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Calibration of SuperCDMS dark matter detectors for low-mass WIMPs MacDonell, Danika 2018

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Calibration of SuperCDMS DarkMatter Detectors for Low-MassWIMPsbyDanika MacDonellB.Sc., The University of Victoria, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2018c© Danika MacDonell 2018The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Calibration of SuperCDMS Dark Matter Detectors for Low-Mass WIMPssubmitted by Danika MacDonell in partial fulfillment of the requirementsfor the degree of Master of Science in Physics.Examining Committee:Scott Oser, Physics and AstronomySupervisorJanis McKenna, Physics and AstronomyAdditional ExamineriiAbstractObservational evidence suggests that the majority of mass in the universetakes the form of non-luminous “dark matter”. The Super Cryogenic DarkMatter Search (SuperCDMS) is a direct-detection dark matter experimentthat searches primarily for a well-motivated dark matter candidate knownas the weakly-interacting massive particle (WIMP). The experiment looksfor an above-background excess of nuclear recoil events in cryogenic solid-state detectors that could be attributed to WIMP-nucleon collisions. Themost recent SuperCDMS run at the Soudan underground laboratory set aworld-leading limit on the spin-independent WIMP-nucleon cross sectionfor a WIMP mass as low as ∼3 GeV/c2, and the next installation of theexperiment at SNOLAB aims to be sensitive to WIMP masses below 1GeV/c2.To better understand the response of solid-state germanium detectors tolow-mass WIMPs, “photoneutron” calibration data was taken at the Soudanlaboratory in Minnesota by passing quasi-monoenergetic neutrons throughSuperCDMS detectors. Gamma rays used in the photoneutron productionprocess create an overwhelmingly dominant background of electron recoilevents in the detector. This gamma background is measured directly withregular “neutron-off” data-taking periods during which the neutron produc-tion mechanism is removed. We compare the observed electron and nuclearrecoil spectra with Geant4-simulated spectra to obtain a model-dependentcalibration of the nuclear recoil energy scale of the detectors. The calibra-tion is performed using a negative log likelihood fit to a parameterized Lind-hard ionization yield model. The fit includes a semi-analytical model of thegamma background component obtained from the neutron-off data.iiiLay SummaryAstronomical observations indicate that the majority of mass in the universetakes the form of “dark matter”. Unlike most of the normal matter thatmakes up planets and stars, dark matter interacts only very weakly withother matter, and does not produce light. For this reason, it has never beendirectly detected.SuperCDMS is an experiment which seeks to detect dark matter particles bymeasuring the tiny amounts of vibrational energy that would be produced bya dark matter particle scattering inside a SuperCDMS detector. We presenta calibration study which aims to improve our estimate of the masses ofdark matter particles that the experiment could measure. We study therelationship between the amount of energy a dark matter particle wouldgive the detector when it scattered – which is related to its mass – and theamount of vibrational energy that the detector would actually measure as aresult.ivPrefaceWhile all discussions in this thesis are my own words, the introductory chap-ters (Chapters 1-3) exclusively discuss work done by others to motivate thesearch for dark matter, and describe how SuperCDMS and other experi-ments perform such a search.Chapter 4 presents my contributions to an analysis of calibration data thatwas taken in 2015 by the SuperCDMS collaboration at the Soudan under-ground laboratory in Minnesota, U.S.A. The analysis work presented inChapter 4 was done in close collaboration with an analysis working groupwithin SuperCDMS. As such, the work is heavily reliant upon that of othermembers of both the analysis group and the wider collaboration, and hasbenefited from the many ideas and suggestions of my collaborators. I wasresponsible for developing the data quality cuts, detector resolution model,and “neutron-off” background model used for the analysis of calibrationdata taken with detectors operating in CDMSlite mode. I was also respon-sible for the development and implementation of the unbinned Lindhardyield extraction technique that is applied to the data to obtain an experi-mental determination of the ionization yield as a function of nuclear recoilenergy.The Lindhard yield extraction technique, and some of the data quality cuts,are built upon early work by Dr. Belina von Krosigk. The high-statisticssimulated neutron energy deposition data, produced by Dr. Anthony Vil-lano using his detailed Geant4 simulation of the calibration setup, forms afundamental input to the Lindhard yield extraction technique. The tech-niques described in Table 4.1 for defining the phonon pulse shape cuts weredeveloped by Mr. Vijay Iyer.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Evidence for Dark Matter . . . . . . . . . . . . . . . . . . . . 11.1 Observational Evidence for Dark Matter . . . . . . . . . . . 21.1.1 Galactic Rotation Curves . . . . . . . . . . . . . . . . 21.1.2 Collisions of Galaxy Clusters . . . . . . . . . . . . . . 41.2 Dark Matter Content of the Universe . . . . . . . . . . . . . 61.2.1 Cosmological Evolution Model of an Expanding Uni-verse . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . 91.2.3 Estimation of ΩM from Galaxy Clusters . . . . . . . 111.2.4 CMB Anisotropies . . . . . . . . . . . . . . . . . . . . 121.3 Composition Hypotheses . . . . . . . . . . . . . . . . . . . . 171.3.1 WIMPs . . . . . . . . . . . . . . . . . . . . . . . . . . 182 WIMP Detection Techniques . . . . . . . . . . . . . . . . . . 222.1 Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . . 232.1.1 Predictions for WIMP-nucleon Interactions on Earth 232.1.2 Background Discrimination Methods . . . . . . . . . 262.2 Accelerator Searches . . . . . . . . . . . . . . . . . . . . . . . 27viTable of Contents2.3 Indirect Detection . . . . . . . . . . . . . . . . . . . . . . . . 293 Overview of the SuperCDMS Experiment . . . . . . . . . . 303.1 Detectors and Experimental Setup . . . . . . . . . . . . . . . 323.2 Data Acquisition and Triggering . . . . . . . . . . . . . . . . 333.3 Detector Operating Modes . . . . . . . . . . . . . . . . . . . 343.3.1 iZIP Mode . . . . . . . . . . . . . . . . . . . . . . . . 343.3.2 CDMSlite Mode . . . . . . . . . . . . . . . . . . . . . 384 Photoneutron Calibration . . . . . . . . . . . . . . . . . . . . 414.1 Background and Motivation . . . . . . . . . . . . . . . . . . 414.1.1 Ionization Yield from Lindhard Theory . . . . . . . . 424.1.2 Existing Ionization Yield Data . . . . . . . . . . . . . 424.2 Calibration Concept and Setup . . . . . . . . . . . . . . . . . 444.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . 474.3.1 Physical Setup . . . . . . . . . . . . . . . . . . . . . . 474.3.2 Data-taking Conditions . . . . . . . . . . . . . . . . . 494.3.3 Trigger and readout settings . . . . . . . . . . . . . . 504.4 Cut Development . . . . . . . . . . . . . . . . . . . . . . . . 514.4.1 Basic Cuts . . . . . . . . . . . . . . . . . . . . . . . . 514.4.2 Quality Cuts . . . . . . . . . . . . . . . . . . . . . . . 534.5 Energy Resolution Model . . . . . . . . . . . . . . . . . . . . 764.5.1 Fit to the Resolution Model . . . . . . . . . . . . . . 784.6 Background Modelling . . . . . . . . . . . . . . . . . . . . . 794.7 Unbinned Lindhard Yield Extraction . . . . . . . . . . . . . 834.7.1 Neutron Spectrum PDF . . . . . . . . . . . . . . . . . 844.7.2 Options for the Lindhard Model Parameters . . . . . 894.7.3 Fit Results and Statistical Uncertainties . . . . . . . 914.7.4 Systematic Uncertainties . . . . . . . . . . . . . . . . 964.8 Final Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 995 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118B.1 1-Parameter Fit . . . . . . . . . . . . . . . . . . . . . . . . . 118B.2 2-Parameter Fit . . . . . . . . . . . . . . . . . . . . . . . . . 118viiTable of ContentsAppendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120viiiList of Tables2.1 Dark Matter Search Modes . . . . . . . . . . . . . . . . . . . 234.1 Summary of phonon pulse shape cuts defined within time blocks 564.2 Lowest median events/bin of all series in each data set, andtypical number of 30s bins per series . . . . . . . . . . . . . . 724.3 Comparison between actual and simulated event passage frac-tions for Sb at 70V . . . . . . . . . . . . . . . . . . . . . . . . 734.4 Goodness of fit measures for the background model of eachdata set, before and after adding the smoothed fit residuals . 834.5 Summary of fit results for Lindhard fit parameters . . . . . . 924.6 Summary of fit uncertainties for Lindhard fit parameters. . . 924.7 Summary of fit results and uncertainties for the modified 2-parameter Lindhard fit, with Emax=7 keVr,nr . . . . . . . . . 1024.8 Summary of χ2 results for each data set in the “neutron re-gion”, compared with the number of degrees of freedom. . . . 103ixList of Figures1.1 Mass and Energy Distribution of the Universe . . . . . . . . . 11.2 Expected Rotation Speed Distribution of a Spiral Galaxywith an Exponentially-Decaying Mass Density . . . . . . . . . 31.3 Measured Rotation Curves of Spiral Galaxies . . . . . . . . . 41.4 Bullet Cluster Measured by the Chandra X-ray Telescope . . 51.5 Variation of Primordial Element Abundances with Baryon-to-Photon Ratio Based on BBN Theory . . . . . . . . . . . . 101.6 Gas Densities of Galaxy Clusters Measured by the ChandraTelescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7 CMB Anisotropies Measured by the Planck Collaboration . . 152.1 WIMP-nucleon cross section exclusion limits from direct de-tection searches for spin-independent coupling . . . . . . . . . 252.2 Comparison between upper bounds on the WIMP-nucleon in-teraction cross section obtained from accelerator vs. directdetection searches . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Projected Sensitivity at SNOLAB . . . . . . . . . . . . . . . . 313.2 SuperCDMS Soudan Detector . . . . . . . . . . . . . . . . . . 323.3 SuperCDMS Detector Channel Layout . . . . . . . . . . . . . 333.4 iZIP Operating Mode . . . . . . . . . . . . . . . . . . . . . . 353.5 iZIP Ionization Yield Discrimination . . . . . . . . . . . . . . 373.6 iZIP Surface Electric Field . . . . . . . . . . . . . . . . . . . . 383.7 CDMSlite Operating Mode . . . . . . . . . . . . . . . . . . . 394.1 Existing ionization yield measurements . . . . . . . . . . . . . 434.2 Spin-independent WIMP-nucleon cross section limits fromCDMSlite runs 1 and 2 . . . . . . . . . . . . . . . . . . . . . . 444.3 Photoneutron calibration concept . . . . . . . . . . . . . . . . 45xList of Figures4.4 Dominant nuclear recoil component of the simulated neutronspectra for two of the three detector and source conditionsused in the photoneutron calibration . . . . . . . . . . . . . . 464.5 Physical setup for the photoneutron calibration . . . . . . . . 474.6 Source box used to deploy the radioactive source . . . . . . . 484.7 Radioactive source placement . . . . . . . . . . . . . . . . . . 494.8 Summary of data-taking periods for the photoneutron cali-bration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.9 Sample application of the pre-pulse standard deviation cut . . 544.10 Sample application of the PTOFdelay cut . . . . . . . . . . . 554.11 Sample plots showing the time block development for theparabolic ∆Glitch and ∆LFN cuts . . . . . . . . . . . . . . . 584.12 Plots showing the correlation between the event rate in thedetector and the second-order ∆LFN parabolic fit coefficient 594.13 Sample series removed by the BadSeries cut, showing its en-ergy variation with respect to the ∆Glitch phonon pulse shapevariable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.14 Y at 25V data passing the basic and above-discussed qualitycuts, showing the location of the charge χ2 cut . . . . . . . . 614.15 Sample raw pulse traces for the charge χ2 cut . . . . . . . . . 624.16 Y at 25V: neutron-on data with outlying charge χ2 highlighted 634.17 Two examples of glitchy pulse traces in the 70V data withnarrow, isolated glitches . . . . . . . . . . . . . . . . . . . . . 644.18 Y at 70V data passing the above-discussed quality cuts, show-ing the location of the charge χ2 cut for the 70V data . . . . 654.19 Variation of phonon χ2 with charge χ2, showing the point atwhich positive correlations become visible between these twoχ2 quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.20 Sample fit to LFN blob for the neutron-off Y at 70V data . . 674.21 Illustration of the method for determining the optimized anal-ysis cutoff (E0)opt . . . . . . . . . . . . . . . . . . . . . . . . . 684.22 Time variation of event rate, averaged over each series, forthe Sb at 70V data. The GoodSeriesRate cut shown in yellowremoves series whose average event rate is significantly abovethat of nearby series. . . . . . . . . . . . . . . . . . . . . . . . 694.23 Example of a series with a period of high trigger rate removedby the TriggerBurst cut . . . . . . . . . . . . . . . . . . . . . 704.24 Sample series showing the application of the TriggerBurst cutat varying precision levels . . . . . . . . . . . . . . . . . . . . 71xiList of Figures4.25 Sample MC series for the Sb at 70V data, with the Trigger-Burst cut applied . . . . . . . . . . . . . . . . . . . . . . . . . 734.26 Plots illustrating the effect of the TriggerGlitch cut . . . . . . 754.27 Gaussian fit to the zero-energy peak of the randoms to obtainthe baseline energy . . . . . . . . . . . . . . . . . . . . . . . . 774.28 Fits to the detector resolution model using standard devia-tions of Gaussian fits to spectral peaks . . . . . . . . . . . . . 784.29 Full background model for the Sb at 70V data . . . . . . . . 824.30 Full background model for the Y at 70V data on a log scale,showing the individual contribution from each analytical com-ponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.31 Electric potential difference between final locations of e-hpairs, as a function of their initial position for detector T5Z2at 70V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.32 Conversion of simulated data from nuclear recoil energy tophonon energy using the Lindhard model . . . . . . . . . . . 874.33 Simulated Y at 70V data, converted to phonon energy accord-ing to Lindhard with k=0.157, and overlaid with the resultingsmoothed neutron PDF . . . . . . . . . . . . . . . . . . . . . 884.34 Preliminary cut efficiency results for the phonon pulse shapecuts in time block 2 . . . . . . . . . . . . . . . . . . . . . . . 894.35 Simulated Y at 70V data overlaid with the resulting smoothedneutron PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.36 Sample neutron PDFs for the 2-parameter Lindhard model . 914.37 Sample Gaussian fits to the distributions of fit results to de-termine the statistical uncertainty on each parameter due tothe simulation statistics . . . . . . . . . . . . . . . . . . . . . 934.38 Correlation between klow and khigh for the 2-parameter fitwith randomness associated with simulation statistics varied . 934.39 Sample Gaussian fits to the distributions of fit results withMonte Carlo data to determine the statistical uncertainty oneach parameter due to the experimental statistics . . . . . . . 954.40 Shift in fit results due to the Fano factor systematic for the1-parameter fit . . . . . . . . . . . . . . . . . . . . . . . . . . 984.41 Best-fitting Lindhard ionization yield model, shown as a func-tion of nuclear recoil energy for the 1-parameter fit . . . . . . 994.42 Best-fitting Lindhard ionization yield model, shown as a func-tion of nuclear recoil energy for the 2-parameter fit . . . . . . 100xiiList of Figures4.43 Plots showing the check done to determine the maximum en-ergy to which the data can be sensitive to changes in theLindhard model. . . . . . . . . . . . . . . . . . . . . . . . . . 1014.44 Comparison between best fits to 2-parameter Lindhard mod-els with original value of Ehigh, vs. Ehigh=7 keVr,nr . . . . . . 1024.45 Sample plots showing the binned experimental neutron-ondata, overlaid with the best-fit PDF based on the combinednegative log likelihood fit. . . . . . . . . . . . . . . . . . . . . 103xiiiAcknowledgementsMy supervisor, Scott Oser, has provided constant support and encourage-ment throughout the past two years. He’s taken the time to understand themany stumbling blocks I’ve encountered while taking my first deep plungeinto a data analysis, and his insightful feedback helped me progress from thefirst simple quality cuts to developing the final ‘yield extraction’ technique.I’m grateful to him for giving me the freedom to follow my work where ittook me, providing guidance and much-needed statistics expertise along theway.I’m also indebted to Belina von Krosigk for the staggering amount of timeshe has put into introducing me to the ideas and techniques that have formedthe backbone of both my DAQ and analysis work on SuperCDMS, andhelping me at every step as I grappled with them. As a result, virtuallyevery aspect of my work has benefited from her support. She is a trulyinspiring person to work with, and I can only hope to one day possess somefraction of her calm and humour in the face of panicked masters studentswho tell her they’ve just clobbered the DAQ computer.I would also like to thank Anthony Villano, Lauren Hsu, and all the othermembers of photoneutron group whose support, feedback, and ideas haveled to many fresh directions, and vastly improved the quality of work in thisthesis.Bill Page has taken a ton of time out of his own graduate work to answer mymany questions and help me get acquainted with all things SuperCDMS, forwhich I am very grateful.Last but very much not least, I would like to thank my family for their loveand support, and for encouraging me to find what inspires me.xivChapter 1Evidence for Dark MatterObservational evidence to date indicates that cold non-baryonic dark matterconstitutes ∼26% of all matter and energy in the universe [1], and ∼85% ofthe total matter content. Despite decades of observational data showing thegravitational effects of dark matter, it has thus far evaded any form of di-rect experimental detection. As such, its composition and non-gravitationalinteractions remain a mystery.Figure 1.1: Mass andEnergy Distribution ofthe Universe, as de-rived from the most re-cent measurements bythe Planck collabora-tionDark Matter26%Dark Energy69%Normal Matter5%This chapter gives an overview of the experimental evidence for the exis-tence and abundance of non-baryonic dark matter in the universe. The lastsection discusses the current leading hypotheses predicting the compositionof dark matter, focusing primarily on WIMPs, which are the primary searchcandidate for the Super Cryogenic Dark Matter Search (SuperCDMS).11.1. Observational Evidence for Dark Matter1.1 Observational Evidence for Dark Matter1.1.1 Galactic Rotation CurvesSome of the earliest and clearest evidence for the existence of dark mat-ter in the universe came from studies of galactic rotation curves. In the1970’s, Rubin et al. [2, 3] found a strong disagreement between measure-ments of the rotational speeds of spiral galaxies, and the expectation fromNewtonian physics applied to luminosity-based estimates of galactic massdistributions.Expectation from Newtonian MechanicsAs early as 1970, it had been observed [4] that the surface brightness distri-butions of spiral galaxies were comprised of an inner spheroidal componentrepresenting the central galaxy bulge, with the surface brightness Ibulge(r)along the major axis varying with distance r according to:Ibulge(r) = I0er1/4 (1.1)and an outer exponential (disk) component with an exponential falloff:Iexp(r) = I0e−αr (1.2)Under the assumption that the galactic matter density roughly follows thebrightness distribution, the surface density distribution µ(r) along the majoraxis of the outer disk would fall off as [4]:µexp(r) = µ0e−αr (1.3)Considering only the exponential disk component modelling the luminosityprofile far from the galactic bulge, and taking the approximation of a zero-thickness disk with surface density µexp(r), the rotational speed distributionv(r) along the axis of the exponential disk is calculated in [4] to yield:v2(r) = pir2Gµ0α[I0(η)K0(η)− I1(η)K1(η)](1.4)21.1. Observational Evidence for Dark Matterwhere η = 12αr, G is the gravitational constant, and In and Kn are modifiedBessel functions. The total mass M of the disk is M = 2piµ0/α2. Trans-forming the radius and rotational speed to the dimensionless forms r˜ = αrand v˜ = v√GMα , Eq. 1.4 becomesv˜(r˜) =√12r˜2[I0(η)K0(η)− I1(η)K1(η)](1.5)Eq. 1.5 is used to plot the expected rotational speed distribution for theexponential disk component, as shown in Fig. 1.2. Under the assumptionthat the galactic matter density follows the luminosity distribution, the rota-tional speed should therefore peak, then fall off smoothly beyond the centralgalaxy bulge.Figure 1.2: Expected dimensionless rotational speed distribution for a spiralgalaxy with an exponentially-decaying surface mass density. We expect apeak near 2r˜ (see text for definition of r˜), followed by a smooth decay.Experimental Rotation CurvesFig. 