HIGH-PRECISION HALF-LIFE MEASUREMENTS FOR
SUPERALLOWED FERMI β DECAYS
A Thesis
Presented to
The Faculty of Graduate Studies
of
The University of Guelph
by
GEOFFREY F. GRINYER
In partial fulfilment of requirements
for the degree of
Doctor of Philosophy
December, 2007
c
Geoffrey
F. Grinyer, 2008
ABSTRACT
HIGH-PRECISION HALF-LIFE MEASUREMENTS FOR
SUPERALLOWED FERMI β DECAYS
Geoffrey F. Grinyer
University of Guelph, 2007
Advisor:
Professor C.E. Svensson
High-precision measurements of the f t values for superallowed Fermi β decays
between 0+ isobaric analogue states have, for decades, provided demanding tests of
the Standard Model description of electroweak interactions. In order to significantly
contribute to these tests experimentally, β decay half-lives and branching ratios must
be determined to overall precisions of ± 0.05% or better, and β decay Q values must
be deduced to at least ± 0.01%. For β decay half-lives in particular, this demanding requirement is generally accomplished using direct β counting techniques. This
method was employed as part of this thesis in order to deduce the half-life of the
superallowed β + emitter
62
Ga using mass-separated radioactive ion beams provided
by the Isotope Seperator and Accelerator (ISAC) facility at TRIUMF. The result
of this analysis, T1/2 (62 Ga)β = 116.100 ± 0.025 ms, is now the single most precise
superallowed half-life ever reported.
In cases where there are large amounts of contaminant or daughter activities, one
must instead rely on a measurement of the half-life using the γ-ray activity. Halflife measurements using the technique of γ-ray photopeak counting have, however,
been previously limited by a systematic bias associated with detector pulse pile-up
effects. While detector pulse pile-up has been qualitatively understood for decades,
there has not been a quantitative description of its effects on half-life measurements
to the level of precision required (± 0.05%) for superallowed Fermi β decay studies.
Using the 8π γ-ray spectrometer, a spherical array of 20 HPGe detectors at ISAC,
a new method was developed that, for the first time, provides the necessary quantitative description of detector pulse pile-up to the required level of precision. This
novel technique has been verified through both a detailed Monte-Carlo simulation
and experimentally using radioactive beams of 26 Na. Following a correction of nearly
30 statistical standard deviations for pulse pile-up, the half-life of
26
Na deduced in
this work, T1/2 (26 Na)γ = 1.07167 ± 0.00055 s, is precise to the level of 0.05% and is in
excellent agreement with the corresponding value, T1/2 (26 Na)β = 1.07128 ± 0.00025 s,
deduced from direct β counting. This study has demonstrated the feasibility of using
the γ-ray counting technique to deduce β decay half-lives to the necessary level of
± 0.05% precision. As an extension to this work, the half-life of the superallowed
β + emitter 18 Ne was determined to be, T1/2 (18 Ne)γ = 1.6656 ± 0.0019 s, a result that
is a factor of four times more precise than the previous world average.
“Weaseling out of things is important to learn. It’s what separates us from the
animals...Except the weasel.” -H.S.
Acknowledgements
There are a large number of people to whom I am deeply indebted and without whose
help none of this would have been possible. I would first like to thank my supervisor,
Carl Svensson, who has had the greatest influence on my fledgling scientific career.
We have spent the better part of six years working together, and without his passion
and tireless efforts, I would not be where I am today. I would also like to thank
my pseudo-supervisors Paul Garrett at Guelph and Gordon Ball at TRIUMF. Paul’s
sense of humour provided the comic relief that is a necessity to making it this far.
We will always remember the potatoes at the golf course restaurant!
I have also had the unique opportunity to work with several people from around
the world including my group at the University of Guelph: Corina Andreoiu, Dipa
Bandyopadhyay, Greg Demand, Paul Finlay, Katie Green, Bronwyn Hyland, Jose
Javier, Kyle Leach, Andrew Phillips, Mike Schumaker, and James Wong. As students
working at a university that is 2000 km from the lab, we sometimes forget the tireless
efforts of the on-site TRIUMF staff that make all of these experiments possible. I
would therefore like to thank the TRIUMF crew: Gordon Ball, Greg Hackman, Colin
Morton, Chris Pearson, and Scott Williams. I am grateful to other TRIUMF staff
members including Pierre Bricault, Marik Dombsky, Matt Pearson, John Behr, Jens
Lassen, in addition to all of the members of the beam development groups. I would
i
also like to thank Martin Smith and Hamish Leslie, in particular, for performing
parallel analysis to my own in the case of the
18
Ne and
26
Na &
62
Ga experiments,
respectively. A special thanks to John Hardy and Ian Towner, the superallowed
gurus, who provided me with up-to-the-minute data that were used in Chapter 1 of
this thesis.
Most of all I would like to thank my wife, Joanna, for her never ending love and
support. My daughter, Daniela, who made the last year of my doctoral studies an
unforgettable one to say the least! Finally, who could forget Cleo the husky?
ii
Contents
Acknowledgements
i
1
1
Introduction
1.1
1.2
Nuclear Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1
Beta-Decay Formalism . . . . . . . . . . . . . . . . . . . . . .
6
1.1.2
Beta Decay Classification . . . . . . . . . . . . . . . . . . . .
8
1.1.3
Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Fermi Theory of Beta Decay . . . . . . . . . . . . . . . . . . . . . .
11
1.2.1
1.3
4
Statistical Rate Function f . . . . . . . . . . . . . . . . . . .
14
Superallowed Fermi Beta Decay . . . . . . . . . . . . . . . . . . . . .
16
1.3.1
Radiative Corrections . . . . . . . . . . . . . . . . . . . . . .
18
1.3.2
Isospin Symmetry Breaking . . . . . . . . . . . . . . . . . . .
20
1.3.3
Corrected f t Values . . . . . . . . . . . . . . . . . . . . . . .
24
1.4
Free Neutron Beta Decay . . . . . . . . . . . . . . . . . . . . . . . .
31
1.5
Pion Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
1.6
CKM Quark Mixing Matrix . . . . . . . . . . . . . . . . . . . . . . .
35
1.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2 Experimental Facilities
41
iii
3
2.1
Isotope Separator and Accelerator (ISAC) . . . . . . . . . . . . . . .
41
2.2
The 8π γ-ray Spectrometer . . . . . . . . . . . . . . . . . . . . . . .
45
2.3
The SCEPTAR array . . . . . . . . . . . . . . . . . . . . . . . . . .
49
2.4
Moving Tape Collector System . . . . . . . . . . . . . . . . . . . . .
51
2.5
General Purpose Station (GPS) . . . . . . . . . . . . . . . . . . . . .
53
2.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
Detector Pulse Pile-up
59
3.1
Quantitative Description of Pulse Pile-up . . . . . . . . . . . . . . . .
61
3.2
Pile-up Probabilities for Radioactive Decay . . . . . . . . . . . . . . .
69
3.3
Rate-Independent Tests
. . . . . . . . . . . . . . . . . . . . . . . . .
75
3.3.1
Detector Solid Angle . . . . . . . . . . . . . . . . . . . . . . .
75
3.3.2
Number of Detectors . . . . . . . . . . . . . . . . . . . . . . .
77
3.3.3
γ-ray Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.4
3.5
Rate-Dependent Refinements
. . . . . . . . . . . . . . . . . . . . . .
82
3.4.1
Pile-up Time Resolution . . . . . . . . . . . . . . . . . . . . .
82
3.4.2
Trigger-Energy Threshold . . . . . . . . . . . . . . . . . . . .
88
3.4.3
Pile-up Detection Energy Threshold . . . . . . . . . . . . . . .
97
3.4.4
Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4 Half-life of
26
Na
107
4.1
Compton Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2
Dead-Time Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2.1
4.3
Dead-Time Corrections for Multi-Detector Arrays . . . . . . . 113
Pile-up Probability Analysis . . . . . . . . . . . . . . . . . . . . . . . 115
iv
4.4
4.5
Half-life Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.4.1
Tests of the Pile-up Method . . . . . . . . . . . . . . . . . . . 126
4.4.2
Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . 133
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5 Half-life of
18
Ne
139
5.1
Pile-up Probability Analysis . . . . . . . . . . . . . . . . . . . . . . . 144
5.2
Half-life Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.2.1
Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . 148
5.2.2
Diffusion and the Half-life of
23
Ne . . . . . . . . . . . . . . . . 156
5.3
Comparison to Previous Results . . . . . . . . . . . . . . . . . . . . . 162
5.4
Present Status of the
5.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6 Half-life of
6.1
62
18
Ne f t and F t values . . . . . . . . . . . . . . 164
Ga
167
Half-life Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.1.1
Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . 180
6.2
Comparison to Previous Results . . . . . . . . . . . . . . . . . . . . . 187
6.3
Present Status of the
6.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
62
Ga f t and F t values . . . . . . . . . . . . . . 189
7 Conclusion and Future Work
7.1
193
Scalar Interactions in Superallowed Decay . . . . . . . . . . . . . . . 194
7.1.1
Half-life of
14
O . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.1.2
Half-life of
10
C . . . . . . . . . . . . . . . . . . . . . . . . . . 200
34
7.2
Half-life of
Ar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.3
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
v
A Dead-time and Pile-up Corrected Decay-curve Fitting
216
Bibliography
219
vi
List of Tables
1.1
Radiative and isospin symmetry breaking corrections for the 13 most
precisely determined superallowed decays. . . . . . . . . . . . . . . .
1.2
Present status of all world-averaged data for the thirteen most precisely
determined superallowed decays. . . . . . . . . . . . . . . . . . . . . .
1.3
25
Present status of the world averaged f t and F t values for the thirteen
most precisely determined superallowed decays. . . . . . . . . . . . .
1.4
21
26
Vector coupling constants GV , up-down CKM matrix elements |Vud|,
and corresponding tests of top-row CKM unitarity deduced from nuclear, neutron, and pion β decays. . . . . . . . . . . . . . . . . . . . .
4.1
37
Run-by-run summary of experimental conditions employed in the 26 Na
half-life experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2
Half-life of
26
Na deduced on a run-by-run basis with statistical errors
and resulting reduced χ2 values for all 14 runs in this analysis. . . . . 125
4.3
Comparison of the half-life of
26
Na obtained as each correction to the
pile-up method was applied to the raw γ-ray gated data. . . . . . . . 127
5.1
Run-by-run summary of experimental conditions employed in the 18 Ne
half-life experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
vii
5.2
Half-life of
18
Ne deduced on a run-by-run basis with statistical errors
and resulting reduced χ2 values for all 15 runs in this analysis. . . . . 150
5.3
Half-life of 23 Ne deduced on a cycle-by-cycle basis with statistical errors
and resulting reduced χ2 values for all 20 cycles in this analysis. . . . 159
5.4
Summary of all high-precision
18
Ne half-life measurements. The new
world average of T1/2 = 1.6670 ± 0.0021 s with a reduced χ2 value of
1.52 is obtained from a weighted average of these 5 measurements. . . 163
6.1
Single cycle timing sequence as determined by a 100 kHz oscillator for
a 3.0 s counting cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.2
Run-by-run summary of the electronic settings used in the 62 Ga experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.3
Run-by-run summary of the electronic settings used in the 62 Ga experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.4
Isobaric contamination in the A = 62 beam deduced from β-γ coincidences between the 20 HPGe detectors of the 8π spectrometer and the
20 plastic scintillators of the SCEPTAR array. . . . . . . . . . . . . . 177
6.5
Differences |∆T | between the best-fit
62
Ga half-life in order to arrive
at an estimate of the systematic uncertainty. . . . . . . . . . . . . . . 186
6.6
Summary of all high-precision
62
Ga half-life measurements. The new
world average of T1/2 = 116.121 ± 0.0021 s with a reduced χ2 value of
1.006 is obtained from a weighted average of these 7 measurements. . 188
viii
List of Figures
1.1
Feynman diagram for β + decay. . . . . . . . . . . . . . . . . . . . . .
1.2
Experimental determinations of the f t values for thirteen superallowed
Fermi β decays between A=10 and A=74. . . . . . . . . . . . . . . .
1.3
18
Values of the isospin symmetry breaking corrections in superallowed
Fermi β decays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
7
23
Comparison of the f t values and F t values for thirteen superallowed
Fermi β decays between A=10 and A=74. . . . . . . . . . . . . . . .
28
2.1
The TRIUMF 520 MeV H− cyclotron. . . . . . . . . . . . . . . . . .
42
2.2
The ISAC experimental hall. . . . . . . . . . . . . . . . . . . . . . . .
44
2.3
The 8π γ-ray spectrometer at TRIUMF-ISAC. . . . . . . . . . . . . .
47
2.4
Schematic diagram of a single 8π germanium detector. . . . . . . . .
48
2.5
The SCEPTAR array at TRIUMF-ISAC. . . . . . . . . . . . . . . . .
50
2.6
The 8π moving tape collector system. . . . . . . . . . . . . . . . . . .
52
2.7
The GPS 4π gas counter and fast tape transport system . . . . . . .
54
2.8
Schematic of the 4π gas counter and fast tape transport system . . .
54
2.9
Schematic diagram of the 4π continuous gas flow proportional counter.
55
2.10 Observed count rate of a
3.1
90
Sr source
. . . . . . . . . . . . . . . . . .
56
Schematic representations of detector pulse pile-up. . . . . . . . . . .
63
ix
3.2
Analytic and simulated pile-up probabilities versus the dimensionless
detector rate x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Simulated decay data curves for a single exponential decay plus a constant background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
71
Simulated bin-by-bin probability of pile-up for a single exponential
decay plus a constant background. . . . . . . . . . . . . . . . . . . . .
3.5
67
72
Simulated decay data set following corrections for pile-up and deadtime effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.6
Simulation of 25 runs with and without the pile-up correction applied.
74
3.7
Pile-up probability versus the detector solid angle. . . . . . . . . . . .
76
3.8
Pile-up probabilities versus the number of detectors employed. . . . .
78
3.9
Simulation of 25 runs comparing γ-ray multiplicities Mγ = 1 and Mγ = 2. 80
3.10 Pile-up probabilities versus the γ-ray decay multiplicity. . . . . . . . .
81
3.11 Simulated pile-up time τp with respect to the master-trigger time for
2 different values of the pile-up time resolution τr . . . . . . . . . . . .
86
3.12 Simulated decay-curve data demonstrating the effect of a non-zero pileup time resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3.13 Analytic and simulated pile-up probabilities versus the dimensionless
detector rate x when the fraction of events that exceed the energy
threshold α is varied. . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
3.14 Analytic functions describing the probability of obtaining a trigger
event D and the probability of obtaining a not-piled-up trigger event
PD as a function of the dimensionless detector rate x =Rτp . . . . . .
93
3.15 Simulated probability of pile-up data and best-fit curve when a nonzero CFD energy threshold is included. . . . . . . . . . . . . . . . . .
x
96
3.16 Simulated decay-curve data data and best-fit curve when a pile-up
efficiency that is less than unity is included. . . . . . . . . . . . . . .
99
3.17 Simulated probability of pile-up data when saturating cosmic-ray events
are included . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.18 Simulated decay-curve data data and best-fit curve when a saturating
events due to cosmic-rays are included. . . . . . . . . . . . . . . . . . 104
26
Na β − decay scheme to the stable daughter
26
4.1
Simplifed
4.2
Not piled-up γ-ray singles spectra from all 20 detectors in the 8π spectrometer following the β decay of
4.3
Na.
