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 Permanent Link:
 http://digital.auraria.edu/AA00007255/00001
Material Information
 Title:
 The Effect of ionization variance on nuclearrecoil dark matter searches
 Creator:
 Matheny, Mitchell Douglas
 Place of Publication:
 Denver, CO
 Publisher:
 University of Colorado Denver
 Publication Date:
 2019
 Language:
 English
Thesis/Dissertation Information
 Degree:
 Master's ( Master of integrated science)
 Degree Grantor:
 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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 All applicable rights reserved by the source institution and holding location.

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THE EFFECT OF IONIZATION VARIANCE ON NUCLEARRECOIL 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 NuclearRecoil Dark Matter Searches Thesis directed by Assistant Professor Amy Roberts
m
ABSTRACT
Direct detection dark matter experiments are increasingly interested in the lowmass dark matter parameter space, but zerobackground lowmass 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 lowmass 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 electronhole 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 â€œFanoâ€ factor. For nuclear recoils the effect can be parameterized by an â€œeffectiveâ€ Fano factor, which has similar definition but a different physical origin. The nuclear recoil â€œeffectiveâ€ Fano factor is shown to be much larger than the electronrecoil 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 ElectronHole 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 logarithmicallyspaced bins from
10110 keV for the electron recoils. Pvalue represents significance for ShapiroWilk and KolmogorovSmirnov tests for normality. Any Pvalue < 0.05 indicates rejection of the null hypothesis that the distribution is normal. 44
2.7 Results of the distribution analysis for all 8 logarithmicallyspaced bins from 10110keV for nuclear recoils. Pvalue represents significance for ShapiroWilk test for normality, any Pvalue < 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 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]............. 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 xray 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. Xaxis represents energy deposited in the detector with the xintercept 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 â€œpromptâ€ 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 (yaxis) 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 spinindependent WIMPnucleon cross section under standard halo assumptions, and the horizontal axis is the WIMP mass,where WIMP is used to mean any lowmass 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 10â€”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 spinindependent WIMPnucleon cross section under standard halo assumptions, and the horizontal axis is the WIMP mass, where WIMP is used to mean any lowmass 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 QQ plot (b) for electron recoils simulated at bin center
11.7 keV.................................................................... 67
2.2 Histogram (a) and QQ plot (b) for electron recoils simulated at bin center
15.7 keV.................................................................... 68
Xlll
2.3 Histogram (a) and QQ plot (b) for electron recoils simulated at bin center
21.3 keV............................................................... 68
2.4 Histogram (a) and QQ plot (b) for electron recoils simulated at bin center
28.8 keV.............................................................. 69
2.5 Histogram (a) and QQ plot (b) for electron recoils simulated at bin center
39.4 keV................................................................ 69
2.6 Histogram (a) and QQ plot (b) for electron recoils simulated at bin center
53.1 keV................................................................ 70
2.7 Histogram (a) and QQ plot (b) for electron recoils simulated at bin center
70.4 keV................................................................ 70
2.8 Histogram (a) and QQ plot (b) for electron recoils simulated at bin center
95.1 keV................................................................ 71
2.9 Histogram (a) and QQ plot (b) for nuclear recoils simulated at bin center
11.7 keV................................................................ 72
2.10 Histogram (a) and QQ plot (b) for nuclear recoils simulated at bin center
15.7 keV................................................................ 73
2.11 Histogram (a) and QQ plot (b) for nuclear recoils simulated at bin center
21.3 keV................................................................ 73
2.12 Histogram (a) and QQ plot (b) for nuclear recoils simulated at vbin center
28.8 keV................................................................ 74
2.13 Histogram (a) and QQ plot (b) for nuclear recoils simulated at bin center
39.4 keV................................................................ 74
2.14 Histogram (a) and QQ plot (b) for nuclear recoils simulated at bin center
53.1 keV................................................................ 75
2.15 Histogram (a) and QQ plot (b) for nuclear recoils simulated at bin center
70.4 keV
75
XIV
2.16 Histogram (a) and QQ 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 â€Fano factorâ€, 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 [!]â€¢
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 â€œdark matterâ€ [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
4Â°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)(^ ~ â€˜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].