1.3 shows a compilation of measured rotational speed distributionsfor spiral galaxies. Rather than the expected decay at large radii, the massdensity increases in the stellar bulge, and then flattens out to radii far beyondthe stellar bulge.31.1. Observational Evidence for Dark MatterFigure 1.3: Measured rotation curves of spiral galaxies. Figure from c©[6]. Credit: Annual Review of Astronomy and Astrophysics by AnnualReviews. Reproduced with permission of Annual Reviews in the formatThesis/Dissertation via Copyright Clearance Center.The constant rotational speeds out to large radii suggest that there mustbe another form of matter in addition to the luminous matter that is notlight-emitting, and whose density falls off much more slowly than the ap-proximately exponential decay of the luminous matter. This non-luminousmatter in the galactic halo is identified as “dark matter”.1.1.2 Collisions of Galaxy ClustersA more recent line of evidence pointing to the existence of dark matter ingalaxies comes from observations of the distribution of luminous and non-luminous matter following the collision of two galaxy clusters. The Bulletcluster, first investigated by Tucker et. al. in 1995 [7] using data fromthe Chandra NASA X-ray observatory, is the result of such a collision. Thismerger presents the possibility of decoupling between the luminous and non-luminous (i.e. dark) matter components of the galaxy clusters, thus present-ing a unique opportunity to study dark matter in galaxy clusters.A study [8] of the Bullet cluster built upon previous work [9, 10] showing thatthe mass of hot intracluster plasma in galaxy clusters exceeds the mass ofstars comprising galaxies in the cluster by a factor of 3-15. When two clusterscollide, it is expected that the relatively sparse stellar components will passone another with negligible interaction, whereas the fluid-like intraclusterplasma will be slowed down due to ram pressure. Therefore, the na¨ıve41.1. Observational Evidence for Dark Matterexpectation would be that the majority of matter, and hence gravitationalpotential, in the cluster merger should reside in the main central componentcomprised of X-ray emitting intrastellar plasma that has been slowed downby ram pressure, with only the relatively low-mass stellar components at theextreme edges.A gravitational potential map of the merger was obtained using weak gravi-tational lensing measurements [8], which operate on the principle that lightis bent by small angles when passing through the gravitational field of themerging galaxy clusters, leading to measurable distortions of galaxies be-hind the clusters. The resulting deflection causes the images of the galaxiesbehind the mass in the merger to stretch preferentially in the direction per-pendicular to the centre of mass of the cluster, and with enough statistics,the direction and degree of stretching of the galaxies behind the cluster canbe used to map out the gravitational potential of the merger.Figure 1.4: Bullet cluster measured by the Chandra X-ray tele-scope. Figure reproduced with permission from [11]. Credit: X-ray:NASA/CXC/CfA/M.Markevitch et al.; Optical: NASA/STScI; Magel-lan/U.Arizona/D.Clowe et al.; Lensing Map: NASA/STScI; ESO WFI;Magellan/U.Arizona/D.Clowe et al.In Fig. 1.4, the resulting gravitational potential map (blue) is superimposedon the distribution of X-ray emitting plasma (pink), showing that the vastmajority of mass in the cluster merger actually follows the stellar compo-nent that passes through effectively unimpeded. This observation providesevidence that the majority of mass in the galaxy clusters is comprised of anon-luminous, gravitationally attractive form of matter that interacts very51.2. Dark Matter Content of the Universeweakly with both itself and normal matter.1.2 Dark Matter Content of the UniverseSection 1.1 presented two examples of striking observational evidence forthe existence of dark matter in galaxies. However, these examples did littleto illuminate the question of exactly how much of the universe is actuallycomposed of dark matter. Nor did they indicate to what extent the darkmatter can be accounted for by the normal “baryonic matter” – protons,neutrons, and bound electrons – that constitutes the vast majority of knownmatter in the universe.Of the numerous and diverse observational studies aimed at answering theabove questions, the three discussed in this section are intended to looselyreflect the chronology of early findings revealing the need for a dominantnon-baryonic dark matter component in the universe, followed by precisionmeasurements of its abundance. Comparison of the present-day measure-ments of light element abundances with predictions from the theory of BigBang Nucleosynthesis offered an early estimate of the baryon content of theuniverse. Studies of galaxy clusters, such as measurements of their gas massfraction, suggested that the baryon content represents only a sub-dominantfraction of the full matter density of the universe. Recent measurementsof anisotropies in the Cosmic Microwave Background (CMB) provide inde-pendent confirmation of the dominance of non-baryonic dark matter, withhigh-precision measurements of the contributions from both baryonic anddark matter to the total mass content of the universe.1.2.1 Cosmological Evolution Model of an ExpandingUniverseThis section provides a brief overview of the mathematical formalism thatdescribes the evolution of the universe on the cosmological scale, largelyfollowing Chapter 18 of [12].Redshift as a Measure of Relative SpeedThe approaching or recessional speed of luminous objects observed withtelescopes – such as stars, galaxies, and nebulae – can be estimated by a61.2. Dark Matter Content of the Universequantity known as “redshift” z, which is defined as the fractional differencebetween the wavelength λe of light emitted at time te, and its wavelengthλ0 at time t0 when it arrives at the telescope:z =λ0 − λeλe(1.6)The redshift is related to the relative speed v of the light-emitting objectvia the Doppler effect, which for non-relativistic objects is simply:v = zc (1.7)where c is the speed of light. A positive redshift thus implies that theluminous object is receding away from Earth.The Friedmann EquationAt the cosmological scale, it is found that the universe can be modelled to agood approximation as a homogeneous and isotropic fluid. In 1929, EdwinHubble demonstrated [13] that the universe is expanding after observing apositive, approximately linear relationship between the redshift – i.e. re-cessional speed – of 24 nebulae, and their independently-measured distancefrom Earth. In the late 1990’s, observational data from Type 1a supernovaewas used to further study the expansion of the universe, and these higher-precision measurements revealed [14, 15] that the universe is expanding atan accelerating rate.The entire evolution of the universe, modelled as a homogeneous, isotropic,expanding fluid, is described by a time-dependent scale factor a(t) thatquantifies the physical distance between any two points separated by a con-stant “coordinate distance”. A constant coordinate distance implies thatthe physical separation distance would remain constant if the universe werenot expanding. The time-evolution of the scale factor is described by theFriedmann equation:a˙2 − 8piρ3a2 = −k (1.8)where ρ is the total energy density of the universe, and k is a constant thatquantifies the overall spacetime curvature of the universe: k = 0 implies71.2. Dark Matter Content of the Universethat spacetime is flat on cosmological scales, k = +1 implies a closed uni-verse with positive curvature, and k = −1 implies an open universe withnegative curvature. The existing astronomical evidence, most recently fromthe Planck mission [1], is strongly in favour of a flat universe (k = 0).Dimensionless Energy DensitySolving for the energy density ρ in the Friedmann equation (Eq. 1.8)yields:ρ =38pia˙2 + ka2(1.9)Setting k = 0, corresponding to a flat universe, and evaluating Eq. 1.9 at thepresent time t0 yields what is called the “critical energy density” ρcrit:ρcrit =38pi(a˙(t0)a(t0))2=38piH20 (1.10)where H0 ≡ a˙(t0)a(t0) is the Hubble constant, which gives a measure of therate of expansion of the universe at the present time. The most recentmeasurement of H0 by the Planck mission [1] yields a value of 66.26±0.98km s−1 Mpc−1.It is common to express the energy density ρX of a particular componentX of the universe’s energy budget in terms of the critical energy densityas:ΩX =ρXρcrit(1.11)With this notation, the sum Ω =∑n Ωn over all contributions to the energydensity in the universe will be Ω=1 for a flat universe, Ω < 1 for an openuniverse (k = −1), and Ω > 1 for a closed universe (k = +1). Under theassumption of a flat universe, the energy density ΩX thus represents thefractional contribution of component X to the combined energy density ofthe universe. This quantity is often reported with an additional factor ofh2 ≡ ( H0100 km s−1 Mpc−1)2.81.2. Dark Matter Content of the Universe1.2.2 Big Bang NucleosynthesisBig Bang Nucleosynthesis (BBN) is believed to have taken place during thefirst ∼ 100s to 20 minutes [16, 17] after the Big Bang, as the temperature ofthe early universe cooled sufficiently to allow the formation of light elements.This process of nucleon production continued until the temperature anddensity of the expanding universe became too low for these nuclear reactionsto occur [18].BBN produced what is known as the “primordial abundance” of 2H (alsoknown as D), 3H, 3He, 4He, and 7Li. The abundances of these light elementsare typically expressed in terms of their number density n relative to theH abundance (eg. D/H ≡ n(D)/n(H)), with the exception of 4He, which isexpressed in terms of its mass density relative to the total baryon mass den-sity: Y ≡ ρ(4He)/ρb. Once stars began to form, the nuclear fusion reactionsbetween atoms within stars, known as “stellar nucleosynthesis”, became thedominant process capable of altering these primordial abundances [16]. Thetheory of BBN provided the earliest estimate of the baryon contributionΩb to the energy density of the universe [16], which has only recently beenovercome in sensitivity by detailed studies of the CMB, discussed in Section1.2.4.The rate of fusion reactions taking place during BBN is predicted to increasewith the density nb of baryons in the universe at the time. The baryonnumber density is typically normalized by the photon number density nγ toobtain the baryon-to-photon ratio η [18]:η =nbnγ(1.12)BBN theory predicts the variation in the primordial abundances of D/H,3He/H, and 7Li/H, and to a small degree Y = ρ(4He)/ρb, with the present-day η, as shown in Fig. 1.5. As such, the theory may be applied to present-day measurements of the primordial abundances of these light elements toobtain an experimental estimate of η. The results obtained from the differentlight element abundances can also be compared for consistency. The mea-sured value of η is combined with measurements of H0, and of nγ deducedfrom the present-day CMB temperature, to estimate the baryon density Ωbof the universe using Equations 1.10-1.12.91.2. Dark Matter Content of the UniverseFigure 1.5: Variation of primordial element abundances with baryon-to-photon ratio based on BBN theory. Yellow boxes show the observed rangeof light element abundances, and the resulting range of baryon-to-photonratio for three of the light elements. The magenta dashed band shows theestimated baryon-to-photon ratio as determined from the D/H and Y mea-surements. The blue dashed band shows most recent measurement of thebaryon-to-photon ratio from measurements of the CMB with the Plancktelescope. Figure reproduced with permission from c© [17].The baryon-to-photon ratio is most sensitive to and best constrained bythe primordial abundance of D. Since BBN is the only significant sourceof D [19], which is destroyed in stellar processes, it is important to ob-tain measurements of the D abundance from regions of space that havenot experienced significant stellar nucleosynthesis, which can be identifiedby their low “metallicity”. Metallicity quantifies the concentration of ele-ments heavier than H and He that would – with the exception of the smallamounts of primordial 7Li – have been produced by stellar nucleosynthesis[20]. Studies of low-metallicity gas clouds known as “damped Lyα systems”[21] can provide a reasonable approximation of the primordial abundance.Fig. 1.5 shows the current constraints on the D/H, 7Li/H, and Y as yel-low boxes, along with the estimated baryon-to-photon ratio obtained fromthe D/H and Y abundances, which yields a current best estimate of [17]101.2. Dark Matter Content of the Universe0.021 ≤ Ωbh2 ≤ 0.024 at a 95% confidence level. Using h=0.6726±0.0098measured by the Planck collaboration [1], we obtain an estimated baryonfraction of 0.046 ≤ Ωb ≤ 0.053, in agreement with the result from morerecent measurements of the CMB.The notable inconsistency of η obtained from the 7Li/H abundance couldsimply be due to systematic errors in estimating the primordial abundance,or it may present a hint of new physics [17] that is not yet explained bystandard BBN theory.1.2.3 Estimation of ΩM from Galaxy ClustersIn addition to providing strong evidence for dark matter through their colli-sions (see Section 1.1.2), galaxy clusters can be used to infer the total massdensity ΩM of the universe from the baryon density Ωb, under the “fairsampling assumption”. The fair sampling assumption states that becausegalaxy clusters gather material from a large region of space, their composi-tion provides a fair sample of the universe [22].While stellar matter composes a fraction of the baryonic matter in galaxyclusters, the majority exists as hot (∼108K), X-ray emitting ionized gas inthe “intracluster medium” (ICM) [9, 10]. Under the fair sampling assump-tion, the total mass density ΩM of the universe is obtained from the baryondensity Ωb and the measured fraction fgas +fstars of luminous baryonic mat-ter in galaxy clusters according to:ΩM =Ωbfgas + fstars(1.13)In the early 1990’s, studies of astronomical data from galaxy clusters [23]found that for a flat (Ω = 1) universe, the baryon fraction Ωb inferred fromthe measured gas mass fraction in the Shapley Supercluster could only agreewith earlier results from BBN (see Section 1.2.2) if one assumed that ΩMincludes a significant non-baryonic dark matter component.In a more recent study, [24], data was compiled for 35 luminous clusterscollected from NASA’s Chandra X-ray observatory. The masses Mgas(r)and Mtot(r) of both the ICM gas and the entire cluster were inferred outto a radius r500 (the radius at which the cluster density is 500 times themass density of the surrounding universe at the given redshift) for each111.2. Dark Matter Content of the Universegalaxy cluster from the density and temperature of the X-ray emitting ion-ized plasma, under the assumption of hydrostatic equilibrium. The resultinggas fractions fgas =MgasMtotshown in Fig. 1.6 were averaged over all galaxiesconsidered.Figure 1.6: Black: gas densities for the galaxy clusters measured by theChandra telescope, as a function of cluster mass. Magenta and blue: com-parison with previous studies. Figure reproduced with permission from [24].Copyright (2013) by Oxford University Press, on behalf of the Royal Astro-nomical Society.Combining the average gas mass fraction of fgas = 0.163±0.032 with anothermeasurement of stellar baryon fractions out to r500 yields fgas + fstars =0.182± 0.032. Under the fair sampling assumption, this result from galaxyclusters suggests that only 18% of matter in the universe is accounted forby normal light-emitting baryonic matter.1.2.4 CMB AnisotropiesFirst discovered in 1964 [25], the cosmic microwave background (CMB) is afaint, approximately isotropic background electromagnetic radiation fillingthe observable universe, with a mean temperature of 2.73 K [26]. Astro-nomical evidence indicates that the CMB was created following a point inthe history of the universe known as the “recombination epoch”. Prior to121.2. Dark Matter Content of the Universerecombination, the temperature of the early universe was too high for pro-tons and electrons to form neutral atoms. In this ionized plasma, there wasa high rate of interactions between photons and free electrons via Thomsonand Coulomb scattering [27]:e− + γ ↔ e− + γAs a result of these interactions, the universe was at that time opaque tophotons. During recombination, the temperature of the universe had cooledto the point that it became energetically favourable for protons and electronsto form neutral atoms. Following recombination, the relatively low interac-tion probability of photons with neutral atoms made the universe effectivelytransparent to photons. The escaped photons which comprise the CMB rep-resent the oldest light in the universe, and offer a rich source of informationfor astronomers studying the conditions of the early universe.In 1992, the Cosmic Background Explorer (COBE) satellite produced thefirst evidence of faint anisotropies in the CMB [26], on the order of 1 in 105.Over its nine-year mission from 2001 to 2009, the Wilkinson MicrowaveAnisotropy Probe (WMAP) spacecraft [28] measured the CMB radiationwith unprecedented precision. The data from WMAP made it possible toproduce a detailed map of these anisotropies, and has only recently beenovercome in precision following the four-year operation of the Planck tele-scope [1] from 2009 to 2013. These anisotropies reflect the gravitationalpotential fluctuations in the early universe from which the large-scale struc-ture of the universe is thought to originate. They are typically representedas fractional variations ∆T from the mean temperature T of the CMB ra-diation as a function of their direction (θ, φ) in the sky [29]:Θ(θ, φ) =∆TT(1.14)To study how the size of the temperature fluctuations varies with theirangular scale, the fluctuations are commonly transformed into their sphericalmultipole moments:Θ`m =∫Θ(θ, φ)Y ∗`m(θ, φ)dΩ (1.15)131.2. Dark Matter Content of the Universewhere for reasonably small curvature, θ = 2pi` , meaning that ` ∼ 102 repre-sents degree-scale fluctuations [29]. The resulting power spectrum〈Θ ∗`m Θ`′m′〉 = δ``′δmm′C` (1.16)gives a measure of how the amplitude of the fluctuations varies as a functionof their angular scale.The ionized plasma that existed prior to recombination can be studied as a“photon-baryon fluid” [29]. Small density fluctuations in the fluid producegravitational wells that attract and compress nearby matter. However, pres-sure forces due to the Coulomb interactions resist the gravitational compres-sion, acting as an effective spring force to produce oscillatory compressionand rarefaction of the fluid. At the time of recombination, photons decou-pled from matter, the pressure forces disappeared, and fluctuations in thetemperature of the escaped photons forming the CMB reflect the densityvariations in the photon-baryon plasma at the time of recombination, withhigher-temperature photons originating from denser regions of the plasma.The angular temperature fluctuation modes ` in the CMB correspond ap-proximately to the Fourier wavenumber k ∼ `η0 of oscillatory modes [27],where η0 is the coordinate distance to the last scattering surface. It canbe shown [27] that for small sections of the sky where curvature can be ne-glected, the power spectrum PΘ of the oscillatory wavenumber k ∼ `η0 maybe expressed in terms of the angular power spectrum as:PΘ(k/η0) ≈ `(`+ 1)2piC` (1.17)When this power spectrum is plotted as a function of multipole moment, asshown in Fig. 1.7 with the most recent results from the Planck collaboration,the amplitude of the power spectrum exhibits clear peaks and troughs as afunction of angular scale. It is found that the particular shape of the angularscale dependence is well described [1] by a flat “ΛCDM” Big Bang cosmol-ogy [30] containing cold (i.e. non-relativistic) dark matter (CDM), and acosmological constant Λ associated with dark energy. Further, the absoluteand relative sizes of the peaks contain information about the composition ofthe early universe, including the total baryon fraction, and the dark mattercomponent prior to and at the time of recombination. The discussion hereof these “acoustic peaks”, and their dependence on the baryon fraction and141.2. Dark Matter Content of the Universedark matter component remains largely qualitative, and follows overviewsfound in [27, 29, 31].Figure 1.7: CMB anisotropies measured by the Planck collaboration. Thered line shows the prediction of the best-fitting ΛCDM model for a flatuniverse (Ω = 1). Credit: [1], reproduced with permission c© ESOThe peaks in the CMB correspond to Fourier oscillation modes of the photon-baryon plasma which reached the extreme maximum of their oscillation atthe time of recombination. Assuming that gravitational compression sets upthe first stage of oscillation, the first peak corresponds to the “fundamental”mode kf that had just reached full compression at recombination, and thesecond peak corresponds to the mode 2kf that had undergone a full cycleof compression and rarefaction. In a perfect oscillating fluid with instanta-neous decoupling at the time of recombination, the power spectrum couldbe expected to follow a squared sinusoid.The first effect that modifies the shape of the CMB peaks is known as“diffusion”. Diffusion arises from the fact that recombination was not in-stantaneous. Instead, during the short recombination period while particleswere combining to form neutral atoms, CMB photons executed a randomwalk as they interacted with the remaining ionized particles. This randomwalk ‘washes out’ peaks with wavelengths comparable to or shorter than thedistances travelled by photons during their random walk, which introducesthe observed exponential damping in the power spectrum at high multi-151.2. Dark Matter Content of the Universepoles.The second effect, known as “baryon loading”, is related to the baryoncontent of the universe at recombination. When the attractive force betweenmassive baryons in the fluid is accounted for, the extremes of compressionand rarefaction become asymmetric, because there is now more force pullingthe fluid into gravitational wells than there is pushing them out. The effectis that peaks in the CMB spectrum corresponding to compressional extremesof oscillation – the first, third, fifth, and so on – become amplified comparedwith their adjacent peaks. The degree to which odd peaks are enhancedrelative to even peaks thus provides information about the baryon densityat recombination.The third effect, known as “radiation driving” makes the CMB spectrumsensitive to the dark matter content of the matter-dominated universe lead-ing up to recombination. Prior to recombination, the universe transitionedfrom the “radiation-dominated era”, during which its behaviour was dom-inated by radiation, to the ongoing “matter dominated era” wherein darkmatter dominates the behaviour of the universe. During radiation domina-tion, potential wells are formed primarily by photon density fluctuations,which vary with the fluid pressure. As the photon-baryon fluid compressesinto these potential wells, the photon density initially increases, deepeningthe well, then stabilizes as pressure pushes back against the compression. Inaddition to the shallowing of the well as pressure pushes back against gravityand reduces the photon density, the expansion of the universe further reducesthe potential depth, such that the pressure force has very little gravitationalpotential to overcome during rarefaction. This radiation driving effect en-hances the peak amplitudes, but is diminished during the matter-dominatedera when the potential wells are formed primarily by pressure-less dark mat-ter.The dark matter density ΩDM during the matter-dominated era has twomeasurable effects on the CMB due to radiation driving. First, the over-all degree to which radiation driving is dampened depends on how muchdark matter was present over time, where a higher dark matter densityinduces more damping, and hence leads to smaller peak amplitudes over-all. Second, since higher-frequency (i.e. higher-multipole) peaks will haveundergone more oscillations during radiation domination, they