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The Effect of ionization variance on nuclear-recoil dark matter searches

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Title:
The Effect of ionization variance on nuclear-recoil dark matter searches
Creator:
Matheny, Mitchell Douglas
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Denver, CO
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University of Colorado Denver
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English

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Master's ( Master of integrated science)
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University of Colorado Denver
Degree Divisions:
College of Liberal Arts and Sciences, CU Denver
Degree Disciplines:
Integrated science
Committee Chair:
Roberts, Amy
Committee Members:
Villano, Anthony
Carey, Varis

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University of Colorado Denver
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THE EFFECT OF IONIZATION VARIANCE ON NUCLEAR-RECOIL DARK MATTER
SEARCHES
by
MITCHELL DOUGLAS MATHENY BS, University of Colorado Denver, 2016
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfilment of the requirements for the degree of Master of Integrated Sciences Integrated Sciences Program
2019


11
This thesis for the Master of Integrated Science degree by Mitchell Douglas Matheny has been approved for the Integrated Sciences Program by
Amy Roberts, Chair Anthony Villano, Advisor Varis Carey, Advisor
Date: August 3rd, 2019


Matheny, Mitchell Douglas (MINS, Integrated Sciences)
The Effect of Ionization Variance on Nuclear-Recoil Dark Matter Searches Thesis directed by Assistant Professor Amy Roberts
m
ABSTRACT
Direct detection dark matter experiments are increasingly interested in the low-mass dark matter parameter space, but zero-background low-mass searches require event separation between the electron and nuclear recoil bands, which requires a proper understanding of detector energy reconstruction.
Previous simulations have shown that we do not entirely understand the ionization efficiency (yield) for electron and nuclear recoils, as the assumption that the distribution for the yield is normally distributed for a true recoil energy is violated. Since the yield distribution may directly affect dark matter low-mass limits, it is crucial we understand how the yield is distributed.
A component to understanding the yield distribution is the variance in the number of electron-hole pairs produced or ionization variance. This effect has been studied relatively infrequently as experiments have been interested in large energy deposits (10 - 100 keV) and could accurately separate electron and nuclear recoil events. For electron recoils, the ionization variance is described by a “Fano” factor. For nuclear recoils the effect can be parameterized by an “effective” Fano factor, which has similar definition but a different physical origin. The nuclear recoil “effective” Fano factor is shown to be much larger than the electron-recoil version above around 10 keV deposited energy.
The form and content of this abstract are approved. I recommend its publication.
Approved: Amy Roberts


IV
TABLE OF CONTENTS
I Dark Matter 1
1.1 Introduction............................................................. 1
1.2 Velocity Profile and Galactic Rotation Curves......................... 1
1.3 Gravitational Lensing.................................................... 3
1.4 Dark Matter Candidates................................................... 5
1.4.1 Axions............................................................ 5
1.4.2 Sterile Neutrinos................................................. 6
1.4.3 Weakly Interacting Massive Particles.............................. 7
1.5 Direct Detection of WIMP Dark Matter .................................... 8
1.6 Lindhard Model.......................................................... 11
1.7 Simulated Recoil Bands and Limits....................................... 13
1.8 Ionization Variance..................................................... 15
II Simulation: Recoil Band Structure 20
2.1 Charge and Phonon Resolution............................................ 20
2.1.1 Recoil Bands from Data........................................... 21
2.2 Simulated Recoil Bands ................................................. 22
2.2.1 Electron Recoils: No Fano Factor................................. 22
2.2.2 Containment fraction............................................. 23
2.2.3 Nuclear Recoil Band.............................................. 27
2.3 Fano Factor in Resolutions.............................................. 32


V
2.4 Fano Factor: Varying Number of Electron-Hole Pairs....................... 37
2.5 Test of Normality........................................................ 42
III Simulated Yield Distribution Analysis 48
3.1 Two Independent Normal Distributions..................................... 49
3.1.1 Model Validation.................................................. 50
3.2 Two Dependent Normal Distributions....................................... 53
IV Impact on Dark Matter Searches 56
4.1 Eq EP Space.............................................................. 56
4.2 WIMP Mass Accessible..................................................... 60
4.2.1 Effect on Dark Matter Limit..................................... 62
Appendices 63
A Simulation Algorithim 64
1.1 Yield Algorithm: VI...................................................... 64
1.2 Yield Algorithm: V2...................................................... 65
B Distribution Plots 67
2.1 Electron Recoils: Fano = 0.13.......................................... 67
2.2 Nuclear Recoils: Er Dependent Fano Factor................................ 72
C Yield Distribution
77


LIST OF TABLES
vi
LIST OF TABLES
2.1 Detector resolution coefficients a, (3, and 7 are found from lines of best fit in
[29] for Detector 1...................................................... 21
2.2 Containment fraction for Electron recoil band. The expected 68% contain-
ment fraction is under the assumption that the yield for electron recoils is normally distributed..................................................... 26
2.3 Containment fraction for Nuclear recoil band simulated with no Fano factor.
The expected 68% containment fraction is under the assumption that the yield
for nuclear is normally distributed...................................... 29
2.4 Containment fraction for electron recoil band with a Fano factor of 0.13. The
expected 68% containment fraction is under the assumption that the yield for electron recoils is normally distributed................................. 39
2.5 la Containment fraction for Nuclear recoil band simulation using EDELWEISS parameterized Fano factor. The expected 68% containment fraction is under the assumption that the yield for nuclear recoils is normally distributed. 39
2.6 Results of the distribution analysis for all 8 logarithmically-spaced bins from
10-110 keV for the electron recoils. P-value represents significance for Shapiro-Wilk and Kolmogorov-Smirnov tests for normality. Any P-value < 0.05 indicates rejection of the null hypothesis that the distribution is normal. 44


2.7 Results of the distribution analysis for all 8 logarithmically-spaced bins from 10-110keV for nuclear recoils. P-value represents significance for Shapiro-Wilk test for normality, any P-value < 0.05 indicates rejection of the null hypothesis that the distribution is normal.......................................


Vlll
LIST OF FIGURES
1.1 Rotation curves of spiral galaxies showing the rotational velocity of astrophys-ical bodies as a function of their distance from the center of the galaxy. Solid
line is the expected velocity, dots are from observations [1]............. 3
1.2 (Left) Optical images from the Magellan telescope with overplotted contours of spatial distribution of mass, from gravitational lensing . (Right) The same contours overplotted over Chandra x-ray data that traces hot plasma in a galaxy. It can be seen that most of the matter resides in a location different from the plasma (which underwent frictional interactions during the merger
and slowed down) [2]...................................................... 4
1.3 Expected WIMP spsctrum for 3 potential masses. X-axis represents energy deposited in the detector with the x-intercept representing the maximum amount of energy transfer possible for the given mass. The shaded region around each
line represents the 90% confidence regions................................ 8
1.4 Sketch of a SuperCDMS iZIP detector showing charge and phonon propaga-
tion. Here, the liberated charge drifts across the detector due to an applied voltage. The liberated charge excites “prompt” phonons (also known as primary phonons) which get collected by phonon sensors....................... 9
1.5 Ionization Yield vs recoil energy from Californium calibration data. Top band represents the electron recoil band, bottom band represents nuclear recoil band. Solid Black bands represent 2.5 11


IX
1.6 Comparison of expected ionization yield to measurements made by several
experiments showing a deviation from theory for most experiments......... 13
1.7 (Top) Simulated electron and nuclear recoil bands and a function of recoil energy (Pr) found in [22], (Bottom) Electron and nuclear recoil bands from Cf calibration data. Blue and red bands are formed by fitting the ionization yield (y-axis) with a Gaussian and calculating the 2.5 and plotting as a function of recoil energy.............................. 14
1.8 Projected exclusion sensitivity for the SuperCDMS SNOLAB direct detection
dark matter experiment. The vertical axis is the spin-independent WIMP-nucleon cross section under standard halo assumptions, and the horizontal axis is the WIMP mass,where WIMP is used to mean any low-mass particle dark matter candidate. The blue dashed curves represent the expected sensitivities for the Si HV and iZIP detectors and the red dashed curves the expected sensitivities of the Ge HV and iZIP detectors [23]....................... 15
1.9 Predicted effective Fano factor for 2 assumptions made by Lindhard. Also shown, measurements made by Doughetry in slicon [21, 26]. Figure courtesy
of Anthony Villano....................................................... 17
1.10 Electron and nuclear recoil bands from data taken during a 252Cf calibration.
Red and blue lines represent 1.625 2.1 Simulated electron recoil band with no Fano factor. Red dashed line represents
the mean yield Y = 1. Black bands represent la containment for electron recoil events from SuperCDMS [30]......................................... 24
2.2 Simulated electron recoil band with data generated using true recoil energies that are logarithmic spaced from 10—110 keV with no Fano factor. Red dashed line represents the mean yield Y = 1. Black bands represent la containment for electron recoil bands from SuperCDMS [30]
25


X
2.3 Containment fraction for electron recoil band with no Fano factor. As shown,
the percent of data within la varies from the expected 68%.............. 27
2.4 Simulated nuclear recoil band with Fano factor = 0. Red dashed line repre-
sents the mean defined by equation 2.13. Black bands represent 1 a containment for nuclear recoil bands from CDMS [30]...................... 29
2.5 Simulated Nuclear recoil band with Fano factor = 0 binned into 8 logarith-
mically spaced bins. Red dashed line represents the mean yield. Black bands represent la containment for nuclear recoil bands from CDMS [30].. 30
2.6 Containment fraction as a function of energy for the simulated nuclear recoil
band with Fano factor = 0. Red dashed lined represents expected containment fraction if the yield is normally distributed..................... 31
2.7 Top: Simulated electron recoil band with Fano factor = 0.13 included in the
resolutions. Black bands represent la containment bands derived from fitting data. Red dashed line represent mean of recoil band. Bottom: Containment fraction for simulated electron recoil band with Fano factor of 0.13.... 34
2.8 Nuclear recoil Fano factor vs. Recoil energy............................ 35
2.9 Top: Simulated nuclear recoil band with Fano factor paramerterized from
[25] included in the resolutions. Black bands represent la containment bands derived from fitting data. Red dashed line represent mean of recoil band. Bottom. Containment fraction for simulated nuclear recoil band with Fano factor parameterized from [25].......................................... 36
2.10 Top. Simulated nuclear recoil band with Fano factor of 0.13. Black bands
represent la containment bands derived from fitting data. Red dashed line represent mean of recoil band. Bottom. Containment fraction for simulated electron recoil band. [25].............................................. 40


2.11 Top. Simulated nuclear recoil band with Fano factor paramerterized from
[25] included in the resolutions. Black bands represent 1 a containment bands derived from fitting data. Red dashed line represent mean of recoil band. Bottom. Containment fraction for simulated nuclear recoil band with fan factor parameterized from [25].............................................
2.12 a) Histogram of the electron recoil band yield for the energy bin centered at
15.74 keV. Overlaid is a Gaussian distribution. As shown, there is evidence
for a positive skew, as the distribution has a slight tail on the right side, b) QQ plot. The upward curve in the data indicates a positive skew. To see the histograms and QQ plots for each of the 8 energy bins, please refer to the Appendix B.................................................................
2.13 a) Histogram of the nuclear recoil band yield for the energy bin centered at
15.74 keV. Overlaid is a Gaussian distribution. As shown, there is evidence
for a positive skew, as the distribution has a slight tail on the right side, b) QQ plot. The upward curve in the data indicates a positive skew. To see the histograms and QQ plots for each of the 8 energy bins, please refer to the Appendix B.................................................................
3.1 a. Containment fraction for electron recoil band. b. Containment fraction for
nuclear recoil band. Both generated with a fano factor included in the charge and phonon resolutions. Black dotted line represents expected containment fraction predicted by the PDF in equation 3.4..............................
3.2 a. Containment fraction for electron recoils, b. containment fraction for
nuclear recoils. Both generated with a fano factor included in the charge and phonon resolutions. Black dotted line represents expected containment fraction predicted by the PDF in equation 3.6..............................


xii
4.1 Simulated total charge energy (Eq) and total phonon energy (EP) plane for electron and nuclear recoils. Simulated for recoil energies ranging from 0 to
20 keV...................................................................... 57
4.2 Eq/Ep plan simulated with a Fano factor of 0. Black band represents electron
recoils, blue band nuclear recoil bands. Orange and black points represent location of the 2a mark for each bin. Data points are for to a line to form lower bound for electron recoils and upper bound for nuclear recoils. The red cross represents the intersection of the 2a bands........................... 58
4.3 Eq/Ep plan simulated with a Fano factor of 100. Black band represents elec-
tron recoils, blue band nuclear recoil bands. Red and black points represent location of the 2a mark for each bin. Data points are for to a line to form lower bound for electron recoils and upper bound for nuclear recoils. The red cross represents the intersection of the 2a bands .......................... 59
4.4 Minimum mass accessible vs nuclear recoil Fano factor. Mass calculated using recoil energy corresponding to the intersection point in figures such as Figure
4.3......................................................................... 61
4.5 Projected exclusion sensitivity for the SuperCDMS SNOLAB direct detection dark matter experiment. The vertical axis is the spin-independent WIMP-nucleon cross section under standard halo assumptions, and the horizontal axis is the WIMP mass, where WIMP is used to mean any low-mass particle dark matter candidate. The blue dashed curves represent the expected sensitivities for the Si HV and iZIP detectors and the red dashed curves the expected
sensitivities of the Ge HV and iZIP detectors [23].......................... 62
2.1 Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center
11.7 keV.................................................................... 67
2.2 Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center
15.7 keV.................................................................... 68


Xlll
2.3 Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center
21.3 keV............................................................... 68
2.4 Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center
28.8 keV.............................................................. 69
2.5 Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center
39.4 keV................................................................ 69
2.6 Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center
53.1 keV................................................................ 70
2.7 Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center
70.4 keV................................................................ 70
2.8 Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center
95.1 keV................................................................ 71
2.9 Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center
11.7 keV................................................................ 72
2.10 Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center
15.7 keV................................................................ 73
2.11 Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center
21.3 keV................................................................ 73
2.12 Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at vbin center
28.8 keV................................................................ 74
2.13 Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center
39.4 keV................................................................ 74
2.14 Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center
53.1 keV................................................................ 75
2.15 Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center
70.4 keV
75


XIV
2.16 Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center
95.1 keV
76


I DARK MATTER
1
CHAPTER I DARK MATTER
1.1 Introduction
Dark matter is one of the most mysterious problems cosmologists are faced with today. If we could understand the origins and properties, we could clarify many observations, and strengthen our limited understanding of the universe. This thesis will focus on understanding how to simulate the electron and nuclear recoil events accurately. As motivation for content covered later in this document, Chapter I will briefly outline the evidence for dark matter, and describe in detail how energy deposited inside a detector is reconstructed. This chapter will also shed light on previous evidence for simulating the nuclear recoil band incorrectly, and introduce how a rarely explored phenomenon, the ”Fano factor”, and how it might affect the projected dark matter limits in terms of a minimum detectable dark matter mass.
1.2 Velocity Profile and Galactic Rotation Curves
A galaxy is a gravitationally bound system of stars, gas, dust, and as we now believe, dark matter. By studying the velocities of stars distributed throughout galaxies, the cosomological community now largely agrees that dark matter in an important component [!]•
The circular velocity of stars can be measured as a function of distance from the center of the galaxy and using the virial theorem, we can get a good approximation of what


I DARK MATTER
2
this profile should look like. The virial theorem states that for a bound system a star gravitationally bound to a galaxy, the average kinetic energy is:
(KE) = -f Where V is the gravitational potential energy of the star and Ms is the mass of start. For a spiral galaxy,
(V)
GMtot
R
(1.2)
where Mtot is the total mass of the galaxy enclosed at the position at the star, and R is the distance that the star is away from the center of the galaxy. Using the standard definition for kinetic energy, the velocity of the star is:
GMi
tot
R
(1.3)
The first observational evidence against this formulation for the velocity of stars was found by Fritz Zwicky while observing the coma cluster. Fie discovered that the virial mass, Mtot) and the luminous mass (found by using the number of galaxies and a mass to light ration conversion) differed by a factor of 500, implying that most of the mass was a form of “dark matter” [2],
For rotation curves, from Equation 1.3, we expect 1 vs oc \J~ji, but looking at Figure 1.1, we can see that this is not the case. The velocity profile flattens instead of dropping as expected. This flattening of the rotation curve indicates that the mass within a radius R from the center of the galaxy obeys a scaling law :
M(R)
VjlatR
G
(1.4)
Tor radi near apparent (visible) edge of the galaxy.


I DARK MATTER
3
Figure 1.1: Rotation curves of spiral galaxies showing the rotational velocity of astrophysical bodies as a function of their distance from the center of the galaxy. Solid line is the expected velocity, dots are from observations [1]
Assuming the mass is spherically distributed
pm
4°i
iirGR2
(1.5)
This indicates that the mass seems to increase linearly as a function of distance beyond what can be visibly seen, and has a density that goes as far from the edge2 [3]. This gave rise to the view that a sphere of dark matter resides in a halo surrounding the disk of visible stars and gas in a galaxy.
1.3 Gravitational Lensing
General Relativity predicts that light should deflect, or bend, around a gravitational potential created by a large mass, thus creating a lens. The simplest example of a gravitational lens is a Schwartzchild lens [4], The angular deflection around a point like mass is
4GM
rc2
(1.6)
2 Our derivation differs slightly from that of the reference as they include a shaping factor where here we assume spherical symmetry.


I DARK MATTER
4
where M is the mass of the lens and r is the distance from observer and the source [5]. Assuming a simple treatment in which the object of interest is directly behind the lens and the same distance away from that the lense that the observer is, we can find the mass of the lens. Leaving the derivation to [3], the mass is
Miens
c2dta.n(0obs)(^ ~ ‘20obs) 4 G
(1.7)
Where d is the distance to the lens, c is the speed of light, G is the gravitational constant, and 0obs is the observed angle of deflection. Experiments have observed that the observed angle of deflection is to great for the amount of luminous mass contained in a galaxy, once again indicating the presence of dark matter. It is interesting to note that this simple treatment of a point like mass breaks down when we consider a continuous mass distribution, but this treatment was well suited for one of the more famous confirmations of General Relativity when Arthur Eddington went on an expedition to measure the deflection of light around the sun in 1919 [6].
6l’5am42s
6I’58'"42‘
Figure 1.2: (Left) Optical images from the Magellan telescope with overplotted contours of spatial distribution of mass, from gravitational lensing . (Right) The same contours overplotted over Chandra x-ray data that traces hot plasma in a galaxy. It can be seen that most of the matter resides in a location different from the plasma (which underwent frictional interactions during the merger and slowed down) [2].
An interesting property of dark matter is that it does not interact with itself. Gravitational lensing presents strong evidence for this observation. A cluster named the “Bullet Cluster” contains the remains of two sub clusters after they collided. Figure 1.2


I DARK MATTER
5
shows the cluster after the merger in the visible spectrum (left plot) and in xray (right plot). Looking at the left plot, there is no evidence of dark matter (as expected), but looking at right plot, we can see that the amount of lensing, shown as the green contours, is displaced from where the heated gas is. This observation indicates that when the two two sub clusters collided, most of the mass (dark matter) passed on by without interacting, reinforcing the notion that dark matter does not interact with itself [2],
1.4 Dark Matter Candidates
The observations mentioned in the previous sections are just a small example of many observations and theories that support the existence of dark matter, all of which rely on gravitational interactions. Though there are several dark matter candidates, whether dark matter interacts via the other fundamental forces is currently pure speculation as no experimental evidence exists to support the claim that it does. Currently, dark matter is thought to be composed of non-baryonic material that has mass and does not interact significantly with radiation or ordinary matter [5]. Many dark matter searches are focused on identifying a particle that is the lightest supersymmetric particle theses dark matter particles are called weakly interactive massive particles (WIMPs). Other candidates such as Axions and sterile neutrinos also exist. In this section, I will give a brief introduction into theses candidates.
1.4.1 Axions
Axions, perhaps a less popular candidate for dark matter, is a spinless, electrically neutral, very light particle which was initially introduced as a solution to the strong charge conjugation parity symmetry problem (strong CP problem)[7]. The particle’s mass, Ma is dependent upon its decay constant fa.


I DARK MATTER
6
Ma
QjieV
1012GeV
fa
Since the energy density of cold axions is
(1.8)
Vah2
(
fa
1012GeV
(1.9)
if the mass is O(10fieV) [8], the axion could be a candidate for cold dark matter as Qah2 could account for the theorized amount of dark matter in the universe.
The Axion Dark Matter experiment (ADMX) is working to detect axions that have converted to photons in a microwave cavity permeated by a magnetic held [9]. With plans to look into the 10fieV range, ADMX results have already excluded low mass axions within the range 1.9/ieV < Ma < 3.69/ieV [10, 11].
1.4.2 Sterile Neutrinos
Sterile Neutrinos are hypothetical particles that are dark matter candidates as they may be heavy and only interact very weakly with other particles. A sterile neutrino J\f, is an example of decaying dark matter. Through its mixing with the ordinary neutrinos, J\f can decay (via Z boson exchange) into three antineutrinos,
M -+v + u + u (1.10)
and a more constraining decay channel where it decays into a neutrino and a photon.
M —> v + 7 (1.11)
To be dark matter, the lifetime of M should be greater than the age of the Universe [12]. This constraint on the life time of M means that the mass of the particle(s) need to be Mdm > 400eU [12].The possible decay channels means that it should be detectable. If the


I DARK MATTER
7
mass of M is constrained to be OAkeV < Mdm < 10keV the energy of the photon should be in x-ray spectrum. Experiments such as XMM-Newton, Chandra, and INTEGRAL are looking for such decays [12, 13, 14], Though J\f has not been found, and upper bound on the mass has been made, Mdm < 4keV.
1.4.3 Weakly Interacting Massive Particles
The search for dark matter via direct detection techniques has been motivated by the popularity of Supersymmetric models with a stable lightest Supersymmetric particle as their dark matter candidate. For the last 20 years, the focus of these searches has been on ‘Weakly Interacting Massive Particles’ or WIMPs. WIMPs are expected to be electrically neutral, have a mass somewhere between lOGeV and 100 TeV and should interact with ordinary matter via the weak nuclear force, giving a small but non negligible coupling to standard model particles [15]. WIMPs are proposed to have decoupled from equilibrium with Standard Model particles once the rate of conversion between WIMPs and other particles became less than the expansion rate of the Universe. Previously, neutrinos were originally proposed as a WIMP candidate, but the three known neutrinos do not have enough mass to account for the current estimated dark matter density [16].
The rate at which WIMPS are expected to interact with nuclei in a detector is given by:
dR NrMr
f§aer) + 7~hcdo
(1.12)
dEr 2 Mw/i2
where Mw is the mass of the wimp, MT is the mass of the target nuclei,// is the reduced mass of the system, Nt is the number of nuclei in the target, and Er is the energy of the nuclear recoil. aSD ,aSI,FSd ,and FSI are the spin dependent and spin independent cross section and nuclear form factors respectively [17]. Thaio is the halo-model form factor, and depends on the the velocities of the WIMPS in the halo which are usually assumed to take


I DARK MATTER
8
on a standard Maxwellian velocity distribution. Figure 1.3 shows the expected recoil spectra for three WIMP masses.
101
&
bO
>
oj
K
10"
10
-l
fL,
5 10_2
1(T3
Recoil Energy [keV]
Figure 1.3: Expected WIMP spsctrum for 3 potential masses. X-axis represents energy deposited in the detector with the x-intercept representing the maximum amount of energy transfer possible for the given mass. The shaded region around each line represents the 90% confidence regions.
1.5 Direct Detection of WIMP Dark Matter
The primary candidate for direct detection dark matter are WIMPS, and they are expected to interact with the nuclei of the detector. Detecting WIMP dark matter requires a highly sensitive particle detector. The SuperCDMS collaboration uses cryogenically cooled interdigitated Z-sensitive ionization and phonon detectors (iZIPs)made of germanium. These detectors are capable of measuring charge liberated and phonons created through the use of electrodes and superconducting transition-edge sensors.
When a particle interacts/deposits energy inside a detector, it has two modes of interaction: interacting with the electrons bound the the target’s atoms (electron recoils)


I DARK MATTER
9
or scattering off the target nuclei (nuclear recoils). In both cases, it liberates electron-holes pairs and produces primary phonons. The electron-hole pairs are drifted across the detector by a applied voltage. The electron-hole pairs collide with germanium nuclei and produce secondary or “Luke Phonons” [18]. A sketch of this process is visualized in Figure 1.4.
Figure 1.4: Sketch of a SuperCDMS iZIP detector showing charge and phonon propagation. Here, the liberated charge drifts across the detector due to an applied voltage. The liberated charge excites “prompt” phonons (also known as primary phonons) which get collected by phonon sensors.
The primary and secondary phonons are detected by aluminum fins that are attached to the transition-edge sensors. Any electron hole pairs that are not trapped in a vacancy within the lattice structure drift across the detector and induce charge on the ionization sensors. Determining the energy deposited inside the detector requires measurements of the charge and phonon energy:
Ep — Er + Eiuke
(1.13)
Eq Ng/fotj
where, Ep is the phonon energy, Epuke is the contribution to the total phonon energy from
secondary phonons created from drifting electrons [18]:
Epake C K Ef jj,.
(1.14)


I DARK MATTER
10
Eq is the charge energy, Ne/h is the number of electron hole pairs created, e7 is the average energy required to liberate 1 electron-hole pair, and V is the voltage applied across to the detector. The number of electron-hole pairs created can be expressed in terms of recoil energy:
Ne/h =------- (1.15)
e7
where Y is the ionization yield. The ionization yield is the fraction of energy given to the electron-hole pairs. For electron recoils Y = 1 and for nuclear recoils the yield is defined by Lindhard in [19] and will be discussed in greater detail in the next section. The reconstructed recoil energy is then:
Er — Ep--------Eq (1.16)
e7
Figure 1.5 shows the ionization yield for both types of interactions from a SuperCDMS iZIP detector. The electron and nuclear recoil events appear in horizontal bands in the plot. For this reason, we often refer to these regions as electron or nuclear recoil bands.
The plot shows the event separation abilities at energies greater than about 10 keV.


