rand msc thesis

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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


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ABSTRACT

GEANT4 SIMULATIONS FOR THE RADON ELECTRIC

DIPOLE MOMENT SEARCH AT TRIUMF

Evan Thomas RandUniversity 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 199Hg 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-ray scattering and

backgrounds in the experimental apparatus and -ray detectors, and are being used

to study the overall sensitivity of the RnEDM experiment at TRIUMF in Vancouver,

B.C.


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Dictated but not read.

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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. Ill always cherish the memories from experiments

and conferences.

Finally I would like to thank my family, who supports me throughout all my adven-

tures. Thanks to my sisters Erin and Lauren, for teasing me relentlessly throughout

my childhood, and subsequently higher education. Your unique way of showing sup-

port 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.

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Contents

Acknowledgements ii

1 Introduction 1

1.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.2 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Atomic EDMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Radon EDM Enhancements . . . . . . . . . . . . . . . . . . . 7

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

2.3 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

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2.4 GRIFFIN Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Geant4 Developments for the RnEDM Experiment 32

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.2 Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 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

3.3 Simulating -Decay Process . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.1 Timing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.2 Particle Emission . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 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

3.5 RnEDMGeant4 Geometry . . . . . . . . . . . . . . . . . . . . . . . 54

3.5.1 Cell and Oven Design. . . . . . . . . . . . . . . . . . . . . . . 54

3.5.2 LaBr3(Ce) Scintillator . . . . . . . . . . . . . . . . . . . . . . 55

3.6 Data Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.6.1 The GUGI Program . . . . . . . . . . . . . . . . . . . . . . . 58

3.6.2 Output Data and Sort Codes . . . . . . . . . . . . . . . . . . 59

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4 Results 60

4.1 -Ray Spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1.1 Efficiencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1.2 223Rn-Decay Spectra . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Frequency Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1 Multipolarity Effects . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.2 Fitting Process . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.3 Statistical Limit. . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3 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

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List of Tables

1.1 Current upper limits on EDMs of the neutron, electron and 199Hg atom 6

3.1 Hexidecimal flags in Geant4output binary data . . . . . . . . . . . 40

3.2 X-ray energies and intensities per 100 Fr K-shell vacancies . . . . . . 48

3.3 Three-dimensional-ray angular distributions for various transtions . 52

4.1 Fit results for the 416 keV-ray M1 transition. . . . . . . . . . . . . 73

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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 4He+ ion. . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Level diagram of 85Rb . . . . . . . . . . . . . . . . . . . . . . . . . . 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

2.10 GRIFFIN detectors in the back configuration . . . . . . . . . . . . . 31

3.1 Cross section of the RnEDM simulated apparatus . . . . . . . . . . . 38

3.2 Geant4simulation of radioactive decay . . . . . . . . . . . . . . . . 43

3.3 particle angular distributions for various degrees of polarization. . . 49

3.4 -ray angular distributions for various multipolarities and spins . . . 53

3.5 Cross section of the EDM cell, oven, magnet and Metal shielding . . 54

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3.6 Cross section of the BrilLanCe 380 LaBr3(Ce) scintillator . . . . . . . 56

3.7 Screenshots of the GUGI program. . . . . . . . . . . . . . . . . . . . 57

3.8 FWHM2 versus -ray energy for a LaBr3(Ce) and a HPGe detector . 58

4.1 Geant4absolute efficiency curve for a ring of GRIFFIN detectors. . 62

4.2 Geant4absolute efficiency curve for the RnEDM apparatus . . . . . 63

4.3 Geant4absolute efficiency curve for various thicknesses ofMetal . 64

4.4 Full 223Rn decay detected by a ring of eight GRIFFIN detectors . . . 65

4.5 The energy gate and time projection of the 416 keVray. . . . . . . 66

4.6 Three-dimensional-ray angular distributions for transitions in 223Fr 68

4.7 The time projections and fits for transitions in 223Fr . . . . . . . . . . 69

4.8 The fit to the 416 keV-ray time projection. . . . . . . . . . . . . . . 70

4.9 Weighted average of 20 precession frequency fits . . . . . . . . . . . . 72

4.10 The sensitivity of the fitted frequency versus the simulatedT2 time . 74

4.11 The sensitivity of the fitted frequency versus the number of counts . . 75

4.12 Geant4

absolute efficiency curve for a ring of BrilLanCe 380 detectors 76

4.13 Full 223Rn 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 decay of 223Rn to 223Fr. . 83





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Chapter 1

Introduction

The search for an atomic electric dipole moment (EDM) in odd-A isotopes of radon

is beginning at TRIUMF, Canadas 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 non-

