates m1, m2, m3, distinct from the three ﬂavor eigenstates νe, νµ, ντ and with non-zero masses.

The relationship between the mass and ﬂavor states is given by the matrix Uej. For example, the electron neutrino, νe, is given by

1 CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 2

3 X νe = Uejmj. (1.1) j=1 While the measurements of flavor oscillations provide information on the neutrino 2 2 mass differences, ∆ij = mi − mj , they do not provide much information on the −3 2 absolute neutrino mass scale. The largest mass difference, ∆23 = (2.4±0.3)×10 eV , does provide a lower limit on the mass of the heaviest neutrino mh > 0.04 eV [9].

However, whether this is m1, m2 or m3 depends on the mass hierarchy, with several experimentally allowed possibilities:

“normal” : m1 ∼ m2 << m3

“inverted” : m3 << m1 ∼ m2

“degenerate” : m1 ∼ m2 ∼ m3

Non-oscillation experiments, such as β-decay end-point measurements, cosmological observations, and neutrino-less double-beta decay have the potential to provide complementary information on the neutrino sector. Experiments investigating the tritium β-decay spectrum near the end-point attempt to measure the electron neutrino mass squared

3 2 X 2 mνe = |Uei| mi , (1.2) i=1 and have set an upper limit on the electron neutrino mass mνe < 2.3 eV [10, 11]. The KATRIN experiment [12] plans to improve the sensitivity of this method by an order of magnitude. Cosmological observations can also be used to weigh neutrinos. Neutrino thermal velocities are non-negligible in the early universe, and lead to a smearing out of over-dense regions. Thus structure formation is inhibited for scales smaller that the free-streaming distance of neutrinos, a parameter dependent on the sum of neutrino masses. Combining results of small scale structure from the Lyman α forest with CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 3

results of large scale structure from the cosmic microwave background, a tight limit on the sum of neutrino masses has been obtained

3 X mi < 0.17eV [13]. (1.3) i=1 However, a more recent study with the same approach derives a less stringent bound of 0.9 eV on the sum of the neutrino masses [14]. For more detail on these and other cosmological constraints on neutrino masses see for example Reference [15].

1.2 Neutrino-less double-beta decay

Double-beta decay is a nuclear process where a nucleus with charge Z and mass number A decays to a nucleus with charge Z+2 and mass number A with the emission of two electrons. This may proceed in several modes, including two neutrino double- beta decay (2νββ) and neutrino-less double-beta decay (0νββ). The two-neutrino mode is a standard second order electroweak process

− − (Z,A) → (Z + 2,A) + e + e +ν ¯e +ν ¯e. (1.4)

This is shown schematically in Figure 1.1a, and involves two neutrons decaying to two protons, two electrons and two electron anti-neutrinos. The energy released by the decay (the Q-value) is distributed between the electrons, anti-neutrinos and, in very small amount, the nuclear recoil in the final state. Double-beta decay can only be observed for nuclei where the single-beta decay is energetically forbidden, shown for the case of 136Xe in Figure 1.2. The single-beta decay to 136Cs is forbidden, while the double-beta decay to 136Ba is allowed. This situation is not uncommon, due to differences in binding energy between even-even and odd-odd nuclei, but the 11 cases with Q-value greater than 2 MeV are of the most experimental interest, namely 48Ca, 76Ge, 82Se, 96Zr, 100Mo, 110Pd, 116Cd, 124Sn, 130Te, 136Xe and 150Nd. Of these, all but 110Pd and 124Sn have been the subject of significant studies of double-beta decay. Nuclei that decay by 2νββ may also decay by 0νββ, CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 4

Figure 1.1: Diagrams for (a) two-neutrino double-beta decay (2νββ), a standard model process, and (b) neutrinoless double-beta decay (0νββ). N and N 0 are the initial and ﬁnal state nuclei respectively.

Figure 1.2: Atomic masses of A = 136 isotopes. Single β-decay from the even-even 136Xe nuclei to the odd-odd 136Cs nuclei is energetically forbidden allowing for the possibility of observing the ββ decay to 136Ba. CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 5

Figure 1.3: Spectra of the sum electron energies, Ke, from 2νββ and 0νββ, normalized to the decay Q-value. The curves are drawn assuming the 0νββ rate is 1% of the 2νββ rate and the 1σ energy resolution is 2%.

(Z,A) → (Z + 2,A) + e− + e−, (1.5) shown schematically in Figure 1.1b. In this process, the right handed anti-neutrino emitted at vertex A must become the left-handed neutrino absorbed at vertex B. A mass term is required to ﬂip the chirality, so the amplitude of the process is dependent on ∼ m/E of the neutrino. Thus, for 0νββ to occur it is necessary that the neutrino is both massive, and its own anti-particle (i.e. it is a Majorana particle, unlike all charged fermions, which are Dirac particles). The observation of 0νββ would hence provide valuable insight into the nature and mass of neutrinos. In 0νββ all the energy is transferred to the two electrons, unlike for 2νββ where the energy is shared with the neutrinos. Thus by measuring the sum energy spectrum of the two electrons the two processes can be distinguished, illustrated in Figure 1.3. The sum electron energy spectrum for 2νββ is a broad distribution, while for 0νββ it is a detector resolution limited peak at the Q-value of the decay. If 0νββ occurs, all three neutrino mass eigenstates participate, so we deﬁne the CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 6

Table 1.1: A list of recent 0νββ experiments and their 90% conﬁdence-level (except 0ν as noted) limits on T1/2.

0ν Isotope Technique T1/2 (yr) Reference 48 22 Ca CaF2 scint. crystals > 5.8 × 10 [18] 76Ge HPGe det. > 1.9 × 1025 [19] 76 +0.44 25 Ge HPGe det. 2.23−0.31 × 10 (1σ) [16] 82Se Thin metal foils and tracking > 3.6 × 1023 [20] 96Zr Thin metal foils and tracking > 9.2 × 1021 [21] 100Mo Thin metal foils and tracking > 1.1 × 1024 [20] 116 116 23 Cd CdWO4 scint. crystals > 1.7 × 10 [22] 130 24 Te TeO2 bolometers > 2.8 × 10 [23] 136Xe Liquid Xe scint. > 4.5 × 1023a [24] 150Nd Thin metal foils and tracking > 1.8 × 1022 [25]

eﬀective Majorana mass hmββi by

2 X X 2 2 αj hmββi = Uejmj = Uej e mj , (1.6) j j

0ν where αj are phases in the matrix Uej. The 0νββ half-life, T1/2, is related to the eﬀective Majorana mass by

0ν −1 0ν 0ν 2 T1/2 = G M hmββi , (1.7) where G0ν is an exactly calculable phase space factor depending on the decay Q-value and the nuclear charge, Z, and M 0ν is a Nuclear Matrix Element (NME). Current 0ν experimental limits on T1/2 are listed in Table 1.1. There is a claim of evidence for the observation of 0νββ in 76Ge [16], but it is controversial [17]. Accurate calculations of the NMEs, which incorporate nuclear structure eﬀects, are important for extracting the eﬀective Majorana mass. In the last few years the reliability of these calculations has improved, with better agreement between the

aFollowing Reference [3] we quote the limit from footnote 4 in Reference [24]. The limit quoted 24 in the main text of Reference [24] (T1/2 > 1.2 × 10 yr) is based on an incorrect interpretation of the 90% exclusion contour from a simulataneous ﬁt of the 0νββ peaks of 136Xe and 134Xe. CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 7

various techniques being used: including the Quasiparticle Random Phase Approxi- mation (QRPA) [26–32], the nuclear shell model [33–35] and the Interacting Boson Model (IBM) [36]. While diﬀerences of about a factor of two are now typical between the techniques, it has been suggested that since the QRPA and IBM are expected to overestimate the NMEs, and the ISM is expected to underestimate them, the true uncertainties in the NMEs are smaller, of the order of 20–30% [37].

The phases αj in Equation 1.6 are unknown and can cause cancellations in hmββi leading to uncertainties in the determination of the neutrino mass. Nevertheless it is possible to place bounds on hmββi dependent on the absolute values of the mixing angles [38]:

X 2 hmββimax = |Uej| mj, (1.8) j 2 hmββimin = max 2 |Uej| mj − hmββimax , 0 . (1.9)

Therefore, if hmββi is measured by the observation of 0νββ and the mixing angles and mass diﬀerences can be determined from neutrino oscillation experiments, then a range of absolute neutrino masses can be determined. Figure 1.4 illustrates the relationship between the allowed range of hmββi and the minimum neutrino mass. Several extra-standard model mechanisms have been proposed for 0νββ, other than the light neutrino exchange described here. For example, 0νββ may be medi- ated by a right-handed lepton current [39]. However, regardless of the mechanism, the occurrence of 0νββ implies that neutrinos are Majorana particles with non-zero mass [40]. This is demonstrated in Figure 1.5, where the diagram contributing to 0νββ is inserted into the neutrino propagator.

1.3 Motivation for 2νββ measurements

1.3.1 2νββ measurements and NMEs

The 2νββ mode has been observed for all isotopes of main experimental interest for 0νββ, with the notable exception (prior to this work) of 136Xe. The most stringent CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 8

Figure 1.4: Allowed range of hmββi as a function of the minimum neutrino mass. The normal, inverted and degenerate hierarchies are in diﬀerent regions of the plot and so can be accessed with varying detector sensitivities. A sensitivity of ∼100 meV covers most of the degenerate hierarchy, ∼10 meV covers most of the inverted hierarchy and ∼1 meV is required to cover most of the normal hierarchy. In the normal hierarchy, cancellations can result in hmββi arbitrarily close to 0 and experimentally inaccessible. CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 9

Figure 1.5: Majorana propagator resulting from 0νββ.

Table 1.2: Experimental half-lives and NME values for 2νββ as recommended by Reference [41] before this work.

2ν 2ν −1 Isotope T1/2 (yr) M (MeV ) 48 +0.6 19 Ca (4.4−0.5) × 10 0.047 ± 0.003 76Ge (1.5 ± 0.1) × 1021 0.140 ± 0.005 82Se (0.92 ± 0.07) × 1020 0.098 ± 0.004 96Zr (2.3 ± 0.2) × 1019 0.096 ± 0.004 100Mo (7.1 ± 0.4) × 1018 0.246 ± 0.007 116Cd (2.8 ± 0.2) × 1019 0.136 ± 0.005 130 +1.2 20b Te (6.8−1.1) × 10 0.034 ± 0.003 136Xe > 1.0 × 1022 < 0.009 150Nd (8.2 ± 0.9) × 1018 0.063 ± 0.003

limit on the 2νββ half-life of 136Xe was provided by Bernabei et al. [24]. The 2νββ 2ν half-life, T1/2, is given by 2ν −1 2ν 2ν 2 T1/2 = G M , (1.10) where G2ν is the phase space factor and M 2ν is the NME. The measured half-lives and nuclear matrix elements for 2νββ are shown in Table 1.2. The 2ν NMEs are calculated with similar techniques to the M 0ν calculations. However, these calculations are especially diﬃcult because this mode is very sensitive to nuclear structure eﬀects. For example, for the previously unmeasured 2νββ of 136Xe, Caurier et al. [43] calculated the 2νββ half-life of 136Xe with the nuclear shell 21 model to be T1/2 = 2.1 × 10 yr, in disagreement with the limit from Bernabei et al., while Staudt et al. [44], with the QRPA, calculated a range of values from 1.50 × 1019

bA more recent measurement of (7.0 ± 0.9) × 1020 yr has been reported [42]. CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 10

22 to 2.11 × 10 yr, depending of the strength of the particle-particle interaction, gpp used in the calculation. As will be discussed below, where a measurement of 2νββ is available to compare to the calculations, it is possible to use the comparison to set parameters of the M 0ν calculations.

1.3.2 Theoretical motivation for 2νββ measurements

There remains signiﬁcant uncertainties in the calculations of the NMEs, M 0ν, in Equation 1.7. Comparisons of measured 2νββ rates with theoretical calculations of M 2ν have the potential to test the theoretical models, and have relevance to the calculations of M 0ν—in particular because the two modes have the same initial and ﬁnal states.

Rodin et al. [27] use measured 2νββ rates to constrain the value of gpp used in 0ν the calculation of M . They show that ﬁxing gpp in this way removes some of the uncertainty in the QRPA calculation of M 0ν. Simkovic et al. [45] outline an alternate formalism to relate the dominant Gamow- 0ν 0ν 2ν Teller part, MGT , of M to the matrix element, Mcl of the 2νββ in the closure 0ν 2ν approximation. The relation is based on the functions CGT and Ccl that describe the dependence of the matrix elements on the distance, r, between the pair of neutrons participating in the double beta decay. The relation they ﬁnd is

Z ∞ 0ν 0ν MGT = CGT (r)dr, (1.11) 0 Z ∞ 2ν 2ν Mcl = Ccl (r)dr, (1.12) 0 0ν 2ν CGT (r) = H(r, E) × Ccl (r), (1.13) where H(r, E) is the neutrino potential. While M 2ν is directly related to the observed 2ν 2νββ rate, Mcl requires additional theoretical calculation. Nevertheless, the authors suggest that this formalism has potential to constrain the 0νββ NMEs by 2νββ measurements. CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 11

1.3.3 Experimental implications of the 2νββ rate for 0νββ searches

Experimentally, 2νββ is a significant and unavoidable background in any search for 0νββ. The rate of 2νββ places a limit on the sensitivity of a 0νββ search for a given detector resolution, even if all other backgrounds can be eliminated. The two decays can only be distinguished by the sum energy of the two electrons, thus their separation relies on good detector energy resolution. This factor impacts the decision of detector technology and design to reach a targeted sensitivity. The 2νββ rate also has implications in the design of a daughter nucleus identification scheme, such as the one planned by EXO for the identification of the 136Ba daughter from the double-beta decay of 136Xe. For example, for a detector with 2000 kg of 136Xe, a 1022 yr 2νββ half-life corresponds to one decay every 36 s. Since the Ba identification is likely to take a similar amount of time, some online analysis is required to select only those events near the Q-value such that the dead-time is not overwhelming. In addition, the Ba ions drift very slowly in LXe, about 1 mm/s, so with an average drift length of 100 mm, there would be ∼3 Ba ions in the LXe at any given time. This must be addressed by spatial selectivity of the Ba identification scheme. The 2νββ can also potentially be used for detector calibration. It provides a source of double-beta decay events uniformly distributed throughout the source medium with which to calibrate the detector response. These decays can be used to measure the efficiency and selectivity of a daughter nucleus tagging system, since they have exactly the same final state nucleus as the 0νββ of interest.

1.4 Previous 136Xe experiments

There have been a number of previous experiments that have searched for the 0νββ and 2νββ modes of 136Xe. The two most sensitive experiments are Bernabei et al. [24] and Gavriljuk et al. [46]. Neither of these experiments claimed to have observed either 136 decay mode, but they put lower limits on the 2νββ half-life of Xe of T1/2 > 1.0 × CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 12

22 21 10 yr and T1/2 > 8.5 × 10 yr respectively. Those limits are remarkable for how long they are, considerably longer than any of the other major double-beta emitters, indicating a matrix element smaller than any of the other isotopes. The longest half- life from any of the other high Q-value double-beta decay nuclei is (1.5±0.1)×1021 yr for 76Ge [47] (see Table 1.2). Considering the first of these experiments in more detail, Bernabei et al. used a liquid xenon scintillation detector with ≈ 6.5 kg of xenon enriched to 68.8% in the isotope 136Xe, and 8823.54 hours observation time (about one year). The authors observe a number of events in the 2νββ window, and they state that the rate of events is significantly higher than when the same detector is filled with xenon enriched in the isotope 129Xe. However, the observed energy spectrum does not match that expected for 2νββ, and the authors suggest that the high rate is because the xenon enriched in 136Xe was contaminated. This suggestion is unsatisfying since they do not speculate on the type of contamination that may be responsible. The authors do not claim any observation of double-beta decay, but they place what they claim to be a conservative lower limit (with no background subtraction) on 22 the half-life of T1/2 > 1.0 × 10 yr (90% C.L). Figure 1.6 shows the energy spectrum from Reference [24] and the expected 2νββ spectrum corresponding to their stated lower limit of the half-life using the Primakoff-Rosen approximation [48]. The half-life lower limit is equivalent to an upper limit on the rate, but since their data is entirely above their limit, it is apparent that their data without any backgroud subtraction do not support their limit, and the authors must have made a mistake. In the other experiment Gavriljuk et al. used two gas proportional counters filled with 9.1 L of xenon gas at a pressure of 14.8 atm. The authors do not state the xenon temperature, but assuming it is 300 K we calculate that each chamber was filled with 0.77 kg of xenon (density from Reference [49]). One counter was filled with enriched xenon (93% 136Xe) and the other was filled with natural xenon with extracted light isotopes (9.2% 136Xe). A series of runs were taken, occasionally exchanging the two xenon gases between the two chambers, with a total live time of 8000 hours. The natural xenon spectrum was subtracted from the enriched xenon spectrum in order to subtract background contributions from the chambers or the environment. CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 13

Figure 1.6: The spectrum of events observed by Bernabei et al. The data (points) has been extracted from Fig. 3 in Reference [24] and the expected spectrum from their quoted limit has been calculated by us (line). CHAPTER 1. NEUTRINOS AND DOUBLE-BETA DECAY 14

For the enriched xenon a total of 4850 events were observed in the 2νββ window before background subtraction, over four 2000 hour data taking runs. After subtracting the natural xenon data they found −64 ± 99(stat) ± 23(sys) counts in the 2νββ 21 window, which they translate into a half-life limit T1/2 > 8.5 × 10 yr (90% C.L). A drawback of this background subtraction technique is that any differences in the efficiency of the detectors with the two gases needs to be negligible. It is not apparent how the authors could have confirmed this to be the case. In addition, it is not clear that the assumption that the two xenon samples have the same contamination is justified; there may be differences such as those suggested in Reference [24].

1.5 EXO

EXO (Enriched Xenon Observatory) is a broad experimental program to search for 0νββ in 136Xe. The ultimate goal of the program is to build a 2–10 ton double-beta decay experiment, possibly incorporating the identification of the final state 136Ba nucleus, to explore the 10 meV neutrino mass range. The ability to identify the daughter Ba nucleus on an event-by-event basis could result in a background-free experiment (except for the background from the 2ν mode with the same final state). The identification of single Ba ions in the presence of a small amount of xenon gas has already been demonstrated [50], although the ability of transferring Ba ions from a detector to a suitable tagging device is still under development. The first phase of the EXO project, EXO-200, is a low background liquid Xe (LXe) time projection chamber (TPC) with 200 kg of Xe enriched to 80% in the isotope 136Xe. EXO-200 does not incorportae Ba identification, but nevertheless is a significant improvement over previous 136Xe double-beta decay experiments in active mass, background reduction, and energy resolution. The EXO-200 detector is described in detail in Chapters 3, 4, 5 and 6. A letter on results from early data taking has been published [51]. The results discussed in that letter are expanded on in Chapters 7 and 8. Chapter 2

Liquid xenon as a detection medium

The use of liquid xenon (LXe) for radiation detection was first investigated by Al- varez in 1968 [52]. In recent years, LXe has been used (along with other liquefied noble gases) increasingly for a number of applications, including searches for Weakly Interacting Massive Particle (WIMP) interactions [53, 54], searches for double-beta decay [24] and other exotic processes [55, 56], for high-energy calorimetry [57, 58], for gamma-ray astronomy [59] and for medical imaging [60]. An overview of LXe detectors is given in Reference [61]. For double-beta decay searches, LXe offers a number of specific advantages, many of them related to its status as a noble liquid.

