GEANT4 SIMULATIONS FOR THE RADON ELECTRIC
DIPOLE MOMENT SEARCH AT TRIUMF
A Thesis
Presented to
The Faculty of Graduate Studies
of
The University of Guelph
by
EVAN THOMAS RAND
In partial fulfilment of requirements
for the degree of
Master of Science
April, 2011
©Evan Thomas Rand, 2011
ABSTRACT
GEANT4 SIMULATIONS FOR THE RADON ELECTRIC
DIPOLE MOMENT SEARCH AT TRIUMF
Evan Thomas Rand
University of Guelph, 2011
Advisors:
Professor C.E. Svensson
The existence of a permanent electric dipole moment (EDM) requires the violation
of time-reversal symmetry (T) or, equivalently, the violation of charge conjugation C
and parity P (CP). Although no particle EDM has yet been found, current theories
beyond the Standard Model, e.g. multiple-Higgs theories, left-right symmetry, and
supersymmetry (SUSY), generally predict EDMs within current experimental reach.
In fact, present limits on the EDMs of the neutron, electron and
199
Hg atom have
significantly reduced the parameter spaces of these models. The measurement of a
non-zero EDM would be the first direct measurement of a violation of time-reversal
symmetry, and it would represent a clear signal of CP violation from physics beyond
the Standard Model. The search for an EDM with radon has an enticing feature.
Recent theoretical calculations predict substantial enhancements in the atomic EDMs
for atoms with octupole-deformed nuclei, making odd-A Rn isotopes prime candidates
for the EDM search. Such measurements require extensive development work and
simulation studies. The Geant4 simulations presented here are an essential aspect
of these developments. They provide an accurate description of γ-ray scattering and
backgrounds in the experimental apparatus and γ-ray detectors, and are being used
to study the overall sensitivity of the RnEDM experiment at TRIUMF in Vancouver,
B.C.
Dictated but not read.
i
Acknowledgements
I would like to take the opportunity to thank a number of people who made this
project possible. Firstly, I would like to thank my supervisor Dr. Carl Svensson,
who guided me throughout my research with patience and much encouragement. I
am truly grateful for the opportunities that I have been given in the Nuclear Physics
Group. I would also like to thank Dr. Paul Garrett for his support, along with
the other members of the Nuclear Physics Group, to name a few: Jack Bangay,
Laura Bianco, Sophie Chagnon-Lessard, Greg Demand, Alejandra Diaz Varela, Ryan
Dunlop, Paul Finlay, Kyle Leach, Andrew Phillips, Michael Schumaker, Chandana
Sumithrarachchi and James Wong. I’ll always cherish the memories from experiments
and conferences.
Finally I would like to thank my family, who supports me throughout all my adventures. Thanks to my sisters Erin and Lauren, for teasing me relentlessly throughout
my childhood, and subsequently higher education. Your unique way of showing support is greatly appreciated, and will be reciprocated. Thanks to my parents Gerry
and Denise for their continuing encouragement throughout my graduate studies. And
to Christie, for putting up with the long hours, travelling and for pretending to be
interested in physics and my research in “time travel”.
ii
Contents
Acknowledgements
ii
1 Introduction
1
1.1
1.2
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.1
CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.1.2
The Standard Model . . . . . . . . . . . . . . . . . . . . . . .
5
Atomic EDMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.1
7
Radon EDM Enhancements . . . . . . . . . . . . . . . . . . .
2 RnEDM Experiment at TRIUMF
11
2.1
The ISAC Facility at TRIUMF . . . . . . . . . . . . . . . . . . . . .
11
2.2
RnEDM Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2.1
Transferring Radioactive Noble Gas Isotopes . . . . . . . . . .
15
Measuring Atomic EDMs . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.3.1
Optical Pumping of Rubidium Vapour . . . . . . . . . . . . .
17
2.3.2
Spin Exchange . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.3.3
RnEDM Measurement . . . . . . . . . . . . . . . . . . . . . .
25
2.3.4
Gamma-Ray Anisotropies . . . . . . . . . . . . . . . . . . . .
26
2.3.5
Statistical Limit . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.3
iii
2.4
GRIFFIN Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . .
3 Geant4 Developments for the RnEDM Experiment
3.1
3.2
3.3
3.4
3.5
3.6
27
32
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.1.1
Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.1.2
Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
Simulation Properties . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.2.1
Geant4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.2.2
Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.2.3
Volumes and Geometry . . . . . . . . . . . . . . . . . . . . . .
37
3.2.4
Physical Processes . . . . . . . . . . . . . . . . . . . . . . . .
38
3.2.5
Simulated Data . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Simulating β-Decay Process . . . . . . . . . . . . . . . . . . . . . . .
40
3.3.1
Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.3.2
Particle Emission . . . . . . . . . . . . . . . . . . . . . . . . .
42
Simulating Angular Distributions . . . . . . . . . . . . . . . . . . . .
47
3.4.1
Beta Particle Anisotropies . . . . . . . . . . . . . . . . . . . .
47
3.4.2
Gamma-Ray Anisotropies . . . . . . . . . . . . . . . . . . . .
49
3.4.3
Tracking m-States . . . . . . . . . . . . . . . . . . . . . . . .
52
RnEDM Geant4 Geometry . . . . . . . . . . . . . . . . . . . . . . .
54
3.5.1
Cell and Oven Design . . . . . . . . . . . . . . . . . . . . . . .
54
3.5.2
LaBr3 (Ce) Scintillator . . . . . . . . . . . . . . . . . . . . . .
55
Data Management . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.6.1
The GUGI Program . . . . . . . . . . . . . . . . . . . . . . .
58
3.6.2
Output Data and Sort Codes . . . . . . . . . . . . . . . . . .
59
iv
4 Results
4.1
4.2
4.3
60
γ-Ray Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.1.1
Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.1.2
223
Rn β-Decay Spectra . . . . . . . . . . . . . . . . . . . . . .
63
Frequency Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.2.1
Multipolarity Effects . . . . . . . . . . . . . . . . . . . . . . .
67
4.2.2
Fitting Process . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.2.3
Statistical Limit . . . . . . . . . . . . . . . . . . . . . . . . . .
74
LaBr3 (Ce) Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
5 Conclusions and Future Directions
79
5.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.2
Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
Appendix A
83
Appendix B
88
Bibliography
103
v
List of Tables
199
1.1
Current upper limits on EDMs of the neutron, electron and
3.1
Hexidecimal flags in Geant4 output binary data . . . . . . . . . . .
40
3.2
X-ray energies and intensities per 100 Fr K-shell vacancies . . . . . .
48
3.3
Three-dimensional γ-ray angular distributions for various transtions .
52
4.1
Fit results for the 416 keV γ-ray M1 transition . . . . . . . . . . . . .
73
vi
Hg atom
6
List of Figures
1.1
An illustration of the parity and time-reversal transformations . . . .
3
1.2
Timeline of EDM experimental upper limits. . . . . . . . . . . . . . .
7
1.3
Double well potential that arises in octupole-deformed nuclei.
. . . .
9
2.1
The ISAC-I Hall at TRIUMF . . . . . . . . . . . . . . . . . . . . . .
13
2.2
Schematic of the prototype on-line noble gas collection apparatus . .
16
2.3
Level diagram for a 4 He+ ion . . . . . . . . . . . . . . . . . . . . . . .
18
2.4
Level diagram of
Rb . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.5
Polarization transfer processes . . . . . . . . . . . . . . . . . . . . . .
24
2.6
The full 16 detector GRIFFIN array . . . . . . . . . . . . . . . . . .
28
2.7
One GRIFFIN/TIGRESS HPGe clover . . . . . . . . . . . . . . . . .
29
2.8
Cross-section of GRIFFIN heads in forward and back configurations .
30
2.9
GRIFFIN detectors in the forward configuration . . . . . . . . . . . .
31
85
2.10 GRIFFIN detectors in the back configuration
. . . . . . . . . . . . .
31
3.1
Cross section of the RnEDM simulated apparatus . . . . . . . . . . .
38
3.2
Geant4 simulation of radioactive decay . . . . . . . . . . . . . . . .
43
3.3
β particle angular distributions for various degrees of polarization. . .
49
3.4
γ-ray angular distributions for various multipolarities and spins . . .
53
3.5
Cross section of the EDM cell, oven, magnet and µMetal shielding . .
54
vii
3.6
Cross section of the BrilLanCe 380 LaBr3 (Ce) scintillator . . . . . . .
56
3.7
Screenshots of the GUGI program. . . . . . . . . . . . . . . . . . . .
57
3.8
FWHM2 versus γ-ray energy for a LaBr3 (Ce) and a HPGe detector .
58
4.1
Geant4 absolute efficiency curve for a ring of GRIFFIN detectors . .
62
4.2
Geant4 absolute efficiency curve for the RnEDM apparatus . . . . .
63
4.3
Geant4 absolute efficiency curve for various thicknesses of µMetal .
64
4.4
Full
Rn decay detected by a ring of eight GRIFFIN detectors . . .
65
4.5
The energy gate and time projection of the 416 keV γ ray. . . . . . .
66
4.6
Three-dimensional γ-ray angular distributions for transitions in
4.7
The time projections and fits for transitions in
Fr . . . . . . . . . .
69
4.8
The fit to the 416 keV γ-ray time projection. . . . . . . . . . . . . . .
70
4.9
Weighted average of 20 precession frequency fits . . . . . . . . . . . .
72
4.10 The sensitivity of the fitted frequency versus the simulated T2 time .
74
4.11 The sensitivity of the fitted frequency versus the number of counts . .
75
223
223
223
Fr
68
4.12 Geant4 absolute efficiency curve for a ring of BrilLanCe 380 detectors 76
4.13 Full
223
Rn decay detected by a ring of eight BrilLanCe 380 detectors .
77
4.14 LaBr3 (Ce) time projection and fit resulting from a large energy gate .
78
A.1 Simulated decay scheme (1 of 5) for the β − decay of
223
Rn to
223
Fr . .
83
A.2 Simulated decay scheme (2 of 5) for the β − decay of
223
Rn to
223
Fr . .
84
A.3 Simulated decay scheme (3 of 5) for the β − decay of
223
Rn to
223
Fr . .
85
A.4 Simulated decay scheme (4 of 5) for the β − decay of
223
Rn to
223
Fr . .
86
A.5 Simulated decay scheme (5 of 5) for the β − decay of
223
Rn to
223
Fr . .
87
viii
Chapter 1
Introduction
The search for an atomic electric dipole moment (EDM) in odd-A isotopes of radon
is beginning at TRIUMF, Canada’s national subatomic physics laboratory located in
Vancouver, British Columbia. The interest in particle and atomic EDMs derives from
the desire to understand the fundamental symmetries of the laws of physics and the
most basic origins of matter in the universe. The measurement of a permanent nonzero particle or atomic EDM would represent the discovery of new physics beyond the
Standard Model of particle physics and may explain the observed asymmetry between
matter and antimatter in the universe.
Despite over 50 years of searching with ever increasing experimental sensitivity, no
permanent non-zero particle or atomic EDM has been detected. However, many current theories for physics beyond the Standard Model, such as multiple-Higgs theories,
left-right symmetry, and supersymmetry (SUSY), predict EDMs within current experimental reach [1]. Present limits on the EDMs of the neutron, electron, and
199
Hg
atom have, in fact, already significantly reduced the allowed parameter spaces of these
models. The search for an EDM in radon is strongly motivated by recent theoretical calculations [2, 3, 4, 5, 6, 7] which predict large enhancements in the observable
1
atomic EDM for atoms in which the nucleus has a non-zero octupole deformation.
1.1
Motivation
The electric dipole moment (EDM), d~ = Σi ei r~i , of a particle or atom in an electric
~ can be described by the following Hamiltonian,
field E
~ .
Ĥ = −d~ · E
(1.1)
The EDM of a particle or atom is a vector quantity that must be either aligned or
~ Therefore d~ can be expressed as αS,
~ where α is
anti-aligned with the total spin S.
a constant of proportionality. The Hamiltonian for a system with an electric dipole
moment may thus be rewritten as
~ ·E
~ .
Ĥ = −αS
(1.2)
The Hamiltonian for this system is odd under both the parity (P̂ ) and time-reversal
(T̂ ) operations, as can be seen through:
P̂ Ĥ
= P̂ (−αŜ · Ê)
T̂ Ĥ
= T̂ (−αŜ · Ê)
= −α(+Ŝ) · (−Ê)
= −α(−Ŝ) · (+Ê)
= +αŜ · Ê
= +αŜ · Ê
= −Ĥ
= −Ĥ
Acting with the parity operator (P̂ ) on this Hamiltonian leaves the spin invariant,
but changes the sign of the electric field. Conversely, acting with the time-reversal
operator (T̂ ) on this Hamiltonian changes the direction of the spin and leaves the
electric field invariant. Both operations change the sign of the original Hamiltonian.
If parity or time reversal was a good symmetry of the laws of physics we would expect
2
P̂
ր
T̂
ց
Figure 1.1: An illustration of the parity and time-reversal transformations operating
on a quantum system with a charge distribution and non-zero spin. Acting with the
parity (P̂ ) operator on this system leaves the spin invariant, but changes the sign of
the electric dipole moment. Conversely, acting with the time-reversal (T̂ ) operator
on this system changes the direction of the spin and leaves the electric dipole moment
invariant. The resulting states on the right-hand side are equivalent to each other
under a rotation of 180◦ . Note that in both cases, the constant of proportionality
~ and d~ has changed sign under the P̂ or T̂ transformations.
between S
to find all particle and atomic EDMs to be zero. The measurement of a permanent
non-zero EDM would represent a violation of both of these symmetries. The violation
of parity symmetry by the weak nuclear force is well known. The direct violation of
time-reversal symmetry, however, has not been detected in any of the currently known
fundamental interactions of nature.
Figure 1.1 illustrates the effect of the P̂ and T̂ transformations on a quantum
system (particle or atom) with a charge distribution and non-zero spin. Acting with
the parity operator (P̂ ) on the quantum system flips the positions of the charges. This
3
~ in the above equations. The
is equivalent to changing the sign of the electric field (E)
P̂ transformation in Figure 1.1 changes the direction of the EDM from the upward
direction to the downward direction. The EDM is now anti-parallel to the total
~ and the magnetic moment ~µ. The magnetic moment, which can be
spin vector S
~ retains its constant of proportionality with the total spin vector
expressed as ~µ = g S
under both the P̂ and T̂ transformations. Acting with the time-reversal operator
(T̂ ) on the quantum system changes the direction of the spin. This is equivalent
~ in the above equations. The T̂
to changing the sign of the total spin vector (S)
~ from the upward direction
transformation in Figure 1.1 flips the total spin vector S
~ and the magnetic moment µ now
to the downward direction. The total spin vector S
point in the downward direction, anti-parallel to the EDM. The resulting quantum
systems on the right-hand side of the figure are equivalent via a rotation of 180◦ , and
~ has changed sign.
in both cases the constant of proportionality between d~ and S
One symmetry left out of this discussion so far is the charge conjugation symmetry (Ĉ), which exchanges particles for anti-particles. This symmetry, in combination
with the P̂ and T̂ form what is called the CP T symmetry. All experimental evidence
to date supports that CP T is a true symmetry of nature, known as the CP T Theorem [8]. As these symmetries were discovered, physicists believed that the law of
physics should be invariant under each of the three symmetries independently. This
view was challenged in 1956 when parity conservation in weak interactions was questioned [9]. Soon after this publication parity was shown to be violated in the β decay
of
60
Co nuclei [10]. This result implied violations in other combinations of C, P and
T in order for CP T to remain a good symmetry of the laws of nature.
4
1.1.1
CP Violation
ˆ
Following the observation of P̂ violations in the β decay, it was believed that CP
was a true symmetry of nature. CP violation was, however, observed in the decay
of K mesons [11] and also more recently in the decay of B mesons [12, 13]. CP
violation implies the existence of T violation via the CP T Theorem. Direct evidence
for time-reversal violation has been suggested in the transition rates of the antikaon
to kaon process, and its reverse, kaon to antikaon. However, these results remain
controversial [14, 15].
In 1967, Andrei Sakharov demostrated [16] that CP violation is essential for baryogenesis, the physical processes required to produce the asymmetry between baryons
and antibaryons in the early universe. In the Standard Model, CP violation enters
via weak interaction flavor mixing represented by the complex phase δCKM of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix and via θQCD , the vacuum expectation
value of the QCD gluon field. These sources of CP violation are, however, not sufficient to account for the observed asymmetry between matter and anti-matter in our
universe. Thus additional sources of CP violation are required and provide a strong
motivation to search for new physics beyond the Standard Model.