. . . . . . . . . . . . . . . . 109
Sum HPGe time spectrum from all 20 detectors in the 8π spectrometer
for a single run with the
4.4
26
Mg. . . . . 108
56
Co source in the array. . . . . . . . . . . . 114
Sum pile-up time spectra from all 20 detectors in the 8π spectrometer
for a single run with the
56
Co source in the array. . . . . . . . . . . . 116
4.5
Number of 1809-keV photopeaks recorded per-cycle for a single run. . 118
4.6
Experimental probability of pile-up spectra for runs with and without
the
4.7
56
Co source.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Run-by-run plot of the maximum probability of pile-up taken at the
start of the decay for each run (t = beam off) in the
4.8
Comparison of the
26
26
Na experiment. 121
Na decay curves obtained from single runs via
a direct β counting a γ-ray photopeak counting determination at the
8π spectrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.9
Pile-up and dead-time corrected decay curve obtained from a single
run following a gate on the 1809-keV transition in
4.10 Half-life of
26
26
Mg. . . . . . . . 124
Na with statistical errors versus run number for all 14
runs in this analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
xi
4.11 Half-life of
26
Na versus the number of leading channels removed. . . . 131
4.12 Half-life of
26
Na versus the γ-ray photopeak gate width.
4.13 Half-life of
26
Na with statistical errors deduced from an average of all
. . . . . . . 132
runs at each common adjustable setting. . . . . . . . . . . . . . . . . 135
18
Ne β + decay to
18
5.1
Decay scheme for
F. . . . . . . . . . . . . . . . . . 140
5.2
Singles spectra of γ rays following the β decay of
5.3
Experimental probability of pile-up spectra for the
5.4
Run-by-run plot of the maximum probability of pile-up taken at the
18
Ne. . . . . . . . . 143
18
18
start of the decay for each run (t = beam off) in the
5.5
Ne experiment. . 146
Ne experiment. 147
Pile-up and dead-time corrected decay curve obtained from a single
18
run following a gate on the 1042-keV transition in
18
F. . . . . . . . . 149
5.6
Half-life of
Ne versus individual run number. . . . . . . . . . . . . . 150
5.7
Half-life measurements of
5.8
Calculated bremsstrahlung yield of
5.9
Pile-up and dead-time corrected decay curve obtained from a single
18
Ne sorted by adjustable electronic setting. 151
17
F and
18
Ne β particles in delrin.
run following a gate directly above the 1042-keV transition in
18
153
F. . . 154
5.10 Deduced half-life of 18 Ne versus the number of leading channels removed.155
5.11 Singles spectra of γ rays following the β decay of
23
5.12 Experimental probability of pile-up spectra for the
5.13 Pile-up and dead-time corrected decay curve for
23
Ne. . . . . . . . . 157
23
Ne experiment. . 158
Ne obtained from a
single run following a gate on the 440-keV transition in
5.14 Half-life of
23
23
Na. . . . . . 158
Ne deduced on a cycle-by-cycle basis. . . . . . . . . . . . 159
5.15 Deduced half-life of 23 Ne versus the number of leading channels removed.161
5.16 Comparison of
18
Ne half-life measurements. . . . . . . . . . . . . . . . 163
xii
62
6.1
Cycle-by-cycle data pre-selection criteria in the
6.2
Summed β-γ coincidence spectra collected in the
6.3
Typical dead-time corrected decay curve from a single
6.4
Half-life of
6.5
Half-life measurements of
62
Ga experiment.
62
Ga experiment.
62
. . 173
. 176
Ga run. . . . 178
Ga versus the experimental run number. . . . . . . . . . 179
62
Ga sorted by adjustable electronic and ex-
perimental settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.6
Deduced half-life of
62
Ga as a function of the number of leading chan-
nels removed from the analysis. . . . . . . . . . . . . . . . . . . . . . 183
62
6.7
Half-life of
Ga versus the detector rate at t = 0. . . . . . . . . . . . 184
6.8
Comparison of all high-precision 62 Ga half-life measurements. The new
world average of T1/2 = 116.121 ± 0.0021 ms with a reduced χ2 value
of 1.006 is obtained from a weighted average of these 7 measurements
and is overlayed for comparison. . . . . . . . . . . . . . . . . . . . . . 188
7.1
Current F t values for the thirteen most precisely determined superallowed decays with a scalar interaction of ± 0.2% overlayed for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
14
O β + decay to
14
7.2
Decay scheme for
7.3
Previous
7.4
Systematic effect between γ-ray and β counting half-life determinations
on the
10
14
N. . . . . . . . . . . . . . . . . . 197
O half-life measurements separated by counting method. . 198
C and
14
O F t values. . . . . . . . . . . . . . . . . . . . . . . 199
10
C β + decay to
7.5
Decay scheme for
7.6
Total simulated β activities for
7.7
Decay scheme for
7.8
Initial attempt to deduce the
34
34
10
Ar and
Ar β + decay to
34
B. . . . . . . . . . . . . . . . . . 201
34
34
Cl decay. . . . . . . . . . . 204
Cl. . . . . . . . . . . . . . . . . 205
Ar via direct β counting with the 20
plastic scintillators of the SCEPTAR array. . . . . . . . . . . . . . . . 206
xiii
7.9
Sum γ-ray singles spectrum obtained with the 8π spectrometer comprised of 10 hours of A = 34 beam. . . . . . . . . . . . . . . . . . . . 208
7.10 Growth and decay curves highlighting the contamination due to
34
Cl
bremsstrahlung radiation . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.11 Typical growth and decay curve obtained from the sum of all 20 SCEPTAR plastic scintillators for a single
xiv
34
Ar run. . . . . . . . . . . . . . 210
Chapter 1
Introduction
High-precision measurements of the f t values for superallowed 0+ → 0+ Fermi β decays between isobaric analogue states have, for decades, provided demanding tests
of the Standard Model description of electroweak interactions [1]. To first order, because neither spin nor orbital angular momentum can be transferred in these decays,
the axial-vector current does not contribute and these transitions can be described
solely in terms of the vector current. The f t values for these decays are therefore
directly related to the vector coupling constant for semi-leptonic weak interactions
GV under the conserved-vector-current (CVC) hypothesis which stipulates that GV
is not re-normalized in the nuclear medium. Once corrected for small (of order 1%)
radiative and isospin symmetry breaking effects, the corrected superallowed f t values
(denoted F t) have confirmed the CVC hypothesis at the level of 1.3×10−4 [2], limited
the existence of possible scalar interactions to the level of 1.3×10−3 [3], and, together
with data from kaon decays, have provided the most demanding test of the unitarity
of the Cabbibo-Kobayashi-Maskawa (CKM) quark mixing matrix [4].
In order to significantly contribute to these tests experimentally, β decay halflives and branching ratios must be determined to precisions of 0.05% or better and
1
β decay Q values must be deduced to at least 0.01%. Similarly, because the theoretical
corrections to the f t values are on the order of 1%, these calculations must be accurate
to within 10% of their central value, a demanding requirement of any theoretical
nuclear-structure dependent model. Even with these stringent demands, by 1990
eight F t values (14 O,
26m
Al,
34
Cl,
38m
K,
42
Sc,
46
V,
50
Mn, and
54
Co) were determined
to a precision of 0.1% or better [5] and their consistency to such a high degree of
accuracy and over such a large mass range was a triumph for both theoretical and
experimental nuclear physics. The eight well studied cases were, however, the most
straightforward to measure because nearly 100% of these β decays proceed directly
to ground state of the stable daughters and, due to their proximity to stability, all of
these nuclides could be readily produced in statistically significant quantities.
In the 17 years that have followed, new experimental techniques, combined with
intense and exotic ion beams being developed at world-class radioactive beam facilities such as the Isotope Separator and Accelerator (ISAC) facility at TRIUMF (Tri
University Meson Facility), have provided new opportunities in superallowed β decay studies. Five new cases
74
10
C (1995),
22
Mg (2003),
34
Ar (2002),
62
Ga (2006), and
Rb (2003) have since been added to the eight previous high-precision F t values and
together they establish the world-average value as F t = 3074.9(12) s (see Sec. 1.3.3),
a factor of 3 improvement in precision over the result from 1990 [5]. Experimentally these decays were previously considered some of the most difficult to measure.
However, with the recent advances of on-line precision Penning traps (in the cases of
62
Ga [6] and 74 Rb [7]), painstaking absolute detector efficiency calibrations (22 Mg [8]),
novel in situ efficiency calibration techniques (10 C [9]), ultra-high sensitivities to weak
non-analogue branches (62 Ga [10],74 Rb [11]), and high-efficiency mass separation techniques (34 Ar [12]) these previously inaccessible superallowed decays are now some of
2
the most precisely determined. All of these techniques, which are now widely applicable across all areas of nuclear physics research, were born out of the necessity to
determine, with high precision, superallowed f t values. Before the accuracy of these
methods could be trusted to such a high-degree of precision, years of calibration and
simulation were performed to ensure that potential sources of systematic uncertainty
were either corrected for, or minimized.
A primary focus of this thesis is the presentation of a new technique that allows,
for the first time, measurements of β decay half-lives using the method of γ-ray
photopeak counting to the level of 0.05% precision. Many nuclear half-lives have
been determined using γ-ray counting techniques, however, there are very few highprecision measurements (< 0.1 %) available because of the potentially large systematic
effects resulting from detector pulse pile-up. While γ-ray detector pulse pile-up has
been qualitatively understood for decades (and even electronic circuits available for
the rejection of pile-up since the early 1960’s [13]) there has not been a quantitative
description of its effects on half-life measurements to the level of precision (0.05%)
necessary for the superallowed-Fermi β decay program.
In this chapter an introduction to β decay in general, and superallowed Fermi β
decay in particular, is given to explain the need for high-precision measurements of
the f t values for this special class of β decays.
3
1.1
Nuclear Beta Decay
In the process of nuclear β decay an unstable nucleus, with atomic number Z and
neutron number N, is transformed into a nucleus that is more stable, with Z ± 1
and N ∓ 1, and is accompanied by the emission of a β particle (e∓ ) and a neutrino
(ν). Because nuclear β decay describes the process of a neutron decaying into a
proton (denoted β − decay) or a proton into a neutron (β + decay), the atomic number
A = Z + N is unchanged. These decay processes can be expressed as,
A
Z XN
→
A
Z+1 YN −1
A
Z XN
→
A
Z−1 WN +1
+ e− + νe ,
+ e+ + νe .
[ β − decay ]
(1.1)
[ β + decay ]
(1.2)
A third decay process, called electron capture or EC decay, competes with β + decay
and involves the capture of an orbital atomic electron by a proton in the nucleus.
This decay process can be expressed as,
A
Z XN
+ e− →
A
Z−1 WN +1
+ νe .
[ EC decay ]
(1.3)
For nuclear β decay or orbital electron capture the Q value, or energy released in the
decay process, can be expressed in terms of the atomic mass differences [14],
Qβ − =
Qβ + =
QEC =
m
m
m
A
Z XN
A
Z XN
A
Z XN
−m
A
Z+1 YN −1
A
Z−1 WN +1
−m
−m
A
Z−1 WN +1
c2 ,
(1.4)
c2 − 2me c2 ,
(1.5)
c2 .
(1.6)
Decay of an unstable parent nucleus via β + and EC decay yields the same daughter
nucleus and the difference between the Q values for these decays is given by,
QEC − Qβ + ≈ 2me c2 = 1.022 MeV.
4
(1.7)
Nuclei in which β + decay is energetically possible may also undergo EC decay. The
reverse is not true, however, as it is possible to have situations in which the QEC value
is positive while the Qβ + value is negative. In these cases electron-capture decay is
the only possible decay mode.
The energy from the β decay of the unstable parent nucleus (A
Z XN ) is released in
the form of kinetic energy (T ) to the decay products. Considering only the 3-body
final-state β ± decays, conservation of energy yields,
Qβ = TD + Te + Tνe ,
(1.8)
where TD , Te , and Tνe are the kinetic energies of the recoiling daughter nucleus, the
β ± particle, and the neutrino, respectively. In the limit where the kinetic energy of
the daughter is neglected (the recoiling daughter nucleus is at least 103 times more
massive than the β particle), the Q value for β decay can be approximated by,
Qβ ≈ Te + Tνe ,
(1.9)
and it follows that the maximum kinetic energy of the β particle (denoted Temax )
occurs when the kinetic energy of the neutrino is zero. In this limit,
Qβ ≈ Temax .
(1.10)
To a good approximation, the β decay Q value is equal to the maximum kinetic
energy of the emitted β particle (or neutrino). For high-precision measurements
the β decay Q values are generally determined either through threshold energies for
nuclear reactions connecting the parent and daughter nuclei of the β decay, or directly
from Eqns. 1.4- 1.6 through differences of high-precision atomic mass measurements.
5
1.1.1
Beta-Decay Formalism
Nuclear β decay is a semi-leptonic process that is governed by the weak interaction
and is mediated through the short-range exchange of massive intermediate charged
bosons (W ± ). A Feynman diagram for a single proton in the nucleus undergoing β +
decay is depicted in Fig. 1.1. In a nuclear system, the initial proton and final neutron
may be one of many nucleons contributing to complex parent and daughter nuclear
wavefunctions. The β decay matrix element between initial and final states Mf,i , must
then be calculated from the initial state nuclear wavefunctions ψP corresponding to
the parent nucleus, and the final state wavefunctions which contain contributions from
the daughter nucleus ψD , the emitted β particle φe , and the neutrino φν . Denoting
b int , the transition matrix element is,
the weak-interaction operator as H
Z
∗ ∗ ∗ b
Mf,i = [ψD
φe φν ]Hint ψP d3 r,
(1.11)
where the integral is over all nucleons as well as the β particle and neutrino cob int is described (in the minimal elecordinates. The weak-interaction operator H
troweak Standard Model) by an equal mixture of vector type (V) and axial-vector
type (A) interactions and weak interaction theory is therefore universally known as
“V-A theory”. Despite decades of searches for contributions from other interaction
types (scalar, pseudoscalar, and tensor) with ever increasing precision there has thus
far been no defect observed in V-A theory. The matrix element can be expressed
as a reduced matrix element |M f,i |2 by removing the strength parameter of the weak
interaction, or weak interaction coupling constant g,
|Mf,i |2 = g 2 |M f,i |2 .
(1.12)
In order to calculate the matrix element (or the reduced matrix element) for nuclear
β decay, the neutrino wavefunctions can be expressed using free-particle plane-wave
6
n
{
<e
d u d
e+
W+
g
gVud
{
d u u
p
Figure 1.1: Feynman diagram for β + decay of a single proton in the nucleus showing
the exchange of an intermediate vector boson W + . The weak-interaction coupling
constant is denoted g at the lepton vertex and gVud at the hadronic vertex.
solutions,
ψν∗ (r) = eiq·r/~,
(1.13)
while the β-particle wavefunction is a free-particle plane wave distorted by the Coulomb
field of the daughter nucleus,
1
ψe∗ (r) = [F (Z, p)] 2 eip·r/~.
(1.14)
In these expressions the Fermi function F (Z, p) describes the Coulomb distortion
of the outgoing β particle wavefunction and p and q are the electron and neutrino
momenta, respectively. In order to complete the calculation of the β decay matrix
element and relate it to decay observables such as the β decay half-life (see Sec. 1.2),
the parent ψP and daughter ψD nuclear wavefunctions must also be calculated. For
this, one must rely on a model that predicts the complex nuclear wavefunctions.
7
1.1.2
Beta Decay Classification
For a β particle of momentum p ∼ 1 MeV/c, for example, the quantity p/~ in the
electron wavefunction of Eqn. 1.14 is only 0.005 fm−1 . The wavefunctions for both
the neutrino and the β particle can therefore be expanded yielding for the matrix
element between initial and final states,
Mf,i
Z
Z
p
i(p + q)
∗ b
3
∗ b
3
= F (Z, p)
ψD Hint ψP d r +
· ψD rHint ψP d r + · · ·
~
(1.15)
Note that the first term (zeroeth order expansion, L = 0) measures the overlap between the parent and daughter nuclear wavefunctions. Applying the parity operator,
which is a reflection of the radial coordinates in 3-dimensions (r → −r), this term is
positive with respect to parity and is therefore zero unless the parities of the parent
and daughter wavefunctions were identical. The second term (first order, L = 1) is
negative under the parity operator because of the additional factor of r in the integrand, and the parent and daughter wavefunctions must therefore have opposite
parity for this term to contribute. In general the β decay selection rule is governed
by πP = πD (-1)L where πP and πD are the parities of the parent and daughter states,
respectively. Allowed decay is defined as L = 0 and must therefore occur between
states of the same parity.