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11
Figure 1.5: Ionization Yield vs recoil energy from Californium calibration data. Top band represents the electron recoil band, bottom band represents nuclear recoil band. Solid Black bands represent 2.5 1.6 Lindhard Model
For SuperCDMS and other charge-based detectors, the question we must answer to reconstruct the energy is: For a nuclear recoil, how much energy is given up to the electronic system? In other words given Er, what is the ionization yield? In the dark matter community, the most widely used model to answer this question is the Lindhard model. The Lindhard model predicts the fraction of energy given to the electronic system, or ionization yield for nuclear recoils of a given initial kinetic energy. A dislodged nucleus (nuclear recoil) will generally stop in a short distance within a solid. Along the stopping path, the nucleus will interact with either the electronic system or other lattice atoms and generally these energy loss mechanisms compete with one another. The average energy loss due to atomic collisions, z/(e) is described by the following: [21]
2
A:e1/2z/(e) = f f (t1/2)(u(e -- v(e) + u(-)) (1.17)
Jo 2 M e e
e
E
a
2 Z2e2
(1.18)


I DARK MATTER
12
Where e is dimensionless energy, a is the Bohr radius, Z is the atomic number, e is elementary charge of an electron and t is a variable representing the energy transfer for a scatter of a nuclear recoil with energy e. Solving this equation using a f(t1/2) derived from the Thomas-Fermi Potential [21] gives the energy loss due to atomic motion that can be approximated by:
l + kg(e) (1>19'
g(e) = 3e0'15 + 0.7e°'6 + e
where k is a constant that is determined by the material of interest. For Germanium, the generally accepted value for k is 0.157. As the quantity of interest is the fraction of energy given to the electrons, or the ionization yield Y, we can use z/(e) to parameterize this quantity:
Y
t - //(e)
e
kg{t)
1 + kg(e)
(1.20)
A plot of measured values of the ionization yield (ionization efficiency in the plot) for nuclear recoils is shown in Figure 1.6.


I DARK MATTER
13
Figure 1.6: Comparison of expected ionization yield to measurements made by several experiments showing a deviation from theory for most experiments.
1.7 Simulated Recoil Bands and Limits
In direct detection dark matter experiments, simulating the electron and nuclear recoil bands is an essential component in understanding detector response. The bands have been previously simulated in [22], Figure 1.7 shows a comparison of simulated bands to bands created from Cf calibration data. Though not a perfect comparison, one can see that the simulated bands (Top) are much narrower than the bands from calibration. Understanding why the bands are narrow is important as it can effect the projected dark matter mass limits. Dark matter mass limits are used to determine what mass, and at what probability, we are expecting to be able to detect dark matter. Figure 1.8 shows current projected SuperCDMS SnoLab limits [23]. The x-axis represent the dark matter mass, and the y-axis represents the cross section, or probability of interaction. The narrow simulation bands impacts this plot by shifting where the limits fall. For example, if the simulated bands are really narrow, this could shift a limit down and to the left for the loweset WIMP masses indicating a stronger limit. This happens because narrow bands indicate that we have the ability to better discriminate between electron and nuclear recoils, meaning that we are


I DARK MATTER
14
more likely to see an interaction for a lower mass dark matter particle, as a low mass particle has a lower maximum possible energy it can deposit. In other words, the better event separation ability, the more confident we are in detecting low-mass WIMP.
The bands simulated in [22] might be to narrow due to not including the ionization variance.
Soudan T1Z2 Net 4 V Data Model
Pr TkeVl
T1ZZ R123 Cf
Figure 1.7: (Top) Simulated electron and nuclear recoil bands and a function of recoil energy (Pr) found in [22], (Bottom) Electron and nuclear recoil bands from Cf calibration data. Blue and red bands are formed by fitting the ionization yield (y-axis) with a Gaussian and calculating the 2.5

I DARK MATTER
15
-1
-2
-3
-4
-5
-6
-7
-8
_Q
CL
C
o
o
0)
00
w
{/)
o
V—
Q
O
0)
0 3
c
1
CL
5
Figure 1.8: Projected exclusion sensitivity for the SuperCDMS SNOLAB direct detection dark matter experiment. The vertical axis is the spin-independent WIMP-nucleon cross section under standard halo assumptions, and the horizontal axis is the WIMP mass,where WIMP is used to mean any low-mass particle dark matter candidate. The blue dashed curves represent the expected sensitivities for the Si HV and iZIP detectors and the red dashed curves the expected sensitivities of the Ge HV and iZIP detectors [23].
1.8 Ionization Variance
The direct-detection community has been concerned with the average ionization yield produced by a recoiling nucleus within a detector. This has largely been accepted as experiments such as CDMS, EDELWEISS and CRESST were interested in large energy deposits (10-100keV) which allowed the ability to distinguish between electron and nuclear recoils accurately. With the parameter space for WIMP shifting to lower masses, and therefore lower energy deposits, the ability to distinguish between electron and nuclear recoils becomes dependent upon the ionization variance.
For electron recoils, the variance in the number of electron hole pairs produced for a single recoil is cyv = \JFNe/hl where F is the Fano factor and Ne/h is the average number of electron hole pairs. The Fano factor is a constant that accounts for the fact that energy


I DARK MATTER
16
loss in particle collision is not purely statistical. If the variance in electron-hole pair production was purely statistical, the variance would follow that of a Poisson distribution &N = \JNe/h. For electron recoils in germanium F = 0.13 [24], For nuclear recoils, the concept of the Fano factor is that same for electron recoils as it accounts for the various ways in which a recoiling nucleus can liberate electron-hole pairs. The key difference is that there is evidence that the Fano factor for nuclear recoils is energy dependent. As far as we know, Lindhard, Doughetry, and Edelweiss [25, 21, 26] are the only ones to make a prediction/measurement for the variation in the number of electron-hole pairs produced. Lindhard predicts a variation in phonon energy fl2, but since the total energy doesn’t vary, this variation must be the same for ionization.
e = u(e) + r](e), (1.21)
where v and rj are phonon and ionization energies, respectively. Using this definition, the ionization yield becomes Y = A Using the definition of the Fano factor used previously = VFN and the definition of fractional variance we can write an effective Fano factor for nuclear recoils.
y/FN _ Q _ Q N rj Ye
NF _ Q2 ~W ~ Y U2
(1.22)
F
yn c
Figure 4.6 shows effective Fano factors for assumptions(approximations) made by
Lindhard in [21] as well as the effective Fano factor measured and predicted by Doughetry for Silicon. As shown, the effective Fano factor is significantly greater than that for electron recoils ( F = 0.13 for Germanium and F = 0.115 for silicon) as there is significant


I DARK MATTER
17
variation in the amount of energy that can be deposited in the phonon system.
Figure 1.9: Predicted effective Fano factor for 2 assumptions made by Lindhard. Also shown, measurements made by Doughetry in slicon [21, 26]. Figure courtesy of Anthony Villano.
Similarly to CDMS, EDELWEISS has the ability to directly measure the ionization yield, accept they use an term called a Quenching factor represented by Q as [25]:
Q
Ej_
Er
(1.23)
V V
Er = (1 H--)Eh----A/
e e
Here, £/ is the ionization energy, which is alalogous to Eq, and Er is the “Heat” Energy which is analogous to EP. Figure 1.10 shows the measured quenching factor Q vs. ER for both the electron and nuclear recoil band. Q is parameterized as:


I DARK MATTER
18
where Q is parameterized as:
Q = aEbr (1.24)
where Er is the recoil energy and a = 0.16, b = 0.18.
1.5
1
O
0.5
0
0 50 100 150 200
ER(keV)
Figure 1.10: Electron and nuclear recoil bands from data taken during a 252Cf calibration. Red and blue lines represent 1.625

I DARK MATTER
19
Edelweiss made measurements for the variance in the quenching factor:
(7,
Q

—Q)V, + (1 +
H
(1.25)
\ tgamma tgamma J
and V is the detector voltage, e is the energy required to create 1 electron-hole pair, aj is the variance in the charge measurement,aj1 is the variance in the heat measurement [25], aj and a2H are parameterized as:
*i(E)2 = (aj)2 + (ajE)2
(1.26)
ME)2 = K)2 + (aHE)2
where aj and a°n are the baseline resolutions and a/ and an are deduced from the resolution of the ionization and heat signals at 122 keV [25].
EDELWEISS found that distribution of data was to wide for what they expected. Atomic scattering, variation in the number of charges created by a nuclear recoil [27] and multiple scattering are expected to give an intrinsic width to the Q distribution for nuclear recoils and thus explain this behavior [28]. To account for this, they added a constant in quadrature to the resolution a2q.
^(E*) = + C2 (1.27)
By fitting data, they determined that a typical value of C is 0.04. In the next chapter, the nuclear recoil Fano factor will be parameterized in terms of the EDELWEISS resolutions.


II SIMULATION: RECOIL BAND STRUCTURE
20
CHAPTER II
SIMULATION: RECOIL BAND STRUCTURE
The goal of this thesis is to investigate the role the Fano factor has on setting dark matter mass limits. To do so, we need first to be able to simulate the bands accurately. In the sections that follow, the electron and nuclear recoil bands are simulated in three ways: First, to confirm the narrowness found in the nuclear recoil band by [22], both bands are simulated with no Fano factor. An in-depth analysis of the resulting distribution of the ionization yield is also investigated. Second, the bands are simulated by including the Fano factor in the resolutions. Lastly, I simulate the bands using a more physically accurate model, by using the fano factor to vary the number of electron hole pairs produced.
2.1 Charge and Phonon Resolution
An important concept to understand when attempting to simulate the electron and nuclear recoil bands is detector resolution as the value of the resolutions can directly affect the measured ionization yield, which is used to distinguish between the two recoil types. In this section, I give a brief description of how the charge and phonon resolutions for the CDMS iZIP detectors are found and how they will be implemented in simulating the electron and nuclear recoil band.
The charge and phonon resolutions for the iZIP detectors where found by fitting Gaussians to peaks located at 0, 10.36, 66.7, 356, and 511 keV. By comparing the 1 a width of each peak to the associated mean peak location, a functional form was fit for both the phonon and charge resolutions as a function of energy [29]:


II SIMULATION: RECOIL BAND STRUCTURE
21
(7p
Oip + [3pEp + jpEp
(2.1)
aq = \Jaq + Pq^Q + Tg-E-Q
where the values for a, (3, and 7 for Table 2.1: Detector resolution coefficients a, (3, and 7 are found from lines of best fit in [29] for Detector 1.
a P 7
dp (Jq 0.155 0.166 9.1 x 10“n 0.0023 0.00051 9.52 x 10“5
2.1.1 Recoil Bands from Data
For electron and nuclear recoils, the distribution for the ionization yield is usually modeled to be normally distributed for a given recoil energy. To aid in the analysis of the shape of the electron and nuclear recoils bands, fitted bands from SuperCDMS are used. The black ler bands pictured in Figure 2.1 are created by fitting the ionization yield from SuperCDMS data with a Gaussian and calculating the mean and standard deviation. The results from the fit are used to fold an upper U(Er) and lower L(Er) functions that define the band that take the form:
U (Er) — fIy(Er) + dy(Er)
L(Er) — Hy(Er) — dy(Er)
(2.2)
where:
fj>y(Er) = aEbr
ay(Er)
\J cEf + e Er
(2.3)
were, a , b c, d, and e are calculated from the fits [30].


II SIMULATION: RECOIL BAND STRUCTURE
22
2.2 Simulated Recoil Bands
Nuclear recoils are the primary way dark matter is expected to interact with atoms inside a detector. But before trying to simulate the nuclear recoil band, it is essential first to understand how to simulate the electron recoil band, as the electron recoil band has the convenient property that the expected ionization yield is independent of the electron recoil energy, specifically Y = 1.
2.2.1 Electron Recoils: No Fano Factor
To simulated the electron recoil band, first a “true” electron recoil energy, Eer is randomly drawn from a normal distribution of energies ranging from 10 — 100 keV. This energy is then used to calculate the “true” phonon energy, charge energy, and average number of electron-hole pairs produced Ne/h- The true phonon and charge energies are calculated assuming a perfect resolution:
Nf
EP
e/h
Ep — Eer V
e/h
(2.4)
Eq -
here Ne/h is the average number of electron-hole pairs produced, V is the applied detector voltage, and e7 is the average amount of energy needed to liberate one electron hole pair. For germanium e7 = 3.32 eV [31]. To determine a measured yield value,the measured phonon energy Ep, and the measured charge energy Eq, it is helpful to have a good understanding of what the distributions for~p and Eq are expected to look like. Both EP and Eq are expected to be normally distributed:


II SIMULATION: RECOIL BAND STRUCTURE
23
f(EQ\aQ,Y,ER) g(Ep\crp, Y, Er) =
(Eq-yer)2
2 =e <3
2
V72
(ep-ii+(^)y]er)2
> 2 (2.5)
7T <7
P
Where / and g are the PDFs for Eq and EP respectively.To simulate these normal distributions the measured charge and phonon energies are calculated by randomly sampling from normal distributions with mean Ep and Eq and standard deviations av and oq respectively:
Ep ~ N(Ep,crp(Er)) Eq ~ N[EQ,aQ(Er))
(2.6)
Using the measured values EP and Eq the measured recoil energy and the measured ionization yield can be calculated
Er
Ep —
EqV
Y
Eq
Er
(2.7)
Here Er is the measured recoil energy. The result of this simulation is shown below in Figure 2.1.
2.2.2 Containment fraction
As mentioned previously, for single recoil energy Er the yield is expected to be normally distributed and therefore 68.27% of the data should be within 1

II SIMULATION: RECOIL BAND STRUCTURE
24
Simulated Electron Recoil Band
0.6
20 40 60 80 100 120 140 160
Er( keV)
Mean 1 £7
Figure 2.1: Simulated electron recoil band with no Fano factor. Red dashed line represents the mean yield Y = 1. Black bands represent la containment for electron recoil events from SuperCDMS [30].
^ N-\U + D\ , ,
%contained — ---------- * 100 (2.8)
Here, N is the total number of data points in a specific energy bin, U is the amount of data above the upper la band, and D is the amount of data below the lower la band. This counting algorithm follows that of a binomial distribution, allowing for a simple derivation of the uncertainty in the containment fraction:
y/Np(l-p)

N
(2.9)


II SIMULATION: RECOIL BAND STRUCTURE
25
Simulated Electron Recoil Band
— - Mean
0 20 40 60 80 100 120
Er( keV)
Figure 2.2: Simulated electron recoil band with data generated using true recoil energies that are logarithmic spaced from 10 — 110 keV with no Fano factor. Red dashed line represents the mean yield Y = 1. Black bands represent la containment for electron recoil bands from SuperCDMS [30].
where N is the total number of data points within a bin, and p is the probability of success, or number of data points that lay inside the upper and lower la containment bands.
V
N-U-D
N
(2.10)
To quantify the symmetry of the distribution of each bin, the contribution from the upper and lower half of the distribution is calculated:
%uP = N ,2U * 100
N
N - 2D
yudown — * 100
(2.11)


II SIMULATION: RECOIL BAND STRUCTURE
26
The containment fraction for each bin is listed in Table 2.2 and visuallized in Figure 2.3. It can be seen that the amount of data present within \a is greater than 68% for all 8 energy bins. On average, the containment fraction differs from the expected value of 68% by 5.8%. This indicates that the distribution has a smaller width than expected based on the resolutions, confirming the observation found in [22], The last two columns in Table 2.2 look at the symmetry of the data.
Table 2.2: Containment fraction for Electron recoil band. The expected 68% containment fraction is under the assumption that the yield for electron recoils is normally distributed.
Energy Bin [KeV] % Containment Expected Percent Prom High Percent From Low
10-13.4 81.96 ±0.41 68 71.50 ±0.67 78.41 ±0.59
13.4-18.1 80.23 ±0.40 71.91 ±0.66 78.32 ±0.59
18.1-24.5 78.49 ±0.42 71.01 ±0.67 78.30 ±0.59
24.5-33.1 75.69 ±0.42 70.24 ±0.68 76.06 ±0.62
33.1-44.8 71.88 ±0.44 70.24 ±0.68 76.06 ±0.62
44.8-60.02 69.46 ±0.43 65.69 ±0.71 69.22 ±0.68
60.6-80.2 70.47 ±0.43 66.52 ±0.73 70.81 ±0.55
80.2-110.0 70.13 ±0.42 68.39 ±0.70 72.39 ±0.66
Percent from high quantifies the amount of data in the upper half of the distribution. Percent from low quantifies the amount of data in the lower half of the distribution. The percent from low being greater than percent from high indicates that the data has a positive skew.


II SIMULATION: RECOIL BAND STRUCTURE
27
la Containment Fraction for Electron Recoils
Recoil Energy [keV]
Figure 2.3: Containment fraction for electron recoil band with no Fano factor. As shown, the percent of data within la varies from the expected 68%.
2.2.3 Nuclear Recoil Band
As the primary candidate for experiments such as SuperCDMS and EDELWEISS are WIMPs, which are expected to interact primarily with the nuclei in a detector, understanding the structure of the nuclear recoil band is important. Simulating the nuclear recoil band is similar to that of the electron recoil band, except the ionization yield in no longer unity and therefore the average number of electron-hole pairs created during a nuclear recoil event is now dependent upon the ionization yield Y.
Nf
YE„
e/h
(2.12)


II SIMULATION: RECOIL BAND STRUCTURE
28
here Enr and Y is defind as the fraction of energy given to the electrons in [19] and originally by Lindhard in [21]:
y= kg(e)
1 + kg(t)
g = 3e0'15 + 0.7e°'6 + 0.6e (2-13)
where k is
k = 0.133ZizH
(2.14)
and e is the reduced energy defined by Lindhard [21] as:
— En
a
a = 0.8853
2Ze2
Q>o
(2.15)
2Zi
where a is a scaled distance, aQ is the bohr radius, Z is the atomic number of both the incoming particle and the target. ( Lindhard expresses e in terms of Z\ and Z2, here we look at the case where Z\ = Z2).
After accounting for the yield, the simulation for the nuclear recoil band is the same as in the previous section (see Equations 2.4-2.7). Figure 2.4 shows the results from simulating nuclear recoils with no Fano factor with energies ranging from 0-160 keV. Unlike the electron recoil band, it is visually obvious that the distribution of data is to narrow as almost all of the data falls within the la containment band. To further confirm this observation, that data is split into logramithcally spaced bins as before (shown in 2.6) and the percent contained within 1

II SIMULATION: RECOIL BAND STRUCTURE
29
QJ
>â– 
c
o
'â– *->
_N
"c
o
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
Recoil Energy (keV)
Figure 2.4: Simulated nuclear recoil band with Fano factor = 0. Red dashed line represents the mean defined by equation 2.13. Black bands represent la containment for nuclear recoil bands from CDMS [30].
Table 2.3: Containment fraction for Nuclear recoil band simulated with no Fano factor. The expected 68% containment fraction is under the assumption that the yield for nuclear is normally distributed.
Energy Bin [KeV] % Containment Expected Percent From High Percent From Low
10-13.4 87.09 ±1.23 68 74.33 ±2.00 79.86 ±2.01
13.4-18.1 90.10 ±1.61 76.89 ±1.93 86.34 ±2.24
18.1-24.5 94.80 ±1.58 81.80 ±1.80 91.47 ±2.10
24.5-33.1 96.91 ±1.36 90.84 ±1.25 97.13 ±2.06
33.1-44.8 98.03 ±1.23 95.62 ±0.90 98.78 ±1.84
44.8-60.02 99.40 ±1.22 98.87 ±0.42 99.83 ±1.86
60.6-80.2 99.50 ±1.27 99.45 ±0.32 99.45 ±1.81
80.2-110.0 99.60 ±1.15 99.07 ±0.41 99.81 ±1.55


II SIMULATION: RECOIL BAND STRUCTURE
30
Q)
>-
c
o
4-J
<13
_N
'c
o
f Yield —— Mean
----- 1 £7
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
Simulated Nuclear Recoil Band Model
20 40 60 80 100 120
Recoil Energy (keV)
Figure 2.5: Simulated Nuclear recoil band with Fano factor = 0 binned into 8 logarithmically spaced bins. Red dashed line represents the mean yield. Black bands represent la-containment for nuclear recoil bands from CDMS [30].


II SIMULATION: RECOIL BAND STRUCTURE
31
la Containment Fraction for Nuclear Recoils
Recoil Energy [keV]
Figure 2.6: Containment fraction as a function of energy for the simulated nuclear recoil band with Fano factor = 0. Red dashed lined represents expected containment fraction if the yield is normally distributed..


II SIMULATION: RECOIL BAND STRUCTURE
32
2.3 Fano Factor in Resolutions
As mentioned in Chapter I, the variance in the number of electron hole pairs produced can be expressed as a product of a constant of variation (the Fano factor) and the average number of electron hole pairs produced Ne/h. While simulating the electron and nuclear recoil bands, we can attempt account for this variation by adding a term to the phonon and charge resolutions:
measured charge and phonon (through the Luke effect) energies.
Looking at Figure 2.7 and comparing with Figure 2.2, we can see no visual evidence
the 60.6 keV point in Figure 2.7 bottom, there appeared to be no difference in the containment fraction. This is not all that surprising. With a flat Fano factor of 0.13, the contribution to the resolution is small.
For the nuclear recoil band, the story is a bit different. As mentioned in Chapter 1, the form of the Fano factor for the nuclear recoil band is thought not to be constant, but energy dependent. As mentioned, EDELWEISS added a constant C to account for the missing variance when comparing their calculated widths and results from data. If we assume that the extra variance in the variance of the quenching factor o2q contributed by C is due to the Fano factor, we can find the Fano factor F as a function of recoil energy and the constants a, b, C.
(2.16)
This equation accounts for the fact that the number of electron-hole pairs affects both the
that the width of the yield has been increased for electron recoils. Aside from a variation in


II SIMULATION: RECOIL BAND STRUCTURE
33
K)2 +
(2.17)
F
C2
where, Gq2 is the intrinsic detector resolutions as shown in equation 1.25. Substituting in
for Q:
F(Er, a, b, C)
C2
(2.18)
Here, a,b and C are constants found from [25]. It is important to note that this formulation of the Fano factor from Edelweiss is an approximation and assumes that the charge and phonon measurements are independent of one another and the yield is normally distributed, which is actually not the case. Figure 2.8 shows that Equation 2.18 yields a form similarity to that predicted by Linhdard in Figure 1.6.
Using this form of the Fano factor in Equation 2.18 yields the results shown in Figure 2.9. When comparing Figure 2.9 the results with Figures 2.5 and 2.6 we can see a significant difference in the width and the containment fraction. The reason for the significant difference, as seen in Figure 2.8, is the fact that the magnitude of the Fano factor increases rapidly with recoil energy and is on average 2 orders of magnitude larger than the electron recoil Fano factor.