zero particle or atomic EDM would represent the discovery of new physics beyond theStandard 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 cur-

rent theories for physics beyond the Standard Model, such as multiple-Higgs theories,

left-right symmetry, and supersymmetry (SUSY), predict EDMs within current ex-

perimental reach [1]. Present limits on the EDMs of the neutron, electron, and 199Hg

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 theoreti-

cal calculations[2,3, 4, 5, 6, 7] which predict large enhancements in the observable

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atomic EDM for atoms in which the nucleus has a non-zero octupole deformation.

1.1 Motivation

The electric dipole moment (EDM), d= ieiri, of a particle or atom in an electric

field Ecan be described by the following Hamiltonian,

H= d E . (1.1)

The EDM of a particle or atom is a vector quantity that must be either aligned or

anti-aligned with the total spin S. Therefore

d can be expressed as

S, where is

a constant of proportionality. The Hamiltonian for a system with an electric dipole

moment may thus be rewritten as

H= S E . (1.2)

The Hamiltonian for this system is odd under both the parity ( P) and time-reversal

(T) operations, as can be seen through:

PH = P(S E) TH = T(S E)

= (+S) (E) = (S) (+E)

= +S E = +S E

= H = H

Acting with the parity operator (P) on this Hamiltonian leaves the spin invariant,

but changes the sign of the electric field. Conversely, acting with the time-reversal

operator (T) 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

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P

T

Figure 1.1: An illustration of the parity and time-reversal transformations operatingon a quantum system with a charge distribution and non-zero spin. Acting with theparity (P) operator on this system leaves the spin invariant, but changes the sign ofthe electric dipole moment. Conversely, acting with the time-reversal (T) operatoron this system changes the direction of the spin and leaves the electric dipole momentinvariant. 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 proportionalitybetween Sand d has changed sign under the P or T transformations.

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 P and T transformations on a quantum

system (particle or atom) with a charge distribution and non-zero spin. Acting with

the parity operator (P) on the quantum system flips the positions of the charges. This

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is equivalent to changing the sign of the electric field ( E) in the above equations. The

Ptransformation in Figure1.1changes the direction of the EDM from the upward

direction to the downward direction. The EDM is now anti-parallel to the total

spin vector S and the magnetic moment . The magnetic moment, which can be

expressed as= g Sretains its constant of proportionality with the total spin vector

under both the P and T transformations. Acting with the time-reversal operator

(T) on the quantum system changes the direction of the spin. This is equivalent

to changing the sign of the total spin vector ( S) in the above equations. The T

transformation in Figure1.1flips the total spin vector S from the upward direction

to the downward direction. The total spin vector Sand the magnetic moment now

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

in both cases the constant of proportionality between dand Shas changed sign.

One symmetry left out of this discussion so far is the charge conjugation symme-

try (C), which exchanges particles for anti-particles. This symmetry, in combination

with the P and Tform what is called theC P Tsymmetry. All experimental evidence

to date supports that CP T is a true symmetry of nature, known as the CP T The-

orem [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 ques-

tioned[9]. Soon after this publication parity was shown to be violated in the decay

of 60Co nuclei [10]. This result implied violations in other combinations ofC,P and

T in order for C P Tto remain a good symmetry of the laws of nature.

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1.1.1 CP Violation

Following the observation ofPviolations in the decay, it was believed that CP

was a true symmetry of nature. CPviolation 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 ofTviolation via the C P 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 CPviolation is essential for baryo-

genesis, 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 CKM of the

Cabibbo-Kobayashi-Maskawa (CKM) matrix and via QCD, the vacuum expectation

value of the QCD gluon field. These sources ofC Pviolation are, however, not suffi-

cient to account for the observed asymmetry between matter and anti-matter in our

universe. Thus additional sources ofCPviolation 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 CKM and QCD. 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 CP-

violating complex phases and predict EDMs within current experimental reach [1].