Similarly if angular momentum is explicitly introduced into the above analysis
by expanding the electron and neutrino plane waves into spherical harmonics, then
the orbital angular momentum carried by the electron and neutrino pair L must
satisfy the vector sum LP = LD + L. Furthermore, because the electron and neutrino
are spin- 21 fermions, intrinsic spin angular momentum carried by the electron and
neutrino pair S must also be included by introducing the Pauli spin matrices to
Eqn. 1.15. The coupling of the electron and neutrino spins to S = 0 is called Fermi
8
(or vector) β decay, while the S = 1 coupling corresponds to Gamow-Teller (or axialvector) β decay. Defining the total angular momentum of the electron and neutrino
pair J = L + S the corresponding angular momentum selection rule is therefore
JP = JD + J.
Allowed Fermi β decay therefore has L = 0, S = 0, and hence J = 0 and thus
cannot change the total angular momentum of the nucleus thus JP = JD . A change
in the total angular momentum of the nucleus by 1~ following an allowed β decay
(e.g. 0+ → 1+ ) must therefore be a pure Gamow-Teller (S = 1) β decay and is
commonly observed. Far more rare are transitions of the pure Fermi type as these can
only occur between states of J = 0 total angular momentum (0+ → 0+ or 0− → 0− ),
because Gamow-Teller decay is then strictly forbidden by the angular momentum
selection rules (~0 6= ~0 + ~1). For all other β decays between states of the same total
spin and parity (e.g. 1+ → 1+ ), Gamow-Teller decay is not forbidden (~1 = ~1 + ~1)
and these transitions are mixtures of Fermi and Gamow-Teller decay. A further
angular-correlation experiment is then required to distinguish the vector and axialvector contributions. This will be discussed in further detail in Sec. 1.4 where the
mixture of Fermi and Gamow-Teller contributions is currently the main limitation in
extracting a high-precision f t value from the β decay of the free neutron ( 21
1.1.3
+
→
1+
).
2
Isospin
Unlike the electromagnetic interaction, the nuclear force (to a good approximation)
does not distinguish between protons and neutrons. It is therefore convenient to
describe these two particles as isobaric spin or “isospin” projections of the same
particle, the nucleon. A nucleon can be described as being either “isospin-up” with
isospin projection tz = + 12 (the neutron), or “isospin-down” with tz = − 21 (the proton),
9
both of which are measured with respect to an arbitrary z-axis in isospin space. A
two nucleon system, for example, can have total isospin T = 0 or 1 corresponding
to the anti-parallel or parallel coupling of the isospin vectors, respectively. Isospin
algebra is thus equivalent to ordinary angular momentum algebra for a system of
spin- 21 particles. The z component of the total isospin for a multi-nucleon system can
be written in terms of the total number of neutrons N and protons Z in the nucleus,
1
Tz = (N − Z).
2
(1.16)
The total isospin of the nucleus T is the vector sum of the isospins of the the individual
nucleons and can therefore take on any value in integer steps between |Tz | and
T = |Tz |, |Tz | + 1, · · · ,
N +Z
.
2
N +Z
,
2
(1.17)
As decided by the symmetry energy term in the semi-empirical formula for nuclear
masses and binding energies, nuclear states with the lowest total isospin T = |Tz |
are generally more bound than states of higher isospin. Almost all nuclei thus have
ground-state isospin T = |Tz | = 12 (N − Z), with the only known exceptions being
certain heavy (A ≥ 34) odd-odd N = Z nuclei for which the pairing energy overcomes
the symmetry energy making the T = 1, I π = 0+ the ground state, rather than the
lowest T = Tz = 0 state [15].
Similar selection rules to those for angular momentum apply to isospin in nuclear
β decay. Fermi decay, for example, is strictly forbidden unless the change in isospin
between the parent and daughter states ∆T is identically 0 while Gamow-Teller decay
can occur when ∆T = −1, 0, +1. The term “superallowed” Fermi β decay is used to
describe allowed (L = 0) pure Fermi (S = 0) decay between states of the same isospin
(∆T = 0), which are isobaric analogue states. These states have identical nucleonic
wavefunctions with the exception of one proton exchanged for a neutron between the
10
parent and the daughter and correspond to members of a multiplet of 2T + 1 states
with the same total isospin T but different values of Tz . Non-analogue pure Fermi
branches are relatively weak (and would be forbidden if isospin was an exact nuclear
symmetry) but are observed connecting states that are not isobaric analogues of each
other, although they do generally have the same total isospin (∆T = 0). The matrix
elements for non-analogue Fermi β decay branches are typically reduced by 3 to 4
orders of magnitude compared to the superallowed Fermi β decay branch. Weak
Fermi β decay branches to non-analogue 0+ states will be discussed in Sec. 1.3.2.
1.2
Fermi Theory of Beta Decay
The decay constant λf,i for nuclear β decay can be derived from Fermi’s Golden
Rule, which is a calculation of the probability (per unit time) to observe a transition
between some initial and final state [16],
λf,i =
where
dn
dE
2π
dn
|Mf,i |2
,
~
dE
(1.18)
is the density of final states, and |Mf,i |2 is the matrix element of the weak
interaction Hamiltonian between the initial and final states given in Eqn.1.11. Expanding the electron and neutrino free-particle exponential wavefunctions (under the
assumption p · r ≪ ~ as discussed above in Sec. 1.1.2) and keeping only the zeroeth
order (L = 0) term is known as the “allowed approximation”. The expectation values of the electron and neutrino wavefunctions under the allowed approximation and
normalized to an arbitrary volume V can therefore be expressed as,
|ψe∗ (r)|2 ≈
1
F (Z, p),
V
|ψν∗ (r)|2 ≈
1
.
V
(1.19)
Under the allowed approximation, the reduced matrix element |M f,i |2 depends only
on the electron momentum through the Fermi function F (Z, p) and is independent
11
of the neutrino momentum. It is customary to remove the Coulomb effects and the
volume-normalization from the matrix element to define,
|Mf,i |2 =
g 2 F (Z, p)
′
|M f,i |2 .
2
V
(1.20)
The number of final states n for a particle of momentum p in 3-dimensions is given
in standard texts [16, 14] and can be expressed as,
Z
Z
1
3
d x d3 p,
n=
(2π~)3
(1.21)
where the first integral in Eqn. 1.21 is over the arbitrary volume V that was chosen
to normalize the wavefunctions, while the second can be expressed as the volume of
p3 ). In differential form, the density
a sphere in 3-dimensional momentum space ( 4π
3
of final states for the electron and neutrino are therefore,
dne
V p2 dp
= 2 3
,
dE
2π ~ dE
dnν
V q 2 dq
= 2 3
.
dE
2π ~ dE
(1.22)
Nuclear β ± decay (as opposed to EC decay) results in a 3-body final state (electron,
neutrino, and recoiling daughter nucleus), thus the neutrino momentum q (or energy
Eν ) can be expressed in terms of the energies of both the electron and the daughter
nucleus using energy conservation (E = Ee +Eν +ED ) and the relativistic energy and
momentum relation, Eν2 = q 2 c2 + m2ν c4 . Neglecting the mass of the neutrino and
kinetic energy of the daughter recoil (as discussed in Sec. 1.1, Eqn. 1.9) one obtains,
q2
dq
(E − Ee )2
=
.
dE
c3
(1.23)
Substituting this result, along with the wavefunctions for the electron and neutrino under the allowed approximation (Eqn. 1.19), and the density of final states
(Eqn. 1.22) into the expression for Fermi’s Golden Rule (Eqn. 1.18), and integrating
over all β particle momenta yields,
′
λf,i
g 2 |M f,i |2
=
2π 3 ~7 c3
pZmax
e
F (Z, p)p2(E − Ee )2 dp,
0
12
(1.24)
where pmax
is the maximum momentum of the β particle. The integral in Eqn. 1.24 is
e
conventionally expressed as a dimensionless quantity through the substitution of the
dimensionless variables W =
Ee
me c2
and ρ =
p
,
me c
that satisfy the relativistic energy-
momentum relation W 2 = ρ2 + 1. Expressing the integrand solely in terms of the
dimensionless energy of the β particle yields,
′
λf,i
g 2 m5e c4 |M f,i |2
=
2π 3 ~7
ZWo
√
F (Z, W )W W 2 − 1(Wo − W )2 dW,
(1.25)
1
where Wo =
Eemax
me c2
≈
Qβ
me c2
+ 1. The decay constant λ between initial and final states
can therefore be expressed as,
′
λf,i
g 2 m5e c4 |M f,i |2
=
f,
2π 3 ~7
(1.26)
where the dimensionless quantity “f ” is known as the “statistical rate function” or
“phase space integral” and replaces the integral in Eqn. 1.25. Evaluation of the
statistical rate function will be discussed in Sec. 1.2.1 of this thesis.
The decay constant λf,i can also be written in terms of the partial half-life t for
decay to the final state of interest which is defined as the total β decay half-life T1/2
divided by the β branching ratio to the final state BR. For β + decays an additional
factor is required to account for the fraction of decays that occur by electron-capture
PEC . The partial half-life for β + decays is therefore,
t=
T1/2
ln2
=
(1 + PEC ) .
λf,i
BR
(1.27)
This expression is also valid for β − decays with PEC = 0 as electron-capture decay
does not compete with β − decays (see Sec. 1.1). Combining the expressions for f and
t defines the “f t value”, which is a convenient way to describe the β decay transition.
For allowed β decays the following result is obtained,
ft =
2π 3 ~7 ln2
.
′
g 2|M f,i |2 m5e c4
1
13
(1.28)
The right hand side of Eqn. 1.28 contains fundamental constants with the exception
′
of the matrix element, |M f,i |2 . Differences between f t values for all allowed β decay
transitions are due solely to differences in nuclear matrix elements and the relative
contributions from Fermi and Gamow-Teller decays contained in them:
′
′
′
g 2|M f,i |2 = GV 2 |M f,i (F )|2 + GA 2 |M f,i (GT )|2,
(1.29)
where GV and GA are the coupling constants for vector and axial-vector nuclear β decay, respectively. The f t values can be deduced experimentally, thus the coupling
′
′
constants GV and GA , and the reduced matrix elements |M f,i (F )|2 and |M f,i (GT )|2
must be computed in order to complete Eqn. 1.28. Theoretical calculations of decay
constants for allowed β decays, requires a model that predicts the nuclear wavefunctions from which the matrix element can be computed. Because any model to describe
the nuclear many-body system is at best an approximation, the resulting calculations
would only be as accurate as the approximations used to describe the wavefunctions.
Superallowed Fermi β decays are, however, a special class of β decays that are of
′
the pure Fermi type (|M f,i (GT )|2 = 0), and occur between nuclear isobaric analogue
′
states for which the calculation of the Fermi matrix element |M f,i (F )|2 is largely independent of the individual nuclear wavefunctions. If the reduced Fermi matrix element
is known exactly, and the f t values are deduced from experiment, then it is possible
to determine the fundamental coupling constant of the weak vector interaction GV .
1.2.1
Statistical Rate Function f
Evaluation of the f t values for nuclear β decay requires the calculation of the statistical rate function f given by,
ZWo
√
f = F (Z, W )W W 2 − 1(Wo − W )2 dW,
1
14
(1.30)
where the Fermi function F (Z, W ) was introduced in Eqn. 1.14 to account for the
Coulomb distortion as the liberated β particle interacts with the Coulomb field of the
daughter nucleus. This function has been evaluated using a relativistic treatment of
the electron wavefunction integrated through the nuclear charge distribution taken, in
a first approximation, as a uniformly charged sphere of radius R and is given by [17],
√
2(γ−1)
|Γ(γ + iη)|2
,
F (Z, W ) ≈ 4L(Z, W ) 2R W 2 − 1
eπη
|Γ(2γ + 1)|2
where γ =
(1.31)
p
√
1 − (αZ)2 , η = ±ZαW/ W 2 − 1 for β − /β + decay and α is the fine
structure constant. The function L(Z, W ) replaces a point nucleus with the uniformly charged sphere of radius R, and accounts for the integration of the electron
Dirac wavefunction through the resulting Coulomb field. For small values of Z an
approximate expression for L(Z, W ) is [17]:
L(Z, W ) = 1 +
13
αZRW (41 − 26γ) αZRγ(17 − 2γ)
(αZ)2 −
−
,
60
15(2γ − 1)
30W (2γ − 1)
(1.32)
although for high-precision determinations of the statistical rate function for superallowed Fermi β decay studies (see Sec. 1.3 below), still higher-order corrections to
L(Z, W ) are required. In addition, further corrections to the Fermi function F (Z, W )
itself are required to account for screening by the atomic electron cloud, a more
realistic nuclear charge distribution, higher-order interactions between the leptonic
wavefunctions of the electron and neutrino with the nucleonic wavefunctions of the
parent and daughter nuclei, and a daughter nucleus recoil correction. All of these
have been calculated [1, 17] and are required for high-precision work. Inserting these
results into Eqn. 1.30 provides a means to deduce the statistical rate function given
measurements of β decay Q values. Note that in the extreme limit of high energy and
15
(or) low Z where F (Z, W ) ≈ 1 the statistical rate function can be approximated by,
f
≈
ZWo
W 2 (Wo − W )2 dW
1
≈
1 5
W ≈
30 o
1
30
Qβ
me c2
5
,
(1.33)
which scales with the Q value to the fifth power. In order to deduce f values to the
same level of precision as partial half-lives, one therefore requires Q-value measurements to be performed with five times more precision than is required for half-life or
branching-ratio measurements.
1.3
Superallowed Fermi Beta Decay
Superallowed Fermi β decays occuring between 0+ isobaric analogue states have, for
decades, provided stringent tests of the Standard Model’s description of electroweak
′
interactions. For this special class of β decays, the reduced matrix element |M f,i (F )|2
can be calculated exactly in the limit that isospin is a good quantum number. As
Fermi β decay involves changing a proton into a neutron (or a neutron into a proton)
while leaving the spatial and spin wavefunctions of the nucleon unchanged, the Fermi
transition operator is equivalent to the isospin ladder operator T̂ ± (for β ± decays).
The matrix element is therefore given by [18],
′
|M f,i (F )|2 = (T ∓ Tz )(T ± Tz + 1),
(1.34)
where Tz is the z-projection of the isospin of the parent nucleus (Eqn. 1.16). For
superallowed Fermi β decays between members of a T = 1 multiplet (Tz = −1, 0, +1),
for example, the reduced matrix element is given by,
′
|M f,i (F )|2 = 2.
(1.35)
Note that this result is exact, to the extent that isospin is an exact symmetry of
the nuclear Hamiltonian, and is independent of the complex parent and daughter
16
nuclear wavefunctions. Substituting this result into the expression for the f t value
(Eqn. 1.28) and noting that, to first order, there is no axial-vector contribution to
these pure Fermi decays one obtains,
ft =
2π 3 ~7 ln2
= constant.
2GV 2 m5e c4
(1.36)
The f t values for all superallowed Fermi β decays between T = 1 isobaric analogue
states are therefore predicted to be constant and independent of nuclear structure.
This is somewhat surprising considering the nucleon involved in the β decay is not
a free particle but is embedded within the nucleus. In deriving this result it was
implicitly assumed that the vector coupling GV is indeed a fundamental constant and
is not re-normalized in the nuclear medium. This important assumption is known
as the Conserved Vector Current (CVC) hypothesis [19], and measurements of the
f t values for superallowed transitions set sensitive limits on its validity. Under the
assumption that CVC is satisfied, these transitions can similarly be used to extract
the fundamental weak interaction coupling constant GV to high precision without any
model-dependent assumptions regarding the individual nucleon wavefunctions.