II SIMULATION: RECOIL BAND STRUCTURE
34
Simulated Electron Recoil Band
---Mean
---- 1 a
6 20 40 60 80 100 120
ER{keV)
1.4
1.2
OJ
>-
.2 1.0
-l—i
03
N
'c
o
0.8
0.6
lcr Containment Fraction for Electron Recoils
~o £ 85- 68.27% J Percent Contain ?d
(0 “ CO c ■a c o u (D 4—1 i [
i
o 75 *4— o 4-J c
65 1 J.Q3 418 1 24 .5 33.1 44 P -8 6C .ecoil En .6 80 ergy [keV] .2 11 ).0
Figure 2.7: Top: Simulated electron recoil band with Fano factor = 0.13 included in the resolutions. Black bands represent 1 a containment bands derived from fitting data. Red dashed line represent mean of recoil band. Bottom: Containment fraction for simulated electron recoil band with Fano factor of 0.13.


II SIMULATION: RECOIL BAND STRUCTURE
35
Nuclear Recoil IFano Factor
Figure 2.8: Nuclear recoil Fano factor vs. Recoil energy.


II SIMULATION: RECOIL BAND STRUCTURE
36
Recoil Energy (keV)
(a)
lcr Containment Fraction for Nuclear Recoils
■O £ 68 — 68.27% J Percent Contain ?d

ru CD "O ro 4-1 c o u ro 4—1
o H— o 4-t c (U ^ 62
Q) Q_
10 03 .418 .1 24 .5 33.1 44 .8 6C .6 80 .2 n ).0
Recoil Energy [keV]
(b)
Figure 2.9: Top: Simulated nuclear recoil band with Fano factor paramerterized from [25] included in the resolutions. Black bands represent la containment bands derived from fitting data. Red dashed line represent mean of recoil band. Bottom. Containment fraction for simulated nuclear recoil band with Fano factor parameterized from [25].


II SIMULATION: RECOIL BAND STRUCTURE
37
2.4 Fano Factor: Varying Number of Electron-Hole Pairs
In the previous section, the extra variance due to the Fano factor was included in the charge and phonon resolutions. Though it is true that the charge and phonon resolutions are affected by this added variance, including the effective Fano factor in that manner is not entirely a physically accurate model. A more physically accurate model would be to vary the amount of electron-hole pairs produced, as a electron or nuclear recoil with a single energy will not produce the same amount of electron-hole pairs each time. In other words, an recoil of 60 keV will produce a different amount of electron-hole pairs as a different recoil of the same energy. To account for this effect, the number of electron hole pairs is sampled from a normal distribution with mean Ne/h and standard deviation cpy.
ATe/h ~ N(Ne/h,aN)
cpv — y FNe/h
(2.19)
After accounting for the variation in Ne/h, the “true” values for Eq and Ep are calculated in the same manner as before:
Ep — Eer + V Ne/h Fq Are//le7
(2.20)
Using this new method of including the Fano factor and following the same method as defined in Equations 2.4-2.7, the electron recoil band is simulated. Figure 2.10 and Table
2.4 show the results of the simulation. Comparing the results from the previous section (Figure 2.7), we can see that there is very little difference in amount of data contained in each bin.
For the nuclear recoil band, it is a different story. Figure 2.11 and Table 2.5 shows the results for the nuclear recoil band. Unlike the electron recoil band, there is a significant difference in the containment fraction when comparing to to Figure 2.9. This is due to the Fano factor increasing as a function of energy and the fact that the “true” Ep and Eq


II SIMULATION: RECOIL BAND STRUCTURE
38
values now vary correctly. In the first version of including the fano factor, by including the fano factor in the resolutions, EP and Eq varied independently, which physically is incorrect. EP and Eq should vary together with Ne/h. In otherwords, if there is an up fluctuation in the number of electron-hole pairs produced, both EP and Eq should fluctuate up. By first varying the number of electron-hole pairs created, we correctly account for this effect. The fact that EP and Eq now vary dependently decreases the possible variation in the yield, and hence the greater containment fraction.
After simulating the electron and nuclear recoil bands with a more physically accurate way of including the Fano factor, it is clear that we do not understand the correct form of the ionization yield distribution. Both SuperCDMS and EDELWEISS assume that the yield is normally distributed, however the containment fraction shows that that is not the case. In the next section, the degree in which the distribution for the yield in the electron and nuclear recoil bands deviates from a normal distribution will be investigated.


II SIMULATION: RECOIL BAND STRUCTURE
39
Table 2.4: Containment fraction for electron recoil band with a Fano factor of 0.13. The expected 68% containment fraction is under the assumption that the yield for electron recoils is normally distributed.
Energy Bin [KeV] % Containment Expected Percent From High Percent From Low
10-13.4 81.70 ±0.41 68 71.27 ±0.67 78.93 ±0.58
13.4-18.1 78.83 ±0.40 72.03 ±0.66 77.86 ±0.59
18.1-24.5 78.19 ±0.42 71.59 ±0.66 77.23 ±0.60
24.5-33.1 74.59 ±0.43 70.22 ±0.67 77.04 ±0.60
33.1-44.8 72.18 ±0.44 68.38 ±0.69 72.64 ±0.67
44.8-60.6 67.82 ±0.45 65.47 ±0.72 70.01 ±0.68
60.6-80.2 70.37 ±0.43 65.88 ±0.72 71.26 ±0.67
80.2-110.0 70.60 ±0.42 66.82 ±0.70 72.88 ±0.65
Table 2.5: la Containment fraction for Nuclear recoil band simulation using EDELWEISS parameterized Fano factor. The expected 68% containment fraction is under the assumption that the yield for nuclear recoils is normally distributed.
Energy Bin [KeV] % Containment Expected Percent From High Percent From Low
10-13.4 74.21 ±1.33 68 61.78 ±2.23 73.30 ±2.01
13.4-18.1 72.59 ±1.41 67.32 ±2.21 66.25 ±2.24
18.1-24.5 72.73 ±1.38 66.79 ±2.22 70.87 ±2.10
24.5-33.1 75.87 ±1.36 69.35 ±2.16 72.58 ±2.06
33.1-44.8 78.16 ±1.23 78.12 ±1.88 79.20 ±1.84
44.8-60.02 79.70 ±1.22 81.96 ±1.77 80.04 ±1.86
60.6-80.2 80.67 ±1.23 76.54 ±1.91 79.19 ±1.81
80.2-110.0 81.60 ±1.15 81.32 ±1.76 83.88 ±1.65


II SIMULATION: RECOIL BAND STRUCTURE
40
Simulated Electron Recoil Band
0 20 40 60 80 100 120
E«( keV)
(a)
la Containment Fraction for Electron Recoils
TJ
C
ro
CO
"O
â– E 77,
03
c
o
U
03
Q
c
cu
U 70.C Cl
— 68.27% J Percent Contain ?d








10.03.418.1 24.5
Recoil Energy [keV]
(b)
Figure 2.10: Top. Simulated nuclear recoil band with Fano factor of 0.13. Black bands represent 1 a containment bands derived from fitting data. Red dashed line represent mean of recoil band. Bottom. Containment fraction for simulated electron recoil band. [25].


II SIMULATION: RECOIL BAND STRUCTURE
41
T3
CD
>
C
o
03
N
£=
o
Yield
---Mean
---- 1 a
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
Simulated Nuclear Recoil Band Model
20 40 60 80 100 120
Recoil Energy (keV)
(a)
la Containment Fraction for Nuclear Recoils
TJ
C
ro
CO
"O
CD
c
c
o
u
03
Q
c
CD
u
CD
Cl
— 68.27% $ Percent Contained









10 03 .418 .1 24 .5 33 .1 44 .8 6C .6 80 .2 11 ).o
Recoil Energy [keV]
(b)
Figure 2.11: Top. Simulated nuclear recoil band with Fano factor paramerterized from [25] included in the resolutions. Black bands represent la containment bands derived from fitting data. Red dashed line represent mean of recoil band. Bottom. Containment fraction for simulated nuclear recoil band with fan factor parameterized from [25].


II SIMULATION: RECOIL BAND STRUCTURE
42
2.5 Test of Normality
In the previous section, the containment fraction shows that the distributions for the various energy bins for the electron and nuclear recoil band do not contain the amount of data expected from a normal distribution. The containment fraction does not however, give a sense to what degree these distributions differ from that of a normal distribution. In this section, this deviation will be explored.
Figure 2.12 and Figure 2.13 shows a histogram and a Q-Q plot 1 of the lowest energy bin (bin center at 15.75 keV) for the electron and nuclear recoil band respectivley. As shown, there is visual evidence for a asymmetry in both the histogram and the Q-Q plots.
To quantify the amount of asymmetry we can use the skew, also known as the 3rd central moment of a normal distribution, using Pearson’s skewness coefficient Gi [32], Pearson’s coefficient compares the sample to that of a symmetric distribution. If the coefficient Gi = 0, there is no skew to the distribution and if Gi deviates from zero, it indicates that the distribution has skew. The amount and direction of skew is indicated by the magnitude and sign of Gi. The larger the magnitude, the larger the skew, a negative coefficient indicates negative skew, where positive coefficient indicated positive skew. Equation 2.21 describes how the skewness is calculated. To summarize, the traditional way to represent Pearson’s skewness coefficient is use the ratio of the 2nd and 3rd moments about the mean i.e m2 = ^ J2n= iO^ — 4:)3 and mz = n Sn=iOr — 4:)2.
r 3
3
ml
iYLiix-*)2
[£ JZ=i(x ~ *)3]~2
The 2nd moment of a normal distribution is also known as the variance. Programs and
(2.21)
packages such as Excel and Numpy in Python use a version that is adjusted for the sample size of the distribution:
1A QQ (quantile-quantile) plot is a probability plot, which is a graphical method for comparing two
probability distributions by plotting their quantities against each other


II SIMULATION: RECOIL BAND STRUCTURE
43
Gi =
n
B—)
t J rr
(2.22)
(n — l)(n — 2) n ^ o
v v 7 n=1
Using Numpy, the skewness in each energy bin is calculated, the results of which are shown in Tables 2.6 and 2.7. As we can see, the Pearson’s Skewness Coefficient confirms the observations from Figure 2.12 and 2.13 that the distributions for the yield are positively skewed. To further quantify the distribution of the yield for electron and nuclear recoils, it is important to look at the Kurtosis. Kurtosis, also know as the 4th central moment of a distribution, is a quantitative way to look at the “tailedness” of the distribution.
k = - £(—)4 <2-23)
n ' a
n= 1
Values for K range from 1 — oo. The larger the value for K, the less normal, or heavy tailed, the distribution becomes. For example, a normal distribution has a value of K = 3
[33]. The results for K shown in Tables 2.6 and 2.7 indicated that the “tailedness” does not deviate much from that of a normal distribution because the percent difference is only 9%. To complete the analysis of the normality of the yield, one should test specifically if the data is from a normal distribution. This can be done using a Shapiro-Wilk and Kolmogorov-Smirnov test for normality. The Shapiro-Wilk test for normality calculates a W statistic and tests whether the sample comes from (specifically) a normal distribution.
W
(EILi cu)s
(2.24)
Zti(x-x)2
Where a are statistics generated a normal distribution with mean zero and variance 1 [34], The Kolmogorov-Smirnov test is a non-parametric alternative the the Shapiro-Wilk test as it only assumes that the data is continuous. Shapiro-Wilk assumes that the data is randomly sampled, continuous, and has homoscedasticity (constant variance). It is important to note that both Shapiro-Wilk and Kolmogorov-Smirnov are sensitive to large sample sizes. For a sample size greater than N 2000, both tests have a tendency to increase


II SIMULATION: RECOIL BAND STRUCTURE
44
the type-1 error. In other words, both tests are extremely sensitive to small deviations
from a normal distribution. Therefore, both tests have a significant chance at rejecting the
null hypothesis ( that the distribution is normal) even when it is true. The sensitivity of
both tests means one must use both the tests and the visual representations of the data
(histogram and QQ plot) to truly determine whether or not the distributions are normal.
Table 2.6: Results of the distribution analysis for all 8 logarithmically-spaced bins from 10-110 keV for the electron recoils. P-value represents significance for Shapiro-Wilk and Kolmogorov-Smirnov tests for normality. Any P-value < 0.05 indicates rejection of the null hypothesis that the distribution is normal..
Energy bin [keV] Gi K Shapiro-Wilk P-Valuesw Kolm-Smirnov P-Valuexs'
10-13.4 0.41 3.5 0.92 < 0.01 0.031 < 0.01
13.4-18.1 0.39 3.2 0.98 < 0.01 0.028 < 0.01
18.1-24.5 0.34 3.3 0.99 < 0.01 0.032 < 0.01
24.5-33.1 0.33 3.7 0.98 < 0.01 0.029 < 0.01
33.1-44.8 0.38 3.0 0.99 < 0.01 0.031 < 0.01
44.8-60.6 0.31 2.9 0.97 < 0.01 0.031 < 0.01
30.6-80.2 0.29 2.9 0.96 < 0.01 0.028 < 0.01
80.2-110.0 0.29 2.8 0.95 < 0.01 0.032 < 0.01
Table 2.7: Results of the distribution analysis for all 8 logarithmically-spaced bins from 10-110 keV for nuclear recoils. P-value represents significance for Shapiro-Wilk test for normality. any P-value < 0.05 indicates rejection of the null hypothesis that the distribution is normal.
Energy bin [keV] Gi K Shapiro-Wilk P-Valuesw Kolm-Smirnov P-Valuexs'
10-13.4 0.27 3.02 0.99 < 0.01 0.022 < 0.01
13.4-18.1 0.19 2.97 0.99 < 0.01 0.028 0.034
18.1-24.5 0.18 3.25 0.99 < 0.01 0.032 0.02
24.5-33.1 0.09 2.82 0.98 < 0.01 0.031 0.26
33.1-44.8 0.38 2.71 0.99 < 0.01 0.038 0.27
44.8-60.6 0.31 2.84 0.98 < 0.01 0.032 0.45
30.6-80.2 0.29 3.07 0.99 < 0.01 0.031 0.26
80.2-110.0 0.29 3.02 0.97 < 0.01 0.029 0.19
The results of the analysis are clear: The ionization yield for the electron and nuclear recoil bands is not normally distributed. There is too much data within \a and the distributions are positively skewed. The reason that the data is not normally distributed is due to the way we calculate the ionization yield. Equation 2.7 is a ratio of two normally


II SIMULATION: RECOIL BAND STRUCTURE
45
distributed random variables. As it turns out, the distribution of the ratio of two normal distributions is not a normal distribution, but a ratio distribution.


II SIMULATION: RECOIL BAND STRUCTURE
46
Electron Recoil Yield DistributiontlO.O, 13.4] kevr fano = 0
4.0
3.5
3.0
â– m 2.5 c u o
u 2.0
1.5
1.0 0.5
0.7 0.8 0.9 1.0 1.1 1.2 1.3
Yield
(a)
0-0 plot 11.7 keV
(b)
Figure 2.12: a) Histogram of the electron recoil band yield for the energy bin centered at 15.74 keV. Overlaid is a Gaussian distribution. As shown, there is evidence for a positive skew, as the distribution has a slight tail on the right side, b) QQ plot. The upward curve in the data indicates a positive skew. To see the histograms and QQ plots for each of the 8 energy bins, please refer to the Appendix B.


II SIMULATION: RECOIL BAND STRUCTURE
47
C
73
O
U
Nuclear Recoil Yield DistributionUO.O, 13.4] kev, fano = EDW
0.15 0.20 0.25 0.30 0.35 0.40 0.45
Ionization Yield
(a)
Q-Q plot for Nuclear Recoils at (10.0r 13.4] keV
(b)
Figure 2.13: a) Histogram of the nuclear recoil band yield for the energy bin centered at 15.74 keV. Overlaid is a Gaussian distribution. As shown, there is evidence for a positive skew, as the distribution has a slight tail on the right side, b) QQ plot. The upward curve in the data indicates a positive skew. To see the histograms and QQ plots for each of the 8 energy bins, please refer to the Appendix B.


Ill SIMULATED YIELD DISTRIBUTION ANALYSIS
48
CHAPTER III
SIMULATED YIELD DISTRIBUTION ANALYSIS
For dark matter searches, confidently knowing the shape in which the probability distribution for the ionization yield takes directly impacts what data in the electron and nuclear recoil band can be accepted or rejected. As mentioned prior, the shape in which the PDF takes has always been assumed to be normal and as shown from the analysis in the previous chapter, this is not the case. The containment fraction for the electron and nuclear recoil band differed from that of a normal distribution for all energies.
In this chapter, an analytical form for the probability distribution of the ionization yield is derived for the last two ways in which the data is simulated in Chapter II. The first is assuming that the charge energy Eq and the phonon energy Ep are independent of one another. In the second, Eq and EP are assumed to be dependent.


Ill SIMULATED YIELD DISTRIBUTION ANALYSIS
49
3.1 Two Independent Normal Distributions
In Chapter II, the yield is simulated by including the fano factor in the charge and phonon resolutions. This way of simulation lets us treat the charge and phonon energies as if they are independent, or not correlated.
Y
E,
Q
EP ~ IE,
Q
Here Ep and Eq are normally distributed:
(3.1)
Ep ~ N(fj,p, ap) Eq ~ N(nq,aq)
(3.2)
Lets consider the yield as a ratio of two random variables such that Y = jj, Where X is just equivalent to Eq so we know it is normally distributed:
X ~ N(fiq, aq) (3.3)
The random variable U is composed of both random variables EP and Eq. The distribution of U can be obtained by using conditional probability.
P(U = u) = P(U = u\X = x)P{X = x) (3.4)
Since this implies probability distribution of U is equivalent to the difference of two normal distributions we have:
U ~ N(p, - kJa‘ + kaff
(3.5)
where k = -r-E-—
e{gamma
If X and U uphold the assumption for Independence and both of there means were


Ill SIMULATED YIELD DISTRIBUTION ANALYSIS
50
zero, this would be a straightforward problem and the distribution would simply be that of a Cauchy distribution. Since X and U are not independent and do not both have mean zero, this problem is a bit more complicated. The result of the calculation gives the PDF of Y as the following:
Py (Yj Ppi Pqi (?pi &q) Ae
A02+02)
+ Bec + Erf(D)
A
1
AYTfq) + (1 + kY)Tfq))
B
°p f^q ap
+ (1 + kY)%)
V^{Y^f + {l + kYf{^))i
(3.6)
C
- (1 + kY)^)2
V <7p <7q V____' <7q '
Y(Y*(%)* + (l + kY)*)
D
Y^ + (l + kY)^^
(7 q K } <7p Vp
The probability distribution function for Y represents the pdf for the ratio distribution. 3.1.1 Model Validation
Now that there is a analytical expression for ionization yield for the electron and nuclear recoil band, we need to see if it matches with the simulated data. To do so, the containment fraction is calculated continuously from 10 - 130 keV by integrating equation
3.6 between the upper and lower \a bounds that are generated in Equation 2.3.


Ill SIMULATED YIELD DISTRIBUTION ANALYSIS
51
ray
Py(Y^ /j>p^ /Jjq., Cp} crx)dY
(3.7)
'-ay
The total area calculated between —ay and ay determines how much data we should expect to see within \a of the mean. Looking at Figure 3.1, we can see that the expected containment fraction matches what we see from the simulation for both electron and nuclear recoils.


Ill SIMULATED YIELD DISTRIBUTION ANALYSIS
52
1 a Containment Fraction for Electron Recoils
82.5 T3 c 03 00 80.0 ~o QJ — Expected-VI — 68.27% $ Percent Contained
\ \ k
V ,\ \ \
'03 4—1 3 ™ 03 4-» 03 ' \ V \ [
\ \ \ \
u— O 4-J \ \ \ V 1 r
1

10 13 .48.124 1 c .5 33.1 44-8 60 Re t Containmen .6 80.2 110.0 coil Energy [keV] (a) t Fraction for Nuclear Recoils
— Expected-Vl — 68.27% J Percent Contained
T3 S 72
03 u CD C
OJ _c 03 4—< i i
t
o 68 U 03 4-J 1 L [
Q 4— o -M 1 1 l \ 1 V t * [
cu u cu 62 { \] * / f"

10 13 .48 .124 .5 33 .1 44 .8 60 .6 80 .2 110.0
Recoil Energy [keV]
(b)
Figure 3.1: a. Containment fraction for electron recoil band. b. Containment fraction for nuclear recoil band. Both generated with a fano factor included in the charge and phonon resolutions. Black dotted line represents expected containment fraction predicted by the PDF in equation 3.4.


Ill SIMULATED YIELD DISTRIBUTION ANALYSIS
53
3.2 Two Dependent Normal Distributions
In the previous section, a PDF for the ionization yield was derived. As shown, the expected containment fraction agreed with what was seen in the simulated data. The issues with the form of the PDF in the last section, is that the assumption that Eq and EP are independent is incorrect. In this section, a PDF for the yield assuming that Eq and Ep are dependent will be derived. The containment fraction for the new PDF will then be compared with the containment fraction calculated from the simulated data in which the Fano factor is included by varying the number of electron-hole pairs produced.
To make the derivation simpler, thinking about the yield as a function of three independent random variables makes for a slightly easier derivation for the probability distribution function:
Y
+ Xq
Er + XP - f Xq
(3.8)
where Er, V and e7 are constants and N,Xq and XP are independent normally distributed random variables distributed as Ne/h ~ N(fiN,a%), Xq ~ Y(0, /CO
fABXQ(a,b,q)dq (3.9)
•CO
where fABXQ(ci,b,q) is the joint distribution function. For a relatively detailed solution to deriving the PDF for dependent ratio distribution, please see Appendix C.
The PDF for the yield is shown in equation 3.10. As with the independent version of the yield, the new PDF is used to compare the containment fraction for both the electron and nuclear recoil band. As shown in Figure 3.2, the expected containment fraction agrees with the data simulated by including the Fano factor in the Ne/h variation.


Ill SIMULATED YIELD DISTRIBUTION ANALYSIS
54
P( Er, Ne/h, CTq, crp, ON)

2AwVk 2y/A
2 YD
A
((^7 + 1 )ctq)2 + (xcrp)2 + (taN)2 2k
B
(7<72Q{ErxtNe/h) + xtNe/h{{^(TQ)2) + (t2p + Er{(T2Q + (eaN)2)
k
(((Ne/hV + Er)crQ)2 + ((Ne/hCrp)2 + (£,r (3.10)
D
B2
4A
C
7 ___ 2 2 i t u2 2 2 i 22 2
k — OpGq + V &Q&N + 6 (7N(7P


Ill SIMULATED YIELD DISTRIBUTION ANALYSIS
55
1 a Containment Fraction for Electron Recoils
— Expected-V2 — 68.27%
~o c 03 CQ c 80.0 ~o \ \j
\ \ \ \ i [
\ \ V \ 1 :
\ \ \ \ \ V
b 4-1 c CD U 70.0 d> Q_ 67.5 i \ \ \ \ J
\] **— _

—
10JB.J8.124.5 33-1 44J 606 802 110.0
Recoil Energy [keV]
(a)
lo Containment Fraction for Nuclear Recoils
Expected-V2 68.27%
T3 C 5 Percent Contained
CD -O 80.0 \ \
_c 03 775 k" J c
O u 03 \ j A * / / i L
03 Q v*— \ i ‘J \ / / V
4-J c <1) /u 0 Q_


10 13 .48 .124 .5 33 .1 44 .8 60 .6 80.2 116.0
Recoil Energy [keV]
(b)
Figure 3.2: a. Containment fraction for electron recoils, b. containment fraction for nuclear recoils. Both generated with a fano factor included in the charge and phonon resolutions. Black dotted line represents expected containment fraction predicted by the PDF in equation 3.6.