1.1.2
The Standard Model
The Standard Model of particle physics does predict the existence of non-zero
particle EDMs through δCKM and θQCD . However, these EDMs are many orders of
magnitude smaller than current experimental sensitivity (see Figure 1.2). On the
other hand, models beyond the Standard Model, such as multiple-Higgs theories,
left-right symmetry and supersymmetry (SUSY), generally include additional CPviolating complex phases and predict EDMs within current experimental reach [1].
5
Current upper limits on the EDMs of the neutron, electron and
199
Hg atom (see
Table 1.1) have, in fact, already significantly reduced the allowed parameter spaces
of these models.
Table 1.1: Current upper limits on the EDMs of the neutron, electron and
atom.
Species
Neutron
Electron
199
Hg Atom
1.2
EDM Upper Limit
< 2.9 × 10−26 e·cm [17]
< 1.6 × 10−27 e·cm [18]
< 3.1 × 10−29 e·cm [19]
199
Hg
C.L.
90%
90%
95%
Atomic EDMs
Measuring an EDM in a neutral atom is complicated by orbiting atomic electrons,
which arrange themselves to exactly cancel the EDM of a point nucleus. Due to the
finite size of the nucleus, however, the screening effect does not completely cancel the
observable atomic EDM. The intrinsic Schiff moment, the lowest order time-reversal
odd moment of a nucleus that is measurable in a neutral atom [5], is a measure of
the difference between the charge and dipole distributions in the intrinsic frame of
the nucleus. It is responsible for inducing the observable atomic EDM in the electron
cloud. The intrinsic Schiff moment is given by
~intr = 1
S
10
Z
1 ~
er ρ(r)~rd r −
dN
6Z
2
3
Z
ρs (r)r 2 d3 r ,
(1.3)
where d~N is the nuclear EDM, ρ(r) is the charge distribution of the nucleus and ρs (r)
the spherically symmetric part of the the charge distribution. It can be thought of as
the difference between the charge and dipole distribution in a nucleus of finite size.
6
Figure 1.2: Experimental upper limits for the EDMs of the neutron, electron and
199
Hg atom as a function of time compared to the ranges predicted for typical parameter in various models of particle physics. Adapted from reference [1].
1.2.1
Radon EDM Enhancements
The search for an atomic EDM with odd-A radon isotopes is motivated by the
predictions of large enhancements in the observable atomic EDM. Recent theoretical
calculations predict an enhancement factor of ∼600 for 223 Rn relative to 199 Hg [2, 3, 4,
5, 6, 7], which is the most sensitive EDM measurement to date [19]. This enhancement
is derived from three sources: octupole deformation of the nucleus, close-lying parity
doublet states in the nucleus, and the large Z of the isotope.
7
Certain neutron-rich isotopes of radon are predicted to have octupole deformed
nuclei [20]. The magnitude of octupole deformation can be described by the parameter
β3 , where β3 measures the presence of the octupole (L = 3) spherical harmonic in
the nuclear shape. The intrinsic Schiff moment of the nucleus is proportional to the
parameter β3 , hence a large octupole deformation gives a large intrinsic Shiff moment.
According to the Schiff theorem [21], the nuclear EDM is “screened” by the orbiting electrons. The observed EDM of the atom is rather induced by the Schiff moment.
The intrinsic Schiff moment of a permanent octupole deformed nucleus can be written
as [3]
Sintr = eZR03
9
√ β2 β3 ,
20π 35
(1.4)
where R0 is the nuclear radius and βL measures the presence of the spherical harmonic
of order L in the nuclear shape.
The second enhancement factor is induced by the existence of close-lying parity
doublet states that also arise from the nuclear octupole deformation [3]. The expectation value for the laboratory Schiff moment is the result of the mixing of these nearby
opposite parity states by a P and T odd interaction (V P T ),
hSlab i = 2α
I
Sintr ,
I +1
(1.5)
where I is the nuclear spin and
α=
hψ − | V P T |ψ + i
,
E+ − E−
(1.6)
where the even- and odd-parity states (ψ + and ψ − ) have energies E + and E − . These
states arise from a breaking of the degeneracy of octupole-deformed states in a doublewell potential as a function of β3 (see Figure 1.3). This is similar to the ammonia
molecule (NH3 ) in molecular physics. The nitrogen atom experiences a double-well
8
Figure 1.3: Illustration of the double well potential as a function of β3 that arises
for octupole-deformed nuclei. The magnitude of octupole (L = 3) deformation is described by the parameter β3 , where β3 measures the presence of the octupole spherical
harmonic in the nuclear shape.
potential with one potential well for the N atom on either side of the H3 plane. The
wave function for the N atom can be either symmetric or anti-symmetric with the
two states of opposite parity representing equal admixtures of the intrinsic states
populated by tunnelling through the H3 plane and split by a small energy difference
∆E associated with the tunnelling process. The same physics applies to octupole
deformed nuclei in which a doublet of states with opposite parity and a small energy
splitting ∆E results from equal admixtures of the two intrinsic states at ±β, and the
tunnelling through a large potential energy barrier at β3 = 0.
To completely describe the P -, T -odd nuclear potential V P T in Equation 1.6, a
two-body interaction is required. However for the purpose of estimating the collective
9
Schiff moment in the laboratory frame, an effective one-body potential is sufficient.
The one-body potential describing the CP -odd nucleon-nucleon interaction can be
expressed as [3]
V P T = −η
3G
√
δ(R0 − r ′ ) ,
3
8π 2mr0
(1.7)
where G is the Fermi constant, r0 is the internucleon distance and η parametrizes the
strength of the P T -odd interaction. From Equation 1.7, the collective Schiff moment
in the laboratory frame can be estimated as [3]
Slab ∼
0.05eβ2 β32 ZA2/3 ηr03
.
|E + − E − |
(1.8)
Equation 1.8 characterizes the Schiff moment in terms of the deformation parameters,
Z and A of the nucleus and the energy splitting between the opposite parity states.
The final enhancement factor derives from the large Z of the radon isotopes. In
1974, M.A. and C. Bouchiat demonstrated that parity-violating interactions in atoms
increase with the atomic number faster than Z 3 [22, 23]. This significant enhancement
encouraged (and continues to motivate) many studies for parity-violation studies in
heavy atoms. An increase of parity-violating interactions in the atom would result
in a larger observed EDM. Thus, heavier atoms are also more favourable for atomic
EDM searches.
These three factors, a collective octupole deformation of the nucleus, close-lying
parity doublet states and the high Z of radon, give certain odd-A radon isotopes a
large enhancement in the observable atomic EDM and thus make radon an excellent
candidate for an EDM search. As noted previously, detailed calculations for 223 Rn [2]
predict an enhancement of the observable atomic EDM by a factor of ∼600 relative
to the
199
Hg atom, which currently has the best atomic EDM upper limit at 3.1 ×
10−29 e·cm [19].
10
Chapter 2
RnEDM Experiment at TRIUMF
Certain odd-A radon isotopes are predicted to exhibit permanent octupole-deformation
and are of particular interest for an EDM experiment as they could have a significantly
enhanced sensitivity to fundamental CP-violating interactions [2]. These isotopes of
radon (221,223,225 Rn) are relatively short lived, (half-lives of ≃ 25 minutes) which make
it challenging to obtain large enough quantities to perform an EDM measurement using standard NMR techniques. Furthermore, these isotopes do not occur naturally
in the decay chains of 238 U and 232 Th. Therefore, the RnEDM experimental program
must be performed at a radioactive ion beam facility, such as TRIUMF, capable of
producing exotic nuclei, rapidily ionizing them, and delivering them to experiments
on timescales that are short compared to their half-lives.
2.1
The ISAC Facility at TRIUMF
TRIUMF is Canada’s national subatomic physics laboratory located on the campus of the University of British Columbia in Vancouver. TRIUMF, TRI-University
11
Meson Facility, is built around a 500 MeV proton cyclotron which provides simultaneously extracted beams with various intensities. Beams of rare isotopes are produced
at the Isotope Separator and ACcelerator (ISAC) facility at TRIUMF. The ISAC
facility uses an Isotope Separation On-Line (ISOL) technique to produce the RareIsotope Beams (RIBs). The ISOL system consists of a primary production beam of
500 MeV protons with an intensity up to 100 µA, a primary production target/ion
source, a high-resolution mass separator and beam transport system.
The production of a RIB at ISAC begins with the target and ion-source modules,
which are housed two floors below the experimental hall and encased in layers of steel
and concrete shielding. A schematic of the ISAC facility is shown in Figure 2.1. The
beam of 500 MeV protons bombards thick layered-foil targets and produce a variety
of exotic nuclides through spallation reactions. Heating the production target causes
the reaction products diffuse through the target material. Once outside the target,
a coupled ion source removes one or more electrons from the atoms, creating ions
which can be directed and accelerated electromagnetically. In principle, any bound
nuclide with proton (Z) and neutron (N) numbers less than or equal to those of the
target material can be produced. However, the division of proton and neutrons in the
spallation products are statistically distributed, favouring the production of isotopes
with N/Z ratios similar to that of the target material, with decreasing production
yields for more exotic isotopes with either larger proton or large neutron excess. In
addition, specific ion sources are most efficient at ionizing particular elements. The
efficiency of the combination of target and ion source is largely dependent on elemental
chemistry. For example, alkai metal elements are readily ionized through a surface
ion source, whereas the noble gas Rn isotopes of interest for the RnEDM experiment
will require either a FEBIAD (forced electron beam induced arc discharge) or an ECR
12
Figure 2.1: A schematic of the ISAC-I Hall at TRIUMF illustrating the locations
of: the accelerated proton beam, the target ion-source modules, high-resolution mass
separator, beam transport system and the RnEDM apparatus.
(electron cyclotron resonance) ion source.
The ionized products are sent to a high-resolution mass separator, which selects
nuclei of a specific charge-to-mass ratio according to the classical expression,
s
1 2m∆V
r=
,
B
q
(2.1)
where r is the radius of the circular orbit, B is the applied magnetic field, m is the
mass of the ionized product (proportional to its mass number A), q is the charge and
∆V is the voltage difference between the ion source and the mass separator. The
voltage difference between the ion source and the mass separator at ISAC is between
30 and 60 kV, therefore a singly-charged ion beam has an energy between 30 and
13
60 keV. A pair of adjustable slits downstream of the magnet are tuned to select the
radius of the charge-to-mass ratio of interest. The resolution of the mass separator,
typically
∆m
m
=
1
,
1000
is able to distinguish between neighbouring isotopes (different
mass number A), however, isobaric and even molecular contamination with the same
total A is possible. These contaminants can be reduced or eliminated by using an ion
source that selectively ionizes specific elements.
TRIUMF is licensed to operate the ISAC facility with proton beam intensities
up to 100 µA on target materials with Z ≤ 82 [24]. Two actinide targets, uranium oxide and uranium carbide, have been tested in the past two years to study
actinide beam production at ISAC. These tests have been conducted with proton
beam intensities of 2 µA for licensing reasons. The uranium-oxide target will remain
limited to approximately 2 µA due to the low operating temperature of the material.
The uranium-carbide target underwent its first tests in December 2010. This target
is projected to operate up to 75 µA using similar techniques as for other carbide
targets (silicon carbide, titanium carbide, zirconium carbine) [25]. These actinide
target developments are essential for the RnEDM experiment. They will not only
extend the range of available nuclei to higher masses, but also increase the achievable
neutron/proton ratio enabling the production of more exotic nuclei at the TRIUMF
ISAC facility.
2.2
RnEDM Apparatus
The RnEDM experimental program is beginning at TRIUMF. The final design
for the radon EDM measurement remains under development, however the apparatus
will be comprised of three basic sections: a target chamber, a transfer chamber and
14
a measurement cell. The RnEDM experiment will implant a beam of Rn ions (likely
221
Rn or
223
Rn) into a thin foil located in the target chamber. The collected Rn
atoms are then transferred into the measurement cell via the transfer chamber using
techniques discussed in the following section.
2.2.1
Transferring Radioactive Noble Gas Isotopes
The process of transferring radioactive noble gas isotopes on-line to a measurement
cell has been shown to be successful at TRIUMF [26]. A prototype noble gas collection
apparatus, shown in Figure 2.2, was tested with 120 Xe as beams of radon isotopes were
not available at the time of the tests. The 120 Xe isotope was chosen due its half-life of
40 minutes, comparable to the roughly 25 minute half-lives of
initial beam of
120
Cs produced the
120
221
Rn and
223
Rn. An
Xe isotopes through β decay; the decay chain
is shown in Equation 2.2. The beam was implanted in a thin zirconium foil for about
two
120
Xe half-lives, after which the remaining
minutes to decay into
120
120
120
Cs atoms were given roughly 10
Xe.
Cs(64 s) → 120 Xe(40 m) → 120 I(81 m) → 120 Te(stable) .
(2.2)
Valve V1 was closed to separate the target chamber from the beam line. Heating the
zirconium foil to about 1350 K released the xenon atoms into the target chamber volume. Opening the V2 valve allowed the xenon gas to diffuse into the transfer chamber,
where the xenon atoms froze onto a pre-cooled coldfinger. After cryopumping, the
V2 valve was closed and V3 to the cell was opened while simultaneously warming the
coldfinger to release the xenon gas. Once the coldfinger was warmed, the V4 valve
was opened which released a ballest volume of N2 gas into the transfer chamber. The
N2 gas expanded and pushed the xenon gas into the measurement cell. The V3 valve
15
Figure 2.2: Schematic of the prototype on-line noble gas collection apparatus tested
at TRIUMF with isotopes of 120 Xe [26].
was then closed to trap the xenon gas in the measurement cell. The trapped radioactive 120 Xe atoms were observed in the measurement cell using high-purity germanium
(HPGe) γ-ray detectors. This process of transferring the xenon from the foil to the
measurement cell was demonstrated to be greater than 40% efficient [26].
Improvements and adjustments have since been made to the initial prototype
design in order to enhance the overall transfer efficiency. In the summer of 2008,
improvements in the coldfinger design and nitrogen system resulted in a transfer
efficiency greater than 90% using radioactive isotopes of
2.3
121,123
Xe [27].
Measuring Atomic EDMs
High-precision measurements will be necessary in order to search for a non-zero
EDM in radon. Magnetic moments in nuclei are measured with great sensitivity using
16
NMR techniques, and supplementing standard NMR techniques with a strong electric
field provides an excellent method to search for an EDM. The process of measuring
an EDM using these techniques begins with polarizing the nuclei.
Nuclear spin polarization of noble gases is possible through spin-exchange collisions with optically pumped alkali-metals [28, 29, 30]. This method has been shown
to be successful in the polarization of radon isotopes (209,223 Rn) at the ISOLDE isotope separator at CERN [31]. Similar polarization studies have been tested and are
continuing to be developed for the RnEDM experiment at TRUMF.
2.3.1
Optical Pumping of Rubidium Vapour
Alkai-metals are commonly used for optical pumping since they have only one
valence electron. This electron can be easily excited with photon wavelengths conveniently in the range where diode lasers are available. The measurement cell in
the RnEDM apparatus will contain an optical pumping region, containing natural
rubidium (85,87 Rb) vapour, radon atoms and roughly 1 atm of N2 gas. The natural
rubidium vapour is the alkali-metal used to polarize the radon nuclei through spinexchange collisions and the N2 gas acts as a buffer. The natural rubidium (85,87 Rb)
atoms have odd-A nuclei and therefore have a non-zero nuclear spin (I) which complicates its level structure. To illustrate the process of optical pumping we will first
consider a much simpler atom with no nuclear spin.
Consider the 4 He+ ion, which is similar to the hydrogen atom without the nuclear
spin. The electron can be in its ground state (1S1/2 ), or given enough energy, it can
occupy the first excited state (2P1/2 ). Placing this atom in an external magnetic field
will cause the energy levels to spilt, known as the Zeeman effect. This is illustrated in
Figure 2.3. Both energy levels have J = 1/2; in the presence of a magnetic field the
17
m = + 1/2
Excited State (2P1/2)
m = _ 1/2
Energy
Excitation
Decay
m = + 1/2
Ground State (1S1/2)
m = _ 1/2
Magnetic Field
Figure 2.3: Level diagram for a 4 He+ ion. The splitting of the energy levels of the
ground state and first excited state increase with the applied magnetic field (not
drawn to scale). Shining positive helicity circularly polarized light on the atom drives
the transition from the m = −1/2 ground state to the m = +1/2 excited state. The
excited state can decay into both ground-state levels. The resulting effect “pumps”
the atom into the m = +1/2 ground state.
energy levels split into two distinct states, “spin up” (m = +1/2) and “spin down”
(m = −1/2) states.