will exhibitgreater enhancement due to radiation driving, such that the exact degree ofhigher-multipole peak amplitude enhancement is also sensitive to the darkmatter density.161.3. Composition HypothesesThe flat ΛCDM model used to fit the CMB power spectrum measuredby the Planck collaboration in Fig. 1.7 incorporates the effects discussedabove, along with others, to parametrize the shape of the peaks in terms ofimportant cosmological constants. The best fit to the ` ≥ 30 data in Fig.1.7 gives a baryon density of Ωb = 0.0491± 0.0015, and a cold dark matterdensity of Ωc = 0.2650±0.0091. For a flat universe, these results imply that26.5% of the universe is composed of dark matter, and only 4.9% of normalbaryonic matter.1.3 Composition HypothesesThe previous section presented three lines of evidence, which together stronglysuggest that the majority of matter in the universe is dark, and that only asmall portion of this dark matter could be accounted for by normal baryonicmatter. There is nothing in the current Standard Models of cosmology orparticle physics [32] that can account for such a significant fraction of themass budget of the universe.Theories known as “modified gravity” have been developed that modify gen-eral relativity [33] in order to avoid the need for dark matter entirely. Othersmodify the standard inflationary cosmology to allow for the development of“primordial black holes” [34] prior to BBN, which would not be accountedfor in the estimates of Ωb based on BBN theory and measurements of theCMB. However, these modified theories have so far suffered significant theo-retical and/or observational hurdles, such as difficulties with using modifiedgravity to explain the evidence for dark matter on cosmological scales, andfine-tuning assumptions associated with the development of primordial blackholes in the early universe [34]. The current most widely accepted dark mat-ter candidates take the form of particles associated with theories beyond theStandard Model (SM) of particle physics.In order to represent a significant contribution to the dark matter contentof the universe, candidate particles must satisfy several basic criteria [35].First, they must be stable on cosmological time scales – otherwise they wouldhave decayed away by now. Second, analysis of structure formation in theuniverse suggests that particle dark matter should be “cold”, meaning thatit was non-relativistic when galaxies began to form. Third, to constitute asignificant fraction of the observed dark matter density ΩCDM, they musthave the correct “relic density” at the time that their interaction rate be-171.3. Composition Hypothesescame negligible with SM particles. Lastly, dark matter must be effectively“collisionless” [36], meaning that the only significant long-range interactionsare gravitational.Of the hypothesized particles satisfying the above criteria, weakly interact-ing massive particles (WIMPs) and axions have for a number of years beenconsidered the most theoretically and experimentally viable particle darkmatter candidates, and have been the subject of many dedicated experi-mental searches.The axion was originally proposed as a part of the Peccei-Quinn [37] solu-tion to the strong CP problem in quantum chromodynamics (QCD). Thestrong CP problem arises from the fact that although the strong interactioncontains a CP-violating term in its Lagrangian, CP violation has not beenobserved in QCD interactions. Axions would have been produced “non-thermally” [35], meaning that they were not in thermal equilibrium withSM particles in the early universe. This allows them to have masses as lowas µeV, much lower [35, 36] than would be allowed for “thermally-produced”dark matter, as will be discussed further in the next section. The axion, andaxion-like particles (ALPs) are predicted to decay into pairs of SM particles[38], thus presenting a possibility for experimental detection.The axion would reside in one of potentially many hidden “dark sectors” [39]that interact through their own particles, forces, and structure, with onlyvery weak couplings to SM particles. More recent theoretically-motivatedcandidates from the dark sector include the “dark photon”, whose couplingswith electrically charged particles through kinetic mixing with the SM pho-ton could provide the possibility for experimental detection [39].The remainder of this chapter will be devoted to WIMPs – the primarysearch candidate of the SuperCDMS experiment – with particular emphasison low-mass WIMP candidates to which SuperCDMS is particularly sensi-tive.1.3.1 WIMPsWIMPs represent a general class of massive, non-baryonic fundamental par-ticles that interact gravitationally with normal matter, and whose interac-tions with SM particles are at or below the weak interaction strength, butnon-negligible [40]. This broad definition could in fact encompass any ofthe axions and dark photon candidates discussed above, but in practice,181.3. Composition HypothesesWIMPs are often considered to have masses in the approximate range of∼10 GeV - TeV predicted by minimal supersymmetric models [32], withlow-mass WIMPs (which may be unrelated to supersymmetry) occupyingthe approximate mass range of several MeV to 10 GeV.Thermal Freeze-Out and the WIMP MiracleAs the abundance of many popular WIMP candidates is predicted to arisefrom “thermal freeze-out” in the early universe [40], the mechanism for ther-mal freeze-out is briefly discussed here. According to the thermal freeze-outhypothesis, WIMPs were in thermal equilibrium with SM particles in thehigh temperature and matter density of the early universe. As the uni-verse expanded and cooled, the interaction rate declined. Once the WIMP-SM interaction rate dropped below the expansion rate of the universe, theWIMPs rapidly “froze out” of thermal equilibrium with the SM particles,effectively fixing the co-moving WIMP density to its present-day value [40].The present-day WIMP dark matter density Ωχh2 is then approximatelygiven by [41]:Ωχh20.12≈ 1〈σann10−36cm2v/c0.1〉 (1.18)For expected WIMP velocities at freeze-out (v/c ≈ 0.1), the interactioncross section of 10−36cm2 corresponding to the present-day dark matterdensity (ΩDMh2 ≈ 0.12) is at a scale typical for interactions mediated bythe weak force. This apparent coincidence was dubbed the “WIMP miracle”,and spurred significant interest in dark matter candidates with weak-scalecross sections and typical weak force masses in the range of ∼10 GeV/c2to several TeV. It has however been argued [42] that the WIMP mass neednot necessarily be constrained to the above “weak force” range in order toproduce weak scale WIMP-SM interaction cross sections.Supersymmetric WIMPsMany supersymmetric models predict a stable particle with weak-scale in-teractions and a mass in the range of ∼10 GeV/c2 to several TeV/c2 aspart of a solution to the “hierarchy problem” [32] in particle physics. The191.3. Composition Hypotheseshierarchy problem is the question of why the physical Higgs boson mass isso much smaller than the Planck mass scale.The apparent naturalness of supersymmetric WIMP candidates arising froma solution to an independent problem, combined with the popularity of su-persymmetry as an extension to the SM, has for many years made it the mostimportant candidate for many experimental dark matter searches. However,dedicated experimental searches have so far failed to find evidence for themost constrained supersymmetric WIMP models [35]. These null results,combined with the absence of evidence for supersymmetry at the LHC[43, 44], have motivated increasing interest in theories predicting WIMPsoutside of the traditional mass range of ∼10 GeV/c2 to several TeV. Thefollowing section discusses in particular theories predicting relatively low-mass dark matter, in the sub-GeV/c2 to ∼10 GeV/c2 range.Low-mass WIMP HypothesesOf the numerous theories that have emerged in recent years with low-massWIMP predictions, some arise from non-traditional extensions of supersym-metry, such as models with an extended Higgs sector [35], or others predict-ing axinos and gravitinos – fermionic partners of the axion and graviton, re-spectively – as dark matter constituents [41]. However, many of the more re-cent low-mass WIMP theories are entirely unrelated to supersymmetry, andmay invoke either thermal or non-thermal production mechanisms.Asymmetric dark matter (ADM) is an example of a non-supersymmetricmodel with low-mass WIMP predictions. The theory allows for various pos-sible mechanisms for interaction with SM particles, and WIMP masses inthe range of ∼5-15 GeV/c2 [46]. In this model, WIMPs are not their ownantiparticle, and the observed abundance of dark matter is assumed to arisenot from the details of its thermal decoupling, but rather from a mechanismanalogous to that which produced the observed baryon-antibaryon asymme-try in the universe. It is argued [46] that this common origin of baryon anddark matter abundances could account for the fact that the observed densi-ties are of similar magnitudes (ΩDM ≈ 5Ωb), a result which may otherwiseappear coincidental.Other potential candidates could arise from the hidden dark sector intro-duced earlier. The massive dark photon, with predicted masses ranging fromMeV/c2 to GeV/c2 depending on the details of its origin and “portal interac-201.3. Composition Hypothesestions” with SM particles [39], would couple through the “vector portal”, oneof four possible dominant interaction portals between SM and dark sectorparticles. Many of the early exclusion limits for dark sector candidates havecome from reanalyses of existing WIMP search data [47]. However, futureexperiments, including a planned search for dark photon absorption eventsin SuperCDMS single-charge sensitive detectors [48], could offer targetedsensitivity to predicted portal interactions.21Chapter 2WIMP DetectionTechniquesThe last chapter presented several lines of observational evidence indicat-ing that ∼85% of the matter in the universe is cold (i.e. non-relativistic)non-luminous “dark matter” that interacts via the gravitational force, butotherwise very weakly – if at all – with either itself or normal matter. Whileobservational data has provided precise determinations of the dark mattermakeup of the universe, it fails to answer some important questions, suchas:• Assuming that dark matter is composed of elementary particles, whatare the properties of these particles (mass, spin, etc.)?• By what mechanisms other than gravity does dark matter interactwith itself and/or other matter?This chapter will discuss the techniques that are employed to detect parti-cle dark matter (DM) in the hopes of understanding its particle propertiesand non-gravitational interactions. These dark matter detection techniquesare divided into three classes – accelerator searches, indirect detection, anddirect detection – which are summarized in Table 2.1, and discussed in moredetail in the next three sections, focusing on WIMP detection. Particularemphasis is given to direct detection, which is the method employed by theSuperCDMS experiment.222.1. Direct DetectionTable 2.1: Summary of the three primary dark matter search modesSearch Mode Summary ExamplesDirect DetectionDirectly measure elasticWIMP-nucleon scattering in alow-background environmentSuperCDMS,EDELWEISS,DEAP 3600Accelerator SearchesSearch for missing total or transversemomentum in high-energy collisions,indicating the production of newstable, neutral particles.ATLAS, CMS,Belle IIIndirect DetectionUse observational data to detectproducts of WIMP annihilations ordecays in regions of the observableuniverse that are expected to have ahigh dark matter densityPAMELAsatellite, HESStelescope2.1 Direct DetectionMost of the WIMP search experiments employ some form of direct detection.For this search mode, detectors are operated in a low-background environ-ment, typically underground, and experiments look for an above-backgroundexcess of nuclear recoil events that could be attributed to WIMP-nucleoncollisions in the detectors.2.1.1 Predictions for WIMP-nucleon Interactions onEarthIt is expected that WIMPs should be present in our Milky Way galaxy withthe appropriate density profile to produce the rotation curves measuredin other galaxies (see Section 1.1.1). Given the expected speed of WIMPsrelative to terrestrial detectors – typically taken to be ∼220 km/s on average[51] – it is expected that the dominant interaction mechanism will be elasticscattering in the detector [51]. For WIMP masses comparable to or largerthan the proton mass (& 100 MeV/c2), it is kinematically favourable forWIMPs to interact via nuclear recoils, rather than electron recoils, in thedetector material.The interaction rate of WIMPs with terrestrial detectors depends on boththe interaction cross section – which is material dependent – and the lo-232.1. Direct Detectioncal WIMP flux. The WIMP flux is set by the local dark matter density(most recently calculated at 0.39 GeV cm−3 [52]), the mean WIMP veloc-ity, the escape velocity vesc ≈ 544 km/s [53] of the Milky Way galaxy, andthe WIMP mass. Therefore, the two unknowns in predicting the rate ofWIMP-nucleon interactions with a given detector material are the WIMPmass mχ and WIMP-nucleon scattering cross section σ0. For this reason,WIMP exclusion limits from direct detection searches are typically shownas upper bounds on the WIMP-nucleon interaction cross section as a func-tion of WIMP mass, as shown in Fig. 2.1 with a compilation of recentexclusion limits from direct detection experiments on “spin-independent”WIMP-nucleon couplings. Spin-independent couplings are characterized byscalar and vector WIMP-nucleon currents [35]. Exclusion limits also existfor “spin-dependent” couplings, which involve axial vector currents [35] andassume that WIMPs have a non-zero spin, but accelerator searches are typ-ically more sensitive to these couplings than direct detection searches (seeSection 2.2).242.1. Direct DetectionFigure 2.1: WIMP-nucleon cross section exclusion limits from direct detec-tion searches for spin-independent coupling. Figure reproduced with per-mission from c© [35].For spin-independent couplings, the WIMP-nucleon scattering cross sectionσ0 is expected to vary with the atomic number A of the target nucleusaccording to [51]: σ0 ∝ A2σχ−p, where σχ−p is the WIMP-proton interactioncross section. Therefore, it is often desirable to select a target material witha high atomic number for direct detection searches, though other details suchas the nuclear form factor F (Q) [51] and background rejection possibilitiescan also affect the choice of target material. It is also important to considerthe range of recoil energies that can be accurately measured by the detector.For non-relativistic scatters, the nuclear recoil energy is related to the WIMP252.1. Direct Detectionmass and scattering angle θr according to [54]:Er = 2v2 µ2mAcos2 θr (2.1)where mA is the mass of the target nucleus, µ = mχmA/(mχ + mA) is theWIMP-nucleus reduced mass, and v is the WIMP speed in the laboratoryrest frame. Depending on the WIMP and target masses, this can result innuclear recoil energies in the range of 0 to ∼100 keV.The lower bound on the WIMP masses to which direct detection searchesare sensitive is in general dictated by the signal to noise ratio of the de-tector(s). As the WIMP mass decreases, the average nuclear recoil energyEr imparted to the nucleus in a WIMP-nucleon collision will also decreasebased on Eq. 2.1, and at some point the measured recoil energy will becomeindistinguishable from detector noise. Near this “noise wall”, it becomesincreasingly difficult to place strong constraints on the interaction cross sec-tion, resulting in the characteristic sharp rise in cross section limits seen atlow WIMP masses in Fig. 2.1.2.1.2 Background Discrimination MethodsFor typical values of σ0 and mχ, predicted signal event rates are on theorder of 1 event year−1 kg−1 [51], much lower than the typical radioactivebackground on Earth’s surface [35]. This background can be reduced byoperating the detectors deep underground to minimize the flux of cosmicrays, surrounding the detector with additional radioactive shielding, andselecting radio-pure materials for the detectors and surrounding structure.In addition, most direct-detection experiments employ some form of back-ground discrimination to minimize the number of background events thatcan be interpreted as WIMP signals. Active detector shielding, such as acosmic ray veto surrounding the detector, is a common method of identify-ing measured events as background. Other discrimination techniques aim toidentify predicted temporal variations in the flux or directionality of signalevents [51].Annual modulations in the WIMP flux are predicted due to the change inrelative velocity between Earth and the galaxy’s WIMP halo. The DAMAcollaboration [55] has reported a 9.3 σ annual modulation in their signalrate with a total of 13 years of data-taking and 350kg of detector material,262.2. Accelerator Searcheswith the possible WIMP masses and cross sections of the signal enclosedby the turquoise DAMA/LIBRA regions in Fig. 2.1. Since the first an-nouncement of the DAMA/LIBRA annual modulation signal, other WIMPsearches have explored the same parameter space, with null results. Yet, aconvincing alternative explanation of the annual modulation has yet to bedemonstrated.Daily modulations in the directionality of WIMP recoil events are predicteddue to Earth’s rotation as it travels through the WIMP halo [51], with anexcess of nuclei recoiling in the direction of the “WIMP wind”. Identifyingthis directionality requires detectors capable of reconstructing the tracks ofmeasured particles, such as the gaseous detectors employed by the DRIFTexperiment [56].The annual and daily modulation effects described above are statistical innature, and can only be resolved with a reasonably large set of candidatesignal events. Another approach is to use expected differences in detectorresponse to WIMP vs. background events to discriminate on an event-by-event basis. This technique is applied in solid state detectors such asSuperCDMS and EDELWEISS, where differing amounts of ionization areexpected for nuclear vs. electron recoil events of the same energy. Theionization signal is compared with the total event energy to discriminatenuclear recoils of interest from the electron recoil background.2.2 Accelerator SearchesAt particle colliders such as the LHC, the WIMP detection strategy is tocollide hadrons or leptons at a sufficient rate and energy such that WIMPScould be produced at a measurable rate. One can then use the fact that thetotal momentum of all particles produced by a collision should nominallysum to 0 in the direction transverse to the beam line. Due to the negli-gible interaction probability of WIMPs with the detector material, WIMPsearches with collider data look for events with an excess of missing momen-tum transverse to the beam line. Such an excess could suggest that one ormore neutral and stable (or at least long-lived) particles passed through thedetector without depositing energy, thus satisfying two of the four criteriaidentified in Section 1.3 to represent a viable dark matter candidate.Since neutrinos also have a very low probability of interacting in the detec-tor, Standard Model (SM) processes that produce neutrinos in their final272.2. Accelerator Searchesstates can also result in missing transverse momentum. Such processes canrepresent a significant background [49] in accelerator WIMP searches.Constraints on WIMP properties obtained from accelerator searches can,using various assumptions [49], be converted to upper limits on the WIMP-nucleon interaction cross section as a function of WIMP mass for ease ofcomparison with direct detection search constraints. Such a comparison isshown in Fig. 2.2, with accelerator constraints determined from LHC datacollected by the ATLAS detector.Figure 2.2: Comparison between upper bounds on the WIMP-nucleon inter-action cross section obtained from accelerator vs. direct detection searches.Left: Constraints for spin-independent WIMP-nucleon coupling. Right:Constraints for spin-dependent coupling. See Section 2.1.1 for definitionof spin-independent and spin-dependent couplings. Reprinted figure withpermission from [49]. Copyright (2012) by the American Physical Society.Comparing the accelerator constraints (red and blue) in Fig. 2.2 with directdetection constraints, it is clear that accelerator searches can offer com-plementary limits. This is particularly true for sub-GeV/c2 WIMP massesrange where the sensitivity of direct detection searches tends to drop offquite steeply due to their low-energy noise threshold.282.3. Indirect Detection2.3 Indirect DetectionIndirect WIMP detection uses observational data to search for unique spec-tral signatures showing an excess of one or more cosmic rays (CRs) (e±, p¯,γ, ν) which cannot be explained by any known astrophysical process. Suchsignatures could be produced by hadronization following the annihilation oftwo WIMPs, forming a pair of SM particles [57]. Indirect detection searchestypically focus on regions of space, such as galaxy clusters and the centreof the Milky Way galaxy [58], that are expected to have a relatively highdensity of dark matter.Since ordinary astrophysical processes produce a significant background ofCRs [58], accurately modelling these background processes is important forthe sensitivity of indirect WIMP searches. Further, it is useful to selectdecay channels and energy ranges with a relatively low flux of CRs frombackground processes [57]. For example, it is common for indirect detectionsearches to focus on antiparticle CRs rather than the corresponding particleflux due to the relatively low background abundance of antiparticles.Indirect searches are particularly suited for studying WIMP masses near orabove the TeV range [57, 58], and thus have the potential to complementboth accelerator searches, as well as direct detection searches which focusprimarily on the sub-GeV to ∼TeV range.29Chapter 3Overview of the SuperCDMSExperimentThe Super Cryogenic Dark Matter Search (SuperCDMS) is an experimentthat aims either to directly detect particle dark matter candidates, or toplace constraints on the properties of proposed particles, such as their massand interaction cross section. The primary search candidate – and the onewhich I will discuss primarily in this thesis – is the hypothetical WIMPparticle.The first generation of direct detection WIMP search experiments were pri-marily sensitive to WIMP masses ranging from a few GeV/c2 to a fewTeV/c2, based on the expectation that the WIMP could be a stable su-persymmetric particle [61]. Not yet having found convincing evidence fromdirect detection experiments for WIMPs in this mass range, nor evidencefor supersymmetry at the LHC [43, 44], recent theoretical