• Xe can serve as both the source and detector medium. Thus large homogeneous detectors are possible, taking full advantage of self-shielding, especially given the relatively high density, and full energy reconstruction.

• Other double-beta decay experiments (for example those using 76Ge or 130Te) require the growth of high-purity crystals. This is not required for a LXe detector, which greatly simpliﬁes scaling up to large masses.

• Continuous puriﬁcation is possible and relatively simple because xenon is a noble element.

15 CHAPTER 2. LIQUID XENON AS A DETECTION MEDIUM 16

• Xe has no long lived isotopes that could be cosmogenically activated. The longest lived unstable Xe isotope is 127Xe with a half-life of 36 days.

• Due to the low isotopic abundances of most double-beta decay candidates, isotopic enrichment is desirable. This is relatively easy for xenon, because it is a gas at room temperature, allowing for enrichment by gas centrifuge with no required chemistry.

• Xe can be easily transferred from one detector to another. Even with the relative ease of Xe enrichment, the enriched Xe represents a substantial fraction of the experimental cost, so the ability to re-use the Xe is a signiﬁcant advantage.

A major disadvantage of using LXe for double-beta decay searches is that the energy resolution thus far achieved is significantly worse than that possible with some of the other detector technologies. Other double-beta decay experiments with 76Ge or 130Te have shown an energy resolution at the Q-value of σ/E ∼ 0.1% [62, 63]. By comparison, the most recent previous LXe double-beta decay experiment [24], detecting Xe scintillation light only, observed resolution at the Q-value of σ/E ∼ 8%. Recent work has demonstrated that LXe detectors that allow detection of both ionization charge and scintillation light can achieve significantly better energy resolution. Work by EXO has demonstrated a resolution of 3% (σ/E) at 570 keV [64] and Aprile et al. have shown a resolution of 1.7% at 662 keV [65]. The Q-value of 136Xe double-beta decay is 2.458 MeV [66], so these results suggest detector resolution much better than 2% (σ/E) at the Q-value should be possible. While this is still significantly worse than other detector types, with sufficiently low background, and in particular with the background rejection possible with the good spatial resolution possible in a LXe TPC, competitive 0νββ searches are possible. For 2νββ searches the energy resolution is less critical because of the broad decay spectrum.

2.1 Ionization in liquid xenon

High-energy radiation interacting in LXe produces a cascade of electrons that can both ionize the Xe, producing ions (Xe+) and electrons, and produce Xe excitons CHAPTER 2. LIQUID XENON AS A DETECTION MEDIUM 17

(Xe∗). By applying an external electric ﬁeld, some of the electrons can be separated from the Xe+ ions before they have a chance to recombine. The fraction of electrons and ions that recombine depends on both the strength of the electric ﬁeld, ε, and the density of ionization, which in turn depends on the type of ionizing radiation. Recombination is discussed in more detail in Section 2.2.

The transfer of deposited energy, E0, into ionization, excitation and sub-excitation electrons is described by the energy balance equation, proposed by Platzman [67]

E0 = NiEi + NexEex + Ni (2.1) where Ni is the number of electron-ion pairs produced with average energy expenditure Ei, Nex is the number of excited atoms produced with average energy expenditure

Eex and is the average kinetic energy of sub-excitation electrons (electrons with in- suﬃcient energy to ionize or excite Xe atoms). The W -value is deﬁned as the average energy required to produce one electron-ion pair

Nex W = Ei + Eex + . (2.2) Ni This value has been measured in LXe to be W = 15.6 ± 0.3 eV [68]. By comparison with the band gap energy in LXe, Eg = 9.28 eV, we see that ∼60% of the energy goes into ionizatoin and ∼40% of the energy is taken by the other two channels, excitation and sub-excitation electrons

2.1.1 Charge collection and xenon purity

The ionization can be detected by either collecting the electrons at an anode electrode and measuring the charge (as is done in EXO-200), or extracting the electrons into a gas phase region and accelerating them to produce electroluminescence that can be detected with photo-detectors, a so-called two-phase detector [69].

The electron drift velocity, vd, has been measured as a function of external electric ﬁeld [70, 71] and is illustrated in Figure 2.1. For the range of electric ﬁelds typically CHAPTER 2. LIQUID XENON AS A DETECTION MEDIUM 18

Figure 2.1: Electron drift velocity in LXe (and solid Xe) as a function of electric ﬁeld. Image from Reference [70].

used (O(1 kV/cm)) vd ∼ 2 mm/µs. Diﬀusion of the cloud of electrons is of minimal importance in LXe detectors. The transverse diﬀusion, δ, is given by

p δ = DT d/vd (2.3) where d is the drift length and the transverse diffusion constant DT has been measured to be 80 cm2/s at 1 kV/cm [72]. Thus over a drift length of 20 cm, the diffusion is only ∼1 mm. Of significant concern in ionization detection is the capture of electrons by electro- negative impurities such as O2,H2O, CO2,N2 and N2O. These and other impurities can attach to drifting electrons preventing them from reaching the anode. The number of surviving electrons can be described by

dN e = −k (ε)N N (2.4) dt S e S CHAPTER 2. LIQUID XENON AS A DETECTION MEDIUM 19

where Ne and NS are the concentrations of electrons and impurity S respectively, kS(ε) is the electric ﬁeld dependent capture rate for S. The electron lifetime, τe is thus given by

1 τe = . (2.5) kS(ε)NS

Very often O2 is the most signiﬁcant electro-negative impurity because of its strong electro-negativity and presence in air. The value of kS(ε) for O2 at 1 kV/cm is 7 × 1010 (mol/L)−1s−1 [73], thus for an oxygen concentration of 1 part-per-billion −8 −4 (ppb) by mass, equivalent to ∼9 × 10 mol/L in LXe, τe ∼ 1.5 × 10 s. This is comparable to the ∼10−4 s drift time for a drift length of ∼20 cm. Ideally the electron lifetime would be signiﬁcantly longer than the maximum drift time, requiring

O2 concentration < 0.1 ppb by mass for a drift length of ∼20 cm.

2.2 Scintillation in liquid xenon

There are two distinct processes whereby ionizing radiation in LXe produces scintillation photons [72]. The ﬁrst process is de-excitation of Xe∗ excitons

Xe∗ + Xe → Xe∗ 2 (2.6) ∗ Xe2 → 2Xe + hv. The second process is recombination of Xe+ ions with electrons and includes the de-excitation of excitons from Equation 2.6

+ + Xe + Xe → Xe2 + − ∗∗ Xe2 + e → Xe + Xe Xe∗∗ → Xe∗ + heat (2.7) ∗ ∗ Xe + Xe → Xe2 ∗ Xe2 → 2Xe + hv. where heat is an infra-red photon [74, 75] and hv is a vacuum ultra-violet (VUV) scintillation photon. The scintillation spectrum has been measured [76] to have a CHAPTER 2. LIQUID XENON AS A DETECTION MEDIUM 20

peak at 178 nm with FWHM of 14 nm. The de-excitation of excitons (Equation 2.6) actually describes two different processes: the decay of the singlet state with a time constant of 4.3 ns, and the decay of the triplet state with a time constant of 22 ns [77]. The recombination process (Equa- tion 2.7) has a non-exponential decay structure [78] with the vast majority occurring within 1 µs. The process in Equation 2.6 has been shown to be largely independent of electric field [77], the process in Equation 2.7 is strongly dependent on electric field since the participation of two charged species (electrons and Xe+ ions) is involved. With no electric field, it is expected that all electrons and Xe+ ions would recombine, whereas for very large electric fields the recombination rate will tend to zero, leaving only Xe∗ de-excitation as a source of scintillation light. Figure 2.2 illustrates the field dependence of final ionization and scintillation yields in LXe due to the field dependence of the recombination [61].

2.3 Energy resolution and correlated ﬂuctuations

Fano [79] demonstrated that the standard deviation, δ, in the number of electron-ion pairs, N, produced by ionizing radiation is not given by Poisson statistics but by the formula

2 2 δ = (N − Ni) = F × Ni (2.8) where Ni is the mean number of electron-ion pairs and F is a constant less than 1, known as the Fano factor, depending on the material. When F = 1 the distribution is Poisson-like. The Fano factor has been calculated for LXe by Doke to be F = 0.059 [72]. The theoretical energy resolution, σE, for an energy deposit E is given by

√ σE = FWE. (2.9)

This resolution, comparable to that possible with a Ge- or Si-detector, has never been CHAPTER 2. LIQUID XENON AS A DETECTION MEDIUM 21

Figure 2.2: Field dependence of ionization and scintillation yields, Q(E) and S(E) respectively, in LXe, relative to the yield at no ﬁeld. Data is shown for electron recoil (ER), nuclear recoil (NR) and alpha events. The yield is dependent on particle type because of diﬀerences in the initial ionization density and possibly in the ratio of excitons to ions. Image reproduced from Reference [61]. CHAPTER 2. LIQUID XENON AS A DETECTION MEDIUM 22

achieved. In Reference [72] an energy resolution of 30 keV was measured for 214Bi 554 keV gamma-rays at the highest ﬁeld of 17 kV/cm. This is a factor of 40 worse than predicted by Equation 2.9. It is believed that variations in the recombination rate are the primary reason for the poor observed resolution in LXe. Egorov et al. [80] proposed that large ﬂuctuations in the number of secondary ionizing electrons (δ-rays) are responsible for variations in the recombination rate. This conclusion is supported by observed anti-correlation in the ionization and scintillation channels and improved energy resolution when the two channels are measured simultaneously and appropriately combined. This improved resolution is illustrated in Figure 2.3 for 570 keV gamma-rays from 207Bi [64]. A microscopic LXe recombination model, which reproduces the anti- correlation between ionization and scintillation, is described in Appendix A. CHAPTER 2. LIQUID XENON AS A DETECTION MEDIUM 23

Figure 2.3: Resolutions for the ionization channel only, σe, and for the combination of both ionization and scintillation channels, σmin. A signiﬁcant improvement in energy resolution is possible with the combined measurement. The image is from Reference [64] and the data is for 570 keV gamma-ray peak from 207Bi. Chapter 3

The EXO-200 liquid xenon TPC

3.1 Ionization detection

The EXO-200 TPC is divided into two drift regions by a central cathode at negative high voltage, set to -8 kV for the initial data taking period. At both ends of the TPC is a charge collection grid, referred to as the anode, composed of two wire planes at 60 degrees to each other. The regions between the cathode and collection grids are referred to as the drift regions. The anodes collect negative charge (electrons) created in the drift region, and having two wire planes at an angle to each other allows for two-dimensional reconstruction of the charge location. Figure 3.1 is a photograph of one half of the TPC, without the central cathode. The plane farthest from the cathode is referred to as the collection plane, and is held at virtual ground. All of the charge is collected on this plane. Closer to the cathode by 6 mm is the induction plane, which shields the collection plane from charge in the drift region, and detects induction signals from electrons drifting past on the way to the collection plane. The length of each of the two drift regions, from the cathode to the collection plane is 198 mm. Each of the four wire planes (two on either end of the TPC), consist of a total of 114 wires, electrically ganged together in sets of three to provide 38 electrical readout channels. The wires making up the collection plane are referred to as u-wires and the wires making up the induction plane are referred to as v-wires. The wires are

24 CHAPTER 3. THE EXO-200 LIQUID XENON TPC 25

Figure 3.1: Photograph on one half of the TPC. The central cathode is not shown. CHAPTER 3. THE EXO-200 LIQUID XENON TPC 26

Figure 3.2: Simulation of the drift and collection of an electron cloud. constructed from photo-etched phosphor bronze, and are of approximately square cross-section with 125 µm thickness and width. Within each plane the wires are spaced by 3 mm, thus each read-out channel relates to a 9 mm wide region in the detector, deﬁning the spacial resolution of the TPC. The drift and collection of an electron cloud are illustrated in Figure 3.2.

3.2 Scintillation detection

In EXO-200 the 178 nm LXe scintillation light is detected by large area avalanche photo-diodes (LAAPDs). A description of the characterization of the LAAPDs is given in Chapter 4 [81]. CHAPTER 3. THE EXO-200 LIQUID XENON TPC 27

3.2.1 LAAPDs

The EXO-200 LAAPDs are used “bare”, i.e. without standard encapsulation, and immersed in the LXe, which acts as an good electrical insulator. This eliminates issues of radioactive background and differential expansion related to the use of the ceramic encapsulation. In addition it allows for a higher packing density and eliminates the front window, resulting in a higher efficiency for VUV photon detection. The LAAPDs are Advanced Photonix (API) unencapsulated dies [82], shown in Fig- ure 3.3. They have 16 mm diameter active area (200 mm2), with an overall physical diameter between 19.6 and 21.1 mm. The thickness varies between 1.32 and 1.35 mm and the mass is 0.5 g. The LAAPDs are mounted on two gold-plated copper platters (Figure 3.4), one at each end of the TPC. The 234 devices on each platter are arranged in a triangular grid with a photosensitive packing fraction of 48%. The LAAPDs are electrically ganged in groups of seven (or, in some special cases, five or six), giving a total of 74 LAAPD electronic readout channels (37 per side). Each group is held in place by a platinum-plated phosphor bronze spring, which also provides electrical contact to the gold-plated cathodes of the LAAPDs. The platters physically locate the LAAPDs, and provide an electrical connection for the cathodes of the devices, through a ring surrounding the active surface. The platters are held at about -1400 V potential, reverse biasing the LAAPDs, and also providing a retarding potential so that electrons are not able to drift past the collection plane, and hence ensuring that all charge is collected by the collection wires. Gain matching of the LAAPDs is achieved with small adjustments of the potential of the LAAPD anodes by applying a “trim” voltage to the signal read-out cables. A total of twelve trim voltage supplies allow for each of twelve sectors of LAAPDs (each containing between 36 and 44 devices) to be supplied with a slightly different bias voltage, correcting for differences in the LAAPDs that would otherwise result in substantial variations in the gains of the devices. CHAPTER 3. THE EXO-200 LIQUID XENON TPC 28

Figure 3.3: Advanced Photonix LAAPD. The ruler indicates the size in cm and inches. The gold-plated cathode is displayed on the right and the active surface surrounded by the gold-plated ring-wafer anode on the left. Note the bevel on the cathode side, providing a longer distance along which to hold the bias high voltage. CHAPTER 3. THE EXO-200 LIQUID XENON TPC 29

Figure 3.4: A partially ﬁlled LAAPD platter. The triangular platinum-plated contact springs have been installed at the top and right; the LAAPDs at the bottom center have yet not had their contact springs installed.

3.2.2 VUV reﬂection

To maximize the photon collection efficiency of the EXO-200 detector, surfaces not covered with LAAPDs are designed for high VUV reflectivity. There are no LAAPDs on the sides of the detector because of the large electric field; Teflon [83] reflectors are mounted in these regions. Teflon is known to be a good diffuse reflector of xenon scintillation light [84], increasing the light collection efficiency by 50–150%, depending on the location of the event in the detector. The Teflon reflectors have an inner diameter of 18.3 cm, defining the radius of the nominal active volume of LXe. In reality, the LXe within a few mm of the Teflon has poor charge collection properties because of electric field non-uniformity and incomplete coverage of the anode (see Section 7.4.2), so the true active volume of LXe is slightly smaller. The front face of the LAAPD support platter is beveled and coated [85] with aluminum and magnesium fluoride in order to reflect photons that do not hit an CHAPTER 3. THE EXO-200 LIQUID XENON TPC 30

LAAPD, further increasing light collection eﬃciency.

3.3 Drift ﬁeld

3.3.1 Field uniformity

A uniform drift field between the cathode and induction plane is required for optimal detector operation. There are two primary reasons for this. First, several detector properties of LXe, such as ionization yield, scintillation yield, and the effect of impurities on electron lifetime, are dependent on electric field (see Chapter 2). We would like the detector properties to be as uniform as possible over the full detector volume. Second, accurate position reconstruction of events requires electrons to drift in straight lines at constant velocity, necessitating a uniform electric field. Field uniformity is achieved with a field cage surrounding the drift volume, shown in Figure 3.1. The field cage for each half-TPC is made from ten coaxial copper rings uniformly spaced between the cathode and anode. The potential of each ring is set via a resistive divider, located just inside the field cage as shown in Figure 3.5, which grades the potential between the cathode and induction plane. The resistors have resistance ∼900 MΩ each. They are made from a resistive paste [86] deposited on a sapphire substrate and are mounted on an acrylic block, with platinum-plated phosphor-bronze spring contacts. Electric field uniformity has been investigated by finite-element analysis with MAXWELL [87]. Variations of the electric field are less than 1% over the full drift region except for very close to the field shaping rings. The region of non-uniformity extends up to 1 cm in from the field cage over most of the drift length, and up to 2 cm in from the field cage within 2 cm of the cathode. None of this volume is included in the analysis fiducial volume.

3.3.2 Induction grid transparency

The induction plane, located in front of the collection plane, is held at a negative voltage. In order for the induction plane to be fully transparent (i.e. for all of the CHAPTER 3. THE EXO-200 LIQUID XENON TPC 31

Figure 3.5: A section of the resistive divider that divides the potential to each of the ten field-shaping rings on each half-TPC. charge to be collected on the collection plane and not on the induction place itself), the electric field between the two wire grids must be a certain amount greater than the field in the drift region. The condition for full transparency is given in the case of wires with circular cross section by Bunemann et al. [88] as

E 1 + ρ P ≥ , (3.1) EQ 1 − ρ where EP is the collection ﬁeld (between the induction and collection planes), EQ is the drift ﬁeld and

2πr ρ = , (3.2) d with r the wire radius and d the wire pitch. If we assume that the wires are circular with diameter 125 µm we get CHAPTER 3. THE EXO-200 LIQUID XENON TPC 32

E P ≥ 1.31 (3.3) EQ for full transparency. Since the detector is not ideal, the electric field ratio is set well above this minimum value, aiming for a ratio of 2.0. An analytic formula for calcu- lating the electric fields EP and EQ for multiple wire grids is given in Reference [89]. For a cathode potential of -8 kV, with the APD plane at -1400 V, we calculate that an induction grid potential of -757 V is required for a field ratio of 2.0, and set the induction plane voltage above this at -780 V. We have confirmed the results of the analytical calculations with a numerical calculation using MAXWELL, incorporating a more realistic geometry including square cross-sections for the wires. With MAXWELL we calculate the electrostatic potential in a region near the induction wires. The condition for full transparency is equivalent to the presence of a local minimum of the electrostatic potential at the induction wire location, since drifting electrons always follow a path of increasing potential and thus can not be collected at a local minimum. With this method, induction grid potentials are scanned to find the minimum required potential for full transparency, and -600 V is found to be sufficient with the cathode at -8 kV. Thus the numerical calculation confirms that -780 V will give full transparency.