IV IMPACT ON DARK MATTER SEARCHES
56
CHAPTER IV
IMPACT ON DARK MATTER SEARCHES
As mentioned in chapter 1, as technology improves, dark matter communities are eliminating the parameter space in which dark matter is expected to be. With experiments such as SuperCDMS looking for low mass WIMP dark matter, the effect of the Fano factor becomes increasingly important, as the Fano factor directly affects the width of the electron and nuclear recoil band. The higher the Fano factor, the wider the bands. The question we ask is: how does the ability to discriminate between electron and nuclear recoils at low energies effected by the Fano Factor? In this section, the effect of the Fano factor on the lowest WIMP mass detectable will be investigated by looking at where 2a containment bands overlap for electron and nuclear recoils as the Fano factor increases.
The intersection of the two bands will dictate threshold of a background free analysis, and thus, the lowest WIMP mass that can be detected with exposure limited data.
This chapter will take a slightly different approach than the previous chapters. The last few chapters have been looking at the yield vs. recoil energy plane. Though the exact analytical distribution for the yield has been derived, the it does not have a well defined mean or standard deviation. For that reason, the following analysis will be carried out in the Eq/Ep plane.
4.1 Eq Ep Space
To look at the effect the Fano factor has on the minimum mass detectable, we need to first look at the data in the Eq/Ep plane. Figure 4.1 shows the total charge energy Eq vs


IV IMPACT ON DARK MATTER SEARCHES
57
total phonon energy EP for electron recoils (pictured in black) and nuclear recoils (pictured in blue) simulated for recoil energies between 0 and 20 keV. One can see that there is clear separation between the bands until lOkeV in both Eq and EP. A property that makes Eq/Ep space easier to analyze is the fact that for a fixed true recoil energy, the two dimensional distribution is a bi-variate normal and therefore has a well defined standard deviation.
Simulated EP Eq Space
Figure 4.1: Simulated total charge energy (Eq) and total phonon energy (EP) plane for electron and nuclear recoils. Simulated for recoil energies ranging from 0 to 20 keV.
To begin the analysis, the Eq/Ep bands are simulated following the procedure outlined in section 2.4, with a constant Fano factor (F = 0.13) for the electron recoil band and a Fano factor with values ranging from zero to one hundred in integer multiplies of 10 for the nuclear recoil band. As a reminder, the resolutions used in this simulation are from


IV IMPACT ON DARK MATTER SEARCHES
58
that of the SuperCDMS projection paper [23].
To see where the bands overlap, each bin is histogrammed and fit with a Gaussian in Eq. The mean and standard deviation are calculated. For the electron recoil band, the lower 2 bound is plotted for each bin and used to form a lower 2a band for the entire energy range by fitting the data points to a line. The same procedure is used for the nuclear recoil band except an upper 2a band is formed. Figure 4.2 and Figure 4.3 show the results of this procedure for two values of the Fano factor ( F=0 and F = 100.) As expected, the location of the intersection point shifts to higher Eq and Ep values - It is interesting to note that the points do not shift much, however.
>
Ep vs. Eq Fano = 0.0
K Intersection ♦ Electron Recoils
Ep [keV]
Figure 4.2: Eq/Ep plan simulated with a Fano factor of 0. Black band represents electron recoils, blue band nuclear recoil bands. Orange and black points represent location of the 2a mark for each bin. Data points are for to a line to form lower bound for electron recoils and upper bound for nuclear recoils. The red cross represents the intersection of the 2a bands.


IV IMPACT ON DARK MATTER SEARCHES
59
EP vs. E0 Fano = 100.0
10 T---------------------------------------------
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
Ep [keV]
Figure 4.3: Eq/Ep plan simulated with a Fano factor of 100. Black band represents electron recoils, blue band nuclear recoil bands. Red and black points represent location of the 2a mark for each bin. Data points are for to a line to form lower bound for electron recoils and upper bound for nuclear recoils. The red cross represents the intersection of the 2a bands


IV IMPACT ON DARK MATTER SEARCHES
60
4.2 WIMP Mass Accessible
Direct detection dark matter experiments such as SuperCDMS and EDELWEISS are looking to move their parameter space to lower dark matter masses. Experiments need to know the lowest energy at which they can distinguish between the electron and nuclear recoil band, because marks the boundary before resolution limited and background limited search. In this section, the effect the Fano factor has on the smallest detectable dark matter mass will be investigated by using the intersection of the 2a bands for electron and nuclear recoil bands in Eq/Ep space.
To calculate the minimum mass detectable, one first needs to calculate the nuclear recoil energy Er of an event using the intersection point of the Eq/Ep space plot:
Er — Ep--------Eq (4-1)
e7
Since we are interested in the minimum WIMP mass detectable, we assume that the nuclear recoil energy associated with the intersection point in Figure 4.3 represents the maximum amount of energy a WIMP can transfer to a germanium nucleus. When a WIMP deposits the maximum energy available, it scatters backward at 180°, which allows the use of ID kinematics ( conservation of energy and momentum) in the lab frame (earth frame):
KEXt = KExt + KEfjf
l\

P-. + Pc,
(4.2)
Xi — x Xf
where KEXi and PXi is the initial kinetic energy and momentum of the WIMP, KEcf and Pcf is the kinetic energy and momentum of the Germanium nucleus after the collision. As we are assuming the germanium nucleus deposits all of its energy, the final kinetic energy of the germanium nucleus is equal to the nuclear recoil energy, \KEcf\ = Er. Using this system of equations, we can solve for the WIMP mass as a function of nuclear recoil energy:


IV IMPACT ON DARK MATTER SEARCHES
61
Minimum Accessable Mass
so
4.8
>
U
£
i/i
in
H3
4.6
^ y = 0.0101x + 4.0012
cl
2 4.4 •

42 -
40 â–  i

20
40 60
Nuclear Recoil Fano Factor
BO
100
Figure 4.4: Minimum mass accessible vs nuclear recoil Fano factor. Mass calculated using recoil energy corresponding to the intersection point in figures such as Figure 4.3.
Mx
MGEr
Vmax \f2McE~r — Er
(4.3)
where Mg is the mass of a Germanium nucleus, and Vmax is approximately the maximum velocity a WIMP can have in the lab frame: Vmax = Vescape + Vearth + Vsoiar. Vmax is around 780km/s [35].
Figure 4.4 shows the minimum mass detectable as a function of Fano factor for a no background search. As expected, the minimum mass detectable increases with increasing Fano factor as we lose the ability to differentiate between electron and nuclear recoils at low energies. But with this model of the Fano Factor, a 2 order of magnitde increase only increases the minimum mass detectable by 1 GeV


IV IMPACT ON DARK MATTER SEARCHES
62
4.2.1 Effect on Dark Matter Limit
The Fano factor has the potential to affect where we can confidently remove parameter space for dark matter. For example, if we look at the silicon iZIP line in Figure 4.5, we can see a kink in the line where the cross section beings to steeply dive. This kink represents the transition between a background limited and a background free interval [23].
-1
-2
-3
-4
-5
-6
-7
-8
-Q
Q.
C
o
o
CJ {ft if) if)
e
u
c
o
0)
0
c
1
CL
5
Figure 4.5: Projected exclusion sensitivity for the SuperCDMS SNOLAB direct detection dark matter experiment. The vertical axis is the spin-independent WIMP-nucleon cross section under standard halo assumptions, and the horizontal axis is the WIMP mass, where WIMP is used to mean any low-mass particle dark matter candidate. The blue dashed curves represent the expected sensitivities for the Si HV and iZIP detectors and the red dashed curves the expected sensitivities of the Ge HV and iZIP detectors [23]
In the background free situation the scientific impact can be improved by increasing the experiment’s run time. However, the background limited situation cannot be improved without a detailed investigation into the background sources and engineering an overall cleaner experiment [36]. This study quantifies how the location of this kink point is effected by the Fano factor. For example: if a measured value of a Fano factor was found to be 40, the minimum mass detectable would be 4.5 GeV. This would shift the kink point on


APPENDIX IMPACT ON DARK MATTER SEARCHES
63
the limit curve to a higher WIMP mass, meaning the background limited situation would be applicable to more parameter space and therefore we would not be able to improve the limit simply by running longer for more of the WIMP mass parameter space. The Fano factor does have an impact on the division of background limited and exposure limited parameter space-but not much. An obvious question to pursue is whether the model used in this study can be verified using experimental data.


APPENDIX A SIMULATION ALGORITHM
64
APPENDIX A
SIMULATION ALGORITHIM
1.1 Yield Algorithm: VI
The following algorithm is used to simulate the fraction of energy given to the electronic system, or yield, as a function of energy for electron and nuclear recoils by including the fano factor in the detector resolutions as discussed in section 2.3
1. Find the true recoil energy.
• Create a uniform distribution of energy between 10 and 150 keV.
• The true recoil energy is then randomly drawn for this distribution.
2. Calculate the average number of electron hole pairs produced N based on the yield Y. See Equation 1.15
• Y is the average yield calculated from Lindhard for a given recoil energy ( for electron recoils Y = 1.)
3. Calculate Ep and Eq based on N (see equations 1.13 and 1.14
4. Calculate (dp)2 and (o°q)2 based on Dan Jardin’s note [29]
5. Add e2FN to (

APPENDIX A SIMULATION ALGORITHM
65
• The fano factor F is added here, as we are not varying the number of electron hole pairs created.
6. Add (eV)2FN to a°P to get a2p = y/(tt°p)2 + (eV)2FN
7. Smear EP and Eq with aP and oq to find EP and Eq
• Create two normal distributions with means EP and Eq and standard deviations aP and oq.
• Randomly draw from these distributions to find EP and Eq respectively.
8. Calculate the ’measured’ recoil energy Er using EP and Eq. See Equation 1.16
9. Calculate the “measured’ yield Y
Y
E.
Q
EP - ^Eq
(1.1)
1.2 Yield Algorithm: V2
The following algorithm is used to simulate the fraction of energy given to the electronic system, or yield, as a function of energy for electron and nuclear recoils by including the fano factor in the variation of electron-hole pair production as discussed in section 2.4.
1. Find the true recoil energy.
• Create a uniform distribution of energy between 10 and 150 keV.
• The true recoil energy is then randomly drawn for this distribution.
2. Calculate the average number of electron hole pairs produced N based on the yield Y. See Equation 1.15
• Y is the average yield calculated from Lindhard for a given recoil energy.


APPENDIX A SIMULATION ALGORITHM
66
3. Randomly draw the number of electron hole pairs produced N
• N is randomly drawn from a normal distribution with a mean of N and a standard deviation VNF, where F is the fano factor.
4. Calculate EP and Eq based on N.
• EP and Eq are considered the ’’true” values.
5. Calculate detector resolutions av and aq
• (jp(Ep) and aq(Eq) are based on the quantities found in Dan Jardin’s note [ref].
6. Smear Ep and Eq with av and aq to find Ep and Eq
• Create two normal distributions with means Ep and Eq and standard deviations av and aq.
• Randomly draw from these distributions to find Ep and Eq respectivly.
7. Calculate the ’measured’ recoil energy Er using Ep and Eq. See equation 1.16
8. Calculate the ’measured’ yield Y
Y
E.
Q
EP - ^Eq
(1.2)


APPENDIX B DISTRIBUTION PLOTS
67
APPENDIX B
DISTRIBUTION PLOTS
Shown below are the histograms and QQ plots for ionization yield for the electron and nuclear recoil bands. Data is generated using version 2 of the simulation ( varying number of electron-hole pairs), described in section 2.4. Data was split using the following bin edges: [10,13.4,18.1,24.5,33.1,44.8,60.4,80.2,110]. Each plot is labeled by it’s bin center.
2.1 Electron Recoils: Fano = 0.13
Q-Q plot for Electron Recoils at (10.0,13.4] keV
Figure 2.1: Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center 11.7
keV


APPENDIX B DISTRIBUTION PLOTS
68
0.8 0.9 1.0 1.1 1.2 1.3
Ionization Yield
(a)
(b)
Figure 2.2: Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center 15.7 keV
Q-Q plot for Electron Recoils at {18.1, 24.5] keV
(a)
(b)
Figure 2.3: Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center 21.3
keV


APPENDIX B DISTRIBUTION PLOTS
69
Q-Q plot for Electron Recoils at (24.5, 33.1] keV
Theoretical quantiles
(a)
(b)
Figure 2.4: Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center 28.8 keV
Q-Q plot for Electron Recoils at (33.1, 44.8] keV
(a)
(b)
Figure 2.5: Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center 39.4
keV


APPENDIX B DISTRIBUTION PLOTS
70
Electron Recoil Yield Distribution^.8, 60.6] kev, fano = 0.13
1--- Gaussian
Ionization Yield
Q-0 plot for Electron Recoils at (44.8, 60.6] keV
Theoretical quantiles
(a)
(b)
Figure 2.6: Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center 53.1 keV
Ionization Yield
(a)
(b)
Figure 2.7: Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center 70.4
keV


APPENDIX B DISTRIBUTION PLOTS
71
7
6
5
2
1
Electron Recoil Yield Distribution(80.2,110.0] kev, fano = 0.13
Q-Q plot for Electron Recoils at (80.2,110.0] keV
(a)
(b)
Figure 2.8: Histogram (a) and Q-Q plot (b) for electron recoils simulated at bin center 95.1 keV


APPENDIX B DISTRIBUTION PLOTS
72
2.2 Nuclear Recoils: Er Dependent Fano Factor
Nuclear Recoil Yield Distribution(10.0.13.4] kev, fano = EDW
7
6
5
i 4
o
u
3
2
1
0.15 0.20 0.25 0.30 0.35 0.40 0.45
Ionization Yield
0-0 plot for Nuclear Recoils at (10.0,13.4] keV
Theoretical quantiles
(a)
(b)
Figure 2.9: Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center 11.7 keV


Count
APPENDIX B DISTRIBUTION PLOTS
73
Nuclear Recoil Yield Distribution(13.4,18.1] kev, fano = EDW
0.15 0.20 0.25 0.30 0.35 0.40 0.45
Ionization Yield
0-0 plot for Nuclear Recoils at (13.4,18.1] keV
(a)
(b)
Figure 2.10: Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center 15.7 keV
0-0 plot for Nuclear Recoils at (18.1, 24.5] keV
Theoretical quantiles
(a)
(b)
Figure 2.11: Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center 21.3
keV


APPENDIX B DISTRIBUTION PLOTS
74
0-0 plot for Nuclear Recoils at (24.5, 33.1] keV
Theoretical quantiles
(a)
(b)
Figure 2.12: Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at vbin center 28.8 keV
0.25 0.30 0.35 0.40
Ionization Yield
(a)
(b)
Figure 2.13: Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center 39.4
keV


APPENDIX B DISTRIBUTION PLOTS
75
Nuclear Recoil Yield Distribution(44.8, 60.6] kev, fano = EDW
0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 Ionization Yield
(a)
0-0 plot for Nuclear Recoils at (44.8, 60.6] keV
Figure 2.14: Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center 53.1 keV
0-0 plot for Nuclear Recoils at (60.6, 80.2] keV
(a)
(b)
Figure 2.15: Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center 70.4
keV


APPENDIX B DISTRIBUTION PLOTS
76
Q-Q plot for Nuclear Recoils at (80.2,110.0] keV
(a)
(b)
Figure 2.16: Histogram (a) and Q-Q plot (b) for nuclear recoils simulated at bin center 95.1 keV


APPENDIX C YIELD DISTRIBUTION
77
APPENDIX C
YIELD DISTRIBUTION
This Appendix describes the derivation for the probability distribution function for the case of varying the number of electron hole pairs as outlined in section 2.4 and 3.2. (model V2)
Prerequisite Information
Properties of Random Variables
• If X and Y are two continuous random variables described by probability density functions fx (x) and fY (y) respectively, then the joint distribution is defined as follows.
fxr (x,y) P X = x p|V = y\ = fx (x\Y = y) fY (y) = fx (x) fy (y\X = x) (3.1)
• If X and Y are two continuous random variables described by probability density functions fx (x) and fy (y) respectively, then the following property holds:
fx (x) = / fxr (x, y) dy
yen Y
(3.2)


APPENDIX C YIELD DISTRIBUTION
78
where Qy is the set of all possible values of the variable Y . This leads to the next property:
• If X , Y , and Z are three continuous random variables described by probability density functions fx (A) , fy (y) , and fz (z) respectively, then the following property holds
fxy (x, y)
z&lz
fxrz (x,y,z) dz
(3-3)
Ratio Distribution Density Function
• If X and Y are two continuous random variables described by probability density functions fx (x) and fy (y) respectively, then the ratio distribution defined as Z = y has a density function that can be calculated through the following formula.
fz(z)= f \t\fxy (zt, t) dt (3.4)
Derivation of Density Function for Yield Variable
In the current model of the Yield, the random variable is given by the following expression:
Y
£N + Xq
e,. + xP + (y xQ
(3.5)
where Er , V , and e are constants and N , Xq , and XP are independent normally distributed variables distributed as N ~ N (/ix, &%) > Xq ~ N (0, u‘q) , and
(—y jy)2
Xp ~ N (0,

APPENDIX C YIELD DISTRIBUTION
79
A variable switch was introduced where A = eN + Xq and B = Er + XP + (7) Xq . Thus:
Y
£N + Xq
Er + XP+(^)XQ
A
B
(3.6)
Using the Ratio distribution density function formula gives the distribution of the Yield as:
fviy)
— CO
Wab (yt,t)dt
(3.7)
Now, the joint distribution Jab (a, b) must be calculated. Using the third listed property of random variables, we have:
fab (a,b)
— CO
}abxq ( (3.8)
In order to calculate the joint distribution Jabxq (a, b, q) , we use the first listed property of random variables, where:
!abxq (a, b, q) = fAB (a, b\XQ = q)Xq (q) (3.9)
Then, because A and B are independent without Xq , the conditional joint distribution can be calculated as:
fAB(a,b\XQ = q) = fA (a\XQ = q)B(b\XQ = q) (3.10)
The remaining conditional variables are then:
A\ (XQ = q) = eN + q ~ N (e/iN + q, £2a2N)
(3.11)
B | {Xq = q) = Er + Xp + (j) q ~ N (.Er + (^) q,a2P)


APPENDIX C YIELD DISTRIBUTION
Then, the probability density functions to plug in are:
fxQ (x) fa (x\XQ = q) fb (x\XQ = q) =
Thus, summarizing previous steps:
1 2CT2 =e <3
gq/2t\
(x-(sliN+q))2
JJV\/27T
2f2 _2
(-(MW)2
T£*/2tt
80
(3.12)
fr — CO
I*I/ab (yM) dt
oo /*oo
1*1 / /abxq (yt,t,q) dqdt
-CO J — CO
— CO

(g) dqdt
(3.13)
/CO /’OO
1*1 / Ja (yt\XQ = q)B {t\XQ = q)XQ (q)
•CO J — CO
fr(y)
1
J
EOPOqON (27t)2
£2_2 £ CTJV
(£t-(£fir + Vq))2
£2ct2,
dqdt
Evaluating this expression gives the result of:


81
P(Er, Ne/h, ctq, crp, an,
,-c
Be
D
B
2A*Vk + 2VAErfi2VA>
((®7 + 1)<7q)2 + (xaP)2 + (taN)2
Qj(j2Q{ErxtNe/h) + xeNe/h((^aQ)2) + a2P + Er(cj2Q + (ecrw)2)
(j _ (((Ne/hV + Er)(jQ)2 + ((Ne/h(jp)2 + (Er(jpi)2)t:2)
(3-14)
D
B2
4A
C
7 2 2 , T ^2 2 2,2 2 2
k — OpOq + V &Q&N + e (7N(7P


82
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THEEFFECTOFIONIZATIONVARIANCEONNUCLEAR-RECOILDARKMATTER SEARCHES by MITCHELLDOUGLASMATHENY BS,UniversityofColoradoDenver,2016 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfullment oftherequirementsforthedegreeof MasterofIntegratedSciences IntegratedSciencesProgram 2019

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ii ThisthesisfortheMasterofIntegratedSciencedegreeby MitchellDouglasMatheny hasbeenapprovedforthe IntegratedSciencesProgram by AmyRoberts,Chair AnthonyVillano,Advisor VarisCarey,Advisor Date:August3rd,2019

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iii Matheny,MitchellDouglasMINS,IntegratedSciences TheEectofIonizationVarianceonNuclear-RecoilDarkMatterSearches ThesisdirectedbyAssistantProfessorAmyRoberts ABSTRACT Directdetectiondarkmatterexperimentsareincreasinglyinterestedinthelow-mass darkmatterparameterspace,butzero-backgroundlow-masssearchesrequireevent separationbetweentheelectronandnuclearrecoilbands,whichrequiresaproper understandingofdetectorenergyreconstruction. Previoussimulationshaveshownthatwedonotentirelyunderstandtheionization eciencyyieldforelectronandnuclearrecoils,astheassumptionthatthedistribution fortheyieldisnormallydistributedforatruerecoilenergyisviolated.Sincetheyield distributionmaydirectlyaectdarkmatterlow-masslimits,itiscrucialweunderstand howtheyieldisdistributed. Acomponenttounderstandingtheyielddistributionisthevarianceinthenumberof electron-holepairsproducedorionizationvariance.Thiseecthasbeenstudiedrelatively infrequentlyasexperimentshavebeeninterestedinlargeenergydeposits-100keV andcouldaccuratelyseparateelectronandnuclearrecoilevents.Forelectronrecoils,the ionizationvarianceisdescribedbyaFano"factor.Fornuclearrecoilstheeectcanbe parameterizedbyaneective"Fanofactor,whichhassimilardenitionbutadierent physicalorigin.Thenuclearrecoileective"Fanofactorisshowntobemuchlargerthan theelectron-recoilversionabovearound10keVdepositedenergy. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:AmyRoberts

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iv TABLEOFCONTENTS IDarkMatter1 1.1Introduction....................................1 1.2VelocityProleandGalacticRotationCurves.................1 1.3GravitationalLensing...............................3 1.4DarkMatterCandidates.............................5 1.4.1Axions...................................5 1.4.2SterileNeutrinos.............................6 1.4.3WeaklyInteractingMassiveParticles..................7 1.5DirectDetectionofWIMPDarkMatter....................8 1.6LindhardModel..................................11 1.7SimulatedRecoilBandsandLimits.......................13 1.8IonizationVariance................................15 IISimulation:RecoilBandStructure20 2.1ChargeandPhononResolution.........................20 2.1.1RecoilBandsfromData.........................21 2.2SimulatedRecoilBands.............................22 2.2.1ElectronRecoils:NoFanoFactor....................22 2.2.2Containmentfraction...........................23 2.2.3NuclearRecoilBand...........................27 2.3FanoFactorinResolutions............................32

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v 2.4FanoFactor:VaryingNumberofElectron-HolePairs.............37 2.5TestofNormality.................................42 IIISimulatedYieldDistributionAnalysis48 3.1TwoIndependentNormalDistributions.....................49 3.1.1ModelValidation.............................50 3.2TwoDependentNormalDistributions......................53 IVImpactonDarkMatterSearches56 4.1 E Q E P Space...................................56 4.2WIMPMassAccessible..............................60 4.2.1EectonDarkMatterLimit.......................62 Appendices63 ASimulationAlgorithim64 1.1YieldAlgorithm:V1...............................64 1.2YieldAlgorithm:V2...............................65 BDistributionPlots67 2.1ElectronRecoils:Fano=0.13..........................67 2.2NuclearRecoils: E r DependentFanoFactor..................72 CYieldDistribution77

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vi LISTOFTABLES 2.1Detectorresolutioncoecients ; ,and arefoundfromlinesofbesttin [29]forDetector1.................................21 2.2ContainmentfractionforElectronrecoilband.Theexpected68%containmentfractionisundertheassumptionthattheyieldforelectronrecoilsis normallydistributed................................26 2.3ContainmentfractionforNuclearrecoilbandsimulatedwithnoFanofactor. Theexpected68%containmentfractionisundertheassumptionthattheyield fornuclearisnormallydistributed........................29 2.4ContainmentfractionforelectronrecoilbandwithaFanofactorof0.13.The expected68%containmentfractionisundertheassumptionthattheyieldfor electronrecoilsisnormallydistributed......................39 2.51 ContainmentfractionforNuclearrecoilbandsimulationusingEDELWEISSparameterizedFanofactor.Theexpected68%containmentfractionis undertheassumptionthattheyieldfornuclearrecoilsisnormallydistributed.39 2.6Resultsofthedistributionanalysisforall8logarithmically-spacedbinsfrom 10-110keVfortheelectronrecoils.P-valuerepresentssignicanceforShapiroWilkandKolmogorov-Smirnovtestsfornormality.AnyP-value < 0.05indicatesrejectionofthenullhypothesisthatthedistributionisnormal......44