work [39, 46] haspredicted WIMPs with masses near or below 10 GeV/c2, some of whichare unrelated to supersymmetry. SuperCDMS is designed to be sensitive tothese low-mass WIMP candidates that would be below the detection thresh-old of most other dark matter search experiments.To attain the low-background environment required to search for rare WIMP-nucleon collisions, the detectors are operated deep underground, therebyminimizing the flux of cosmic rays. Having recently established a world-leading limit on the spin-independent WIMP-nucleon cross section for aWIMP mass as low as ∼3 GeV/c2 [74] following four years of operation780m underground at the Soudan laboratory in Minnesota, the experimentwill continue the low-mass WIMP search ∼2km underground in the SNO-LAB facility near Sudbury, Ontario. Fig. 3.1 shows the projected sensitivityof the SuperCDMS experiment at SNOLAB.30Chapter 3. Overview of the SuperCDMS ExperimentFigure 3.1: Projected sensitivity of the SuperCDMS experiment at SNO-LAB. Reprinted figure with permission from [67]. Copyright (2017) by theAmerican Physical SocietyThe experiment uses solid-state silicon and germanium detectors that de-tect both tiny vibrations in the crystal lattice, called phonons, and ionization(electron-hole) pairs produced when an incident particle scatters elasticallyoff a nucleus in the detector [59]. The electron-hole pairs are measured byaluminum electrodes patterned as shown in Fig. 3.2 on either detector face,which at Soudan were amplified and read out by Field Effect Transistors(FETs). These FETs will be replaced with High Electron Mobility Transis-tors at SNOLAB due to their reduced readout noise and power dissipation[60].In order to detect the minute phonon signals generated by low-energy parti-cles scattering in the detector, aluminum collection fins on the two detectorfaces are coupled to tungsten conductors cooled to their superconductingtransition temperature, on the order of 50 mK [62]. At this temperature, asmall increase in temperature caused by an incident phonon will result in adrastic change in the conductor’s resistance. For this reason, the detectoris known as a “transition-edge sensor” (TES). The change in resistance canbe measured by a change in the current flowing through the conductors.The current signal is further amplified by a superconducting quantum inter-ference device (SQUID) circuit. The SQUID outputs a measurable voltagesignal pulse in response to a magnetic flux though its superconducting loopinduced by the change in current [63].The analysis discussed in Chapter 4 uses calibration data collected from313.1. Detectors and Experimental Setupgermanium SuperCDMS detectors at Soudan, which were an upgrade to theolder CDMS II detectors also operating at Soudan [68].3.1 Detectors and Experimental SetupThe SuperCDMS Soudan Ge detectors, having the approximate dimensionsof a hockey puck, are stacked into vertical arrays of three detectors knownas “towers”, of which five were running at Soudan. The detector stackingis advantageous for the rejection of nuclear recoil events due to neutrons,because neutrons will often scatter in multiple detectors, whereas a WIMPwould only be expected to scatter once due to the low interaction crosssection [69].The detector towers are located in a low-radioactivity copper cryostat, whichis coupled via nested tubes known as the “C-stem” to a helium dilutionrefrigerator to maintain a base temperature between 40 and 50 mK.The overburden of 780 m of rock, or 2090 meters water equivalent, reducesthe flux of cosmic ray muons by a factor of 5×104 [69]. Additional shieldingis provided by successive layers of copper (∼0.5 cm), lead (22.5 cm), andpolyethylene (50cm) shielding [69] surrounding the cryostat, as shown inFig. 4.5 in Chapter 4, to reduce the background of gammas and neutronsfrom radioactive and cosmogenic sources that would reduce the sensitivityof the experiment. Surrounding this shielding is an active veto to tag cosmicrays, which consists of 5.2 cm thick plastic scintillator panels read out byphotomultiplier tubes [69, 72].Figure 3.2: Sample SuperCDMS iZIP detector used at Soudan. Figure fromSuperCDMS collaboration standard public plots [70].323.2. Data Acquisition and TriggeringWhen a pulse is measured by the phonon and ionization sensors, the firststage of SQUID and FET amplification, respectively, takes place withinthe cryostat [72]. The signals are then carried to the room-temperaturedata acquisition electronics, discussed in the next section, via copper-kaptoncables.3.2 Data Acquisition and TriggeringFig. 3.3 shows the channel layout of the SuperCDMS Soudan detectorsused for the photoneutron analysis in Chapter 4. The phonon sensors oneach detector face are divided into four channels, with three inner chan-nels labelled B, C, and D, and an outer channel labelled A. Each phononchannel produces a separate pulse, which is subsequently read out by thedata acquisition electronics. The relative pulse shapes and amplitudes inthe four phonon channels enable some degree of position sensitivity, whichis important for defining fiducial volume cuts. There are two charge chan-nels, divided into an inner disk and an outer ring, each of which produces aseparate pulse.Figure 3.3: SuperCDMS Detector Channel Layout. Figure from Super-CDMS collaboration public website [71].Following a measured recoil event in a detector, the six electrical signals– four phonon and two ionization – arriving from the SQUID and FETamplifiers are read into the detector’s room temperature front end board333.3. Detector Operating Modes(FEB) for further amplification. From the FEB, the signals are directed tothe Receiver/Trigger/Filter board, which passes them through a low-passfilter and digitizes them. Trigger information is determined separately forthe phonon and ionization signals in the RTF board by summing each signalover all channels and comparing the summed waveforms with adjustable highand low thresholds. The resulting high and low trigger bits are fed into thetrigger logic board (TLB), which makes and records the final trigger decisionbased on the set trigger condition [72]. The trigger condition could be, forexample, one of the following:• passing the phonon low threshold (Plow)• passing both phonon and charge low thresholds, but failing the phononhigh threshold (Plow Qlow, NOT Phi)3.3 Detector Operating ModesSuperCDMS detectors can be operated in two distinct modes, known as“iZIP” (interleaved Z-sensitive Ionization Phonon) and “CDMSlite” (Cryo-genic Dark Matter Search low ionization threshold experiment). Of the two,the CDMSlite operating mode is a more recent development, and will be theprimary mode discussed in the remainder of this thesis.3.3.1 iZIP ModeIn iZIP mode, the detector is operated with a 4V bias across the ionizationcollectors on either detector face, and both faces are used for both chargeand phonon signal readout. The 4V bias voltage is optimized to be as lowas possible while enabling full charge collection.343.3. Detector Operating ModesFigure 3.4: iZIP operating mode. Electron-hole pairs produced by a particlescattering in the detector drift across the 4V bias to charge collectors on ei-ther detector face. The grounded phonon sensors measure the primary recoilphonons, and any additional phonons produced by the drifting electron-holepairs. Reproduced from c© [64], with the permission of AIP Publishing.The iZIP detectors employ two background rejection techniques, both ofwhich are important for their sensitivity as dark matter detectors. The firstis ionization yield discrimination, and the second is surface event rejection.Both are notably unavailable in the CDMSlite operating mode.Ionization Yield DiscriminationIonization yield – or simply “yield” – gives a measure of the total energyof electron-hole pairs ionized by a particle recoiling in the detector. It isquantified by the ratio of the measured ionization energy Ee/h to the totalenergy Er of the recoil. SuperCDMS follows the conventional practice ofcalibrating the ionization energy scale such that this yield Y =Ee/hEris onaverage unity for electron recoils.By simultaneously collecting primary ionization and phonon signals, theiZIP detectors have the ability to measure the energy of a recoil event inthe detector both in terms of the measured ionization energy Ee/h and thephonon energy Ep. The ionization energy is calibrated for electron recoilsusing peaks in the measured spectrum of electron recoil events from a 133Ba353.3. Detector Operating Modesγ calibration source. Using these same events, the phonon energy scale iscalibrated relative to the ionization energy such that, after correcting for theLuke-Neganov gain discussed in Section 3.3.2, the resulting primary phononenergy measured for each event matches the calibrated ionization energy[65]:Y =Ee/hEp − eVb Ee/h≡ 1 for electron recoil events (3.1)The term eVb Ee/h in the denominator represents the energy of the “Luke-Neganov” phonons discussed further in Section 3.3.2, where e is the absolutevalue of the fundamental electric charge, Vb is the 4V bias across the detector,and  is the energy required to ionize a single electron-hole pair.The yield measurement is important for the sensitivity of iZIP detectors be-cause it makes it possible to discriminate between nuclear recoils of interestfor the WIMP search and the electron recoil background. Such discrimi-nation is possible because nuclear recoil events have a significantly lowerfraction of the initial recoil energy going into ionization compared with elec-tron recoils, and can hence be distinguished from the electron recoil back-ground by their lower ionization yield. Fig. 3.5 shows the discriminationbetween electron and nuclear recoil events based on the measured yield withdata taken during activation of the detectors by a 252Cf γ and neutronsource.363.3. Detector Operating ModesFigure 3.5: Discrimination between electron recoil (upper band) and nu-clear recoil events (lower band) produced by γ and neutron interactions,respectively, due to activation of an iZIP detector by a 252Cf γ and neutronsource. Reprinted figure with permission from [69]. Copyright (2004) by theAmerican Physical Society.Surface Event RejectionAs shown in Fig. 3.4, the phonon collectors (blue lines) and charge collectors(narrow grey lines) are interleaved in iZIP detectors, with a 1mm pitch [64].The aluminum charge collectors are biased at +2V on one detector face and-2V on the other, and the phonon sensors are grounded. This interleavedpattern of phonon and charge collectors is an improvement over CDMS IIdetectors, which had phonon sensors on one side and charge sensors on theother [64]. The motivation for this improvement was to reject a backgroundknown as “surface events”. Surface events occur near the top and bottomof the detector, and represented a background in past generations of theexperiment [64] because electronic carrier trapping at the surface preventedthe full collection of their ionization signal [66]. This incomplete ionizationcollection made it possible to mistake the surface events for nuclear recoilsdue to the reduced ionization yield measurement.373.3. Detector Operating ModesFigure 3.6: Electric field near the surface of an iZIP detector produced bythe pattern of interleaved biased charge collectors and grounded phononcollectors. Reprinted from [64], with the permission of AIP Publishing.The interleaved pattern of biased charge collectors and grounded phononsensors addresses the surface event issue by creating the unique shallowelectric field pattern shown in Fig. 3.6 in the first ∼2mm of the detectornearest the surface, while maintaining an approximately uniform electricfield in the bulk of the detector. This surface field pattern ensures that anelectron-hole pair created by an event near the detector surface is fully col-lected at that surface, with one charge collected at the biased electrode, andthe other at the grounded phonon sensor. Since the otherwise uniform elec-tric field ensures symmetric charge collection for electron-hole pairs createdin the bulk of the detector, surface events can be identified and rejected bythe absence of charge collected on the opposite detector surface.3.3.2 CDMSlite ModeIn the CDMSlite operating mode, the detectors are biased at 25-70V, muchhigher than is needed for full charge collection. The motivation for increas-ing the bias voltage is to take advantage of the “Luke-Neganov” effect [75] tolower the energy detection threshold of recoil events in the detector, therebyextending the reach of CDMS detectors to lower WIMP masses. The lowerenergy threshold enables SuperCDMS to test extensions of the StandardModel that predict low-mass (∼1-10 GeV/c2) WIMP candidates [74]. Theextended reach comes at a cost of sacrificing two significant background dis-383.3. Detector Operating Modescrimination tools available to iZIP detectors, namely the yield discriminationand surface event rejection discussed in Section 3.3.1.Figure 3.7: CDMSlite operating mode.The Neganov-Luke EffectThe CDMSlite operating mode achieves sensitivity to lower-energy recoilevents than would be possible in iZIP mode because the relatively high biasvoltage can amplify the primary – and potentially sub-threshold – ioniza-tion signal into a large measurable phonon signal. The phonon signal isa combination of the primary phonons produced by the recoil event, and“Luke-Neganov phonons” emitted when the work done by the electric fieldin drifting the electron-hole pairs to the charge collectors is converted intothermal energy.The energy ELN of the Luke-Neganov phonons varies linearly [75] with boththe number N of electron-hole pairs produced by the primary recoil event,and the bias voltage Vb across the detector:ELN = NeVb (3.2)where e is the absolute value of the fundamental electric charge. N, theaverage number of electron-hole pairs created, is given by:N = YEr(3.3)where Y is the ionization yield discussed previously in Section 3.3.1 anddefined to be unity for electron recoils, Er is the energy of the recoil event,393.3. Detector Operating Modesand  is the average energy required to produce a single electron-hole pair. is generally taken to be 3 eV / pair [74] for Ge. The total measurablephonon energy Ep is hence:Ep = Er + ELN = Er(1 + YeVb)(3.4)The ionization yield Y for nuclear recoils, which will be discussed further inChapter 4, has measured values within the range of 0.1-0.3 in the recoil en-ergy range below 10 keV of interest for CDMSlite. At a nominal CDMSlitebias voltage of 70V, the ratio between the energy of the initial recoil eventand the energy of the Neganov-Luke phonons is 2.5 < 70Y3 < 7 for nuclearrecoils. This has two important implications for the detector’s WIMP sen-sitivity. First, it means that this operating mode can amplify the primaryionization signal of a WIMP-nucleon into a measurable phonon signal withup to 7 times the energy of the initial recoil event. Second, that the pri-mary phonon signal is effectively overwhelmed by the larger Luke-Neganovphonon signal, making it impossible to calculate the ionization yield of theinitial recoil event, and hence the yield-based discrimination between nu-clear and electron recoil events available to iZIP detectors is not possible forCDMSlite detectors.Data Acquisition Modifications for CDMSlite OperationStandard iZIP detectors were not designed to operate at a bias voltage above10V, so detectors operated in CDMSlite mode are instrumented with customelectronics that keep one detector face at the desired operating voltage andthe other detector face at ground [73]. The custom electronics at the biaseddetector face were not designed to read out signals, so all signal readoutis done from the grounded detector face. The one-sided readout has theeffect of both halving the total phonon signal, and making it impossibleto employ the surface event rejection achieved by the two-sided iZIP-modereadout. This will not be a limitation at SNOLAB, where the high-voltage(HV) detectors will be instrumented with specialized electronics which willenable two-sided signal readout [67].40Chapter 4Photoneutron Calibration4.1 Background and MotivationFor a candidate WIMP-nucleon scatter event in a dark matter detector,the mass mχ of the incident WIMP is calculable from the recoil angle θrand the energy Er of the recoiling nucleon (see Eq. 2.1). Therefore, anaccurate determination of the recoil energy is needed to determine the dis-tribution of possible WIMP masses that could have produced a candidatesignal event.It is thus important to understand the relationship between the phononenergy detected by the TESs, and the recoil energy of the struck targetnucleus. This relationship can be calibrated to ∼ 2% precision for electronrecoil events in SuperCDMS detectors by passing γ rays from a radioactivesource such as 133Ba through the detectors and comparing peaks in the mea-sured spectrum with known γ lines from the source. However, the absenceof characteristic peaks in neutron spectra makes such a calibration relativelydifficult for neutron recoil events.In principle, the phonon energy scale due to nuclear recoils can be relatedto the electron recoil scale according to Eq. 3.4, where the ionization yieldY defined in Section 3.3.1 is calibrated to unity for electron recoils. Thiscalibration is described for iZIP detectors in Section 3.3.1. For CDMSlitedetectors, the phonon energy scale is calibrated directly for electron recoilsusing peaks in spectra from a radioactive source, taking Y≡1 in Eq. 3.4.The nuclear recoil scale Er,nr is thus related to the electron recoil scale Er,eeaccording to:Er,nr = Er,ee1 + eVb1 + Y (Er,nr)eVb(4.1)414.1. Background and Motivationwhere Y (Er,nr) is the energy-dependent ionization yield of a nuclear recoilevent. The other terms in Eq. 4.1 are known, and described below Eq.3.4. For iZIP detectors, Y (Er,nr) can be measured directly at recoil energiesabove ∼10 keVr,nr, as shown in Fig. 3.5. However, as discussed in Section3.3.2, CDMSlite detectors have no ability to measure yield at all. Therefore,an independent determination of the ionization yield is required for the fullCDMSlite nuclear recoil energy range, and below ∼10 keV for iZIP-modedetectors. This independent determination could come from theory, exper-iment, or a combination thereof. This chapter focuses on an experimentalmeasurement of the CDMSlite nuclear recoil spectrum for SuperCDMS Gedetectors in the low-energy range of ∼0.4 to 10 keVr,nr.4.1.1 Ionization Yield from Lindhard TheoryLindhard theory [76] provides a semi-empirical prediction for the ionizationyield of a nuclear recoil Y (Er,nr) as a function of nuclear recoil energy Er,nrfor a material of mass number A and atomic number Z:Y (Er,nr) = kg()1 + kg()(4.2)where:g() = 20.15 + 0.70.6 +  = 11.5Er,nrZ−7/3and the k value from Lindhard theory is nominally given by:knom = 0.133Z2/3A−1/2 (4.3)which ranges from 0.156 to 0.160 for stable isotopes of Ge.4.1.2 Existing Ionization Yield DataExisting measurements of the ionization yield – also known as “ionizationefficiency” – are shown for Ge in Fig. 4.1. With a few exceptions, the ex-perimental measurements tend to agree with the Lindhard model prediction424.1. Background and Motivationabove ∼50 keVr,nr, but begin to deviate significantly in the region near andbelow 10 keVr,nr, resulting in a large uncertainty in the ionization yield inthis low recoil energy range.Figure 4.1: Existing ionization yield measurements. Measurements generallybegin to agree with the Lindhard model, with k ≈0.159, above ∼50 keVr,nr.Reprinted from [77], Copyright (2012), with permission from Elsevier.The most recently published CDMSlite WIMP search using Run 2 dataproduced a world-leading limit on spin-independent WIMP-nucleon crosssections down to ∼3 GeV/c2 [74] by pushing the low-energy noise threshold(discussed in Section 2.1.1) of the detectors down to nuclear recoil energies of∼0.5 keVr,nr. The upcoming installation of the SuperCDMS experiment atSNOLAB expects to further lower the noise threshold down to 0.04 keVr,nr[67] to probe WIMP masses as low as ∼0.5 GeV/c2. However, determina-tions of the WIMP mass in this low energy region are limited by the largeuncertainty in the ionization yield of Ge for nuclear recoil energies below∼10 keVr,nr. Previous and ongoing analyses of CDMSlite data use the Lind-hard yield model, with k in Eq. 4.2 allowed to range from 0.1 to 0.2 toaccount for the spread and uncertainty of existing yield data in the recoilenergy range below ∼10 keVr,nr. The resulting uncertainty in the energyscale dominates the uncertainty band on the WIMP-nucleon cross sectionlimit, as shown in Fig. 4.2. The photoneutron calibration aims to constrainthe low-energy ionization yield of SuperCDMS detectors, thus reducing the434.2. Calibration Concept and Setupuncertainty on SuperCDMS sensitivity limits.Figure 4.2: Spin-independent WIMP-nucleon cross section limits fromCDMSlite runs 1 and 2, with uncertainty bands dominated by the energyscale uncertainty. Reprinted figure with permission from [74]. Copyright(2018) by the American Physical Society.4.2 Calibration Concept and SetupThe concept of the photoneutron calibration is to pass approximately mono-energetic neutrons through SuperCDMS detectors, as illustrated in Fig. 4.3,and compare the resulting nuclear recoil spectrum, measured in phonon en-ergy (keVt), with a simulated nuclear recoil spectrum (measured in keVr,nr)to calibrate the phonon energy scale.444.2. Calibration Concept and SetupFigure 4.3: Photoneutron calibration conceptThe neutrons are produced by passing γ rays from a radioactive source– either 124Sb or 88Y – through a 9Be wafer. Due to the instability ofthe 9Be nucleus, incident γ rays with energy Eγ above the “photoneutronproduction” threshold Eth=1.666 MeV [78] of9Be, and below 2.5 MeV [79]can induce “photo-disintegration” of the nucleus via a (γ, n) reaction. Inthis reaction, the absorption of the γ ray by the nucleus is followed by therelease of a neutron.The energy of the emitted neutron is approximately mono-energetic, andgiven theoretically by [80]:En =AA− 1[Eγ − Eth −E2γ(1862 MeV)(A− 1)]+ δ(θ) (4.4)where A is the mass number of the target nucleus, and θ is the angle of theemitted neutron relative to the incident γ. δθ represents the small angulardependence of the neutron energy, and is given by:δθ = Eγ[2(A− 1)Eγ − Eth931A3]1/2cos θ (4.5)In practice, the energy variation due to the angular term δ(θ) can be mini-mized by allowing only a small solid angle to reach the detector.124Sb has strong γ lines above Eth at 1.691 MeV (47.49% absolute intensity)and 2.091 MeV (5.498% absolute intensity), both of which contribute to454.2. Calibration Concept and Setupphotoneutron production. 