6lâ€™5am42s
6Iâ€™58'"42â€˜
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 xray 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 â€œBullet Clusterâ€ 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 nonbaryonic 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 particleâ€™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 â€”> 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 xray spectrum. Experiments such as XMMNewton, 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 â€˜Weakly Interacting Massive Particlesâ€™ 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
fÂ§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 halomodel 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. Xaxis represents energy deposited in the detector with the xintercept 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 Zsensitive 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 transitionedge sensors.
When a particle interacts/deposits energy inside a detector, it has two modes of interaction: interacting with the electrons bound the the targetâ€™s atoms (electron recoils)
I DARK MATTER
9
or scattering off the target nuclei (nuclear recoils). In both cases, it liberates electronholes pairs and produces primary phonons. The electronhole pairs are drifted across the detector by a applied voltage. The electronhole pairs collide with germanium nuclei and produce secondary or â€œLuke Phononsâ€ [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 â€œpromptâ€ 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 transitionedge 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 â€” 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 electronhole pair, and V is the voltage applied across to the detector. The number of electronhole 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 electronhole 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 â€” EpEq (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.
I DARK MATTER
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 chargebased 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 ThomasFermi Potential [21] gives the energy loss due to atomic motion that can be approximated by:
l + kg(e) (1>19'
g(e) = 3e0'15 + 0.7eÂ°'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 xaxis represent the dark matter mass, and the yaxis 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 lowmass 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 (yaxis) 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
Vâ€”
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 spinindependent WIMPnucleon cross section under standard halo assumptions, and the horizontal axis is the WIMP mass,where WIMP is used to mean any lowmass 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 directdetection 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 (10100keV) 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 electronhole 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 electronhole 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 electronhole pairs produced. Lindhard predicts a variation in phonon energy fl2, but since the total energy doesnâ€™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)EhA/
e e
Here, Â£/ is the ionization energy, which is alalogous to Eq, and Er is the â€œHeatâ€ 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
â€”Q)V, + (1 +
H
(1.25)
\ tgamma tgamma J
and V is the detector voltage, e is the energy required to create 1 electronhole 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 aÂ°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 indepth 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 + TgEQ
where the values for a, (3, and 7 for

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THEEFFECTOFIONIZATIONVARIANCEONNUCLEARRECOILDARKMATTER 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 TheEectofIonizationVarianceonNuclearRecoilDarkMatterSearches ThesisdirectedbyAssistantProfessorAmyRoberts ABSTRACT Directdetectiondarkmatterexperimentsareincreasinglyinterestedinthelowmass darkmatterparameterspace,butzerobackgroundlowmasssearchesrequireevent separationbetweentheelectronandnuclearrecoilbands,whichrequiresaproper understandingofdetectorenergyreconstruction. Previoussimulationshaveshownthatwedonotentirelyunderstandtheionization eciencyyieldforelectronandnuclearrecoils,astheassumptionthatthedistribution