Illuminating the atom with laser light of the correct frequency will induce transitions from either of the ground-state levels to either of the excited-state levels. If
the laser light is circularly polarized with positive helicity (σ + ) along the axis of
the B field, the transition must satisfy the selection rule ∆m = +1. This selection
rule derives from the conservation of angular momentum along the B axis, since the
positive helicity circularly polarized light carries a quantum of angular momentum
(~). The m = −1/2 ground state can be driven to the m = +1/2 excited state, but
the m = +1/2 ground state can not be excited since there is no m = +3/2 state
to transition to. Although the atoms can relax into either of the ground-state levels
18
Occupied
Orbitals
Fine
Structure
Hyperfine
Structure
Zeeman
Effect
2
P3/2
mF
5p
F=3
+3
2
P1/2
_3
F=2
_2
+2
D1
D2
F=3
+3
2
5s
S1/2
_3
F=2
_2
+2
Figure 2.4: Level diagram of
85
Rb (not drawn to scale).
(the selection rule for this process is ∆m = ±1 or 0 since the emitted photon can
have any polarization), only the m = −1/2 level can absorb the incident polarized
photons. The result is that the atoms become trapped in the m = +1/2 ground-state
level, this process is called “optical pumping” and can produce a very high degree of
polarization with almost all atoms occupying the m = +1/2 magnetic substate of the
1S1/2 ground state.
Natural rubidium (85,87 Rb) vapour will be used in the RnEDM experiment to
polarize the radon nuclei through spin-exchange collisions. The occupied electronic
orbitals of the rubidium atom in its ground state are
1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 5s .
The first 36 electrons are in closed sub-shells which gives zero total angular momentum. The valence electron in the 5s orbital behaves similarly to the above simplified
19
scenario with 4 He+ . The first excited state for this valence electron is the 5p level.
The angular momentum of the valence electron is given by
J=L+S
(2.3)
where L is the orbital angular momentum and S is the spin angular momentum. To
completely describe the total angular momentum of the atom, we must also consider
the nuclear spin I. The total angular momentum of the atom, F, is given by
F=J+I
(2.4)
where the nuclear spin for
85
Rb is I = 5/2 and for
87
Rb is I = 3/2. There are
four principle interactions which determine the energy levels in rubidium. In order of
decreasing strength, they are: the Coulomb interaction, the spin-orbit interaction, the
hyperfine interaction and the Zeeman effect. These interactions and the corresponding
splittings for the case of
85
Rb are illustrated in Figure 2.4.
The Hamiltonian for the Coulomb interaction is characterized by the principle
quantum number (n) and the orbital quantum number (l). It is given by
Ho =
p2
Ze2
−
.
2m
r
(2.5)
The typical energy scale for the Coulomb interaction is about 1 eV.
The first leading order perturbation to Ho is due to the interaction between the
spin magnetic moment of the electron and the magnetic moment produced by the
orbit of the electron around the nucleus. This term is called the spin-orbit coupling
and its Hamiltonian is given by,
Hso =
ge Ze2 1
L·S ,
4m2 c2 r 3
(2.6)
where ge is the electronic g-factor. The splittings which result from this interaction
are called the fine structure and have an energy scale of approximately 10−4 eV. These
20
energy levels are separated according to the value of J. The 5s orbital does not split,
as shown in Figure 2.4, since l = 0. The energy difference between the 2 S1/2 and
2
P1/2 levels coresponds roughly to a 794.8 nm wavelength and is called the D1 line.
The D2 line coresponds to the energy difference between the 2 S1/2 and 2 P3/2 levels
and has a wavelength of about 780 nm.
The next correction to Ho is due to the interaction between the magnetic moment
of the electron and the nuclear magnetic moment. These splittings are called the
hyperfine structure and are given by the following Hamiltonian,
Hhf
Ze2 gN 1
1
21
=
,
S · − I∇ + ∇(I · ∇)
2mMN c2 4π
r
r
(2.7)
where gN is the nuclear g-factor. These levels split according to the total angular
momentum F .
85
Rb has a nuclear spin of I = 5/2, thus the allowed values of F are
5/2 − 1/2 = 2 and 5/2 + 1/2 = 3, as shown in Figure 2.4. These splittings are on the
order of 10−6 eV.
The final splitting is due to the Zeeman effect, also shown in the above simplified
example. The splitting occurs in the presence of a weak external magnetic field (B).
Classically, the magnetic moment of a particle with charge q and angular momentum
L is given by,
µ=
q
L.
2mc
(2.8)
Extending this result into quantum mechanics, the magnetic moment due to the total
electronic angular momentum is given by,
µJ = −gJ
e
J,
2mc
(2.9)
and the magnetic moment of the atom is given by,
µF = −gF
e
F,
2mc
(2.10)
21
where the gJ and gF are the Landé g-factors. These factors arise from the addition
of angular momentum operators. The Landé g-factors are given by,
gJ = 1 +
J(J + 1) + S(S + 1) − L(L + 1)
,
2J(J + 1)
(2.11)
and
gF = gJ
F (F + 1) + J(J + 1) − I(I + 1)
.
2F (F + 1)
(2.12)
The energy splittings for an atom in a weak magnetic field is given by HB = −µF · B,
which results in the following first-order perturbation,
HB = gF µB mF Bz ,
(2.13)
where µB is the Bohr magneton and mF ~ is the eigenvalue of the Fz operator. The
energy splittings are approximately linear for small external magnetic fields. The
splittings of the 2 S1/2 , F = 3 state in
85
Rb is 1.9293 × 10−9 eV/Gauss. As stated
above, the linear relationship between the energy levels and the applied magnetic field
is only valid for small magnetic fields. With a large applied magnetic field the size of
the Zeeman splittings become comparable to the hyperfine energy difference. At this
point the quantum mixing between the Zeeman splittings and the hyperfine states
needs to be accounted for, i.e. the eigenvalues of the Hamiltonian H = Hhf + HB
need to be solved. This result is known as the Breit-Rabi formula:
Ehf
1
E(F = I±1/2, mF ) = −
±
2(2I + 1) 2
The pumping process for the 5s
85
r
2
+
Ehf
4mF
gJ µB BEhf + (gJ µB B)2 . (2.14)
2I + 1
Rb electron is similar to the simplified example
discussed above for 4 He+ . Illuminating the
85
Rb atom with laser light tuned to the
D1 line will drive transitions from any of the 2 S1/2 ground states into any of the
2
P1/2 excited states. If the laser light is circularly polarized with positive helicity
22
(σ + ) along the axis of the B field, the transition must satisfy the selection rule
∆mF = +1. Because there is no mF = +4 magnetic substate in the 2 P1/2 excited
state, see Figure 2.4, any electrons in the mF = +3 magnetic substate of the 2 S1/2
ground state will remain trapped there. As the 2 P1/2 excited states relax through
photon emission the probability of populating the mF = +3 magnetic substate of
the 2 S1/2 ground state is determined by the selection rule ∆m = ±1 or 0 for photon
emission. After several S1/2 - P1/2 - S1/2 pumping cycles almost all the atoms will
end up trapped in the mF = +3 magnetic substate of the 2 S1/2 level, and thus, the
mF = +3 ground state has been optically “pumped” into an ensemble of atoms with
a very high degree of polarization.
2.3.2
Spin Exchange
Optically pumped alkali-metal atoms can polarize noble-gas atoms via spin-exchange
interactions. The Hamiltonian that describes the interaction between alkali-metal and
noble-gas atoms is [29]
H = AI · S + γN · S + αK · S + gs µB B · S + gI µB B · I + gK µB B · K + · · · , (2.15)
where S is the spin of the alkai-metal valence electron, I is the alkai-metal nuclear
spin, K is the noble-gas nuclear spin, N is the rotational angular momentum of the
formed alkai-metal noble-gas van der Waals molecule, and B is the external magnetic
field.
The transfer of angular momentum can occur while the atoms are bound in shortlived van der Waals molecules or by simple binary collisions between atoms, shown
in Figure 2.5. For light noble gases, such as 3 He, binary collisions dominate the
transfer of angular momentum and the contribution from van der Waals molecules is
23
(a) Formation and
breakup of an alkalimetal/noble-gas van
der Waals molecule
(b) Binary collision
between an alkalimetal atom and a
noble-gas atom
Figure 2.5: Polarization transfer process.
negligible. For heavier noble gases, such as Xe and Rn, the contributions of van der
Waals molecules dominate over the contribution of binary collisions at low pressures.
The pressure of the buffer gas participates in the creation and destruction of the
formed alkai-metal noble-gas van der Waals molecule (Figure 2.5). At high pressures
(multiatmosphere pressures) the collisions from N2 greatly suppress the lifetime of
the formed van der Waals molecule. Thus binary collisions dominate the transfer of
angular momentum.
The RnEDM experiment will use high pressures in the measurement cell, where
the N2 gas also plays a role in the optical pumping process. The N2 gas will nonradiatively de-excite (“quenche”) the excited rubidium atoms before they can reradiate a photon. This avoids radiation trapping, where a photon is emitted by one
24
atom and absorbed by another. Radiation trapping can destroy the polarization of
rubidium atoms since the emitted photons can have any polarization (∆m = ±1 or 0).
This can excite an electron in the trapped mF = +3 ground state and thus depolarize
the atom.
2.3.3
RnEDM Measurement
Once radon nuclei have been polarized inside the cell, an EDM will be sought
via NMR techniques. The measurement cell containing the radon nuclei will be
located inside coils generating a magnetic field. The radon nuclear spins will be
polarized along the magnetic field axis by the optical pumping and spin-exchange
processes described above. Applying high voltage to integrated electrodes in the
measurement cell will generate an electric field parallel or anti-parallel to the direction
of the magnetic field. With the application of an RF pulse, the polarized radon nuclei
will begin to precess about the magnetic and electric field axis at a frequency
~ω± = 2µB ± 2dE ,
(2.16)
where µ is the magnetic moment, B is the magnetic field, d is the electric dipole
moment and E is the electric field. The +(−) corresponds to the electric field oriented
parallel (anti-parallel) to the magnetic field. The EDM,
d=
~∆ω
,
4E
(2.17)
is extracted through a measurement of the change in precession frequency ∆ω between
the two different electric field orientations.
25
2.3.4
Gamma-Ray Anisotropies
The short half-life of the radon isotopes of interest make it difficult to obtain
sufficient quantities to observe an EDM using standard NMR techniques. Therefore,
the RnEDM experiment will measure the precession frequency by detecting the γ
radiation from the decaying radon nuclei. The angular distribution of the γ radiation
is dependent on the angle with respect to the precessing polarization vector of the
nuclear spins. In addition to the γ rays, the emitted β particles from the decay of
the polarized Rn nuclei also have an anisotropy that would enable the precession
frequency to be measured. However, these β particles will be strongly scattered and
absorbed by the glass in the measurement cell and oven. In order to use the β particles
as a means to observe an EDM signal, β detectors would need to be integrated into
the measurement cell. Hence, the first stage of the RnEDM experiment will use the
γ-ray anisotropies to measure the precession frequency.
2.3.5
Statistical Limit
The sensitivity of the EDM measurement relies on detecting a small change in
the precession frequencies between the two different electric field orientations. The
statistical limit to the precision of this measurement can be readily calculated [32].
The change in the precession frequency (∆ω = 4dE/~) will signal a non-zero EDM
in radon. The precision of this measurement for N detected γ rays is given by [32]
s
2
1
δ∆ω =
,
(2.18)
2
T2 A (1 − B)2 N
where T2 is the spin-decoherence time, A is the analyzing power for a measurement
that detects a change of counts ∆N = AN, and B is the fraction of N due to
background. For a ring of eight high-efficiency germanium detectors (see Section 2.4)
26
an average of 120 kHz photopeak count rate is possible. Thus, for 100 days of counting
we would expect approximately N = 1 × 1012 photopeak counts, with negligable
background when gated on the γ-ray photopeaks. Studies with 209 Rn at Stony Brook
[32] have demonstrated 30 seconds for the spin-decoherence time, and a value of 0.2 for
the analyzing power is typical for γ-ray anisotropies. With an electric field of 5 kV/cm
we calculate the sensitivity for the EDM measurement (σd ) to be approximately
1×10−26 e-cm using the γ-ray anisotropy technique. For the β asymmetry method, we
expect larger backgrounds, B ≈ 0.2, but a significantly higher count-rate capability.
The count rate in the β detectors will only be limited by the decay rate of the available
number of radon nuclei, allowing a total detected count rate of order 5 MHz. An EDM
sensitivity of 2 × 10−27 e-cm is thus expected for 100 days of counting using the betaasymmetry technique [32].
These estimates are statistical limits and do not account for the realistic sources
of backgrounds in the actual γ-ray experiment caused by bremsstrahlung production
from the stopping β-particles and Compton scattering of photons both into and out
of the γ-ray detectors which can lead to backgrounds underneath the γ-ray photopeaks with different angular distributions than the γ-rays of interest. The detailed
simulations presented in this thesis are motivated by the need to study the realistic
signal for the precession frequencies that will ultimately determine the sensitivity of
the RnEDM measurement using the γ-ray anisotropy technique (see Chapter 3).
2.4
GRIFFIN Spectrometer
To observe the γ radiation in the RnEDM experiment the recently funded GRIFFIN spectrometer will be utilized. GRIFFIN stands for Gamma-Ray Infrastructure
27
For Fundamental Investigations of Nuclei. The GRIFFIN array, shown in Figure 2.6,
will be comprised of 16 unsegmented large-volume clover-type high-purity germanium
(HPGe) detectors with full Compton-suppression shields constructed of bismuth germanate (Bi4 Ge3 O12 ), commonly referred to as BGO. Each of the sixteen GRIFFIN
clover detectors will consist of four individual HPGe cyrstals cut to meet along flat
edges (Figure 2.7), to enable efficient packing of the detectors.
Figure 2.6: The full 16 detector GRIFFIN array.
The BGO suppression shields around each detector will be comprised of front,
side and back suppression shields. The front shields may be pulled back or pushed
fully forward, giving the array two different configurations illustrated in Figure 2.8.
When the front shields are pulled back, see Figure 2.9, the detector is in its highest
efficiency mode with the HPGe detectors close-packed with an inner radius of 11.0 cm.
When the front shields are pushed forward, see Figure 2.10, the detector is in its
fully suppressed mode optimizing the peak-to-total ratio. In this configuration the
suppression shields are close-packed with the HPGe detector faces at a radius of
28
Figure 2.7: One GRIFFIN/TIGRESS HPGe clover comprised of four individual crystals. Each crystal is 90 mm in length and 60 mm in diameter and has an efficiency
of 40% relative to a standard 3”× 3” NaI(Tl) crystal for 662 keV γ rays.
14.5 cm. The front suppressors in this configuration also have the option of being
collimated with a dense metal (hevimet). The dense metal, composed mostly of
tungsten, prevents γ rays from directly hitting the front suppression shields.
The GRIFFIN spectrometer is very similar in external geometry to its sister array TIGRESS (TRIUMF-ISAC Gamma-Ray Escape Suppressed Spectrometer) [33],
which currently resides in the ISAC-II experimental hall at TRIUMF. The primary
difference between the two designs is that a TIGRESS detector has each HPGe crystal electrically segmented into 8 segments. This results in 32 electrical signals (plus
four addition core contacts) per clover, whereas a GRIFFIN detector will have only
4 electrical signals (the four core contacts) per clover. The principle reason for TIGRESS’s highly segmented crystals is to provide three-dimensional localization of
γ-ray interactions inside the crystals [34], which is necessary for experiments with
the accelerated radioactive ion beams at ISAC-II. GRIFFIN, on the other hand, will
reside in the ISAC-I hall and will be used primary as a decay spectrometer with
low-energy radioactive ion beams. The segmentation of the TIGRESS detectors is
29
(a) Back detector configuration
(b) Forward detector configuration
Figure 2.8: Geant4 renderings of two GRIFFIN detector heads cross-sectioned in
both forward and back configurations.
therefore not necessary for GRIFFIN.
A full Geant4 simulation for TIGRESS detectors has been extensively modelled
and verified by measurements of peak-to-total ratios and relative efficiencies [35, 36].
This provided an excellent foundation to build a simulation of the RnEDM experiment using GRIFFIN detectors. In terms of Geant4 simulation, both detector
systems may be modelled identically. Essentially, the performance of a GRIFFIN
detector in terms of γ-ray interactions in the detector materials should be identical
to an unsegmented TIGRESS detector. From this fully verified Geant4 detector
framework for TIGRESS, a realistic Geant4 simulation for the RnEDM experiment
at TRIUMF involving 8 GRIFFIN detectors in a ring geometry (see Figures 2.9 and
2.10) was developed as part of the work presented in this thesis.