88Y has a single strong line at 1.8361 MeV. The124Sb and 88Y sources are experimentally found [81] to produce neutrons ofenergy 24±3 keV and 151±8 keV, respectively.In principle, there should be a sharp and easily recognizable cutoff at themaximum kinetically allowable nuclear recoil energy given the energy of theincident neutrons, which could be used for calibration. This high-energycutoff is visible in Fig. 4.4, in the dashed red “single-scatter” componentof the simulated nuclear recoil spectra developed for the analysis. However,given the size of the detectors, there is a reasonably high probability thata single incident neutron will induce multiple scatter events. This presentsa problem, because the detector cannot in general resolve multiple scatterevents, which means that the amplitude of a single pulse read out from thedetector represents the sum of all energy depositions, and may add up toan energy greater than the kinematic cutoff. As such, the multiple scatterevents have the effect of blurring the sharp spectral cutoff, as shown by theblue dashed “multiple-scatter” component in Fig. 4.4.Instead of searching for a sharp cutoff, the nuclear recoil energy scale is cali-brated as follows: a Geant4 simulation of the experimental setup, developedby another member of the analysis group, is used to generate a spectrum ofrecoil energies due to the neutron flux. The simulated spectrum is convertedto phonon energy using a parameterized Lindhard model, and the resultingspectral shape is compared with that of the experimental spectrum to fit foroptimized Lindhard model parameters, as described in Section 4.7.Figure 4.4: Dominant nuclear recoil component of the simulated neutronspectra for two of the three detector and source conditions used in the pho-toneutron calibration. Please see Fig. 4.32 left for a sample plot of the fullneutron spectrum due to both nuclear and electron recoils.464.3. Experimental Setup4.3 Experimental Setup4.3.1 Physical SetupThe physical setup for the photoneutron calibration is illustrated by thesimulated side-view in Fig. 4.5. The radioactive source is placed in a sourceholder above the vacuum cryostat containing the five towers of detectors,with a lead block immediately underneath to reduce the flux of γ rays fromthe source. The vast majority of γ rays pass through the wafer withoutinteracting (on the order of one (γ, n) reaction per ∼105 γs). For this reason,it was necessary to alternate the data-taking conditions between:a) placing the 9Be wafer underneath the source to collect data with boththe neutrons and the γ background (i.e. “neutron-on”), andb) removing the wafer to collect data with the γ background alone, whichcan be used for spectral subtraction (i.e. “neutron-off”).Figure 4.5: Physical setup for the photoneutron calibrationPreliminary studies found that, in the absence of the lead shielding, thedeadtime-limited DAQ would be collecting nearly all γs, and very few neu-trons. The thickness of the lead shielding was thus optimized to reduce theflux of γ rays to a manageable level without excessive degradation of theneutron spectrum due to neutrons interacting in the lead blocks.The veto panels, as well as the polyethylene and lead lids that are normally474.3. Experimental Setupplaced above the cryostat during WIMP search runs, were removed for thecalibration, and mylar film was placed above the cryostat to prevent radonfrom seeping into it.The source holder is designed to contain the source and the optional 9Bewafer. The holder is mounted on top of several layers of lead shielding,on a source bridge above the outer vacuum cryostat. The source bridge ismounted on a translation stage (see Fig. 4.6 right) which offers one degreeof translational freedom.Figure 4.6: Source box used to deploy the radioactive source. Left: Leadshielding is wrapped in aluminum foil and placed underneath the source.Right: The source box is mounted on a translation stage during data-taking.The source is coupled to the end of a rod, which is screwed into a metalbar. The source can then be easily removed and replaced in a consistentlocation simply by unscrewing the bar that the rod is attached to, and laterscrewing it back into place (see Fig. 4.7 right). The source is placed directlyabove a circular hole that the 9Be wafer (the circular black wafer below thesource in Fig. 4.7 left) can fit snugly into to collect neutron-on calibrationdata.484.3. Experimental SetupFigure 4.7: Radioactive source placement. Left: The source was placed onthe end of a source rod, which was screwed into the source holder. Right:The source bar allows the source to be removed and replaced in a consistentlocation4.3.2 Data-taking ConditionsThe photoneutron data was taken over a 5 month period. Of the five towersof detectors in the vacuum cryostat, two were used to take data in CDMSlitemode for the photoneutron calibration: T2Z1 in Tower 2, and T5Z2 in Tower5. There were three distinct data-taking conditions (i.e. “data sets”) usedfor CDMSlite-mode detectors:• Operating detector T5Z2 at a 70V bias with the 124Sb source in place→ “Sb at 70V”• Operating detector T5Z2 at a 70V bias with the 88Y source in place→ “Y at 70V”• Operating detector T2Z1 at a 25V bias with the 88Y source in place→ “Y at 25V”Fig. 4.8 summarizes the periods during which each of the above three con-ditions were applied over the full course of the photoneutron calibrationrun.494.3. Experimental SetupFigure 4.8: Summary of data-taking periods for the photoneutron calibra-tionEach week of data-taking is divided approximately equally between:a) placing the 9Be wafer beneath the source to collect data with both theneutrons and γ background (known as “neutron-on” data), andb) removing the wafer to collect the γ background alone (known as “neutron-off” data).Data is collected in “series” with nominal lengths ranging from 50 minutesto 3 hours. Each series is given a unique 11-digit identifier called a “Series-Number”, which encodes the following information:• which facility the data was collected from (SuperCDMS has severaltest facilities in addition to the underground lab)• the time and date at which the series started.4.3.3 Trigger and readout settingsFor typical calibration runs with the source located above the detectors,the high interaction cross section makes it such that an incident particleis very likely to interact at least once in each of the three detectors in atower. Therefore, traces and trigger bits are typically read out from allthree detectors in the tower to determine the “trigger threshold” for eachdetector – i.e. the minimum measured recoil energy down to which a triggerin another detector in the tower is consistently accompanied by a trigger inthe detector under consideration.The high event rate produced by the γ background motivated two atypicaltrigger and readout settings for the photoneutron data-taking:504.4. Cut Development• Data was not taken in full-tower readout mode described above. Ratherthan collecting waveforms from every detector in the tower when aglobal trigger was produced, waveforms were only saved from the de-tectors that produced a trigger.• A high-energy (Phi) trigger veto was applied in addition to the usuallow-energy (Plow) trigger. The effect of this phi trigger was to preventhigh-energy recoil events from being saved by the DAQ system, sincethe analysis is only interested in the relatively low-energy neutronspectrum below 10 keVr,nr.Since the absence of full-tower readout prevented a calculation of the trig-ger threshold using the photoneutron calibration data, this calculation wasinstead done using calibration data that had previously been taken witha 252Cf activation source. The trigger threshold was calculated for eachdetector by another member of the analysis group.4.4 Cut Development4.4.1 Basic CutsVarious “basic cuts”, summarized below, are applied to each data set toremove events that are considered unsuitable for the analysis. The basiccuts are characterized as being easily defined without the need for significantanalysis or tuning.NotEmpty CutFor a given trigger or set of triggers produced by one or more of the detectors,the DAQ system decides, depending on the readout mode, which detectors tocollect raw traces from for later processing. If a raw trace was not collectedfor a given event from the detector under consideration, this event is removedfrom the analysis of this detector using the NotEmpty cut.NotRandom CutDuring each series, “random” events – or simply “randoms” – are periodi-cally collected. Unlike typical saved events, these randoms do not correspond514.4. Cut Developmentto a detector trigger. Randoms are used for data processing to evaluate thenoise environment during each series. However, since they are not triggeredby physical scatter events, they are removed from the analysis with theNotRandom cut.BaseTemp CutThe operating base temperature of the detectors lies in the approximaterange of 35 to 55 mK. The base temperature is normally stored for eachevent, and may be used to make small corrections to the energy estimate. Ifthe base temperature is not read out properly – as signified by a value ≤ 0– its energy cannot be properly corrected. Such events are conservativelyremoved with the BaseTemp cut.High-Voltage CutThe energy scale of the detector varies with the bias voltage Vb across thedetector according to Eq. 3.4. Therefore, the analysis relies on an accu-rate, consistent bias voltage, so any events for which the voltage outputreported by the power supply deviates from the nominal operating voltageare removed by the high-voltage cut.BiasFlashTime CutEach data-taking run nominally begins by “flashing” infrared LED lightthrough the detector, which “neutralizes” the detector crystal – i.e. liber-ates charges that were trapped in crystal impurities. If the detector is notneutralized, electron-hole pairs produced by a recoil event may not be fullycollected, which results in a reduced ionization yield estimate. The timeduring which a detector is biased following the most recent LED flash isrecorded by the DAQ as the “BiasFlashTime”.The BiasFlashTime cut is designed to remove any events that were recordedafter the nominal run length following the most recent detector flash, whichcould happen if the detector was for some reason not flashed before the nextrun.524.4. Cut DevelopmentZero Current and Current Leakage CutsIt is expected that a small amount of leakage current (on the order of nA)will leak from the detector’s voltage supply due to a parasitic resistance toground. This leakage current is measured from the power supply and savedfor each event. The leakage current reduces the bias voltage across thedetector, and this effect is corrected for during processing when the energyof an event is estimated.As with the base temperature measurement, events for which the currentleakage was not properly readout – as signified by a leakage current of 0 –will not have been properly corrected, and are therefore removed by the zerocurrent cut.Events with an excessive amount of leakage current – where the excess ap-pears to be associated with collecting data too soon after the last LED flash– are removed with the current leakage cut, with a maximum allowableleakage of 3.0 nA.4.4.2 Quality CutsQuality cuts require some degree of analysis and tuning. These cuts aredescribed below, with particular emphasis on the cuts or aspects of cutsthat I developed.Pre-pulse Standard Deviation CutRaw phonon pulse traces, with a total length of 6553.6 µs, include ∼950 µsprior to the trigger. This “pre-pulse region” is used during processing tocalculate the pre-pulse standard deviation, which provides a measure of thenoise level in the detector at the time of the pulse.A pre-pulse standard deviation cut is applied individually to each week ofneutron-on and neutron-off data to remove data for which the pre-pulsestandard deviation in any of the four phonon channels varies too stronglyfrom the typical value.For each week of neutron-on or neutron-off data, the distribution of pre-pulse standard deviations in each phonon channel (A, B, C, and D) is fitwith a Gaussian distribution, as shown in Fig. 4.9, and any events lying534.4. Cut Developmentmore than four standard deviations away from the centroid of the Gaussianfit in any of the four phonon channels are removed from the analysis.Figure 4.9: Sample application of the pre-pulse standard deviation cut. AGaussian distribution is fit to the distribution of pre-pulse standard devia-tions in phonon channel A for a sample week of the neutron-off Y data (i.e.“YBlank”). Data lying more than ±4σ from the centroid of the Gaussiandistribution is removed by the cut.Phonon Good Start Time (i.e. Phonon Delay) CutThe reconstructed start-time of an event is calculated as the optimized timedelay (i.e. “phonon delay”) relative to the global trigger obtained whenthe raw phonon trace is fit to a phonon pulse template using the “optimalfilter” (OF) fitting method. The Phonon Good Start Time cut is designedto eliminate events whose reconstructed start-time occurs too close to theedges of the allowable OF search window to accurately measure the pulse.The OF search window ranges from -200µs to +100µs relative to the globaltrigger.The cut, shown for the Sb at 70V data set in Fig. 4.10, is defined as:−195µs < phonon delay < 35µsNote that the asymmetry in terms of the proximity of the upper and lowercutoffs to the OF search window edges arises from the fact that most of theportion of the pulse relevant for the OF fitting algorithm occurs after thereconstructed start-time.544.4. Cut DevelopmentFigure 4.10: Sample application of the PTOFdelay cut to data taken withthe 124Sb source at a 70V bias (i.e. Sb at 70V). The upper and lowercut bounds are shown as red horizontal lines. Data passing the basic cutsdescribed in Section 4.4.1 is shown in blue, and data additionally passingthe phonon delay cut is shown in orange.Phonon Pulse Shape CutsThe phonon pulse shape cuts are comprised of three cuts that make use ofχ2 estimates obtained from fitting each event’s phonon pulse trace to one ofthree pulse templates during processing. These χ2 estimates are summarizedas follows:• Phonon χ2 (χ2p): Based on the fit to the “good” signal pulse template,from which the best-fit pulse amplitude is used to obtain the phononenergy estimate.• Glitch χ2 (χ2g): Uses the “glitch” pulse template, which is designed torepresent pulses produced by electronic glitches in the detector readoutelectronics, rather than true signal events. This template is producedby averaging many pulses identified as electronic glitches, and is muchmore sharply peaked than the good signal pulse.• LFN χ2 (χ2lf): Uses the “low frequency noise” (LFN) pulse template.This template is designed to represent pulses produced by or con-taminated with persistent LFN in the detectors. As with the glitchtemplate, the LFN template is obtained by averaging many pulsesidentified with low-frequency noise. The LFN is particularly prob-lematic near the low-energy trigger threshold. In this region, the vastmajority of triggered events are due to LFN, resulting in what’s known554.4. Cut Developmentas the “low-frequency noise blob”.My primary involvement in the phonon pulse shape cuts was the devel-opment of “time blocks” in which the ∆Glitch and ∆LFN cuts are mostappropriately defined. Table 4.1 briefly summarizes the methodology usedto define the cuts within each time block.Table 4.1: Summary of phonon pulse shape cuts defined within time blocks.CutQuantitiesUsed forDefinitionPurpose SummaryPhononχ2χ2p, Total PhononEnergy (Ep)Identify instrumentalbackground orpileup events basedon a poor fit to thephonon pulse shapetemplate, asindicated by a highχ2p.For each data set (Sb at 70V,etc.), data is binned in Ep. In eachEp bin, the cumulative densityfunction (CDF) is computed in χ2p,to obtain the 97% CDF point.These 97% CDF points are fit witha third-order polynomial to obtainan energy-dependent upper boundon χ2p.∆Glitch∆Glitch≡(χ2g − χ2p), EpIdentify instrumentalbackground eventsarising fromelectronic glitches,as indicated by arelatively good fit tothe glitch templatecompared with thegood phonontemplate.Within each time block, the datais binned in energy. In each bin,the mean and RMS values of∆Glitch are calculated, and themean + 2.58×(RMS) points are fitwith a parabola to obtain anenergy-dependent upper bound on∆Glitch. A flat cut in ∆Glitch isalso placed to minimize thecontribution from thelow-frequency noise blob.∆LFN∆LFN≡(χ2lf − χ2p), EpIdentify instrumentalbackground eventsarising fromlow-frequency noise,as indicated by arelatively good fit tothe LFN templatecompared with thegood phonontemplate.Within each time block, theprocedure is analogous to thatused to define the ∆Glitch cut.The phonon pulse shape cut definitions described in Table 4.1 were devel-oped by another member of the analysis group.564.4. Cut DevelopmentThe remaining discussion in this section will pertain to the developmentof the time blocks within which the finalized ∆Glitch and ∆LFN cuts aredefined.Time Block Development for the ∆Glitch and ∆LFN CutsAs described in Table 4.1, the ∆Glitch and ∆LFN cuts are defined to removeoutliers for which these respective quantities lie beyond a certain percentileabove the main distribution. It is known from previous SuperCDMS analy-ses that the detector noise environment can vary over time. Since the noiselevel, as measured on a series-by-series basis using series randoms, is usedduring processing in computing the χ2 quantities, a time variation in themeasured noise environment can result in a time variation in typical χ2 es-timates. It is therefore important to define these χ2 cuts within time blocksduring which the location of the main distribution does not shift signifi-cantly in terms of these quantities. Otherwise, the cut will effectively bemuch harsher in some periods than others, which is undesirable.Although one would also expect the phonon χ2 cut to be sensitive to timevariations in the noise environment, it was decided based on previous Su-perCDMS analyses that it would be sufficient to define this cut within thethree main data sets (Sb at 70V, etc.).The second-order coefficient of a parabolic fit to the high-energy (4 keVt-400keVt) Ep distribution of ∆LFN and ∆Glitch was used to indicate ap-propriate time blocks for the parabolic portion of the ∆LFN and ∆Glitchcuts. These distributions were fit on a series-by-series basis, since the noiseenvironment is computed for each series.A parabolic least-squares fit to the ∆LFN distribution is shown for a sampleseries in Fig. 4.11 left, and the second-order ∆Glitch parabolic coefficientsare plotted together for all series in Fig. 4.11 right. The distribution ofsecond-order ∆LFN coefficients, not shown, is very similar in appearance.Due to this similarity, the time blocks, shown as vertical dashed lines in Fig.4.11 right, are identical for the ∆Glitch and ∆LFN cuts.574.4. Cut DevelopmentFigure 4.11: Sample plots showing the time block development for theparabolic ∆Glitch and ∆LFN cuts. Left: Quadratic least-squares fit to the∆LFN distribution for a sample series. Right: Time variation of second-order ∆Glitch parabola coefficients, showing the locations of time blocksbased on periods in which the bulk population in the second-order paraboliccoefficient remains reasonably constant. The colours of the data points rep-resent the condition in which the data was taken – for example, the “SbBeat 70V” shown in blue was taken at a 70V voltage bias, with 124Sb sourceand 9Be wafer in place.It is worth noting in Fig. 4.11 right that the second-order parabola coefficientgradually decreases within each time block. This pattern appears to arisefrom a correlation between the second-order parabolic coefficients and theevent rate in the detector as the source decays. The correlation is illustratedin Fig. 4.12, and discussed in the figure caption.584.4. Cut DevelopmentFigure 4.12: Plots showing the correlation between the event rate in the de-tector and the second-order ∆LFN parabolic fit coefficient. Left: The timevariation in event rate as a function of event time shows some clear correla-tions when compared with the variation in 2nd-order parabola coefficientsshown in 4.12 right. Right: The correlation is seen more directly by plot-ting the variation in the second-order parabola coefficient with event rate.The slight shift in behaviour of the 25V data compared with the 70V datacould be due at least in part to the fact that the 25V data was taken witha different detector (T2Z1 rather than T5Z2), and at a different operatingvoltage.A possible explanation for the observed correlation between the event rateand the second-order parabolic coefficient is as follows: if the event rate ishigh enough to produce pulses in a significant number of the randoms used toevaluate the noise environment, the calculated noise level would be positivelycorrelated with the event rate. Since the χ2 decreases as the expected datavariation increases, the sizes of the calculated χ2 quantities should in generalbe negatively correlated with the calculated noise level. As such, the overallamplitude of the χ2 values, and linear combinations thereof – and hence theamplitude of the second-order fit coefficient – would be negatively correlatedwith the event rate. This correlation was not investigated further, however,as it was not considered directly relevant for the analysis.594.4. Cut DevelopmentBadSeries CutThe BadSeries cut removes three series that were deemed during the courseof the analysis to be unusable due to their exceptionally low noise environ-ment, as measured using the randoms discussed earlier. In principle, anunusually low measured noise environment does not necessarily imply thatthe data itself is problematic - in fact, raw event traces sampled from thesethree series appeared quite normal.The issue arises from the fact that the measured noise environment affectsthe calculation of the phonon χ2 quantities, discussed above in relation to thephonon pulse shape cuts, such that series with a highly atypical measurednoise environment tend not to follow the main distribution of events in thephonon χ2, ∆Glitch, and ∆LFN planes, as shown for one of the series inFig. 