3.4 TPC vessel

The TPC, including the detector planes, field cage and Teflon reflectors, is housed in a thin copper pressure vessel. The vessel is constructed from a welded cylindrical section containing the TPC, with seven legs welded to one side, which are in turn mounted onto the inner cryostat vessel hatch. Six of these legs, which are rectangular in cross-section and support the TPC, contain signal read-out cables, ground cables and bias cables for the induction grids and LAAPD platters. Two of the legs are also the xenon inlet and outlet paths. The seventh leg contains the cathode voltage cable, designed for use up to -75 kV. The TPC during installation into the cryostat is shown in Figure 3.4. CHAPTER 3. THE EXO-200 LIQUID XENON TPC 33

Figure 3.6: The fully welded EXO-200 TPC being installed into the copper cryostat. The seven legs containing the signal and high-voltage cables are visible extended to the left of the cylindrical TPC body. CHAPTER 3. THE EXO-200 LIQUID XENON TPC 34

The main cylindrical section is 1.5 mm thick, with an inner diameter of 39.62 cm. The ends ﬂare out to a diameter of 45.47 cm to allow for the larger diameter of the detection planes. By keeping the diameter of the central section as small as possible, maximum use can be made of the limited quantity of extremely valuable enriched xenon.

3.5 Signal cables

The EXO-200 front-end electronics are located outside the copper cryostat and are maintained at room temperature. Signals from the collection wires, induction wires and LAAPD gangs are carried to the electronics by an array of cables. These flexible cables are constructed by photo-etching [90] polyimide-coated copper strips purchased from Nippon Steel Chemical Co [91]. The cables travel through the six rectangular legs of the TPC seen in Figure 3.4. In each leg, a cable “bundle” is routed, consisting of a number of signal conductors as well as conducting planes for shielding and in some cases for supplying the necessary potentials of the induction plane (up to -4 kV) and the LAAPD platters (-1.4 kV). For each half-TPC, one of the three legs is dedicated to the u-wire cable bundle, one to the v-wire cable bundle and one to the LAAPD cable bundle. The cable bundles also include a number of Teflon insulating strips. The cables pass though a pair of epoxy vacuum feedthroughs, the inner one of which is cryogenic, formed in copper flanges at the cryostat hatches, so as to maintain the vacuum integrity of the TPC vessel. Figure 3.5 shows one of the cable bundles attached to the inner (cryogenic) copper feedthrough flange. Outside the cryostat, the copper traces of the cables are coated with a coverlay (Pyralux FR Coverlay [92]) to provide additional insulation and durability. The coverlay was not used on the sections of the cables inside the TPC legs to minimize radiological impurities close to the TPC, but is also not necessary because of the additional insulation provided by liquid xenon and the mechanical protection of the TPC legs. CHAPTER 3. THE EXO-200 LIQUID XENON TPC 35

Figure 3.7: One of the signal cable bundles. The bundle is made from alternating signal ﬂex-cables and Teﬂon insulators.

3.6 Detector component preparation

Special care was taken during assembly to protect detector materials from contamination. Before installation, all detector components received thorough surface treatment, to remove any radioactive surface contaminants and to degrease surfaces for LXe purity requirements. The treatment typically consisted of degreasing rinses in acetone and ethanol, an acid rinse in either nitric or hydrochloric acid, and finally a deionized-water rinse. In addition, Teflon parts were baked in dry N2 atmosphere at 150◦C to drive out volatile impurities, and all plastics (except for the polyethylene of the HV cable insulation) were stored under dry N2 for more than 6 months before the detector was filled with LXe. Chapter 4

LAAPD characterization

In recent years Large Area Avalanche PhotoDiodes (LAAPDs) have been discussed as photodetectors of scintillation light [93–95]. LAAPDs are compact, semiconductor devices with high quantum efficiency (QE) from the infrared to the vacuum ultra- violet (VUV). Made mostly of high purity silicon, very lightweight, and fabricated in a clean room environment, they can be produced with low intrinsic radioactive contamination [96]. Hence they are well suited to the detection of scintillation light in ultra-low background experiments, especially when the wavelength to be detected is in regions of the spectrum that are problematic for photomultiplier tubes (PMTs). This is the case for noble elements as the scintillation light is in the VUV region [76]. Although several groups working on the detection of low energy, rare events have been investigating LAAPDs [97, 98], EXO-200 is the first such detector to make wide use of them. LAAPDs are silicon devices consisting of a p-n junction. Light photons are absorbed on the active surface in the p-region of the diode and produce electron-hole pairs. The electric field drives the primary electrons to a multiplication region, where the field strength is above the threshold for the production of new electron-hole pairs by impact ionization. Multiplication gains of order 100 are typical, increasing more than exponentially with the applied voltage. The devices used in EXO-200 are made by growing the p-type epitaxial layer on n-type neutron transmutation doped silicon [99]. A second silicon wafer, ring-shaped and gold-plated, is bonded by a layer

36 CHAPTER 4. LAAPD CHARACTERIZATION 37

of aluminum to form the anode. The edge of the device is beveled and coated with a polyimide paint to improve the breakdown characteristics at the edges. Extensive radioactivity measurements were made of both complete and partial LAAPDs [96]. Following these measurements, the aluminum used to bond the ring wafer to the rest of the device was found to be unsatisfactory and was replaced with a special type of ultra-pure aluminum [100].

4.1 LAAPD handling and initial qualiﬁcation

A total of 851 production LAAPDs were purchased for EXO-200 and produced by API between September 2006 and June 2008. Great care has been taken at all times in the handling of these LAAPDs, both to protect the unencapsulated diodes from damage and to maintain the ultra-low radioactive background and xenon purity requirements of EXO-200. In particular, unencapsulated diodes are adversely affected by humidity, so the devices are transported in a dry-nitrogen filled container and stored in a dry box purged with boil-off nitrogen. All testing of the production LAAPDs is done in a clean room. A number of laboratory tests were performed to characterize the properties of unencapsulated LAAPDs. In particular, effort was put into understanding the noise, gain and QE of the devices, and their behavior as a function of temperature. This initial testing of pre-production LAAPDs, not to be installed in EXO-200, was done in a single-device setup with commercial electronics [101]. In addition, all 851 production LAAPDs have been individually characterized under expected operating conditions. This was done in a multiple-LAAPD test setup with custom electronics.

4.2 Multiple-LAAPD test setup

The testing and characterization of the production LAAPDs is done in a vacuum chamber maintained at better than 10−6 torr by a turbomolecular pump backed by a dry scroll pump. Figure 4.1 shows a schematic of the test chamber and Figure 4.2 displays a photograph of the system. Sixteen devices, mounted face down on a 9.5 mm CHAPTER 4. LAAPD CHARACTERIZATION 38

thick copper disk, are tested simultaneously in the chamber. The copper disk connects together the anodes for all 16 LAAPDs and maintains them at a constant and equal temperature. The back surface of each device is contacted by a beryllium copper spring, mounted, with nylon supports, on a second, removable copper disk. These cathode connections reverse bias each device at ∼1400 V for a gain of ∼100. A 16 conductor flex cable transmits signals to the front-end electronics card (FEC) outside the vacuum chamber (Figure 4.3). The temperature of the devices is maintained near the operating temperature of EXO-200 (170 K) for the testing. The LAAPDs are cooled by boil-off nitrogen gas, flowing through a copper tube brazed to the circumference of the lower copper mounting disk. The temperature of the nitrogen gas is regulated by a two-stage resistive heater attached to a section of the cooling tube external to the vacuum chamber. A 150 W heater provides coarse temperature control. A 20 W heater, controlled by a PID feedback circuit reading out a resistance temperature detector (RTD) on the LAAPD mounting disk, provides fine temperature control. This configuration achieves cool down from room temperature to 170 K in approximately two hours and temperature stability better than 0.2 K. It is important that the temperature be held within a very tight range because of the strong dependence of LAAPD gain on temperature, and a fast cool-down is important because of the production nature of the measurements. Ceramic insulators are used to electrically isolate the vacuum chamber from the boil-off nitrogen source and from the vacuum pump system. Two radioactive sources mounted below the copper disk are used to measure certain parameters of the LAAPDs. A 5 µCi 55Fe source provides 5.9 keV X-rays to measure the absolute gain of the devices, by total absorption directly in the junction. X-rays incident on the device junction generate electron-hole pairs, with an average of one pair produced for each 3.66 eV of incident energy [102]. Thus, a 5.9 keV X-ray from the 55Mn de-excitation generates ∼1600 electron-hole pairs in the active region. Measuring the gain in this way assumes that the gain is linear in the number of electron-hole pairs produced. This is not completely correct, as X-rays produce high charge densities, decreasing the local electric field and producing local heating. The non-linearity of the gain has been measured [99], and the difference in gain between CHAPTER 4. LAAPD CHARACTERIZATION 39

Figure 4.1: Test chamber design. VUV photons from the xenon scintillation light source (XSLS) and X-rays from the 55Fe source are detected by the downward facing LAAPDs. CHAPTER 4. LAAPD CHARACTERIZATION 40

Figure 4.2: The vacuum chamber of the multiple-LAAPD test setup. The copper mounting disks are at the top of the picture. The XSLS is in the middle, the cooling tubes on the right, and the 16 conductor ﬂex cable on the left. The pumping manifold can be seen at the bottom right and in the background. CHAPTER 4. LAAPD CHARACTERIZATION 41

Figure 4.3: Electrical schematic of the test system up to the preampliﬁers on the FEC.

5.9 keV X-rays and VUV photons is less than 1% at a gain of 100. A xenon scintillation light source (XSLS), made by enclosing a 1 nCi 148Gd alpha source within a cell filled to 3.3 atm with xenon gas, produces VUV photons. The scintillation light resulting from the alpha decays produces the characteristic spectrum of xenon, peaked at 178 nm (7 eV photons), so that the QE can be measured in the most relevant way. The top of the cell is fitted with a quartz window and the scintillation light generated is enhanced by a conical Teflon reflector inside the xenon cell. While 55Fe events deposit their energy into individual LAAPDs, events from the XSLS consist of a broad fan of VUV photons producing signals in all LAAPDs. Therefore the QE measure is a relative one, normalized to a “standard” LAAPD that is installed in the central position.

4.2.1 Data acquisition

The multiple-LAAPD test setup is read out using a partial prototype of the EXO-200 Data Acquisition (DAQ) system. Signals from the LAAPDs are digitized on a custom- made 16 channel Front End Card (FEC). Each channel consists of a pre-ampliﬁer, an analog shaping circuit, and a 12 bit charge analog-to-digital converter (ADC). The CHAPTER 4. LAAPD CHARACTERIZATION 42

output of the preamplifiers is sampled and digitized at 1 MHz. The digitized output from all 16 channels is transmitted over an optical cable to a Trigger Electronics Module (TEM). This module maintains a buffer of 1024 digitization samples and stores the trigger logic. Once a trigger is issued an event is created, consisting of 2048 samples from each of the 16 channels, centered on the time of the trigger. Thus each event contains approximately 2 ms of data. The event data is stored on a computer and is analyzed offline. The data acquisition rate is limited to 80 events per second by the transfer rate to the computer. Since the two radioactive sources produce different signatures, the DAQ is triggered separately for 55Fe events and XSLS events. An 55Fe trigger is issued if the signal from any one channel crosses a pre-defined threshold and an XSLS trigger is issued if the sum of all 16 channels crosses a threshold.

4.2.2 Test procedure

All production LAAPDs are characterized for leakage current, noise, gain and QE in sets of 15. In each batch, the central (16th) location on the copper support is occupied by an LAAPD designated as “standard” and left in place for many measurement cycles. This LAAPD provides information on the system stability. The LAAPDs are ﬁrst subjected to a break-in cycle consisting of at least two cool-downs from room temperature to 169 K, one warm-up from 169 K to room temperature, and 24 hours maintained at 169 K. During this break-in, the leakage currents of the devices are measured, both at room temperature and at 169 K. Fol- lowing this, measurements at 169 K of noise, gain and QE require the accumulation of a few thousand events per device. The full data acquisition for a set of 16 LAAPDs takes 2–3 hours once the temperature has stabilized. Of the 851 LAAPDs, about 180 were found to be either unstable or very noisy at 169 K and biased for a gain of 100. It was not possible to conduct the full suite of measurements on these devices and they are not included in the noise, gain and QE results presented in this paper. CHAPTER 4. LAAPD CHARACTERIZATION 43

5500 Gain 5000 250 200 4500 150 4000 100

Noise (electrons) 50 3500

3000

2500

2000 160 180 200 220 240 260 280 Temperature (K)

Figure 4.4: Dark noise as a function of temperature for fixed gains from 50 to 250 for one specific LAAPD. The base value of ∼2600 electrons is the result of the capacitance on the input of the preamplifier; the excess noise above 250 K is due to leakage current.

4.3 Noise

Figure 4.4 shows the measured dark noise as a function of temperature for ﬁxed gains of 50, 100, 150, 200 and 250 for a typical LAAPD in the single-device setup. Electronic noise has been subtracted, and the RMS widths are given in electron equivalent charge. The noise performance of the device improves with decreasing temperature down to ∼250 K, with no observable change below that. The intrinsic device noise includes contributions from both the capacitance and the leakage current. We have investigated each of these separately. The capacitance, responsible for part of the noise, decreases markedly as the bias voltage increases because of the change in depletion depth. Typical devices have a capacitance of ∼1 nF at 50 V and ∼200 pF at 1400 V. Figure 4.5 shows the leakage current as a function of temperature for a single LAAPD. The leakage current drops over four orders of magnitude, from ∼300 nA at room temperature to ∼20 pA at 218 K. This is the primary reason the device noise improves with decreasing temperature, with leakage current typically becoming CHAPTER 4. LAAPD CHARACTERIZATION 44

102

1 Leakage Currnt (nA)

10-1

220 230 240 250 260 270 280 290 300 Temperature (K)

Figure 4.5: Leakage current as a function of the temperature for a typical device at ﬁxed gain of 100. a negligible contribution to the noise below 250 K. However, we have observed many devices with anomalously high leakage current. All devices were tested by the manufacturer at room temperature to ensure that the leakage current at a gain of 100 does not exceed 1 µA. Despite this, our testing found a substantial number of devices failing this criterion, indicating deterioration after the testing at the manufacturer’s facility. While it is unclear what causes such deterioration, we have observed no sig- niﬁcant change in the leakage current after the LAAPDs are stored in our facility. As part of the acceptance for EXO-200 we have also measured leakage currents at 169 K, a more interesting parameter that the manufacturer was unable to test. A number of devices exhibit leakage currents much greater than 1 nA at 169 K, in some cases with no indication of high leakage current at room temperature. We have found that anomalously high leakage current at 169 K is often correlated with noisy or unstable devices. In the 16 device setup, the noise of the LAAPDs is determined by an analysis of the ADC output when there is no trigger. The RMS of the output around the baseline is calculated for a large number of events and averaged to give a dark noise measurement, CHAPTER 4. LAAPD CHARACTERIZATION 45

00 500 1000 1500 2000 2500 3000 3500 4000 Noise (electrons)

Figure 4.6: Noise of the EXO-200 production LAAPDs measured at 169 K and a gain of 100. A total of 667 devices are represented in the histogram. shown in Figure 4.6 for all devices. The noise of most devices lies between 700 and 1500 electrons. Although in this case the contribution of the electronics has not been subtracted, the measured noise is substantially lower than that measured in the single-device setup because the preampliﬁer is better matched to the capacitance of these LAAPDs. Devices showing electronic noise greater than 1500 electrons are considered unsuitable for installation in EXO-200.

4.4 Energy resolution

For accurate characterization of the LAAPDs gain and QE it is important to achieve good energy resolution in the 55Fe and XSLS spectra. Figure 4.7 shows an example pulse height spectrum for the X-rays from the 55Fe source for a single LAAPD. The source produces X-rays of 5.90 keV (89%) and of 6.49 keV (11%). The two peaks are not fully resolved even at low temperatures where the device noise is low. The energy resolution is determined by ﬁtting the main peak of this spectrum to a Gaussian distribution. Figure 4.8 shows the resulting energy resolution as a function of gain CHAPTER 4. LAAPD CHARACTERIZATION 46

120 Counts 100

0 0 100 200 300 400 500 600 700 800 Channel

Figure 4.7: Pulse height spectrum of the 55Fe source for a typical device at 207 K. The secondary bump on the upper side of the 5.90 keV peak is the 6.49 keV peak. both close to room temperature and at 197 K. It appears that the best resolution is obtained at a gain of ∼100 irrespective of the temperature. 2 The variance in the LAAPD output signal expressed in electrons, σN , is given by

2 2 2 2 2 2 2 σN = G σn + nehσG + nehσNU + σnoise. (4.1)

The ﬁrst two terms describe the statistical variance in the number of primary 2 photoelectrons and the gain respectively. Here G is the gain, σn is the variance in the number of primary photoelectrons, neh is the number of primary photoelectrons and 2 σG is the statistical variance in the single electron gain. In addition, σNU represents the gain non-uniformity in the diode volume and σnoise is the dark noise of the diode- preampliﬁer system. The variance in the primary electrons is determined by the Fano factor, f, with a value of about 0.1 in silicon [103]

2 σn = nehf. (4.2)

Following [99] we describe the statistical variance in the gain by introducing an CHAPTER 4. LAAPD CHARACTERIZATION 47

7.5 293 K /E (%)

E 197 K σ 7

6.5

5.5

4.5

40 50 100 150 200 250 300 350 400 450 500 Gain

Figure 4.8: Energy resolution of the 5.90 keV X-rays from the 55Fe source at 293 K and 197 K for a typical device. The best resolution is seen at gain of ∼100 for both temperatures. excess noise factor, F

σ2 F = 1 + G , (4.3) G2 with F = 2.0 at a gain of 100 [104, 105]. Thus Eqn. 4.1 becomes

2 2 2 2 2 σN = G [nehf + neh(F − 1)] + nehσNU + σnoise. (4.4)

This can be rewritten in terms of the energy released in the device, E = neh, where is average energy required to produce one e–h pair (3.66 eV for silicon). This

σE leads to the equation for energy resolution, E

σ 2 1 2 E = (f + F − 1) + σ2 + σ2 . (4.5) E E G2 NU E2G2 noise 55 2 For 5.90 keV Mn X-rays, we calculate E (f +F −1) ' (2.6%) . Figure 4.9 shows the observed energy resolution at 169 K for the production LAAPDs at a gain of 2 ∼100. The dark noise term, σnoise, is known to be negligible at 169 K, so we interpret CHAPTER 4. LAAPD CHARACTERIZATION 48

00 2 4 6 8 10 12 14 σ E/E (%)

Figure 4.9: Energy resolution of the 5.90 keV X-ray line for 667 EXO-200 production LAAPDs.

2 the peak energy resolution of ∼5.5% as due to the non-uniformity term, σNU, as was already noticed for the same type of devices in [104]. Events from the XSLS consist of 150–300 VUV photons detected by each LAAPD, depending on the device location on the mounting disk. Since the photons strike the device at random locations, gain non-uniformities are averaged out. However, the new terms: σph accounting for the Poisson distribution in the number of detected photons, σscint accounting for the photon statistics in the scintillation process and

σsource deriving from the fact that the source is not point-like and hence diﬀerent events are seen under slightly diﬀerent solid angles, contribute to the variance in the primary electrons. Indeed, 148Gd was chosen as the α source because of its low energy (3.3 MeV), producing relatively localized scintillation tracks. Figure 4.10 shows the observed XSLS energy resolution at 169 K at a gain of ∼100. In most cases the observed energy resolution is in the range 9–15%. LAAPDs tested in the central position of the mounting disk receive the most photons (∼300 per event) and have the best XSLS resolution (9–10%); LAAPDs tested in the outside positions receive fewer photons (∼150 per event) and consequently have worse resolution (13– 15%). There are some devices with XSLS resolution above 15%, but the majority of CHAPTER 4. LAAPD CHARACTERIZATION 49

00 5 10 15 20 25 30 σ E/E (%)

Figure 4.10: Energy resolution of the XSLS events for 667 EXO-200 production LAAPDs. these are either particularly noisy or have unusually low QE. In fact, although XSLS resolution is not used as a selection criterion, only two devices eventually selected as suitable for installation in EXO-200 have XSLS resolution above 15.5%, and none above 16.5%.