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vii 2.7Resultsofthedistributionanalysisforall8logarithmically-spacedbinsfrom 10-110keVfornuclearrecoils.P-valuerepresentssignicanceforShapiroWilktestfornormality.anyP-value < 0.05indicatesrejectionofthenull hypothesisthatthedistributionisnormal....................44

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viii LISTOFFIGURES 1.1Rotationcurvesofspiralgalaxiesshowingtherotationalvelocityofastrophysicalbodiesasafunctionoftheirdistancefromthecenterofthegalaxy.Solid lineistheexpectedvelocity,dotsarefromobservations[1]..........3 1.2LeftOpticalimagesfromtheMagellantelescopewithoverplottedcontours ofspatialdistributionofmass,fromgravitationallensing.RightThesame contoursoverplottedoverChandrax-raydatathattraceshotplasmaina galaxy.Itcanbeseenthatmostofthematterresidesinalocationdierent fromtheplasmawhichunderwentfrictionalinteractionsduringthemerger andsloweddown[2]................................4 1.3ExpectedWIMPspsctrumfor3potentialmasses.X-axisrepresentsenergydepositedinthedetectorwiththex-interceptrepresentingthemaximumamount ofenergytransferpossibleforthegivenmass.Theshadedregionaroundeach linerepresentsthe90%condenceregions....................8 1.4SketchofaSuperCDMSiZIPdetectorshowingchargeandphononpropagation.Here,theliberatedchargedriftsacrossthedetectorduetoanapplied voltage.Theliberatedchargeexcitesprompt"phononsalsoknownasprimaryphononswhichgetcollectedbyphononsensors.............9 1.5IonizationYieldvsrecoilenergyfromCaliforniumcalibrationdata.Topband representstheelectronrecoilband,bottombandrepresentsnuclearrecoil band.SolidBlackbandsrepresent2 : 5 tstoeachband.[20]........11

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ix 1.6Comparisonofexpectedionizationyieldtomeasurementsmadebyseveral experimentsshowingadeviationfromtheoryformostexperiments......13 1.7TopSimulatedelectronandnuclearrecoilbandsandafunctionofrecoil energyPrfoundin[22].BottomElectronandnuclearrecoilbandsfrom Cfcalibrationdata.Blueandredbandsareformedbyttingtheionization yieldy-axiswithaGaussianandcalculatingthe2 : 5 pointfromthemean andplottingasafunctionofrecoilenergy....................14 1.8ProjectedexclusionsensitivityfortheSuperCDMSSNOLABdirectdetection darkmatterexperiment.Theverticalaxisisthespin-independentWIMPnucleoncrosssectionunderstandardhaloassumptions,andthehorizontalaxis istheWIMPmass,whereWIMPisusedtomeananylow-massparticledark mattercandidate.Thebluedashedcurvesrepresenttheexpectedsensitivities fortheSiHVandiZIPdetectorsandthereddashedcurvestheexpected sensitivitiesoftheGeHVandiZIPdetectors[23]................15 1.9PredictedeectiveFanofactorfor2assumptionsmadebyLindhard.Also shown,measurementsmadebyDoughetryinslicon[21,26].Figurecourtesy ofAnthonyVillano.................................17 1.10Electronandnuclearrecoilbandsfromdatatakenduringa 252 Cfcalibration. Redandbluelinesrepresent1 : 625 %containmentlinesformedusing Gaussiantstothequencingfactor[25].....................18 2.1SimulatedelectronrecoilbandwithnoFanofactor.Reddashedlinerepresents themeanyield Y =1 : Blackbandsrepresent1 containmentforelectronrecoil eventsfromSuperCDMS[30]...........................24 2.2Simulatedelectronrecoilbandwithdatageneratedusingtruerecoilenergies thatarelogarithmicspacedfrom10 )]TJ/F15 11.9552 Tf 9.53 0 Td [(110keVwithnoFanofactor.Reddashed linerepresentsthemeanyield Y =1 : Blackbandsrepresent1 containment forelectronrecoilbandsfromSuperCDMS[30].................25

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x 2.3ContainmentfractionforelectronrecoilbandwithnoFanofactor.Asshown, thepercentofdatawithin1 variesfromtheexpected68%..........27 2.4SimulatednuclearrecoilbandwithFanofactor=0.Reddashedlinerepresentsthemeandenedbyequation2.13.Blackbandsrepresent1 containmentfornuclearrecoilbandsfromCDMS[30]..................29 2.5SimulatedNuclearrecoilbandwithFanofactor=0binnedinto8logarithmicallyspacedbins.Reddashedlinerepresentsthemeanyield.Blackbands represent1 containmentfornuclearrecoilbandsfromCDMS[30]......30 2.6Containmentfractionasafunctionofenergyforthesimulatednuclearrecoil bandwithFanofactor=0.Reddashedlinedrepresentsexpectedcontainment fractioniftheyieldisnormallydistributed....................31 2.7Top:SimulatedelectronrecoilbandwithFanofactor=0.13includedinthe resolutions.Blackbandsrepresent1 containmentbandsderivedfromtting data.Reddashedlinerepresentmeanofrecoilband.Bottom:Containment fractionforsimulatedelectronrecoilbandwithFanofactorof0.13......34 2.8NuclearrecoilFanofactorvs.Recoilenergy...................35 2.9Top:SimulatednuclearrecoilbandwithFanofactorparamerterizedfrom [25]includedintheresolutions.Blackbandsrepresent1 containmentbands derivedfromttingdata.Reddashedlinerepresentmeanofrecoilband. Bottom.ContainmentfractionforsimulatednuclearrecoilbandwithFano factorparameterizedfrom[25]..........................36 2.10Top.SimulatednuclearrecoilbandwithFanofactorof0.13.Blackbands represent1 containmentbandsderivedfromttingdata.Reddashedline representmeanofrecoilband.Bottom.Containmentfractionforsimulated electronrecoilband.[25]..............................40

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xi 2.11Top.SimulatednuclearrecoilbandwithFanofactorparamerterizedfrom [25]includedintheresolutions.Blackbandsrepresent1 containmentbands derivedfromttingdata.Reddashedlinerepresentmeanofrecoilband. Bottom.Containmentfractionforsimulatednuclearrecoilbandwithfan factorparameterizedfrom[25]..........................41 2.12aHistogramoftheelectronrecoilbandyieldfortheenergybincenteredat 15.74keV.OverlaidisaGaussiandistribution.Asshown,thereisevidence forapositiveskew,asthedistributionhasaslighttailontherightside.b QQplot.Theupwardcurveinthedataindicatesapositiveskew.Toseethe histogramsandQQplotsforeachofthe8energybins,pleaserefertothe AppendixB.....................................46 2.13aHistogramofthenuclearrecoilbandyieldfortheenergybincenteredat 15.74keV.OverlaidisaGaussiandistribution.Asshown,thereisevidence forapositiveskew,asthedistributionhasaslighttailontherightside.b QQplot.Theupwardcurveinthedataindicatesapositiveskew.Toseethe histogramsandQQplotsforeachofthe8energybins,pleaserefertothe AppendixB....................................47 3.1a.Containmentfractionforelectronrecoilband.b.Containmentfractionfor nuclearrecoilband.Bothgeneratedwithafanofactorincludedinthecharge andphononresolutions.Blackdottedlinerepresentsexpectedcontainment fractionpredictedbythePDFinequation3.4..................52 3.2a.Containmentfractionforelectronrecoils.b.containmentfractionfor nuclearrecoils.Bothgeneratedwithafanofactorincludedinthecharge andphononresolutions.Blackdottedlinerepresentsexpectedcontainment fractionpredictedbythePDFinequation3.6.................55

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xii 4.1Simulatedtotalchargeenergy E Q andtotalphononenergy E P planefor electronandnuclearrecoils.Simulatedforrecoilenergiesrangingfrom0to 20keV.......................................57 4.2 E Q / E P plansimulatedwithaFanofactorof0.Blackbandrepresentselectron recoils,bluebandnuclearrecoilbands.Orangeandblackpointsrepresent locationofthe2 markforeachbin.Datapointsarefortoalinetoform lowerboundforelectronrecoilsandupperboundfornuclearrecoils.Thered crossrepresentstheintersectionofthe2 bands................58 4.3 E Q / E P plansimulatedwithaFanofactorof100.Blackbandrepresentselectronrecoils,bluebandnuclearrecoilbands.Redandblackpointsrepresent locationofthe2 markforeachbin.Datapointsarefortoalinetoform lowerboundforelectronrecoilsandupperboundfornuclearrecoils.Thered crossrepresentstheintersectionofthe2 bands...............59 4.4MinimummassaccessiblevsnuclearrecoilFanofactor.Masscalculatedusing recoilenergycorrespondingtotheintersectionpointinguressuchasFigure 4.3..........................................61 4.5ProjectedexclusionsensitivityfortheSuperCDMSSNOLABdirectdetection darkmatterexperiment.Theverticalaxisisthespin-independentWIMPnucleoncrosssectionunderstandardhaloassumptions,andthehorizontalaxis istheWIMPmass,whereWIMPisusedtomeananylow-massparticledark mattercandidate.Thebluedashedcurvesrepresenttheexpectedsensitivities fortheSiHVandiZIPdetectorsandthereddashedcurvestheexpected sensitivitiesoftheGeHVandiZIPdetectors[23]...............62 2.1HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter 11.7keV......................................67 2.2HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter 15.7keV......................................68

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xiii 2.3HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter 21.3keV......................................68 2.4HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter 28.8keV......................................69 2.5HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter 39.4keV......................................69 2.6HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter 53.1keV......................................70 2.7HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter 70.4keV......................................70 2.8HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter 95.1keV......................................71 2.9HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter 11.7keV......................................72 2.10HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter 15.7keV......................................73 2.11HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter 21.3keV......................................73 2.12HistogramaandQ-Qplotbfornuclearrecoilssimulatedatvbincenter 28.8keV......................................74 2.13HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter 39.4keV......................................74 2.14HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter 53.1keV......................................75 2.15HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter 70.4keV......................................75

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xiv 2.16HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter 95.1keV......................................76

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IDARKMATTER1 CHAPTERI DARKMATTER 1.1Introduction Darkmatterisoneofthemostmysteriousproblemscosmologistsarefacedwithtoday. Ifwecouldunderstandtheoriginsandproperties,wecouldclarifymanyobservations,and strengthenourlimitedunderstandingoftheuniverse.Thisthesiswillfocuson understandinghowtosimulatetheelectronandnuclearrecoileventsaccurately.As motivationforcontentcoveredlaterinthisdocument,ChapterIwillbrieyoutlinethe evidencefordarkmatter,anddescribeindetailhowenergydepositedinsideadetectoris reconstructed.Thischapterwillalsoshedlightonpreviousevidenceforsimulatingthe nuclearrecoilbandincorrectly,andintroducehowararelyexploredphenomenon,the "Fanofactor",andhowitmightaecttheprojecteddarkmatterlimitsintermsofa minimumdetectabledarkmattermass. 1.2VelocityProleandGalacticRotationCurves Agalaxyisagravitationallyboundsystemofstars,gas,dust,andaswenowbelieve, darkmatter.Bystudyingthevelocitiesofstarsdistributedthroughoutgalaxies,the cosomologicalcommunitynowlargelyagreesthatdarkmatterinanimportantcomponent [1]. Thecircularvelocityofstarscanbemeasuredasafunctionofdistancefromthe centerofthegalaxyandusingthevirialtheorem,wecangetagoodapproximationofwhat

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IDARKMATTER2 thisproleshouldlooklike.Thevirialtheoremstatesthatforaboundsystemastar gravitationallyboundtoagalaxy,theaveragekineticenergyis: h KE i = )]TJ/F33 11.9552 Tf 10.494 8.087 Td [(M s 2 h V i .1 Where V isthegravitationalpotentialenergyofthestarand M s isthemassofstart.Fora spiralgalaxy, h V i = )]TJ/F33 11.9552 Tf 10.494 8.088 Td [(GM tot R .2 where M tot isthetotalmassofthegalaxyenclosedatthepositionatthestar,and R isthe distancethatthestarisawayfromthecenterofthegalaxy.Usingthestandarddenition forkineticenergy,thevelocityofthestaris: v 2 s = GM tot R .3 Therstobservationalevidenceagainstthisformulationforthevelocityofstarswas foundbyFritzZwickywhileobservingthecomacluster.Hediscoveredthatthevirial mass, M tot ,andtheluminousmassfoundbyusingthenumberofgalaxiesandamassto lightrationconversiondieredbyafactorof500,implyingthatmostofthemasswasa formofdarkmatter"[2]. Forrotationcurves,fromEquation1.3,weexpect 1 v s / q 1 R ,butlookingatFigure 1.1,wecanseethatthisisnotthecase.Thevelocityproleattensinsteadofdroppingas expected.ThisatteningoftherotationcurveindicatesthatthemasswithinaradiusR fromthecenterofthegalaxyobeysascalinglaw: M R = v 2 flat R G .4 1 forradinearapparentvisibleedgeofthegalaxy.

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IDARKMATTER3 Figure1.1:Rotationcurvesofspiralgalaxiesshowingtherotationalvelocityofastrophysical bodiesasafunctionoftheirdistancefromthecenterofthegalaxy.Solidlineistheexpected velocity,dotsarefromobservations[1] . Assumingthemassissphericallydistributed R = v 2 flat 4 GR 2 .5 Thisindicatesthatthemassseemstoincreaselinearlyasafunctionofdistancebeyond whatcanbevisiblyseen,andhasadensitythatgoesas 1 R 2 farfromtheedge 2 [3].This gaverisetotheviewthatasphereofdarkmatterresidesinahalosurroundingthediskof visiblestarsandgasinagalaxy. 1.3GravitationalLensing GeneralRelativitypredictsthatlightshoulddeect,orbend,aroundagravitational potentialcreatedbyalargemass,thuscreatingalens.Thesimplestexampleofa gravitationallensisaSchwartzchildlens[4].Theangulardeectionaroundapointlike massis = 4 GM rc 2 .6 2 Ourderivationdiersslightlyfromthatofthereferenceastheyincludeashapingfactorwhereherewe assumesphericalsymmetry.

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IDARKMATTER4 where M isthemassofthelensandristhedistancefromobserverandthesource[5]. Assumingasimpletreatmentinwhichtheobjectofinterestisdirectlybehindthelensand thesamedistanceawayfromthatthelensethattheobserveris,wecanndthemassof thelens.Leavingthederivationto[3],themassis M lens = c 2 d tan obs )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 obs 4 G .7 Wheredisthedistancetothelens, c isthespeedoflight, G isthegravitationalconstant, and obs istheobservedangleofdeection.Experimentshaveobservedthattheobserved angleofdeectionistogreatfortheamountofluminousmasscontainedinagalaxy,once againindicatingthepresenceofdarkmatter.Itisinterestingtonotethatthissimple treatmentofapointlikemassbreaksdownwhenweconsideracontinuousmass distribution,butthistreatmentwaswellsuitedforoneofthemorefamousconrmationsof GeneralRelativitywhenArthurEddingtonwentonanexpeditiontomeasurethe deectionoflightaroundthesunin1919[6]. Figure1.2:LeftOpticalimagesfromtheMagellantelescopewithoverplottedcontours ofspatialdistributionofmass,fromgravitationallensing.RightThesamecontours overplottedoverChandrax-raydatathattraceshotplasmainagalaxy.Itcanbeseen thatmostofthematterresidesinalocationdierentfromtheplasmawhichunderwent frictionalinteractionsduringthemergerandsloweddown[2]. Aninterestingpropertyofdarkmatteristhatitdoesnotinteractwithitself. Gravitationallensingpresentsstrongevidenceforthisobservation.Aclusternamedthe BulletCluster"containstheremainsoftwosubclustersaftertheycollided.Figure1.2

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IDARKMATTER5 showstheclusterafterthemergerinthevisiblespectrumleftplotandinxrayright plot.Lookingattheleftplot,thereisnoevidenceofdarkmatterasexpected,but lookingatrightplot,wecanseethattheamountoflensing,shownasthegreencontours, isdisplacedfromwheretheheatedgasis.Thisobservationindicatesthatwhenthetwo twosubclusterscollided,mostofthemassdarkmatterpassedonbywithoutinteracting, reinforcingthenotionthatdarkmatterdoesnotinteractwithitself[2]. 1.4DarkMatterCandidates Theobservationsmentionedintheprevioussectionsarejustasmallexampleofmany observationsandtheoriesthatsupporttheexistenceofdarkmatter,allofwhichrelyon gravitationalinteractions.Thoughthereareseveraldarkmattercandidates,whetherdark matterinteractsviatheotherfundamentalforcesiscurrentlypurespeculationasno experimentalevidenceexiststosupporttheclaimthatitdoes.Currently,darkmatteris thoughttobecomposedofnon-baryonicmaterialthathasmassanddoesnotinteract signicantlywithradiationorordinarymatter[5].Manydarkmattersearchesarefocused onidentifyingaparticlethatisthelightestsupersymmetricparticlethesesdarkmatter particlesarecalledweaklyinteractivemassiveparticlesWIMPs.Othercandidatessuch asAxionsandsterileneutrinosalsoexist.Inthissection,Iwillgiveabriefintroduction intothesescandidates. 1.4.1 Axions Axions,perhapsalesspopularcandidatefordarkmatter,isaspinless,electrically neutral,verylightparticlewhichwasinitiallyintroducedasasolutiontothestrongcharge conjugationparitysymmetryproblemstrongCPproblem[7].Theparticle'smass, M A is dependentuponitsdecayconstant f a .

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IDARKMATTER6 M A =6 eV 10 12 GeV f a .8 Sincetheenergydensityofcoldaxionsis a h 2 = f a 10 12 GeV 2 .9 ifthemassis O eV [8],theaxioncouldbeacandidateforcolddarkmatteras a h 2 couldaccountforthetheorizedamountofdarkmatterintheuniverse. TheAxionDarkMatterexperimentADMXisworkingtodetectaxionsthathave convertedtophotonsinamicrowavecavitypermeatedbyamagneticeld[9].Withplans tolookintothe10 eV range,ADMXresultshavealreadyexcludedlowmassaxionswithin therange1 : 9 eV
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IDARKMATTER7 massof N isconstrainedtobe0 : 4 keV M DM 10 keV theenergyofthephotonshould beinx-rayspectrum.ExperimentssuchasXMM-Newton,Chandra,andINTEGRALare lookingforsuchdecays[12,13,14].Though N hasnotbeenfound,andupperboundon themasshasbeenmade, M DM 4 keV: 1.4.3 WeaklyInteractingMassiveParticles Thesearchfordarkmatterviadirectdetectiontechniqueshasbeenmotivatedbythe popularityofSupersymmetricmodelswithastablelightestSupersymmetricparticleas theirdarkmattercandidate.Forthelast20years,thefocusofthesesearcheshasbeenon `WeaklyInteractingMassiveParticles'orWIMPs.WIMPsareexpectedtobeelectrically neutral,haveamasssomewherebetween10GeVand100TeVandshouldinteractwith ordinarymatterviatheweaknuclearforce,givingasmallbutnonnegligiblecouplingto standardmodelparticles[15].WIMPsareproposedtohavedecoupledfromequilibrium withStandardModelparticlesoncetherateofconversionbetweenWIMPsandother particlesbecamelessthantheexpansionrateoftheUniverse.Previously,neutrinoswere originallyproposedasaWIMPcandidate,butthethreeknownneutrinosdonothave enoughmasstoaccountforthecurrentestimateddarkmatterdensity[16]. TherateatwhichWIMPSareexpectedtointeractwithnucleiinadetectorisgiven by: dR dE r = N T M T 2 M w 2 SI F 2 SI E R + SD F 2 SD E R halo .12 where M w isthemassofthewimp, M T isthemassofthetargetnuclei, isthereduced massofthesystem, N t isthenumberofnucleiinthetarget,and E R istheenergyofthe nuclearrecoil. SD , SI , F SD ,and F SI arethespindependentandspinindependentcross sectionandnuclearformfactorsrespectively[17]. halo isthehalo-modelformfactor,and dependsonthethevelocitiesoftheWIMPSinthehalowhichareusuallyassumedtotake

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IDARKMATTER8 onastandardMaxwellianvelocitydistribution.Figure1.3showstheexpectedrecoil spectraforthreeWIMPmasses. Figure1.3:ExpectedWIMPspsctrumfor3potentialmasses.X-axisrepresentsenergy depositedinthedetectorwiththex-interceptrepresentingthemaximumamountofenergy transferpossibleforthegivenmass.Theshadedregionaroundeachlinerepresentsthe90% condenceregions. 1.5DirectDetectionofWIMPDarkMatter TheprimarycandidatefordirectdetectiondarkmatterareWIMPS,andtheyare expectedtointeractwiththenucleiofthedetector.DetectingWIMPdarkmatterrequires ahighlysensitiveparticledetector.TheSuperCDMScollaborationusescryogenically cooledinterdigitatedZ-sensitiveionizationandphonondetectorsiZIPsmadeof germanium.Thesedetectorsarecapableofmeasuringchargeliberatedandphonons createdthroughtheuseofelectrodesandsuperconductingtransition-edgesensors. Whenaparticleinteracts/depositsenergyinsideadetector,ithastwomodesof interaction:interactingwiththeelectronsboundthethetarget'satomselectronrecoils

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IDARKMATTER9 orscatteringothetargetnucleinuclearrecoils.Inbothcases,itliberateselectron-holes pairsandproducesprimaryphonons.Theelectron-holepairsaredriftedacrossthedetector byaappliedvoltage.Theelectron-holepairscollidewithgermaniumnucleiandproduce secondaryorLukePhonons"[18].AsketchofthisprocessisvisualizedinFigure1.4. Figure1.4:SketchofaSuperCDMSiZIPdetectorshowingchargeandphononpropagation. Here,theliberatedchargedriftsacrossthedetectorduetoanappliedvoltage.Theliberated chargeexcitesprompt"phononsalsoknownasprimaryphononswhichgetcollectedby phononsensors. Theprimaryandsecondaryphononsaredetectedbyaluminumnsthatareattached tothetransition-edgesensors.Anyelectronholepairsthatarenottrappedinavacancy withinthelatticestructuredriftacrossthedetectorandinducechargeontheionization sensors.Determiningtheenergydepositedinsidethedetectorrequiresmeasurementsofthe chargeandphononenergy: E P = E r + E luke E Q = N e=h .13 where, E P isthephononenergy, E Luke isthecontributiontothetotalphononenergyfrom secondaryphononscreatedfromdriftingelectrons[18]: E Luke = eVN e=h ; .14

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IDARKMATTER10 E Q isthechargeenergy, N e=h isthenumberofelectronholepairscreated, istheaverage energyrequiredtoliberate1electron-holepair,and V isthevoltageappliedacrosstothe detector.Thenumberofelectron-holepairscreatedcanbeexpressedintermsofrecoil energy: N e=h = YE r .15 where Y istheionizationyield.Theionizationyieldisthefractionofenergygiventothe electron-holepairs.Forelectronrecoils Y =1andfornuclearrecoilstheyieldisdenedby Lindhardin[19]andwillbediscussedingreaterdetailinthenextsection.The reconstructedrecoilenergyisthen: E r = E P )]TJ/F33 11.9552 Tf 13.466 8.088 Td [(V E Q .16 Figure1.5showstheionizationyieldforbothtypesofinteractionsfromaSuperCDMS iZIPdetector.Theelectronandnuclearrecoileventsappearinhorizontalbandsinthe plot.Forthisreason,weoftenrefertotheseregionsaselectronornuclearrecoilbands. Theplotshowstheeventseparationabilitiesatenergiesgreaterthanabout10keV.