Presently there are thirteen superallowed f t values that have been determined to
experimental precisions of 0.3% or better (10 C,
42
Sc,
46
V,
50
Mn,
54
Co,
62
Ga, and
74
14
O,
22
Mg,
26m
Al,
34
Cl,
34
Ar,
38m
K,
Rb) and are plotted in Fig. 1.2. Although there
does appear to be some fluctuations about a constant value, the f t values for these
thirteen decays are all within approximately 1% of each other. The f t values for
β decay in general span nearly 20 orders of magnitude and thus the small deviations
in the f t values observed between these thirteen superallowed decays agree to first
order with the CVC hypothesis. Corrections to the f t values resulting primarily from
charge dependent sources such as radiative bremsstrahlung processes as the charged
β particle is emitted from the positively charged nucleus, and the assumption that
17
3100
3090
3080
74
62
ft (s)
3070
22
3060
34
Mg
34
Cl
38m
K
42
14
10
3030
3020
0
C
5
O
Rb
Ar
46
V
54
3050
3040
Ga
Sc
50
Co
Mn
26m
Al
10
20
15
25
30
35
40
Z of daughter
Figure 1.2: Experimental determinations of the f t values for thirteen superallowed
Fermi β decays between A=10 and A=74.
isospin is an exact nuclear symmetry are required in order to further improve tests of
the CVC hypothesis.
1.3.1
Radiative Corrections
There are two main corrections that must be applied to the f t values resulting from
the need to incorporate small (∼ 1%) charge-dependent radiative effects. The first
correction (denoted δR′ ) results from the fact that, once created, the charged β particle
interacts with the Coulomb field of the daughter nucleus. This effect is a function only
of the total charge of the daughter nucleus Z and the electron energy and thus, while it
depends on the particular nucleus, it is independent of nuclear structure. Because no
18
model assumptions need to be made, the nucleus independent radiative corrections
δR′ have been calculated to order Zα2 and estimated to Z 2 α3 [20, 21, 22]. These
corrections are on the order of 1.5% [1] and are considered to be “very reliable” [23].
In deriving the superallowed f t values between 0+ isobaric analogue states it
was shown in Sec. 1.1.2 that these decays are pure vector transitions as the axialvector interaction is forbidden to participate due to angular momentum selection
rules. Although angular momentum is an exact nuclear symmetry (unlike isospin), it
is possible for the axial-vector current to cause a nucleon to undergo a spin change.
This can then be followed by virtual photon (or virtual Z0 boson) exchange with
the departing β particle, which can change it back. The entire process therefore has
the appearance of a pure vector transition but was mediated through an axial-vector
weak interaction in higher-order. This process can either act on the same nucleon,
and is therefore transition and nucleus independent (discussed below), or can occur in
different nucleons and will therefore depend on the structure of the particular nucleus.
The nuclear-structure dependent part of the axial-vector corrections (denoted δN S )
have been evaluated using shell-model calculations with a Woods-Saxon plus Coulomb
potential [23] and are generally at least a factor of five smaller than δR′ . The total
radiative correction δR is the sum of these two effects,
δR = δR′ + δN S .
(1.37)
A summary of the values δR′ and δN S from the most recent calculation [24] are listed
in Table 1.1.
A final correction is required to account for radiative effects that are not dependent
on the specific nuclear decay. This correction is denoted ∆VR and contains contributions from axial-vector spin-exchanges on the same nucleon as discussed above, as
19
well as higher-order loops involving virtual W ± and Z0 boson exchange with the polarized fermion vacuum [25, 26, 27]. Recently, a new calculation of this correction
factor (estimated to order Z 2 α3 ) was published [28] that was able to better control
hadronic uncertainties in the short distance loops. This result has led to an increase
in the precision of ∆VR by a factor of 2. The present value [24] is,
∆VR = 2.361 ± 0.038%,
(1.38)
and is independent of both the particular transition and the details of nuclear structure. Because this radiative correction describes the interaction of a single quark with
the polarized fermion vacuum, this factor also applies to the decay of a free neutron,
and pion decay which will be discussed in Secs. 1.4 and 1.5 of this thesis.
1.3.2
Isospin Symmetry Breaking
Isospin is not an exact symmetry of the nuclear Hamiltonian and another small correction must therefore be applied to the matrix element of Eqn. 1.35 which describes
superallowed Fermi β decay between isobaric analogue states of a T = 1 multiplet.
Because the breaking of perfect isospin symmetry can only lead to a reduced overlap
between the neutron and proton wavefunctions, the correction (denoted δC ) must act
to reduce the overall matrix element:
′
|M f,i (F )|2 = 2(1 − δC ).
(1.39)
Isospin symmetry breaking is also manifested as weak Fermi β decay branches to
non-analogue 0+ states in the daughter and the total correction for isospin symmetry
breaking δC is typically expressed as a sum of two components [29],
δC = δIM + δRO .
20
(1.40)
Table 1.1: Radiative and isospin symmetry breaking corrections for the 13 most precisely determined superallowed decays. † Calculations using two distinct interactions
were performed in Ref. [34] for A ≥ 62 and the ranges obtained are shown for 62 Ga
and 74 Rb.
Decay
10
C
14
O
22
Mg
26m
Al
34
Cl
34
Ar
38m
K
42
Sc
46
V
50
Mn
54
Co
62
Ga
74
Rb
δR′ (%)
1.679(4)
1.543(8)
1.466(17)
1.478(20)
1.443(32)
1.412(35)
1.440(39)
1.453(47)
1.445(54)
1.445(62)
1.443(71)
1.459(87)
1.498(120)
δN S (%)
δR (%)
−0.357(35) 1.322(35)
−0.247(50) 1.296(51)
−0.237(20) 1.229(26)
+0.012(20) 1.490(28)
−0.081(15) 1.362(35)
−0.181(15) 1.231(38)
−0.096(15) 1.344(42)
+0.033(20) 1.486(51)
−0.037(7) 1.408(54)
−0.038(7) 1.407(62)
−0.025(7) 1.418(71)
−0.036(20) 1.423(89)
−0.061(20) 1.437(122)
(TH)δC (%)
(OB)δC (%)
0.180(18)
0.320(25)
0.265(14)
0.270(14)
0.635(36)
0.640(41)
0.620(45)
0.490(42)
0.425(32)
0.505(36)
0.610(43)
1.380(155)
1.430(404)
0.15(9)
0.15(9)
0.21(9)
0.30(9)
0.57(9)
0.39(9)
0.59(9)
0.42(9)
0.38(9)
0.35(9)
0.44(9)
1.26-1.32†
0.91-1.05†
The first term δIM accounts for the fact that if isospin symmetry were exact then the
only 0+ state populated in the daughter would be the isobaric analogue state while
all other decays to 0+ states in the daughter would be strictly forbidden. The second
term δRO is the dominant contribution to the total isospin symmetry breaking correction and accounts for imperfect overlap of the single nucleon radial wavefunctions
between the neutron and proton due to binding energy and Coulomb effects.
As there can be many 0+ states in the daughter, the first term δIM can be approximately expanded into the contributions from each of the non-analogue states,
1
2
δIM ≈ δIM
+ δIM
+ ···,
(1.41)
where mixing of different isospins (T = 0, 2, 3, · · ·) into the dominant T = 1, I π = 0+
states breaks the exact identity of the above expression. Calculations of δIM are
21
performed using a shell model calculation which computes the nucleon wavefunctions
and hence the matrix elements for the analogue and all non-analogue β branches. For
0+ states within the β decay Q value window, this correction can also be deduced
experimentally by measuring the β branching ratios to the analogue (BR0 ) and nonanalogue (BRi ) states. Considering a simple case where the daughter has only a
single non-analogue 0+ state with a branching ratio BR1 and squared matrix element
1
2δIM
(1 − δRO ), in addition to the dominant decay branch to the analogue state BR0
1
with squared matrix element 2(1 − δIM
)(1 − δRO ), the β branching ratio to the non-
analogue state can be expressed using Eqn. 1.36,
BR1 = BR0
1
t0
f1 δIM
f1 1
≈ BR0
,
≈ BR0 δIM
1
t1
f0 1 − δIM
f0
(1.42)
1
where the assumption that δIM
≪ 1 was used to obtain the final simplified expression.
This result can be generalized to the case of many non-analogue states yielding,
i
δIM
≈
f0 BRi
.
fi BR0
(1.43)
In deriving the above expressions it was implicitly assumed that the total correction
for isospin symmetry breaking δC could be factorized using [30],
1 − δC = 1 − δIM − δRO ≈ (1 − δIM )(1 − δRO ),
(1.44)
which is valid for small (< 1%) values of δIM and δRO of interest here. Under these assumptions it is clear from Eqn. 1.43 that measurements of the non-analogue branching
ratios BRi and the β decay Q values (which are required to calculate the statistical
i
rate functions) to the analogue and non-analogue states, allow the δIM
to be deduced
approximately from experiment.
Because theoretical calculations of isospin mixing δIM and imperfect radial overlap δRO corrections require predictions of the nucleon wavefunctions, both of these
22
2.0
Woods-Saxon (TH)
Hartree-Fock (OB)
62
70
Ga
Br
Calculated δC (%)
1.5
66
30
1.0
42
38
18
0.5
14
10
0
26m
C
5
54
Al
46
Si
34
Cl
34
42
Ar
15
74
Co
Rb
V
Sc
50
Mg
10
As
Ti
K
26
O
Ca
38m
Ne
22
0.0
S
Mn
20
25
30
35
40
Z of Daughter
Figure 1.3: Values of the isospin symmetry breaking corrections in superallowed
Fermi beta decays. There is qualitative agreement between the Woods-Saxon (TH)
calculations and the Hartree-Fock model (OB), however, a small systematic effect
between the two methods lead to an increased uncertainty in the absolute corrections. Woods-Saxon data were taken from [23], while the Hartree-Fock data are
taken from [29, 32, 33, 34].
corrections, and thus the total theoretical corrections δC , are highly dependent on the
nuclear model employed. One set of calculations are performed by Towner and Hardy
(TH) and use a shell-model diagonalization with a Woods-Saxon plus a Coulomb potential [23, 31]. A second and independent calculation was performed by Ormand
and Brown (OB) and used a self-consistent Hartree-Fock method to deduce the single
particle wavefunctions [29, 32, 33, 34]. A summary of the isospin symmetry breaking
corrections δC obtained from both models is listed in Table 1.1 for the thirteen most
precisely determined superallowed β decay f t values. Although both models show
23
qualitative agreement over many of the superallowed decays, as shown in Fig. 1.3,
there is a small systematic discrepancy in the absolute magnitude for these corrections
with the OB δC corrections systematically smaller than the TH values.
Although the discrepancies between isospin symmetry breaking corrections arise
primarily from the two different interactions used in the calculations, it is not clear
which of these two models provides a better description of the effects of isospin symmetry breaking in superallowed Fermi β decay. Tests of the theoretical predictions
can, however, be performed experimentally as was discussed above in the context of
the isospin mixing δIM corrections. These experimental measurements are criticial to
discriminate between the two theoretical models for δC and considerable effort has
therefore recently focused on the nuclei in Fig. 1.3 that show the greatest model dependencies (14 O, 18 Ne, 30 S, 34 Ar, for example) or have the largest absolute corrections
(all decays with A ≥ 62).
1.3.3
Corrected f t Values
With the small corrections for radiative and isospin symmetry breaking effects, the
corrected f t values (denoted F t) for superallowed Fermi β decays between T = 1
isobaric analogue states can be expressed as [1]:
F t = f t(1 + δR )(1 − δC ) =
2π 3 ~7 ln2
= constant.
2GV 2 (1 + ∆VR )me 5 c4
(1.45)
As discussed in Sec. 1.2 of this thesis, experimental determination of the f t value
requires measurements of the β decay half-life and the superallowed β branching
ratios in order to deduce the partial half-life t, while the statistical rate function f
depends on the decay energy or Q value. Because several new measurements have
been published since the previous superallowed world data evaluation performed in
24
Table 1.2: Present status of all world-averaged data for the thirteen most precisely
determined superallowed decays.
Decay
10
C
O
22
Mg
26m
Al
34
Cl
34
Ar
38m
K
42
Sc
46
V
50
Mn
54
Co
62
Ga
74
Rb
14
f
T1/2
(ms)
BR
(%)
2.3009(12)
19290(12)
1.4646(19)
42.775(23)
70620(14)
99.375(65)
418.42(17) 3875.2(24)
53.16(12)
478.250(94) 6345.0(19)
>99.997
1996.39(41) 1526.55(44)
>99.988
3414.2(14) 843.81(40)
94.45(25)
3298.10(33) 924.33(27) 99.9671(44)
4470.6(12) 680.72(26) 99.9941(14)
7209.7(33) 422.50(11) 99.9848(42)
10731.8(17) 283.21(11) 99.9423(30)
15750.3(32) 193.271(62) 99.996(30)
26401.6(83) 116.121(21) 99.8590(80)
47285(108) 64.776(43)
99.50(10)
PEC
(%)
ft
(s)
0.297
0.088
0.069
0.082
0.080
0.069
0.085
0.099
0.101
0.107
0.111
0.137
0.194
3039.48(471)
3042.47(265)
3052.24(722)
3036.96(108)
3050.02(113)
3052.47(821)
3052.11(96)
3046.40(139)
3049.62(161)
3044.36(125)
3047.59(148)
3074.32(114)
3084.28(796)
early 2005 [1], it was necessary to re-calculate the f t values for the thirteen most
precise cases. The f t values were calculated from all of the experimental data given
in the previous survey [1] and updated to include new measurements of the halflives of
14
O [35],
34
measurements for
for
26m
Al [42],
42
Cl [12],
14
34
O [39],
Sc [42],
46
Ar [12],
38m
50
Mn [36], and
K [40] and
V [42, 43],
62
62
62
Ga [37, 38], β branching ratio
Ga [10, 41], and Q-value measurements
Ga [6], and
74
Rb [44]. The f t value for
62
Ga
decay can only now be added to the twelve previous high-precision f t values as new
experimental measurements on all three necessary quantities have been published
since the last survey.
Inclusion of the new Q-value results for
46
V was critical in
the present analysis because the value of Ref. [43], which was one of the first to
use a precision Penning trap to deduce a superallowed Q value, obtained a result
that disagreed with an older (3 He,t) transfer reaction experiment [45] at the level
25
Table 1.3: Present status of the world averaged f t and F t values for the thirteen
most precisely determined superallowed decays. † Case-by-case uncertainties in the
OB δC corrections have been adopted to be equal to those of TH to avoid double
counting an overall systematic uncertainty in these calculations already included in
the values of Table 1.1.