4.13.Figure 4.13: Sample series removed by the BadSeries cut, showing its energyvariation with respect to the ∆Glitch phonon pulse shape variable. Theseries stands out in this plane, such that a ∆Glitch cut that is well tunedfor the main distribution of events would be poorly tuned for this particularseries.The unusual distribution of these series in the phonon χ2 planes has twonegative effects:1. It was found that the RMS and CDF points discussed in Table 4.1are shifted by these series to the extent that the phonon pulse shape604.4. Cut Developmentcut definitions would become significantly skewed if the series were notremoved.2. The phonon pulse shape cut definitions developed based on the maindistribution of events in each time block would not target these seriesin the desired manner.Charge χ2 CutThe charge χ2 cut is a flat cut on the χ2 quantity computed when fittingthe summed charge pulse to the “good charge pulse” template. The χ2is summed over both the inner and outer charge channels. The cut wasdeveloped for each of the three CDMSlite data sets – Sb at 70V, Y at 70V,and Y at 25V – with the aim of removing potential events arising due toelectronic glitches in the detector rather than true recoil events that mayhave been missed by the phonon pulse shape cuts.Y at 25VThe Y at 25V data set, taken with the T2Z1 detector, is plotted in chargeχ2 vs. charge energy space in Fig. 4.14 left. The distribution shows aclear main population of events at relatively low charge χ2, with a sparsedistribution of low-energy events going up to high charge χ2.Figure 4.14: Y at 25V data passing the basic and above-discussed qualitycuts, showing the location of the charge χ2 cut. Left: Shown in charge χ2vs. charge energy space. Right: Charge χ2 distribution. The exact cutplacement is chosen to be near the shoulder of this distribution.614.4. Cut DevelopmentThe nominal cut in charge χ2 space was placed based on two considera-tions:1. in what range of charge χ2 do raw sample charge traces begin to appearvisually glitchy, like the charge trace in Fig. 4.15 right, and2. at what point does the charge χ2 distribution exhibit a clear shoulder.Figure 4.15: Sample raw pulse traces for the charge χ2 cut. The top panelin each plot shows the raw phonon trace, and the bottom panel shows theraw charge trace. Left: “good” charge trace, without any glitchy behaviour.Right: “bad” charge trace, with clear glitchy behaviour.Visual inspection of the raw traces revealed that glitchy behaviour likethat shown in Fig. 4.15 right began in the range of 6000 < charge χ2 <7000.Based on the above considerations, the cut for the Y at 25V data is nominallydefined as:Charge χ2 < 6250However, there is a conspicuous population of data points, highlighted inred in Fig. 4.16, that appears shifted upward compared with the maindistribution. It was found that this population originates from a singleseries in the neutron-on Y at 25V data.624.4. Cut DevelopmentFigure 4.16: Y at 25V: neutron-on data with outlying charge χ2 highlightedin red. A non-negligible number of events without charge pulse glitcheswould be removed from this series if the nominal cut were applied to it.An inspection of the raw charge traces from this series revealed that none ofthe traces for events in this series with charge χ2 above the nominal cut lo-cation appear glitchy, so this particular series was exempted from the chargeχ2 cut to avoid unnecessary removal of data. It is assumed that the shift incharge χ2 arises from a change in the measured noise environment duringthat series that affected the χ2 calculation during processing. Therefore, thefinal cut definition for the Y at 25V data is:Charge χ2 < 6250, or SeriesNumber==11510061304Sb at 70V and Y at 70VThe charge-channel behaviour is quite different in the 70V data sets, whichwere taken with the T5Z2 detector. In this detector, it is found for manyevents that the outer (QO) charge channel exhibits sparse, narrow, andisolated glitches, an example of which is shown in Fig. 4.17 left.634.4. Cut DevelopmentFigure 4.17: Two examples of glitchy pulse traces in the 70V data. Left:These glitches are narrow, isolated, and confined to the QO channel. TheQI and phonon pulse traces appear reasonable. Right: This trace has acombination of isolated glitches in the QO channel, and electronic glitchessuch as those seen in Fig. 4.15 right. The electronic glitches are correlatedbetween the QO and QI channels, and the phonon channel does not appearto have a significant pulse.These isolated glitches are suspected to be unrelated to true electronicglitches in the detector, arising instead from issues associated with the outercharge channel readout on T5Z2. This suspicion arises mainly from the fol-lowing observations:1. The isolated QO channel glitches do not correspond to glitches in theinner charge channel (QI) pulse trace, whereas such a correspondencebetween QI and QO glitches is typically seen for electronic glitchessuch as that shown in Fig. 4.15 right.2. The presence of the isolated glitches does not tend to be correlatedwith visually glitchy or irregular behaviour in the phonon channel.This correlation is often seen for the electronic glitches found in the Yat 25V data.The isolated glitches in the QO channel present a problem for the cut defi-nition because they tend to smear the total charge χ2 estimates to relativelyhigh values, as shown in Fig. 4.18 left. This creates a degeneracy in chargeχ2 space between events of concern with true electronic glitches, and eventswith the isolated QO channel glitches that do not appear to be correlatedwith glitchy behaviour in either of the QI charge channel or the phononchannels, and are therefore not considered to be indicative of glitchy phonontraces.644.4. Cut DevelopmentFigure 4.18: Y at 70V data passing the above-discussed quality cuts, showingthe location of the charge χ2 cut for the 70V data. Right: Charge χ2distribution. Due to the isolated QO glitch events described in the text, thedistribution does not exhibit the clear shoulder that is seen for the 25V datain Fig. 4.14.As shown in Fig. 4.18 right, the charge χ2 distribution for the 70V data doesnot exhibit a shoulder such as that used to guide the choice of cut location inthe Y at 25V data. Instead, the approach for setting the cut location of the70V data is as follows: since the purpose of the charge χ2 cut is to removepulses that do not represent real physics events, but which managed to makeit through the phonon pulse shape cuts, one would expect some degree ofpositive correlation between the charge and phonon χ2 quantities prior tothe application of quality cuts, in the region where the charge χ2 cut isapplicable. To verify this approach, the charge and phonon χ2 quantities areplotted together for the Y at 25V data in Fig. 4.19 left, with only the basiccuts applied. Indeed, the region of charge χ2 at which positive correlationsbecome apparent roughly corresponds to the cut location obtained by theindependent methods discussed previously. An analogous plot for the Sb at70V data is shown in Fig. 4.19 right.654.4. Cut DevelopmentFigure 4.19: Variation of phonon χ2 with charge χ2, showing the point atwhich positive correlations become visible between these two χ2 quantities.Left: Verification that the cut location chosen for the Y at 25V data roughlycorresponds to the point at which positive correlations are seen between thecharge and phonon χ2. Right: Cut location for the Sb at 70V data, usingthe charge χ2 for which positive correlation becomes visible to guide thechoice of cut location.Based on the above criterion, the charge χ2 cut is placed as follows for the70V data:Charge χ2 < 50, 000Low-Frequency Noise Blob CutLow-frequency noise (LFN) represents a particularly significant backgroundfor the analysis at low energies near the trigger threshold, where events arepredominantly triggered by LFN and electronic glitches, rather than truerecoils in the detector. A sample “LFN blob” is shown in ∆Glitch space forthe neutron-off (“blank”) data in Fig. 4.20 left.The noise blob is effectively removed by the ∆Glitch cut down to a cer-tain energy, below which the noise blob becomes degenerate with the maindistribution of events.A study was done to evaluate how much the low-energy analysis cutoff wouldneed to be increased above the trigger threshold in order to be confident thatthe low-frequency noise events would not affect analysis results at a signifi-cant level given the statistical uncertainty associated with the experimental664.4. Cut Developmentspectra. The study was performed by first identifying a region in the low-energy spectrum above which the shoulder of the LFN blob spectrum isreasonably well fit with a decaying exponential y(E):y(E) = Ae−(E−Emin)/E1/2 +B (4.6)where E is the phonon energy (with the usual subscript p removed for nota-tional convenience) Emin is minimum energy of the fitting region, E1/2 is thedecay constant of the exponential, A is the amplitude of the exponential fit,and B represents the approximately constant background of electron recoilevents.Figure 4.20: Sample fit to LFN blob for the neutron-off Y at 70V data.Left: Energy distribution of the ∆Glitch variable, in which the LFN blobis found to be best separated from the main distribution. Right: Phononenergy spectrum, with the shoulder of the LFN blob fit with the decayingexponential defined in Eq. 4.6.Using the pure exponential (i.e. non-background) portion of y(E), the ideais to find the energy E0 below which these LFN blob events can significantlyaffect the analysis result. Above E0, the contribution of LFN blob events tothe energy spectrum should become negligible compared with the scale ofPoisson uncertainty associated with the spectrum of true recoil events. Themethod is illustrated in Fig. 4.21, and detailed below.674.4. Cut DevelopmentFigure 4.21: Illustration of the method for determining the optimized analy-sis cutoff (E0)opt. Top: For values of E0 above (E0)opt, the noise blob model(magenta) integrated from E0 out to∞ (Iblob) becomes negligible comparedwith the Poisson uncertainty√NBe (navy points below (E0)opt and limegreen points above) associated with the integrated number of neutron-onevents between E0 and E0 + E1/2 (beige region). Bottom: The exact pointabove which Iblob is considered negligible compared with√NBe occurs whenthe ratio Iblob√NBedrops below 13 .For a given candidate value of E0, the LFN model is integrated from E0 outto ∞:Iblob =∫ ∞E0Ae−(E−Emin)/E1/2dE (4.7)and the total number of events is compared with the Poisson uncertainty√NBe associated with the number NBe of neutron-on events in a region ∆Eabove the candidate analysis threshold E0, where ∆E – nominally set toE1/2 – is intended to represent an energy range over which the noise blob isan important component of the spectrum.Therefore, the optimum analysis threshold (E0)opt is set as the highest can-didate value for which:Iblob√NBe<13(4.8)The cutoff ratio of 13 was chosen to be conservatively less than 1.684.4. Cut DevelopmentThe GoodSeriesRate and TriggerBurst CutsGoodSeriesRate CutThe GoodSeriesRate and TriggerBurst cuts are intended to target periodsof data-taking that exhibit unusually high event rates, since the rate of γevents in the photoneutron calibration should in theory decay reasonablysmoothly as the source decays. As shown in Fig. 4.22, the GoodSeriesRatecut removes entire series whose average trigger rate is significantly abovethat of nearby series, as this would suggest that many of the events areinstrumental background. The location of the GoodSeriesRate cut is chosenfor each data set by fitting the event rate decay with an exponential, andvertically shifting the exponential until the cut visually removes the obviousoutliers.Figure 4.22: Time variation of event rate, averaged over each series, for theSb at 70V data. The GoodSeriesRate cut shown in yellow removes serieswhose average event rate is significantly above that of nearby series.TriggerBurst CutThe idea of the TriggerBurst cut is similar to that of the GoodSeriesRate,except that it operates within each individual series. The idea is to removeperiods within a series during which the rate of events passing the basic cutsis too far above the typical event rate in the series.Fig. 4.23 shows an example of a series with a period of unusually high eventrate highlighted in purple that is removed by even the most conservativeTriggerBurst cut considered. The period at the end accumulates eventsmuch more quickly than the remainder of the series, indicating that this is aperiod with an unusually high trigger rate (i.e. a TriggerBurst period).694.4. Cut DevelopmentFigure 4.23: Example of a series with a period of high trigger rate removedby the TriggerBurst cut.An iterative approach was developed to remove periods of unusually highevent rate within a series, as described below:1. First, the events in the series are binned according to the time atwhich they were measured by the DAQ (also referred to as their “unixtime”) into bins approximately 30 seconds in width. Within each bin,an average rate is calculated as the total number of events in thebin, divided by total livetime of events in the bin passing the otherbasic and quality cuts. The idea is then to remove bins that representsignificant outliers due to their high event rate.2. The TriggerBurst cut will iteratively remove the bin with the highestevent rate, each time re-calculating the mean over the remaining binsuntil the mean value x¯ stabilizes to within the required precision p,expressed in % as:p = 100%x¯i − x¯i−1x¯i−1(4.9)where x¯i is the mean calculated after removing the ith outlying bin.Once the precision drops below the required level at the nth iteration,the cut is set at a level that removes outlying bins up to the (n-1)thiteration. The cut can then be tuned by varying the required precisionlevel of the mean. Fig. 4.24 shows an example of a series with severalbins that would be removed by the TriggerBurst cut, depending onthe precision level.704.4. Cut DevelopmentFigure 4.24: Sample series showing the application of the TriggerBurst cutat varying precision levels. The bins removed by each precision level arehighlighted with a marker whose colour corresponds to the precision levellisted in the legend, and the dashed line of the same colour shows the eventrate level above which bins are removed for the given precision requirement.Note that, in some cases, several different precision levels will produce thesame cut, in which case the largest precision level that produces the cut isshown.Beyond this basic cut-setting algorithm, there is an additional subtlety aris-ing from the fact that not all series were taken for the same amount of time.With 30s bin widths, this means that not all series have the same numberof bins. It can be shown (see Appendix A) that the shift ∆x¯ in the mean x¯due to an outlier varies on average with the number of bins N as:∆x¯ ∝ 1N(4.10)To address this issue, the candidate precision level p100 is chosen for serieswith 100 bins, and the actual required precision levels pM for a series withM bins is defined as:pM =p100M100(4.11)The exact precision level requirement for the TriggerBurst cut is tuned foreach data set by producing corresponding Monte Carlo (MC) data sets withthe typical 30s bin width and Poisson-distributed event numbers, and apply-ing the TriggerBurst cut to the MC data sets to compare the cut’s passagefraction.714.4. Cut DevelopmentThe event rate, and hence the mean number of events per bin, decreases bothbetween and within each data set due to the decay of the sources. In orderto make the harshest possible estimate of how low the passage fraction couldbe for Poisson-distributed events in a given data set, the median number ofevents per bin was determined for each series in the data set with at least50 bins. The lowest median from all the series was then used as the meanvalue of the parent Poisson distribution of events-per-bin when producingthe MC histograms for that data set. The results are listed for each dataset in Table 4.2.Table 4.2: Lowest median events/bin of all series in each data set, andtypical number of 30s bins per series. Both values were used for producingthe MC data sets. Note that the number of bins in the 25V data is triplethat of the 70V data because the run lengths for these series were 3h ratherthan 1h.data setMinimumEvents/BinTypical Number of BinsPer SeriesSb at 70V 9 100Y at 70V 28 100Y at 25V 20 300Fig. 4.25 shows a sample MC series for the Sb at 70V data, with several binsremoved by the TriggerBurst cut. Table 4.3 shows a comparison of eventpassage fractions for the cut between the experimental data with basic andother quality cuts pre-applied, and the MC data set. There are two criteriaused to select the chosen precision level highlighted in Table 4.3.1. The passage fraction for Poisson-distributed MC event number his-tograms should be very high (above 99%), thus indicating that thecut is not removing a significant amount of normal Poisson fluctua-tion.2. While achieving a very high MC passage fraction, it is also desirableto maximize the difference between MC passage fraction and the pas-sage fraction of the actual data, as this would suggest that the cut isactually removing periods of high trigger-rate bursts, rather than justnormal Poisson fluctuations.724.4. Cut DevelopmentFigure 4.25: Sample MC series for the Sb at 70V data, with the TriggerBurstcut applied as described in the caption of Fig. 4.24.Table 4.3: Comparison between actual and simulated event passage fractionsfor Sb at 70V, with the chosen precision level highlighted in bold font. Notethat the precision level listed in the left-most column refers to the nominalprecision level p100 for a series with 100 bins (see Eq. 4.11 for the conversionto the true precision level for a given series).TriggerBurst CutsExperimentalEvent PassageFraction (%)SimulatedEvent PassageFraction (%)Difference inPassageFractions (%)None 100.00 100.00 0.00+2.0% Precision 99.58 99.999 +0.42+1.75% Precision 99.28 99.997 +0.72+1.5% Precision 98.71 99.976 +1.27+1.25% Precision 98.71 99.835 +1.13+1.0% Precision 95.79 99.044 +3.25+0.85% Precision 93.62 97.361 +3.74The precision levels for the Y at 70V and Y at 25V data were set by ananalogous procedure to 0.85% and 1.0%, respectively.TriggerGlitch CutThe TriggerGlitch cut is applied in addition to the ∆Glitch cut as a meansof identifying and removing pulses produced by electronic glitches. When734.4. Cut Developmentan event is triggered, the DAQ system records which detectors triggeredin both the charge and phonon channels within 1ms of the global trig-ger time. This information is combined during processing to produce the“ntrigp/q”quantities, defined as the total number of running detector sides– of which there are 12 for the photoneutron analysis – that produced aphonon (ntrigp) or charge (ntrigq) trigger. The ∆ntrig quantity is thenconstructed as the difference between the number of phonon and chargetriggers:∆ntrig = ntrigp− ntrigq (4.12)Since a pulse produced by a recoil event should ideally produce triggersin both the charge and phonon channels, the idea of the cut is to removeevents for which too many detectors produced a phonon trigger without anaccompanying charge trigger.As shown in Fig. 4.26 top, events with relatively high values of ∆ntrig standout in several parameters, most notably the x and y partition parameters,which are calculated using the relative pulse amplitudes in the three innerphonon channels to give a coarse estimate of the location of the event in thedetector.744.4. Cut DevelopmentFigure 4.26: Top: The red population of events in the upper right corner ofthe x vs. y partition with high ∆ntrig do not follow the main distribution,but are partially degenerate with it. This population is fully removed bythe TriggerGlitch cut. Bottom: Events with high ∆ntrig often exhibit thesesharp, square phonon traces in all detectors, which from visual inspectionare clearly not good signal events.Examining raw pulse traces, it is found that events with ∆ntrig≥ 4 canproduce short, square-shaped pulses such as the one shown in Fig. 4.26bottom that are visually identified as instrumental artifacts, but which passthe existing basic and quality cuts. These events are removed with the754.5. Energy Resolution Modelfollowing TriggerGlitch cut:∆ntrig < 4 (4.13)4.5 Energy Resolution ModelThe total energy resolution σT of SuperCDMS detectors varies with recoilenergy, and is generally modelled by the quadrature sum of three contribu-tions:σT (Eee) =√σ20 + FEee + (AEee)2 (4.14)where:• Eee is the energy of the recoil event, assuming an electron recoil, withenergy measured in “keVee”.• σ0 is the baseline resolution at Eee = 0 due to the detector electronics.• F is the “Fano factor” that gives a measure of the variance in thenumber Ne/h of e-h pairs ionized by a single recoil. For sufficientlyhigh statistics, the rms statistical fluctuation in N is given by [75]√FNe/h.• AEee is an empirical term that models any detector effects such asposition dependence.In order to fit this model to the photoneutron data for each of the twoCDMSlite detectors, it is necessary to have measurements of the energyresolution at as many recoil energies as possible. These are obtained using“post-Cf” low-background WIMP search data that was collected shortlyafter a 252Cf neutron activation of the detectors.During neutron activation, neutrons are captured by 70Ge nuclei in thedetector to form 71Ge, which is unstable, and decays via electron capture ofinner K, L, or M shell electrons to 71Ga with a half-life of 11 days. Followingan electron capture, an outer electron drops down to fill the inner electronshell vacancy, producing one or more characteristic X rays with energies of10.37 keVee, 1.30 keVee, and 0.16 keVee, which respectively correspond to K,L, and M shell vacancies. The post-252Cf data therefore has spectral peaks764.5. Energy Resolution Modelat these three characteristic X ray energies, and the measured widths ofthese peaks represent the detector energy resolution at these energies.The original phonon energy scale (keVt) is converted to keVee by fittingGaussian distributions to the L and K shell spectral peaks using a simpleleast-squares fitting routine, and comparing the centroids of the fitted Gaus-sian distributions to the known peak energies to obtain a conversion factor.The Gaussian distributions g(Eee) are given by:g(Eee) = Ae− (Eee−(Eee)0)22σ2 +B (4.15)where A, B, and (Eee)0 are fit parameters.The baseline resolution σ0 is measured using the reconstructed energies ofrandom traces from the low-background data, which should be 0 since theydo not in general contain pulses. The peak in the energy spectrum of therandoms is fit with a Gaussian distribution, as shown for T2Z1 at 25V inFig. 