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Current upper limits on the EDMs of the neutron, electron and 199Hg atom (see

Table1.1) 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 199Hgatom.

Species EDM Upper Limit C.L.Neutron


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Figure 1.2: Experimental upper limits for the EDMs of the neutron, electron and199Hg atom as a function of time compared to the ranges predicted for typical param-eter 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 of600 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.

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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

3, where 3 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

parameter3, hence a large octupole deformation gives a large intrinsic Shiff moment.

According to the Schiff theorem[21], the nuclear EDM is screened by the orbit-

ing 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 = eZ R30

9

20

3523 , (1.4)

whereR0is the nuclear radius and Lmeasures 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 expecta-

tion value for the laboratory Schiff moment is the result of the mixing of these nearby

opposite parity states by a P andTodd interaction (VPT),

Slab = 2 II+ 1

Sintr , (1.5)

whereIis the nuclear spin and

=| VPT |+

E+ E

, (1.6)

where the even- and odd-parity states (+ and) have energies E+ andE. These

states arise from a breaking of the degeneracy of octupole-deformed states in a double-

well potential as a function of3 (see Figure 1.3). This is similar to the ammonia

molecule (NH3) in molecular physics. The nitrogen atom experiences a double-well

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Figure 1.3: Illustration of the double well potential as a function of 3 that arisesfor octupole-deformed nuclei. The magnitude of octupole (L= 3) deformation is de-scribed by the parameter3, where3measures the presence of the octupole sphericalharmonic in the nuclear shape.

potential with one potential well for the N atom on either side of the H 3 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

Eassociated with the tunnelling process. The same physics applies to octupole

deformed nuclei in which a doublet of states with opposite parity and a small energysplitting Eresults from equal admixtures of the two intrinsic states at , and the

tunnelling through a large potential energy barrier at 3= 0.

To completely describe the P-, T-odd nuclear potential VPT in Equation1.6, a

two-body interaction is required. However for the purpose of estimating the collective

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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]

VPT = 3G8

2mr30(R0 r) , (1.7)

whereG is the Fermi constant, r0 is the internucleon distance and parametrizes the

strength of theP T-odd interaction. From Equation1.7, the collective Schiff moment

in the laboratory frame can be estimated as [3]

Slab 0.05e223 ZA

2/3r30

|E+

E

| . (1.8)

Equation1.8characterizes 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 Zof the radon isotopes. In

1974, M.A. and C. Bouchiat demonstrated that parity-violating interactions in atoms

increase with the atomic number faster thanZ3 [22,23]. This significant enhancement

encouraged (and continues to motivate) many studies for parity-violation studies inheavy 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 of600 relative

to the 199Hg atom, which currently has the best atomic EDM upper limit at 3.1

1029 ecm[19].

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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 (221,223,225Rn) are relatively short lived, (half-lives of 25 minutes) which make

it challenging to obtain large enough quantities to perform an EDM measurement us-

ing standard NMR techniques. Furthermore, these isotopes do not occur naturallyin the decay chains of 238U and 232Th. 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 Canadas national subatomic physics laboratory located on the cam-

pus of the University of British Columbia in Vancouver. TRIUMF, TRI-University

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Meson Facility, is built around a 500 MeV proton cyclotron which provides simultane-

ously 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 Rare-

Isotope Beams (RIBs). The ISOL system consists of a primary production beam of

500 MeV protons with an intensity up to 100 A, 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

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Figure 2.1: A schematic of the ISAC-I Hall at TRIUMF illustrating the locationsof: the accelerated proton beam, the target ion-source modules, high-resolution massseparator, 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,

r= 1

B

2mV

q , (2.1)

wherer 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),qis 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

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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 mm = 11000 , is able to distinguish between neighbouring isotopes (different

mass number A), however, isobaric and even molecular contamination with the same

totalA 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 A on target materials with Z 82 [24]. Two actinide targets, ura-

nium 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 A for licensing reasons. The uranium-oxide target will remain

limited to approximately 2A 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 A 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

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a measurement cell. The RnEDM experiment will implant a beam of Rn ions (likely

221Rn or 223Rn) 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 Figure2.2,was tested with 120 Xe as beams of radon isotopes were

not available at the time of the tests. The 120Xe isotope was chosen due its half-life of

40 minutes, comparable to the roughly 25 minute half-lives of 221Rn and 223Rn. An

initial beam of 120Cs produced the 120Xe isotopes through decay; the decay chain

is shown in Equation2.2. The beam was implanted in a thin zirconium foil for about

two 120Xe half-lives, after which the remaining 120Cs atoms were given roughly 10

minutes to decay into 120Xe.