4.5 Gain

Figure 4.11 and 4.12 show the gain of an LAAPD as a function of bias voltage and temperature. These data demonstrate the necessary requirements to maintain stable gain. At a gain of 100, corresponding to ∼1400 V bias, and at 169 K, the slope in the gain is approximately 1.5% V−1. The gain also increases rapidly with decreasing device temperature, approximately 5% K−1 near the operating temperature of 170 K. Hence to achieve a gain stability below 1%, the variation in voltage (temperature) must be below 1 V (0.2 K). Gain measurements are made for each production LAAPD at a series of bias voltages, typically over the approximate gain range 50-200. We construct a gain CHAPTER 4. LAAPD CHARACTERIZATION 50

Gain 300 169 K 293 K 250

200

150

100

0 1300 1400 1500 1600 1700 1800 Bias Voltage (V)

Figure 4.11: Gain versus bias voltage near room temperature (data supplied by the manufacturer) and at 169 K (EXO data) for a typical device.

220 200 Gain 180 160 140 120 100 80 60 40 20 0 160 180 200 220 240 260 280 Temperature (K)

Figure 4.12: Gain versus temperature at a ﬁxed bias voltage (1439 V) for a typical device. CHAPTER 4. LAAPD CHARACTERIZATION 51

0 1360 1380 1400 1420 1440 1460 1480 Voltage

Figure 4.13: Bias voltage required to operate at a gain of 100 for 667 EXO-200 production LAAPDs. curve from these measurements, and determine the required bias voltage to operate at a gain of 100 for each LAAPD. Figure 4.13 shows the distribution of voltages required to produce the same gain of 100 for all EXO-200 production LAAPDs. Gain uniformity in EXO-200 is obtained through a combination of device matching (in groups of ∼40 units with the same bias voltage) and bias voltage tuning (for each such group).

4.6 VUV response

We determine the relative QE for xenon scintillation light by measuring the response to XSLS events of each production LAAPD with respect to a reference LAAPD. Corrections are applied for the previously measured device gain and the location dependent XSLS brightness. The brightness at each of the 16 LAAPD locations is calibrated by comparing the response to the XSLS signal for a set of LAAPDs in multiple positions. A histogram of the relative QE of the production LAAPDs is shown in Figure 4.14. CHAPTER 4. LAAPD CHARACTERIZATION 52

90 80 70 60 50 40 30 20 10

00 0.2 0.4 0.6 0.8 1 1.2 1.4 Relative QE

Figure 4.14: Relative QE for 667 EXO-200 production LAAPDs.

We can see that the majority of the devices (∼96%) have relative QE between 0.8 and 1.2. The devices with relative QE below 0.8 are not used in EXO-200. Surface cleanliness is critical for maintaining good VUV response, and we have observed the effects of surface contamination on two sets of 15 devices. Significant deterioration in VUV QE for these 30 devices was noticed following accidental high voltage discharges in the test chamber. The devices were otherwise unaffected, and in most cases their response to VUV light partially recovered after ultrasonic cleaning in methanol and deionized water followed by a vacuum bake at 160◦ C for 24 hours. This suggests that surface contamination was indeed the cause of the poor VUV response, although the nature of the contamination and the mechanism by which it was released during the electrical breakdown episodes remains unclear.

4.7 Stability

The stability of the multiple-LAAPD test setup is monitored by keeping the same device in the central position over many testing cycles. These central LAAPDs (a total of three have been used over the course of two years of testing), as well as CHAPTER 4. LAAPD CHARACTERIZATION 53

120 LAAPD #16 118 LAAPD #43 LAAPD #31 116

LAAPD Gain 114 112 110 108 106 104 0 100 200 300 400 500 600 Time (days)

Figure 4.15: Variation of reference LAAPD gain at a bias of 1420 V and at 169 K. Note the zero suppression. The horizontal axis shows time since testing began in November 2006. Three different reference LAAPDs were used over the course of two years of testing, each with slightly different gains—explaining the jumps at days 100 and 390—but for each of the three no systematic gain fluctuations are observed. several others that were characterized a second time several months after the initial characterization, also allow us to measure the stability of the devices themselves over time. Figure 4.15 displays the measured gain of the central LAAPDs over time. No systematic changes are evident and the gain of each device is constant to within 2%. Chapter 5

External detector components

The LXe TPC is housed within a double-walled copper cryostat, which is filled with HFE-7000 as both a heat transfer fluid and radiation shield. The cryostat is surrounded by a lead shield. Three fluid systems service the detector, providing Xe to the TPC, HFE-7000 to the cryostat and refrigerant to cool the cryostat and external Xe condenser. The experiment is located in a class 100 clean room at the Waste Isolation Pilot Plant (WIPP) near Carlsbad New Mexico.

5.1 The cryostat and HFE system.

The double-walled copper cryostat is shown in Figure 5.1 and can also be seen in Figure 3.4. The inner and outer cryostat vessels are both made from copper plates with wall thickness 2.7 cm. The cross-section is a regular dodecagon with height 149.2 cm for the inner vessel and 166.2 cm for the outer vessel. The length of the inner vessel is 152.2 cm and the length of the outer vessel is 165 cm. The inner vessel is wrapped in a five-layer blanket of aluminized polyester film [106] and the space between the two vessels is maintained at high vacuum to minimize heat transmission from the outer vessel to the inner vessel. The inner cryostat vessel is filled with 3M Novec Engineered Fluid HFE-7000 [107], with inlet and outlet ports at the bottom and top of the front face of the cryostat respectively. HFE-7000 is in liquid phase at both room temperature and at EXO-200

54 CHAPTER 5. EXTERNAL DETECTOR COMPONENTS 55

Figure 5.1: The copper cryostat. A temporary steel hatch, later replaced with a copper one, is shown here. CHAPTER 5. EXTERNAL DETECTOR COMPONENTS 56

operating temperature (168 K), greatly simplifying its handling. When not is use the HFE is stored in an external dewar and it is transferred between the dewar and the cryostat pneumatically with 99.999% grade bottled nitrogen. A 100 L ballast bottle, kept ∼50% full of liquid HFE, is used to regulate the HFE pressure. Thermo-electric coolers mounted to the outside of the ballast bottle regulate the temperature, and hence the vapor pressure, of the HFE in the ballast bottle, which is directly connected to the cryostat. Three heat exchangers for cooling the inner vessel are made from copper plates welded to the inner cryostat. Cooling is provided by Polycold PFC-672HC refrig- erators [108] connected to the heat exchangers via vacuum insulated lines. The refrigerant fluid is a proprietary hydro-fluorocarbon/argon mixture. Each refrigerator, connected to one heat exchanger, is capable of extracting 1500 W from the inner vessel. Openings are cut into the front of both cryostat vessels—83.6 cm diameter for the inner vessel and 106.1 cm for the other vessel. The openings are closed with 2.5 cm thick copper hatches. The cable feedthrough legs of the LXe TPC are TIG welded to the inner vessel hatch. Elastomer o-rings could not be used for sealing the surfaces at LXe temperature, including the large seal between the inner vessel and inner hatch, so custom made “U-mega” gaskets made by Jetseal [109] were used. The gasket is made from phosphor-bronze and is coated with 125 µm of indium. The signal cable feedthroughs on both hatches are formed in copper flanges using a two-component epoxy from MasterBond [110]. Acrylic cups hold the epoxy during fabrication. Figure 5.2 shows the two feedthroughs for one of the signal cables after being installed on the inner hatch but prior to installation of the outer hatch. Fig- ure 3.5 shows the the inner feedthrough from the opposite side before being installed on the inner hatch, also showing the “U-mega” gasket for sealing the feedthrough flange to the inner vessel hatch.

5.2 Xenon handling system

The Xe handing system has two primary functions: CHAPTER 5. EXTERNAL DETECTOR COMPONENTS 57

Figure 5.2: A set of cable feedthroughs installed on the inner vessel hatch. (1) The inner vessel cable flange. (2) A temporary bracket holding the outer flange before it is attached to the outer hatch. (3) The outer vessel cable flange. (4) Teflon strain relief.

1. Transfer of the Xe from high pressure gas bottles to the LXe vessel and back.

2. Continuous re-circulation of the Xe through gas phase puriﬁers.

A simplified schematic of the Xe system is shown in Figure 5.3. Xenon is transferred from the gas bottles to the LXe vessel via the feed system. The feed system consists of two redundant paths, each consisting of a regulator to reduce the pressure from ∼800 to 30 psia, and a computer controlled block-valve/proportional-valve combination to regulate the flow. The bleed system includes two redundant two-stage compressors to return Xe to the gas bottles. The compressors remove gas from a 50 L buffer volume, which is maintained at ∼650 Torr. Xe from the TPC is transferred to the buffer volume by a another pair of computer controlled block-valve/proportional- valve sets. The feed and bleed systems together are computer controlled to regulate the pressure in the LXe TPC, matching it at all times to the HFE pressure to minimize stress on the thin copper walls of the TPC vessel. CHAPTER 5. EXTERNAL DETECTOR COMPONENTS 58

Figure 5.3: Simpliﬁed schematic of the Xe handing system.

The Xe recirculation loop includes the LXe TPC, a Xe heater, a Xe pump, two hot zirconium purifiers [111] and a Xe condenser. The Xe heater consists of a heating element inside a copper cylinder, through which the Xe flows. The Xe pump, a novel magnetically coupled piston pump [112], is described in more detail in Ap- pendix B. The Xe condenser is composed of a 6.6 m copper tube wound around a copper block. The copper block is cooled via a second coil by a Telemark TVP-2000 refrigerator [113]. Between the Xe coil and the cooling coil a heating element reg- ulates the temperature, measured by ten thermocouples in various locations. Both the Xe heater and Xe condenser are housed in vacuum insulated cannisters, and are connected to the TPC by vacuum insulated transfer lines. Three Gas Purity Monitors (GPMs) [114] measure the Xe purity at different locations in the recirculation loop. By means of the recirculation loop, Xe is continuously circulated through the purifiers, removing electro-negative impurities. These impurities affect the LXe electron lifetime and may originate from outgassing of the TPC internal components (in particular the plastics used in construction—Teflon, acrylic and polyimide) and outgassing of the Xe system plumbing. The largest source of impurities is the Xe bottles themselves, so the feed system taps into the recirculation loop immediately before the purifiers, so the gas is immediately purified. CHAPTER 5. EXTERNAL DETECTOR COMPONENTS 59

5.3 Underground installation, shielding and muon veto

EXO-200 is installed underground at the WIPP facility near Carlsbad, New Mexico, located at a depth of 2150 ft (1600 m.w.e) in a salt deposit. Direct counting of the salt walls shows a contamination of 124 ± 2 ng/g of 40K, 66 ± 2 ng/g of 232Th and 27 ± 2 ng/g of 238U. The LXe TPC is protected from this external radioactivity, and radioactivity from the experiment support systems, by a 25 cm thick lead shield surrounding the cryostat. The double-walled copper cryostat and minimum 50 cm of HFE provide significant additional shielding from external radiation and from the lead shield. The lead shield is constructed from interlocking bricks to avoid any direct line-of- sight path through the shield. However, penetrations through the front wall of the lead shield for the signal cables and Xe transfer lines introduce line-of-sight openings. For this reason, a second lead wall, which the signal cables and Xe transfer lines do not pass through, is installed 10 cm in front of the first. This second lead wall was installed in October 2011, after the initial data taking period. An old-air purge, to provide radon-free air to gaps between the lead bricks and gaps between the cryostat and lead wall, was also commissioned in October 2011, so was not operational during the data collection described here. An active muon veto detector is installed to reject events resulting from the flux of muons at WIPP, measured to be 3.1×10−7 s−1cm−2sr−1 [115]. The detector consists of 30 plastic scintillator panels, provided by the KARMEN collaboration, approximately 350 cm × 65 cm × 5 cm in size. Each is read out by eight photo-multiplier tubes (PMTs). The panels are installed outside the clean room, surrounding the detector on four sides. The efficiency of the muon veto has been measured to be 95.9%. Chapter 6

Data acquisition electronics and calibration system

6.1 Electronics

A schematic diagram of the EXO-200 data acquisition electronics (DAQ) is shown in Figure 6.1. Eighteen Front End Cards (FECs) continuously digitize signals from the u-wires, v-wires and LAAPD gangs. The digitized signals are optically transmitted to the Trigger Event Module (TEM), which assembles events and transmits them to the PC for storage. The distribution board issues commands and clock signals to the FECs. The FECs are located in two copper boxes (“E-boxes”), one for each half of the TPC outside of the cryostat, shown in Figure 6.2. Signals transmitted to the FECs via the signal cables are amplified, shaped and digitized, shown schematically in Figure 6.3. The shaper circuit consists of two integrators, each with time constant 3 µs, and two differentiators with time constant 10 µs. The amplifier also has an intrinsic differentiation time of 60 µs. These shaping times are adjustable by replacing components on the FECs, and different shaping times will be used for subsequent data collection. The signals are digitized by a 12-bit 1 MHz Analog to Digital Converter (ADC).

60 CHAPTER 6. DATA ACQUISITION AND CALIBRATION SYSTEM 61

Figure 6.1: Schematic of the EXO-200 electronics.

Figure 6.2: The two copper E-boxes located in front of the cryostat (which is not visible). CHAPTER 6. DATA ACQUISITION AND CALIBRATION SYSTEM 62

Figure 6.3: EXO-200 signal read-out scheme.

Based on the trigger conditions, the TEM assembles the signals into “events”, consisting of 2048 samples from all 76 u-wire channels, 76 v-wire channels and 74 LAAPD channels. Events are assembled from 1024 µs of data prior to the trigger time and another 1024 µs after the trigger time, so that the trigger time is in the mid-point of the event waveforms. Events can be triggered by any of the u-wire, v-wire or LAAPD signals crossing a pre-set threshold or the sum of all u-wire, v-wire or LAAPD signals crossing a threshold. In addition, forced triggers are requested with pre-set time intervals.

6.2 Calibration system

The response of the detector to ionizing radiation is calibrated with external γ sources placed near the TPC. Three radioactive isotopes, 228Th, 60Co and 137Cs, are used in detector calibration, with two strengths for each isotope (listed in Table 6.1) [116]. No calibrations with the 137Cs sources were done in the data period included in the analysis described here; the detector was calibrated with the 228Th and 60Co sources. CHAPTER 6. DATA ACQUISITION AND CALIBRATION SYSTEM 63

Table 6.1: γ calibration sources. The source strengths are as of September 1st, 2009 and are accurate to ±3% (99% C.L) according to the manufacturer.

Isotope Weak source (kBq) Strong source (kBq) γ lines (keV) Half-life (yr) 137Cs 2.820 13.14 637 30.2 60Co 0.530 7.060 1173, 1332 5.27 228Th 1.417 34.04 2614 1.91

Table 6.2: The ﬁve standard calibration source locations. Coordinates are given as (x, y, z) where the origin is the center of the TPC, x is away from the cryostat hatches, y is up and z is along the detector axis.

Source position name Nominal coordinates (cm) px (25.4, 0, 0) py (0, 25.4, 0) pz (0, 0, 30.4) ny (0, -25.4, 0) nz (0, 0, -30.4)

The calibration sources are inserted near the TPC in a 3/8 inch copper tube, which passes through feedthroughs in the outer and inner cryostat vessels. The tube loops around the TPC vessel, allowing sources to be placed near all sides of the detector, except in the direction of the cryostat hatches. The calibration tube and standard source locations are shown in Figure 6.4. The standard source positions are in ﬁve of the six Cartesian directions from the center of the TPC, and are labeled as in Table 6.2. Three of the source positions are on the plane of the TPC cathode (px, py and ny) and the other two positions are behind the two TPC anodes (pz and nz). CHAPTER 6. DATA ACQUISITION AND CALIBRATION SYSTEM 64

Figure 6.4: Drawing of the TPC vessel, welded to the inner vessel hatch, and calibration tube, in which calibration sources are inserted. Blue points indicate the standard source positions. Green points are source positions studied for potential use. Chapter 7

Reconstruction and detector calibration

7.1 Event Reconstruction

The EXO-200 TPC allows for full 3-dimensional reconstruction of events. The two wire grids, crossed at a 60 degree angle to each other, provide two dimensions (u and v) of the event location, and the third dimension, z, is given by the drift time, measured by the time delay of the charge collection relative to the scintillation signal. An EXO-200 event consists of 2.048 ms-long waveforms (2048 samples digitized at 1 MHz) from all channels (76 u-wires, 76 v-wires and 74 LAAPD channels).

7.1.1 u-wire reconstruction

For each of the u-wire waveforms, collection pulses are identified with a matched-filter technique, described in Appendix C. Once signals have been identified, each one is fit to a simulated template waveform, where the simulation includes charge drift and signal shaping. A typical u-wire signal is shown in Figure 7.1.

65 CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 66

Figure 7.1: A typical u-wire signal.

7.1.2 Clustering

The result of the u-wire pulse identiﬁcation and signal ﬁtting is a set of u-wire signals with amplitude, time and channel information. We introduce the concept of a “cluster” to describe a localized energy deposit, even if the charge is split and collected on two neighboring u-wires. The cluster concept is critical in the separation of events from γ-backgrounds (which typically scatter multiple times in the LXe and result in multiple-clusters) from β-decay or ββ-decay events (where the the β particles result in a well-localized energy deposit and a single-cluster event). Collection signals on neighboring wires are combined into a single cluster if they are separated by less than a clustering time, tcl=10 µs. The value of tcl is based on a comparison of data to Monte Carlo simulations, as described in Section 7.4.1.

7.1.3 v-position assignment

The v-wires see induction signals from charge drifting by. In order to assign a v- position to each charge cluster, we compare each v-wire waveform to a template waveform with start time ﬁxed based on the u-wire arrival time. The v-position of CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 67

the cluster is simply determined by the v-wire with the best match to the template waveform.

7.1.4 LAAPD reconstruction

A pulse search is done for the LAAPD signals with a matched filter as for the u- wires. In this case, the waveforms of all 37 channels in each LAAPD plane are added together and the matched-filter search, and template signal fitting, are performed on the two LAAPD sum waveforms. Once a set of signals are identified and fit for each plane, any signals from the two planes that are within 3 µs of each other are combined into one “dual plane” signal. Signals on one plane with no associated signal on the other plane are retained as “single plane” signals. An additional fitting step involves fitting all individual LAAPD channels with signal start times fixed based on the LAAPD sum fits. It is necessary to fix the times in these individual channel fits since in many cases the signal-to-noise is very low on a single channel.

7.1.5 3D event location

For each cluster, 2D event location is given by the u and v-wires. If, in addition, an LAAPD signal is found preceding the charge collection, within the maximum drift time, then that is used to establish the true time of the event. The time diﬀerence between the LAAPD signal and the u-wire signal, divided by the drift velocity, gives the third dimension, z, of the cluster.