Decay
10
C
14
O
22
Mg
26m
Al
34
Cl
34
Ar
38m
K
42
Sc
46
V
50
Mn
54
Co
62
Ga
74
Rb
ft
(s)
δR
(%)
TH δC
(%)
OB δC †
(%)
3039.5(47) 1.322(35) 0.180(18) 0.150(18)
3042.5(27) 1.296(51) 0.320(25) 0.150(25)
3052.2(72) 1.229(26) 0.265(14) 0.210(14)
3037.0(11) 1.490(28) 0.270(14) 0.300(14)
3050.0(11) 1.362(35) 0.635(36) 0.570(36)
3052.5(82) 1.231(38) 0.640(41) 0.380(41)
3052.1(10) 1.344(42) 0.620(45) 0.590(45)
3046.4(14) 1.486(51) 0.490(42) 0.420(42)
3049.6(16) 1.408(54) 0.425(32) 0.380(32)
3044.4(12) 1.407(62) 0.505(36) 0.350(36)
3047.6(15) 1.418(71) 0.610(43) 0.440(43)
3074.3(11) 1.423(89) 1.380(155) 1.290(155)
3084.3(80) 1.437(122) 1.430(404) 0.980(404)
Average F t
χ2 /ν
TH F t
(s)
OB F t
(s)
3074.1(49)
3072.0(32)
3081.6(73)
3073.9(15)
3071.9(19)
3070.3(84)
3074.0(21)
3076.5(25)
3079.4(25)
3071.6(25)
3071.9(29)
3075.0(55)
3084(15)
3075.0(49)
3077.3(32)
3083.3(73)
3073.0(15)
3073.9(19)
3078.3(85)
3074.9(21)
3078.7(24)
3080.8(25)
3076.4(25)
3077.2(29)
3077.8(56)
3098(15)
3074.0(8)
0.86
3075.8(8)
1.17
of 3σ. Because the older result of Ref. [45] appeared in a publication that included
Q value measurements for six other superallowed decays, an intense experimental
effort has been launched [42, 46, 47] to re-measure all of the superallowed Q values
using precision Penning traps. Recently, new measurements have been performed for
26m
46
Al [42],
42
Sc [42], and
74
Rb [44], in addition to an independent confirmation of
V [42], all of which are in generally good agreement with previous results effectively
ruling out a widespread systematic effect. The updated world averaged values for
the statistical rate functions f , β decay half-lives T1/2 , superallowed branching ratios
BR, electron-capture fractions PEC [1], and the f t values (calculated using the partial
26
half-life t defined in Eqn. 1.27) are presented in Table 1.2.
With an updated data set for the thirteen most precisely determined superallowed f t values one can now apply the nucleus-dependent corrections for radiative bremsstrahlung processes and isospin symmetry breaking discussed in Sec. 1.3.1
and 1.3.2, respectively in order test the CVC hypothesis. The corrected F t values
using both the isospin symmetry breaking corrections from the Woods-Saxon calculation (TH) and those of the self-consistent Hartree-Fock calculations (OB) are
summarized in Table 1.3. The corrected F t values using the Woods-Saxon isospin
symmetry breaking corrections are compared in Fig. 1.4 with the uncorrected experimental f t values. Note that because the uncertainties in the isospin symmetry
breaking corrections of Ormand and Brown already included an estimate of the systematic uncertainty in the model, the OB corrected F t values were calculated from
the OB central δC values with the case-by-case TH δC uncertainties. If one were to
instead use the isospin symmetry breaking corrections of Ormand and Brown with
their overall quoted uncertainties, the result would lead to an increased estimate of
the systematic uncertainty between these calculations because of double counting the
systematic effect.
With the corrections applied to each of the thirteen superallowed decays the 1%
fluctuations in the experimental f t values are removed providing a strong confirmation
of the CVC hypothesis. In fact, a weighted average of the thirteen corrected F t values
using the Woods-Saxon corrections for isospin symmetry breaking (TH) yields,
F t(TH) = 3074.00(75) s,
χ2 /ν = 0.86.
(1.46)
This result is a powerful vindication of both the theoretical corrections and the CVC
hypothesis as the f t values listed in Table 1.3 are comprised of more than 125 independent measurements [2] and encompass the large mass range 10 ≤ A ≤ 74. If
27
3100
3090
3080
74
62
ft (s)
3070
22
3060
34
Mg
34
Cl
38m
K
42
3040
10
3030
3020
0
O
C
Rb
Ar
46
V
54
3050
14
Ga
Sc
50
Co
Mn
26m
Al
10
5
20
15
25
30
35
40
Z of daughter
3110
Ft
3105
= 3074.0 ± 0.8 s
2
χ /ν = 0.86
3100
74
Rb
3095
Ft (s)
22
3090
Mg
46
3085
34
3080
Ar
42
V
62
Sc
54
3075
3070
26m
10
3065
3060
0
C 14
O
10
Al
34
Cl
Ga
38m
K
50
Co
Mn
20
30
40
Z of daughter
Figure 1.4: Comparison of the f t values and F t values for thirteen superallowed
Fermi β decays between A=10 and A=74.
28
instead the isospin symmetry breaking corrections from the Hartree-Fock calculations
(OB) are used the corrected F t value is,
F t(OB) = 3075.75(75) s,
χ2 /ν = 1.17,
(1.47)
which is an equally powerful result, albeit with a somewhat larger reduced χ2 value.
For twelve degrees of freedom, the probability of obtaining a reduced χ2 value that
exceeds 1.17 is approximately 25% [48] and thus one cannot discriminate between
the theoretical isospin symmetry breaking corrections solely by statistical means.
Unfortunately, while both of these results are in agreement with the CVC hypothesis
and both are precise to the level of 0.03%, the systematic effect between the two
calculations of the isospin symmetry breaking corrections has manifested itself in
the corrected F t values as a difference of 1.75 s or nearly three times the statistical
uncertainty.
In order to address the systematic effect of the isospin symmetry breaking corrections in the overall average corrected F t value, a general procedure has been
adopted [1]. This method involves taking the unweighted average of the corrected
F t values from both calculations as the overall mean F t and combining this with a
systematic uncertainty equal to half of the spread between them [1]. Adopting this
method the overall average corrected F t value is given by,
F t = 3074.88(75)F t(87)δC s,
= 3074.9(12) s
(1.48)
where the first uncertainty is due to the experimental f t values in addition to the caseby-case uncertainties from the radiative and isospin symmetry breaking corrections
from the Woods-Saxon calculations of Towner and Hardy. The second uncertainty is
the estimated systematic effect resulting from the discrepancy in the calculations for
29
isospin symmetry breaking effects. Combining the statistical and systematic uncertainties in quadrature yields an overall uncertainty of 1.2 s. It should be noted that
in the previous superallowed data evaluation only the nine historically well-measured
superallowed f t values were used to calculate the systematic uncertainty. The result
yielded an estimated systematic of 0.9 s [1], which is statistically equivalent to the
value derived here with the inclusion of
22
Mg,
34
Ar,
62
Ga, and
74
Rb. Removing these
four decays from the calculation decreases the systematic uncertainty estimate only
marginally from 0.87 s to 0.82 s.
Using the fundamental constants from the Particle Data Group [4], one can express
the constants contained in Eqn. 1.45 as,
K
2π 3 ~ln2
=
= (8120.278 ± 0.004) × 10−10 GeV−4 s.
(~c)6
(me c2 )5
(1.49)
With the transition-independent radiative correction ∆VR = 2.361(38)% from Eqn. 1.38,
the vector coupling constant GV deduced from superallowed Fermi β decay between
isobaric analogue states is,
GV
= 1.13577(14)F t(16)δC (21)∆VR × 10−5 GeV−2 ,
3
(~c)
= 1.13577(30) × 10−5 GeV−2 ,
(1.50)
where the first uncertainty combines the experimental f t values with the case-bycase uncertainties in the radiative and isospin symmetry breaking corrections, the
second is due to the systematic difference between calculations of isospin symmetry
breaking, and the third is due to the 1.6% uncertainty associated with the transitionindependent radiative correction ∆VR . The total uncertainty is obtained from the
quadrature sum of these three contributions. Although the superallowed decays permit the extraction of the vector coupling constant at the level of 0.03%, this precision
is not limited by experiment but rather by theory with the dominant contribution
30
coming from the uncertainty in the nucleus-independent radiative correction ∆VR .
This uncertainty is based on a conservative estimate [28] and can, in principle, be
reduced further by refining estimates of the intermediate-energy QCD loop effects.
The dominant uncertainty in the extraction of GV could therefore soon be due to the
systematic discrepancy associated with the isospin symmetry breaking corrections.
There is therefore considerable interest in studying pion and free neutron β decay for
which there are no nuclear-structure dependent corrections.
1.4
Free Neutron Beta Decay
The decay of a free neutron can also be used to deduce the weak vector coupling
constant for pure Fermi decays GV . Because neutron β decay,
n → p + e− + νe
(1.51)
occurs between free spin- 12 fermions, the decay is not purely vector as in the 0+ → 0+
superallowed Fermi decays discussed above. As a result, the total matrix element will
be a mixture of vector and axial-vector components,
g
′
2
′
|M f,i |2
= GV
2
GA 2
′
′
2
2
|M (GT )| ,
|M f,i (F )| +
GV 2 f,i
(1.52)
′
where |M f,i (F )|2 and |M f,i (GT )|2 are the reduced matrix elements for the Fermi and
Gamow-Teller contributions to the overall β decay and GA and GV are the axial-vector
and vector coupling constants, respectively. The radiative corrections for neutron
decay are similar to those used in superallowed nuclear β decay, however, because
the neutron is a free particle there are no nuclear-structure dependent corrections to
consider. The f t value for neutron decay can therefore be expressed analogously to
31
Eqn. 1.45 [2, 49]:
f τn (1 + δR′ ) =
GV
2
K/ln2
,
(1 + ∆VR )(1 + 3λ2 )
(1.53)
where τn is the neutron mean lifetime, δR′ is the same transition-dependent radiative
correction as discussed above evaluated for the case of a neutron [23] (note that
there is no δN S equivalent), K is the constant given by Eqn. 1.49, and λ contains the
admixture of the vector and axial-vector contributions to neutron decay given by [49],
1
λ=
2
GA (1 + ∆A
R)
1
GV (1 + ∆VR ) 2
.
(1.54)
The radiative correction ∆VR (Eqn. 1.38) is the same as is used in the superallowed
decays and corrects for the small contribution of the axial-vector current to pure vector
transitions as well as QCD loop effects. Similarly the correction ∆A
R is required to
correct for the influence of the vector current in pure axial-vector decays. The factor of
3 in front of the λ2 term is obtained from summing over all possible spin orientiations
for the Gamow-Teller matrix element.
Although there are no corrections required for nuclear structure dependencies,
the f t value for the case of neutron decay requires experimental determinations of
the statistical rate function f , the neutron mean lifetime τn , and the asymmetry
parameter λ. Using the recommended values of λ = −1.2690(28) [2], the neutron
half-life T1/2 = 885.7(8) s [4], and the phase space integral evaluated for neutron
decay f (1 + δR′ ) = 1.71489(15) [50, 51] one obtains,
GV
(n) = 1.13676(52)f t(21)∆VR (208)λ × 10−5 GeV−2 ,
(~c)3
= 1.1368(22) × 10−5 GeV−2 ,
(1.55)
for the vector coupling constant derived from free neutron decay, where the first
uncertainty is obtained from the experimental f t value, the second is due to the
32
uncertainty in the radiative correction ∆VR , and the third is from the experimental
asymmetry parameter λ. This result agrees with the value of GV obtained from the
superallowed decays (Eqn. 1.50) but is 7 times less precise.
As the overall uncertainty in the GV deduced from neutron β decay is presently
limited by experimental measurements of the asymmetry parameter λ, new and more
precise measurements are warranted before a true comparison can be made to the
value deduced from the superallowed decays. Furthermore, a recent measurement
of the neutron mean lifetime τn = 878.5(8) s [52] disagrees with the world average
by nearly 9 statistical standard deviations. Although the Particle Data Group has
simply rejected this measurement on the basis of its departure from the average, it
should be noted that the world average is itself dominated by the single measurement
of Ref. [53]. With the lack of precision in the asymmetry parameter λ combined with
the recent ambiguities in the neutron half-life, extraction of GV from neutron decay,
at least for the time being, cannot compete with superallowed nuclear β decay.
1.5
Pion Beta Decay
A third method to extract the weak vector coupling constant GV is from pion β decay,
π + → π 0 + e+ + νe .
(1.56)
Pion decay is, like neutron decay, free of the nuclear structure dependent corrections
that limit the uncertainty in the superallowed F t values. Furthermore it is a 0− → 0−
pure vector decay and there is therefore no axial-vector to vector ratio term λ which
is the present limitation in extracting GV precisely from neutron decay. However,
because the branching ratio for this decay mode is on the order of 10−8 , extraction
of GV from pion decay suffers from severe statistical limitations. The f t value for
33
pion decay can be expressed as [2, 49, 50]:
f t(1 + δR′ ) =
2GV
2
K
.
(1 + ∆VR )
(1.57)
Substitution of the radiative correction ∆VR from Eqn. 1.38, the constant K from
Eqn. 1.49, the value of the statistical rate function evaluated for the case of the pion
f (1+δR′ ) = 1770.4(28) [50], the pion mean lifetime τπ = 2.6033(5)×10−8 s [4], and the
most precise measurement of the branching ratio BR = 1.040(6)×10−8 [54] yields,
GV
(π) = 1.13655(91)f t(21)∆R (350)BR × 10−5 GeV−2 ,
(~c)3
= 1.1366(36) × 10−5 GeV−2 ,
(1.58)
for the vector coupling constant. This value agrees with, but is an order of magnitude
less precise than, the value deduced from the superallowed Fermi β decays and is
presently limited by the overall precision in the branching ratio.
Although the vector coupling constant GV can be extracted by measurements
of either nuclear, neutron, or pion β decays, superallowed Fermi β decays presently
yield the most precise method. The weighted average of GV deduced by all three
of these methods differs from the superallowed value by only 0.002%. It should be
noted, however, that the precision in the superallowed result is presently limited
by theoretical uncertainties whereas the neutron and pion decay results are limited
by significant experimental difficulties. With a further reduction anticipated in the
nucleus-independent correction ∆VR , the systematic differences between the nuclearstructure dependent isospin symmetry breaking corrections must be resolved in order
to further constrain the value of GV . Eventually, however, because ∆VR is common to
all three β decays this calculated correction may ultimately limit the overall precision
with which GV can be determined by any method.
34
1.6
CKM Quark Mixing Matrix
The ratio of the vector coupling constant GV to the Fermi coupling constant from
purely leptonic muon decay GF defines the up-down element of the Cabibbo-KobayashiMaskawa (CKM) quark mixing matrix, a 3×3 matrix that relates the weak interaction
quark eigenstates (d′ , s′ , b′ ) to the mass eigenstates (d, s, b) through the relation:
 

 
′
 d   Vud Vus Vub

 
 ′  
 s  =  Vcd Vcs Vcb

 

 
Vtd Vts Vtb
b′
 d 
 
 
 s .
 
 
b
(1.59)
In the Standard Model, the CKM matrix is a unitary transformation in three dimensional quark “flavour space”. One condition of unitarity is that the sum of
the squares of the elements of any row or column must equal unity. The most
precise test to date comes from the top row for which Vud , deduced from superallowed Fermi β decay, is the dominant term. With the Fermi coupling constant
GF /(~c)3 = 1.16637(1)×10−5 GeV−2 [4] from muon decay, and the value for the vector coupling constant GV deduced from superallowed decays (Eqn. 1.50) one obtains,
Vud =
GV
GF
= 0.97376(12)F t(14)δC (18)∆VR ,
= 0.97376(26),
(1.60)
for the up-down element of the CKM matrix. From the vector coupling constants
deduced from free neutron and pion decay given by Eqns. 1.55 and 1.58, respectively
the corresponding values for Vud are,
Vud (n) =
GV (n)
= 0.97461(44)f t(18)∆VR (178)λ ,
GF
= 0.9746(18),
Vud (π) =
(1.61)
GV (π)
= 0.97444(78)f t(18)∆VR (300)BR ,
GF
= 0.9744(31),
35
(1.62)
which are less precise than, but in agreement with, the value of Vud deduced from
superallowed Fermi β decay.