120Cs(64 s) 120Xe(40 m) 120I(81 m) 120Te(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 vol-

ume. 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

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Figure 2.2: Schematic of the prototype on-line noble gas collection apparatus testedat TRIUMF with isotopes of 120Xe[26].

was then closed to trap the xenon gas in the measurement cell. The trapped radioac-

tive 120Xe atoms were observed in the measurement cell using high-purity germanium

(HPGe) -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 121,123Xe [27].

2.3 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

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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 colli-

sions with optically pumped alkali-metals [28,29, 30]. This method has been shown

to be successful in the polarization of radon isotopes ( 209,223Rn) at the ISOLDE iso-

tope 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 con-

veniently in the range where diode lasers are available. The measurement cell in

the RnEDM apparatus will contain an optical pumping region, containing natural

rubidium (85,87Rb) 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 spin-

exchange collisions and the N2 gas acts as a buffer. The natural rubidium (85,87Rb)

atoms have odd-A nuclei and therefore have a non-zero nuclear spin ( I) which com-

plicates its level structure. To illustrate the process of optical pumping we will first

consider a much simpler atom with no nuclear spin.

Consider the 4He+ 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

Figure2.3. Both energy levels have J= 1/2; in the presence of a magnetic field the

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Magnetic Field

Energy

= 1/2

= 1/2

= 1/2

= 1/2

_

+

_

+

Excitation Decay

Excited State (2P1/2

)

Ground State (1S1/2

)

m

m

m

m

Figure 2.3: Level diagram for a 4He+ ion. The splitting of the energy levels of theground state and first excited state increase with the applied magnetic field (notdrawn to scale). Shining positive helicity circularly polarized light on the atom drivesthe transition from the m = 1/2 ground state to the m= +1/2 excited state. Theexcited state can decay into both ground-state levels. The resulting effect pumpsthe 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 tran-

sitions from either of the ground-state levels to either of the excited-state levels. If

the laser light is circularly polarized with positive helicity (+) 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

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OrbitalsFine

Structure StructureHyperfine

EffectZeeman

F

Occupied

m

3+

_

+

_

+

+

_

_

2

3

2

2

3

3

2

D1

5s

2P

1/2

2P

3/2

5p

F = 3

F = 2

F = 3

F = 2

2S

1/2

D2

Figure 2.4: Level diagram of 85Rb (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 them = +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 (85,87Rb) 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

1s22s22p63s23p63d104s24p65s.

The first 36 electrons are in closed sub-shells which gives zero total angular momen-

tum. The valence electron in the 5s orbital behaves similarly to the above simplified

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scenario with 4He+. 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)

whereLis 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 spinI. The total angular momentum of the atom, F, is given by

F= J + I (2.4)

where the nuclear spin for 85Rb is I = 5/2 and for 87Rb 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 85Rb are illustrated in Figure2.4.

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

2mZ e

2

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= geZe

2

4m2c21

r3L S , (2.6)

wherege is the electronic g-factor. The splittings which result from this interaction

are called the fine structure and have an energy scale of approximately 104 eV. These

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energy levels are separated according to the value ofJ. The 5s orbital does not split,

as shown in Figure 2.4, since l = 0. The energy difference between the 2S1/2 and

2P1/2 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 2S1/2 and 2P3/2 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= Ze2gN

2mMNc2

1

4

S

I2 1

r

+

(I

)

1

r , (2.7)

where gN is the nuclear g-factor. These levels split according to the total angular

momentum F. 85Rb has a nuclear spin ofI= 5/2, thus the allowed values ofF are

5/2 1/2 = 2 and 5/2 + 1/2 = 3, as shown in Figure2.4. These splittings are on the

order of 106 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 qand angular momentum