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IDARKMATTER11 Figure1.5:IonizationYieldvsrecoilenergyfromCaliforniumcalibrationdata.Topband representstheelectronrecoilband,bottombandrepresentsnuclearrecoilband.SolidBlack bandsrepresent2 : 5 tstoeachband.[20] 1.6LindhardModel ForSuperCDMSandothercharge-baseddetectors,thequestionwemustanswerto reconstructtheenergyis:Foranuclearrecoil,howmuchenergyisgivenuptothe electronicsystem?Inotherwordsgiven E r ,whatistheionizationyield?Inthedark mattercommunity,themostwidelyusedmodeltoanswerthisquestionistheLindhard model.TheLindhardmodelpredictsthefractionofenergygiventotheelectronicsystem, orionizationyieldfornuclearrecoilsofagiveninitialkineticenergy.Adislodgednucleus nuclearrecoilwillgenerallystopinashortdistancewithinasolid.Alongthestopping path,thenucleuswillinteractwitheithertheelectronicsystemorotherlatticeatomsand generallytheseenergylossmechanismscompetewithoneanother.Theaverageenergyloss duetoatomiccollisions, isdescribedbythefollowing:[21] k 1 = 2 0 = Z 2 0 dt 2 t 3 2 f t 1 = 2 )]TJ/F33 11.9552 Tf 13.401 8.088 Td [(t )]TJ/F15 11.9552 Tf 12.623 0 Td [( + t .17 = E a 2 Z 2 e 2 .18

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IDARKMATTER12 Where isdimensionlessenergy, a istheBohrradius, Z istheatomicnumber, e is elementarychargeofanelectronand t isavariablerepresentingtheenergytransferfora scatterofanuclearrecoilwithenergy .Solvingthisequationusinga f t 1 = 2 derivedfrom theThomas-FermiPotential[21]givestheenergylossduetoatomicmotionthatcanbe approximatedby: = 1+ kg g =3 0 : 15 +0 : 7 0 : 6 + .19 where k isaconstantthatisdeterminedbythematerialofinterest.ForGermanium,the generallyacceptedvalueforkis0.157.Asthequantityofinterestisthefractionofenergy giventotheelectrons,ortheionizationyield Y ,wecanuse toparameterizethis quantity: Y = )]TJ/F15 11.9552 Tf 12.623 0 Td [( = kg 1+ kg .20 Aplotofmeasuredvaluesoftheionizationyieldionizationeciencyintheplotfor nuclearrecoilsisshowninFigure1.6.

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IDARKMATTER13 Figure1.6:Comparisonofexpectedionizationyieldtomeasurementsmadebyseveralexperimentsshowingadeviationfromtheoryformostexperiments. 1.7SimulatedRecoilBandsandLimits Indirectdetectiondarkmatterexperiments,simulatingtheelectronandnuclearrecoil bandsisanessentialcomponentinunderstandingdetectorresponse.Thebandshavebeen previouslysimulatedin[22].Figure1.7showsacomparisonofsimulatedbandstobands createdfromCfcalibrationdata.Thoughnotaperfectcomparison,onecanseethatthe simulatedbandsToparemuchnarrowerthanthebandsfromcalibration.Understanding whythebandsarenarrowisimportantasitcaneecttheprojecteddarkmattermass limits.Darkmattermasslimitsareusedtodeterminewhatmass,andatwhatprobability, weareexpectingtobeabletodetectdarkmatter.Figure1.8showscurrentprojected SuperCDMSSnoLablimits[23].Thex-axisrepresentthedarkmattermass,andthey-axis representsthecrosssection,orprobabilityofinteraction.Thenarrowsimulationbands impactsthisplotbyshiftingwherethelimitsfall.Forexample,ifthesimulatedbandsare reallynarrow,thiscouldshiftalimitdownandtotheleftforthelowesetWIMPmasses indicatingastrongerlimit.Thishappensbecausenarrowbandsindicatethatwehavethe abilitytobetterdiscriminatebetweenelectronandnuclearrecoils,meaningthatweare

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IDARKMATTER14 morelikelytoseeaninteractionforalowermassdarkmatterparticle,asalowmass particlehasalowermaximumpossibleenergyitcandeposit.Inotherwords,thebetter eventseparationability,themorecondentweareindetectinglow-massWIMP. Thebandssimulatedin[22]mightbetonarrowduetonotincludingtheionization variance. Figure1.7:TopSimulatedelectronandnuclearrecoilbandsandafunctionofrecoilenergy Prfoundin[22].BottomElectronandnuclearrecoilbandsfromCfcalibrationdata. Blueandredbandsareformedbyttingtheionizationyieldy-axiswithaGaussianand calculatingthe2 : 5 pointfromthemeanandplottingasafunctionofrecoilenergy.

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IDARKMATTER15 Figure1.8:ProjectedexclusionsensitivityfortheSuperCDMSSNOLABdirectdetection darkmatterexperiment.Theverticalaxisisthespin-independentWIMP-nucleoncross sectionunderstandardhaloassumptions,andthehorizontalaxisistheWIMPmass,where WIMPisusedtomeananylow-massparticledarkmattercandidate.Thebluedashed curvesrepresenttheexpectedsensitivitiesfortheSiHVandiZIPdetectorsandthered dashedcurvestheexpectedsensitivitiesoftheGeHVandiZIPdetectors[23]. 1.8IonizationVariance Thedirect-detectioncommunityhasbeenconcernedwiththeaverageionizationyield producedbyarecoilingnucleuswithinadetector.Thishaslargelybeenacceptedas experimentssuchasCDMS,EDELWEISSandCRESSTwereinterestedinlargeenergy deposits10-100keVwhichallowedtheabilitytodistinguishbetweenelectronandnuclear recoilsaccurately.WiththeparameterspaceforWIMPshiftingtolowermasses,and thereforelowerenergydeposits,theabilitytodistinguishbetweenelectronandnuclear recoilsbecomesdependentupontheionizationvariance. Forelectronrecoils,thevarianceinthenumberofelectronholepairsproducedfora singlerecoilis N = q F N e=h ,where F istheFanofactorand N e=h istheaveragenumber ofelectronholepairs.TheFanofactorisaconstantthataccountsforthefactthatenergy

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IDARKMATTER16 lossinparticlecollisionisnotpurelystatistical.Ifthevarianceinelectron-holepair productionwaspurelystatistical,thevariancewouldfollowthatofaPoissondistribution N = q N e=h .ForelectronrecoilsingermaniumF=0.13[24].Fornuclearrecoils,the conceptoftheFanofactoristhatsameforelectronrecoilsasitaccountsforthevarious waysinwhicharecoilingnucleuscanliberateelectron-holepairs.Thekeydierenceisthat thereisevidencethattheFanofactorfornuclearrecoilsisenergydependent.Asfaraswe know,Lindhard,Doughetry,andEdelweiss[25,21,26]aretheonlyonestomakea prediction/measurementforthevariationinthenumberofelectron-holepairsproduced. Lindhardpredictsavariationinphononenergy 2 ,butsincethetotalenergydoesn'tvary, thisvariationmustbethesameforionization. = + ; .21 where and arephononandionizationenergies,respectively.Usingthisdenition,the ionizationyieldbecomes Y = .UsingthedenitionoftheFanofactorusedpreviously N = p F N andthedenitionoffractionalvariance N N ,wecanwriteaneectiveFano factorfornuclearrecoils. p F N N = = Y NF N 2 = 2 Y 2 2 F = N 1 Y 2 2 .22 Figure4.6showseectiveFanofactorsforassumptionsapproximationsmadeby Lindhardin[21]aswellastheeectiveFanofactormeasuredandpredictedbyDoughetry forSilicon.Asshown,theeectiveFanofactorissignicantlygreaterthanthatfor electronrecoilsF=0.13forGermaniumandF=0.115forsiliconasthereissignicant

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IDARKMATTER17 variationintheamountofenergythatcanbedepositedinthephononsystem. Figure1.9:PredictedeectiveFanofactorfor2assumptionsmadebyLindhard.Alsoshown, measurementsmadebyDoughetryinslicon[21,26].FigurecourtesyofAnthonyVillano. SimilarlytoCDMS,EDELWEISShastheabilitytodirectlymeasuretheionization yield,accepttheyuseantermcalledaQuenchingfactorrepresentedby Q as[25]: Q = E I E R E R =+ V E H )]TJ/F33 11.9552 Tf 13.151 8.088 Td [(V E I .23 Here, E I istheionizationenergy,whichisalalogousto E Q ,and E H istheHeat"Energy whichisanalogousto E P .Figure1.10showsthemeasuredquenchingfactor Q vs. E R for boththeelectronandnuclearrecoilband. Q isparameterizedas:

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IDARKMATTER18 where Q isparameterizedas: Q = aE b r .24 where E r istherecoilenergyand a =0 : 16 ;b =0 : 18. Figure1.10:Electronandnuclearrecoilbandsfromdatatakenduringa 252 Cfcalibration. Redandbluelinesrepresent1 : 625 %containmentlinesformedusingGaussiantsto thequencingfactor[25].

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IDARKMATTER19 Edelweissmademeasurementsforthevarianceinthequenchingfactor: 2 Q = 1 E 2 R + V gamma Q 2 2 I ++ V gamma 2 Q 2 2 H .25 and V isthedetectorvoltage, istheenergyrequiredtocreate1electron-holepair, 2 I is thevarianceinthechargemeasurement, 2 H isthevarianceintheheatmeasurement[25], 2 I and 2 H areparameterizedas: I E 2 = o I 2 + a I E 2 H E 2 = o H 2 + a H E 2 .26 where o I and o H arethebaselineresolutionsand a I and a H arededucedfromthe resolutionoftheionizationandheatsignalsat122keV[25]. EDELWEISSfoundthatdistributionofdatawastowideforwhattheyexpected. Atomicscattering,variationinthenumberofchargescreatedbyanuclearrecoil[27]and multiplescatteringareexpectedtogiveanintrinsicwidthtotheQdistributionfornuclear recoilsandthusexplainthisbehavior[28].Toaccountforthis,theyaddedaconstantin quadraturetotheresolution 2 Q : Q E R = q o Q 2 E R + C 2 .27 Byttingdata,theydeterminedthatatypicalvalueofCis0.04.Inthenextchapter,the nuclearrecoilFanofactorwillbeparameterizedintermsoftheEDELWEISSresolutions.

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IISIMULATION:RECOILBANDSTRUCTURE20 CHAPTERII SIMULATION:RECOILBANDSTRUCTURE ThegoalofthisthesisistoinvestigatetheroletheFanofactorhasonsettingdark mattermasslimits.Todoso,weneedrsttobeabletosimulatethebandsaccurately.In thesectionsthatfollow,theelectronandnuclearrecoilbandsaresimulatedinthreeways: First,toconrmthenarrownessfoundinthenuclearrecoilbandby[22],bothbandsare simulatedwithnoFanofactor.Anin-depthanalysisoftheresultingdistributionofthe ionizationyieldisalsoinvestigated.Second,thebandsaresimulatedbyincludingtheFano factorintheresolutions.Lastly,Isimulatethebandsusingamorephysicallyaccurate model,byusingthefanofactortovarythenumberofelectronholepairsproduced. 2.1ChargeandPhononResolution Animportantconcepttounderstandwhenattemptingtosimulatetheelectronand nuclearrecoilbandsisdetectorresolutionasthevalueoftheresolutionscandirectlyaect themeasuredionizationyield,whichisusedtodistinguishbetweenthetworecoiltypes.In thissection,Igiveabriefdescriptionofhowthechargeandphononresolutionsforthe CDMSiZIPdetectorsarefoundandhowtheywillbeimplementedinsimulatingthe electronandnuclearrecoilband. ThechargeandphononresolutionsfortheiZIPdetectorswherefoundbytting Gaussianstopeakslocatedat0,10.36,66.7,356,and511keV.Bycomparingthe1 width ofeachpeaktotheassociatedmeanpeaklocation,afunctionalformwastforboththe phononandchargeresolutionsasafunctionofenergy[29]:

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IISIMULATION:RECOILBANDSTRUCTURE21 p = q p + p E P + p E 2 P q = q q + q E Q + q E 2 Q .1 wherethevaluesfor ; ,and for p and q areparametersthatwerettedforeach SuperCDMSdetector.TheresultsforiZIPDetector1areshowninTable2.1. Table2.1:Detectorresolutioncoecients ; ,and arefoundfromlinesofbesttin[29] forDetector1. p 0.155 9 : 1 10 )]TJ/F31 7.9701 Tf 6.587 0 Td [(11 0.00051 q 0.166 0.0023 9 : 52 10 )]TJ/F31 7.9701 Tf 6.586 0 Td [(5 2.1.1 RecoilBandsfromData Forelectronandnuclearrecoils,thedistributionfortheionizationyieldisusually modeledtobenormallydistributedforagivenrecoilenergy.Toaidintheanalysisofthe shapeoftheelectronandnuclearrecoilsbands,ttedbandsfromSuperCDMSareused. Theblack1 bandspicturedinFigure2.1arecreatedbyttingtheionizationyieldfrom SuperCDMSdatawithaGaussianandcalculatingthemeanandstandarddeviation.The resultsfromthetareusedtondanupper U E r andlower L E r functionsthatdene thebandthattaketheform: U E r = Y E r + Y E r L E r = Y E r )]TJ/F33 11.9552 Tf 11.955 0 Td [( Y E r .2 where: Y E r = aE b r Y E r = p cE d r + e E r .3 were, a , bc , d ,and e arecalculatedfromthets[30].

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IISIMULATION:RECOILBANDSTRUCTURE22 2.2SimulatedRecoilBands Nuclearrecoilsaretheprimarywaydarkmatterisexpectedtointeractwithatoms insideadetector.Butbeforetryingtosimulatethenuclearrecoilband,itisessentialrst tounderstandhowtosimulatetheelectronrecoilband,astheelectronrecoilbandhasthe convenientpropertythattheexpectedionizationyieldisindependentoftheelectronrecoil energy,specically Y =1. 2.2.1 ElectronRecoils:NoFanoFactor Tosimulatedtheelectronrecoilband,rstatrue"electronrecoilenergy, E er is randomlydrawnfromanormaldistributionofenergiesrangingfrom10 )]TJ/F15 11.9552 Tf 11.955 0 Td [(100 keV: This energyisthenusedtocalculatethetrue"phononenergy,chargeenergy,andaverage numberofelectron-holepairsproduced N e=h : Thetruephononandchargeenergiesare calculatedassumingaperfectresolution: N e=h = E er E P = E er + V N e=h E Q = N e=h .4 here N e=h istheaveragenumberofelectron-holepairsproduced,Vistheapplieddetector voltage,and istheaverageamountofenergyneededtoliberateoneelectronholepair. Forgermanium =3 : 32 eV [31].Todetermineameasuredyieldvalue,themeasured phononenergy ~ E P ,andthemeasuredchargeenergy ~ E Q ,itishelpfultohaveagood understandingofwhatthedistributionsfor~ P and ~ E Q areexpectedtolooklike.Both ~ E P and ~ E Q areexpectedtobenormallydistributed:

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IISIMULATION:RECOILBANDSTRUCTURE23 f ~ E Q j Q ;Y;E R = 1 q 2 2 Q e ~ E Q )]TJ/F35 5.9776 Tf 5.756 0 Td [(YE R 2 2 2 Q g ~ E P j P ;Y;E R = 1 p 2 2 P e ~ E P )]TJ/F32 5.9776 Tf 5.756 0 Td [([1+ eV Y ] E R 2 2 2 P .5 Where f and g arethePDFsfor E Q and E P respectively.Tosimulatethesenormal distributionsthemeasuredchargeandphononenergiesarecalculatedbyrandomly samplingfromnormaldistributionswithmean E p and E q andstandarddeviations p and q respectively: ~ E P N )]TJ/F15 11.9552 Tf 8.212 -6.662 Td [( E P ; p E r ~ E Q N )]TJ/F15 11.9552 Tf 8.212 -6.661 Td [( E Q ; Q E r .6 Usingthemeasuredvalues ~ E P and ~ E Q themeasuredrecoilenergyandthemeasured ionizationyieldcanbecalculated ~ E r = ~ E P )]TJ/F15 11.9552 Tf 15.883 11.11 Td [(~ E Q V Y = ~ E Q ~ E r .7 Here ~ E r isthemeasuredrecoilenergy.Theresultofthissimulationisshownbelowin Figure2.1. 2.2.2 Containmentfraction Asmentionedpreviously,forsinglerecoilenergy E r theyieldisexpectedtobe normallydistributedandtherefore68 : 27%ofthedatashouldbewithin1 ofthemean Y =1.Toinvestigatetheamountofsimulateddatacontainedwithin1 ,thedatais generatedusingtruerecoilenergiesthatarelogarithmicspacedfrom10 )]TJ/F15 11.9552 Tf 11.955 0 Td [(110keVas shownin2.2andthecontainmentfractioniscalculatedbycomparingtheamountofdata withinthe1 .

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IISIMULATION:RECOILBANDSTRUCTURE24 Figure2.1:SimulatedelectronrecoilbandwithnoFanofactor.Reddashedlinerepresents themeanyield Y =1 : Blackbandsrepresent1 containmentforelectronrecoileventsfrom SuperCDMS[30]. % contained = N )-222(j U + D j N 100.8 Here,Nisthetotalnumberofdatapointsinaspecicenergybin, U istheamountofdata abovetheupper1 band,and D istheamountofdatabelowthelower1 band.This countingalgorithmfollowsthatofabinomialdistribution,allowingforasimplederivation oftheuncertaintyinthecontainmentfraction: c = p Np )]TJ/F33 11.9552 Tf 11.955 0 Td [(p N .9

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IISIMULATION:RECOILBANDSTRUCTURE25 Figure2.2:Simulatedelectronrecoilbandwithdatageneratedusingtruerecoilenergiesthat arelogarithmicspacedfrom10 )]TJ/F15 11.9552 Tf 12.193 0 Td [(110keVwithnoFanofactor.Reddashedlinerepresents themeanyield Y =1 : Blackbandsrepresent1 containmentforelectronrecoilbandsfrom SuperCDMS[30]. where N isthetotalnumberofdatapointswithinabin,and p istheprobabilityofsuccess, ornumberofdatapointsthatlayinsidetheupperandlower1 containmentbands. p = N )]TJ/F33 11.9552 Tf 11.955 0 Td [(U )]TJ/F33 11.9552 Tf 11.955 0 Td [(D N .10 Toquantifythesymmetryofthedistributionofeachbin,thecontributionfromthe upperandlowerhalfofthedistributioniscalculated: % up = N )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 U N 100 % down = N )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 D N 100 .11

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IISIMULATION:RECOILBANDSTRUCTURE26 ThecontainmentfractionforeachbinislistedinTable2.2andvisuallizedinFigure2.3.It canbeseenthattheamountofdatapresentwithin1 isgreaterthan68%forall8energy bins.Onaverage,thecontainmentfractiondiersfromtheexpectedvalueof68%by5 : 8%. Thisindicatesthatthedistributionhasasmallerwidththanexpectedbasedonthe resolutions,conrmingtheobservationfoundin[22].ThelasttwocolumnsinTable2.2 lookatthesymmetryofthedata. Table2.2:ContainmentfractionforElectronrecoilband.Theexpected68%containment fractionisundertheassumptionthattheyieldforelectronrecoilsisnormallydistributed. EnergyBin[KeV] %Containment Expected PercentFromHigh PercentFromLow 10-13.4 81.96 0 : 41 68 71.50 0 : 67 78.41 0 : 59 13.4-18.1 80.23 0 : 40 . 71.91 0 : 66 78.32 0 : 59 18.1-24.5 78.49 0 : 42 . 71.01 0 : 67 78.30 0 : 59 24.5-33.1 75.69 0 : 42 . 70.24 0 : 68 76.06 0 : 62 33.1-44.8 71.88 0 : 44 . 70.24 0 : 68 76.06 0 : 62 44.8-60.02 69.46 0 : 43 . 65.69 0 : 71 69.22 0 : 68 60.6-80.2 70.47 0 : 43 . 66.52 0 : 73 70.81 0 : 55 80.2-110.0 70.13 0 : 42 . 68.39 0 : 70 72.39 0 : 66 Percentfromhighquantiestheamountofdataintheupperhalfofthedistribution. Percentfromlowquantiestheamountofdatainthelowerhalfofthedistribution.The percentfromlowbeinggreaterthanpercentfromhighindicatesthatthedatahasa positiveskew.

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IISIMULATION:RECOILBANDSTRUCTURE27 Figure2.3:ContainmentfractionforelectronrecoilbandwithnoFanofactor.Asshown, thepercentofdatawithin1 variesfromtheexpected68%. 2.2.3 NuclearRecoilBand AstheprimarycandidateforexperimentssuchasSuperCDMSandEDELWEISSare WIMPs,whichareexpectedtointeractprimarilywiththenucleiinadetector, understandingthestructureofthenuclearrecoilbandisimportant.Simulatingthenuclear recoilbandissimilartothatoftheelectronrecoilband,excepttheionizationyieldinno longerunityandthereforetheaveragenumberofelectron-holepairscreatedduringa nuclearrecoileventisnowdependentupontheionizationyield Y . N e=h = YE nr .12

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IISIMULATION:RECOILBANDSTRUCTURE28 here E nr and Y isdendasthefractionofenergygiventotheelectronsin[19]and originallybyLindhardin[21]: Y = kg 1+ kg g =3 0 : 15 +0 : 7 0 : 6 +0 : 6 .13 where k is k =0 : 133 Z 2 3 A )]TJ/F32 5.9776 Tf 7.782 3.258 Td [(1 2 .14 and isthereducedenergydenedbyLindhard[21]as: = E nr a 2 Ze 2 a =0 : 8853 a o 2 Z 1 3 .15 where a isascaleddistance, a o isthebohrradius,Zistheatomicnumberofboththe incomingparticleandthetarget.Lindhardexpresses intermsof Z 1 and Z 2 ,herewe lookatthecasewhere Z 1 = Z 2 . Afteraccountingfortheyield,thesimulationforthenuclearrecoilbandisthesameas intheprevioussectionseeEquations2.4-2.7.Figure2.4showstheresultsfrom simulatingnuclearrecoilswithnoFanofactorwithenergiesrangingfrom0-160keV.Unlike theelectronrecoilband,itisvisuallyobviousthatthedistributionofdataistonarrowas almostallofthedatafallswithinthe1 containmentband.Tofurtherconrmthis observation,thatdataissplitintologramithcallyspacedbinsasbeforeshownin2.6and thepercentcontainedwithin1 iscalculated.TheresultsinTable2.3andFigure2.6 conrmthevisualobservationastheaveragecontainmentfractionis91 : 7%.

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IISIMULATION:RECOILBANDSTRUCTURE29 Figure2.4:SimulatednuclearrecoilbandwithFanofactor=0.Reddashedlinerepresents themeandenedbyequation2.13.Blackbandsrepresent1 containmentfornuclearrecoil bandsfromCDMS[30]. Table2.3:ContainmentfractionforNuclearrecoilbandsimulatedwithnoFanofactor.The expected68%containmentfractionisundertheassumptionthattheyieldfornuclearis normallydistributed. EnergyBin[KeV] %Containment Expected PercentFromHigh PercentFromLow 10-13.4 87.09 1 : 23 68 74.33 2 : 00 79.86 2 : 01 13.4-18.1 90.10 1 : 61 . 76.89 1 : 93 86.34 2 : 24 18.1-24.5 94.80 1 : 58 . 81.80 1 : 80 91.47 2 : 10 24.5-33.1 96.91 1 : 36 . 90.84 1 : 25 97.13 2 : 06 33.1-44.8 98.03 1 : 23 . 95.62 0 : 90 98.78 1 : 84 44.8-60.02 99.40 1 : 22 . 98.87 0 : 42 99.83 1 : 86 60.6-80.2 99.50 1 : 27 . 99.45 0 : 32 99.45 1 : 81 80.2-110.0 99.60 1 : 15 . 99.07 0 : 41 99.81 1 : 55

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IISIMULATION:RECOILBANDSTRUCTURE30 Figure2.5:SimulatedNuclearrecoilbandwithFanofactor=0binnedinto8logarithmicallyspacedbins.Reddashedlinerepresentsthemeanyield.Blackbandsrepresent1 containmentfornuclearrecoilbandsfromCDMS[30].