30
(a) One GRIFFIN detector in the HPGe
forward configuration
(b) Ring of eight GRIFFIN
detectors close-packed in forward configuration
Figure 2.9: Geant4 renderings of the GRIFFIN detectors in the forward configuration (highest efficiency mode).
(a) One GRIFFIN detector in the
HPGe back configuration
(b) Ring of eight GRIFFIN detectors close-packed in
back configuration
Figure 2.10: Geant4 renderings of the GRIFFIN detectors in the back configuration
(optimized peak-to-total).
31
Chapter 3
Geant4 Developments for the
RnEDM Experiment
This chapter presents the development of the Geant4 simulations for the RnEDM
experiment at TRIUMF. The goal of the Geant4 simulations is to provide an accurate description of γ-ray scattering and backgrounds in the experimental apparatus
and γ-ray detectors. From this Geant4 framework, realistic measurements for the
RnEDM experiment can be simulated and ultimately test the sensitivity of the apparatus. The Geant4 simulations presented in this thesis are an essential aspect of
the development towards an EDM measurement.
32
3.1
3.1.1
Introduction
Previous Work
The foundation of the RnEDM simulations was built on the existing Geant4
simulations of the TRIUMF-ISAC Gamma-Ray Escape Suppressed Spectrometer (TIGRESS) [35]. The geometry of the TIGRESS Geant4 simulation was constructed
from design drawings of the detector and suppression shields. The performance of
the simulation was validated with measurements conducted with prototype TIGRESS
γ-ray detectors [36]. As discussed in Section 2.4, the GRIFFIN γ-ray spectrometer
is very similar in external geometry to its sister spectrometer TIGRESS. The primary difference between the two γ-ray spectrometers is the crystal segmentation.
The outer electrical contact of a TIGRESS crystal is highly segmented, to allow for
a three-dimensional localization of γ-ray interactions inside the crystal. GRIFFIN,
on the other hand, will be used primarily as a decay spectrometer with low-energy
radioactive ion beams and the crystal segmentation is therefore not necessary. The
performance of the GRIFFIN γ-ray spectrometer will, however, be very similar to
that of an unsegmented TIGRESS γ-ray spectrometer. As electrical contact segmentation for TIGRESS has been incorporated at the analysis, rather than simulation,
stage, the fully verified Geant4 simulation framework for TIGRESS can, in fact, be
used as a Geant4 simulation of GRIFFIN.
3.1.2
Modifications
A significant number of modifications were made in the previous Geant4 code
used with TIGRESS for the RnEDM studies presented in this thesis. Firstly, the
code structure was initially “hard-coded” to only allow for one detection system in
33
the simulation. For many decay-spectroscopy experiments, multiple detection systems are used to detect a variety of particles including: γ rays, internal conversion
electrons, β particles and α particles. The code structure was therefore rewritten in
an object oriented manner in order to increase flexibility and give the user the option
of including multiple detection systems and data streams. Furthermore, writing the
code in an object oriented fashion made it much more transferable, allowing easy
integration into other Geant4 simulations.
Following the code restructuring it was noted that the layered volume geometries
in Geant4 were no longer being handled properly. In the original TIGRESS simulation if two geometries, composed of different materials, were overlapping in the threedimensional physical space, the last constructed geometry would fill any overlapping
space and this method was frequently used to create volumes inside other volumes. In
the rebuilt code, the two geometries would simply occupy the same space. To resolve
this, the proper method of volume subtraction was employed. In this method physical geometries are “cut” using similar geometric shapes. This avoided all overlapping
volumes and ensured that all geometries were composed of the proper materials with
correct densities.
3.2
Simulation Properties
The RnEDM simulation at its most fundamental level is a Monte Carlo simulation of particle interactions in matter. The particle interactions are handled by the
Geant4 toolkit [37], which uses well-developed electromagnetic processes to Monte
Carlo the interaction of γ rays, electrons and positrons in matter.
The β-decay simulation developed for the RnEDM experiment selected particle
34
energies, branching ratios, emission times and angular distributions from calculated
probability distributions. Random numbers were generated from a uniform distribution between [0.0, 1.0) to select from these probability distributions. The random
numbers were generated with the srand48 and drand48 functions from the C++ standard library, where the srand48 function was seeded on the system clock to ensure
initial randomness during each use.
3.2.1
Geant4
Geant4 is a C++ toolkit used to simulate the passage of particles through matter
[37]. Encompassing an abundant set of physical models, Geant4 handles geometry,
physical processes and particle interactions. Geant, which is usually pronounced like
the French word géant (giant), stands for “GEometry ANd Tracking”. The Geant
code was originally developed by CERN for high-energy physics simulations. Today
Geant4 is the leading computation method for high-energy, nuclear and accelerator
physics simulations, as well as a growing influence in the medical and space science
fields.
Geant4 simulations are built upon well-developed Monte Carlo models which
are tested and maintained by the Geant4 collaboration of scientists and software
engineers [38]. Geant4 users construct three-dimensional simulation environments
by defining volumes and materials. From simple geometrical shapes, such as cubes,
spheres, cylinders and cones, complex geometries are built, where the materials are defined by combinations of atomic elements. Inside the three-dimensional environment,
users fire particles (α, β, γ, neutron, proton, etc.) with defined energies and directions.
Interactions which deposit energy in defined materials lead to the recording of energy,
time, and position information. Interactions that lead to the cascading production of
35
energetic particles, such as a γ-ray above 1.022 MeV producing an electron-positron
pair or the slowing of an energetic β particle emitting bremsstrahlung photons, lead
to all of the cascading particles with mean paths greater than a user defined cut
off, which was set to 10µm in this work, being tracked by Geant4. The simulations presented in this thesis were performed using Geant4 version 9.2 patch 4
(geant4.9.2.p04 ).
3.2.2
Materials
All materials in Geant4 are generated from user-defined atomic elements. The
elements are defined based on atomic numbers and masses. From these basic building
blocks larger volumes of solids, gases and liquids are generated. The majority of
materials used in the simulations were already created and verified in a previous
work [35]. Three significant additions were made to the materials class, namely,
PYREX glass, µMetal shielding and cerium-doped lanthanum bromide crystal.
PYREX glass is used in the construction of the EDM cell and oven design. The
simulated glass was formulated with specifications from the Corning company [39].
PYREX glass has a density of 2.23 g/cm3 and is composed of (by weight) 80.6%
silicon dioxide (SiO2 ), 13.0% boron trioxide (B2 O3 ), 4.0% sodium oxide (Na2 O), 2.3%
aluminium oxide (Al2 O3 ) and 0.1% miscellaneous traces. In these simulations the
miscellaneous traces were composed of equal parts of potassium oxide (K2 O), calcium
oxide (CaO), dichlorine (Cl2 ), magnesium oxide (MgO) and iron three oxide (Fe2 O3 ).
µMetal is a nickel-iron alloy which has a very high magnetic permeability. The
high magnetic permeability makes µMetal an effective magnetic shield. The RnEDM
experiment is extremely sensitive to magnetic field fluctuations and µMetal will be
36
used around the measurement cell in order to maintain a static magnetic field. Unfortunately, this shielding between the EDM cell and the γ-ray detectors will also
attenuate the emitted γ rays from the decaying radon nuclei; thereby lowering the
over-all efficiency of the system. This effect was studied in the RnEDM simulations
and is presented in Chapter 4. The simulated µMetal was formulated from specifications given by The MuShield Company© [40]. µMetal has a density of 8.747 g/cm3
and is composed of (by weight) 80.0% nickel, 14.93% iron, 4.20% molybdenum, 0.50%
manganese, 0.35% silicon and 0.02% carbon.
The simulated cerium-doped lanthanum bromide crystal was used in the construction of Saint-Gobain’s BrilLanCe 380 detector. The performance of this scintillator
and its role in the RnEDM experiment is discussed in Section 3.5.2. According to
Saint-Gobain, their cerium-doped lanthanum bromide (LaBr3 (Ce)) crystal has a density of 5.08 g/cm3 with approximately 5.0% cerium [41, 42].
3.2.3
Volumes and Geometry
Complex volumes and geometries are constructed from combinations of much
simpler shapes (cubes, spheres, cylinders, cones, etc.). Each shape may have its own
defined material (see Section 3.2.2). From these basic building blocks complicated
geometries, such as the RnEDM experimental setup shown in Figure 3.1, can be
constructed.
The entire Geant4 experimental geometry was built in a large cube defined as
the “experimental hall”. The “experimental hall” was usually constructed of vacuum,
but could also be defined as air or any other material. Each component of the detector
and experimental setup was then added into this volume, defined around a point of
origin in the three-dimensional space. The origin was reserved for particle emission
37
Figure 3.1: Cross section of the RnEDM simulated apparatus. The EDM cell and
oven are surround by a ring of GRIFFIN γ-ray detectors in their highest efficiency
mode.
due to the symmetric construction of the detectors.
In Geant4, volumes which record information or “hits” are defined as “sensitive”.
A hit is a snapshot of a physical interaction or an accumulation of interactions inside
a sensitive volume. The hit energy, time, and position of the interaction can be
recorded and written to an output file.
3.2.4
Physical Processes
The particles involved in the RnEDM simulations were photons, electrons and
positrons, thus the physical processes describing their interactions were entirely electromagnetic. Only the decay products (γ rays, β particles and internal conversion
electrons) were simulated.
Interaction probabilities for γ rays within a material were calculated by Geant4’s
built-in electromagnetic processes [37].
The possible physical processes included
photoelectric absorption, Compton scattering and pair production. Similarly, the
built-in Geant4 models were used for the interaction probabilities of electrons and
positrons. The possible physical processes included multiple scattering, ionization,
bremsstrahlung creation and annihilation for positrons.
38
3.2.5
Simulated Data
The code was written so that the output from the Geant4 simulations was written
as binary files, containing all the interaction energy, time, and position information
within the simulation. The advantage of writing information in binary files versus
ASCII is its compactness. For example, the number 1000000000 in ASCII would
require 10 bytes to store (10 characters long), whereas if it were represented as an
unsigned binary it would only use 4 bytes. The sheer number of simulations needed
to achieve desirable statistics made use of this compactness. The other option, of
course, was to write histograms and spectra directly from the Geant4 simulation.
This method does not preserve the correlations between event energies, times and
locations that are included in the list-mode binary event data and often desirable
during the analysis phase.
The binary stream was written in small segments which correspond to individual
events. One event represents one β-decay process including the subsequent γ decays
and internal conversion processes until the daughter nucleus reaches its ground state.
Every event in the output stream begins with the hexidecimal flag 0x8000 and ends
with 0xFFFF. The binary data between these flags encompasses all the information
relating to a single decay, including the energy deposited inside detectors, position
information of the γ ray interactions, pair production processes, bremsstrahlung processes, and their interaction times. See Table 3.1 for a list of all flags used in the data
stream.
39
Table 3.1: Hexidecimal flags in Geant4 output binary data
Flag Type
Start Flag
Normal Event Data
0x8000
Photon Position Data
0x8010
Pair Production Photon 1 Position Data
0x8020
Pair Production Photon 2 Position Data
0x8030
Bremsstrahlung Photon Position Data
0x8040
Timing Data
0x8050
3.3
End Flag
0xFFFF
0xFFE0
0xFFD0
0xFFC0
0xFFB0
0xFFA0
Simulating β-Decay Process
The simulations presented in this thesis focused on the β decay of
223
Rn to
223
Fr,
a likely candidate for the RnEDM search at TRIUMF. A simulation of the entire βdecay process was developed as part of this thesis. This required a significant amount
of input data including the spin, parity and half-life of the parent nucleus, the proton
number of the daughter nucleus, the Q-value for the β decay, level spins, parties, γray energies, intensities, multipolarities, mixing ratios, internal conversion coefficients
and X-ray shell vacancies in the daughter nucleus. Unfortunately, the level structure
of
223
Fr [43] is not known to a very high precession. In fact, a significant number of
the observed γ rays from an experiment at ISOLDE [43] could not be placed in the
derived level structure. For the purposes of simulation those γ rays were given their
own energy level with the appropriate intensity, see Appendix A on page 83 for the
complete simulated β decay scheme.
The decay mode of 223 Rn is 100% β − decay. The simulation of this decay began by
emitting a β particle (electron) into the Geant4 geometry. The energy and angular
distribution of the β particle are described in Sections 3.3.2 and 3.4.1 respectfully.
Following the initial β decay, the daughter nucleus may exist in an excited state. The
excited daughter nucleus will then undergo γ decay and/or internal conversion to
40
lower its total energy. Internal conversion occurs when the excited nucleus interacts
with an electron in one of the lower atomic orbitals (K, L, etc), causing the electron
to be ejected from the atom. This process becomes dominate for higher Z nuclei and
is more probable for transitions of lower energy. The γ decay and internal conversion
processes are competitive. For any excited state the probabilities for either process
are given by
Pγ =
1
α
, PIC =
,
1+α
1+α
(3.1)
where α is the total internal conversion coefficient.
Internal conversion coefficients used in the RnEDM simulations were calculated
with the BrIcc v2.2b program [44]. The BrIcc program calculates theoretical internal
conversion coefficients based on a relativistic self-consistent Dirac-Fock model [44].
Given the proton number and energy of the transition, the program outputs the total
and individual shell conversion coefficients for multipolarities up to L = 5. For mixed
transitions involving multipolarities L1 and L2, such as an M1+E2 transition, the
conversion coefficient can be calculated by [44]
α=
δ 2 αL2 + αL1
(1 + δ 2 )
(3.2)
where δ, the mixing ratio, is defined as
δ=
hJf ||L2 ||Ji i
.
hJf ||L1 ||Ji i
(3.3)
The total internal conversion coefficient is defined as a sum of all the individual shell
internal conversion coefficients, such that αTot = αK + αL + αM + · · · . Thus, the
probabilities for an electron to be emitted from a particular shell is easily calculated
from the BrIcc output. For example, the probability for an electron to be emitted
from the K shell is simply αK /αTot . Simulating the atomic shell structure enabled
41
an accurate description of the electron energy and resulting X-ray energy. X rays are
produced when electrons transition from higher orbitals to the lower vacant orbitals.
The simulation of the X-rays are described in detail in Section 3.3.2.
3.3.1
Timing
The time of each individual β-decay event from the ensemble of
223
Rn nuclei was
generated using Monte Carlo from the input half-life and number of nuclei remaining. The time differences between successive events in a radioactive decay can be
simulated using a well known Monte Carlo method for sampling from an exponential
distribution [45]. The time intervals are given by
∆t =
−1
ln(1 − χ) ,
N ′λ
where λ is the decay constant (λ =
(3.4)
ln(2)
),
t1/2
N ′ is the remaining number of nuclei
and χ is a random number generated from a uniform distribution between [0.0, 1.0).
Keeping track of these small time differences generated the absolute time of each
decay event in the simulation. Figure 3.2 illustrates the decay of one hundred million
223
Rn nuclei detected by a ring of GRIFFIN detectors and confirms the validity of
the Monte Carlo by fitting the decay with a maximum likelihood function resulting
in excellent agreement of the input 24.3 minute half-life.
3.3.2
Particle Emission
Particle emission was handled by the Geant4 Particle Gun. The Particle Gun
was passed the type of particle to be emitted (photon, electron or positron), its energy, initial position and momentum direction. The emission direction generally had
two main user options which were selected in the GUGI program before running the
42
10000
2
χred = 1.21
1000
Counts
t1/2 = 24.300(6) min
100
10
0
50
100
Time (min)
150
200
Figure 3.2: Geant4 simulation of the decay of 108 223 Rn nuclei detected by a ring of
eight GRIFFIN detectors in their highest efficiency mode with the radiation emitted
isotropically into 4π. The times were binned into seconds and the function used to
− ln(2)t
fit was y = A1 + A2 e A3 , where A1 is due to background, A2 is initial count rate
and A3 is the half-life (t1/2 ).
simulation. The two options for the emission of radiation were an isotropic distribution or an angle dependent distribution based on the initial and final states of the
nucleus and the orientation of the nuclear spin in space, described by its magnetic
substate populations.
To generate an isotropic distribution, emitting the radiation evenly into a solid
angle of 4π, the following equations for θ and φ were utilized:
φ = 2πχ1
θ = cos−1 (1.0 − {χ2 [1.0 − cos(α)]})
Where χi are random numbers generated using the drand48 function and α is the
maximum angle of emission (α = π for emission into 4π radians).
43
The second option for particle emission was an angle dependent distribution based
on the orientation of the nucleus. The orientation of the nuclei in the RnEDM experiment refers to the polarization along a time-dependent axis of quantization rotating
with the polarized ensemble of Rn nuclei, such that the population of the m = J
sublevel is significantly higher than that of the other 2J sublevels. The 223 Rn nuclear
spins are initially aligned along the magnetic field axis. Following the application
of an RF pulse, these spins precess in the plane of the detectors as described in
Section 2.3.3.