L is given by,

= q

2mcL . (2.8)

Extending this result into quantum mechanics, the magnetic moment due to the total

electronic angular momentum is given by,

J= gJ e2mc

J, (2.9)

and the magnetic moment of the atom is given by,

F = gF e2mc

F , (2.10)

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where the gJ and gF are the Lande g-factors. These factors arise from the addition

of angular momentum operators. The Lande g-factors are given by,

gJ= 1 +J(J+ 1) + S(S+ 1) L(L+ 1)

2J(J+ 1)

, (2.11)

and

gF =gJF(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 = F B,

which results in the following first-order perturbation,

HB =gFBmFBz , (2.13)

whereB is the Bohr magneton and mFis the eigenvalue of the Fz operator. The

energy splittings are approximately linear for small external magnetic fields. The

splittings of the 2S1/2, F = 3 state in 85Rb is 1.9293 109 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 HamiltonianH = Hhf+HB

need to be solved. This result is known as the Breit-Rabi formula:

E(F =I

1/2, mF) =

Ehf

2(2I+ 1)1

2E2hf+ 4mF

2I+ 1

gJBBEhf+ (gJBB)2 . (2.14)

The pumping process for the 5s 85Rb electron is similar to the simplified example

discussed above for 4He+. Illuminating the 85Rb atom with laser light tuned to the

D1 line will drive transitions from any of the 2S1/2 ground states into any of the

2P1/2 excited states. If the laser light is circularly polarized with positive helicity

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(+) 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 2P1/2 excited

state, see Figure2.4, any electrons in the mF = +3 magnetic substate of the 2S1/2

ground state will remain trapped there. As the 2P1/2 excited states relax through

photon emission the probability of populating the mF = +3 magnetic substate of

the 2S1/2 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 2S1/2 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 + N S + K S + gsBB S + gIBB I + gKBB 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 short-

lived van der Waals molecules or by simple binary collisions between atoms, shown

in Figure 2.5. For light noble gases, such as 3He, binary collisions dominate the

transfer of angular momentum and the contribution from van der Waals molecules is

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(a) Formation andbreakup of an alkali-metal/noble-gas vander Waals molecule

(b) Binary collisionbetween an alkali-metal atom and anoble-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 (Figure2.5). 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 non-

radiatively de-excite (quenche) the excited rubidium atoms before they can re-

radiate a photon. This avoids radiation trapping, where a photon is emitted by one

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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 trappedmF = +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

= 2B 2dE , (2.16)

where is the magnetic moment, B is the magnetic field, d is the electric dipole

moment andEis the electric field. The +() corresponds to the electric field oriented

parallel (anti-parallel) to the magnetic field. The EDM,

d=

4E , (2.17)

is extracted through a measurement of the change in precession frequency between

the two different electric field orientations.

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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

radiation from the decaying radon nuclei. The angular distribution of the radiation

is dependent on the angle with respect to the precessing polarization vector of the

nuclear spins. In addition to therays, the emitted particles from the decay of

the polarized Rn nuclei also have an anisotropy that would enable the precession

frequency to be measured. However, these particles will be strongly scattered and

absorbed by the glass in the measurement cell and oven. In order to use the particles

as a means to observe an EDM signal, detectors would need to be integrated into

the measurement cell. Hence, the first stage of the RnEDM experiment will use the

-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 (= 4dE/) will signal a non-zero EDM

in radon. The precision of this measurement for N detected rays is given by [32]

= 2

T2

1

A2(1 B)2N , (2.18)

whereT2 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)

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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 -ray photopeaks. Studies with 209Rn 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-ray anisotropies. With an electric field of 5 kV/cm

we calculate the sensitivity for the EDM measurement (d) to be approximately

11026 e-cm using the-ray anisotropy technique. For theasymmetry method, we

expect larger backgrounds, B 0.2, but a significantly higher count-rate capability.

The count rate in thedetectors 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 1027 e-cm is thus expected for 100 days of counting using the beta-

asymmetry technique [32].