The value of Vud must be combined with the values of Vus = 0.2257(21) [4], which
is determined from semi-leptonic neutral and charged Kℓ3 decays, K 0 → π − ℓ+ νℓ and
K + → π 0 ℓ+ νℓ , where the leptons ℓ are electrons or muons, and Vub = 0.00431(30) [4]
whose value is small enough that when squared (|Vub |2 ∼ 10−5 ) is entirely negligible in
the unitarity sum. Combining the value of Vud deduced from the superallowed decays
(Eqn. 1.60) with the presently accepted world averages of Vus and Vub , the unitarity
test of the top row of the CKM quark mixing matrix yields,
|Vud |2 + |Vus |2 + |Vub |2 = 0.9992(5)Vud (9)Vus ,
= 0.9992(11),
(1.63)
which satisfies unitarity at the level of 0.7 standard deviations. The unitarity test
can also be performed using the values of Vud deduced from neutron and pion decays
and the results,
|Vud(n)|2 + |Vus |2 + |Vub |2 = 1.0008(36)Vud (9)Vus ,
= 1.0008(37),
(1.64)
|Vud (π)|2 + |Vus |2 + |Vub |2 = 1.0005(60)Vud (9)Vus ,
= 1.0005(61),
(1.65)
are both evidently in agreement with unity at the quoted level of precision. A summary of the values of Vud and the corresponding tests of CKM unitarity obtained in
this analysis for all three β decay types is listed in Table 1.4.
The present value of Vus which dominates the uncertainty in the unitarity sum
of the top row of CKM matrix elements (when GV deduced from the superallowed
36
Table 1.4: Vector coupling constants GV , up-down CKM matrix elements |Vud |, and
corresponding tests of top-row CKM unitarity deduced from nuclear, neutron, and
pion β decays.
Decay
Nuclear
Neutron
Pion
GV ×10−5
GeV−2
Vud
1.13577(30)
1.1368(22)
1.1366(36)
0.97376(26)
0.9746(18)
0.9744(31)
P
|Vui|2
i
0.9992(11)
1.0008(37)
1.0005(61)
data is used in the unitarity sum) has been under intense scrutiny over the past three
years. In the 2004 evaluation of the Particle Data Group the adopted value of Vus
was Vus = 0.2196(26) [55] and, with the present analysis of Vud from the superallowed
P
decays, would lead to a violation of the unitarity test |Vui |2 = 0.9965 ± 0.0012
i
by nearly 3 standard deviations. This apparent discrepancy prompted new measure-
ments of Kℓ3 decays [56, 57, 58], which were all consistent with each other, and have
led to a 2.6% increase to the value Vus = 0.2257(21) that has now been adopted
in the 2006 evaluation [4]. Although this new value apparently restores unitarity, a
theoretical correction to Vus to account for SU(3) quark-flavour symmetry breaking
(similar to the SU(2) isospin symmetry breaking δC correction in nuclear β decay) is
required. The presently accepted calculated value for this symmetry breaking in the
semi-leptonic Kℓ3 decays is given by, f+ (0) = 0.961 ± 0.008 [59], a calculation from
1984 that has been confirmed by a recent calculation performed in 2005 [60]. However, other calculations in the last 4 years obtained f+ (0) = 0.974 ± 0.012 [61, 62].
If the latter calculations are accepted the value of |Vus | would be decreased to the
Vus = 0.2227 ± 0.0029 and, when combined with the value of Vud from nuclear β decay,
P
would lead to |Vui |2 = 0.9978 ± 0.0014, again violating unitarity at 1.6 standard
i
deviations. Although all three of the modern experimental measurements of Vus are
37
in agreement, the calculation of f+ (0) remains the dominant source of uncertainty in
Vus and which calculation is to be adopted has not yet been resolved.
If the value of Vub = 0.0043 ± 0.0003 [4] is removed from the above calculations of
the unitarity sums, the corresponding values listed in Table 1.4 would only be reduced
by 2×10−5 . The up-bottom element is therefore insignificant at the present level of
0.1% precision.
A deviation from CKM unitarity would have the important consequence of requiring new physics, such as an unknown fourth quark generation, right-handed weak
interactions, or additional interactions types (scalar, psuedoscalar, or tensor interactions). Although the present data seem to suggest that unitarity is satisfied, further
improvements in the precision of Vud , which is a by-far the most precisely determined
CKM matrix element, clearly remain desirable. Independent of the extraction of Vud ,
the superallowed F t values provide the most stringent tests of the CVC hypothesis
and set strict limits on the existence of scalar currents in the Standard Model. With
the ambiguities in the neutron half-life and with no significant improvements to the
dominant experimental uncertainties of the neutron asymmetry parameter λ and the
pion branching ratio expected in the immediate future, the best chance for improving
these demanding tests of the Standard Model rests with the superallowed decays.
1.7
Summary
Improving the overall precision in Vud is a major pursuit in experimental nuclear
physics. Experimentally this can be most readily accomplished through high-precision
measurements of the f t values for superallowed Fermi β decays between isobaric analogue states. However, because the systematic discrepancy between two independent
38
calculations of the isospin symmetry breaking corrections is anticipated to soon dominate the overall uncertainty in GV deduced from the superallowed data, the first
experimental objective is to attempt to discriminate between the two models employed. Recent work has therefore focused on the Tz = 0 superallowed emitters such
as
62
Ga [6, 10, 37, 103] and
74
Rb [7, 11, 64] in the A ≥ 62 region where large δC
corrections (> 1%) are predicted. Similar tests of the isospin symmetry breaking
corrections can be achieved through measurements of the Tz = −1 superallowed decays in the 14 ≤ A ≤ 42 region, where the discrepancy between the δC calculations
is either enhanced (14 O,
30
S,
34
Ar, see Fig. 1.3) or, in certain cases (18 Ne,
42
Ti), the
Woods-Saxon calculations are anomalously large while the Hartree-Fock calculations
are not available.
Although the experimental f t values are the smallest contributor to the overall
uncertainty in Vud , expanding the number of precisely determined superallowed F t
values will continue to have a significant impact. Of the thirteen most precisely
determined F t values only 4 are Tz = −1 superallowed emitters (10 C,
34
Ar) and, with the exception of
14
14
O, 22 Mg, and
O, have all been added in the past 12 years. As
the Tz = −1 decays are further from stability than the well-known Tz = 0 cases, in
general have unstable daughters, and exhibit multiple Gamow-Teller decay branches,
high precision measurements represent a considerable experimental challenge. Novel
experimental techniques are, however, being developed to deal with these difficulties.
In this thesis, a new method has been developed for the measurement of superallowed β decay half-lives to high-precision (≤ 0.05%) using the technique of γ-ray
photopeak counting. Although γ-ray photopeak counting has been used to measure
nuclear half-lives in the past, there are relatively few high-precision measurements
available due to the systematic effects of detector pulse pile-up. Half-lives of many of
39
the Tz = 0 superallowed emitters with A ≤ 62 have already been determined to the
required level of precision through the technique of β counting which is favoured for
these decays because there is generally no β contamination from the stable Tz = +1
daughters. The half-life determinations for the Tz = −1 superallowed emitters, however, have proven to be a considerable experimental challenge because the daughters
are radioactive and yield unwanted grown-in and decay contamination in the resulting β-decay time spectrum. These problems are compounded further if the initial
samples cannot be produced free of other radioactive species which can introduce
further time-dependent β activities to the composite β-decay spectrum. A review of
the experimental facilities employed during this thesis work is presented in Chapter 2.
In Chapter 3 an introduction to detector pulse pile-up in γ-ray counting experiments
is presented and leads to the first quantitative description of these effects to the level
of precision (0.05%) required by the superallowed Fermi β decay program. This technique is demonstrated in Chapter 4 with a half-life determination of
26
Na using the
8π γ-ray spectrometer at TRIUMF’s ISAC facility. In Chapter 5 the half-life determination of
18
Ne, the first high-precision superallowed half-life measurement via γ-ray
photopeak counting is presented. In Chapter 6, an introduction to A ≥ 62 superallowed decays is discussed in the context of a high-precision half-life measurement for
62
Ga using the method of direct β counting. This work has led to the most precise
superallowed half-life measurement ever reported, and makes important contributions
to constraining the systematic discrepancy between two independent calculations of
the isospin symmetry breaking corrections to the experimental f t values. Conclusions
and future work are discussed in Chapter 7.
40
Chapter 2
Experimental Facilities
2.1
Isotope Separator and Accelerator (ISAC)
The ISAC facility at the Tri-Universities Meson Facility (TRIUMF) in Vancouver
B.C., is one of the premier experimental facilities in the world to conduct research
with radioactive ion beams [65]. The primary driver is the TRIUMF 520 MeV cyclotron [66] which provides intense beams of up to 100µA of protons to ISAC, in addition to simultaneous secondary beams of π mesons and muons, for use in a variety
of multidisciplinary experiments in chemistry, solid-state physics, biology, medicine,
and nuclear physics. A photograph of the cyclotron, taken in the spring of 1972 during its construction, is shown in Fig. 2.1. A unique feature of this cyclotron is the
acceleration of negative H− ions. The main advantage for the use of H− ions is the
increased proton extraction efficiency which is simplified once the negative ions are
passed through thin carbon foils which strip the two atomic electrons leaving positively charged protons that bend in the opposite direction in the applied magnetic
field. This high extraction efficiency permits a continuous delivery of proton beams
simultaneously delivered to multiple experimental stations at independent energies
41
Figure 2.1: The TRIUMF 520 MeV H− cyclotron and staff during its construction in
the spring of 1972 (Courtesy of TRIUMF).
and intensities [66].
The ISAC radioactive ion-beam facility is located on a dedicated 100 µA beam
line from the TRIUMF main cyclotron and has been delivering radioactive ion beams
since 1998 [67]. The 500 MeV protons bombard thick layered-foil targets and produce
radioisotopes through spallation of the target nuclei. The reaction products diffuse to
the target surface and are ionized in a coupled ion source. The target and ion-source
modules are housed 2 floors below the experimental hall and are encased within
2 m of steel and between 2 and 4 m of concrete shielding [67]. To date, various
ion sources at ISAC have been employed to provide radioactive ion beams to the
experimental facilities and include surface ion sources [68, 69], a 2.45 GHz electroncyclotron-resonance (ECR) source [70], the TRIUMF-ISAC Laser Ionization Source
(TRILIS) [71, 72], and a Forced Electron Beam-Induced Arc Discharge (FEBIAD) ion
42
source [70]. The ionized reaction products are removed from the ion source with an
electric field and are transported to a mass separator. The mass separator at ISAC
is a 45 degree magnetic bend between two electrostatic quadrupoles [65, 68]. Ions
are deflected in circular orbits based on their mass-to-charge ratio according to the
classical expression,
1
r=
B
s
2m∆V
,
q
(2.1)
where r is the radius of the circular orbit, B is the applied magnetic field in the
separator, m is the mass of the particular ion (which can be alternatively expressed
as the atomic mass number A), q is the ionic charge, and ∆V is the voltage difference
between the ion source and the mass separator. At ISAC the voltage difference is
∆V = 30-60 kV and thus singly charged 1+ ion beams have energies between 30 and
60 keV. Following the mass separator, the beam passes through a set of adjustable
slits that are tuned to the radius of the particular singly-charged ion of interest. The
mass separator at ISAC is generally operated in a “low-resolution”, high-transmission
mode with resolving power ∆m/m ≈ 1/1000. While able to distinguish between
neighbouring isotopes (with different mass numbers A), this mode does generally
leave the possibility for isobaric (and molecular) contamination within a given mass
number. In the experiments that will be discussed in the following chapters of this
thesis, careful attention was paid to isobaric contamination due to
26m
Al in the
26
Na
beam, 18 F and H17 F in the 18 Ne beam, and 62 Cu, 62m Co, and 62g Co in the 62 Ga beam.
Although, in principle, any atom (up to and including the target materials atomic
mass) can be produced from the target and ionized in the coupled ion source, specific
combinations of target material and ion source are more efficient at ionizing particular
elements and this efficiency depends primarily on the elemental chemistry. For example, a surface ionization source at ISAC uses a rhenium filament heated to 2400 K [68]
43
Figure 2.2: Schematic representation of the ISAC experimental hall. The 8π γray spectrometer and the SCEPTAR array of plastic scintillators, in addition to
a 4π gas proportional counter and fast tape transport system (located under the
TITAN platform), are the major experimental facilities used in this work (Courtesy
of TRIUMF).
which provides sufficient energy to ionize the alkali metal species such as Li, Na, K,
Rb, and Cs. For noble gas beams, a 2.45 GHz ECR source was developed [70] which
provided the necessary higher ionization energies for the noble gas species such as Ne,
Ar, Kr, Xe, and Rn. The TRILIS laser ion source uses one laser tuned to a specific
frequency of an atomic transition in the element of interest, followed by ionization
to the continuum with a second laser and thus particular elements can be selectively
ionized through resonant excitations [71, 72]. For the 62 Ga beams discussed in Chapter 6 of this thesis, the technique of resonant ionization improved the
62
Ga yield by a
factor of 40 over surface ionization. Orders of magnitude changes in radioactive ionbeam yields can therefore be realized by choosing suitable combinations of production
target and ion source.
44
Following ionization and mass separation, high-intensity radioactive beams are
delivered to several experimental facilities in the ISAC hall. A schematic layout of the
ISAC facility is depicted in Fig. 2.2. From the yield station, beams of 30-60 keV can
either be sent directly to several low-energy experiments such as the 8π spectrometer,
or they can be accelerated (if the mass to charge ratio is A/q ≤ 30) to energies of up
to 1.7 MeV per nucleon through a radio frequency quadrupole (RFQ) and drift-tube
linear accelerators (DTL) for use in nuclear astrophysics experiments. Beams can also
be sent to the ISAC-II facility, a recent upgrade to ISAC, that can accelerate beams to
even higher energies of ≤ 6.5 MeV per nucleon. These energies are above the Coulomb
barrier for many targets allowing for a variety of nuclear reaction experiments [73].
The three main experimental facilities employed in this thesis are a spherical array of 20 Compton-suppressed high-purity germanium (HPGe) detectors called the 8π
γ-ray spectrometer. Located inside the 8π spectrometer and surrounding the target
chamber is an array of 20 plastic scintillators called the Scintillating Electron Positron
Tagging Array (SCEPTAR). For high-precision half-life experiments, a 4π gas proportional counter and fast tape transport system was utilized and is located under
the TITAN platform in Fig. 2.2 at the general purpose station (GPS). The calibration
and use of these instruments for performing high-precision half-life measurements for
superallowed Fermi β decays form the basis of this thesis.
2.2
The 8π γ-ray Spectrometer
The 8π γ-ray spectrometer [74, 75, 76] is a spherically symmetric array consisting of
20 Compton-suppressed HPGe detectors covering approximately 13% of the 4π solid
angle. Each cylindrical HPGe crystal has a diameter of 5.3 cm and has a nominal
45
relative efficiency of 25% compared to that of a 7.5 cm×7.5 cm NaI crystal. The full
array has an absolute photopeak efficiency of approximately 1.0% at 1.3 MeV. Photographs of the 8π spectrometer are shown in Fig. 2.3. The geometry of the 8π is that
of a truncated icosahedron (soccer ball) with the 20 HPGe detectors located in the
positions of the 20 hexagons. Each detector is equipped with tungsten heavy-metal
(HM) collimators to prevent γ rays from directly striking the bismuth germanate
(BGO) Compton-suppression shields. The collimators are covered with 2 cm thick
plastic (delrin) to minimize bremsstrahlung production from high-energy β particles
(up to ∼ 7.5 MeV for
26
Na β − decay, see Chapter 4). A schematic of a single 8π
detector is shown in Fig. 2.4.