The angle of emission, θ, with respect to the axis of quantization can be generally
expressed in powers of cos(θ) [46],
W (θ) =
X
Ak cosk (θ) ,
(3.5)
k=0,1,2,...
where Ak are the angular-distribution coefficients and k is even for γ radiation and
odd for β radiation. The exact form of W (θ) for β and γ radiation is described in
detail in Section 3.4.
β Particle Emission
In the process of β decay, the emitted β particle has a range of kinetic energies
between zero and the Q-value of the β decay. This energy spectrum results from the
existence of the neutrino or anti-neutrino in the decay processes shown below:
β − decay:
A N
ZX
→
A
N −1
Z+1 Y
+ e− + ν¯e
β + decay:
A N
ZX
→
A
N +1
Z−1 Y
+ e+ + νe
Through the conservation of energy, the electron (or positron) and anti-neutrino (or
neutrino) share the kinetic energy of the decay, which generates an energy distribution
for both particles. The spectral intensity for the electron (or positron) may be written
44
as [47]
I(E) =
G
2π 3 ~7 c6
| Mif |2 (Te2 + 2Te me c2 )1/2 (Q − Te )2 (Te + me c2 )F (Z, Te ) , (3.6)
where G is a constant representing the strength of the weak interaction, Mif is the
transition matrix element, Te is the kinetic energy of the electron (or positron), Z is
the proton number of the daughter and F (Z, Te ) is called the Fermi function.
The Fermi function accounts for Coulomb effects between the emitted electron or
positron and the charge of the daughter nucleus; due to the opposite charges of the
electron and positron their spectral intensities differ. For example in a semi-classical
view of β − decay, the electron created in the decay of the neutron is held back by the
attractive Coulomb force with the daughter nucleus decreasing the average energy
of the emitted electrons. In β + decay, on the other hand, the positively-charged
positron created by the decay of the proton is repelled by the Coulomb force with the
daughter nucleus increasing the average energy of the emitted positrons.
The shape of the spectral intensity was calculated in the RnEDM simulations for
every β branch to accurately describe the energy of the emitted electrons from the
decay of
223
Rn into
223
Fr. An analytic approximation of the Fermi function was used
to calculate the shape of the spectrum, given by [48]
s
4π(1
+
s)
2φη−2s+
2s−2
2
2
s−1/2
2
2
6(s +η )
F (Z, E) ∼
(2pρ)
(s + η )
e
,
=
[(2s)!]2
(3.7)
where s = [1 − (Ze2 /~c)2 ]1/2 , ρ = R/(~/mc), R is the nuclear radius approximated
as R = r◦ A1/3 with r◦ = 1.2 fm and A the nuclear mass number, p is the electron (or
positron) momentum (in units of mc), η = ±Ze2 /~ν (positive sign for β − decay and
negative for β + decay) and ν is the electron (or positron) velocity. This approximation
does not account for electron screening, however this correction is important only at
very low energies (on the order of 100 keV or less) [48].
45
γ-Ray and Internal Conversion Electron Emission
As described in Section 3.3, the γ-decay and internal conversion processes are
competitive. Level structure information is required to accurately describe their probabilities. At any particular excited state in the daughter nucleus there are N number
of γ decays which can populate lower energy levels. The probability for a particular
transition is given by (1 + α)Iγ /Isum , where α is the internal conversion coefficient,
P
Iγ is the measured γ-ray intensity and Isum = N (1 + αN )IγN , such that the total
decay probability for that excited state is normalized to 100%. Through Monte Carlo
techniques, a decay branch is selected and the probabilities of γ decay versus internal
conversion are calculated from Equations 3.1, and again selected by Monte Carlo.
If the selected process is γ decay, the energy of the emitted γ ray is simply read
from the input data. If the selected process is internal conversion, the energy of the
emitted electron is the γ-ray energy minus the electron binding energy for the electron
shell. The shell is determined from the probabilities given from the individual shell
internal conversion coefficients. Atomic electron binding energies [49] were directly
coded into the Geant4 simulation, resulting in accurate internal conversion electron
energies for shells K to N, and a general code that can be used in cases other than
the
223
Rn to
223
Fr decay studied here.
X-ray Emission
Following the emission of an internal conversion electron there exists a low lying
vacancy in the atomic shell structure of the daughter atom (223 Fr). An electron in a
higher orbital will prefer to occupy this lower energy state. As it transitions into the
vacancy energy is released in the form of a X ray.
The simulation of the X-ray emissions had two main user options. The first option
46
was to use an average X-ray energy per vacancy. These tables for all elements were
coded directly into the Geant4 simulation. The other more accurate option was to
provide an input file of the X-ray energies and intensities per shell vacancy. This
method was important to accurately simulate Fr K shell vacancies as the energy
of the X rays approach 100 keV, which are easily measured and resolved within
the GRIFFIN γ-ray detectors. The input data for Fr K shell vacancies is given in
Table 3.2. Careful inspection of these data reveals the intensities total to greater than
100% (per 100 K-shell vacancies). For every K vacancy one or more X-rays may be
emitted. For example, an electron from the L shell can drop down and occupy the
K shell vacancy, while leaving a new vacancy in the L shell. Another electron will
cascade down from the M shell to fill the L shell vacancy and so on. This process
was incorporated into the Geant4 simulations to provide an accurate simulation of
the entire X-ray cascade following internal conversion.
3.4
Simulating Angular Distributions
3.4.1
Beta Particle Anisotropies
The directional distribution of β particles from aligned nuclei has the form [50]
W (θ) = 1 + Aβ P cos(θ),
(3.8)
where Aβ is the beta-asymmetry correlation coefficient, P is the degree of polarization
and θ is the angle of emission relative to the polarization axis. The beta-asymmetry
correlation coefficient depends on the initial and final spins and parities of the nuclear
levels, as well as the relative contributions of Fermi and Gamow-Teller β decay for
mixed transitions. These input data are not known for many of the β decay branches
47
Table 3.2: X-ray energies and intensities per 100 Fr K-shell vacancies [49]. The total
intensity is 130.42 per 100 K-shell vacancies.
Initial K Shell Vacancy
Final L Shell Vacancy
Energy (keV) Intensity (%)
86.105
45.8
83.231
27.9
82.496
0.0675
97.474
10.70
100.214
4.01
96.815
5.58
100.548
0.10
98.069
0.358
100.972
0.84
101.118
0.114
of
223
Initial L Shell Vacancy
Final M Shell Vacancy
Energy (keV) Intensity (%)
12.031
14.6
11.896
1.63
14.770
9.9
14.443
3.74
14.978
0.106
14.319
0.102
14.967
0.683
13.877
0.251
17.302
2.25
17.635
0.038
17.800
0.041
17.839
0.47
13.255
0.247
10.381
0.89
Rn. An average β asymmetry correlation coefficient over all branches of
223
Rn
decay has been estimated as Aβ ≈ 0.45 [32]. As the β particle anisotropies are not
used as a signal in the current work, and only have a minor effect associated with the
anisotropy of the bremsstrahlung photons produced by the stopping of the β particles,
this average β asymmetry parameter was used for all
223
Rn β decay branches in the
simulations presented here.
The degree of polarization, assuming a spin-temperature distribution of the magnetic substates, can be written as [27]
P =
eβ − 1
.
eβ + 1
(3.9)
where β is the spin-temperature parameter. This parameter is related to the ratios
of the populations of neighbouring magnetic substates by [27]
eβ =
ρm
.
ρm−1
(3.10)
48
1.5
1.4
P = 100%
P = 75%
P = 50%
P = 25%
P = 0%
1.3
1.2
W(θ)
1.1
1
0.9
0.8
0.7
0.6
0.5
0
π
4
π
2
π
3π
4
θ
5π
4
3π
2
7π
4
2π
Figure 3.3: β particle angular distributions for various degrees of polarization. The
beta-asymmetry correlation coefficient was simulated to be 0.45.
As the degree of polarization reduces from its maximum value of 100% to 0% the
angular distribution of the electrons becomes more isotropic as shown in Figure 3.3.
3.4.2
Gamma-Ray Anisotropies
The theory of angular distributions of γ radiation from oriented nuclei is well
established [51, 52, 53]. The work from H.A. Tolhoek and J.A.M. Cox [51] derives
formula for oriented nuclear spins assuming pure multipole transitions. For a more
general approach, T. Yamazaki [52] has outlined the general theory and tabulated
coefficients for γ-ray angular distributions for aligned and partially-aligned nuclei
for pure and mixed multipolarities. Alignment, polarization and orientation can have
various definitions. In this work and the work of Yamazaki they are defined as follows.
Alignment refers to symmetric populations of magnetic substates about m = 0, thus
49
for perfectly aligned nuclei with integer nuclear spin P (m = 0) = 100%, where P is
the population of the m-states. Polarization, on the other hand, has P (m = J) >>
P (m 6= J) and P (m = J) = 100% for perfect polarization. Finally, orientation is the
completely general case, it can refer to any non-uniform population of m-states. In
the RnEDM simulations the m-states are tracked at each step, thus these populations
are known exactly. Given this information we can described any degree of orientation
using Yamazaki’s theory (see Appendix B for the MATLAB version of the γ-ray
angular distribution code).
In Yamazaki’s notation, the angular-distribution function for a transition Ji → Jf ,
where J is defined as the spin of the nuclear state, is expressed as
W (θ) = 1 + A2 P2 (cos θ) + A4 P4 (cos θ) ,
(3.11)
where Ak are the angular-distribution coefficients, Pk are Legendre polynomials, and
θ is the angle of emission relative to the alignment axis. For fully aligned nuclei, the
angular-distribution coefficients are given the notation Amax
k . For this ideal case,
Amax
=
k
1 2
f
(J
,
L
,
L
,
J
)+2δf
(J
,
L
,
L
,
J
)+δ
f
(J
,
L
,
L
,
J
)
, (3.12)
k
j
1
1
i
k
j
1
2
i
k
j
2
2
i
1 + δ2
where δ is the mixing ratio (defined in Equation 3.3) and
fk ≡ Bk (Ji )Fk (Jf , L1 , L2 , Ji ) .
(3.13)
The term fk can be broken up into a statistical tensor Bk (J),



(2J + 1)1/2 (−1)J (J0J0|k0) for integral spin,
Bk (J) =

 (2J + 1)1/2 (−1)J− 21 (J 1 J 1 |k0) for half-integral spin.
2 2
1/2
Fk (Jf , L1 , L2 , Ji ) ≡ (−1)Jf −Ji−1 (2L1 + 1)(2L2 + 1)(2Ji + 1)
× (L1 1L2 − 1|k0)W (JiJi L1 L2 ; kJf ) ,
50
and (3.14)
(3.15)
where (a b c d | e f ) are Clebsch-Gordan coefficients and W is a Racah coefficient.
Racah coefficients can be expressed in terms of Wigner 6 j-symbols,
W (a,b,c,d,e,f ) = (−1)(a+b+d+c)
(
abc
def
)
,
(3.16)
and equivalently, combinations of Clebsch-Gordan coefficients. In the RnEDM simulations Clebsch-Gordan coefficients were calculated using a developed algorithm yielding high computation efficiency and accuracy [54].
In more realistic scenarios, where the nuclei are not fully aligned, the angulardistribution coefficients can be written as
Ak (Ji , L1 , L2 , Jf ) = αk (Ji )Amax
k (Ji , L1 , L2 , Jf ) ,
(3.17)
where αk (Ji ) is the attenuation coefficient of the alignment. This coefficient may be
represented as
αk (Ji ) ≡
ρk (Ji )
,
Bk (Ji )
(3.18)
where ρk (J) is a statistical tensor defined as
ρk (J) = (2J + 1)1/2
X
m
(−1)J−m (JmJ − m|k0)Pm (J) .
(3.19)
Yamazaki discusses the experimentally justifiable assumption that partial alignment
may be represented by a Gaussian of m-sates about m = 0 characterized by a parameter σ which is the half-width of the Gaussian distribution, given by
2
2
e−m /2σ
Pm (J) = P
,
−m′2 /2σ2
m′ e
(3.20)
where σ = 0 defines the fully aligned state. As stated earlier, the nuclear m-states
are known at each step during the RnEDM Geant4 simulations. Therefore we may
describe any degree of orientation by explicitly inputting the m-state populations
51
Pm (J). Table 3.3 illustrates three-dimensional γ-ray angular distributions for various
L transitions, and Figure 3.4 gives a few examples of angular distributions for E2 and
M1 multipolarities.
Table 3.3: γ-ray angular distributions for various L transitions. The angular momen~ is aligned in the “upward” direction.
tum vector (J)
Ji
Jf
Various L Transitions
L=1
L=2
“Stretched” L Transitions
L
Ji = 5, Jf = Ji − L
5 5
2
5 4
3
5 6
4
3.4.3
Tracking m-States
The magnetic substates are tracked at each step of the simulation. To begin
the tracking process, the parent
223
Rn m-state populations are calculated from the
52
2
2
Ji = 5/2, Jf = 5/2
Ji = 7/2, Jf = 5/2
Ji = 7/2, Jf = 3/2
1.5
Normalized W(θ)
Normalized W(θ)
1.5
1
0.5
0
Ji = 5/2, Jf = 5/2
Ji = 7/2, Jf = 5/2
Ji = 5/2, Jf = 5/2 .
(mixed
M1/E2 δ=0.6)
.
1
0.5
0
π
2
π
θ
3π
2
0
π
π
2
0
π
θ
(a) E2 multipolarities
3π
2
π
(b) M1 and M1+E2 multipolarities
Figure 3.4: γ-ray angular distributions for various multipolarities and spins. The
input required to generate γ-ray angular distributions includes the initial and final
nuclear spins, the multipolarity of the transition (δ if mixed) and the m-state populations. The angular distributions shown here are for perfectly polarized nuclei, such
that m = Ji .
current polarization (see Equations 3.9 and 3.10). The β-decay branch is determined
using Monte Carlo and β-branch probabilities; angular momentum coupling between
the
223
Rn ground state and the excited state in
223
Fr is used to calculate resulting
probabilities for m-state populations. From these populations an m-state is chosen
using Monte Carlo. Similarly for γ decays, the angular momentum coupling between
the two states is used to calculate probabilities of m-state populations. At each step
of the decay an m-state is found. As shown in the previous section, the m-state
information is used to determine the angular distribution of γ-ray radiation.
53
Figure 3.5: Cross section of the simulated EDM cell, oven, magnet and µMetal shielding. The cell and cylindrical oven wall composed of PYREX glass with the oven caps
constructed of delrin. The simulated magnetic coil was designed to fit tightly around
the EDM oven, with a thickness matching that of 24 AWG wire. The thickness of
the µMetal was user-defined in the GUGI program.
3.5
RnEDM Geant4 Geometry
The simulated geometry for the RnEDM experiment consists of a ring of eight
GRIFFIN detectors around an EDM cell and oven (see Figure 3.1). The eight GRIFFIN detectors are close-packed in their highest efficiency mode as shown in Figure 2.9.
3.5.1
Cell and Oven Design
The design for the RnEDM experiment has not been finalized. The experiment
will, however, consist of a few basic components; including a cell, oven, magnetic coil
and magnetic shielding. The spherical cell and cylindrical oven were both constructed
of PYREX glass in the Geant4 simulation. The cell was located inside the oven and
was 1 mm thick with an inner radius of 1 cm. The oven was 5 mm thick with an inner
54
of radius 10 cm, allowing the GRIFFIN detectors to be close-packed in their highest
efficiency mode with an inner radius of 11 cm. The oven caps were constructed of
delrin and were 22 cm in diameter and 1 cm thick. The optional solenoidal magnet
was constructed of pure copper and designed to tightly fit around the oven. The
thickness of the cylinder was equivalent to 24 AWG wire. Finally, the µMetal was
given a user-defined thickness and it was placed tightly around the magnet. A cross
section of this setup is shown in Figure 3.5. While this does not represent the final
design of the RnEDM experiment, it does include the key components that will scatter
and absorb γ rays with variable thickness that enable scattering studies as a function
of component thickness.
3.5.2
LaBr3(Ce) Scintillator
The LaBr3 (Ce) scintillator material is used in the construction of Saint-Gobain’s
BrilLanCe 380 detector. The BrilLanCe 380 scintillator has excellent fast timing
properties (∼200 ps) and offers a better energy resolution when compared to a standard NaI(Tl) detector [41]. The impressive count rate capability of LaBr3 (Ce) could
give a strong advantage over HPGe in the RnEDM experiment. The sensitivity of the
EDM measurement will be statistics limited and will be determined by the number of
photopeak events detected. The higher count rate capability of LaBr3 (Ce) could thus
lead to greater sensitivity. However, to observe the EDM signal, the time structure
of a single γ-ray transition needs to be monitored precisely, and the energy resolution
of HPGe is much better than LaBr3 (Ce). Using LaBr3 (Ce) to identify individual
transitions in the decay of 223 Rn may be difficult or impossible. Geant4 simulations
of the BrilLanCe 380 detector were performed as part if this work (Section 4.3), to
study the sensitivity achieved with LaBr3 (Ce) energy resolution.