These estimates are statistical limits and do not account for the realistic sources

of backgrounds in the actual-ray experiment caused by bremsstrahlung production

from the stopping -particles and Compton scattering of photons both into and out

of the -ray detectors which can lead to backgrounds underneath the -ray photo-

peaks with different angular distributions than the -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 -ray anisotropy technique (see Chapter 3).

2.4 GRIFFIN Spectrometer

To observe theradiation in the RnEDM experiment the recently funded GRIF-

FIN spectrometer will be utilized. GRIFFIN stands for Gamma-Ray Infrastructure

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For Fundamental Investigations of Nuclei. The GRIFFIN array, shown in Figure2.6,

will be comprised of 16 unsegmented large-volume clover-type high-purity germanium

(HPGe) detectors with full Compton-suppression shields constructed of bismuth ger-

manate (Bi4Ge3O12), 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 (Figure2.7), 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 Figure2.9,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

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Figure 2.7: One GRIFFIN/TIGRESS HPGe clover comprised of four individual crys-tals. Each crystal is 90 mm in length and 60 mm in diameter and has an efficiencyof 40% relative to a standard 3 3 NaI(Tl) crystal for 662 keV 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 rays from directly hitting the front suppression shields.

The GRIFFIN spectrometer is very similar in external geometry to its sister ar-

ray 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 crys-

tal 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 TI-

GRESSs highly segmented crystals is to provide three-dimensional localization of

-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

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(a) Back detector configura-tion

(b) Forward detector configuration

Figure 2.8: Geant4 renderings of two GRIFFIN detector heads cross-sectioned inboth forward and back configurations.

therefore not necessary for GRIFFIN.

A full Geant4simulation 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 exper-

iment 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-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 Geant4simulation for the RnEDM experiment

at TRIUMF involving 8 GRIFFIN detectors in a ring geometry (see Figures 2.9and

2.10) was developed as part of the work presented in this thesis.

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(a) One GRIFFIN detector in the HPGeforward configuration

(b) Ring of eight GRIFFINdetectors close-packed in for-ward configuration

Figure 2.9: Geant4renderings of the GRIFFIN detectors in the forward configura-tion (highest efficiency mode).

(a) One GRIFFIN detector in theHPGe back configuration

(b) Ring of eight GRIF-FIN detectors close-packed inback configuration

Figure 2.10: Geant4renderings of the GRIFFIN detectors in the back configuration(optimized peak-to-total).

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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 Geant4simulations is to provide an accu-

rate description of-ray scattering and backgrounds in the experimental apparatus

and -ray detectors. From thisGeant4

framework, realistic measurements for the

RnEDM experiment can be simulated and ultimately test the sensitivity of the ap-

paratus. The Geant4simulations presented in this thesis are an essential aspect of

the development towards an EDM measurement.

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3.1 Introduction

3.1.1 Previous Work

The foundation of the RnEDM simulations was built on the existing Geant4

simulations of the TRIUMF-ISAC Gamma-Ray Escape Suppressed Spectrometer (TI-

GRESS) [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

-ray detectors [36]. As discussed in Section2.4, the GRIFFIN -ray spectrometer

is very similar in external geometry to its sister spectrometer TIGRESS. The pri-

mary difference between the two -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-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 -ray spectrometer will, however, be very similar to

that of an unsegmented TIGRESS -ray spectrometer. As electrical contact segmen-

tation for TIGRESS has been incorporated at the analysis, rather than simulation,

stage, the fully verified Geant4simulation framework for TIGRESS can, in fact, be

used as a Geant4simulation 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

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the simulation. For many decay-spectroscopy experiments, multiple detection sys-

tems are used to detect a variety of particles including: rays, internal conversion

electrons, particles and 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 Geant4simulations.

Following the code restructuring it was noted that the layered volume geometries

in Geant4were no longer being handled properly. In the original TIGRESS simula-

tion if two geometries, composed of different materials, were overlapping in the three-

dimensional 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 physi-

cal 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 simula-

tion 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 ofrays, electrons and positrons in matter.