7.2 Detector Calibration

7.2.1 LAAPD gain calibration

We measure the gains of the individual LAAPD gangs by injecting pulses from an external laser pulser into the TPC via optical ﬁbers installed with the LAAPD cable bundle. The LAAPD gains are not used extensively in this initial charge-only analysis, CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 68

but for future analyses incorporating both ionization and scintillation for improved detector resolution it is important to calibrate the LAAPD response as accurately as possible. The LAAPD gain measurement procedure is described in Appendix D.

7.2.2 Charge injection calibration

Individual u-wire channels are calibrated with an HP-33120 precision pulser. Four different pulse sizes are passed through a precision 1:1000 attenuator, and one of two precisely measured capacitors with values 20.18 pF and 20.10 pF. Thus, well known charges are injected into each u-wire channel and the output recorded by the DAQ, providing a measurement of the channel gains in electrons per ADC count (e/ADC). These calibrations are done with the FECs disconnected from the TPC signal cables. The FEC cards are also loaded with on-board pulsers for calibration while the boards are connected to the TPC. These pulsers are not absolutely calibrated, but are used to check for gain stability over time and to correct for the load capacitance of the TPC. The capacitances of the u-wire signal channels, including the cables and the wires themselves, are 27–32 pF with some variation due to differences in cable and wire lengths. The pre-amplifiers have equivalent capacitance, given by the feedback capacitance multiplied by the amplifier gain [117], of 10–100 nF, thus the load of the cables is expected to introduce a < 0.3% correction to the gains. This correction is measured with two calibration runs with the onboard pulsers: one when the boards are connected to the TPC, and one in the setup used for the precision calibration—disconnected from the TPC. The TPC load correction is found to be 0.1–0.3%. The results of the u-wire gain calibrations, corrected for the TPC load, are shown in Figure 7.2.

7.2.3 Drift velocity measurements

The electron drift velocity is used to calculate the z-position of a cluster from the drift time. Drift velocity is measured with data from the 228Th calibration source runs. It is dependent on the electric ﬁeld, so must be measured with the same cathode voltage as used for the data to be analyzed. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 69

Figure 7.2: Measurements of the wire gains in e/ADC. Top: the u-wires in the southern anode grid, numbered 0–37. Bottom: the u-wires in northern anode grid, numbered 76–113. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 70

Figure 7.3: The distribution of drift times for 228Th calibration source clusters. Here the drift time is the time between the earliest found scintillation signal in the event and each charge cluster. The drift velocity is determined from the maximum drift time, where the event rate drops sharply. The high event rate from the 228Th source results in many accidental coincidences and for a significant number of clusters the correct associated scintillation signal was not identified by reconstruction, hence the flat components of the background. Fewer of these accidental coincidences have drift time less than zero then drift time greater that the maximum drift time because the first scintillation signal is more likely near the beginning of the event.

The drift time is determined from the maximum time for events reaching the anode. Events at the cathode will have the longest drift time, therefore the drift velocity is given by the maximum drift time divided by the cathode-to-anode distance. A histogram of drift times is plotted in Figure 7.3, and the maximum drift time (the end-point of the histogram) is taken to be the time at which the frequency of events drops to one-third of the maximum frequency. Figure 7.4 shows measurements of drift velocity at a number of electric fields. At the drift field of the initial data set, 376 V/cm, the measured drift velocity is 1.70 mm/µs. This method is also used to check for varying drift velocities in different u-v regions of the detector, and we see no significant spatial dependence of the drift velocity. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 71

Figure 7.4: Measurements of the drift velocity in the TPC for several electric ﬁelds.

7.2.4 Purity measurements

Electronegative impurities in the LXe attach to drifting electrons, reducing the col- coll lected charge signal. The collected charge, Ne , is given by the equation

coll 0 −tD/τe Ne = Ne e (7.1)

0 where Ne is the original charge, tD is the drift time, and τe is the electron lifetime.

Thus we can recover the original charge if τe is known. Figure 7.5 shows data from one 228Th source calibration run, demonstrating that the data follows Equation 7.1.

Good energy resolution requires either extremely good LXe purity, such that τe tD for all events, or an accurate measurement of τe so that all events can be corrected for charge loss.

Optimized energy resolution

To measure the electron lifetime we ﬁnd the value of τe that gives the best energy resolution for the γ lines in either the 228Th or 60Co energy spectra when the data CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 72

Figure 7.5: Reconstructed energy and drift time of single-cluster events from one 228Th source calibration. The ﬁducial cut from Section 8.3.1 has been applied so all 228 events have ∼15 µs < tD < ∼105 µs, and the Th source is located near the cathode so there are more events with longer drift times. The data displays the exponential dependence on tD and the plotted curve is given by Equation 7.1 with τe=250 µs at the approximate energy of the 2614 keV peak. The low-energy data does not display the same exponential decay because of threshold eﬀects. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 73

Figure 7.6: Energy resolution of the 2614 keV 228Th γ line for a set of possible electron-lifetime corrections. The minimum in the resolution curve corresponds to the correct value of τe. are corrected by Equation 7.1. Since the γs from the calibration sources sample the entire detector, only using the true value of τe to correct the charge will give well defined peaks in the spectra, and the value of τe that gives the best energy resolution is a good measurement of electron lifetime. Single-cluster events located within the fiducial volume (as defined in Section 8.7.3) are selected and all events are corrected for a series of purity guesses. The full-energy peak is fit with a Gaussian plus an error function (or 2 Gaussians plus two error functions in the case of 60Co). For each purity guess the resolution of the full-energy peak is plotted as a function of 1/τe, and a parabola fit to the data to find the minimum energy resolution (Figure 7.6). The minimum energy resolution corresponds to the best value for τe.

Purity variations

We observe that τe varies with time, as is illustrated in Figure 7.7. The time variation of τe is ﬁt to a piecewise polynomial function to give a purity correction for all times CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 74

Figure 7.7: Variation of the electron lifetime, τe, over time. The electron lifetime is between 210 and 310 µs over the full data taking period. Blue data points are for the 60Co sources and red data points are for the 228Th sources.

during the data taking period. The ﬁt function of τe is given by

( −284.569 + 53.6978t − 1.88664t2 + 0.0269101t3 − 0.000133772t4 : t < 58 τe = 14068.5 − 908.011t + 21.8864t2 − 0.230994t3 + 0.00090631t4 : t > 58 (7.2) where t is time in days since 01:00 MDT April 30, 2011. This ﬁt is also shown in Figure 7.7. The division of the data into two periods corresponds to an increase in the Xe recirculation rate on day 58 (see Figure B.6). There is a noticeable change in the slope of the purity at this time. The spread of the points about the ﬁtted calibration curve gives rise to an uncertainty in the calibrated lifetime of 2 µs. This value is propagated through as an uncertainty to the energy calibrations described below. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 75

7.2.5 Energy scale calibration

Peak ﬁts to purity-corrected source data

Energy scale calibration of the detector is done with the known energy γs from the calibration sources. The 228Th decay chain has a strong γ line with energy 2614 keV from the decay of 208Tl, and the 60Co decay has two strong γ lines with energies 1173 keV and 1332 keV. In addition, pair-production interactions of the 2614 keV γs produce 511 keV positron annihilation photons to give another calibration energy. We also use the pair-production peak at 2614 − 2 × 511 = 1592 keV for calibration, but since it is a different type of interaction a different energy scale is obtained, as will be described below. The calibration source runs are typically taken without a physics trigger. Instead, forced triggers are requested at the maximum rate events can be processed by the DAQ (∼ 50 Hz). The sources are strong enough that a large fraction of events triggered this way include γ interactions. Strong-source calibration runs in particular have many events with multiple γ interactions. To avoid any ambiguity in assignment of charge clusters to scintillation signals, if any two scintillation signals are separated by less than 150 µs from each other (a time interval slightly longer than the maximum drift time), both are excluded from the calibration (along with their associated charge clusters). Events within the standard fiducial volume (as defined in Section 8.7.3) are used in the analysis. The energy scale of single-cluster events and multiple-cluster events are calibrated separately. For the pair-production lines at 511 and 1592 keV, only single-cluster events are used because of the way these events are selected. Pair production events from the 228Th source data are selected by first identifying three-cluster events. Then, the cluster with the largest energy deposition is identified, and the angle, θ, that is made from the smaller clusters going through the largest is calculated. Events with cos θ = −1, i.e. back-to-back emission, are selected as pair production events. The largest energy deposition is used for the 1592 keV calibration point, and the two smaller clusters are used for the 511 keV calibration point. The calibration data is purity corrected using Equations 7.1 and 7.2. Figure 7.8 CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 76

shows fits to the 2614 keV peak in the 228Th spectrum and the two peaks in the 60Co spectrum for single-cluster events. The fit functions are a Gaussian peak plus an error function for 228Th and two Gaussians plus two error functions for 60Co. The peak positions are measured over a number of calibration runs taken at different times, including runs with both source strengths for each isotope and all five standard source locations surrounding the TPC. The distributions of these peak fits can be seen in Figure 7.9. There is some unexplained position or time dependent variation, and in particular there appears to be a difference in the measured values between calibrations performed with the sources at the cathode versus either anode. The error bars on the points are a combination of systematic errors (from the uncertainty in the purity correction) and statistical errors. We take the spread of the points for each calibration energy to represent the uncertainty in the measurement of the peak position for that energy.

Energy calibration

Figure 7.10 shows the results for the four single-cluster γ peaks ﬁt to a linear energy calibration. The best ﬁt line relating the “true” energy to the reconstructed energy is:

Etrue = 0.916 ± 0.007 × Erecon + 85 ± 7, (7.3) where the uncertainty in the best fit is illustrated by the two plotted curves. Figure 7.11 shows the percentage deviation of the points from the best fit line. Also plotted are the multiple-cluster points and the pair production point, relative to the single-cluster linear calibration. It is evident that these additional points fall outside of the allowed range of calibrations. The offset of the 1592 keV peak from the single-cluster γ calibration is particularly important because these pair production interactions are topologically very similar to 136Xe ββ events. We understand this offset to be the result of different detector responses for different event topologies, and we observe the same offset in simulation for all β-like events when compared with single-cluster γ events. It should be noted CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 77

Figure 7.8: Peak ﬁts of the single-cluster 228Th (top) and 60Co (bottom) spectra. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 78

Figure 7.9: Variances of the peak position measurements for the 228Th (top) and 60Co (bottom) sources at both anode and cathode positions. For the 60Co source both the 1173 and 1332 keV peak calibrations are plotted. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 79

Figure 7.10: Energy calibration of single-cluster γ events. Error bars are smaller than the markers. The curves represent the 1σ band of allowable calibrations. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 80

Figure 7.11: Comparison of the pair-production and multiple-cluster peaks to the calibration curve determined from single-cluster events. This suggests the need to include multiple calibrations as a function of that topology. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 81

that γ events identified by reconstruction as single-cluster are actually mostly multiple Compton scatter events that can not be resolved with the detector position resolution, rather than single photo-absorption events, so they are generally not β-like. Figure 7.12 shows a comparison of β- and γ-like events from simulation. It is clear that all β-like events exhibit a different energy response than single-cluster γ events, and we can use the ratio of the energy response for β- and γ-like events from simulation to determine an energy calibration curve for β-like events. This curve is plotted in Figure 7.13 relative to the single-cluster calibration, and we see that the pair-production 1592 keV peak lies on this predicted curve. We also see in Figure 7.12 that selecting only the photo-electric interactions of the γs gives a response similar to that for β-like events, as expected since these events are topologically equivalent to β events. We suspect a combination of several detector effects is responsible for the different response to γ- and β-like events: induction signals from concentrated charge deposits producing larger induction signals on neighboring collection wires, which are more likely to be wrongly identified as charge collection signals; ballistic deficit in the signals from the more diffuse γ events giving a lower peak amplitude; and missing energy from γ clusters below the detector threshold. The first two of these effects can be reduced with less aggressive filtering on the signals, which allows better discrimination between induction and collection signals and reduces ballistic deficit. The signal filtering was changed in September 2011, which is expected to reduce the offset for future data. The multiple-cluster peaks are fit with a separate linear fit, shown also in Fig- ure 7.13. The multiple-cluster best fit line is:

Etrue = 0.939 ± 0.015 × Erecon + 90 ± 24. (7.4)

7.2.6 Energy resolution

The energy resolution of the calibration peaks, σE, is determined from the peak ﬁts. The values are plotted in Figure 7.14, along with a parameterized model that includes √ a term independent of energy, a term that scales with E and a term that scales CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 82

Figure 7.12: Ratio of reconstructed energy to true energy for a number of simulated signals. Shown are 136Xe ββ events, single β events (from the decay of 234mPa), single- cluster events from a 228Th source and photo-electric interactions from the 228Th. A ∼4% shift in energy response between β-like and γ-like events is evident. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 83

Figure 7.13: The three different energy scales: β-like events (thick line), single-cluster γ events (uncertainty band illustrated by thin solid lines) and multiple-cluster γ events (dotted lines). The single and multiple-cluster calibrations are linear fits to the data; the β-like calibration reflects the offset in simulation between β and γ events, and the 1592 keV pair-production peak falls right on the line. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 84

Figure 7.14: Energy resolution for the calibration points with a parameterized ﬁt. The three components of the ﬁt are plotted individually. with E:

2 2 2 2 2 σE = σ0 + σ1E + σ2E . (7.5)

We do not attempt to ascribe physical meaning to the parameters σ1, σ2 and σ3.

7.3 Detector threshold

Reconstruction efficiency for low energy events is dominated by the efficiency for re- constructing the LAAPD scintillation signal. Measuring the LAAPD reconstruction efficiency as a function of event energy is challenging because if the scintillation signal is not reconstructed, it is impossible to determine the drift time and to correct CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 85

the energy for τe. To conservatively estimate the threshold, we apply a very crude purity correction, assuming that all events from cathode calibration source runs are located at the cathode and have the maximum drift time, while all events from anode calibration source runs are located at the anode and have zero drift time. Figure 7.15 displays plots of the probability for detecting the scintillation signal as a function of the ionization energy, both before and after applying the crude purity correction. These data were taken some time after the other calibration runs, when the purity had improved (τe = 800 µs), so that errors introduced by the crude purity correction are smaller. In the most conservative case, that is the cathode runs after “purity correction”, the scintillation detection efficiency approaches 100% for E > 700 keV. We choose to apply a 720 keV energy threshold for the low background data analysis to avoid the region where the efficiency is not well understood. The energy threshold obtained in this way is conservative not only because we over- correct for purity, but also because many of the events for which the scintillation signal is missed are either very close to the cathode, or very close to one of the anodes—where shadowing and other geometric effects reduce the scintillation detection efficiency. Those regions are excluded by a fiducial cut in the analysis of the low-background data, so the threshold in the fiducial volume is likely lower than this analysis indicates. The same fiducial cut could not be applied here for the same reason it is not possible to apply a more accurate purity correction—the z coordinate is unknown for events with no reconstructed scintillation signal.

7.4 Comparison of calibration data to Monte Carlo simulations

Interactions in the EXO-200 detector are simulated with the Geant4 toolkit [118]. To verify the accuracy of the detector simulation, data from the calibration sources is compared to data from the Monte Carlo simulation. Where necessary, these comparisons are used to set some detector parameters in the simulation. First, the energy scale of the Monte Carlo is corrected in a similar manner as CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 86

Figure 7.15: Scintillation reconstruction efficiency. The black curves are for the anode source positions, while the red are for two of the cathode positions. Left: the uncorrected charge scale efficiencies. The difference in the two anode curves indicates a different scintillation threshold for the two TPC halves. Right: the charge signal has been corrected by the worst case purity (assuming all events where at the cathode for those source positions) for τe=800 µs. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 87

the real source calibrations. The energy resolution, as a function of energy, is also corrected according to Equation 7.5. The simulation does not accurately predict the LXe resolution, so Gaussian smearing with the appropriate energy dependent σ is added to the reconstructed energy of each simulated event, so as to reproduce the energy resolution shown in Figure 7.14.

7.4.1 Clustering time

One of the most powerful aspects of the EXO-200 detector is the separation of single- and multiple-cluster events, because the multiple-cluster spectrum heavily constrains the contribution of γ-backgrounds to the single-cluster spectrum. Therefore, it is particularly important that the simulation accurately predicts the multiplicity of the data. This is checked by looking at the ratio of single-cluster to multiple-cluster events from the calibration source runs, both for data and simulation.

Multiplicity is dependent on the clustering time, tcl, and choosing too small a value for tcl results in significant differences in multiplicity between data and simulation. The reason is illustrated in Figure 7.16, where clear differences between data and simulation in the time separation of related signals on neighboring wires can be seen. The related signals are a combination of charge sharing and induction, and it is the induction signals, which precede the collection signals by several µs, that can result in relatively large time separations between related signals. Ideally, induction signals would be excluded for optimal energy resolution, but it is not possible to differentiate the induction and charge sharing in a consistent way in both data and simulation, so we are forced to combine all related signals by setting a fairly loose clustering time of tcl=10 µs. This choice of tcl ensures good agreement between data and simulation at the expense of a slight degradation of the γ-background rejection. Combining induction signals and charge sharing signals in this way also slightly degrades the energy resolution, which is not critical for this 2νββ analysis. It is also partially responsible for the three different energy scales observed for different event topologies. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 88

Figure 7.16: The amplitude ratio of signals on neighboring u-wires versus the time separation of the signals. E1 and t1 are the amplitude and time of the larger of the two signals, E2 and t2 relate to the smaller signal. Only events where exactly two wires had reconstructed signals are shown. Left: data with the 60Co calibration source. Right: a simulation of the 60Co source. It is evident that a 10 µs clustering time is necessary to combine all related signals on neighboring wires in both data and simulation.

7.4.2 Detector dead volume

We expect there are regions of the LXe inside the Teflon reflectors from which ionization is not fully collected. Near the field cage the drift trajectories are not perfectly parallel to the detector axis, and some of the charge will drift into the reflectors. Also, the u-v anode grid does not fully cover the region inside the reflectors. Neither of these effects, in part due to mechanical imperfections, are expected to be accurately modeled by the simulation. The fiducial cut applied to the data excludes this region of poor charge collection. However, since the non-fiducial LXe acts as a Compton-veto, differences between simulation and data in the non-fiducial region are still important. Figure 7.17 shows the radial distribution of events from the 228Th source for data and simulation. There is a substantial discrepancy between the two at large radius, illustrating the inefficient charge collection. To address this, we simply ignore charge clusters with R > 168 mm (i.e. within 15 mm of the Teflon reflectors) in both data and simulation, so that differences in this region are irrelevant. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 89

Figure 7.17: Radial distributions of events for the weak thorium source for data (black) and true position from the simulation (red). The curves are normalized to have the same integral. The simulation disagrees with the data for R ≥ 170 mm, conﬁrming that there is ineﬃcient charge collection in this region. CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 90

Table 7.1: Percentage diﬀerence in the weak source activities as measured by the TPC and from the source certiﬁcates for three cathode source positions.

Position 228Th 60Co px -2.6% 8.2% py 6.9% -1.9% ny 3.9% -1.3%

7.4.3 Charge detection threshold

Event multiplicity is aﬀected by the charge reconstruction threshold, which can not be accurately measured because it is well below 500 keV, where we do not have an energy calibration. Thus, the charge detection threshold is considered to be a free parameter of the simulation, and is adjusted to improve agreement between simulation and data. A charge detection threshold of 90 keV in simulation gives the best overall agreement in the ratio of single- to multiple-cluster events.