55
Figure 3.6: Cross section of Saint-Gobain’s BrilLanCe 380 LaBr3 (Ce) scintillator.
Saint-Gobain’s BrilLanCe 380 detector contains a 76.2 mm×76.2 mm cylindrical
crystal of LaBr3 (Ce). The full design of the BrilLanCe 380 detector is confidential,
therefore a simple can construction was used for the Geant4 simulation. It consisted
of a 76.2 mm×76.2 mm cylindrical crystal of LaBr3 (Ce) with an aluminium can
surrounding the scintillator material. The can wall and back plate were 1 mm thick
and the can face was 0.5 mm. Simulated vacuum was placed between the crystal and
can, such that the scintillator material and Al can were not touching. The vacuum
gap was simulated to be 1 mm thick, with the exception of the detector face where it
was only 0.5 mm thick. Figure 3.6 shows a Geant4 rendering of the BrilLanCe 380
detector.
56
(a) Screenshot showing the detector options under “Experimental Options”
(b) Screenshot showing the equipment options under “Experimental Options”
Figure 3.7: Screenshots of the GUGI program.
57
3000
15
2500
y = a + bx + cx
10
2
1500
2
a = 1.10
b = 0.00183744
c = 0.0000007
2
2
2
y = a + bx + cx
FWHM (keV )
2000
FWHM (keV )
2
a = 1.7006116
b = 0.5009382
c = 6.5451219e-05
1000
5
500
0
0
500
1000
1500
γ-ray Energy (keV)
2000
2500
3000
(a) BrilLanCe 380 LaBr3 (Ce) scintillator
0
0
500
1000
1500
γ-ray Energy (keV)
2000
2500
3000
(b) GRIFFIN HPGe detector
Figure 3.8: The square of the full-width at half-maximum (FWHM2 ) versus γ-ray
energy for a BrilLanCe 380 and a GRIFFIN detector. The resulting fits were used
in the calculations of realistic energy resolutions for the LaBr3 (Ce) scintillator and
HPGe detector.
3.6
Data Management
3.6.1
The GUGI Program
The Geant4 User Generated Input (GUGI) program was written as part of this
work to help manage the simulation input data in a user-friendly manner. The program was written in C + + with the Qt (version 4.5.3) framework, taking advantage
of the Qt GUI toolkit. Its purpose was to allow the user to easily change simulation
parameters. It separates static input, the input that will usually never change (e.g.
the 223 Fr level scheme), from variable input, the input that the user would most likely
want to change between runs (e.g. the degree of initial polarization). The program
links together these two inputs into one file which is read into the Geant4 simulation.
See Figure 3.7 for a screenshot of the GUGI program.
58
3.6.2
Output Data and Sort Codes
The simulated data, described in Section 3.2.5, was written to binary files. The
binary files were sorted to generate spectra and matrices files. The energy resolution of
the detectors was modelled during the sorting process, since the statistical variations
and electronic effects that give rise to the resolution can not be modelled directly by
Geant4 . The energies output from Geant4 were idealized, within 1 keV bins. To
facilitate a comparison between experiment and simulation, a resolution function was
applied, specific to the detector type.
The relationship between the square of the full-width at half-maximum (FWHM)
and γ-ray energy was determined for each detector by fitting a second order polynomial. These coefficients for the TIGRESS/GRIFFIN detectors were previously
calculated and verified [35] and are shown in Figure 3.8(b). The coefficients for the
BrilLanCe 380 detectors were fit to performance measurements given by Saint-Gobain.
Figure 3.8(a) illustrates the measurements and the derived fit. The energy resolution
values derived from these functions were converted into standard deviations, and applied to the idealized energies. The resulting energies were chosen from a Gaussian
distribution with the idealized energy as the mean [35].
59
Chapter 4
Results
This chapter presents the results from the RnEDM Geant4 simulations and outlines the sensitivity of the experiment. The sensitivities are derived from the simulated
precession frequencies fit with a function which describes the physics involved in the
simulation using a chi-squared minimization method. This method used a maximum
likelihood technique specific for counting experiments which are based on the Poisson
distribution. The achievable EDM sensitivity according to the Geant4 simulations
is discussed.
4.1
γ-Ray Spectroscopy
The results from the Geant4 simulations were analyzed using γ-ray spectroscopy
techniques. One dimensional γ-ray energy spectra and two dimensional γ-ray energytime matrices were constructed from the Geant4 simulated data. Preserving all
energies, times and interaction locations in the output binary data files allowed for
a complete reconstruction of the data into spectra and matrices during the sorting
process. Sorting the data had three main tasks, firstly to remove any rejected hits
60
(user-defined option), secondly to apply a realistic energy resolution response function
for the detectors (see Section 3.6.2), and finally to generate the output spectrum and
matrix files for subsequent analysis.
The GRIFFIN detectors can be operated in two different modes to accept “good”
events, namely, the suppressed and unsuppressed mode. In the unsuppressed mode,
all γ-ray energies deposited in the detectors are accepted as valid hits and recorded
in the output files. In the suppressed mode, however, a GRIFFIN detector which
registered a γ-ray event in a germanium crystal in coincidence with any of the BGO
detectors surrounding that detector would not be included in the output files. This
mode significantly reduces the background in the spectrum by suppressing Compton
scattering in which a γ-ray scatters out of an HPGe detector without depositing its
full energy. All the analysis presented in this Chapter was conducted in the suppressed
mode.
4.1.1
Efficiencies
The first characteristic studied using the Geant4 simulations was the absolute γray photopeak efficiency for the RnEDM experiment. The absolute efficiency of a ring
of eight GRIFFIN detectors in both the forward and back configurations is presented
in Figure 4.1. The detectors in their highest efficiency mode have an efficiency of
26.3% at 100 keV and 9.7% at 1 MeV, while in the fully suppressed mode they have
an efficiency of 17.5% at 100 keV and 6.5% at 1 MeV. The remainder of the simulations
were performed with the GRIFFIN detectors in the forward position to maximize the
number of counts detected. The efficiency of the entire RnEDM apparatus is lower
relative to Figure 4.1 due to the increased probability of γ-ray scattering inside the
EDM cell, oven, magnet and shielding. Figure 4.2 studies the decrease in efficiency
61
30
Forward Configuration
Back Configuration
26.3%
Absolute Efficiency (%)
25
20
17.5%
15
9.7%
10
6.5%
5
0
10
100
γ-ray Energy (keV)
1000
10000
Figure 4.1: Geant4 generated absolute efficiency curve for a ring of eight GRIFFIN detectors in the highest efficiency mode (forward configuration) and the fully
suppressed mode (back configuration).
due to various components of the apparatus. Figure 4.3 illustrates the decrease in
efficiency with increasing thicknesses of µMetal shielding.
The efficiency of the RnEDM experiment is extremely important as it is directly
related to the sensitivity of measurement. The precision of the EDM measurement
is limited by the number of photopeak counts detected. The statistical limit for the
frequency measurement [32], introduced in Section 2.3.5, is
s
2
1
,
δ∆ω =
T2 A2 (1 − B)2 N
(4.1)
where T2 is the spin-decoherence time, A is the analyzing power for a measurement
that detects a change of counts ∆N = AN, and B is the fraction of N due to
background. The linearity of δ∆ω (or equivalently σf ) with T2−1 and N −1/2 was
studied using the Geant4 simulations and is discussed in Section 4.2.3.
62
30
RnEDM Components
Absolute Efficiency (%)
25
no cell, oven or magnet
cell and oven only
magnet only
cell, oven and magnet
20
15
10
5
0
10
100
γ-ray Energy (keV)
1000
10000
Figure 4.2: Geant4 generated absolute efficiency curves for a ring of eight GRIFFIN
detectors in the highest efficiency mode (forward configuration) including various
components of the RnEDM apparatus.
4.1.2
223
Rn β-Decay Spectra
The β decay of
223
Rn into
223
Fr was simulated to include all known β decay
branches and subsequent γ-ray, conversion electron and X-ray decays. The full γray spectrum for the β decay of
223
Rn is presented in Figure 4.4(a). The Geant4
simulation consisted of 1.2 billion β decay events from an initial sample of 8 × 1010
nuclei inside the EDM cell. The EDM cell, oven, magnet and 1 mm of µMetal shielding
were included in the simulation. The initial polarization was assumed to be 100% and
both the spin-relaxation (T1 ) and the spin-decoherence (T2 ) times were simulated to be
30 seconds, resulting in a combined depolarization time of 15 seconds. The simulated
decay scheme (See Appendix A) includes 136 levels and 294 γ-ray transitions. The
inclusion of the internal conversion process is shown to be important as the resulting
X-rays dominate the spectrum at low energies.
63
Absolute Efficiency (%)
15
µMetal Thickness
0.10 cm
0.25 cm
0.50 cm
0.75 cm
1.00 cm
10
5
0
100
1000
γ-ray Energy (keV)
10000
Figure 4.3: Geant4 generated absolute efficiency curves for a ring of eight GRIFFIN
detectors in the highest efficiency mode (forward configuration) including the full
RnEDM setup (EDM cell, oven and magnet) with various thicknesses of µMetal.
Timing information was extracted from γ-ray energy-time matrices. Figure 4.4(b)
illustrates a three-dimensional view of a small energy and time slice for one detector
in the simulation presented in Figure 4.4(a). The dimension of the γ-ray energy-time
matrices were 4096 by 4096, where the γ-ray energies were binned into 1 keV bins
resulting in a maximum energy of about 4.1 MeV. The resolution of the time bins
was flexible and thus optimized for the length of each simulation. Generally the times
were binned into 10 ms, resulting in just over 40 seconds of simulation time.
4.2
Frequency Signal
The frequency signal can be clearly seen in the three-dimensional Figure 4.4(b).
To generate a spectrum of this signal, energy gates were placed around the γ-ray
64
11
10
9
8
7
6
5
4
3
2
1
0
1e+06
Counts (e+05)
Linear
Logarithmic
Counts
1e+05
10000
1000
100
10
1
0
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700
γ-ray Energy (keV)
(a)
(b)
Figure 4.4: a) γ-ray energy spectrum and b) γ-ray energy-time matrix from the β
decay of 1.2 billion 223 Rn nuclei from an initial 8 × 1010 nuclei located inside the EDM
cell surround by a ring of eight GRIFFIN detectors. The EDM cell, oven, magnet and
1 mm of µMetal shielding were included in the simulation. The initial polarization
was 100% and both the spin-relaxation (T1 ) and the spin-decoherence (T2 ) times were
simulated to be 30 seconds, resulting in a combined depolarization time of 15 seconds.
65
1400
24000
6
5
Counts
1300
1200
23000
22000
1100
Counts
Counts (e+05)
21000
4
3
0
5
10
20
15
25
Seconds (500ms Time Bins)
30
1000
900
2
800
1
700
0
300
350
400
γ-ray Energy (keV)
450
500
1
2
3
4
5
(a)
7
8
9
10 11
6
Seconds (20ms Time Bins)
12
13
14
15
(b)
Figure 4.5: The a) energy gate and b) time projection of the 416 keV γ ray for two antiparallel GRIFFIN detectors. The precession frequency generated from the polarized
radon nuclei is evident in the time projection. The depolarization of the radon nuclei
results in an increase in counts over the short time scale of the depolarization time
(T1 and T2 ).
photopeak and the time spectrum was projected. Figure 4.5 illustrates the gating and
projection process with the 416 keV photopeak. All slicing and projection operations
were performed with the RadWare software package [55].
Figure 4.5(b) illustrates the frequency signal for two anti-parallel detectors. The
counts in the facing detectors are summed together to increase the statistics in the
plot. Due to the symmetry of the γ ray angular distributions the two detectors observe
statistically identical signals. The 416 keV γ ray is an M1 transition with Ji = Jf = 25 .
The three-dimensional representation of the γ-ray angular distribution for such a
transition is shown in Figure 4.6. The shape and behaviour of the frequency signal is
dependent on the angular distribution of the γ radiation. Section 4.2.1 investigates
the precession frequencies for various transitions found in the
66
223
Rn decay.
4.2.1
Multipolarity Effects
The shape of the γ ray angular distribution has a significant impact on the observed signal in the GRIFFIN detectors. Figure 4.6 gives examples of some common γ
ray angular distributions in the 223 Rn decay. Figure 4.7 illustrates the resulting signal
and fits for various multipolarities. The simulations presented in Figure 4.7 consisted
of two million 223 Rn decay events from an initial two billion nuclei. The decay scheme
was modified to enhance the 592 keV transition, such that the 592 keV γ ray was
emitted in every decay. The multipolarity of the 592 keV transition was modified to
reflect that of the multipolarity of interest. In these simulations the EDM cell, oven,
magnet and µMetal were absent. The initial polarization of the
223
Rn nuclei was
100% with a spin-decoherence time (T2 ) of 10 seconds. The precession frequency was
simulated to be 1 Hz and the ring of eight GRIFFIN detectors were in their highest
efficiency mode in the forward position.
The resulting fits were good, with reduced χ2 values (see Figure 4.7) of approximately one. The observed precession frequencies are twice the input value due to
the symmetric nature of the γ ray angular distributions about 180◦ . The exception
is the Ji =
7
2
to Jf =
5
2
E2 transition which is four times the input frequency due
to its four lobed angular distribution. The second term in the fitting Equation 4.2
−x
(A3 e A4 ) is shown to be necessary as the average intensity varies on the timescale of
the depolarization as the angular distribution becomes isotropic. A positive polarization intensity (A3 ) gives an overall decreasing intensity on the time scale of the
depolarization time, whereas a negative polarization intensity gives an overall increasing intensity on the time scale of the depolarization time. The polarization intensity
(A3 ) is positive for “dumbbell” shaped γ-ray angular distributions and negative for
“donut” shaped γ-ray angular distributions.
67
Figure 4.6: Three-dimensional γ-ray angular distributions for various E2 and M1
~ is aligned in
transitions generated by Geant4. The angular momentum vector (J)
the “upward” direction.
68
Figure 4.7: The time projections and fits of various E2 and M1 transitions. The
EDM cell, oven, magnet and µMetal was not included in the simulation. The initial
polarization of the radon nuclei was 100% and the spin-decoherence time (T2 ) was
simulated to be 10 seconds. Over the short time period of the spin-decoherence time,
the average count rate detected by the two anti-parallel GRIFFIN detectors increase
or decrease depending on the shape of the γ-ray angular distribution (see Table 4.6).
69
1400
24000
1300
Counts
2
χred
= 1.04
f = 1.9999(2) Hz
1200
23000
22000
21000
Counts
1100
0
5
10
20
15
25
Seconds (500ms Time Bins)
30
1000
900
800
700
0
1
2
3
4
5
7
8
9
10
6
Seconds (20ms Time Bins)
11
12
13
14
15
Figure 4.8: The fit to the 416 keV γ-ray time projection presented in Figure 4.5 for
two anti-parallel GRIFFIN detectors. The resulting reduced chi-squared indicates a
good fit and the precession frequency agrees with the simulated value of 2 Hz.
4.2.2
Fitting Process
Precession frequencies were extracted through fitting the data to a representative
function and using a χ2 minimization method [56]. This method utilized a maximum
likelihood approach tailored for counting experiments based on Poisson statistics.
The function used to fit the data was
y = A2 e
− ln(2)x
A1
−x
+ A3 e A4 + A7 sin(1A5 (2πx) + A6 )
−x
+ A8 sin(3A5 (2πx) + A6 ) + A9 sin(5A5 (2πx) + A6 ) + · · · e A4 ,
where Ai are the fit parameters. The first term (A2 e
− ln(2)x
A1
(4.2)
) describes the radioactive
decay of the radon nuclei, where A1 is the half-life in seconds and A2 is the intensity in
−x
counts per second. The second term (A3 e A4 ) describes the average intensity (A3 ) as
a function of the depolarization time (A4 ). Depending on the shape of γ ray angular
70
distribution, this average intensity can increase or decrease over the depolarization
time. The 416 keV M1 transition shown in Figure 4.5(b) clearly illustrates this effect.