The data aquisition system of the 8π spectrometer has been upgraded [73] in order
to provide event-by-event time stamping for every γ ray, an essential component in
tracking and correcting the effects of detector pulse pile-up to high precision. Two
identical signals from the HPGe detector preamplifiers are used. The energy signal is amplified by an Ortec 572 spectroscopy amplifier with shaping times adjusted
between 1.0 and 6.0 µs. The outputs from the amplifiers are digitized with Ortec
AD114 14-bit analogue-to-digital converters (ADC’s). The ADC’s are peak sensing, and are operated in a mode that yields zero-suppressed, non-overflow-suppressed
data. The second signal from the preamplifier (the time signal) is amplified by a
timing-filter amplifier and discriminated with Ortec 583b constant-fraction discriminators (CFD’s). The fast outputs from the CFD’s are subsequently used as input
into LeCroy 4516 logic modules. One of the outputs from the logic modules is sent
to a LeCroy 3377 32-channel multi-hit time-to-digital converter (TDC) that provides
intra-event nanosecond timing of the γ-rays. A common stop to the TDC is derived
from the delayed master trigger. Other elements of the time signal include separate
46
Figure 2.3: The 8π γ-ray spectrometer at TRIUMF-ISAC. (Top) An aerial view of
the 20 HPGe detector array as seen from an overhead maintenance crane (courtesy
of M.R. Pearson). (Bottom) One hemisphere (10 detectors) of the 8π γ-ray spectrometer. Two centimetre thick plastic (delrin) covers each detector and minimizes
bremsstrahlung production from high energy β particles.
47
Figure 2.4: Schematic diagram of a single 8π germanium detector.
LeCroy 3377 TDC’s that are used for (i) a BGO time to provide Compton-suppression
and (ii) a pile-up indicator that uses the inhibit outputs from the spectroscopy amplifiers. The presence of one or more hits in the pile-up TDC with a corresponding
time in the TDC of the same HPGe detector is used to indicate that the event
recorded by that detector was piled-up. The final crucial element in the aquisition
system is a LeCroy 2367 universal logic module (ULM) which consists of three 32-bit
latching scalars. The first scalar counts pulses from a Stanford Research Systems
high-precision 10 MHz ± 0.1 Hz temperature-stabilized oscillator providing a global
time stamp for every event. The second scalar counts pulses from the same 10 MHz
oscillator that remain after a veto by the aquisition dead time which also veto’s the
48
master trigger for the same period of time after each trigger event, and the third
counts the number of master triggers. In this way the dead time can be calculated
on an event-by-event basis using differences in the changes of the recorded number of
clock ticks between the first two scalars in the ULM. When rates permit, the array
can be operated in γ-singles mode where every event recorded by the HPGe, BGO
suppressors, and pile-up TDC’s are written to disk and all gating, suppression, coincidence timing, and pile-up rejection windows are set in software during the offline
analysis to provide the maximum degree of flexibility.
2.3
The SCEPTAR array
The Scintillating Electron Positron Tagging Array (SCEPTAR) is an array of 20 plastic scintillators covering approximately 80% of the 4π solid angle. The detectors are
arranged into two rings of five trapezoidal pieces and two rings of five rectangular
pieces and are positioned so that one plastic scintillator covers the same solid angle
subtended by one of the HPGe detectors of the 8π [74]. Each plastic scintillator is
1.6 mm in thickness and thus β particles in excess of ≈ 500 keV are not stopped in
the array and SCEPTAR therefore primarily acts as a ∆E detector. Light is collected
from the scintillator edges and transported using light guides of approximately 25 cm
in length to the the photo-multiplier tubes that are located outside of the 8π array.
The photomultiplier tubes for the upstream half of SCEPTAR are visible in Fig. 2.5.
In spectroscopy experiments, coincidences between β particles recorded in SCEPTAR
and γ rays registered in the HPGe detectors of the 8π provide a powerful tool to suppress room background allowing measurements of γ-ray transitions with beam rates as
low as 2 ions per second [78]. Other experiments have been performed that have used
49
Figure 2.5: The SCEPTAR array at TRIUMF-ISAC. This picture shows the upstream
hemisphere (10 detectors) of SCEPTAR which is located inside the 8π spectrometer
and surrounds the beam implantation location.
the γ-ray activity in coincidence with the β activity to measure β decay branches as
low as 10−5 [10]. In this thesis, SCEPTAR was used solely to provide measurements
of possible isobaric beam contaminants in the
18
Ne half-life determination presented
in Chapter 5. The use of SCEPTAR as a β counter for high-precision half-life studies
is also being investigated.
A signal from each of the 20 SCEPTAR photomultiplier tubes is fed into a fast
Phillips Scientific 776 preamplifier that provides a factor of 10 amplification of the
pulse height. One of the outputs of the preamplifier is delayed by ∼ 720 ns and input
into a LeCroy 4300 fast encoding read-out amplifier (FERA) for charge-to-digital
conversion. The other output of the preamplifier is used for timing, and is sent to an
Ortec 935 CFD. The CFD outputs are fed into individual channels of a 32-channel
50
input SIS3801 virtual machine environment (VME) multi-channel scaler (MCS), and
into CAEN 894 discriminators which are then sent to 32-channel multi-hit LeCroy
3377 TDC’s. As with the germanium data stream, a second LeCroy 2367 ULM is
used to provide a global time stamp for every β event. In most applications, due
primarily to the large solid angle subtended by the array, SCEPTAR is operated in
a “scaled-down” β singles mode, where only a small fraction (typically 1 in 10) of
the singles events are written to disk while the MCS’s multiscale all singles events in
addition to β-γ coincidence events individually. Both the SCEPTAR and 8π triggers
are input into LeCroy 2365 octal logic units that can be programmed to handle
the master trigger logic. Considering only the 8π and SCEPTAR arrays, users can
choose to collect experimental data using γ singles (or scaled γ singles), β singles
(or scaled β singles), β-γ coincidences, γ-γ coincidences, or any combination of these
simultaneously. Trigger selections can be changed through input to the data aquisition
computer interface.
2.4
Moving Tape Collector System
Another recent addition to the 8π spectrometer was the installation of a moving
continuous-loop (approximately 120 m in length) in-vacuum tape collector system [73,
74, 76, 77]. The radioactive ion beam is implanted in the centre of the 8π target
chamber on a 40 µm thick mylar-backed iron-oxide collector tape as shown in Fig. 2.6.
Beam pulsing and tape movement intervals are controlled through a Jorway controller
in CAMAC (computer automated management and control databus). The Jorway is
controlled from the main aquisition computer where tape cycle times, dwell times,
tape movement length, and beam on and off times can all be modified at the aquisition
51
Figure 2.6: (Top) Photograph of the downstream half of SCEPTAR (10 detectors)
with the moving collector tape passing through the centre of the target chamber.
(Bottom) Longer-lived activities can be moved outside of the 8π array where the
collector tape is stored in a loose pile in a shielded box.
52
computer terminal.
In the
26
Na experiment discussed in Chapter 4 for example, tape cycling times
of 1-1-30-1 were used which corresponded to counting background activity for 1.0 s
before turning the beam on for 1.0 s to build up a sample of
beam was then turned off and the decay of
26
26
Na on the tape. The
Na was counted for 30.0 s (∼ 30 half-
lives). This was followed by a 1.0 s delay while the tape was moved bringing a fresh
section of tape into the array while simultaneously removing any unwanted longlived contaminant activities that may have been present in the beam. The unwanted
long-lived activities are stored outside the array, behind a lead wall, in a shielded
tape-storage box shown in Fig. 2.6. To avoid unnecessary tension and unwanted
tape breaks (which can only be fixed by breaking vacuum), the tape is not held on
spools but rather is loosely piled as shown in Fig. 2.6. For the
18
Ne experiment
discussed in Chapter 5 of this thesis, the tape system proved invaluable as the
half-life is only T1/2 = 1.6656(19) s [79], while the daughter,
18
18
Ne
F, has a half-life of
T1/2 = 109.77(5) minutes [80]. Without the ability to cycle the tape after the decay
of every
18
Ne sample, the
18
F decay activity inside the target chamber would have
continually increased throughout the experiment.
2.5
General Purpose Station (GPS)
High-precision half-life measurements using the technique of direct β counting are
performed using a 4π gas proportional counter and fast tape transport system. A
photograph and a schematic representation of the apparatus are shown in Figs. 2.7
and 2.8, respectively. Radioactive beams are implanted onto a 25 mm wide aluminized mylar tape (denoted by a solid circle in Fig 2.8) under vacuum. The beam
53
Figure 2.7: The GPS 4π gas counter and fast tape transport system.
Figure 2.8: Schematic of the 4π gas counter and fast tape transport system.
54
Figure 2.9: Schematic diagram of the 4π continuous gas flow proportional counter
used in the 62 Ga half-life measurement (see Chapter 6).
is then turned off and the sample is rapidly moved out of vacuum and into a 4π
continuous-flow gas-proportional counter, where the beta particles from the decays
of the sample are recorded for approximately 20 half-lives. The distance between the
beam implantation site and the centre of the proportional counter is approximately
36 cm as indicated in Fig. 2.8. The tape speed can be altered between 2 and 4 metres
per second thus the total time required to move the sample from the implantation
site to the detector varies between 90 and 180 ms. Once the counting of the sample
is completed, the beam is turned on again implanting a second sample onto a second region of the tape and the cycle is repeated. Unlike the moving tape collector
used with the 8π spectrometer (Sec. 2.4), the tape system at GPS is almost entirely
at atmospheric pressure with the exception of the implantation site which is under
vacuum.
The proportional counter at GPS was machined out of low background copper.
55
2.0
90
Sr
Count Rate (arbitrary units)
1.8
Voltage Plateau
1.6
1.4
1.2
1.0
0.8
2200
2300
2400
2500
2600
Detector Voltage (V)
2700
2800
2900
Figure 2.10: Observed count rate of a 90 Sr source placed at the centre of the proportional counter. The count rate is approximately constant over the “voltage plateau”
region which extends from 2600 V to 2800 V for this detector.
It consists of two cylindrical gas cells separated by a space of 0.25 mm to allow
the aluminized mylar tape to pass through. Methane at 1 atmosphere pressure is
continuously pumped through the gas cells at a rate of ∼ 0.5 cc/min. Within each
cell is a 13 µm diameter gold plated anode wire which runs through the centre [81].
The cells are protected by 1 mg cm−2 nickel foil windows, which prevent air from
entering and contaminating the methane within the cells. The transport tape with
the implanted sample is stopped, sandwiched between the two gas cells. Decays from
either side of the tape are recorded by the two halves which cover almost the entire
4π solid angle. A schematic diagram of the 4π gas proportional counter used in this
work is given in Fig. 2.9. The ideal voltages across the gas counter are those for which
the observed count rate of a long-lived test source is approximately independent of
56
the applied voltage. This voltage range is known as the “voltage plateau” [82]. A
plot of the observed count rate of a
90
Sr source placed at the centre of the nickel foil
windows of this detector is shown in Fig. 2.10. The voltage plateau is estimated to
lie between 2600 and 2800 V.
The signal from the gas counter is sent to a Phillips 300 MHz bipolar preamplifier.
The preamp pulse is fed into an Ortec 579 fast filter amplifier (FFA) which provides 20
times amplification and is operated at high gain for optimal high-rate pulse processing.
An Ortec 436 100 MHz discriminator is used to reject electronic noise and generates
a NIM logic pulse which is separated and fed into two LeCroy 222N non-retriggerable
gate-and-delay generators which generate fixed dead-times per-event intentionally set
to be long (3-4 µs) compared to all other series dead times in the system. Two
separate and independent multi-channel scalars were used to bin the decay data into
250 bins of adjustable bin-time widths from the two gate and delay generatores. The
first MCS is a Data Design Corporation IS10A integrating MCS and the second is a
LeCroy 3521A MCS. Beam pulsing, tape movement intervals, and dwell times were
controlled through a Jorway controller in CAMAC. An external time standard was
provided by a Stanford Research Systems model DS335 1 MHz ± 1 Hz temperaturestablized precision laboratory clock scaled to 100 kHz.
2.6
Summary
A broad experimental program in Superallowed Fermi β decay studies using the 4π
gas proportional counter and fast tape system in addition to the 8π γ-ray spectrometer with the SCEPTAR scintillator array is presently a major research pursuit of
the ISAC facility at TRIUMF. The 4π gas counter and fast tape system is used for
57
high-precision half-life determinations using the method of direct β counting where
this technique is favoured. For many of Tz = −1 superallowed decays, however, statistically significant fractions of β decays yield γ-ray activity. In addition, because the
daughters of the Tz = −1 decays are the Tz = 0 superallowed decays, significant and
unwanted daughter activities can contaminate the resulting β counting spectrum. For
these cases, half-life determinations via the γ-ray activity would be ideal. There has,
however, not been previous half-life measurements of this kind performed to the level
of precision required (± 0.05%) for the superallowed program. The 8π spectrometer
was recommissioned for this purpose and the calibration of this instrument for highprecision superallowed half-life measurements forms the basis of this thesis. Because
γ-ray half-life determinations utilize photopeak energies to discriminate between different β-decays, rate-dependent photopeak losses due to multiple γ rays entering the
detector within a short time interval can systematically bias the deduced half-life.
The rate-dependent and systematic loss of photopeaks is referred to as detector pulse
pile-up, a process which has been qualitatively understood for decades. The first
quantitative description of these effects to the level of 0.05% are discussed in Chapter 3 of this thesis, and is then used to perform high-precision half-life measurements
for
26
Na and
18
Ne in Chapters 4 and 5, respectively.
58
Chapter 3
Detector Pulse Pile-up
Measurements of the f t values for superallowed Fermi β decays to overall precisions
of 0.1% are necessary to constrain new physics beyond the Standard Model [1]. To
achieve this goal, measurements of the β-decay half-lives of these nuclei to ± 0.05%
or better are desired. While many nuclear half-lives have been determined via γ-ray
counting techniques, there are very few high-precision measurements (< 0.1%) that
use this method because of the potentially large systematic effects resulting from
detector pulse pile-up. While γ-ray detector pulse pile-up has been qualitatively
understood for decades, there has not been a quantitative description of its effects on
half-life measurements to the level of precision (± 0.05%) necessary for superallowed
Fermi β decay studies. Procedures to minimize these effects have been adopted [83]
including performing experiments with different counting rates and, if the half-lives
deduced at all rates fluctuated less than the statistical uncertainties, the effects of pileup were concluded to be within the quoted uncertainty. If the fluctuations exceeded
the statistical uncertainty, the data obtained at the highest rates were assumed to
be affected by pulse pile-up and were removed from the data set. With the benefits
59
over β counting such as improved peak-to-background and decay selectivity that γray counting has to offer, combined with recent advances in electronics and data
throughput, a procedure to correct, rather than discard, the highest-statistics data is
desirable.
Early attempts at γ-ray detector pile-up corrections were performed in the halflife determination of the superallowed decay of
14
O [84]. The procedure recorded the
number of events per time slice that had occurred in the energy region just above the
main photopeak which, after a background subtraction, gave the number of events
that had piled out of the 2.3 MeV γ-ray photopeak from the daughter
14
N. The
highest rates at the start of the counting period, however, were not allowed to exceed
1500 Hz, and the method employed to perform the pile-up correction was “somewhat
painstaking” [84]. Following a 2σ correction for detector pulse pile-up effects, the halflife of 14 O was determined to be T1/2 = 70.684 ± 0.077 s [84]. In a recent determination
of the
14
O half-life using a germanium detector at even lower rates (∼ 500 Hz) the
result, T1/2 = 70.560 ± 0.049 s [85], includes a 0.1σ correction which contains both the
dead-time and pile-up rate-dependent corrections. In an even more recent attempt
to remeasure this half-life via γ-ray photopeak counting [86], a suitable method for
performing a pile-up correction to the level of precision required could not be found
and the “use of a germanium detector was abandoned” [86]. Instead, a β counting
technique was employed to determine the half-life, T1/2 = 70.641 ± 0.020 s [86], a result
that is in agreement with the earlier γ-ray measurement of Ref. [84] but disagrees
with Ref. [85] at the level of 1.6σ. The most recent determination of the
14
O half-life
through β counting found, T1/2 = 70.696 ± 0.052 s [35], a result that agrees with
the high-precision result of [86] but is nearly 3 times less precise because of reactionproduct contamination in the composite β-decay spectrum. This example highlights
60
the inconsistencies that are present in the γ-ray counting half-life determinations
for some of the superallowed decays. Furthermore, while the β counting half-life
determinations do not suffer from pile-up effects, they are themselves often limited by
the presence of unwanted contaminant activities. These limitations in high-precision
half-life determinations could, in principle, be alleviated with the introduction of a
general γ-ray photopeak counting procedure that precisely accounts for the systematic
effects of detector pulse pile-up [87].