7.4.4 Spectral comparison

Figure 7.18 shows a comparison of Monte Carlo simulation and calibration data for the weak 228Th source. The comparison to the weak source avoids pile-up effects that are present with the strong calibration sources. The overall agreement is quite good, although there is some discrepancy in the peak shapes. To assess the agreement in multiplicity assignment, the spectrum is divided into 500 keV energy regions, and the ratio of single- to multiple-cluster events in each region is compared. In all regions the disagreement between data and simulation is less than 7%, so we assign a 7% uncertainty in this ratio. Similar comparisons have been made with the weak 60Co and 228Th calibration sources in all three cathode source positions. The overall event rates are compared to the expected rates based on the known source activities from the source certificates. The results of the rate comparisons are shown in Table 7.1. The excellent agreement confirms that the detector is well understood and that the Monte Carlo simulation accurately reproduces the detector response to external CHAPTER 7. RECONSTRUCTION AND DETECTOR CALIBRATION 91

Figure 7.18: Successful comparison of simulation to data for a weak 228Th source run. Left: single-cluster events. Right: multiple-cluster events. There is no renormaliza- tion of the simulation to match the data, instead the simulation is normalized to the source activity determined from the source certiﬁcate.

γ-radiation. The rates have not been compared for the anode source positions since the TPC vessel end-caps and anode planes, which the γs must pass through, are complicated and have not been modeled with suﬃcient accuracy in the simulation. ββ events, produced uniformly in the LXe volume, are easier to model faithfully than any of the γ sources located outside the detector. Chapter 8

Fits to low-background data

The data presented here were collected between May 21, 2011 and July 9, 2011. The majority of this time was devoted to “low-background” data taking to search for the 2νββ of 136Xe, but about two hours of every day was devoted to detector calibration using the 60Co and 228Th sources. Each day, a specific source was placed at one of the five standard positions near the TPC. Probability density functions (PDFs) are fit to the low background data with an unbinned maximum likelihood fit. The PDFs are generated from Monte Carlo simulations of the 2νββ signal and possible backgrounds. All of the PDFs include an energy spectrum for both single-cluster and multiple-cluster events with the ratio of the two spectra as given by the simulation. The fit is simultaneous in both the single and multiple-cluster spectra and is done with the Roofit framework [119].

8.1 Data selection

High-quality, low-background, physics runs are selected by four criteria.

1. Livetime: The livetime of the runs must closely match the run-time. To measure livetime, a forced trigger at 0.1 Hz is implemented in addition to the physics trigger and a count of the number of forced trigger events gives the livetime. A few runs were found to have livetime signiﬁcantly less that the run time. This

92 CHAPTER 8. FITS TO LOW-BACKGROUND DATA 93

is attributed to a malfunctioning DAQ and these runs are excluded from the analysis.

2. Purity: Only runs with a measured electron lifetime > 200 µs are included. This includes all runs after May 20, 2011.

3. Muon veto: Runs where the muon veto was not running optimally are removed. Stability of the muon veto is evaluated by looking at the rate of muon events in each panel that are coincident with a muon event in at least one other panel. This coincidence requirement gives a more stable measurement than a measurement of the rate in each panel, which is strongly aﬀected by electronic noise, possible light leaks and the locations of calibration sources.

4. Electronic noise: A few runs were taken during periods of signiﬁcant mainte- nance operations near the detector—potentially resulting in electrical or acous- tic interference and an increase in the electronic noise. These runs are excluded from the analysis.

8.2 Background model

The background model consists of a set of PDFs of the energy spectra for both multiple-cluster and single-cluster events from each potential source of background. During the design and construction of EXO-200, a database of radiopurity measurements was created of all materials installed inside the Pb shield, the Pb shield itself, and the salt in the cavern walls. The database also includes eﬃciencies from Monte Carlo simulations to calculate the event rate in the TPC from each background source. From this database, many of whose entries are only upper limits, we determine which components may contribute signiﬁcantly to the total background. These measurements focus on the three naturally occurring radioactive isotopes: 40K, 232Th and 238U. Other potential sources of background are also considered for the background model, including Rn and Kr dissolved in the LXe and cosmogenic backgrounds. CHAPTER 8. FITS TO LOW-BACKGROUND DATA 94

8.2.1 Radioactive impurities in the TPC vessel and components

The radiopurity measurements and Monte Carlo simulations indicate that 40K, 232Th and 238U in the TPC copper vessel, the copper calibration source tubing, the signal read-out cables and the LAAPDs are possible background contributors. We have found that it is not necessary to use diﬀerent PDFs for each of these locations since they all give very similar background spectra. For our background model it is suﬃcient to consider all of these as the same component, which we call “TPC vessel”.

8.2.2 Cosmogenic activation of the TPC copper vessel

We also consider cosmogenic activation of the TPC copper vessel, and add to the background model 137Cs, 60Co, 54Mn and 65Zn in the TPC vessel. Other possible cosmogenic isotopes, such as 58Co and 59Fe, have half-lives that are too short (71 and 45 days respectively) to be signiﬁcant given the time the copper has been underground, shielded from cosmic rays.

8.2.3 Radiation sources external to the cryostat

The low-background data set was taken before the installation of an old-air purge to remove 222Rn from the air inside the Pb shield, so there may be some background from external 222Rn. Also, because the Pb wall installation was not complete, there may be some background from 40K, 232Th and 238U in the salt cavern. For practical reasons, these backgrounds are all simulated as if they originate on the surface of the cryostat, which requires much less processing time than simulations originating in the salt walls. For the 232Th and 238U decay chains we only simulate the highest energy decays within the chains, 208Tl and 214Bi respectively, since those decays dominate the background contribution. The dominant decay in the 222Rn decay chain is also 214Bi, so it is not necessary to generate a separate PDF for the external 222Rn. CHAPTER 8. FITS TO LOW-BACKGROUND DATA 95

8.2.4 Dissolved 222Rn and 220Rn

We expect a small amount of Rn gas to be emitted from components of the Xe gas system. Since Rn is a noble gas it is not removed by the purifiers (and in fact the purifiers may be a source of Rn); so some Rn is dissolved in the LXe. There are two naturally occurring Rn isotopes, 220Rn and 222Rn, produced as part of the 232Th and 238U decay chain, respectively. One of these, 220Rn, has a half-life of only 56 s, so likely decays before reaching the TPC if produced externally to the detector, while 222Rn has a half-life of 3.8 days. We measure the amount of both Rn isotopes by looking for coincident events in the low-background data. For the 220Rn decay chain, we look for the 220Rn-216Po α-α coincidence with half-life 145 ms. We observe no excess of events consistent with 220Rn, so we are able to constrain the 220Rn contamination to be < 0.04 µBq/kg (90% C.L), a negligible amount. For the 222Rn decay chain (Figure 8.1), we look for both the 214Bi-214Po β-α coincidence with half-life 164 µs, and the 222Rn-218Po α-α coincidence with half-life 3.1 minutes. In both cases, we see evidence of 222Rn contamination, an observation confirmed by an α-spectroscopy analysis performed using only scintillation signals. The measured contamination of 222Rn is 4.5 ± 0.5 µBq/kg, and this is included in the background model. Simulating the 222Rn decay chain, to generate the background PDF, is more complicated than the other background components because the 222Rn and its progeny are mobile. The observed 214Bi-214Po β-α events are about two-thirds on the cathode and one-third distributed uniformly throughout the bulk LXe, indicating that some of the 222Rn progeny remain positively charged and drift in the electric field to the cathode. We simulate the first part of the 222Rn decay chain, i.e. the decays of 222Rn, 218Po and 214Pb, as occurring uniformly throughout the LXe, including both the active LXe inside the field cage and the inactive LXe outside the field cage. The 214Bi and 214Po decays in the active LXe as simulated as one-third uniformly in the LXe and two-thirds on the cathode. For the 214Bi and 214Po decays in the inactive LXe we assume that they are uniformly distributed. From most locations in the inactive LXe, CHAPTER 8. FITS TO LOW-BACKGROUND DATA 96

Figure 8.1: 222Rn decay chain. ion drift trajectories terminate at a surface within about 1 cm, so the 222Rn progeny decay in close proximity to their parent.

8.2.5 85Kr

85 Kr is a radioactive ﬁssion product with Qβ=687 keV and half-life 10.8 yr that is present in the atmosphere and generally contaminates Xe, as a fraction of natural Kr. The EXO-200 enriched Xe was measured [120, 121] to contain 25 ± 3 × 10−12 g/g of Kr. However, since we only consider the spectrum > 720 keV, above the end-point of the 85Kr decay, it is not necessary to include 85Kr decay in the background model.

8.2.6 Cosmogenic activation of the Xe

For the TPC copper vessel, we only need to consider cosmogenic activation prior to installation underground, since the activation rate underground is negligible. For the Xe itself, since there is much more of it, and any activated isotopes would be produced CHAPTER 8. FITS TO LOW-BACKGROUND DATA 97

in the bulk LXe, we also consider activation in situ. The most signiﬁcant activation products are 135Xe and 137Xe, from neutron capture on 134Xe and 136Xe respectively.

8.2.7 List of background PDFs

The list of PDFs included in the background model is:

1. 40K in the TPC vessel

2. 232Th and progeny in the TPC vessel

3. 238U and progeny in the TPC vessel

4. 137Cs in the TPC vessel

5. 60Co in the TPC vessel

6. 54Mn in the TPC vessel

7. 65Zn in the TPC vessel

8. 40K external to the cryostat

9. 208Tl external to the cryostat

10. 214Bi external to the cryostat

11. 222Rn dissolved in the LXe

12. 137Xe dissolved in the LXe

13. 135Xe dissolved in the LXe

All backgrounds are simulated with Geant4. Most of these are unconstrained in the maximum likelihood ﬁt, but 222Rn dissolved in the LXe is constrained to agree with the measured value of 4.5 ± 0.5 µBq/kg. CHAPTER 8. FITS TO LOW-BACKGROUND DATA 98

8.3 Event selection

The following event selection cuts are applied.

8.3.1 Fiducial cut

The standard ﬁducial volume used in this analysis is

20 mm < |z| < 162 mm (8.1) r < 153 mm (8.2) where z is the axial co-ordinate and r is the radial co-ordinate. This excludes events within 30 mm of the Teflon reflectors (located at r = 183 mm) or the induction grid (located at z = 192 mm), or within 20 mm of the cathode (located at z = 0 mm). The cut excludes regions of the detector known to have poor charge collection and variations in electric field. Also, by excluding the edges of the detector, backgrounds are reduced, in particular those from 222Rn progeny on the cathode. Cross checks of alternative fiducial volumes are performed in Section 8.4.2. The fiducial cut is applied to all clusters. That is, if any cluster is reconstructed outside the fiducial volume, the event is excluded from the analysis. The fiducial cut also excludes events with any clusters reconstructed in unphysical locations or with unphysical drift times.

8.3.2 Scintillation coincidence

Events with more than one reconstructed scintillation signal are excluded. In particular, this cut excludes 214Bi-214Po β-α coincidences if both decays are observed. Since the 214Po half-life is only 164 µs, it is rare for these two decays to be split into separate events. CHAPTER 8. FITS TO LOW-BACKGROUND DATA 99

Figure 8.2: Ratio of ionization to scintillation in arbitrary units for α and β-decays from 214Bi-214Po β-α coincidence events. The black points are the β events (ﬁrst decay in coincidence) and the red points are the α events (second decay). The dashed line shows a clear division between the two events types. The three red points close to the line are coincidences where the drift time is greater than the decay time and charge clusters are mis-assigned between the two events by reconstruction. This kind of mix-up can not happen for single events, thus these points are not indicative of the likelihood of mis-identifying α events as β-like.

8.3.3 Alpha cut

We take advantage of the large diﬀerence in scintillation and ionization yields for α particles and electron-like events to exclude α events. The ratio of scintillation to ionization provides a very clean α rejection, as shown in Figure 8.2 where events from the 214Bi-214Po β-α coincidence, which are known to be from either β-decay or α-decay, are compared.

8.3.4 Muon veto

The muon veto removes muon events, spallation neutron events and bremsstrahlung events. Events within 5 µs of a muon veto event are excluded. CHAPTER 8. FITS TO LOW-BACKGROUND DATA 100

Table 8.1: Results of the maximum likelihood ﬁt. The amplitudes reported are the total amplitude in both single and multiple-cluster spectra in the simultaneous ﬁt.

Component Amplitude (counts) 2νββ 3887±73 40K TPC vessel 387±50 238U TPC vessel 47±77 232Th TPC vessel 33±78 60Co TPC vessel 169±71 54Mn TPC vessel 126±34 65Zn TPC vessel 108±50 222Rn in LXe 11±2 208Tl external 165±97

8.4 Fit results

The single and multiple-cluster events that pass all cuts are shown in Figure 8.3. The energies of the events have been corrected by Equations 7.3 and 7.4, for single- and multiple-cluster events, respectively. There are 4347 single-cluster events and 585 multiple-cluster events, a ratio 7.4 to 1. The result of the maximum likelihood fit to 2νββ and background PDFs is also shown in Figure 8.3, and the residual of the fit is shown in Figure 8.4. As can be seen, the γ backgrounds are readily identified in the multiple-cluster spectrum, while the single-cluster spectrum is dominated by a large structure consistent with the 2νββ decay of 136Xe. The amplitudes from the fit are summarized in Table 8.1. PDFs that fit to negligible amplitudes have been removed, although alternate choices of which PDFs to include have been tested (Section 8.7.4). Note that the fit is simultaneous in single- and multiple-cluster spectra, so the amplitudes presented in Table 8.1 are for the sum of the amplitudes of single- and multiple-clusters. Taking only single-cluster events into account the signal-to-background ratio is 9.4 to 1. CHAPTER 8. FITS TO LOW-BACKGROUND DATA 101

45 300 40

250 35

30 200 25 150 20 counts /50 keV counts /50 keV 100 15 10 50 5

0 0 1000 1500 2000 2500 1000 1500 2000 2500 energy (keV) energy (keV)

Figure 8.3: Energy distributions of the low-background data. Left: single-cluster spectrum. Right: multiple-cluster spectrum. Note the different scales. The result of the maximum likelihood fit to the model of 2νββ (hatched blue) and backgrounds (lines) is also shown. The largest background in the fit is from 40K - TPC vessel (red, dashed). Also shown are 208Tl - external (black), 60Co - TPC vessel (purple, dashed), 54Mn - TPC vessel (light blue), 65Zn - TPC vessel (green, dashed), 232Th - TPC vessel (purple) and 238U - TPC vessel (orange). Although the fit is unbinned, the combined χ2/degrees of freedom for the binning shown here is 85/90. Multi Site Residual Distribution

30 10 20

10 5

0 0

-10 -5 residual counts/50 keV residual counts/50 keV -20

-30 -10

1000 1500 2000 2500 1000 1500 2000 2500 energy (keV) energy (keV)

Figure 8.4: Residuals from the ﬁt to the low background data. Left: single-cluster residual spectrum. Right: multiple-cluster residual spectrum. CHAPTER 8. FITS TO LOW-BACKGROUND DATA 102

Figure 8.5: Fit results vs time of the 2νββ signal and the largest background component (40K). No signiﬁcant time variation is observed.

8.4.1 Time variation

To check for time variation the data is divided into 2 day time intervals, and each is fit in the same way as the full data set. The fit results of the 2νββ signal and the largest background component (40K) are shown in Figure 8.5. No significant time variation is observed, confirming that the signal is stable in time, as expected.

8.4.2 Fit results versus ﬁducial cut

The analysis is repeated with alternative fiducial cuts to check for any non-uniformity in the 2νββ signal. The fiducial cut is varied by a standoff distance d as:

20 mm < |z| < 192 mm − d (8.3) r < 183 mm − d (8.4) where d is varied from 30 mm (the standard ﬁducial cut as given in Equation 8.2) up to 130 mm. Figure 8.6 shows the ﬁt amplitudes of the 2νββ and 40K signals versus CHAPTER 8. FITS TO LOW-BACKGROUND DATA 103

Figure 8.6: Rate of 2νββ and 40K vs standoﬀ distance. As expected 2νββ does not vary, while 40K decreases with increasing standoﬀ distance. d. We see that the rate of 2νββ per kg of xenon is consistent within errors for all values of d, while the 40K signal decays as expected with increasing distance from the detector edges.

8.5 2νββ detection eﬃciency

The 2νββ PDF is used to calculate the fraction of the 2νββ spectrum within the energy window of the fit, E > 720 keV. The fraction of simulated events in the fiducial volume with energy above threshold, 65.0%, is the energy window efficiency. Two other efficiencies result from fluctuations in electronic noise being identified as spurious charge or scintillation signals. A real event with an extra, spurious, scintillation signal will be excluded because of the multiple scintillation signal cut (Section 8.3.2). A real event with an extra, spurious, charge signal will be excluded because, with very high probability, the drift time calculated for such a signal will be unphysical. The rate of spurious signals is evaluated by looking at the 0.1 Hz forced triggers, since the rate of real interactions is negligible in these events. We find that 4.3% of forced triggers have reconstructed scintillation signals and 2.1% have reconstruced CHAPTER 8. FITS TO LOW-BACKGROUND DATA 104

charge signals, and there is a total 6.3% loss in efficiency due to spurious signals in either scintillation or charge channels. The overall detector efficiency is given by the combination of these efficiencies as = 60.9%.

8.6 2νββ half-life result

The 2νββ half-life, T1/2 is related to the event rate by the formula

NXeT T1/2 = ln(2) (8.5) N2ν 136 where NXe is the number of Xe atoms, T is the experiment live-time, = 60.9% is the detector eﬃciency and N2ν = 3887 is the observed number of 2νββ decays. Livetime is determined from the 0.1 Hz forced trigger as T = 752.66 hours, compared with a run-time of 753.55 hours. The number of 136Xe atoms is given by

V ρηN N = fid A (8.6) Xe M where Vfid is the fiducial volume, ρ is the LXe density, η = 80.6% is the isotopic abundance and M = 135.6 g/mol is the atomic mass of the enriched Xe. The volume 3 of xenon is given by the fiducial cut from Equation 8.2 to be Vfid = 20886 cm . The density of natural Xe at 167 K is 2.929 g/cm3 [49], and correcting this for the atomic mass of the enriched Xe compared with that of natural Xe (131.3 g/mol) we get a 3 26 density of enriched Xe of ρ = 3.026 g/cm . This gives NXe = 2.26 × 10 . The measured half-life of the 2νββ decay in 136Xe given by Equation 8.5 is:

21 T1/2 = (2.11 ± 0.04(stat) ± 0.21(sys)) × 10 yr. (8.7)

The NME is calculated from Equation 1.10, with the calculated phase space factor G2ν = 1.28 × 10−18 yr−1MeV2 from Reference [122] CHAPTER 8. FITS TO LOW-BACKGROUND DATA 105

Table 8.2: Systematic uncertainties in the 2νββ half-life.

Source Percentage uncertainty Energy scale 1.8% Fiducial volume 9.3% Multiplicity assignment 3.0% Background model 0.6% Total 10.0%

M 2ν = 0.019 ± 0.001 MeV−1. (8.8)

8.7 Systematic uncertainties

Systematic uncertainties are summarized in Table 8.2. These are added in quadrature to give the total systematic uncertainty in Equation 8.7. The evaluation of these uncertainties is outlined below.