Even though the number of nuclei inside the cell, and hence the total activity is
decreasing, the observed average count rate increases over the depolarization time
scale (which is short compared to the
223
Rn half-life of 24.3 minutes). This change
in the average count rate occurs because the “donut shaped” distribution depolarizes
into an isotropic distribution that has a larger average γ-ray intensity in the plane of
the eight GRIFFIN detectors. This effect is further studied in Figures 4.6 and 4.7.
The remaining terms describe the precession, where A5 is the frequency, A6 is the
phase and A7 , A8 , A9 , · · · are the intensities of odd-sine oscillations. A maximum of
14 odd-sine terms were included into the fitting program (up to A20 ), however, often
only the first few odd-sine terms were non-zero. In this case all other sine terms fixed
to zero and removed from the fit.
Figure 4.8 illustrates the fit to the 416 keV time projection data given in Figure 4.5(b). In the fit the half-life was “fixed” to its simulated value of 1458 seconds
and the remaining parameters were left “free” to be fitted by the program. The fit
resulted in a good reduced χ2 of 1.04 and a precession frequency which agrees with
the simulated value of 2 Hz. The input precession frequency was 1 Hz, but due to the
symmetric nature of the γ-ray angular distribution the observed frequency is twice
the input value. The fitted parameters are summarized in Table 4.1. It should be
noted that the analyzing power A in Equation 4.1, which that detects a change of
counts ∆N = AN, was estimated as A = 0.2 in the calculated statistical limits for
the RnEDM experiment [32]. Figure 4.8 validates that estimate as the ratio of the
signal amplitude to the background is approximately 0.2.
From this single measurement we can calculate the resulting sensitivity in the
71
2.003
Weighted Average: 2.00000(8) Hz
χ2red = 1.869
2.002
Frequency (Hz)
2.001
2.000
1.999
1.998
1.997
1.996
0
100
200
300
400
500
γ-ray Energy (keV)
600
700
800
Figure 4.9: The resulting frequency from the weighted average of 20 precession frequency fits for various photopeaks. The fits were generated from the data set present
in Figure 4.4 for two anti-parallel GRIFFIN detectors.
EDM measurement using Equation 2.17. The sensitivity is given by,
σd =
π~
σf .
2E
(4.3)
Given an electric field of 5 kV/cm and σf = 0.0002 Hz we find a sensitivity in the EDM
measurement of 4.14 × 10−23 ecm for two anti-parallel GRIFFIN detectors, detecting
the 416 keV γ-ray from the decay of 1.2 × 109
223
Rn nuclei with a spin-depolarization
time of 15 seconds.
This fitting process was repeated for every significant photopeak which produced
a frequency signal in the
223
Rn decay. The weighted averaged of these measurements
is given in Figure 4.9. The weighting function used was
wi =
1
,
σi2
(4.4)
72
Table 4.1: The fit results for the 416 keV γ-ray M1 transition given in Figure 4.8
Description
Half-life
Primary Intensity
Polarization Intensity
Depolarization Time
Precession Frequency
Phase
1st Signal Intensity
2nd Signal Intensity
3rd Signal Intensity
4th Signal Intensity
5th Signal Intensity
6th Signal Intensity
7th Signal Intensity
10th Signal Intensity
11th Signal Intensity
Parameter
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A12
A13
A16
A17
Fixed/Free
Fixed
Free
Free
Free
Free
Free
Free
Free
Free
Free
Free
Free
Free
Free
Free
Value
1458 sec
46921(98) counts/sec
−2816(170) counts/sec
16.7(5) sec
1.9999(2) Hz
4.71(2) rad
8422(157) counts/sec
160(107) counts/sec
117(112) counts/sec
264(120) counts/sec
−153(131) counts/sec
216(148) counts/sec
−259(177) counts/sec
−683(366) counts/sec
−720(575) counts/sec
where the average is given by,
Pn
(xi /σi2 )
x̄ = Pi=1
,
n
2
i=1 (1/σi )
(4.5)
and its variance is given by,
1
.
2
i=1 (1/σi )
σx̄2 = Pn
(4.6)
The resulting sensitivity for 20 different γ-rays is 1.65×10−23 ecm for two anti-parallel
GRIFFIN detectors and a spin-depolarization time of 15 seconds. This accounts for
a total of approximately 13 million counts in two anti-parallel GRIFFIN detectors.
The RnEDM experiment is planned to run for a total of 100 days, which gives an
estimate of 1 ×1012 photopeak counts in our GRIFFIN detectors [32], and is expected
to achieve a spin-depolarization time of at least 30 seconds. Section 4.2.3 calculates
the achievable sensitivity for 100 days of counting.
73
6
5
linear regression:
y = 0.005926 x - 1.3452e-04
σf (e-04 Hz)
4
3
2
1
0
0.05
0.06
0.07
0.08
-1
-1
T2 (sec )
0.09
0.1
Figure 4.10: The sensitivity of the fitted frequency versus the simulated spindecoherence time (T2 ). The linear behaviour validates the statistical precision equation (see Equation 4.1) for the RnEDM measurement.
4.2.3
Statistical Limit
The validity of Equation 4.1 was explored using Geant4. According to Equation 4.1 the sensitivity in the frequency improves linearly with N
−1
2
and T2−1 , where N
is the number of counts and T2 is the spin-decoherence time. Figure 4.10 confirms the
linearity of the sensitivity in the frequency with the inverse of the spin-decoherence
times T2 . In these simulations the most intense 592 keV γ ray was fit over 30 seconds
with varying spin-decoherence times ranging from 10 to 20 seconds. The fits were
similar to the example given in Table 4.1 with the spin-decoherence times fixed to the
simulated values. Similarly, Figure 4.11 confirms the linearity of the sensitivity with
N −1/2 . In these simulations the full
223
Rn decay was simulated up to a maximum
number of 4.5 billion events from an initial 9 × 1010 nuclei, this resulted in just over
74
12
linear regression:
y = 0.22432 x - 1.0887e-06
10
σf (e-04 Hz)
8
6
4
2
0
0
1
2
N
3
-1/2
(e-03 counts )
4
5
6
-1/2
Figure 4.11: The sensitivity of the fitted frequency versus the number of counts fitted.
The linear behaviour validates the statistical precision equation (see Equation 4.1)
for the RnEDM measurement.
100 seconds of simulation time. Of the 14 signal intensities in the fitting program,
only the first signal intensity (A7 ) was fit in the analysis. The EDM cell, oven, magnet
and µMetal were not included in these simulations.
Figure 4.11 can be extrapolated to estimate the sensitivity that will be achieved
by the RnEDM experiment. Estimates for 100 days of counting with the GRIFFIN
detectors running at a total count rate of 120 kHz results in 1×1012 photopeak counts.
With a stable electric field of 5 kV/cm an EDM sensitivity of 4.64 × 10−26 ecm is
calculated from the slope in Figure 4.11 if the spin-decoherence time is 15 seconds
as used in these simulations. Increasing the electric field beyond 5 kV/cm and increasing the spin-decoherence time beyond the 15 seconds simulated here through
the design of the EDM measurement cell will each improve the EDM sensitivity linearly. Taking advantage of the predicted EDM enhancement factor of ∼ 600 for
75
25
20.48%
Absolute Efficiency (%)
20
15
10
6.92%
5
0
10
100
γ-ray Energy (keV)
1000
10000
Figure 4.12: Geant4 generated absolute efficiency curve for a ring of eight BrilLanCe
380 detectors.
223
Rn relative to
199
Hg, the RnEDM experiment would need a sensitivity on the or-
der 1 × 10−26 ecm [27], to be competitive with the current
199
Hg measurement of
d(199 Hg) ≤ 3.1 × 10−29 ecm [19].
4.3
LaBr3(Ce) Detectors
The cerium-doped lanthanum bromide detectors (Saint-Gobain’s BrilLanCe 380
detectors) could offer a unique advantage to the RnEDM experiment. The fast timing
properties of LaBr3 (Ce) (∼200 ps) provides a drastically increased count rate capability compared to HPGe detectors. However, the compromise is in energy resolution,
as shown in Figures 3.8(a) and 3.8(b) for a direct comparison between the energy
resolution of a BrilLanCe 380 detector and a GRIFFIN detector. Unlike high-purity
germanium detectors the cerium-doped lanthanum bromide detectors will not be able
76
Linear
Counts (e+04)
15
10
5
0
1e+05
Logarithmic
Counts
10000
1000
100
10
1
0
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700
γ-ray Energy (keV)
Figure 4.13: γ-ray energy spectrum from the β decay of 120 million 223 Rn nuclei from
an initial 8 billion nuclei located inside the EDM cell surround by a ring of eight
BrilLanCe 380 detectors. The EDM cell, oven, magnet and 1 mm of µMetal shielding
were included in the simulation. The initial polarization was 100% and both the spinrelaxation (T1 ) and the spin-decoherence (T2 ) times were simulated to be 30 seconds,
resulting in a combined depolarization time of 15 seconds.
to resolve most individual photopeaks in the
223
Rn decay spectrum. Therefore the
time projections which produce the frequency signal will be “contaminated” with
other photopeaks.
The absolute efficiency of eight BrilLanCe 380 detectors in a ring configuration
is presented in Figure 4.12; the detectors have an efficiency of 20.5% at 100 keV
and 6.9% at 1 MeV. Figure 4.13 illustrates the γ ray singles spectrum from the β
decay of 120 million
223
Rn nuclei, from an initial 8 billion nuclei inside the EDM cell.
The EDM cell, oven, magnet and 1 mm of µMetal shielding were also included in
this simulation. The simulation had identical properties to that of the simulation of
77
3400
200
f = 2.0010(2) Hz
Counts
3200
2
χred
= 1.001
3000
2800
Counts
2600
0
5
10
20
15
25
Seconds (500ms Time Bins)
30
150
100
0
1
2
3
4
5
7
8
9
10
6
Seconds (20ms Time Bins)
11
12
13
14
15
Figure 4.14: The fit of the 565 keV to 614 keV energy gate time projection for two
anti-parallel BrilLanCe 380 LaBr3 (Ce) detectors.
Figure 4.4(a). The figure shows that not all individual photopeaks are resolved with
the LaBr3 (Ce) detectors. Figure 4.14 illustrates the time projection and fit of a gate
set between 565 and 614 keV. Although the gate encompasses many photopeaks the
frequency signal is still clearly visible and further exploration of the use of the highrate capabilities of LaBr3 (Ce) scintillators in the RnEDM experiment is warranted.
78
Chapter 5
Conclusions and Future Directions
5.1
Conclusions
The Geant4 simulations presented in this work accurately describe the physics
involved in the RnEDM experiment at TRIUMF. The radioactive decay simulation
package developed as part of this thesis is capable of simulating any β-decaying nucleus given the half-life, the β-decay branching ratio to the ground state, Q-value,
daughter level spins and parities, γ-ray energies, intensities, mixing ratios, internal
conversion coefficients, and the resulting X-ray energies and intensities. Along with
reproducing the entire decay scheme and realistic β, γ, electron and X-ray energies
and probabilities, the package also simulates the angular distribution of the emitted β particles and γ rays. This is a highly versatile radioactive decay simulation
package that will find many applications, not only in further studies of the RnEDM
experiment, but also the entire decay spectroscopy research program with GRIFFIN at ISAC. The RnEDM apparatus (cell, oven, magnet and µMetal shielding) was
included in Geant4 to realistically simulate the γ-ray scattering and backgrounds
present in the experimental apparatus. The GRIFFIN γ-ray spectrometer (based on
79
a fully verified Geant4 framework for the TIGRESS array [35]) was simulated to
the highest degree of reality. The GUGI program was written to allow for a simple graphical manipulation of the input simulation parameters, allowing the user to
quickly and effectively study many different aspects of the RnEDM simulation. The
entire simulation package provides a powerful tool for further studies of the RnEDM
experiment at TRIUMF.
The RnEDM experiment will achieve an extremely sensitive measurement that demands expertise in atomic and nuclear physics alongside advances in technology. The
Geant4 simulations for the RnEDM experiment validate the feasibility of measuring
an atomic EDM using NMR techniques and γ-ray detection. The EDM measurement
of
223
Rn would need a sensitivity on the order 1 × 10−26 ecm [27], to be competitive
with the current 199 Hg measurement of d(199 Hg) ≤ 3.1 ×10−29 ecm [19], while taking
advantage of the predicted EDM enhancement factor of ∼ 600 for
199
223
Rn relative to
Hg [2]. The results presented in Chapter 4 show that with 1 × 1012 photopeak
counts (equivalent to 100 days of counting [32]) detected in the GRIFFIN detectors,
a sensitivity of 4.64 × 10−26 ecm would be reached with an electric field of 5 kV/cm
and a spin-decoherence time of 15 seconds. Increases in both E and/or T2 through
the EDM cell design will each improve the achievable sensitivity linearly.
5.2
Future Directions
The Geant4 framework for the RnEDM experiment can be applied to many
different applications beyond the ones studied in this work. Firstly, the β decay
package that handles particle emission is a very general code. Any β process can be
80
simulated given sufficient parent and daughter nucleus information. A natural extension to the simulation would be the option of including of radioactive contamination.
The first tests with the ISAC uranium-carbide target in December 2010 indicated
an overwhelming presence of francium. Alkali metals, such as francium, are readily
ionized through surface ionization. It is very likely that the RnEDM experiment will
have some contamination from francium. Including radioactive contaminates into
the Geant4 simulation could help study the contamination effect on the observed
precession frequencies, and thus the EDM sensitivity.
The final design for the RnEDM measurement remains under development. The
Geant4 simulations can be used to study the effect of γ-ray scattering and backgrounds in the experimental apparatus for various materials and geometries. The
optimization of the materials and geometries of the RnEDM apparatus can lead to
improved detection efficiencies.
Finally, the simulation of Saint-Gobain’s BrilLanCe 380 LaBr3 (Ce) detector can
be improved by modeling the true aluminum construction around the crystal and
including the radioactive nature of
138
La.
138
La is a naturally occurring radioisotope
(0.09% abundance [41]) with a long half-life of 1.05 × 1011 years[49]. It decays with
a 66.4% probability of electron capture into an excited state in
138
Ba, which in turn
decays by the emission of a 1426 keV γ ray. The remaining 33.6% decays via β − decay
into an excited state in
138
Ce, which in turn decays by the emission of a 789 keV γ
ray in coincidence with the emitted electron [49]. Therefore the LaBr3 (Ce) detectors contain internal radiation peaks which can be incorporated into the Geant4
simulations for added realism.
The search for particle and atomic EDMs is an important area of physics research today, as it directly probes the fundamental symmetries of the laws of physics.
81
Current upper limits on the EDMs of the neutron, electron and
199
Hg have already
significantly reduced the allowed parameter spaces of models beyond the Standard
Model of particle physics. A measurement of a permanent non-zero atomic EDM
in radon would represent the discovery of new physics beyond the Standard Model
of particle physics and could explain the observed asymmetry between matter and
antimatter in the universe.
82
Appendix A
Figure A.1: Simulated decay scheme [1 of 5] for the β − decay of 223 Rn to 223 Fr. The
decay scheme was constructed from experimental data [43, 57]; incomplete information was replaced with conservative estimates. The level diagrams shown here were
used for simulation purposes, they do not represent the current knowledge of the excited states in 223 Fr. For unlabelled multipolarities, the lowest order L was assumed.
Figures adapted from reference [57].
83
Figure A.2: Simulated decay scheme [2 of 5] for the β − decay of 223 Rn to 223 Fr. The
decay scheme was constructed from experimental data [43, 57]; incomplete information was replaced with conservative estimates. The level diagrams shown here were
used for simulation purposes, they do not represent the current knowledge of the excited states in 223 Fr. For unlabelled multipolarities, the lowest order L was assumed.
Figures adapted from reference [57].
84
Figure A.3: Simulated decay scheme [3 of 5] for the β − decay of 223 Rn to 223 Fr. The
decay scheme was constructed from experimental data [43, 57]; incomplete information was replaced with conservative estimates. The level diagrams shown here were
used for simulation purposes, they do not represent the current knowledge of the excited states in 223 Fr. For unlabelled multipolarities, the lowest order L was assumed.
Figures adapted from reference [57].
85
Figure A.4: Simulated decay scheme [4 of 5] for the β − decay of 223 Rn to 223 Fr. The
decay scheme was constructed from experimental data [43, 57]; incomplete information was replaced with conservative estimates. The level diagrams shown here were
used for simulation purposes, they do not represent the current knowledge of the excited states in 223 Fr. For unlabelled multipolarities, the lowest order L was assumed.
Figures adapted from reference [57].
86
Figure A.5: Simulated decay scheme [5 of 5] for the β − decay of 223 Rn to 223 Fr. The
decay scheme was constructed from experimental data [43, 57]; incomplete information was replaced with conservative estimates. The level diagrams shown here were
used for simulation purposes, they do not represent the current knowledge of the excited states in 223 Fr. For unlabelled multipolarities, the lowest order L was assumed.