fortheyieldisnormallydistributedforatruerecoilenergyisviolated.Sincetheyield distributionmaydirectlyaectdarkmatterlowmasslimits,itiscrucialweunderstand howtheyieldisdistributed. Acomponenttounderstandingtheyielddistributionisthevarianceinthenumberof electronholepairsproducedorionizationvariance.Thiseecthasbeenstudiedrelatively infrequentlyasexperimentshavebeeninterestedinlargeenergydeposits100keV andcouldaccuratelyseparateelectronandnuclearrecoilevents.Forelectronrecoils,the ionizationvarianceisdescribedbyaFano"factor.Fornuclearrecoilstheeectcanbe parameterizedbyaneective"Fanofactor,whichhassimilardenitionbutadierent physicalorigin.Thenuclearrecoileective"Fanofactorisshowntobemuchlargerthan theelectronrecoilversionabovearound10keVdepositedenergy. 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:VaryingNumberofElectronHolePairs.............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 2.6Resultsofthedistributionanalysisforall8logarithmicallyspacedbinsfrom 10110keVfortheelectronrecoils.PvaluerepresentssignicanceforShapiroWilkandKolmogorovSmirnovtestsfornormality.AnyPvalue < 0.05indicatesrejectionofthenullhypothesisthatthedistributionisnormal......44
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vii 2.7Resultsofthedistributionanalysisforall8logarithmicallyspacedbinsfrom 10110keVfornuclearrecoils.PvaluerepresentssignicanceforShapiroWilktestfornormality.anyPvalue < 0.05indicatesrejectionofthenull hypothesisthatthedistributionisnormal....................44
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viii LISTOFFIGURES 1.1Rotationcurvesofspiralgalaxiesshowingtherotationalvelocityofastrophysicalbodiesasafunctionoftheirdistancefromthecenterofthegalaxy.Solid lineistheexpectedvelocity,dotsarefromobservations[1]..........3 1.2LeftOpticalimagesfromtheMagellantelescopewithoverplottedcontours ofspatialdistributionofmass,fromgravitationallensing.RightThesame contoursoverplottedoverChandraxraydatathattraceshotplasmaina galaxy.Itcanbeseenthatmostofthematterresidesinalocationdierent fromtheplasmawhichunderwentfrictionalinteractionsduringthemerger andsloweddown[2]................................4 1.3ExpectedWIMPspsctrumfor3potentialmasses.Xaxisrepresentsenergydepositedinthedetectorwiththexinterceptrepresentingthemaximumamount 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 yieldyaxiswithaGaussianandcalculatingthe2 : 5 pointfromthemean andplottingasafunctionofrecoilenergy....................14 1.8ProjectedexclusionsensitivityfortheSuperCDMSSNOLABdirectdetection darkmatterexperiment.TheverticalaxisisthespinindependentWIMPnucleoncrosssectionunderstandardhaloassumptions,andthehorizontalaxis istheWIMPmass,whereWIMPisusedtomeananylowmassparticledark 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.TheverticalaxisisthespinindependentWIMPnucleoncrosssectionunderstandardhaloassumptions,andthehorizontalaxis istheWIMPmass,whereWIMPisusedtomeananylowmassparticledark mattercandidate.Thebluedashedcurvesrepresenttheexpectedsensitivities fortheSiHVandiZIPdetectorsandthereddashedcurvestheexpected sensitivitiesoftheGeHVandiZIPdetectors[23]...............62 2.1HistogramaandQQplotbforelectronrecoilssimulatedatbincenter 11.7keV......................................67 2.2HistogramaandQQplotbforelectronrecoilssimulatedatbincenter 15.7keV......................................68
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xiii 2.3HistogramaandQQplotbforelectronrecoilssimulatedatbincenter 21.3keV......................................68 2.4HistogramaandQQplotbforelectronrecoilssimulatedatbincenter 28.8keV......................................69 2.5HistogramaandQQplotbforelectronrecoilssimulatedatbincenter 39.4keV......................................69 2.6HistogramaandQQplotbforelectronrecoilssimulatedatbincenter 53.1keV......................................70 2.7HistogramaandQQplotbforelectronrecoilssimulatedatbincenter 70.4keV......................................70 2.8HistogramaandQQplotbforelectronrecoilssimulatedatbincenter 95.1keV......................................71 2.9HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter 11.7keV......................................72 2.10HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter 15.7keV......................................73 2.11HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter 21.3keV......................................73 2.12HistogramaandQQplotbfornuclearrecoilssimulatedatvbincenter 28.8keV......................................74 2.13HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter 39.4keV......................................74 2.14HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter 53.1keV......................................75 2.15HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter 70.4keV......................................75