8.7.1 Energy scale

The systematic uncertainty associated with uncertainty in the energy calibration is evaluated with a Monte Carlo method, sampling calibrations within the uncertainty range illustrated in Figure 7.13. For each calibration the fit is redone, and the result is weighted by the likelihood value. The distribution of 2νββ fit results is displayed in Figure 8.7, and the width of the distribution (0.2%) contributes to the systematic uncertainty. We also account for the uncertainty in the analysis threshold because of energy scale uncertainty. This results in a 1.8% uncertainty in the total 2νββ rate, much larger than the uncertainty in the fit amplitude. CHAPTER 8. FITS TO LOW-BACKGROUND DATA 106

Figure 8.7: Distribution of the ﬁt amplitude of the 2νββ PDF for a large number of randomly sampled energy calibrations. The result of each ﬁt is weighted by its likelihood. CHAPTER 8. FITS TO LOW-BACKGROUND DATA 107

8.7.2 Multiplicity assignment

The Monte Carlo method is also used to quantify the uncertainty due to the uncertainty in the ratio of single- to multiple-cluster events in the 2νββ PDF. We vary the multiplicity ratio in the PDF and weight the results by the likelihood value, giving a 3.0% uncertainty in the 2νββ rate. For this proceduce the allowed multiplicity ratio is unconstained, even though comparisons of calibration source data to simulations show agreement in the ratio of single-cluster to multiple-cluster events always better than 7%.

8.7.3 Fiducial volume

One measure of the uncertainty in the fiducial volume is provided by the rate comparisons of the calibration sources to simulated data in Table 7.1. These comparisons are done with the same fiducial volume as the low-background data, and they over- sample the edges of the detector where an error in the position determination would be most significant. Thus, we can be confident that the uncertainty in the fiducial volume is less than the maximum difference seen in those comparisons, i.e. 8.2%. A second method uses the relative rate of calibration source events in different detector volumes. The number of events inside a set of radial fiducial cuts, ranging from r < 153 mm to r < 25 mm, is compared to simulation. For each radial fiducial cut the fractional difference between data and simulation is noted, and the fiducial volume uncertainty is taken as half the spread in fractional differences. This method returns a fiducial volume uncertainty of 9.3%, and this is the value that we use.

8.7.4 Background model

The fit shown in Figure 8.3 does not include all background PDFs listed in Section 8.2. PDFs that fit to negligible contributions have been excluded. A number of test fits are done with different sets of background PDFs to evaluate the effect of the specific choice of PDFs in the fit. The 2νββ amplitude varies by no more than 0.6% as the backgrounds are changed, so we assign this as the systematic uncertainty associated with the choice of background model. CHAPTER 8. FITS TO LOW-BACKGROUND DATA 108

The stability of the 2νββ result is due to the fact that none of the backgrounds ﬁt the dominant feature in the single-cluster spectrum. All γ backgrounds result in many more multiple-cluster events than are observed, as can be seen with the example of the 228Th calibration source spectrum in Figure 7.18. Changing the background model simply rearranges the ﬁt amongst the γ backgrounds with little change to the 2νββ amplitude.

8.8 Alternative hypotheses

We consider alternative sources of the large signal seen in the single site spectrum. We have already seen that this signal is constant throughout the LXe (Section 8.4.2), and that it is a broad featureless distribution. Thus, the source of the signal must be distributed throughout the LXe. We also observe that the signal is predominantly single-cluster, a very important consideration. The signal can not be the result of a γ background, since, as we have seen with the 228Th calibration source (Figure 7.18), γ backgrounds of sufficient energy give more multiple-cluster than single-cluster events. We can therefore limit our consideration of alternate explanations for the signal to β emitters dissolved in the LXe, with a Q-value near that for 136Xe ββ, and without associated γs of significant intensity. Table 8.3 is a list of all β-decays with Q-value in the range 1800–2300 keV, and half-live > 5 days (or that are progeny of an isotope with half-life > 5 days). The Q-value range chosen gives the best spectral match to the 136Xe 2νββ with Q-value of 2458 keV. We can eliminate many of the candidates from consideration since they have a high rate of associated γs, leaving only 90Y, 188Re and 234mPa. Furthermore, 234mPa is produced as part of the 238U decay chain, so we can place a limit on it based on the number of 4.27 MeV αs from the decay of 238U. We observe < 10 counts of the 238U decay, so conclude that 234mPa is not a significant background. We generate PDFs for the decays of 90Y and 188Re dissolved in the LXe. Additional test fits are performed with each of these PDFs separately included. At 90% C.L., the 2νββ rate is reduced by less than 7% by the inclusion of 90Y, and by less than CHAPTER 8. FITS TO LOW-BACKGROUND DATA 109

Table 8.3: β-decays that could potentially mimic the observed 2νββ signal. The listed half-lives are either for the isotope or the parent with longer half-life if applicable.

Isotope Parent Half-life Q-value (keV) 182Ta 114 d 1814 160Tb 72 d 1835 154Eu 8 yr 1969 94Nb 20,000 yr 2045 134Cs 754 d 2059 90Y 90Sr 29 yr 2280 188Re 188W 70 d 2120 194Ir 194Os 6 yr 2234 228Ra 228Ac 6 yr 2127 234mPa 234Th 24 d 2195

30% by the inclusion 188Re. These results show that even in the unlikely case that the LXe is contaminated by these isotopes uniformly in volume and time, the discovery of 136Xe 2νββ is robust. It is not appropriate to incorporate these unusual isotopes into the background model, or increase the systematic uncertainty to account for them, because of the extremely low likelihood of their presence in what is otherwise known to be a very clean detector. Chapter 9

Conclusions

The initial data taking of EXO-200 has provided a clear discovery of the 2νββ in 136Xe 21 with measured half-life of T1/2 = (2.11 ± 0.04(stat) ± 0.21(sys)) × 10 yr. The measurement is the result of a maximum likelihood ﬁt to 752.66 hours of low-background data with 2.26 × 1026 atoms of 136Xe. Currently the uncertainty is dominated by systematic uncertainties, and improving the precision will require a more accurate determination of the ﬁducial volume.

The measured T1/2 is signiﬁcantly lower than the lower limits quoted in [24] 22 21 and [46] (T1/2 > 1.0 × 10 yr and T1/2 > 8.5 × 10 yr respectively). Our result is in agreement with some theoretical predictions. For example Caurier et al. [43] 21 predicted a half-life of T1/2 = 2.1 × 10 yr with a nuclear shell model calculation. Our result is the longest half-life of any of the directly measured 2νββ emitters and corresponds to the smallest nuclear matrix element of 0.019 ± 0.001 MeV−1.

110 Appendix A

A microscopic LXe recombination model for EXO-200

A microscopic model of recombination in LXe is incorporated as an option into the EXO-200 Geant4 simulation. The model is based on earlier work by Dahl [123], and has been compared to data from a small LXe detector. Future comparisons of the model to EXO-200 data may give insight into the detector response and an improved understanding of the detector sensitivity.

A.1 Previous work

Langevin [124] described the physics of recombination of electrons and holes. In this model, once an electron-hole pair are close enough to each other that the electrostatic potential between them overcomes diﬀusion they recombine. The critical radius, re, called the Onsager or Coulomb radius, is given by

e2 = kT (A.1) 4πre where e is the electron charge, is the permittivity of the material and kT is the thermal energy of the electron-hole pair. The recombination rate, R, is the rate at which electrons and holes drift into each others’ Onsager radius, proportional to the

111 APPENDIX A. LXE RECOMBINATION MODEL 112

densities of the positive and negative charges, N+ and N−

R = αN+N−, (A.2) where α is the recombination coefficient. This coefficient is equal to the flux of charge carriers across the surface of the sphere at radius re, which is equal to the drift velocity, vd, given by

e vd = µ 2 (A.3) 4πre where µ is the combined mobility of the charges. Thus we get

µe R = N N . (A.4) + − Jaﬀe [125] considered recombination in ion tracks, writing the diﬀerential equations

∂N + = D ∇2N − µ ε · ∇N − αN N (A.5) ∂t + + + + + − ∂N − = D ∇2N − µ ε · ∇N − αN N (A.6) ∂t − + − − + − where D± are the diffusion constants for the two charge carriers, µ± are the mobilities and ε is the electric field. For a liquid xenon TPC we can consider the case where the positive charge carriers have negligible mobility, diffusion can be ignored, and we have a constant electric field in the -z direction (that is vd = µ−ε). Thus the differential equations can be simplified to

∂N + = −αN N (A.7) ∂t + − ∂N ∂N − = −v − − αN N (A.8) ∂t d ∂z + − as shown by Thomas and Imel [126]. For the initial condition of equal positive and negative charge distributions

(N−(t = 0) = N+(t = 0) = N0) as taken by Kramers [127] this can be solved, giving APPENDIX A. LXE RECOMBINATION MODEL 113

N (x, y, z) N (t, x, y, z) = 0 (A.9) + α R z 0 0 1 + dz N0(x, y, z ) vd z−vdt

The total charge escaping recombination, ne, can be found by taking t → ∞ and integrating over all space

Z N (x, y, z) n = d3x 0 . (A.10) e α R z 0 0 1 + dz N0(x, y, z ) vd −∞

Thomas and Imel choose as the initial condition Ni ions distributed uniformly in a box of side 2a:

( Ni 8a3 |x| , |y| , |z| < a N0 = (A.11) 0 otherwise which gives the solution to Equation A.10 as

n 1 e = ln(1 + ξ), where (A.12) Ni ξ

Niα ξ = 2 . (A.13) 4a vd Thomas and Imel find that this fits the observed field dependence of α particles in −1 liquid argon and xenon if ξ ∝ ε . This is interpreted as vd ∝ ε with α and a independent of the electric field, although this is known to be incorrect (see Figure 2.1), so the interpretation must be different. Although Thomas and Imel considered track lengths shorter than a, Dahl extended the model to longer track lengths by assuming that for each partial energy deposit along a track, the ionization is distributed uniformly within a box of side length 2a. Dahl’s extension of the Thomas-Imel model successfully reproduces the mean recombination seen in electron and nuclear recoils below 100 keV of deposited energy, although it does not accurately reproduce the fluctuations in recombination rate, consistently falling short of the measured variance by a factor of three. The model parameters fit by Dahl, a and α/vd, are give in Table A.1. APPENDIX A. LXE RECOMBINATION MODEL 114

Table A.1: Recombination model parameters given by Dahl. The errors are statistical errors only.

α Drift ﬁeld, ε (V/cm) a (nm) 100 2 a vd 4060 ± 190 201(3) 3.17(5) 1951 ± 86 244(4) 3.71(5) 876 ± 36 484(4) 2.96(3) 522 ± 23 648(8) 3.35(3) 60 ± 5 2111(18) 3.39(2)

A.2 A recombination model for EXO-200

We apply a very similar modified Thomas-Imel recombination model as Dahl to simulate recombination in EXO-200. First, energy deposition topologies are determined by GEANT4 Monte Carlo simulations. GEANT4 accurately simulates event topologies for gamma-ray and electron interactions in the energy range of most interest for EXO-200, that is ∼100 keV to 3 MeV, including Compton scattering, pair-production, photo-electric absorption, ionization and bremsstrahlung. The initial ionization distribution is computed by taking each element of the ionization track as determined by the GEANT4 simulation and uniformly distributing the ionization from that element in a box of side length 2a. The electron survival probability is then numerically calculated, by Equation A.10, with this initial distribution, with recombining electrons producing scintillation photons. The model parameters are interpolated from Table A.1, although these can be adjusted in future if necessary. An initial exciton ratio, Nex = 0.06 as used by Dahl Ni for electron recoils, is also included in the model. The initial exciton ratio for α tracks is set to Nex = 0.9, the value Dahl finds to be the best fit to nuclear recoil data, which Ni is expected to be more similar to α interactions than the electron recoil data. The model is compared to data from a ∼1.5 L LXe chamber, with simultaneous readout of ionization and scintillation and a 207Bi calibration source. Results from that data have previously been reported [64], demonstrating for the first time anti-correlated fluctuations between scintillation and ionization in LXe. Figure A.1 APPENDIX A. LXE RECOMBINATION MODEL 115

illustrates comparisons between the model and the data at two values of the drift ﬁeld (500 V/cm and 1000 V/cm). The general features of the data are well reproduced by the model, including the anti-correlated ﬂuctuations between scintillation and ionization. A quantitative comparison is not possible because some parameters of the 1.5 L chamber, such as calibrations for both the photon and electron channels, are poorly known. APPENDIX A. LXE RECOMBINATION MODEL 116

80 80

70 70

60 60

electrons) 50 electrons) 50 3 3

40 40

30 30

20 20 Final ionization (10 Final ionization (10 10 10

0 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Final scintillation (103 photons) Final scintillation (103 photons)

80 80

70 70

60 60

electrons) 50 electrons) 50 3 3

40 40

30 30

20 20 Final ionization (10 Final ionization (10 10 10

0 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Final scintillation (103 photons) Final scintillation (103 photons)

Figure A.1: Comparison of the recombination model (left) to data from the 1.5 L chamber (right), with 500 V/cm (top) and 1000 V/cm (bottom) drift fields. The simulated data are smeared by the detector electronic noise in the ionization channel and statistical fluctuations for the number of detector photons. The two peaks from the 207Bi 570 keV and 1064 keV γs, with anti-correlated fluctuations between scintillation and ionization, are clearly visible in all cases. Appendix B

A magnetically coupled piston xenon pump

EXO-200 requires a reliable, ultra-clean, pump with superior containment properties to recirculate the xenon through the purifiers and back into the detector. Available pump technologies, such as diaphragm and bellows pumps, can be made with clean materials. For example, the MUNU experiment [128] used a bellows pump and the XENON10 experiment [61] used a diaphragm pump. However, for these designs, a failure of a bellows or a diaphragm results in a breach of the gas system, necessitating double containment and leak detection to avoid gas loss and contamination. Even with double containment, a failure results in down time and risk of contamination of the system during the repair. A pump that is both extremely clean and robust with respect to gas containment is desirable. The xenon pump is used in three distinct modes of operation. Firstly, prior to initial liquefaction xenon gas is recirculated through the purifiers to purge impurities from the xenon vessel and gas system. Secondly, during initial liquefaction the xenon pump forces gas though a xenon condenser for accelerated condensation. Thirdly, and most importantly, when the vessel is full of liquid, gaseous xenon evaporated by the Xe heater is pumped through the purifiers and re-condenses in the Xe condenser, establishing continuous purification of the xenon during detector operations. Double containment, all-metal bellows pumps were tested but were found to be

117 APPENDIX B. A MAGNETICALLY COUPLED PISTON XENON PUMP 118

plagued by leaks after periods of operation greater than a month, even when selecting long-lifetime bellows and carefully designing the stroke for durability. The pump described here makes use of magnetic coupling between a pair of permanent magnets, one inside the process fluid (gas Xe) and the other in air outside the gas system, eliminating the need for flexible components isolating the process fluid from the atmosphere. This eliminates the possibility of contamination of the xenon gas due to component failure, limiting the effects of long term wear to a possible decrease of pumping efficiency. The drive mechanism, located completely outside the xenon system, moves an external neodymium ring magnet [129] back and forth over a stainless steel cylinder. Inside the cylinder a piston, containing a sealed cylindrical neodymium magnet [130], moves in phase with the external magnet and pumps the xenon gas. The magnets are axially magnetized with opposing polarity. It is critical that the xenon pump itself is not a source of contamination, so it has been constructed such that the only materials exposed to xenon gas are stainless steel, titanium, and a carefully selected polymer used for the piston seals. Gas containment is insured exclusively by static, all-metal ConFlat R [131] and VCR R [132] seals. No lubricants are used inside the pump.

B.1 Design and construction

A schematic view of the pump is shown in Figure B.1 and photographs of the pump and piston are shown in Figure B.2. The pump body is made from a honed thick- walled non-magnetic 316 stainless-steel cylinder, 40 cm long with inside diameter of 6.5 cm. The inner and outer diameters of the cylinder are chosen to tightly match the dimensions of commercial permanent magnets. The cylinder is welded onto 4-5/8" ConFlat R ﬂanges. The external ring magnet is mounted on a linear actuator [133] driven by a stepper motor [134]. This conﬁguration allows programmable movement of the external magnet—and thus the internal piston—up to speeds greater than 15 cm/s, and, if desired, programmable acceleration and deceleration. Optical position sensors are installed so that the drive can seek the starting location at startup APPENDIX B. A MAGNETICALLY COUPLED PISTON XENON PUMP 119

OUTLET PUMP OUT PUMP OUT PARTICULATE PORT VALVE FILTER OUTLET PORT PUMP OUT VALVE PUMP OUT VALVE

RING INLET MAGNET REED INLET OUTLET PORT REED INNER MAGNET REED VALVE BLOCK

UHMWPE UHMWPE GASKETS GASKETS DRIVE MOTOR

INLET

Figure B.1: Schematic diagram of the magnetically coupled piston pump. One of the reed valve blocks, made from 4-5/8" ConFlat R flanges, is shown separately at the top right. and during operation to correct for drifts. The piston consists of a welded stainless steel cannister with the cylindrical neodymium magnet sealed inside. On both ends plastic piston gaskets are mounted with stainless steel retaining and tensioning rings. The gaskets create a seal between the piston and the honed cylinder. The cannister could not be TIG welded because of the strong magnetic field and was instead laser welded [135]. Fasteners and tools used in the assembly are made of titanium for the same reason. Inlet and outlet ports with reed valve assemblies, made with 50 µm thick stainless steel stock, ensure that gas flow is unidirectional. The seats of the reed valves are APPENDIX B. A MAGNETICALLY COUPLED PISTON XENON PUMP 120

Figure B.2: Top: Photograph of the assembled xenon pump. Bottom: Photograph of the piston removed from the pump. APPENDIX B. A MAGNETICALLY COUPLED PISTON XENON PUMP 121

made by optically lapping the appropriate regions of a double-sided ConFlat R blank in which holes were drilled. As the piston moves through the cylinder, gas is pulled in through the inlet on one end and forced out through the outlet on the other end. The two inlets and outlets are plumbed in parallel so that flow is produced in both directions of travel of the piston. Ports directly accessing the cylinder volume (bypassing the reed valves) allow for more efficient pump-out of air from the interior prior to integration in the xenon system. A sufficiently strong coupling between the magnets is essential for the pump to achieve its required performance. If the coupling is not strong enough, the gas pressure acting on the piston will decouple the piston magnet from the external magnet, disabling the pump (although if this happens the magnets immediately re-couple on the return movement of the external magnet). The design goal was for the pump to operate up to a differential pressure of 750 torr while achieving flows in excess of 10 SLPM. The maximum differential pressure possible with our pump was calculated for three arrangements of commercially available magnets using the MAXWELL [87] software suite. For the piston magnet to remain coupled to the ring magnet, the pressure acting on the piston must be less than the maximum magnetostatic restoring force on the piston. Figure B.3 shows the calculated restoring force for three magnet configurations. For all three configurations the inner piston magnet has a diameter of 5.1 cm, the outer ring magnet has inner diameter of 7.6 cm and outer diameter of 10.2 cm. The length of both magnets is varied from 2.5–7.6 cm between the three configurations. The calculated maximum restoring force is 310 N for the 2.5 cm long magnets, 480 N for the 5.1 cm long magnets and 490 N for the 7.6 cm long magnets. From this we calculate the maximum pressure as 730, 1130 and 1150 torr for the 2.5, 5.1 and 7.6 cm long magnets respectively. A second design challenge is maintaining an adequate seal between the piston and pump cylinder without contaminating the xenon. While for this design a leaky seal does not compromise the integrity of the gas circuit, it does reduce the efficiency of the pump. The appropriate choice of piston ring material is critical, as is the design of the seals. The pump relies on a dynamic seal between the piston and stainless APPENDIX B. A MAGNETICALLY COUPLED PISTON XENON PUMP 122