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IISIMULATION:RECOILBANDSTRUCTURE31 Figure2.6:Containmentfractionasafunctionofenergyforthesimulatednuclearrecoil bandwithFanofactor=0.Reddashedlinedrepresentsexpectedcontainmentfractionif theyieldisnormallydistributed..

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IISIMULATION:RECOILBANDSTRUCTURE32 2.3FanoFactorinResolutions AsmentionedinChapterI,thevarianceinthenumberofelectronholepairsproduced canbeexpressedasaproductofaconstantofvariationtheFanofactorandtheaverage numberofelectronholepairsproduced N e=h .Whilesimulatingtheelectronandnuclear recoilbands,wecanattemptaccountforthisvariationbyaddingatermtothephonon andchargeresolutions: p = q p + p E P + p E 2 P + V 2 N e=h F q = q q + q E Q + q E 2 Q + 2 N e=h F .16 Thisequationaccountsforthefactthatthenumberofelectron-holepairsaectsboththe measuredchargeandphononthroughtheLukeeectenergies. LookingatFigure2.7andcomparingwithFigure2.2,wecanseenovisualevidence thatthewidthoftheyieldhasbeenincreasedforelectronrecoils.Asidefromavariationin the60.6keVpointinFigure2.7bottom,thereappearedtobenodierenceinthe containmentfraction.Thisisnotallthatsurprising.WithaatFanofactorof0.13,the contributiontotheresolutionissmall. Forthenuclearrecoilband,thestoryisabitdierent.AsmentionedinChapter1, theformoftheFanofactorforthenuclearrecoilbandisthoughtnottobeconstant,but energydependent.Asmentioned,EDELWEISSaddedaconstant C toaccountforthe missingvariancewhencomparingtheircalculatedwidthsandresultsfromdata.Ifwe assumethattheextravarianceinthevarianceofthequenchingfactor 2 Q contributedby C isduetotheFanofactor,wecanndtheFanofactorFasafunctionofrecoilenergyand theconstants a;b;C .

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IISIMULATION:RECOILBANDSTRUCTURE33 2 Q = o Q 2 + E R Q + 2 V E R Q 2 + 2 V 2 E R Q 3 F F = C 2 E R Q + 2 V E R Q 2 + 2 V 2 E R Q 3 .17 where, o Q 2 istheintrinsicdetectorresolutionsasshowninequation1.25.Substitutingin forQ: F E r ;a;b;C = C 2 a E 1 )]TJ/F35 5.9776 Tf 5.756 0 Td [(b R + 2 Va 2 E 1 )]TJ/F32 5.9776 Tf 5.756 0 Td [(2 b R + 2 V 2 a 3 E 1 )]TJ/F32 5.9776 Tf 5.756 0 Td [(3 b R .18 Here, a;b and C areconstantsfoundfrom[25].Itisimportanttonotethatthis formulationoftheFanofactorfromEdelweissisanapproximationandassumesthatthe chargeandphononmeasurementsareindependentofoneanotherandtheyieldisnormally distributed,whichisactuallynotthecase.Figure2.8showsthatEquation2.18yieldsa formsimilarllytothatpredictedbyLinhdardinFigure1.6. UsingthisformoftheFanofactorinEquation2.18yieldstheresultsshowninFigure 2.9.WhencomparingFigure2.9theresultswithFigures2.5and2.6wecanseea signicantdierenceinthewidthandthecontainmentfraction.Thereasonforthe signicantdierence,asseeninFigure2.8,isthefactthatthemagnitudeoftheFano factorincreasesrapidlywithrecoilenergyandisonaverage2ordersofmagnitudelarger thantheelectronrecoilFanofactor.

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IISIMULATION:RECOILBANDSTRUCTURE34 Figure2.7:Top:SimulatedelectronrecoilbandwithFanofactor=0.13includedinthe resolutions.Blackbandsrepresent1 containmentbandsderivedfromttingdata.Red dashedlinerepresentmeanofrecoilband.Bottom:Containmentfractionforsimulated electronrecoilbandwithFanofactorof0.13.

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IISIMULATION:RECOILBANDSTRUCTURE35 Figure2.8:NuclearrecoilFanofactorvs.Recoilenergy.

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IISIMULATION:RECOILBANDSTRUCTURE36 a b Figure2.9:Top:SimulatednuclearrecoilbandwithFanofactorparamerterizedfrom[25] includedintheresolutions.Blackbandsrepresent1 containmentbandsderivedfromtting data.Reddashedlinerepresentmeanofrecoilband.Bottom.Containmentfractionfor simulatednuclearrecoilbandwithFanofactorparameterizedfrom[25].

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IISIMULATION:RECOILBANDSTRUCTURE37 2.4FanoFactor:VaryingNumberofElectron-HolePairs Intheprevioussection,theextravarianceduetotheFanofactorwasincludedinthe chargeandphononresolutions.Thoughitistruethatthechargeandphononresolutions areaectedbythisaddedvariance,includingtheeectiveFanofactorinthatmanneris notentirelyaphysicallyaccuratemodel.Amorephysicallyaccuratemodelwouldbeto varytheamountofelectron-holepairsproduced,asaelectronornuclearrecoilwitha singleenergywillnotproducethesameamountofelectron-holepairseachtime.Inother words,anrecoilof60keVwillproduceadierentamountofelectron-holepairsasa dierentrecoilofthesameenergy.Toaccountforthiseect,thenumberofelectronhole pairsissampledfromanormaldistributionwithmean Ne=h andstandarddeviation N . N e=h N )]TJ/F15 11.9552 Tf 8.84 -6.662 Td [( N e=h ; N N = q F N e=h .19 Afteraccountingforthevariationin N e=h ,thetrue"valuesfor E Q and E P arecalculated inthesamemannerasbefore: E P = E er + VN e=h E Q = N e=h .20 UsingthisnewmethodofincludingtheFanofactorandfollowingthesamemethodas denedinEquations2.4-2.7,theelectronrecoilbandissimulated.Figure2.10andTable 2.4showtheresultsofthesimulation.Comparingtheresultsfromtheprevioussection Figure2.7,wecanseethatthereisverylittledierenceinamountofdatacontainedin eachbin. Forthenuclearrecoilband,itisadierentstory.Figure2.11andTable2.5showsthe resultsforthenuclearrecoilband.Unliketheelectronrecoilband,thereisasignicant dierenceinthecontainmentfractionwhencomparingtotoFigure2.9.Thisisduetothe Fanofactorincreasingasafunctionofenergyandthefactthatthetrue" E P and E Q

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IISIMULATION:RECOILBANDSTRUCTURE38 valuesnowvarycorrectly.Intherstversionofincludingthefanofactor,byincludingthe fanofactorintheresolutions, E P and E Q variedindependently,whichphysicallyis incorrect. E P and E Q shouldvarytogetherwith N e=h .Inotherwords,ifthereisanup uctuationinthenumberofelectron-holepairsproduced,both E P and E Q should uctuateup.Byrstvaryingthenumberofelectron-holepairscreated,wecorrectly accountforthiseect.Thefactthat E P and E Q nowvarydependentlydecreasesthe possiblevariationintheyield,andhencethegreatercontainmentfraction. Aftersimulatingtheelectronandnuclearrecoilbandswithamorephysicallyaccurate wayofincludingtheFanofactor,itisclearthatwedonotunderstandthecorrectformof theionizationyielddistribution.BothSuperCDMSandEDELWEISSassumethatthe yieldisnormallydistributed,howeverthecontainmentfractionshowsthatthatisnotthe case.Inthenextsection,thedegreeinwhichthedistributionfortheyieldintheelectron andnuclearrecoilbandsdeviatesfromanormaldistributionwillbeinvestigated.

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IISIMULATION:RECOILBANDSTRUCTURE39 Table2.4:ContainmentfractionforelectronrecoilbandwithaFanofactorof0.13.The expected68%containmentfractionisundertheassumptionthattheyieldforelectronrecoils isnormallydistributed. EnergyBin[KeV] %Containment Expected PercentFromHigh PercentFromLow 10-13.4 81.70 0 : 41 68 71.27 0 : 67 78.93 0 : 58 13.4-18.1 78.83 0 : 40 . 72.03 0 : 66 77.86 0 : 59 18.1-24.5 78.19 0 : 42 . 71.59 0 : 66 77.23 0 : 60 24.5-33.1 74.59 0 : 43 . 70.22 0 : 67 77.04 0 : 60 33.1-44.8 72.18 0 : 44 . 68.38 0 : 69 72.64 0 : 67 44.8-60.6 67.82 0 : 45 . 65.47 0 : 72 70.01 0 : 68 60.6-80.2 70.37 0 : 43 . 65.88 0 : 72 71.26 0 : 67 80.2-110.0 70.60 0 : 42 . 66.82 0 : 70 72.88 0 : 65 Table2.5:1 ContainmentfractionforNuclearrecoilbandsimulationusingEDELWEISS parameterizedFanofactor.Theexpected68%containmentfractionisundertheassumption thattheyieldfornuclearrecoilsisnormallydistributed. EnergyBin[KeV] %Containment Expected PercentFromHigh PercentFromLow 10-13.4 74.21 1 : 33 68 61.78 2 : 23 73.30 2 : 01 13.4-18.1 72.59 1 : 41 . 67.32 2 : 21 66.25 2 : 24 18.1-24.5 72.73 1 : 38 . 66.79 2 : 22 70.87 2 : 10 24.5-33.1 75.87 1 : 36 . 69.35 2 : 16 72.58 2 : 06 33.1-44.8 78.16 1 : 23 . 78.12 1 : 88 79.20 1 : 84 44.8-60.02 79.70 1 : 22 . 81.96 1 : 77 80.04 1 : 86 60.6-80.2 80.67 1 : 23 . 76.54 1 : 91 79.19 1 : 81 80.2-110.0 81.60 1 : 15 . 81.32 1 : 76 83.88 1 : 65

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IISIMULATION:RECOILBANDSTRUCTURE40 a b Figure2.10:Top.SimulatednuclearrecoilbandwithFanofactorof0.13.Blackbands represent1 containmentbandsderivedfromttingdata.Reddashedlinerepresentmean ofrecoilband.Bottom.Containmentfractionforsimulatedelectronrecoilband.[25].

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IISIMULATION:RECOILBANDSTRUCTURE41 a b Figure2.11:Top.SimulatednuclearrecoilbandwithFanofactorparamerterizedfrom[25] includedintheresolutions.Blackbandsrepresent1 containmentbandsderivedfromtting data.Reddashedlinerepresentmeanofrecoilband.Bottom.Containmentfractionfor simulatednuclearrecoilbandwithfanfactorparameterizedfrom[25].

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IISIMULATION:RECOILBANDSTRUCTURE42 2.5TestofNormality Intheprevioussection,thecontainmentfractionshowsthatthedistributionsforthe variousenergybinsfortheelectronandnuclearrecoilbanddonotcontaintheamountof dataexpectedfromanormaldistribution.Thecontainmentfractiondoesnothowever, giveasensetowhatdegreethesedistributionsdierfromthatofanormaldistribution.In thissection,thisdeviationwillbeexplored. Figure2.12andFigure2.13showsahistogramandaQ-Qplot 1 ofthelowestenergy binbincenterat15.75keVfortheelectronandnuclearrecoilbandrespectivley.As shown,thereisvisualevidenceforaasymmetryinboththehistogramandtheQ-Qplots. Toquantifytheamountofasymmetrywecanusetheskew,alsoknownasthe3rd centralmomentofanormaldistribution,usingPearson'sskewnesscoecient G i [32]. Pearson'scoecientcomparesthesampletothatofasymmetricdistribution.Ifthe coecient G i =0,thereisnoskewtothedistributionandif G i deviatesfromzero,it indicatesthatthedistributionhasskew.Theamountanddirectionofskewisindicatedby themagnitudeandsignof G i .Thelargerthemagnitude,thelargertheskew,anegative coecientindicatesnegativeskew,wherepositivecoecientindicatedpositiveskew. Equation2.21describeshowtheskewnessiscalculated.Tosummarize,thetraditionalway torepresentPearson'sskewnesscoecientisusetheratioofthe2 nd and3 rd moments aboutthemeani.e m 2 = 1 n P n n =1 x )]TJ/F15 11.9552 Tf 12.68 0 Td [( x 3 and m 3 = 1 n P n n =1 x )]TJ/F15 11.9552 Tf 12.68 0 Td [( x 2 . G i = m 3 m 3 2 2 = 1 n P n n =1 x )]TJ/F15 11.9552 Tf 12.68 0 Td [( x 2 [ 1 n P n n =1 x )]TJ/F15 11.9552 Tf 12.679 0 Td [( x 3 ] 3 2 .21 The2 nd momentofanormaldistributionisalsoknownasthevariance.Programsand packagessuchasExcelandNumpyinPythonuseaversionthatisadjustedforthesample sizeofthedistribution: 1 AQQquantile-quantileplotisaprobabilityplot,whichisagraphicalmethodforcomparingtwo probabilitydistributionsbyplottingtheirquantitiesagainsteachother

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IISIMULATION:RECOILBANDSTRUCTURE43 G i = n n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 n )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 n n X n =1 x )]TJ/F15 11.9552 Tf 12.68 0 Td [( x 3 .22 UsingNumpy,theskewnessineachenergybiniscalculated,theresultsofwhichare showninTables2.6and2.7.Aswecansee,thePearson'sSkewnessCoecientconrms theobservationsfromFigure2.12and2.13thatthedistributionsfortheyieldarepositively skewed.Tofurtherquantifythedistributionoftheyieldforelectronandnuclearrecoils,it isimportanttolookattheKurtosis.Kurtosis,alsoknowasthe4thcentralmomentofa distribution,isaquantitativewaytolookatthetailedness"ofthedistribution. K = 1 n n X n =1 x )]TJ/F15 11.9552 Tf 12.68 0 Td [( x 4 .23 Valuesfor K rangefrom1 )-222(1 .Thelargerthevaluefor K ,thelessnormal,orheavy tailed,thedistributionbecomes.Forexample,anormaldistributionhasavalueof K =3 [33].Theresultsfor K showninTables2.6and2.7indicatedthatthetailedness"does notdeviatemuchfromthatofanormaldistributionbecausethepercentdierenceisonly 9%.Tocompletetheanalysisofthenormalityoftheyield,oneshouldtestspecicallyif thedataisfromanormaldistribution.ThiscanbedoneusingaShapiro-Wilkand Kolmogorov-Smirnovtestfornormality.TheShapiro-Wilktestfornormalitycalculatesa W statisticandtestswhetherthesamplecomesfromspecicallyanormaldistribution. W = P n i =1 a i x i 2 P n i =1 x )]TJ/F15 11.9552 Tf 12.68 0 Td [( x 2 .24 Where a arestatisticsgeneratedanormaldistributionwithmeanzeroandvariance1[34]. TheKolmogorov-Smirnovtestisanon-parametricalternativethetheShapiro-Wilktestas itonlyassumesthatthedataiscontinuous.Shapiro-Wilkassumesthatthedatais randomlysampled,continuous,andhashomoscedasticityconstantvariance.Itis importanttonotethatbothShapiro-WilkandKolmogorov-Smirnovaresensitivetolarge samplesizes.ForasamplesizegreaterthanN2000,bothtestshaveatendencytoincrease

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IISIMULATION:RECOILBANDSTRUCTURE44 thetype-1error.Inotherwords,bothtestsareextremelysensitivetosmalldeviations fromanormaldistribution.Therefore,bothtestshaveasignicantchanceatrejectingthe nullhypothesisthatthedistributionisnormalevenwhenitistrue.Thesensitivityof bothtestsmeansonemustuseboththetestsandthevisualrepresentationsofthedata histogramandQQplottotrulydeterminewhetherornotthedistributionsarenormal. Table2.6:Resultsofthedistributionanalysisforall8logarithmically-spacedbinsfrom 10-110keVfortheelectronrecoils.P-valuerepresentssignicanceforShapiro-Wilkand Kolmogorov-Smirnovtestsfornormality.AnyP-value < 0.05indicatesrejectionofthenull hypothesisthatthedistributionisnormal.. Energybin[keV] G i K Shapiro-Wilk P-Value SW Kolm-Smirnov P-Value KS 10-13.4 0.41 3.5 0.92 < 0 : 01 0.031 < 0 : 01 13.4-18.1 0.39 3.2 0.98 < 0 : 01 0.028 < 0 : 01 18.1-24.5 0.34 3.3 0.99 < 0 : 01 0.032 < 0 : 01 24.5-33.1 0.33 3.7 0.98 < 0 : 01 0.029 < 0 : 01 33.1-44.8 0.38 3.0 0.99 < 0 : 01 0.031 < 0 : 01 44.8-60.6 0.31 2.9 0.97 < 0 : 01 0.031 < 0 : 01 30.6-80.2 0.29 2.9 0.96 < 0 : 01 0.028 < 0 : 01 80.2-110.0 0.29 2.8 0.95 < 0 : 01 0.032 < 0 : 01 Table2.7:Resultsofthedistributionanalysisforall8logarithmically-spacedbinsfrom 10-110keVfornuclearrecoils.P-valuerepresentssignicanceforShapiro-Wilktestfornormality.anyP-value < 0.05indicatesrejectionofthenullhypothesisthatthedistributionis normal. Energybin[keV] G i K Shapiro-Wilk P-Value SW Kolm-Smirnov P-Value KS 10-13.4 0.27 3.02 0.99 < 0 : 01 0.022 < 0 : 01 13.4-18.1 0.19 2.97 0.99 < 0 : 01 0.028 0 : 034 18.1-24.5 0.18 3.25 0.99 < 0 : 01 0.032 0 : 02 24.5-33.1 0.09 2.82 0.98 < 0 : 01 0.031 0 : 26 33.1-44.8 0.38 2.71 0.99 < 0 : 01 0.038 0 : 27 44.8-60.6 0.31 2.84 0.98 < 0 : 01 0.032 0 : 45 30.6-80.2 0.29 3.07 0.99 < 0 : 01 0.031 0 : 26 80.2-110.0 0.29 3.02 0.97 < 0 : 01 0.029 0 : 19 Theresultsoftheanalysisareclear:Theionizationyieldfortheelectronandnuclear recoilbandsisnotnormallydistributed.Thereistoomuchdatawithin1 andthe distributionsarepositivelyskewed.Thereasonthatthedataisnotnormallydistributedis duetothewaywecalculatetheionizationyield.Equation2.7isaratiooftwonormally

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IISIMULATION:RECOILBANDSTRUCTURE45 distributedrandomvariables.Asitturnsout,thedistributionoftheratiooftwonormal distributionsisnotanormaldistribution,butaratiodistribution.

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IISIMULATION:RECOILBANDSTRUCTURE46 a b Figure2.12:aHistogramoftheelectronrecoilbandyieldfortheenergybincenteredat 15.74keV.OverlaidisaGaussiandistribution.Asshown,thereisevidenceforapositive skew,asthedistributionhasaslighttailontherightside.bQQplot.Theupwardcurve inthedataindicatesapositiveskew.ToseethehistogramsandQQplotsforeachofthe8 energybins,pleaserefertotheAppendixB.

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IISIMULATION:RECOILBANDSTRUCTURE47 a b Figure2.13:aHistogramofthenuclearrecoilbandyieldfortheenergybincenteredat 15.74keV.OverlaidisaGaussiandistribution.Asshown,thereisevidenceforapositive skew,asthedistributionhasaslighttailontherightside.bQQplot.Theupwardcurve inthedataindicatesapositiveskew.ToseethehistogramsandQQplotsforeachofthe8 energybins,pleaserefertotheAppendixB.

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IIISIMULATEDYIELDDISTRIBUTIONANALYSIS48 CHAPTERIII SIMULATEDYIELDDISTRIBUTIONANALYSIS Fordarkmattersearches,condentlyknowingtheshapeinwhichtheprobability distributionfortheionizationyieldtakesdirectlyimpactswhatdataintheelectronand nuclearrecoilbandcanbeacceptedorrejected.Asmentionedprior,theshapeinwhich thePDFtakeshasalwaysbeenassumedtobenormalandasshownfromtheanalysisin thepreviouschapter,thisisnotthecase.Thecontainmentfractionfortheelectronand nuclearrecoilbanddieredfromthatofanormaldistributionforallenergies. Inthischapter,ananalyticalformfortheprobabilitydistributionoftheionization yieldisderivedforthelasttwowaysinwhichthedataissimulatedinChapterII.Therst isassumingthatthechargeenergy E Q andthephononenergy E P areindependentofone another.Inthesecond, E Q and E P areassumedtobedependent.

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IIISIMULATEDYIELDDISTRIBUTIONANALYSIS49 3.1TwoIndependentNormalDistributions InChapterII,theyieldissimulatedbyincludingthefanofactorinthechargeand phononresolutions.Thiswayofsimulationletsustreatthechargeandphononenergiesas iftheyareindependent,ornotcorrelated. Y = E Q E p )]TJ/F34 7.9701 Tf 13.847 4.708 Td [(V E Q .1 Here E P and E Q arenormallydistributed: E p N p ; p E Q N q ; q .2 Letsconsidertheyieldasaratiooftworandomvariablessuchthat Y = X U ,Where X is justequivalentto E Q soweknowitisnormallydistributed: X N q ; q .3 Therandomvariable U iscomposedofbothrandomvariables E P and E Q .The distributionof U canbeobtainedbyusingconditionalprobability. P U = u = P U = u j X = x P X = x .4 Sincethisimpliesprobabilitydistributionof U isequivalenttothedierenceoftwonormal distributionswehave: U N p )]TJ/F33 11.9552 Tf 11.955 0 Td [(k x ; q 2 p + k 2 q .5 where k = V f gamma If X and U upholdtheassumptionforIndependenceandbothoftheremeanswere

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IIISIMULATEDYIELDDISTRIBUTIONANALYSIS50 zero,thiswouldbeastraightforwardproblemandthedistributionwouldsimplybethatof aCauchydistribution.Since X and U arenotindependentanddonotbothhavemean zero,thisproblemisabitmorecomplicated.TheresultofthecalculationgivesthePDFof Y asthefollowing: P Y Y; p ; q ; p ; q = Ae 1 2 q q 2 + p p 2 + Be C + Erf D A 1 Y 2 p q ++ kY 2 p q B p q Y q q p q ++ kY p p p 2 Y 2 p q 2 ++ kY 2 p q 3 2 C Y p p p q )]TJ/F15 11.9552 Tf 11.955 0 Td [(+ kY q q 2 Y Y 2 p q 2 ++ kY 2 D Y q q ++ kY p p q p q 2 Y 2 ++ kY 2 q p 2 .6 Theprobabilitydistributionfunctionfor Y representsthepdffortheratiodistribution. 3.1.1 ModelValidation Nowthatthereisaanalyticalexpressionforionizationyieldfortheelectronand nuclearrecoilband,weneedtoseeifitmatcheswiththesimulateddata.Todoso,the containmentfractioniscalculatedcontinuouslyfrom10-130keVbyintegratingequation 3.6betweentheupperandlower1 boundsthataregeneratedinEquation2.3.

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IIISIMULATEDYIELDDISTRIBUTIONANALYSIS51 Z Y )]TJ/F34 7.9701 Tf 6.586 0 Td [( Y P Y Y; p ; q ; p ; x dY .7 Thetotalareacalculatedbetween )]TJ/F33 11.9552 Tf 9.298 0 Td [( Y and Y determineshowmuchdataweshould expecttoseewithin1 ofthemean.LookingatFigure3.1,wecanseethattheexpected containmentfractionmatcheswhatweseefromthesimulationforbothelectronand nuclearrecoils.

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IIISIMULATEDYIELDDISTRIBUTIONANALYSIS52 a b Figure3.1:a.Containmentfractionforelectronrecoilband.b.Containmentfractionfor nuclearrecoilband.Bothgeneratedwithafanofactorincludedinthechargeandphonon resolutions.Blackdottedlinerepresentsexpectedcontainmentfractionpredictedbythe PDFinequation3.4.