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xiv 2.16HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter 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 overplottedoverChandraxraydatathattraceshotplasmainagalaxy.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 thoughttobecomposedofnonbaryonicmaterialthathasmassanddoesnotinteract 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 beinxrayspectrum.ExperimentssuchasXMMNewton,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 isthehalomodelformfactor,and dependsonthethevelocitiesoftheWIMPSinthehalowhichareusuallyassumedtotake
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IDARKMATTER8 onastandardMaxwellianvelocitydistribution.Figure1.3showstheexpectedrecoil spectraforthreeWIMPmasses. Figure1.3:ExpectedWIMPspsctrumfor3potentialmasses.Xaxisrepresentsenergy depositedinthedetectorwiththexinterceptrepresentingthemaximumamountofenergy transferpossibleforthegivenmass.Theshadedregionaroundeachlinerepresentsthe90% condenceregions. 1.5DirectDetectionofWIMPDarkMatter TheprimarycandidatefordirectdetectiondarkmatterareWIMPS,andtheyare expectedtointeractwiththenucleiofthedetector.DetectingWIMPdarkmatterrequires ahighlysensitiveparticledetector.TheSuperCDMScollaborationusescryogenically cooledinterdigitatedZsensitiveionizationandphonondetectorsiZIPsmadeof germanium.Thesedetectorsarecapableofmeasuringchargeliberatedandphonons createdthroughtheuseofelectrodesandsuperconductingtransitionedgesensors. Whenaparticleinteracts/depositsenergyinsideadetector,ithastwomodesof interaction:interactingwiththeelectronsboundthethetarget'satomselectronrecoils
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IDARKMATTER9 orscatteringothetargetnucleinuclearrecoils.Inbothcases,itliberateselectronholes pairsandproducesprimaryphonons.Theelectronholepairsaredriftedacrossthedetector byaappliedvoltage.Theelectronholepairscollidewithgermaniumnucleiandproduce secondaryorLukePhonons"[18].AsketchofthisprocessisvisualizedinFigure1.4. Figure1.4:SketchofaSuperCDMSiZIPdetectorshowingchargeandphononpropagation. Here,theliberatedchargedriftsacrossthedetectorduetoanappliedvoltage.Theliberated chargeexcitesprompt"phononsalsoknownasprimaryphononswhichgetcollectedby phononsensors. Theprimaryandsecondaryphononsaredetectedbyaluminumnsthatareattached tothetransitionedgesensors.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 energyrequiredtoliberate1electronholepair,and V isthevoltageappliedacrosstothe detector.Thenumberofelectronholepairscreatedcanbeexpressedintermsofrecoil energy: N e=h = YE r .15 where Y istheionizationyield.Theionizationyieldisthefractionofenergygiventothe electronholepairs.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 ForSuperCDMSandotherchargebaseddetectors,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 theThomasFermiPotential[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].Thexaxisrepresentthedarkmattermass,andtheyaxis representsthecrosssection,orprobabilityofinteraction.Thenarrowsimulationbands impactsthisplotbyshiftingwherethelimitsfall.Forexample,ifthesimulatedbandsare reallynarrow,thiscouldshiftalimitdownandtotheleftforthelowesetWIMPmasses indicatingastrongerlimit.Thishappensbecausenarrowbandsindicatethatwehavethe abilitytobetterdiscriminatebetweenelectronandnuclearrecoils,meaningthatweare
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IDARKMATTER14 morelikelytoseeaninteractionforalowermassdarkmatterparticle,asalowmass particlehasalowermaximumpossibleenergyitcandeposit.Inotherwords,thebetter eventseparationability,themorecondentweareindetectinglowmassWIMP. Thebandssimulatedin[22]mightbetonarrowduetonotincludingtheionization variance. Figure1.7:TopSimulatedelectronandnuclearrecoilbandsandafunctionofrecoilenergy Prfoundin[22].BottomElectronandnuclearrecoilbandsfromCfcalibrationdata. BlueandredbandsareformedbyttingtheionizationyieldyaxiswithaGaussianand calculatingthe2 : 5 pointfromthemeanandplottingasafunctionofrecoilenergy.