4.27, and σ0 is taken to be the standard deviation of the fitted Gaussiandistribution.Figure 4.27: Gaussian fit to the zero-energy peak of the randoms to obtainthe baseline energyThe resolutions at the L, K, and M shell γ peaks are also fit with Gaussiandistributions to obtain the energy resolution at each shell energy. In the caseof the 70V M shell peak, it was not possible to fit a Gaussian distributiondue to limited statistics.774.5. Energy Resolution Model4.5.1 Fit to the Resolution ModelThe resolutions obtained at the baseline, M shell, L shell, and K shell arefit with the energy resolution model to obtain the model parameters. Thebaseline resolution σ0 is fixed to the value obtained from the standard devi-ation of the Gaussian fit described in Section 4.5.1 to reduce the number ofdegrees of freedom, given the limited number of data points available. Thefits to the resolution model are shown for both detectors (T5Z2 and T2Z1)in Fig. 4.28.Figure 4.28: Fits to the detector resolution model using standard deviationsof Gaussian fits to spectral peaks. Left: T5Z2 biased at 70V. Right: T2Z1biased at 25V.The fit is done using the orthogonal distance regression (ODR) method [82].The major benefit of the ODR method in this instance is that the fit can beweighted by the data covariance matrices in both the x and y coordinates.Since the uncertainty in the energy is non-negligible in this case – especiallyfor the T2Z1 data shown on the right side of Fig. 4.28 – it is desirable toaccount for it in the fit.The error band ∆σT (Eee) is calculated as follows using the parameter co-variance matrix returned by the fit:∆σT (Eee) =√√√√(∂σT∂A)2var(A) +(∂σT∂(F))2var(F) + 2(∂σT∂A)(∂σT∂(F))cov(A,F)(4.16)784.6. Background Modellingwhere∂σT∂A=AEeeσT,∂σT∂(F)=Eee2σT(4.17)4.6 Background ModellingThe negative log likelihood fit to the Lindhard model, discussed in Section4.7, is unbinned to avoid any bias associated with the choice of binning. Itis therefore not possible to directly subtract the neutron-off γ backgroundand fit to the resulting experimental neutron spectrum, since doing so wouldrequire binning the data. Instead, a background model is developed whichcan be incorporated into the probability density function (PDF) used forthe likelihood fit.The neutron-off γ background model is comprised of four components, whichare listed below.1. Compton StepsThe low-energy steps corresponding to each of the Ge electron shells are pre-dicted to arise in the low-energy γ background spectrum due to the minimumenergy required to free electrons from each shell. For example, incident γrays with energy below the K shell binding energy of 11.1 keV will only beable to free electrons from the L, M, and N electron shells, and will conse-quently have a slightly lower probability of depositing energy in the detectorthan γ rays above this binding energy.It has previously been shown [83] that these steps can be resolved in Su-perCDMS detectors operating in CDMSlite mode. For the CDMSlite datataken at 70V, only the L, K, and M shell Compton steps have energy abovethe analysis threshold. For the 25V data, the M shell step is expected tooccur below the analysis threshold, and so is excluded from the model.The steps are modelled by two (25V) or three (70V) error functions – oneat each step – on top of a flat background. For the 25V data, the Comptonmodel is:fC,25V(Eee) = a0{1 + SLfst(Eee, µL)}+ (4.18)+SKa0{1 + SLfst(Eee, µL)}fst(Eee, µK)794.6. Background Modellingwhere S is the fractional step amplitude, and a0 is the flat background level.The step sizes and background level are left as fit parameters. µ is thecentral energy of the step, which was determined for each step by anothermember of the collaboration using a Geant4 simulation. fst is the errorfunction step:fst(Eee, µ) =12[1 + erf(Eee − µ√2 · σ(µ))](4.19)σ(µ) is the detector energy resolution at the centroid of the step, and is notfit for, but instead obtained from the resolution model discussed in Section4.5.For the 70V data, the model additionally includes an M shell step:fC,70V(Eee) = a0{1 + SMfst(Eee, µM )}+ (4.20)+SLa0{1 + SMfst(Eee, µM )}fst(Eee, µL)++SKa0[{1 + SMfst(Eee, µM )}fst(Eee, µL)]fst(Eee, µK)2. Polynomial ComponentA polynomial component of order 3 is included to model any large-scalevariation in addition to the theoretical Compton step background due toenergy variation in the both the “cut efficiencies” and in the fraction ofinstrumental background events that are removed by the data cuts. Theefficiency of a cut is defined as the fraction of true signal events producedby recoils in the detector that are retained after the cut is applied. Thepolynomial component is given by:fp(Eee) = a1(Eee) + a2(Eee)2 + a3(Eee)3 (4.21)3. Post-Cf PeaksA 252Cf detector calibration ended nine days before data-taking began for thephotoneutron calibration – less than the 11-day half life of electron capturesfollowing neutron activation (see Section 4.5 for further details of the neutronactivation). The fractional electron capture probabilities for each shell of71Ge were calculated by another member of the collaboration using the804.6. Background Modellingmethod of [84] to be BK = 87.57%, BL = 10.53%, and BM = 1.78% for theK, L, and M shells, respectively. Using this information, an additional termmodelling the post-252Cf electron capture peaks is added to the backgroundmodel:fCf(Eee) = AK{Gauss(Eee, EK , σ(EK))+ (4.22)+BLBKGauss(Eee, EL, σ(EL)) +BMBKGauss(Eee, EM , σ(EM ))}whereGauss(Eee, Epeak, σ(Epeak)) = e−(Eee−Epeak)2/(2σ(Epeak)2) (4.23)AK , the amplitude of the K shell peak, is the only fit parameter. EK = 10.37keVee, EL = 1.30 keVee, and EM = 0.16 keVee are the energies of eachpeak, and the peak widths σ(Epeak) are obtained from the resolution modeldiscussed in Section 4.5. The M shell peak is excluded from the post-252Cfmodel for the Y at 25V data, because the M shell peak energy, convertedto phonon energy using Eq. 3.4 with Y = 1, is below the analysis thresholdfor this data set.4. Smoothing ComponentIt was found that the background model obtained from the analytical com-ponents described above cannot alone yield an acceptably good fit to theexperimental data, as indicated by a low p-value based on the calculatedfit χ2 and degrees of freedom, as well as visible structure in the fit resid-uals. Careful checks were done to rule out the possibility of appreciablecontributions to the γ background from:• primary X-rays or electrons from the radioactive source, or• secondary particles produced as decay products following γ activationof the Ge crystals or surrounding material by γ rays from the radioac-tive source.Having ruled out these possibilities, the imperfect fit results could otherwisearise from energy dependence in the instrumental backgrounds, which can814.6. Background Modellingonly be modelled empirically. This empirical model is obtained by smoothingthe fit residuals.After converting the energy scale to electron recoil energy (keVee) and fittingthe neutron-off spectrum to the sum of the above two or three components,the energy scale is converted back to phonon energy (keVt), and the fitresiduals are smoothed using a Gaussian filter [85]. The smoothing σ of theGaussian filter is set such that:1. The χ2 over degrees of freedom is reasonably close to 1 after addingthe smoothed residuals to the analytical fit, and2. any obvious structure in the residuals is removed.The resulting smoothed residuals are added to the existing model. The fullbackground model including the smoothed residuals from the analytical fitis shown for the Sb at 70V data in Fig. 4.29.Figure 4.29: Full background model for the Sb at 70V data, with thesmoothed residuals added to analytical fit.Fig. 4.30 shows the background model for the Y at 70V data on a log scale,with each analytical contribution shown separately.824.7. Unbinned Lindhard Yield ExtractionFigure 4.30: Full background model for the Y at 70V data on a log scale,showing the individual contribution from each analytical component.Table 4.4 lists the χ2 over degrees of freedom, and the resulting p-valuesfor each data set, before and after adding the smoothed residuals to thefit.Table 4.4: Goodness of fit measures for the background model of each dataset, before and after adding the smoothed fit residualsData Setχ2 / dofBeforeSmoothingp-valueBeforeSmoothingχ2 / dofAfterSmoothingp-value AfterSmoothingSb at 70V 792.9 / 676 0.00122 694.5 / 676 0.303Y at 70V 750.4 / 680 0.0311 686.0 / 680 0.428Y at 25V 734.2 / 679 0.0698 695.0 / 679 0.3274.7 Unbinned Lindhard Yield ExtractionAfter calculating the background model with the neutron-off data via themethod described in Section 4.6 above, the neutron-on data is fit to a pa-rameterized Lindhard yield model by minimizing the summed negative loglikelihood of all events given a probability density function (PDF). The PDFis comprised of both the pre-calculated background model, and a neutroncomponent whose shape varies with the value(s) of the Lindhard model pa-rameter(s).834.7. Unbinned Lindhard Yield ExtractionThe summed negative log likelihood function − lnL is given by:− lnL = −3∑data set D=1ND∑i=1ln{pD[(Ep)i)]}(4.24)where ND is the number of events in data set D, and pD(Ep) is the energy-dependent PDF for the Dth data set, given by:pD(Ep) = (FD)n(pD(Ep))n + (1− (FD)n)(pD(Ep))b (4.25)(FD)n represents the fractional contribution of the neutron PDF (pD(Ep))nto the full PDF pD(Ep) for data set D. The neutron PDF is discussedfurther in the next section, and normalized such that it integrates to 1 overall phonon energies. (pD(Ep))b is the background model, which was obtainedas described in Section 4.6 prior to performing the fit, and is normalizedto unit integral. The values of (FD)n for each of the three data sets areleft as free parameters in the fit. − lnL is minimized using the MINUITminimization package.4.7.1 Neutron Spectrum PDFThe neutron spectrum PDF for each data set is obtained from a Geant4simulation of the electron and nuclear recoil energy depositions producedby the neutron flux alone, without the γ background. This simulation wasdeveloped by another member of the analysis group. The simulated elec-tron and nuclear recoil energies are converted to phonon energy based ontheir ionization yield Y (Er), which is calculated for nuclear recoils using theparameterized Lindhard model.Conversion of Simulated Energy Depositions to PhononEnergyFor a set of energy depositions induced by an incident particle scattering inthe detector – collectively referred to as an “event” – the average number ofe-h pairs Ne/h(Er) due to each energy deposition is calculated as:Ne/h(Er) = Y (Er)Er(4.26)844.7. Unbinned Lindhard Yield ExtractionIf the energy deposition is due to an electron recoil, Y ≡ 1. If it is due to anuclear recoil, the yield is calculated according to the Lindhard model (Eq.4.2) with k in the model parameterized by one or more parameters.To account for the physical variance in Ne/h associated with the fitted Fanofactor in the detector resolution model (see Section 4.5), a random number issampled from a Gaussian distribution centred at 0 with standard deviation√FNe/h and added to Ne/h for each energy deposition.After calculating Ne/h and accounting for its variance, the total phononenergy of each event is calculated as:ELuke = eV (x, y, z)Ne/h (4.27)where V (x, y, z) is the total electric potential traversed by an e-h pair pro-duced by a recoil event at a point (x, y, z) in the detector. For the analysispresented in this thesis, V (x, y, z) is set to the nominal operating voltage ofthe detector (either 70V or 25V).The true value of V (x, y, z) may, however, be less than the nominal op-erating voltage for some events due to non-uniformities in the detector’selectric field. The electric potential reduction is simulated, as shown fordetector T5Z2 in Fig. 4.31, as a function of position in the detector, forboth T5Z2 at 70V and T2Z1 at 25V. These electric potential simulationswere developed by another group in the collaboration, and incorporated intonew photoneutron simulation files by another member of the analysis group.The simulation files with the voltage map incorporated were not ready touse in time to incorporate into this thesis, but will be used in the finalanalysis.854.7. Unbinned Lindhard Yield ExtractionFigure 4.31: Electric potential difference between final locations of e-h pairs,as a function of their initial position for detector T5Z2 at 70V. Reprintedfigure with permission from [74]. Copyright (2018) by the American PhysicalSociety.To obtain the total phonon energy of an event, the primary and Luke phononenergies are summed over all the energy depositions of the event:Ep =N depositions∑i=1(Er)i + (ELuke)i (4.28)Before applying the remaining components of the detector energy resolution,the phonon energy is converted to the electron-equivalent energy scale thatwas used when calculating the resolution model in Section 4.5:Eee =Ep1 + eVb(4.29)For each event, a random number is sampled from a Gaussian distribu-tion centred at 0 with standard deviation√σ20 + (AEee)2, and added tothe event’s electron-equivalent energy Eee. The last step is to convert theresolution-smeared energies back to the phonon energy scale by invertingEq. 4.29.Fig. 4.32 shows the conversion from nuclear recoil energy to phonon energy,with the full resolution applied, assuming a Lindhard ionization yield modelwith k = 0.157.864.7. Unbinned Lindhard Yield ExtractionFigure 4.32: Conversion from nuclear recoil energy to phonon energy usingthe Lindhard model. Left: Original energy spectrum for the simulated Sb at70V data. Right: Converted to phonon energy using the method describedin this section, with Y (Er,nr) defined by the Lindhard model with k = 0.157.Smoothed PDF ConstructionTo construct a smoothed PDF, the simulated phonon energies are binned,and the resulting spectrum is smoothed using a Savitzky-Golay filter [86].The smoothing window size and polynomial order of the filter are chosen toprovide as much smoothing as possible while maintaining reasonable agree-ment with the simulated data, as measured by the proximity of the χ2 tothe number of bins in the spectrum – i.e. the degrees of freedom (dof). Thesmoothed histogram is then fit with a cubic spline to produce a functionalform, and normalized over the region from the analysis threshold up to themaximum histogram bin. Fig. 4.33 shows an example of the smoothed PDFfor a conversion to phonon energy with k = 0.157.874.7. Unbinned Lindhard Yield ExtractionFigure 4.33: Simulated Y at 70V data, converted to phonon energy accordingto Lindhard with k=0.157, and overlaid with the resulting smoothed neutronPDFApplication of Cut EfficienciesWhen performing the NLL fits in the final analysis, the energy-dependentcut efficiency E(Ep) for all quality cuts will be applied to the neutron andbackground PDFs, such that the full PDF in Eq. 4.25 represents the trueshape of the experimental data with all cuts applied. The cut efficiencieshave not yet been calculated, so it is not yet possible to incorporate them intothe fits. They will be determined separately for the neutron-on and neutron-off data of each data set by another member of the analysis group. Since theneutron-off background PDF is obtained from the neutron-off data, it willfirst be multiplied by the inverse neutron-off cut efficiency function 1Eb(Ep)to remove any effects on its shape due to the neutron-off cut efficiency. Thefull neutron+background PDF will then be multiplied by the neutron-on cutefficiency function En(Ep) for comparison with the neutron-on data.The cut efficiency functions are not yet ready for inclusion in the fits thatproduce the final results presented in Tables 4.5-4.7, but a preliminary cutefficiency function has been produced for the combined phonon pulse shapecuts in one of the time blocks, as shown in Fig. 4.34. As a check, thispreliminary efficiency function was applied as a proxy for the cut efficiencyin all of the time blocks. The fit results from this check are consistent withinuncertainty with those shown in Tables 4.5-4.7, which were obtained withthe cut efficiency taken to be 100% everywhere. This result suggests thatthe final efficiencies will not result in a significant change in the extractedparameters.884.7. Unbinned Lindhard Yield ExtractionFigure 4.34: Preliminary cut efficiency results for the phonon pulse shapecuts in time block 2, smoothed with a Gaussian filter [85] to obtain a pre-liminary cut efficiency function. As a check, this preliminary cut efficiencyfunction is used as a proxy for the cut efficiency in all the time blocks.4.7.2 Options for the Lindhard Model ParametersTwo parameterizations of the Lindhard model were considered to model theexperimental data, described below.Single-Parameter ModelThe fit with a single-parameter Lindhard model determines the energy-independent k value in the model, in addition to the three neutron contri-bution fractions (FD)n, that minimize the full negative log likelihood − lnLin Eq. 4.24.Fig. 4.35 shows some sample PDFs from a range of k values within theconstrained fitting range of 0.05 < k < 0.25.894.7. Unbinned Lindhard Yield ExtractionFigure 4.35: Sample neutron PDFs for the 1-parameter Lindhard model,produced from the Y at 25V simulation dataTwo-Parameter ModelThe two-parameter Lindhard model adds a linear modification to the k valueover a limited energy range Elow < Ep < Ehigh in which the experimentaldata can possibly be sensitive to variations in the nuclear recoil ionizationyield:k(Er,nr) =klow if Er,nr < Elowklow +khigh−klowEhigh−Elow (Er,nr − Elow) if Elow ≤ Er,nr ≤ Ehighkhigh if Er,nr > Ehigh(4.30)where Elow is defined as the minimum value of the analysis threshold con-verted to nuclear recoil energy, assuming k = 0.25, over all three data setscontributing to the fit. Note that k = 0.25 is the highest value of klow allowedin the fit. This value is chosen because the analysis threshold decreases inNR energy as k increases, so the highest possible k value yields the lowestpossible Elow value of 0.390 keVr,nr. Ehigh is simply the maximum nuclearrecoil energy of the simulated data from all contributing data sets, and isequal to 54.3 keVr,nr.Preliminary fits with mock experimental data sets suggested that the ex-perimental statistics were insufficient to attempt a fit with more than twoparameters in the Lindhard model.904.7. Unbinned Lindhard Yield ExtractionFig. 4.36 shows some sample PDFs from a range of klow and khigh valueswithin the constrained fitting ranges of 0.02 < klow < 0.25 and 0.02 <khigh < 1.6.Figure 4.36: Sample neutron PDFs for the 2-parameter Lindhard model,produced from the Y at 70V simulation data4.7.3 Fit Results and Statistical UncertaintiesAssuming that the MINUIT minimizer accurately determines the parametervalues that yield the global minimum, there are three separate contributionsto the statistical uncertainty in the resulting fit parameters. These are dueto the limited statistics in:• the Geant4-simulated neutron spectrum,• the experimental neutron-on and neutron-off spectra, and• the simulated events used to calculate the cut efficiency function.The uncertainties and covariances are summarized in Table 4.6 for the threesources of uncertainty that have to date been evaluated.Fit Results and Simulation StatisticsDue to limited simulation statistics, the randomness associated with theapplication of the detector energy resolution model, described in Section4.7.1, can affect the exact shape of the neutron PDF. The resulting statisticaluncertainty is estimated by performing the fit 500 times, each time with adifferent random seed associated with the application of energy resolution.914.7. Unbinned Lindhard Yield ExtractionThe resulting distribution of fit results is fit with a Gaussian distributionto estimate the central value and width of the distribution. The fittedparameter value is set to the centroid of the fitted Gaussian distribution,and the statistical uncertainty is set to its standard deviation. The fittedvalues of the Lindhard parameters are listed in Table 4.5. Note that, in thecase of the 2-parameter fit for khigh and klow, the results listed in Tables4.5 and 4.6 are not final. The final results are reported in Section 4.8 afterreducing Ehigh in Eq. 4.30 such that the linear variation occurs over a moreconstrained energy range. The motivation for reducing the energy range oflinear k variation is also discussed in Section 4.8.Table 4.5: Summary of fit results for Lindhard fit parameters. Please seeTable 4.6 below for the fit uncertainties.k klow khigh0.1053 0.05856 1.023Table 4.6: Summary of fit uncertainties for Lindhard fit parameters.Uncertainty ∆k ∆klow ∆khigh ρ(klow, khigh)Stat. (Simulation) 0.00090 0.0019 0.025 -0.861Stat. (Experimental) 0.0034 0.0057 0.12 -0.635Sys. (Fano Factor)(+0.000021,-0.0023)(+0.000,-0.0012)(+0.018,-0.0026)-0.989Fig. 4.37 shows some sample distributions of fit results.924.7. Unbinned Lindhard Yield ExtractionFigure 4.37: Sample Gaussian fits to the distributions of fit results to de-termine the statistical uncertainty on each parameter due to the simulationstatistics. Left: Distribution of khigh from the 2-parameter Lindhard fit.Right: Distribution of the fitted neutron fraction for the Y at 70V data setfrom the 1-parameter fit.The covariance between khigh and klow in the 2-parameter fit is estimated bycalculating the Pearson correlation coefficient ρ based on their fitted values,as shown in Fig. 4.38, and using:cov(klow, khigh) = ρ∆(klow)∆(khigh) (4.31)Figure 4.38: Correlation between klow and khigh for the 2-parameter fit withrandomness associated with simulation statistics allowed to varyExperimental StatisticsIn addition to the uncertainty due to simulation statistics, there is alsosome statistical uncertainty arising from the limited statistics in both the934.7. Unbinned Lindhard Yield Extractionneutron-on and neutron-off spectra.Neutron-On StatisticsTo evaluate the statistical uncertainty arising from the experimental statis-tics in the neutron-on spectrum, 500 sets of mock Monte Carlo (MC) neutron-on data sets are produced, in which the total number of events is the sameas for the experimental data. Each MC data set is comprised of events dueto the neutrons incident on the detector (i.e. the “neutron component”),and events due to the γ rays produced by the radioactive source (i.e. the“background component”).The neutron component of each MC data set D is produced by randomlyselecting events, without replacement, from the Geant4 simulation of neu-tron scatter events in the detector. The number of events selected from eachdata set D is equal to (FD)nNtot, where Ntot is the total number of eventsin the experimental neutron-on spectrum, and (FD)n is the fitted value ofthe fractional contribution of neutron events to data set D. The total energydeposition of each event is converted to phonon energy using the methoddescribed in Section 4.7.1, where the energy-dependent yield function inEq. 4.26 is evaluated using the parameterized Lindhard yield model, withthe Lindhard k parameter(s) set to the best-fitting value(s) listed in Table4.5.The background component is comprised of[1− (FD)n]Ntot events for eachdata set, with the phonon energy of each event sampled from the data set’snormalized background PDF.Neutron-Off StatisticsDespite the smoothing of the neutron-off spectrum, the exact shape ofthe resulting background PDF is somewhat sensitive to the experimentalneutron-off statistics. The resulting statistical uncertainty in the fit resultsis estimated by randomly varying the number of events in each bin of theneutron-off spectrum used to obtain the background PDF. The number ofevents in each bin is sampled randomly from a Poisson