The -decay simulation developed for the RnEDM experiment selected particle

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energies, branching ratios, emission times and angular distributions from calculated

probability distributions. Random numbers were generated from a uniform distri-

bution 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++ stan-

dard 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, Geant4handles geometry,

physical processes and particle interactions. Geant, which is usually pronounced like

the French word geant (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 de-

fined by combinations of atomic elements. Inside the three-dimensional environment,

users fire particles (,,, 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

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energetic particles, such as a -ray above 1.022 MeV producing an electron-positron

pair or the slowing of an energetic 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 10m in this work, being tracked by Geant4. The simu-

lations 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, Metal 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 (B2O3), 4.0% sodium oxide (Na2O), 2.3%

aluminium oxide (Al2O3) and 0.1% miscellaneous traces. In these simulations the

miscellaneous traces were composed of equal parts of potassium oxide (K2O), calcium

oxide (CaO), dichlorine (Cl2), magnesium oxide (MgO) and iron three oxide (Fe

2O

3).

Metal is a nickel-iron alloy which has a very high magnetic permeability. The

high magnetic permeability makes Metal an effective magnetic shield. The RnEDM

experiment is extremely sensitive to magnetic field fluctuations and Metal will be

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used around the measurement cell in order to maintain a static magnetic field. Un-

fortunately, this shielding between the EDM cell and the -ray detectors will also

attenuate the emitted 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 Chapter4. The simulated Metal was formulated from specifica-

tions given by The MuShield Company [40]. Metal 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 construc-

tion of Saint-Gobains 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 den-

sity 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

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Figure 3.1: Cross section of the RnEDM simulated apparatus. The EDM cell andoven are surround by a ring of GRIFFIN -ray detectors in their highest efficiencymode.

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 elec-

tromagnetic. Only the decay products ( rays, particles and internal conversion

electrons) were simulated.

Interaction probabilities forrays within a material were calculated by Geant4s

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.

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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 -decay process including the subsequent 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 theray interactions, pair production processes, bremsstrahlung pro-

cesses, and their interaction times. See Table3.1for a list of all flags used in the data

stream.

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Table 3.1: Hexidecimal flags in Geant4output binary data

Flag Type Start Flag End FlagNormal Event Data 0x8000 0xFFFF

Photon Position Data 0x8010 0xFFE0

Pair Production Photon 1 Position Data 0x8020 0xFFD0Pair Production Photon 2 Position Data 0x8030 0xFFC0

Bremsstrahlung Photon Position Data 0x8040 0xFFB0Timing Data 0x8050 0xFFA0

3.3 Simulating -Decay Process

The simulations presented in this thesis focused on the decay of 223Rn to 223Fr,

a likely candidate for the RnEDM search at TRIUMF. A simulation of the entire -

decay 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 decay, level spins, parties, -

ray energies, intensities, multipolarities, mixing ratios, internal conversion coefficients

and X-ray shell vacancies in the daughter nucleus. Unfortunately, the level structure

of 223Fr [43]is not known to a very high precession. In fact, a significant number of

the observed rays from an experiment at ISOLDE [43] could not be placed in the

derived level structure. For the purposes of simulation thoserays were given their

own energy level with the appropriate intensity, see Appendix A on page 83for the

complete simulated decay scheme.

The decay mode of223 Rn is 100% decay. The simulation of this decay began by

emitting aparticle (electron) into the Geant4geometry. The energy and angular

distribution of the particle are described in Sections 3.3.2 and 3.4.1 respectfully.

Following the initialdecay, the daughter nucleus may exist in an excited state. The

excited daughter nucleus will then undergo decay and/or internal conversion to

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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 decay and internal conversion

processes are competitive. For any excited state the probabilities for either process

are given by

P= 1

1 + , PIC=

1 + , (3.1)

where 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]

=2L2+ L1

(1 + 2) (3.2)

where, the mixing ratio, is defined as

=Jf||L2||JiJf||L1||Ji . (3.3)

The total internal conversion coefficient is defined as a sum of all the individual shell

internal conversion coefficients, such that Tot = K+L + M+ . 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 K/Tot. Simulating the atomic shell structure enabled

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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 Section3.3.2.