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IDARKMATTER15 Figure1.8:ProjectedexclusionsensitivityfortheSuperCDMSSNOLABdirectdetection darkmatterexperiment.TheverticalaxisisthespinindependentWIMPnucleoncross sectionunderstandardhaloassumptions,andthehorizontalaxisistheWIMPmass,where WIMPisusedtomeananylowmassparticledarkmattercandidate.Thebluedashed curvesrepresenttheexpectedsensitivitiesfortheSiHVandiZIPdetectorsandthered dashedcurvestheexpectedsensitivitiesoftheGeHVandiZIPdetectors[23]. 1.8IonizationVariance Thedirectdetectioncommunityhasbeenconcernedwiththeaverageionizationyield producedbyarecoilingnucleuswithinadetector.Thishaslargelybeenacceptedas experimentssuchasCDMS,EDELWEISSandCRESSTwereinterestedinlargeenergy deposits10100keVwhichallowedtheabilitytodistinguishbetweenelectronandnuclear 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.Ifthevarianceinelectronholepair productionwaspurelystatistical,thevariancewouldfollowthatofaPoissondistribution N = q N e=h .ForelectronrecoilsingermaniumF=0.13[24].Fornuclearrecoils,the conceptoftheFanofactoristhatsameforelectronrecoilsasitaccountsforthevarious waysinwhicharecoilingnucleuscanliberateelectronholepairs.Thekeydierenceisthat thereisevidencethattheFanofactorfornuclearrecoilsisenergydependent.Asfaraswe know,Lindhard,Doughetry,andEdelweiss[25,21,26]aretheonlyonestomakea prediction/measurementforthevariationinthenumberofelectronholepairsproduced. 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, istheenergyrequiredtocreate1electronholepair, 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.Anindepthanalysisoftheresultingdistributionofthe 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 numberofelectronholepairsproduced 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 istheaveragenumberofelectronholepairsproduced,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 1013.4 81.96 0 : 41 68 71.50 0 : 67 78.41 0 : 59 13.418.1 80.23 0 : 40 . 71.91 0 : 66 78.32 0 : 59 18.124.5 78.49 0 : 42 . 71.01 0 : 67 78.30 0 : 59 24.533.1 75.69 0 : 42 . 70.24 0 : 68 76.06 0 : 62 33.144.8 71.88 0 : 44 . 70.24 0 : 68 76.06 0 : 62 44.860.02 69.46 0 : 43 . 65.69 0 : 71 69.22 0 : 68 60.680.2 70.47 0 : 43 . 66.52 0 : 73 70.81 0 : 55 80.2110.0 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 longerunityandthereforetheaveragenumberofelectronholepairscreatedduringa 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 intheprevioussectionseeEquations2.42.7.Figure2.4showstheresultsfrom simulatingnuclearrecoilswithnoFanofactorwithenergiesrangingfrom0160keV.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 1013.4 87.09 1 : 23 68 74.33 2 : 00 79.86 2 : 01 13.418.1 90.10 1 : 61 . 76.89 1 : 93 86.34 2 : 24 18.124.5 94.80 1 : 58 . 81.80 1 : 80 91.47 2 : 10 24.533.1 96.91 1 : 36 . 90.84 1 : 25 97.13 2 : 06 33.144.8 98.03 1 : 23 . 95.62 0 : 90 98.78 1 : 84 44.860.02 99.40 1 : 22 . 98.87 0 : 42 99.83 1 : 86 60.680.2 99.50 1 : 27 . 99.45 0 : 32 99.45 1 : 81 80.2110.0 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 Thisequationaccountsforthefactthatthenumberofelectronholepairsaectsboththe 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:VaryingNumberofElectronHolePairs Intheprevioussection,theextravarianceduetotheFanofactorwasincludedinthe chargeandphononresolutions.Thoughitistruethatthechargeandphononresolutions areaectedbythisaddedvariance,includingtheeectiveFanofactorinthatmanneris notentirelyaphysicallyaccuratemodel.Amorephysicallyaccuratemodelwouldbeto varytheamountofelectronholepairsproduced,asaelectronornuclearrecoilwitha singleenergywillnotproducethesameamountofelectronholepairseachtime.Inother words,anrecoilof60keVwillproduceadierentamountofelectronholepairsasa 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 denedinEquations2.42.7,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 uctuationinthenumberofelectronholepairsproduced,both E P and E Q should uctuateup.Byrstvaryingthenumberofelectronholepairscreated,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 1013.4 81.70 0 : 41 68 71.27 0 : 67 78.93 0 : 58 13.418.1 78.83 0 : 40 . 