Figure B.3: MAXWELL calculations of the restoring force acting on the piston magnet versus displacement. The maximum restoring force is the force at which the magnets will decouple. steel cylinder, and since the ultra-pure xenon gas is in contact with the piston, no lubricants can be used. To meet radiological purity requirements, only low radon outgassing materials free of other radioactive contaminants are suitable. Friction between the pump wall and seals must be low, and the seals must be durable. Of particular concern is the possibility of particulate generated by the piston rings contaminating the xenon. We primarily considered polymers as seal materials and have investigated poly- tetrafluoroethylene (PTFE), fluorinated ethylene propylene (FEP), perfluoroalkoxy (PFA), polyether ether ketone (PEEK) and ultra-high-molecular-weight polyethylene (UHMWPE) [136]. Samples of these five materials were tested for wear by spinning under a calibrated pressure a part, fashioned as a tube, against an optically polished quartz plate. Wear was assessed by the amount of particulate generated in the test. For this test to be realistic both the sample material and quartz plate have to be thoroughly degreased. The spinning velocity was set to simulate the linear velocity expected in the gasket. It was found that UHMWPE exhibited the lowest wear, followed by PFA then PTFE and FEP. PEEK performed very poorly in this test. Test APPENDIX B. A MAGNETICALLY COUPLED PISTON XENON PUMP 123

pistons were made out of PFA, PTFE and UHMWPE and operated in a prototype pump, confirming that the latter produces the least particulate. We experimented with a number of different profiles for the piston rings. The rings must seal the piston to the cylinder at speeds up to 15 cm/s and the friction must be kept to a small fraction of the magnet decoupling force. The rings must also accommodate thermal expansion from frictional heating while maintaining a constant radial pressure, and must center the piston with tight tolerance because of the unstable radial equilibrium provided by the magnetic coupling. To achieve this the plastic sealing surface is radially sprung by forcing a conical stainless steel element into it by means of an axial force applied by a spring washer. This method, illustrated in Figure B.4, also allows for adjustment of the radial force that then remains constant until the sealing surface is completely worn out. In the pump in operation at EXO-200 an axial force of 20 N is provided by the spring washer against the 45◦ conical surface. A simple calculation gives a radial linear pressure of 1.0 N/cm along the sealing surface, although this is reduced somewhat by hoop stress of the gasket lip. While the two piston rings are designed and adjusted to mechanically perform as similarly as possible, one of them is vented through, so that the seal only occurs on one side and there is no trapped volume between the two rings. As shown in Figure B.5, prototypes of the pump relying only on the elastic properties of the plastics to provide radial sealing provided inferior performance, as expected.

To minimize outgassing from the UHMWPE gaskets, they were baked in a N2- purged oven for 200 hours at 93◦C before installation. A 0.003 µm particulate filter [137] is installed at the output of the pump to capture any particulate that may be generated. Gas from the pump also passes through the hot zirconium purifiers before returning to the xenon vessel. Purity is measured with a gas purity monitor [114] at the output of the purifiers and it is found to be sufficient for the operation of a liquid xenon detector with a '20 cm drift distance. It is anticipated that the piston rings will wear over time and may need replacing after several months, although this has not been necessary in the first 6000 hours of operation. APPENDIX B. A MAGNETICALLY COUPLED PISTON XENON PUMP 124

LASER WELD NEODYMIUM MAGNET

CONICAL PRESSURE SURFACE

TI SHOULDER BOLT

UHMWPE WAVE SPRING GASKETS

Figure B.4: A view of the piston with sprung UHMWPE gaskets. Radial pressure is applied to the gaskets by axially pressing a conical stainless steel part into it by means of a wavy spring washer. The neodymium magnet is sealed inside the stainless steel cannister. APPENDIX B. A MAGNETICALLY COUPLED PISTON XENON PUMP 125

B.2 Performance characterization

Two largely identical magnetically coupled piston pumps were built as part of the xenon recirculation system in the EXO-200 experiment. The initial prototype (Pump 1), with 2.5 cm long external and internal magnets and no gasket tensioning mechanism, has been operated with two different sets of PFA piston rings (designated A and B, and differentiated primarily by a slight difference in diameter in an attempt to optimize the sealing simply using the elasticity of the plastics). The production device (Pump 2), designed for higher differential pressure, is made with 5.1 cm long magnets and uses sprung UHMWPE gaskets. Also in this case two sets of gaskets with slightly different diameters (designated C and D) were tried. The pump is typically operated at 14 strokes/min, corresponding to a linear velocity of 8 cm/s in the middle of the stroke. The efficiency of the pumps is investigated by comparing the measured mass flow rate to the nominal one calculated as the volumetric displacement of the piston, as shown in Figure B.5. Both imperfect radial seals and back flow though the reed valves degrade the efficiency. The data in Figure B.5 was taken with inlet pressure near 760 torr, but we have observed increasing efficiency as the inlet pressure is increased. We have also consistently observed higher efficiency when pumping room air in bench tests of the pumps when not installed in EXO-200—an effect that is not understood. The performance of Pump 1 with un-sprung PFA gaskets critically depends on the exact tolerance of the gaskets and, in fact, the efficiency of the looser gaskets of 1B substantially improves after a warm-up period (and, presumably, expansion). This is illustrated by the point in Figure B.5 for Pump 1, resulting from gasket warm-up in the first few hours of operation. Pumping efficiency for Pump 1 also increases with increases in room temperature for the same reason. Tighter gaskets (1A) have better initial performance but they wear out in a few days of continuous pumping (not shown in Figure B.5). In the case 1B the efficiency decreased by 80% after 1500 hours of operation from gasket wear. The sprung gaskets made of UHMWPE provide consistent performance at all times and are durable. APPENDIX B. A MAGNETICALLY COUPLED PISTON XENON PUMP 126

Figure B.5: The measured xenon flow rate through the two pumps and four total sets of gaskets at inlet pressures of approximately 760 torr. The dotted line represents 100% efficiency. Pump 1B shows a dramatic improvement in efficiency after some running time (about two hours), so an additional point is plotted after allowing the gaskets to warm up. This warmup effect is much less significant for Pump 1A, and negligible for Pump 2. The maximum flow rate in Pump 1 is limited by the coupling force between the magnets, while for Pump 2 magnet decoupling occurs at much higher differential pressure and higher flow rates are possible.

The performance of the production pump with the D gaskets over more than 6 months of detector operations is shown in Figure B.6. At times there appears to be a slow degradation of the pumping efficiency; for example between day 10 and day 50 the average flow rate decreased by ∼15% with no change in pump speed. However, at other times there is no noticable efficiency change over long periods of time. After more than 6000 hours of operation the efficiency has decreased by less than 30%. For Pump 1 the maximum pumping rate is limited ultimately by the differential pressure at which the piston will decouple from the external magnet. The differential pressure at decoupling was measured to be at least 400 (600) torr for case 1A (1B). The difference between the two gaskets is attributed to additional force due to friction from the tighter fitting A gaskets. The decoupling regime has not been studied extensively for Pump 2. For the C APPENDIX B. A MAGNETICALLY COUPLED PISTON XENON PUMP 127

Figure B.6: The measured flow rate, produced by the production pump, through the xenon recirculation loop over more than 6 months of detector operations, including the period of low-background data taking described in Chapter 8. The band in the measured flow is due to fluctuations within each cycle of pump operation. The pump speed was adjusted a number of times, as can be seen by the changes in flow rate. APPENDIX B. A MAGNETICALLY COUPLED PISTON XENON PUMP 128

gaskets, low pumping efficiency at high piston speeds limits the maximum pumping rate. For the D gaskets we have successfully operated with differential pressures up to 750 torr, although this is rarely required in EXO-200 operations. This is sufficient to recirculate xenon at more than 16 SLPM through the EXO-200 recirculation loop with typical inlet pressures of 700–1000 torr. Appendix C

Signal extraction with a matched ﬁlter

One of the major challenges in event reconstruction in EXO-200 is identifying charge collection or scintillation signals from the waveforms in the presence of noise. This is particularly difficult for low energy signals (which may be part of a higher energy event so are important to identify), or if the detector noise performance is less than ideal, as is the case for the LAAPD signals. The algorithm used is matched filtering. It can be shown that for a linear time invariant system with an output r(t) = s(t)+n(t) where s(t) is the signal of interest and n(t) is white Gaussian noise (WGN), the optimum filter to maximize signal-to-noise is the matched filter, equivalent to a convolution with the known signal pulse s(t). Here is a brief outline of the proof, making use of Fourier transforms. We wish to maximize the signal-to-noise ratio (SNR):

2 ∞ 2 R H(f)S(f)ej2πftdf |so(t)| −∞ SNR = = , (C.1) 2 N0 R ∞ 2 |no(t)| 2 −∞ |H(f)| df where so(t) and no(t) are the output signal and noise after ﬁltering at some time t, H(f) and S(f) are the Fourier transforms of the ﬁlter and signal respectively, and

N0 2 is the frequency independent Fourier transform of the noise. We can use the Schwarz inequality to show

129 APPENDIX C. SIGNAL EXTRACTION WITH A MATCHED FILTER 130

2 R ∞ H(f)S(f)ej2πftdf R ∞ 2 R ∞ 2 −∞ |H(f)| df |S(f)| df SNR = ≤ −∞ −∞ , (C.2) N0 R ∞ 2 N0 R ∞ 2 2 −∞ |H(f)| df 2 −∞ |H(f)| df with equality, i.e. maximized SNR, when

∗ H(f) = k S(f)ej2πft = kS∗(f)e−j2πft, (C.3) where k is an arbitrary constant. This is equivalent to a convolution in the time domain, referred to as a matched filter. In practice the assumption of WGN does not hold. EXO-200 has colored noise since the noise is passed through the shapers and we also have some very non-white pickup noise. We can still use a matched filter, by first passing the waveform through a whitening filter, which is simply done by dividing the Fourier transform of the waveform, R(f), by the magnitude of the the noise frequency spectrum, |N(f)|. Then we can apply the matched filter to the “whitened” response, R0 (f)

0 R(f) R (f) = . (C.4) |N(f)| The matched ﬁlter also needs to be the “whitened” version

0 H(f) H (f) = . (C.5) |N(f)| So the combined ﬁlter we need to apply to the waveforms is

∗ 00 S (f) H (f) = . (C.6) N 2(f) Since the time oﬀset of the signal is unknown, we scan the output of the ﬁlter in the time domain, searching for signals above a set threshold. Appendix D

In-situ LAAPD gain calibration

We measure the gains of the individual LAAPD gangs by injecting pulses from the external laser pulser into the TPC, through the optical ﬁbers that are installed with the LAAPD cable bundles.

D.1 Laser pulser system

The lasers for LAAPD gain calibrations are 5 mW, 406 nm laser diodes with a 1 m single-mode fiber pigtail (Thorlabs part #LPS-406-FC). The lasers are driven by a custom made laser diode driver capable of outputting 0–60 mA with a pulse width of 10 ns to 5 µs. The driver is triggered by the EXO-200 Trigger Electronics Module (TEM). There are two laser pulsers, one for each side of the TPC. The fiber pigtails are coupled to 20 m long single-mode patch cables which are in turn coupled to the unjacketed multi-mode fibers, which inject the pulses into the TPC. The injection fibers are inserted into Teflon diffusers inside the TPC, installed on the LAAPD platters, taking the place of one LAAPD on each side of the TPC. Thus the laser light from each pulse is diffused throughout the TPC. Figure D.1 shows the Teflon diffuser with 406 nm laser light being injected via a fiber during a table-top test of the system.

131 APPENDIX D. IN-SITU LAAPD GAIN CALIBRATION 132

Figure D.1: A Teﬂon diﬀuser being illuminated from behind by the laser pulser during a bench test. D.2 Gain measurements

To measure the LAAPD gain we run the laser pulser and compare the output of each gang with the LAAPDs biased at operating voltage (∼1400 V), to the output of the LAAPDs with only 310 V bias. This measurement makes use of a well known property of the LAAPD gain curve to measure the operating gain with a comparative measurement, even though the laser pulser system is uncalibrated. At low bias voltage the LAAPDs act as unity gain devices over a broad voltage range (∼300–500 V), since the internal field is insufficient for avalanche gain. We can determine the gain at all other bias voltages by comparison to this reference point. The DAQ does not have sufficient dynamic range for this measurement to be done directly on all LAAPD channels simultaneously. Thus, a total of four measurements are taken for each gang, with different combinations of laser pulse power and LAAPD bias voltage, and the results combined to produce the LAAPD operating gain measurement. APPENDIX D. IN-SITU LAAPD GAIN CALIBRATION 133

First, the LAAPD response is measured for all gangs with 310 V bias and a specific power setting on the laser pulser. Second, the bias on all gangs was raised (to 1360 V for the gangs in the South LAAPD plane and 1380 V for the LAAPDs in the North LAAPD plane – slightly below the final operating voltage) while keeping the laser pulse power the same. Some of the LAAPD gangs saturate at this higher gain, but others do not and thus we are able to measure the response of those gangs at this combination of pulser power and bias voltage. Third, the power of the laser pulses is reduced by approximately a factor of four (by decreasing the pulse width) such that none of the LAAPD gangs saturate, while keeping the bias voltage constant. We are now able to accurately measure the relative power of the laser pulses for the two settings by comparing the ratio of the response of the non-saturating gangs. With this information, it is possible to determine the gains of all the LAAPD gangs by a comparison of the response between the third and first measurements, even for the gangs that saturate in the second measurement. Fourth, and finally, we raise the voltage of each LAAPD gang to its nominal operating voltage (between 1365 and 1417 V), chosen to match the gains throughout the detector (using the results of the first three measurements).

D.3 Corrections

We need to account for signal losses in the DAQ chain to accurately determine a gain value for each gang. Figure D.2 displays a simplified schematic of the LAAPD signal readout. Two separate effects reduce the signal observed. First, the negative signal pulse, −qn is shared between the capacitance of the trim voltage decoupling capacitor, CD, and the capacitance of the LAAPD gang itself, Cn. Since the value of CD (10 nF) is ten times larger than Cn (∼1 nF) at full bias this is about a 10% effect. Second, the positive signal pulse, +qn from each LAAPD gang is partially returned to ground via all the LAAPD pre-amplifiers, subtracting from the negative signals. Since the platter decoupling capacitor value (CB = 250 nF) is about seven times larger than the combined parallel capacitance of all 37 LAAPD gangs on each platter, this is about a 15% effect. Both of these effects become more significant at APPENDIX D. IN-SITU LAAPD GAIN CALIBRATION 134

Figure D.2: Simplied schematic of the LAAPD signal readout. lower bias voltage, since the LAAPD capacitance is about twice as large at 310 V than it is at 1400 V. Figure D.3 shows the capacitance of a single LAAPD as a function of bias voltage.

Consider the signal, Sn, for a single LAAPD gang, n. This is primarily due the signal from that LAAPD gang itself, but also receives contribution from signal return of all the other LAAPD gangs on the same plane since the plane is shared. The signal from the gang itself comes from the charge, −qn, at the anode, which is split between the capacitance of the LAAPD gang itself, Cn, and the decoupling capacitor, CD. This this contribution is

CD Sn(1) = −qn . (D.1) CD + Cn The positive charges from all LAAPD gangs return to ground mostly through the bypass capacitor, CB, but some fraction returns via the other LAAPD gangs. Thus the signal receives a contribution from each other gangs’s signal. The fraction of positive signal from gang j that returns through gang n is APPENDIX D. IN-SITU LAAPD GAIN CALIBRATION 135

900 800 700 600 500 Capacitance (pF) 400 300 200 100 0 200 400 600 800 1000 1200 1400 1600 1800 Bias Voltage (V)

Figure D.3: Capacitance measurements of a single LAAPD.

−1 1 + 1 Cn CD −1 , (D.2) P 1 1 CB + + k Ck CD

CD although as with the signal −qn, this needs to be reduced by the factor to CD+Cn account for the signal loss in the LAAPD gang n. Thus the measured signal receives the second contribution due to +qj signals from other gangs

−1 1 + 1 X Cn CD CD Sn(2) = −1 qj. (D.3) P 1 1 CD + Cn j CB + + k Ck CD

The contribution from the −qj signals from all other LAAPD gang anodes also contributes a small amount, so we include it for completeness. This is the same as the contribution from the +qj signals, but opposite sign and reduced reduced by the factor Cj since most of this charge is returned through its own pre-amp. CD+Cj APPENDIX D. IN-SITU LAAPD GAIN CALIBRATION 136

−1 1 + 1 X Cn CD CD −Cj Sn(3) = −1 qj . (D.4) P 1 1 CD + Cn CD + Cj j CB + + k Ck CD Adding the contributions from Equations D.1, D.3 and D.4 we get

−1 1 + 1 CD CD X Cn CD CD+Cn Cj Sn = −qn + −1 qj − qj (D.5) CD + Cn P 1 1 CD + cj j CB + + k Ck CD −1 1 + 1 CD CD X Cn CD CD+Cn CD = −qn + −1 qj (D.6) CD + Cn P 1 1 CD + Cj j CB + + k Ck CD

The sum of the signal from all LAAPDs in one plane, Ssum is thus

 −1  1 + 1 CD C Cn CD CD+Cn C X  D X D  Ssum = −qn + −1 qj (D.7)  CD + Cn P 1 1 CD + Cj  n CB + + j k Ck CD −1 1 + 1 CD X CD X Cn CD CD+Cn X CD = −qj + −1 qj (D.8) CD + Cj P 1 1 CD + Cj j n CB + + j k Ck CD where we have switched the index from n to j in the ﬁrst term. Now replacing the index n with k we get

 −1  P 1 + 1 CD C k C CD CD+C X D  k k  Ssum = −qj 1 − −1 (D.9) CD + Cj  P 1 1  j CB + + k Ck CD Deﬁning

−1 X 1 1 ζ = + , (D.10) Ck CD k APPENDIX D. IN-SITU LAAPD GAIN CALIBRATION 137

and combining Equations D.6 and D.9 we get the expression

 −1 −1 ζ CD 1 + 1 CD CD CD+Ck Cn CD CD+Cn Sn = −qn + Ssum 1 −  (D.11) CD + Cn CB + ζ CB + ζ

which can be rearranged to give the true signal, qn in terms of the measurable quan- tities Sn and Ssum

  −1 −1  ζ CD 1 + 1 CD Cn CD+Ck Cn CD CD+Cn −qn = 1 + Sn − Ssum 1 −   CD  CB + ζ CB + ζ 

(D.12)   −1  1 + 1 CD C Cn CD CD+Cn n    = 1 + Sn − Ssum (D.13) CD   CD  CB + ζ − ζ CD+Ck  −1  1 + 1 C Cn CD n   = Sn 1 + − Ssum (D.14) CD  CD  CB + ζ 1 − CD+Ck  −1  1 + 1 C Cn CD = S 1 + n − S   (D.15) n sum C2C CD  P k D  CB + 2 k (Ck+CD)

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