3.3.1 Timing

The time of each individual-decay event from the ensemble of 223Rn nuclei was

generated using Monte Carlo from the input half-life and number of nuclei remain-

ing. 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=1N

ln(1 ) , (3.4)

where is the decay constant ( = ln(2)t1/2

), N is the remaining number of nuclei

and 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. Figure3.2illustrates the decay of one hundred million

223Rn 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 en-

ergy, initial position and momentum direction. The emission direction generally had

two main user options which were selected in the GUGI program before running the

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0 50 100 150 200Time (min)

10

100

1000

10000

Counts

red

= 1.212

t1/2

= 24.300(6) min

Figure 3.2: Geant4simulation of the decay of 108 223Rn nuclei detected by a ring ofeight GRIFFIN detectors in their highest efficiency mode with the radiation emittedisotropically into 4. The times were binned into seconds and the function used to

fit was y = A1+ A2e ln(2)t

A3 , where A1 is due to background, A2 is initial count rateand A3 is the half-life (t1/2).

simulation. The two options for the emission of radiation were an isotropic distribu-

tion 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 4, the following equations for and were utilized:

= 21

= cos1(1.0 {2[1.0 cos()]})

Where i are random numbers generated using the drand48 function and is the

maximum angle of emission (= for emission into 4 radians).

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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 exper-

iment 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 223Rn 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

Section2.3.3.

The angle of emission, , with respect to the axis of quantization can be generally

expressed in powers of cos()[46],

W() =

k=0,1,2,...

Akcosk() , (3.5)

where Ak are the angular-distribution coefficients and k is even for radiation and

odd for radiation. The exact form ofW() for and radiation is described in

detail in Section3.4.

Particle Emission

In the process ofdecay, the emitted particle has a range of kinetic energies

between zero and the Q-value of the decay. This energy spectrum results from the

existence of the neutrino or anti-neutrino in the decay processes shown below:

decay: AZXN

AZ+1Y

N1 + e + e

+ decay: AZXN AZ1YN+1 + e+ + e

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

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as[47]

I(E) = G

237c6| Mif|2 (T2e + 2Temec2)1/2(Q Te)2(Te+ mec2)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 andF(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

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 + 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 branch to accurately describe the energy of the emitted electrons from the

decay of 223Rn into 223Fr. An analytic approximation of the Fermi function was used

to calculate the shape of the spectrum, given by [48]

F(Z, E) =4(1 + s)[(2s)!]2

(2p)2s2(s2 + 2)s1/2e

22s+ s

6(s2+2)

, (3.7)

where s = [1 (Ze2/c)2]1/2, = R/(/mc), R is the nuclear radius approximated

asR = rA1/3 with r= 1.2 fm andA the nuclear mass number, p is the electron (or

positron) momentum (in units ofmc), = Ze2/(positive sign for decay and

negative for + decay) and 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].

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-Ray and Internal Conversion Electron Emission

As described in Section 3.3, the -decay and internal conversion processes are

competitive. Level structure information is required to accurately describe their prob-

abilities. At any particular excited state in the daughter nucleus there are Nnumber

ofdecays which can populate lower energy levels. The probability for a particular

transition is given by (1 +)I/Isum, where is the internal conversion coefficient,

I is the measured -ray intensity and Isum =N(1 + N)IN, 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 ofdecay versus internal

conversion are calculated from Equations3.1, and again selected by Monte Carlo.

If the selected process is decay, the energy of the emitted ray is simply read

from the input data. If the selected process is internal conversion, the energy of the

emitted electron is the-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 Geant4simulation, 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 223Rn to 223Fr 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 (223Fr). 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

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was to use an average X-ray energy per vacancy. These tables for all elements were

coded directly into the Geant4simulation. 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 -ray detectors. The input data for Fr K shell vacancies is given in

Table3.2. Careful inspection of these data reveals the intensities total to greater than

100% (per 100 K-shell vacancies). For everyKvacancy one or more X-rays may be

emitted. For example, an electron from the L shell can drop down and occupy the

Kshell 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 Geant4simulations 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 ofparticles from aligned nuclei has the form [50]

W() = 1 + APcos(), (3.8)

whereA is the beta-asymmetry correlation coefficient,Pis the degree of polarization

and 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 decay for

mixed transitions. These input data are not known for many of thedecay branches

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0 3 5 3 7 2

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

W()

P = 100%P = 75%P = 50%P = 25%

P = 0%

4 4 4 422

Figure 3.3: particle angular distributions for various degrees of polarization. Thebeta-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 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

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