72.03 0 : 66 77.86 0 : 59 18.124.5 78.19 0 : 42 . 71.59 0 : 66 77.23 0 : 60 24.533.1 74.59 0 : 43 . 70.22 0 : 67 77.04 0 : 60 33.144.8 72.18 0 : 44 . 68.38 0 : 69 72.64 0 : 67 44.860.6 67.82 0 : 45 . 65.47 0 : 72 70.01 0 : 68 60.680.2 70.37 0 : 43 . 65.88 0 : 72 71.26 0 : 67 80.2110.0 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 1013.4 74.21 1 : 33 68 61.78 2 : 23 73.30 2 : 01 13.418.1 72.59 1 : 41 . 67.32 2 : 21 66.25 2 : 24 18.124.5 72.73 1 : 38 . 66.79 2 : 22 70.87 2 : 10 24.533.1 75.87 1 : 36 . 69.35 2 : 16 72.58 2 : 06 33.144.8 78.16 1 : 23 . 78.12 1 : 88 79.20 1 : 84 44.860.02 79.70 1 : 22 . 81.96 1 : 77 80.04 1 : 86 60.680.2 80.67 1 : 23 . 76.54 1 : 91 79.19 1 : 81 80.2110.0 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. Figure2.12andFigure2.13showsahistogramandaQQplot 1 ofthelowestenergy binbincenterat15.75keVfortheelectronandnuclearrecoilbandrespectivley.As shown,thereisvisualevidenceforaasymmetryinboththehistogramandtheQQplots. 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 AQQquantilequantileplotisaprobabilityplot,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.ThiscanbedoneusingaShapiroWilkand KolmogorovSmirnovtestfornormality.TheShapiroWilktestfornormalitycalculatesa 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]. TheKolmogorovSmirnovtestisanonparametricalternativethetheShapiroWilktestas itonlyassumesthatthedataiscontinuous.ShapiroWilkassumesthatthedatais randomlysampled,continuous,andhashomoscedasticityconstantvariance.Itis importanttonotethatbothShapiroWilkandKolmogorovSmirnovaresensitivetolarge samplesizes.ForasamplesizegreaterthanN2000,bothtestshaveatendencytoincrease
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IISIMULATION:RECOILBANDSTRUCTURE44 thetype1error.Inotherwords,bothtestsareextremelysensitivetosmalldeviations fromanormaldistribution.Therefore,bothtestshaveasignicantchanceatrejectingthe nullhypothesisthatthedistributionisnormalevenwhenitistrue.Thesensitivityof bothtestsmeansonemustuseboththetestsandthevisualrepresentationsofthedata histogramandQQplottotrulydeterminewhetherornotthedistributionsarenormal. Table2.6:Resultsofthedistributionanalysisforall8logarithmicallyspacedbinsfrom 10110keVfortheelectronrecoils.PvaluerepresentssignicanceforShapiroWilkand KolmogorovSmirnovtestsfornormality.AnyPvalue < 0.05indicatesrejectionofthenull hypothesisthatthedistributionisnormal.. Energybin[keV] G i K ShapiroWilk PValue SW KolmSmirnov PValue KS 1013.4 0.41 3.5 0.92 < 0 : 01 0.031 < 0 : 01 13.418.1 0.39 3.2 0.98 < 0 : 01 0.028 < 0 : 01 18.124.5 0.34 3.3 0.99 < 0 : 01 0.032 < 0 : 01 24.533.1 0.33 3.7 0.98 < 0 : 01 0.029 < 0 : 01 33.144.8 0.38 3.0 0.99 < 0 : 01 0.031 < 0 : 01 44.860.6 0.31 2.9 0.97 < 0 : 01 0.031 < 0 : 01 30.680.2 0.29 2.9 0.96 < 0 : 01 0.028 < 0 : 01 80.2110.0 0.29 2.8 0.95 < 0 : 01 0.032 < 0 : 01 Table2.7:Resultsofthedistributionanalysisforall8logarithmicallyspacedbinsfrom 10110keVfornuclearrecoils.PvaluerepresentssignicanceforShapiroWilktestfornormality.anyPvalue < 0.05indicatesrejectionofthenullhypothesisthatthedistributionis normal. Energybin[keV] G