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IIISIMULATEDYIELDDISTRIBUTIONANALYSIS53 3.2TwoDependentNormalDistributions Intheprevioussection,aPDFfortheionizationyieldwasderived.Asshown,the expectedcontainmentfractionagreedwithwhatwasseeninthesimulateddata.The issueswiththeformofthePDFinthelastsection,isthattheassumptionthat E Q and E P areindependentisincorrect.Inthissection,aPDFfortheyieldassumingthat E Q and E P aredependentwillbederived.ThecontainmentfractionforthenewPDFwillthenbe comparedwiththecontainmentfractioncalculatedfromthesimulateddatainwhichthe Fanofactorisincludedbyvaryingthenumberofelectron-holepairsproduced. Tomakethederivationsimpler,thinkingabouttheyieldasafunctionofthree independentrandomvariablesmakesforaslightlyeasierderivationfortheprobability distributionfunction: Y = N e=h + X Q E r + X P )]TJ/F34 7.9701 Tf 13.847 4.707 Td [(V X Q .8 where E r ;V and areconstantsand N;X Q and X P areindependentnormallydistributed randomvariablesdistributedas N e=h N N ; 2 N , X Q N ; 2 Q ,and X P N ; 2 P . Usingavariablesubstitution, A = N + X Q and B = E r + X P )]TJ/F34 7.9701 Tf 13.151 4.707 Td [(V X Q ,then Y = A B .We canthenrepresentthepdffortheyieldas: F AB a;b = Z 1 f ABX Q a;b;q dq .9 where f ABX Q a;b;q isthejointdistributionfunction.Forarelativelydetailedsolutionto derivingthePDFfordependentratiodistribution,pleaseseeAppendixC. ThePDFfortheyieldisshowninequation3.10.Aswiththeindependentversionof theyield,thenewPDFisusedtocomparethecontainmentfractionforboththeelectron andnuclearrecoilband.AsshowninFigure3.2,theexpectedcontainmentfractionagrees withthedatasimulatedbyincludingtheFanofactorinthe N e=h variation.

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IIISIMULATEDYIELDDISTRIBUTIONANALYSIS54 P )]TJ/F15 11.9552 Tf 9.382 -9.683 Td [(E r ;N e=h ; Q ; P ; N ; = e )]TJ/F35 5.9776 Tf 5.756 0 Td [(C 2 A p k + Be D 2 p A Erf B 2 p A A x V +1 Q 2 + x P 2 + N 2 2 k B V 2 Q E r x N e=h + x N e=h V Q 2 + 2 P + E r 2 Q + N 2 k C N e=h V + E r Q 2 + N e=h P 2 + E r N 2 2 2 k D B 2 4 A )]TJ/F33 11.9552 Tf 11.955 0 Td [(C k 2 P 2 Q + V 2 2 Q 2 N + 2 2 N 2 P .10

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IIISIMULATEDYIELDDISTRIBUTIONANALYSIS55 a b Figure3.2:a.Containmentfractionforelectronrecoils.b.containmentfractionfornuclear recoils.Bothgeneratedwithafanofactorincludedinthechargeandphononresolutions. BlackdottedlinerepresentsexpectedcontainmentfractionpredictedbythePDFinequation 3.6.

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IVIMPACTONDARKMATTERSEARCHES56 CHAPTERIV IMPACTONDARKMATTERSEARCHES Asmentionedinchapter1,astechnologyimproves,darkmattercommunitiesare eliminatingtheparameterspaceinwhichdarkmatterisexpectedtobe.Withexperiments suchasSuperCDMSlookingforlowmassWIMPdarkmatter,theeectoftheFanofactor becomesincreasinglyimportant,astheFanofactordirectlyaectsthewidthofthe electronandnuclearrecoilband.ThehighertheFanofactor,thewiderthebands.The questionweaskis:howdoestheabilitytodiscriminatebetweenelectronandnuclear recoilsatlowenergieseectedbytheFanoFactor?Inthissection,theeectoftheFano factoronthelowestWIMPmassdetectablewillbeinvestigatedbylookingatwhere2 containmentbandsoverlapforelectronandnuclearrecoilsastheFanofactorincreases. Theintersectionofthetwobandswilldictatethresholdofabackgroundfreeanalysis,and thus,thelowestWIMPmassthatcanbedetectedwithexposurelimiteddata. Thischapterwilltakeaslightlydierentapproachthanthepreviouschapters.The lastfewchaptershavebeenlookingattheyieldvs.recoilenergyplane.Thoughtheexact analyticaldistributionfortheyieldhasbeenderived,theitdoesnothaveawelldened meanorstandarddeviation.Forthatreason,thefollowinganalysiswillbecarriedoutin the E Q / E P plane. 4.1 E Q E P Space TolookattheeecttheFanofactorhasontheminimummassdetectable,weneedto rstlookatthedatainthe E Q / E P plane.Figure4.1showsthetotalchargeenergy E Q vs

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IVIMPACTONDARKMATTERSEARCHES57 totalphononenergy E P forelectronrecoilspicturedinblackandnuclearrecoilspictured inbluesimulatedforrecoilenergiesbetween0and20keV.Onecanseethatthereisclear separationbetweenthebandsuntil10keVinboth E Q and E P .Apropertythatmakes E Q / E P spaceeasiertoanalyzeisthefactthatforaxedtruerecoilenergy,thetwo dimensionaldistributionisabi-variatenormalandthereforehasawelldenedstandard deviation. Figure4.1:Simulatedtotalchargeenergy E Q andtotalphononenergy E P planefor electronandnuclearrecoils.Simulatedforrecoilenergiesrangingfrom0to20keV. Tobegintheanalysis,the E Q / E P bandsaresimulatedfollowingtheprocedure outlinedinsection2.4,withaconstantFanofactorF=0.13fortheelectronrecoilband andaFanofactorwithvaluesrangingfromzerotoonehundredinintegermultipliesof10 forthenuclearrecoilband.Asareminder,theresolutionsusedinthissimulationarefrom

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IVIMPACTONDARKMATTERSEARCHES58 thatoftheSuperCDMSprojectionpaper[23]. Toseewherethebandsoverlap,eachbinishistogrammedandtwithaGaussianin E Q .Themeanandstandarddeviationarecalculated.Fortheelectronrecoilband,the lower2boundisplottedforeachbinandusedtoformalower2 bandfortheentire energyrangebyttingthedatapointstoaline.Thesameprocedureisusedforthe nuclearrecoilbandexceptanupper2 bandisformed.Figure4.2andFigure4.3showthe resultsofthisprocedurefortwovaluesoftheFanofactorF=0andF=100.As expected,thelocationoftheintersectionpointshiftstohigher E Q and E P values-Itis interestingtonotethatthepointsdonotshiftmuch,however. Figure4.2: E Q / E P plansimulatedwithaFanofactorof0.Blackbandrepresentselectron recoils,bluebandnuclearrecoilbands.Orangeandblackpointsrepresentlocationofthe2 markforeachbin.Datapointsarefortoalinetoformlowerboundforelectronrecoilsand upperboundfornuclearrecoils.Theredcrossrepresentstheintersectionofthe2 bands.

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IVIMPACTONDARKMATTERSEARCHES59 Figure4.3: E Q / E P plansimulatedwithaFanofactorof100.Blackbandrepresentselectron recoils,bluebandnuclearrecoilbands.Redandblackpointsrepresentlocationofthe2 markforeachbin.Datapointsarefortoalinetoformlowerboundforelectronrecoilsand upperboundfornuclearrecoils.Theredcrossrepresentstheintersectionofthe2 bands

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IVIMPACTONDARKMATTERSEARCHES60 4.2WIMPMassAccessible DirectdetectiondarkmatterexperimentssuchasSuperCDMSandEDELWEISSare lookingtomovetheirparameterspacetolowerdarkmattermasses.Experimentsneedto knowthelowestenergyatwhichtheycandistinguishbetweentheelectronandnuclear recoilband,becausemarkstheboundarybeforeresolutionlimitedandbackgroundlimited search.Inthissection,theeecttheFanofactorhasonthesmallestdetectabledark mattermasswillbeinvestigatedbyusingtheintersectionofthe2 bandsforelectronand nuclearrecoilbandsin E Q / E P space. Tocalculatetheminimummassdetectable,onerstneedstocalculatethenuclear recoilenergy E r ofaneventusingtheintersectionpointofthe E Q / E P spaceplot: E r = E P )]TJ/F33 11.9552 Tf 13.466 8.088 Td [(V E Q .1 SinceweareinterestedintheminimumWIMPmassdetectable,weassumethatthe nuclearrecoilenergyassociatedwiththeintersectionpointinFigure4.3representsthe maximumamountofenergyaWIMPcantransfertoagermaniumnucleus.WhenaWIMP depositsthemaximumenergyavailable,itscattersbackwardat180 ,whichallowstheuse of1Dkinematicsconservationofenergyandmomentuminthelabframeearthframe: ~ KE i = ~ KE f + ~ KE G f ~ P i = ~ P f + ~ P G f .2 where KE i and P i istheinitialkineticenergyandmomentumoftheWIMP, KE G f and P G f isthekineticenergyandmomentumoftheGermaniumnucleusafterthecollision.As weareassumingthegermaniumnucleusdepositsallofitsenergy,thenalkineticenergy ofthegermaniumnucleusisequaltothenuclearrecoilenergy, j KE G f j = E r .Usingthis systemofequations,wecansolvefortheWIMPmassasafunctionofnuclearrecoilenergy:

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IVIMPACTONDARKMATTERSEARCHES61 Figure4.4:MinimummassaccessiblevsnuclearrecoilFanofactor.Masscalculatedusing recoilenergycorrespondingtotheintersectionpointinguressuchasFigure4.3. M = M G E r V max p 2 M G E r )]TJ/F33 11.9552 Tf 11.955 0 Td [(E r .3 where M G isthemassofaGermaniumnucleus,and V max isapproximatelythemaximum velocityaWIMPcanhaveinthelabframe: V max = V escape + V earth + V solar :V max isaround 780 km=s [35]. Figure4.4showstheminimummassdetectableasafunctionofFanofactorforano backgroundsearch.Asexpected,theminimummassdetectableincreaseswithincreasing Fanofactoraswelosetheabilitytodierentiatebetweenelectronandnuclearrecoilsat lowenergies.ButwiththismodeloftheFanoFactor,a2orderofmagnitdeincreaseonly increasestheminimummassdetectableby1GeV

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IVIMPACTONDARKMATTERSEARCHES62 4.2.1 EectonDarkMatterLimit TheFanofactorhasthepotentialtoaectwherewecancondentlyremoveparameter spacefordarkmatter.Forexample,ifwelookatthesiliconiZIPlineinFigure4.5,wecan seeakinkinthelinewherethecrosssectionbeingstosteeplydive.Thiskinkrepresents thetransitionbetweenabackgroundlimitedandabackgroundfreeinterval[23]. Figure4.5:ProjectedexclusionsensitivityfortheSuperCDMSSNOLABdirectdetection darkmatterexperiment.Theverticalaxisisthespin-independentWIMP-nucleoncross sectionunderstandardhaloassumptions,andthehorizontalaxisistheWIMPmass,where WIMPisusedtomeananylow-massparticledarkmattercandidate.Thebluedashed curvesrepresenttheexpectedsensitivitiesfortheSiHVandiZIPdetectorsandthered dashedcurvestheexpectedsensitivitiesoftheGeHVandiZIPdetectors[23] Inthebackgroundfreesituationthescienticimpactcanbeimprovedbyincreasing theexperiment'sruntime.However,thebackgroundlimitedsituationcannotbeimproved withoutadetailedinvestigationintothebackgroundsourcesandengineeringanoverall cleanerexperiment[36].Thisstudyquantieshowthelocationofthiskinkpointis eectedbytheFanofactor.Forexample:ifameasuredvalueofaFanofactorwasfoundto be40,theminimummassdetectablewouldbe4.5GeV.Thiswouldshiftthekinkpointon

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APPENDIXIMPACTONDARKMATTERSEARCHES63 thelimitcurvetoahigherWIMPmass,meaningthebackgroundlimitedsituationwould beapplicabletomoreparameterspaceandthereforewewouldnotbeabletoimprovethe limitsimplybyrunninglongerformoreoftheWIMPmassparameterspace.TheFano factordoeshaveanimpactonthedivisionofbackgroundlimitedandexposurelimited parameterspace-butnotmuch.Anobviousquestiontopursueiswhetherthemodelused inthisstudycanbeveriedusingexperimentaldata.

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APPENDIXASIMULATIONALGORITHIM64 APPENDIXA SIMULATIONALGORITHIM 1.1YieldAlgorithm:V1 Thefollowingalgorithmisusedtosimulatethefractionofenergygiventothe electronicsystem,oryield,asafunctionofenergyforelectronandnuclearrecoilsby includingthefanofactorinthedetectorresolutionsasdiscussedinsection2.3 1.Findthetruerecoilenergy. Createauniformdistributionofenergybetween10and150keV. Thetruerecoilenergyisthenrandomlydrawnforthisdistribution. 2.Calculatetheaveragenumberofelectronholepairsproduced N basedontheyield Y .SeeEquation1.15 Y istheaverageyieldcalculatedfromLindhardforagivenrecoilenergyfor electronrecoils Y =1. 3.Calculate E P and E Q basedon N seeequations1.13and1.14 4.Calculate o P 2 and o Q 2 basedonDanJardin'snote[29] 5.Add 2 F N to o Q 2 toget Q = q o Q 2 + 2 F N

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APPENDIXASIMULATIONALGORITHIM65 Thefanofactor F isaddedhere,aswearenotvaryingthenumberofelectron holepairscreated. 6.Add eV 2 F N to o P toget 2 P = p o P 2 + eV 2 F N 7.Smear E P and E Q with P and Q tond ~ E P and ~ E Q Createtwonormaldistributionswithmeans E P and E Q andstandard deviations P and Q . Randomlydrawfromthesedistributionstond ~ E P and ~ E Q respectively. 8.Calculatethe'measured'recoilenergy ~ E r using ~ E P and ~ E Q .SeeEquation1.16 9.Calculatethemeasured'yield Y Y = ~ E Q ~ E P )]TJ/F34 7.9701 Tf 13.847 4.707 Td [(V ~ E Q .1 1.2YieldAlgorithm:V2 Thefollowingalgorithmisusedtosimulatethefractionofenergygiventothe electronicsystem,oryield,asafunctionofenergyforelectronandnuclearrecoilsby includingthefanofactorinthevariationofelectron-holepairproductionasdiscussedin section2.4. 1.Findthetruerecoilenergy. Createauniformdistributionofenergybetween10and150keV. Thetruerecoilenergyisthenrandomlydrawnforthisdistribution. 2.Calculatetheaveragenumberofelectronholepairsproduced N basedontheyield Y .SeeEquation1.15 Y istheaverageyieldcalculatedfromLindhardforagivenrecoilenergy.

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APPENDIXASIMULATIONALGORITHIM66 3.Randomlydrawthenumberofelectronholepairsproduced N N israndomlydrawnfromanormaldistributionwithameanof N anda standarddeviation p NF ,whereFisthefanofactor. 4.Calculate E P and E Q basedon N . E P and E Q areconsideredthe"true"values. 5.Calculatedetectorresolutions p and q p E P and q E Q arebasedonthequantitiesfoundinDanJardin'snote[ref]. 6.Smear E P and E Q with p and q tond ~ E P and ~ E Q Createtwonormaldistributionswithmeans E P and E Q andstandard deviations p and q . Randomlydrawfromthesedistributionstond ~ E P and ~ E Q respectivly. 7.Calculatethe'measured'recoilenergy ~ E r using ~ E P and ~ E Q .Seeequation1.16 8.Calculatethe'measured'yield Y Y = ~ E Q ~ E P )]TJ/F34 7.9701 Tf 13.847 4.708 Td [(V ~ E Q .2

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APPENDIXBDISTRIBUTIONPLOTS67 APPENDIXB DISTRIBUTIONPLOTS ShownbelowarethehistogramsandQQplotsforionizationyieldfortheelectronand nuclearrecoilbands.Dataisgeneratedusingversion2ofthesimulationvaryingnumber ofelectron-holepairs,describedinsection2.4.Datawassplitusingthefollowingbin edges:[10,13.4,18.1,24.5,33.1,44.8,60.4,80.2,110].Eachplotislabeledbyit'sbincenter. 2.1ElectronRecoils:Fano=0.13 Figure2.1:HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter11.7 keV

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APPENDIXBDISTRIBUTIONPLOTS68 a b Figure2.2:HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter15.7 keV a b Figure2.3:HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter21.3 keV

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APPENDIXBDISTRIBUTIONPLOTS69 a b Figure2.4:HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter28.8 keV a b Figure2.5:HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter39.4 keV

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APPENDIXBDISTRIBUTIONPLOTS70 a b Figure2.6:HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter53.1 keV a b Figure2.7:HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter70.4 keV

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APPENDIXBDISTRIBUTIONPLOTS71 a b Figure2.8:HistogramaandQ-Qplotbforelectronrecoilssimulatedatbincenter95.1 keV

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APPENDIXBDISTRIBUTIONPLOTS72 2.2NuclearRecoils: E r DependentFanoFactor a b Figure2.9:HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter11.7 keV

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APPENDIXBDISTRIBUTIONPLOTS73 a b Figure2.10:HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter15.7 keV a b Figure2.11:HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter21.3 keV

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APPENDIXBDISTRIBUTIONPLOTS74 a b Figure2.12:HistogramaandQ-Qplotbfornuclearrecoilssimulatedatvbincenter 28.8keV a b Figure2.13:HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter39.4 keV

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APPENDIXBDISTRIBUTIONPLOTS75 a b Figure2.14:HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter53.1 keV a b Figure2.15:HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter70.4 keV

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APPENDIXBDISTRIBUTIONPLOTS76 a b Figure2.16:HistogramaandQ-Qplotbfornuclearrecoilssimulatedatbincenter95.1 keV

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APPENDIXCYIELDDISTRIBUTION77 APPENDIXC YIELDDISTRIBUTION ThisAppendixdescribesthederivationfortheprobabilitydistributionfunctionforthe caseofvaryingthenumberofelectronholepairsasoutlinedinsection2.4and3.2.model V2 PrerequisiteInformation PropertiesofRandomVariables If X and Y aretwocontinuousrandomvariablesdescribedbyprobabilitydensity functions f X x and f Y y respectively,thenthejointdistributionisdenedas follows. f XY x;y P X = x Y = y ! = f X x j Y = y f Y y = f X x f Y y j X = x .1 If X and Y aretwocontinuousrandomvariablesdescribedbyprobabilitydensity functions f X x and f Y y respectively,thenthefollowingpropertyholds: f X x = Z y 2 Y f XY x;y dy .2

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APPENDIXCYIELDDISTRIBUTION78 where Y isthesetofallpossiblevaluesofthevariable Y .Thisleadstothenext property: If X , Y ,and Z arethreecontinuousrandomvariablesdescribedbyprobability densityfunctions f X x , f Y y ,and f Z z respectively,thenthefollowingproperty holds f XY x;y = Z z 2 Z f XYZ x;y;z dz .3 RatioDistributionDensityFunction If X and Y aretwocontinuousrandomvariablesdescribedbyprobabilitydensity functions f X x and f Y y respectively,thentheratiodistributiondenedas Z = X Y hasadensityfunctionthatcanbecalculatedthroughthefollowingformula. f Z z = Z t 2 Y j t j f XY zt;t dt .4 DerivationofDensityFunctionforYieldVariable InthecurrentmodeloftheYield,therandomvariableisgivenbythefollowing expression: Y = "N + X Q E r + X P + )]TJ/F34 7.9701 Tf 6.675 -4.977 Td [(V " X Q .5 where E r , V ,and " areconstantsand N , X Q ,and X P areindependentnormally distributedvariablesdistributedas N N N ; 2 N , X Q N )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(0 ; 2 Q ,and X P N ; 2 P .Thus,theknowndensityfunctionsare f N x = 1 N p 2 e )]TJ/F31 7.9701 Tf 7.782 4.523 Td [( x )]TJ/F35 5.9776 Tf 5.756 0 Td [( N 2 2 2 N , f X Q x = 1 Q p 2 e )]TJ/F35 5.9776 Tf 10.726 3.258 Td [(x 2 2 2 Q ,and f X P x = 1 P p 2 e )]TJ/F35 5.9776 Tf 10.653 3.259 Td [(x 2 2 2 P .

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APPENDIXCYIELDDISTRIBUTION79 Avariableswitchwasintroducedwhere A = "N + X Q and B = E r + X P + )]TJ/F34 7.9701 Tf 6.675 -4.977 Td [(V " X Q . Thus: Y = "N + X Q E r + X P + )]TJ/F34 7.9701 Tf 6.675 -4.977 Td [(V " X Q = A B .6 UsingtheRatiodistributiondensityfunctionformulagivesthedistributionoftheYieldas: f Y y = Z 1 j t j f AB yt;t dt .7 Now,thejointdistribution f AB a;b mustbecalculated.Usingthethirdlistedproperty ofrandomvariables,wehave: f AB a;b = Z 1 f ABX Q a;b;q dq .8 Inordertocalculatethejointdistribution f ABX Q a;b;q ,weusetherstlistedproperty ofrandomvariables,where: f ABX Q a;b;q = f AB a;b j X Q = q X Q q .9 Then,because A and B areindependentwithout X Q ,theconditionaljointdistribution canbecalculatedas: f AB a;b j X Q = q = f A a j X Q = q B b j X Q = q .10 Theremainingconditionalvariablesarethen: A j X Q = q = "N + q N )]TJ/F33 11.9552 Tf 5.48 -9.684 Td [(" N + q;" 2 2 N B j X Q = q = E r + X P + )]TJ/F34 7.9701 Tf 6.675 -4.977 Td [(V " q N )]TJ/F33 11.9552 Tf 5.479 -9.684 Td [(E r + )]TJ/F34 7.9701 Tf 6.675 -4.977 Td [(V " q; 2 P .11

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APPENDIXCYIELDDISTRIBUTION80 Then,theprobabilitydensityfunctionstopluginare: f X Q x = 1 Q p 2 e )]TJ/F35 5.9776 Tf 10.726 3.258 Td [(x 2 2 2 Q f A x j X Q = q = 1 " N p 2 e )]TJ/F31 7.9701 Tf 7.782 4.524 Td [( x )]TJ/F31 7.9701 Tf 5.756 -0.498 Td [( " N + q 2 2 " 2 2 N f B x j X Q = q = 1 P p 2 e )]TJ/F15 11.9552 Tf 7.782 5.519 Td [( x )]TJ/F15 11.9552 Tf 5.756 -1.494 Td [( E r + V " q 2 2 2 P .12 Thus,summarizingprevioussteps: f Y y = Z 1 j t j f AB yt;t dt = Z 1 j t j Z 1 f ABX Q yt;t;q dqdt = Z 1 j t j Z 1 f AB yt;t j X Q = q X Q q dqdt = Z 1 j t j Z 1 f A yt j X Q = q B t j X Q = q X Q q f Y y = 1 " P Q N 3 2 Z 1 j t j Z 1 e )]TJ/F32 5.9776 Tf 7.782 3.258 Td [(1 2 q 2 2 Q + yt )]TJ/F31 7.9701 Tf 5.756 -0.498 Td [( " N + q 2 " 2 2 N + "t )]TJ/F32 5.9776 Tf 5.756 0 Td [( "E r + Vq 2 " 2 2 P ! dqdt .13 Evaluatingthisexpressiongivestheresultof:

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81 P )]TJ/F33 11.9552 Tf 5.479 -9.683 Td [(E r ;N e=h ; Q ; P ; N ; = e )]TJ/F34 7.9701 Tf 6.587 0 Td [(C 2 A p k + Be D 2 p A Erf B 2 p A A = x V +1 Q 2 + x P 2 + N 2 2 k B = V 2 Q E r x N e=h + x N e=h V Q 2 + 2 P + E r 2 Q + N 2 k C = N e=h V + E r Q 2 + N e=h P 2 + E r N 2 2 2 k D = B 2 4 A )]TJ/F33 11.9552 Tf 11.955 0 Td [(C k = 2 P 2 Q + V 2 2 Q 2 N + 2 2 N 2 P .14

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