distribution, whosemean value is set to the true number of events in the bin.Evaluating the Uncertainty Due to Experimental StatisticsEach of the 500 MC data sets is fit using the same method applied to theexperimental data, but with the background PDF varied randomly as de-scribed above. The resulting distribution of fit results is fit with a Gaussiandistribution, and the uncertainty on each parameter due to experimental944.7. Unbinned Lindhard Yield Extractionstatistics is taken as the standard deviation of the fitted Gaussian distribu-tion. Fig. 4.39 shows some sample distributions, with Gaussian fits.Figure 4.39: Sample Gaussian fits to the distributions of fit results withMonte Carlo data to determine the statistical uncertainty on each param-eter due to the experimental statistics. Left: Distribution of k from the1-parameter Lindhard fit. Right: Distribution of klow from the 2-parameterfit.The covariance between klow and khigh is determined using the same methodas described above for the statistical uncertainty due to simulation statis-tics.Statistical Uncertainty Associated with the Cut EfficiencyFunctionThe energy-dependent cut efficiency function will be calculated by anothermember of the analysis group by creating simulated phonon pulses that re-semble good signal events within each data series. After applying all energy-dependent cuts to the simulated events, the total number N of events simu-lated in each data set, and the number X of events passing the cuts withinthe data set will be binned according to their re-constructed phonon en-ergy. Within each bin, the probability p that an event will pass the energy-dependent cuts is estimated as:p =xn(4.32)where x is the number of events that pass the cuts, out of a total of nsimulated events in the bin. Based on this measurement, the distribution of954.7. Unbinned Lindhard Yield Extractionevents k passing the cuts for repeated random samples of n simulated eventsis assumed to follow a binomial distribution, given by:B(k;n, p) =(nk)pk(1− p)n−k (4.33)To propagate the statistical uncertainty associated with the actual numberx of events passing the energy-dependent cuts within each energy bin, 500Monte Carlo samples of the binned passage fraction information will beproduced, in which the number of events k passing the cuts in a given binwill be re-sampled according to the above binomial distribution. For eachMonte Carlo sample, the cut efficiency function will be re-calculated, and thenegative log likelihood fit will be repeated with all other random variableskept fixed. The width of the resulting distribution of fit results will be usedto estimate of the statistical uncertainty associated with the cut efficiencyfunction.4.7.4 Systematic UncertaintiesThere are two sources of systematic uncertainty that need to be accountedfor. First, there is uncertainty in the Geant4 neutron simulation associatedwith the exact form of the energy-dependent neutron-nucleon elastic scat-tering cross section in the Ge detectors. Second, there is some uncertainty inthe value of the Fano factor obtained from the data-driven resolution modeldiscussed in Section 4.5.Neutron-Ge Elastic Scattering Cross SectionThe Geant4 simulation requires as part of its input the energy-dependentneutron-nucleon scattering cross section of the incident neutrons with “nat-ural” Ge. The Ge making up the detector is considered “natural” becausethe relative abundance of different Ge isotopes is unchanged from that ofnatural Ge ore. The uncertainty in the energy-dependent neutron-nucleonscattering cross section in the detector must be accounted for in the simula-tion, particularly in the “resonance region” of ∼2-50 keV. To do so, randomcross section files were obtained for all the stable isotopes of Ge from theTENDL-2017 nuclear database, where the randomness of each cross sectionis associated with randomly varying all the nuclear data used to compute the964.7. Unbinned Lindhard Yield Extractioncross section. The Geant4 neutron simulation will be repeated with severaldifferent random cross section files for each Ge isotope. The negative loglikelihood fit will be repeated for each of the simulations, and the resultingdistribution of fit results will be used to estimate the systematic uncertaintydue to the elastic scattering cross section.Fano FactorAs discussed in Section 4.7.1, the Fano factor F is used as part of the ap-plication of detector energy resolution to the simulated neutron spectrum,which is predominantly comprised of nuclear recoil events. As describedin Section 4.5, the value of F is obtained experimentally for each detectorfrom a fit to the SuperCDMS resolution model, and has an associated fituncertainty. There are, however, sources of uncertainty associated with Fin addition to the fit uncertainty.Electron RecoilsIt has long been suspected that the Fano factor obtained from the fit to theSuperCDMS data-driven resolution model (Eq. 4.14) does not fully repre-sent the true physical Fano factor, which other measurements have indicatedshould be ∼0.13 [87, 88]. The reason is that the position dependence term(AEee)2 may absorb some of the information that should be represented bythe Fano term FEee. Indeed, if we take  = 3 eV, the fitted values of F,based on Fig. 4.28, are 0.19±0.07 and 0.27±0.02 for detector T5Z2 at 70Vand detector T2Z2 at 25V, respectively. The first is only barely consistentwith 0.13, and the second is very inconsistent.Nuclear RecoilsResults from a past measurement [90] of neutron scattering in an Si(Li) de-tector offer some hint that the Fano factor for nuclear recoils in Si couldbe significantly higher than the measured value of ∼ 0.12 [89] for electronrecoils. Using the results of this experiment, calculations discussed in Ap-pendix C suggest that the nuclear recoil Fano factor for Si could be as highas 6.1 within the approximate energy region in which the data is determinedin Section 4.8 to be sensitive to the Lindhard model. Comparable measure-ments have not yet been performed for Ge, but based on the results of the Simeasurements, we consider the possibility that the Fano factor for nuclearrecoils in Ge could be similarly large compared with the electron recoil Fanofactor.974.7. Unbinned Lindhard Yield ExtractionSince the events used to calculate the resolution model were predominantlyelectron recoils, the resulting Fano term may therefore be an underestimateof the value that would be appropriate for nuclear recoils.Evaluating the Systematic Uncertainty Due to the Fano FactorTo account for both the fit uncertainty and the sources of uncertainty dis-cussed above with respect to the electron recoil and nuclear recoil Fanofactors – FER and FNR, respectively – the fit is done twice, under each ofthe following conditions:1. For each data set, FER = FNR=min(0.13, Ffit−∆Ffit). Ffit is the valueof F obtained from the best fit to the resolution model, and ∆Ffit isthe fit uncertainty. The maximum allowable value of 0.13 is intendedto accommodate the possibility of a systematic shift in the fitted Fanofactor from its true physical value, as discussed above.2. For each data set, FER = Ffit + ∆Ffit and FNR = 10. The upper valueof 10 is chosen for FNR to be on the same order as – but conservativelylarger than – the maximum value calculated in Appendix C for Si basedon [90], within our energy region of interest.The resulting shift in the fit results, shown in Fig. 4.40, is taken to be thesystematic uncertainty associated with the Fano factor.Figure 4.40: Shift in fit results due to the Fano factor systematic for the1-parameter fit. The light beige region on either side of the vertical dashedblack line represents the 1σ statistical uncertainty band on the original fitresult.984.8. Final Results4.8 Final ResultsThe fitted Lindhard model parameters and their respective uncertainties arepropagated to the ionization yield Y (Er,nr) vs. nuclear recoil energy planein Fig. 4.41 and 4.42. The propagation of model parameter uncertaintiesand covariances to Y (Er,nr) is discussed in Appendix B.Figure 4.41: Best-fitting Lindhard ionization yield model, shown as a func-tion of nuclear recoil energy for the 1-parameter fit. The statistical uncer-tainty due to experimental statistics is highly dominant compared with thatdue to simulation statistics, such that the additional uncertainty due to sim-ulation statistics (green band) is barely visible. The beige region shows thefull energy range in which there is neutron simulation data for at least onedata set.994.8. Final ResultsFigure 4.42: Best-fitting Lindhard ionization yield model, shown as a func-tion of nuclear recoil energy for the 2-parameter fit. Again, the statisticaluncertainty due to experimental statistics is strongly dominant.The 2-parameter fit result shown in Fig 4.42 rises well above the reasonablerange of nuclear recoil ionization yields, given the general agreement of otherexperimental results with the Lindhard theory prediction with k ≈ 0.156−0.160 above ∼50 keVr,nr (see Fig. 4.1). After some investigation, it wasfound that this unreasonable high-energy behaviour arises from the factthat the neutron spectrum PDFs exhibit very little variation with Lindhardmodel parameters at high energy (see Fig. 4.35 and 4.36), and are thereforeeffectively insensitive to the model at high energy.To determine the point at which the fit results would become insensitiveto changes in the Lindhard model, the 2-parameter model with the best-fit klow and khigh parameters is modified such that above a certain energyEmax, the k value drops back down to klow, as shown in Fig. 4.43 left.To avoid introducing any bias from the details of the experimental datasets, the testing is done on one of the Monte Carlo data sets that had beenproduced with the best-fit values of klow and khigh to evaluate the statisticaluncertainty due to experimental statistics, as described in Section 4.7.4.For several different choices of Emax, the negative log likelihood − lnL iscalculated and plotted, as shown in Fig. 4.43 right, for a range of khigh values,with all other fit parameters set to their best-fitting values, listed in Table4.5. The goal is to determine the approximate value of Emax above which− lnL is not appreciably affected – compared with the total uncertainty onkhigh – by the new step feature above Emax.1004.8. Final ResultsFigure 4.43: Plots showing the check done to determine the maximum energybelow which the data can be sensitive to changes in the Lindhard model.Left: Sample plot showing the modified 2-parameter Lindhard model withEmax =15 keVr,nr. Right: Likelihood scans of khigh for several values ofEmax, showing that the likelihood variation becomes degenerate comparedwith the fit uncertainty by 7 keVr,nr.Based on the check shown in Fig. 4.43 right, it is expected that the fit re-sults should become effectively insensitive to changes in the Lindhard modelabove ∼7 keVr,nr. Therefore, the 2-parameter model shown in Eq. 4.30is updated such that Ehigh = 7 keVr,nr, rather than the maximum energyup to which the simulated neutron spectrum contains nuclear recoil events(54.3 keVr,nr). The fit was repeated with this model, and the fit results withthe two models are compared in Fig. 4.44, confirming that the change tothe model above 7 keVr,nr does not affect the shape of the best-fit modelwithin uncertainty below this high-energy linear variation cutoff. The fitresults and uncertainties for the modified 2-parameter model with Emax = 7keVr,nr are listed in Table 4.7.1014.8. Final ResultsFigure 4.44: Comparison between best fits to 2-parameter Lindhard modelswith original value of Ehigh, vs. Ehigh=7 keVr,nrTable 4.7: Summary of fit results and uncertainties for the modified 2-parameter Lindhard fit, with Emax=7 keVr,nr.Information klow khigh ρ(klow, khigh)Best Fit 0.0575 0.177 N/AStat. Unc. (Simulation) 0.0018 0.0025 -0.712Stat. Unc. (Experimental) 0.0069 0.013 -0.505Sys. Unc. (Fano Factor)(-0.0017,+0.00044)(-0.00063,+0.00042)-0.901Given the strong linear k variation found with the 2-parameter fit, it appearsthat the data prefers some energy-dependent variation in k rather than asingle k value in this region. The relative goodness of fit of the 1-parameterand 2-parameter models is evaluated by binning the experimental data andcalculating the fit χ2 over degrees of freedom for each data set. Fig. 4.45shows some sample plots for this comparison.1024.8. Final ResultsFigure 4.45: Sample plots showing the binned experimental neutron-on data,overlaid with the best-fitting PDF based on the combined negative log like-lihood fit. The χ2 goodness-of-fit parameter is computed in the “neutronregion”, which is the full energy region in which any of the three simulatedneutron spectra contain nuclear recoil events. The χ2 is compared with thenumber of degrees of freedom (dof), where the dof is set as the number ofbins in the neutron region, minus the total number of parameters fit for inthe negative log likelihood fit. For the most part, the fit uncertainty shownin cyan is too small to see. Left: Y at 70V neutron-on data, overlaid withthe best-fit PDF from the combined fit to the 1-parameter Lindhard model.Right: Y at 25V neutron-on data, overlaid with the best-fit PDF from thecombined fit to the 2-parameter Lindhard model.Table 4.8 summarizes the χ2 calculations for all three data sets, as well asthe total result for all three data sets combined. The significant reductionin the total χ2 compared with the number of degrees of freedom in goingfrom the 1-parameter to the 2-parameter fit suggests that the addition oflinear variation to the k value significantly improves the model’s ability torepresent the experimental data within the range of ∼0.39 to 7 keVr,nr.Table 4.8: Summary of χ2 results for each data set in the “neutron region”,compared with the number of degrees of freedom. See the caption of Fig.4.45 for the definition of the “neutron region”. The 2-parameter fit uses thereduced energy range of ∼0.39 to 7 keVr,nr.Fitχ2 / dof (Sbat 70V)χ2 / dof (Yat 70V)χ2 / dof (Yat 25V)χ2 / dof(Total)1-Parameter 212.3 / 146 273.6 / 146 168.3 / 146 654.2 / 4382-Parameter 206.3 / 145 194.1 / 144 155.3 / 145 555.8 / 4341034.8. Final ResultsHowever, the relatively high χ2 values compared with the number of degreesof freedom for both fits suggests that the fit remains imperfect, which maybe due to one or more of the following factors:1. Further refinements to the neutron simulation, and implementation ofthe cut efficiencies and voltage map in the negative log likelihood fitmay be needed to better model the data.2. Limitations in the ability of the parameterized Lindhard models toaccurately represent the true energy dependence of the ionization yieldin this region.3. Imperfect modelling of the source-induced γ background by the neutron-off spectrum. The assumption that the neutron-off background spec-trum “perfectly” – i.e. up to statistical fluctuations – represents theγ background induced by the radioactive source seems reasonable inprinciple, especially given that the data-taking was alternated weeklybetween neutron-on and neutron-off conditions.While changes such as varying the thickness of lead shielding weremade to the experimental conditions prior to switching the data-takingbetween source types or detectors, we are not currently aware of anychanges made while a given source was in place that would have sys-tematically altered the shape of the γ background between neutron-onand neutron-off data-taking. However, recent work has suggested thatthe above-statistics fluctuations modelled by residual smoothing maynot be fully time-independent from week-to-week, which could lead toimperfect agreement between the neutron-on and neutron-off gammabackgrounds.4. Imperfect simulation of the neutron spectrum. While great care wastaken to ensure that the simulation details reflect the experimentalsetup as closely as possible, it is difficult to reject the possibility thatsystematic differences between the simulated and experimental condi-tions, or systematic errors associated with Geant4 library data, couldhave produced some systematic discrepancy between the physical andsimulated neutron spectra.Currently, the only systematic uncertainty that will be considered inassociation with the Geant4 library data is in the neutron-Ge elasticscattering cross section (see Section 4.7.5). This uncertainty has notyet been evaluated.104Chapter 5ConclusionsThis thesis has presented an analysis of energy scale calibration data mea-sured with SuperCDMS detectors. Preliminary results from a model-dependentnegative log likelihood fit of the experimental energy spectrum to the summedbackground and simulated neutron spectra indicate that the data is sensi-tive to the energy-dependent ionization yield in the approximate nuclearrecoil energy range of 0.39 to 7 keVr,nr. Within this range, leaving k inthe Lindhard ionization yield model as an energy-independent fit parameteryields a best-fitting k value of 0.1053 ± 0.0035+0.000021−0.0023 , which is relativelylow compared with the range of 0.156-0.160 predicted by Lindhard theoryand confirmed by experiment above ∼50 keVr,nr.When k is allowed to vary linearly within the sensitivity range of 0.39 keVr,nr <Ep < 7 keVr,nr, it is found that the data prefers a significant k variationwithin this range, rising from a low-energy value of 0.0575 ± 0.0072+0.00044−0.0017at 0.39 keVr,nr, to a high-energy value of 0.177±0.013+0.00042−0.00063 at 7 keVr,nr.The observed energy dependence of the fitted k value is not unexpected giventhe poor agreement of results from past ionization yield measurements inGe with the Lindhard model in this low energy region.Several refinements of the analysis are in progress, such as inclusion of thecut efficiency functions and detector voltage maps into the negative log like-lihood fit. 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Physical Review, 91(5), 1079-1084.doi:10.1103/physrev.91.1079115Appendix AThis appendix proves the assertion made in Section 4.4.2 when discussingthe TriggerBurst cut that the shift ∆x¯x¯ in the mean x¯ due to an outlier varieson average with the number of bins N as:∆x¯ ∝ 1N(A.1)The mean x¯ of the set of N-1 data points without the outlier is givenby:x¯ =1N − 1N−1∑i=1xi (A.2)Suppose the outlying data point, located a distance d above the mean,is added back to the data set. The mean value x¯new of the data set isnow:x¯new =1N(N−1∑i=1xi + [d+ x¯])=1N(N−1∑i=1xi +[d+1N − 1N−1∑i=1xi])(A.3)x¯new =1N(1 +1N − 1)N−1∑i=1xi +dN=1N − 1N−1∑i=1xi +dN= x¯+dN(A.4)The change in the mean value of the data set due to the addition of theoutlier is therefore:∆x¯ = x¯new − x¯ = dN(A.5)116Appendix A.So on average, the expectation is that if the number of points in the sampleis changed from N1 to N2, the relative change in the mean due to an outliershould vary as:∆x¯N2 = ∆x¯N1(dN2)(dN1) = ∆x¯N1N1N2 (A.6)117Appendix BThis appendix discusses the propagation of the experimental uncertaintyand covariance associated with the Lindhard yield model parameters to theenergy-dependent ionization yield.B.1 1-Parameter FitFor the 1-parameter fit, the energy-independent k value is the only freeparameter in the Lindhard model, given by Eq. 4.2.The propagation of uncertainty in the 1-parameter fit is relatively straight-forward, since there is no covariance to consider. In this case, the uncertaintyin k is propagated to ionization yield as:∆Y (Er,nr) =∂Y∂k∆k =g(1 + kg)− kg2(1 + kg)2∆k (B.1)where ∆k is the combined uncertainty on the Lindhard model parameterfrom all sources.B.2 2-Parameter FitIn the 2-parameter fit, k is allowed to vary linearly with energy over alimited energy range from its low-energy value of klow to its high-energyvalue of khigh, as shown in Eq. 4.27.With two model parameters, it is necessary to consider the covariance C=cov(klow, khigh)118B.2. 2-Parameter Fitbetween the model parameters:∆k(Er,nr) =∆klow if Er,nr < Elow√(∂k∂klow∆klow)2+(∂k∂khigh∆khigh)2+ 2(∂k∂klow)(∂k∂khigh)C if Elow ≤ Er,nr ≤ Ehigh∆khigh if Er,nr > Ehigh(B.2)where∂k∂klow= 1− Er,nr − ElowEhigh − Elow (B.3)and∂k∂khigh=Er,nr − ElowEhigh − Elow (B.4)119Appendix CThis appendix describes how the excess width of spectral peaks reported in[90] for neutrons scattering in Si crystals may be interpreted as an indicationthat the Fano factor for nuclear recoils in Si could be significantly larger thanthe electron recoil Fano factor of ∼ 0.12 [89] for recoil energies up to 7.50keV.In the experiment described in [90], a Si(Li) crystal detector is bombardedwith neutrons of all energies up to a maximum set by the energy of theproton beam used for neutron production. “Edges” appear in the result-ing ionization spectra due to resonances in the neutron-Si elastic scatter-ing cross section. These edges are transformed into spectral peaks using asimple differentiation filter. The expected widths of these peaks take intoaccount the electronics noise and the scattering-resonance widths. The ob-served widths, to which small (O(1%)) reductions are applied to correct forany oversmoothing of the differentiation filter, are consistently larger thanexpected, as reported in Table 1 of the publication.Assuming that all other possible sources of fluctuation were appropriatelyaccounted for to obtain the expected peak widths, the excess fluctuationsproducing unexpectedly large widths could potentially be explained by con-sidering the variation in the number of electron-hole pairs produced by aneutron-Si nuclear recoil event, as quantified by the nuclear recoil Fano fac-tor.The excess width σexc is determined from the quadrature difference of theobserved (σob) and expected (σexp) widths.σexc =√σ2ob − σ2exp (C.1)Since the widths are reported in the publication as full width at half maxi-mum (FWHM), the following conversion factor is first applied to obtain the120Appendix C.equivalent 1-σ standard deviation:σ =FWHM2√2 ln 2(C.2)The energy required to ionize a single electron-hole pair in Si is approxi-mately 3.6 eV [91]. Therefore, an observed ionization energy Ei would beexpected to arise from N ionized electron-hole pairs on average, accordingto:N =Er3.6 eV(C.3)The excess energy fluctuation σexc would correspond to a fluctuation ∆Nin the number of ionized electron-hole pairs, given by:∆N =σexc3.6 eV(C.4)∆N is then related to the Fano factor FNR for nuclear recoils in Si accordingto:∆N =√FNRN =⇒ FNR = (∆N)2N(C.5)Since it was determined in Section 4.8 that the data is only sensitive to theLindhard model up to ∼7 keV, only the two lowest-energy results reportedat 4.15 keV and 7.50 keV in Table 1 of [90] are considered. Using thereported values of Ei, σobs, and σexp, the Fano factors are calculated usingthe above procedure to obtain FNR = 2.52 at 4.15 keV and FNR = 6.10 at7.50 keV. The trend of increasing FNR with recoil energy continues for thethree calibration points above 7.50 keV.121

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