Figures adapted from reference [57].
87
Appendix B
MATLAB Code: γ-ray angular distributions
for any transition and degree of orientation.
1
% Alpha.m
2
function out = Alpha(k,ji,pops)
3
rho = Rho(k,ji,pops);
4
b = B(k,ji);
5
% zero checking //////////
6
if (rho == 0 | | b == 0)
7
out = 0;
8
return;
9
end
10
% //////////
11
out = (rho)/(b);
1
% Amax.m
2
function out = Amax(k,ji,L1,L2,jf,∆)
3
if
4
5
== 0
∆
out = (1/(1+∆ˆ2))*(f(k,jf,L1,L1,ji));
else
88
out = (1/(1+∆ˆ2))*(f(k,jf,L1,L1,ji)+2*∆*f(k,jf,L1,L2,ji)+...
6
((∆)ˆ2)*f(k,jf,L2,L2,ji));
7
8
end
1
% B.m
2
function out = B(k,j)
3
J = abs(j);
4
if (mod(2*j,2) == 0 | | j == 1) % integral spin
out = ((2*J+1)ˆ(1/2))*((−1)ˆ(J))*ClebschGordan(J,0,J,0,k,0);
5
6
else % half−integral spin
7
out = ((2*J+1)ˆ(1/2))*((−1)ˆ(J−1/2))*...
8
ClebschGordan(J,1/2,J,−1/2,k,0);
9
end
1
% ClebschGordan.m
2
% returns the Clebsch−Gordan coefficient <j1,m1,j2,m2 | j,m>
3
function cg = ClebschGordan(j1,m1,j2,m2,j,m)
4
% Check Conditions //////////
5
if ( 2*j1
floor(2*j1) | | 2*j2
6=
6
| | 2*m1
7
| | 2*m
6=
6=
6=
floor(2*j2) | | 2*j
floor(2*m1) | | 2*m2
6=
6=
floor(2*j) ...
floor(2*m2)...
floor(2*m) )
8
error('All arguments must be integers or half−integers.');
9
return;
10
end
11
if m1 + m2
6=
m
12
%warning('m1 + m2 must equal m.');
13
cg = 0;
89
return;
14
15
end
16
if ( j1 − m1
6=
floor ( j1 − m1 ) )
17
%warning('2*j1 and 2*m1 must have the same parity');
18
cg = 0;
19
return;
20
end
21
if ( j2 − m2
6=
floor ( j2 − m2 ) )
22
%warning('2*j2 and 2*m2 must have the same parity');
23
cg = 0;
24
return;
25
end
26
if ( j − m
6=
floor ( j − m ) )
27
%warning('2*j and 2*m must have the same parity');
28
cg = 0;
29
return;
30
end
31
if j > j1 + j2 | | j < abs(j1 − j2)
32
%warning('j is out of bounds.');
33
cg = 0;
34
return;
35
end
36
if abs(m1) > j1
37
%warning('m1 is out of bounds.');
38
cg = 0;
39
return;
40
end
41
if abs(m2) > j2
42
%warning('m2 is out of bounds.');
43
cg = 0;
90
return;
44
45
end
46
if abs(m) > j
47
%warning('m is out of bounds.');
48
cg = 0;
49
return;
50
end
51
% //////////
52
term1 = (((2*j+1)/factorial(j1+j2+j+1))*factorial(j2+j−j1)*...
53
factorial(j+j1−j2)*factorial(j1+j2−j)*factorial(j1+m1)*...
54
factorial(j1−m1)*factorial(j2+m2)*factorial(j2−m2)*...
55
factorial(j+m)*factorial(j−m))ˆ(0.5);
56
sum = 0;
57
for k=0:1:99
if ( (j1+j2−j−k < 0) | | (j−j1−m2+k < 0) | | (j−j2+m1+k < 0)...
58
| | (j1−m1−k < 0) | | (j2+m2−k < 0) )
59
else
60
term = factorial(j1+j2−j−k)*factorial(j−j1−m2+k)*...
61
62
factorial(j−j2+m1+k)*factorial(j1−m1−k)*...
63
factorial(j2+m2−k)*factorial(k);
if (mod(k,2) == 1)
64
term = −1*term;
65
66
end
67
sum = sum + 1/term;
end
68
69
end
70
cg =term1*sum;
71
% Reference: An Effective Algorithm for Calculation of the C.G.
72
% Coefficients Liang Zuo, et. al.
73
% J. Appl. Cryst. (1993). 26, 302−304
91
1
% F.m
2
function out = F(k,jf,L1,L2,ji)
3
% zero checking //////////
4
CG = ClebschGordan(L1,1,L2,−1,k,0);
5
if CG == 0
6
out = 0;
7
return;
8
end
9
W = RacahW(ji,ji,L1,L2,k,jf);
10
if W == 0
11
out = 0;
12
return;
13
end
14
% //////////
15
out = ((−1)ˆ(jf−ji−1))*(((2*L1+1)*(2*L2+1)*(2*ji+1))ˆ(1/2))*CG*W;
1
% f.m
2
function out = f(k,jf,L1,L2,ji)
3
% zero checking //////////
4
b = B(k,ji);
5
if b == 0
6
out = 0;
7
return;
8
end
9
% //////////
10
out = b*F(k,jf,L1,L2,ji);
92
1
% LegendreP.m
2
function P = LegendreP(n,x)
3
if n == 0
P = 1;
4
5
elseif n == 2
P = (1/2)*(3*(x)ˆ2−1);
6
7
elseif n == 4
P = (1/8)*(35*(x)ˆ4−30*(x)ˆ2+3);
8
9
elseif n == 6
P = (1/16)*(231*(x)ˆ6−315*(x)ˆ4+105*(x)ˆ2−5);
10
11
elseif n == 8
P = (1/128)*(6435*(x)ˆ8−12012*(x)ˆ6+6930*(x)ˆ4−1260*(x)ˆ2+35);
12
13
elseif n == 10
P = (1/256)*(46189*(x)ˆ10−109395*(x)ˆ8+90090*...
14
(x)ˆ6−30030*(x)ˆ4−3465*(x)ˆ2−63);
15
16
else
17
error('Legendre polynomial not found.');
18
return;
19
end
1
function out = Plot3DSurface(data)
2
resTheta = 5;
3
resPhi = 5;
4
out = zeros(181,1);
5
outRes = zeros((180)/resTheta,1);
6
resSurface = zeros((180/resTheta),(360/resPhi));
7
sum = 0;
8
for i=0:1:180
93
for j=0:1:360
9
sum = sum + data(i+1,j+1);
10
end
11
12
end
13
for i=0:1:180
for j=0:1:360
14
out(i+1) = out(i+1) + data(i+1,j+1)/sum;
15
end
16
17
end
18
indexTheta = 1;
19
for i=0:1:180
if(i == indexTheta*resTheta && i
20
6=
180)
21
outRes(indexTheta) = outRes(indexTheta) + out(i+1);
22
indexTheta = indexTheta + 1;
23
end
24
outRes(indexTheta) = outRes(indexTheta) + out(i+1);
25
end
26
for i=0:1:(180/resTheta)−1
for j=0:1:(360/resPhi)−1
27
28
surfarea = 1;
29
resSurface(i+1,j+1) = outRes(i+1)/surfarea;
end
30
31
end
32
r = resSurface;
33
xx = zeros(size(resSurface));
34
yy = zeros(size(resSurface));
35
zz = zeros(size(resSurface));
36
for i=0:1:((180/resTheta)−1)
37
38
for j=0:1:(360/resPhi)
if(j == (360/resPhi)) % make it the same as theta = 0;
94
xx(i+1,j+1) = r(i+1,1)*sin((i*resTheta*pi/180.0)+...
39
((resTheta/2)*pi/180.0))*cos((0*pi/180.0));
40
yy(i+1,j+1) = r(i+1,1)*sin((i*resTheta*pi/180.0)+...
41
((resTheta/2)*pi/180.0))*sin((0*pi/180.0));
42
zz(i+1,j+1) = r(i+1,1)*cos((i*resTheta*pi/180.0)+...
43
((resTheta/2)*pi/180.0));
44
else
45
46
xx(i+1,j+1) = r(i+1,j+1)*sin((i*resTheta*pi/180.0)+...
47
((resTheta/2)*pi/180.0))*cos((j*resPhi*pi/180.0));
48
yy(i+1,j+1) = r(i+1,j+1)*sin((i*resTheta*pi/180.0)+...
49
((resTheta/2)*pi/180.0))*sin((j*resPhi*pi/180.0));
50
zz(i+1,j+1) = r(i+1,j+1)*cos((i*resTheta*pi/180.0)+...
((resTheta/2)*pi/180.0));
51
end
52
end
53
54
end
55
clf
56
h1 = surf(xx,yy,zz);
57
set(h1,'facealpha',0.85);
58
colormap cool
59
light
60
lighting phong
61
axis tight equal off
62
% xlabel('Normalized Units')
63
% ylabel('Normalized Units')
64
% zlabel('Normalized Units')
65
view(40,30)
66
camzoom(1.5)
67
fh = figure(1); % returns the handle to the figure object
68
set(fh, 'color', 'white'); % sets the color to white
95
1
% RacahW.m
2
function out = RacahW(a,b,c,d,e,f)
3
out = ((−1)ˆ(a+b+d+c))*Wigner6j(a,b,e,d,c,f);
1
% Rho.m
2
function out = Rho(k,ji,pops)
3
sum = 0;
4
J = abs(ji);
5
for mi=−J:1:J
6
mdex = mi+J;
7
M = mi;
8
pop = pops(mdex+1);
9
sum = sum + ((−1)ˆ(J−M))*ClebschGordan(J,M,J,−1*M,k,0)*pop;
10
end
11
out =((2*J+1)ˆ(1/2))*sum;
1
% Wigner3j.m
2
function out = Wigner3j(j1,j2,j3,m1,m2,m3)
3
% error checking //////////
4
if ( 2*j1
6=
floor(2*j1) | | 2*j2
6=
floor(2*j2) | | 2*j3
6=
...
floor(2*j3)...
5
| | 2*m1
6=
floor(2*m1) | | 2*m2
6
| | 2*m3
6=
floor(2*m3) )
6=
floor(2*m2)...
7
error('All arguments must be integers or half−integers.');
8
return;
9
10
end
if m1 + m2 + m3
6=
0
96
11
%warning('m1 + m2 + m3 must equal zero.');
12
out = 0;
13
return;
14
end
15
if ( j1 + j2 + j3
6=
floor(j1 + j2 + j3) )
16
%warning('2*j1 and 2*m1 must have the same parity');
17
out = 0;
18
return;
19
end
20
if j3 > j1 + j2 | | j3 < abs(j1 − j2)
21
%warning('j3 is out of bounds.');
22
out = 0;
23
return;
24
end
25
if abs(m1) > j1
26
%warning('m1 is out of bounds.');
27
out = 0;
28
return;
29
end
30
if abs(m2) > j2
31
%warning('m2 is out of bounds.');
32
out = 0;
33
return;
34
end
35
if abs(m3) > j3
36
%warning('m3 is out of bounds.');
37
out = 0;
38
return;
39
end
40
% //////////
97
41
out = ((−1)ˆ(j1−j2−m3))/((2*j3+1)ˆ(1/2))*...
ClebschGordan(j1,m1,j2,m2,j3,−m3);
42
1
% Wigner6j.m
2
function sum = Wigner6j(J1,J2,J3,J4,J5,J6)
3
% error checking //////////
4
if J3 > J1 + J2 | | J3 < abs(J1 − J2)
warning('first J3 triange condition not satisfied. J3 > J1 + ...
5
J2 | | J3 < abs(J1 − J2)');
6
sum = 0;
7
return;
8
end
9
if J3 > J4 + J5 | | J3 < abs(J4 − J5)
warning('second J3 triange condition not satisfied. J3 > J4 + ...
10
J5 | | J3 < abs(J4 − J5)');
11
sum = 0;
12
return;
13
end
14
if J6 > J2 + J4 | | J6 < abs(J2 − J4)
warning('first J6 triange condition not satisfied. J6 > J2 + ...
15
J4 | | J6 < abs(J2 − J4)');
16
sum = 0;
17
return;
18
end
19
if J6 > J1 + J5 | | J6 < abs(J1 − J5)
20
warning('second J6 triange condition not satisfied. J6 > J1 + ...
J5 | | J6 < abs(J1 − J5)');
21
sum = 0;
22
return;
98
23
end
24
% //////////
25
j1 = J1;
26
j2 = J2;
27
j12 = J3;
28
j3 = J4;
29
j = J5;
30
j23 = J6;
31
sum = 0;
32
for m1=−j1:1:j1
for m2=−j2:1:j2
33
for m3=−j3:1:j3
34
for m12=−j12:1:j12
35
for m23=−j23:1:j23
36
for m=−j:1:j
37
sum = sum + ((−1)ˆ(j3+j+j23−m3−m−m23))*...
38
39
Wigner3j(j1,j2,j12,m1,m2,m12)*...
40
Wigner3j(j1,j,j23,m1,−m,m23)*...
41
Wigner3j(j3,j2,j23,m3,m2,−m23)*...
42
Wigner3j(j3,j,j12,−m3,m,m12);
end
43
end
44
end
45
end
46
end
47
48
end
1
% W Plot2DGammaAngularDistributions.m
2
function [xy] = W Plot2DGammaAngularDistributions(ji,jf,L1,L2,∆,...
99
minTheta,stepTheta,maxTheta,pops)
3
4
Ji = abs(ji);
5
Jf = abs(jf);
6
Alpha2 = Alpha(2,Ji,pops);
7
Alpha4 = Alpha(4,Ji,pops);
8
A2max = Amax(2,Ji,L1,L2,Jf,∆);
9
A4max = Amax(4,Ji,L1,L2,Jf,∆);
10
A2 = Alpha2*A2max;
11
A4 = Alpha4*A4max;
12
sum = 0;
13
theta =minTheta:stepTheta:maxTheta;
14
xy=zeros(length(theta),2);
15
for i=1:1:length(theta)
16
xy(i,1) = theta(i);
17
xy(i,2) = 1 + A2*LegendreP(2,cos(theta(i))) +...
A4*LegendreP(4,cos(theta(i)));
18
sum = sum + sin(theta(i)).*xy(i,2)*stepTheta;
19
20
end
21
disp( sprintf( 'Amax%d
= %d', 2,A2max ) );
22
disp( sprintf( 'Amax%d
= %d', 4,A4max ) );
23
disp( sprintf( 'MyAlpha%d = %d', 2,Alpha2 ) );
24
disp( sprintf( 'MyAlpha%d = %d', 4,Alpha4 ) );
25
disp( sprintf( 'Integral = %d', sum ) );
26
colors={'−+k';'−−or';':*b';'−.xk';'−sr';'−−db';':ˆk';'−.vr';'−>b'};
27
plot(xy(:,1),xy(:,2),char(colors(1)))
28
hold on
29
set(gca,'XTick',0:pi/4:pi)
30
set(gca,'XTickLabel',{'0','pi/4','pi/2','3*pi/2','pi'})
31
xlabel('\theta')
32
ylabel('W')
100
33
if L1 == L2 &&
∆
== 0
title( sprintf( 'Angular Profile Partially Aligned Nuclei For ...
34
L%i Transition with Ji = %d and Jf = %d', L1,ji,jf ) );
35
else
title( sprintf( 'Angular Profile For Partially Aligned Nuclei ...
36
For L%i, L%i (∆ = %d) Transition with Ji = %d and Jf = ...
%d', L1,L2,∆,ji,jf) );
37
end
38
% Reference: Tables of Coefficients for Angular Distribution of
39
% Gamma Rays From Aligned Nuclei, T. Yamazaki
40
% Nuclear Data A (1967), 3, 1−23
1
% W Plot3DGammaAngularDistributions.m
2
function [xy] = W Plot3DGammaAngularDistributions(ji,jf,L1,L2,∆,pops)
3
minTheta=0;
4
stepTheta=pi/180;
5
maxTheta=pi;
6
theta =minTheta:stepTheta:maxTheta;
7
data = W Plot2DGammaAngularDistributions(ji,jf,L1,L2,∆,minTheta,...
stepTheta,maxTheta,pops);
8
9
10
data3D=zeros(length(theta),361);
for i=1:1:length(theta)
for j=1:1:361
11
data3D(i,j) = data(i,2);
12
end
13
14
end
15
Plot3DSurface(data3D);
16
% Reference: Tables of Coefficients for Angular Distribution of
17
% Gamma Rays From Aligned Nuclei, T. Yamazaki
101
18
% Nuclear Data A (1967), 3, 1−23
102
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