i K ShapiroWilk PValue SW KolmSmirnov PValue KS 1013.4 0.27 3.02 0.99 < 0 : 01 0.022 < 0 : 01 13.418.1 0.19 2.97 0.99 < 0 : 01 0.028 0 : 034 18.124.5 0.18 3.25 0.99 < 0 : 01 0.032 0 : 02 24.533.1 0.09 2.82 0.98 < 0 : 01 0.031 0 : 26 33.144.8 0.38 2.71 0.99 < 0 : 01 0.038 0 : 27 44.860.6 0.31 2.84 0.98 < 0 : 01 0.032 0 : 45 30.680.2 0.29 3.07 0.99 < 0 : 01 0.031 0 : 26 80.2110.0 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 containmentfractioniscalculatedcontinuouslyfrom10130keVbyintegratingequation 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 Fanofactorisincludedbyvaryingthenumberofelectronholepairsproduced. 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 dimensionaldistributionisabivariatenormalandthereforehasawelldenedstandard 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 valuesItis 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.TheverticalaxisisthespinindependentWIMPnucleoncross sectionunderstandardhaloassumptions,andthehorizontalaxisistheWIMPmass,where WIMPisusedtomeananylowmassparticledarkmattercandidate.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 parameterspacebutnotmuch.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 includingthefanofactorinthevariationofelectronholepairproductionasdiscussedin 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 ofelectronholepairs,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:HistogramaandQQplotbforelectronrecoilssimulatedatbincenter11.7 keV
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APPENDIXBDISTRIBUTIONPLOTS68 a b Figure2.2:HistogramaandQQplotbforelectronrecoilssimulatedatbincenter15.7 keV a b Figure2.3:HistogramaandQQplotbforelectronrecoilssimulatedatbincenter21.3 keV
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APPENDIXBDISTRIBUTIONPLOTS69 a b Figure2.4:HistogramaandQQplotbforelectronrecoilssimulatedatbincenter28.8 keV a b Figure2.5:HistogramaandQQplotbforelectronrecoilssimulatedatbincenter39.4 keV
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APPENDIXBDISTRIBUTIONPLOTS70 a b Figure2.6:HistogramaandQQplotbforelectronrecoilssimulatedatbincenter53.1 keV a b Figure2.7:HistogramaandQQplotbforelectronrecoilssimulatedatbincenter70.4 keV
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APPENDIXBDISTRIBUTIONPLOTS71 a b Figure2.8:HistogramaandQQplotbforelectronrecoilssimulatedatbincenter95.1 keV
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APPENDIXBDISTRIBUTIONPLOTS72 2.2NuclearRecoils: E r DependentFanoFactor a b Figure2.9:HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter11.7 keV
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APPENDIXBDISTRIBUTIONPLOTS73 a b Figure2.10:HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter15.7 keV a b Figure2.11:HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter21.3 keV
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APPENDIXBDISTRIBUTIONPLOTS74 a b Figure2.12:HistogramaandQQplotbfornuclearrecoilssimulatedatvbincenter 28.8keV a b Figure2.13:HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter39.4 keV
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APPENDIXBDISTRIBUTIONPLOTS75 a b Figure2.14:HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter53.1 keV a b Figure2.15:HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter70.4 keV
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APPENDIXBDISTRIBUTIONPLOTS76 a b Figure2.16:HistogramaandQQplotbfornuclearrecoilssimulatedatbincenter95.1 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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