3.1
Quantitative Description of Pulse Pile-up
The time intervals between successive γ rays from a radioactive source arriving at a
detector are exponentially distributed according to the instantaneous average detector rate R. There is therefore some probability that multiple electronic pulses will
interfere with each other or “pile-up”. The detector system is then unable to properly determine the energy of all of the events involved in the pile-up. For half-life
determinations via γ-ray photopeak counting, the data with the highest statistics is
thus most affected. If one assumes that a minimum time is required to extract the
energy information from a single event, then the probability of pile-up should increase
as this “pile-up time” τp is increased. If the time between two pulses is less than τp ,
the events have incorrect energies and are lost from a subsequent γ-ray photopeak
energy gate. For high-purity germanium (HPGe) detectors, the pile-up time τp is
related to the shaping time of the spectroscopy amplifiers and one can reduce pile-up
by decreasing the shaping times. This apparent gain is, however, offset by a loss in
the energy resolution which leads to a decrease in the peak-to-background and decay
selectivity. Pile-up effects must therefore be understood for all detector rates and
61
pile-up times in order to maximize the usefulness of the γ-ray counting method.
Given a constant absolute detector rate R, the mean number of events in a time
interval ∆t is given by R∆t, and the probability of observing n events in this time
interval follows the Poisson distribution,
(R∆t)n e−R∆t
P (n; ∆t) =
.
n!
(3.1)
For detector pile-up it is assumed that 1 or more events recorded within τp of the
event of interest will ruin the energy determination of that event. Defining “post-pileup” as the probability that the pile-up is caused by events arriving after the event of
interest, and t = 0 as the time that the event of interest is recorded by the detector
(Fig. 3.1a), the probability that this event is not post-piled-up (Ppost ) corresponds to
the probability that exactly 0 events are recorded in the time interval τp after the
event and is given by Eqn. 3.1 with n = 0,
Ppost = P (0; τp ) = e−Rτp .
(3.2)
The probability that the event is post-piled-up (Ppost ) is the probability that 1 or
more events occur in the pile-up time interval τp and is obtained from the fact that
the Poisson distribution is normalized,
Ppost = P (≥ 1; τp ) = 1 − Ppost = 1 − e−Rτp .
(3.3)
This is not yet a complete description of detector pulse pile-up because there exists
the possibility that the event of interest is piled-up by an event that came before, in
a process defined as “pre-pile-up”. In the limit where there is no dead time, every
event triggers the aquisition system and, from symmetry arguments, the probability
of obtaining 1 or more events in the time interval τp before a given pulse is identical
to the probability of obtaining 1 or more events in the time interval τp after that
62
Jp
a) PostPile-up
Jd
t=0
Jp
)t
b) PrePile-up
Jd
Jd
t=0 t
Jr Jp-Jr
c) PostPile-up
with Jr
d) PrePile-up
with Jr
Jd
t=0
Jr Jp-Jr
)t
Jd
Jd
t=0 t
Figure 3.1: Schematic representations of a) post-pile-up and b) pre-pile-up where τp
and τd represent the pile-up and non-extendible aquisition dead times respectively
(see Sec. 3.1). With the introduction of a pile-up time resolution τr (τr < τp ) the
probability of detecting c) post-pile-up and d) pre-pile-up is reduced compared to the
true probability of post- and pre-pile-up (see Sec. 3.4.1).
63
pulse. The existence of an electronic dead time following each trigger event breaks
this symmetry. If a trigger is recorded by the detector and is followed by a dead
time lasting for a time interval τd where τd ≥ τp (typical experimental conditions),
then trigger pre-pile-up can only occur if the previous γ ray that piled-up the trigger
was itself not a trigger. Rather than a minimum of two γ rays that were required
for post-pile-up, one requires no less than three γ rays for pre-pile-up when τd ≥ τp .
A schematic representation of this process is shown in Fig. 3.1b, where for pre-pileup t = 0 is defined as the release of the dead time of the previous trigger. If the
next trigger were to occur at a time t where 0 < t < τp then this trigger has some
probability of being pre-piled-up. If t > τp then these triggers cannot be pre-piled-up.
The differential probability of obtaining a trigger in a time interval between t and
t+dt can be expressed from Eqn. 3.3,
dPT rig (t) = Re−Rt dt.
(3.4)
Note that integrating this result from t = 0 to t = τp returns the probability of postpile-up given in Eqn. 3.3. For pre-pile-up to occur, 1 or more events must occur in the
dead time of the previous trigger (at t ≤ 0), and these events must have been recorded
less than the pile-up time before the next trigger (∆t = τp − t). The probability that
the trigger recorded at time t where 0 ≤ t ≤ τp is pre-piled-up can therefore be
expressed as,
Ppre =
Zτp
P (≥ 1; ∆t = (τp − t))dPT rig (t),
0
Zτp
(1 − e−R(τp −t) )Re−Rt dt,
=
0
= 1 − e−Rτp (1 + Rτp ).
(3.5)
The probability of an event not being pre-piled-up is similarly obtained from the
64
normalization condition,
Ppre = 1 − Ppre = e−Rτp (1 + Rτp ).
(3.6)
Although it is the existence of the dead time that destroys the symmetry between
pre- and post-pile-up, the expressions derived are all independent of the actual length
of this dead time providing it is longer than the pile-up time τd ≥ τp . Because pile-up
times are typically much shorter than event read-out dead times in γ-ray counting
experiments, this condition does not limit the utility of the analytic results presented
here.
A γ-ray photopeak can, of course, only be lost once due to pile-up. For example,
a trigger can be pre-piled-up by one event and then post-piled-up by a later event but
the trigger itself has only been lost once. One must therefore combine the probabilities
of pre- and post-pile-up into the four possible combinations of trigger pile-up. A
trigger can be uniquely described as being: (i) neither pre- nor post-piled-up, (ii) prepiled-up but not post-piled-up, (iii) post-piled-up but not pre-piled-up, or (iv) both
pre- and post-piled-up. Of these 4 possibilities, only (i) corresponds to obtaining
a trigger that is free of detector pulse pile-up. Combining the detector rate R and
the pile-up time τp into a single dimensionless variable x = Rτp , the probability of
obtaining a trigger that is not piled-up (P) can be written as,
P = Ppre Ppost = e−2x (1 + x).
(3.7)
The other three outcomes represent different forms of detector pulse pile-up and can
each be expressed separately in terms of the dimensionless detector rate x,
Ppre Ppost = e−x (1 − e−x )(1 + x) ,
Ppre Ppost = e−x 1 − e−x (1 + x) ,
Ppre Ppost = (1 − e−x ) 1 − e−x (1 + x) .
65
(3.8)
(3.9)
(3.10)
Because these three possibilities completely describe all forms of trigger pile-up, the
total probability of obtaining a piled-up event (P) is given by the sum of Eqns 3.8
to 3.10,
P = Ppre Ppost + Ppre Ppost + Ppre Ppost ,
= 1 − e−2x (1 + x).
(3.11)
a result that is consistent with the imposed normalization condition,
P + P = 1.
(3.12)
The analytic description of detector pulse pile-up in the presence of a fixed dead time
can therefore be expressed entirely in terms of a single parameter, the dimensionless
detector rate x = Rτp and is independent of the length of the dead time per event
providing τd ≥ τp .
The analytic pile-up expressions derived in Eqns. 3.7-3.11 are plotted versus the
dimensionless detector rate in Fig. 3.2 up to x = 5.0 which corresponds to a probability of 99.97% that a detected trigger is piled-up. Because post-pile-up requires
a minimum of two γ rays while pre-pile-up requires at least three, the probability
an event is post-piled-up but not pre-piled-up (Ppre Ppost , Eqn. 3.8) is always larger
than the case of pre-pile-up but not post-pile-up (Ppre Ppost , Eqn. 3.9). Similarly, the
probability an event is both pre- and post-piled-up (Ppre Ppost , Eqn. 3.10) requires a
minimum of four γ rays and is therefore even less probable at low rates. As the rate is
increased, the average time between events can become significantly less than τp , and
the detector(s) are then completely saturated rendering Ppre Ppost the most probable
outcome at the highest rates.
In order to confirm the universality of the pile-up equations derived above, a Monte
Carlo simulation was developed to test the analytic expressions over a wide variety of
66
1.0
(a)
(e)
(d)
Probability
0.8
0.6
Analytic
Monte Carlo
(b)
0.4
0.2
(c)
0.0
0
1
2
3
4
5
Dimensionless Detector Rate x = Rτp
Figure 3.2: Plot of the analytic (lines) and Monte Carlo (circles) pile-up probability
curves versus the dimensionless detector rate x = Rτp . The curves represent the
probabilities of (a) not piled-up (P), (b) post- but not pre-piled-up (Ppre Ppost ), (c) prebut not post-piled-up (Ppre Ppost ), (d) pre- and post-piled-up (Ppre Ppost), and (e) total
pile-up (P).
simulated experimental conditions and detector rates. The simulation tracked pile-up
losses based on the assumption that a trigger event that is not piled-up always leads
to a proper measurement of the γ-ray energy, while a trigger that is piled-up always
prevents the proper extraction of the γ-ray energy. The simulation assumed that a
pile-up did not occur if the time interval between successive events exceeded τp , while
pile-up would always occur if the time interval was less than or equal to τp . Refinements to these assumptions are considered in Sec. 3.4. A fixed non-extendible dead
time was similarly included by defining a time τd for which no triggers could occur
within τd following the previous trigger, and the next event in the decay sequence
67
that entered the detector after τd from the previous trigger would always be the next
trigger. Decay data were generated through the calculation of time intervals between successive events randomly drawn from an exponential probability distribution
according to the procedure outlined in Ref. [48]. For better comparison with experiment, running totals of quantities such as triggers that were not piled-up, piled-up,
dead, or summed (for decay multiplicity > 1) were binned into adjustable-width bin
times.
The first test was to ensure the accuracy of the pile-up expressions derived above
(Eqns. 3.7-3.11). Presented in Fig. 3.2 (circles) are the results of several simulations
each made with a pile-up time of τp = 5.0 µs. The detector rate was varied in
steps of 104 counts/s between R = 104 and R = 106 which spans the dimensionless
detector rate from x = 0.05 to x = 5.0. A dead time of 20.0 µs was arbitrarily
chosen to satisfy the above criteria τd ≥ τp . Each data point shown in Fig. 3.2 is
the average result of 250 simulations with at least 104 events per simulation. The
pile-up equations derived in Sec. 3.1 and the results of the simulation are in perfect
agreement and cover the entire range of detector rates up to a total probability of
pile-up of 99.97 %. Additional simulations were performed with varying pile-up and
dead times and confirmed the independence of all results on the dead time provided
τd ≥ τp . The family of curves presented in Fig. 3.2 and described by Eqns. 3.7-3.11
are thus concluded to be universal and provide exact analytic expressions with which
to predict the probability of pile-up providing the detector rate R and the pile-up
time τp are known.
68
3.2
Pile-up Probabilities for Radioactive Decay
An extension of the constant absolute detector rate analysis above to that of a timedependent radioactive decay is straightforward. The time-dependent dimensionless
detector rate for a radioactive source with decay constant λ in the presence of a
constant background, for example, can be written as,
x(t) = Aτp e−λt + Bτp ,
(3.13)
where A and B are the detector rates from the radioactive source at t = 0 and the
constant background, respectively. The pile-up time τp has again been used to convert
the detector rate to a dimensionless quantity. The pile-up probabilities (Eqns. 3.73.11) were derived using a constant rate x, but without any loss of generality can
be used to describe the instantaneous pile-up probabilities from a time-dependent
source. Equations 3.7 to 3.11 are therefore valid for all rates with the substitution
x = x(t). For the case of a single radioactive decay with a constant background x(t)
is given by Eqn. 3.13, however, this expression can be modified to incorporate any
number of radioactive decay components and daughter activities that may be present
in the experiment.
The Monte Carlo simulation described in Sec. 3.1 was modified to simulate a radioactive decay plus a constant background component. From the number of piled-up
triggers pi and not piled-up triggers ni in the ith time bin, the bin-by-bin probabilities
of pile-up and not pile-up were obtained from the ratios,
pi
,
ni + pi
ni
,
=
ni + pi
Pi =
(3.14)
Pi
(3.15)
where ni + pi is the total number of trigger events in that particular bin. Note
that in the presence of background and contaminant decays, or if the γ-ray decay
69
multiplicity exceeds 1, the quantity ni would count all events that are not piled-up
which corresponds to a situation where no energy gate is taken on any particular
γ ray. In order to apply a γ-ray gate in the simulation a third index gi tracked only
not piled-up triggers that were due to the primary γ ray from the decay of interest.
The not piled-up-and-gated data gi is thus a subset of the total number of not piledup events ni . There is no γ-ray-gated pile-up equivalent for pi because, by definition,
pile-up has prevented the accurate determination of the energy of each of the events
involved in the pile-up.
A sample of simulated data showing the distinction between the total not piled-up
events ni , the γ-ray-gated not piled-up events gi and the total piled-up events pi is
shown in Fig. 3.3 for a single decay with a constant background using a pile-up time
τp = 5.0 µs and a dead-time τd = 30.0 µs. The simulation was performed with an
input half life of T1/2 = 1.0 s and initial rates for the decay component and constant
background of A = 6.7 × 104 s−1 and B = 1.0 × 103 s−1 . In terms of dimensionless
rates these correspond to Aτp = 0.335 and Bτp = 0.005, respectively. Application of
the γ-ray gate is demonstrated in Fig. 3.3 by the removal of the constant background
component. It should be noted that this is the raw output data from the simulation
and has not been corrected for either dead-time or pile-up effects.
From these data the bin-by-bin probability of pile-up Pi was generated according
to Eqn. 3.14 and is plotted in Fig. 3.4. With a total dimensionless rate in the first
time bin of x1 = Aτp + Bτp = 0.340 one expects a pile-up probability of ∼ 32 % from
Eqn. 3.11 which agrees with the simulated ratio shown in Fig. 3.4. At later times
(t > 15 s) the piled-up events are due entirely to the constant background and at a
rate of Bτp = x = 0.005 the probability of pile-up from Eqn. 3.11 is expected to be
only ∼ 0.5 % which also agrees with the bin-by-bin ratios of Fig. 3.4.
70
3
Counts per 100 ms
10
(a) ni
2
10
(b) gi
(c) pi
1
10
0
10
0
2
4
6
8
10
12
14
16
18
20
Time (s)
Figure 3.3: Simulated decay data curves for a single decay (simulated T1/2 = 1.0 s)
plus a constant background showing (a) the number of not piled-up events ni in each
time bin, (b) the number of not piled-up events that satisfy the γ-ray energy gate,
and (c) the number of piled-up events pi . The γ-ray gate removes the background
component and demonstrates the improved peak to background that is possible with
γ-ray counting techniques. These data have not been corrected for dead-time or
pile-up effects.
Combining the analytic expression for the probability of pile-up (Eqn. 3.11) with
the time-dependent decay rate for a single exponential plus a constant background
(Eqn.3.13) yields the appropriate function that should be used to describe the timedependent distribution of these data,
− ln2
t
a
x(t) = a1 e
2
+ a3 ,
P(t) = 1 − e−2x (1 + x).
71
(3.16)
Probability of pile-up: pi / (ni + pi)
0.35
0.30
Best-fit parameters:
Aτp = a1 = 0.331 ± 0.008
0.25
T1/2 = a2 = 1.01 ± 0.03 s
0.20
Bτp = a3 = 0.005 ± 0.001
0.15
0.10
0.05
0.00
0
2
4
6
8
10
12
14
16
18
20
Time (s)
Figure 3.4: The bin-by-bin probability of pile-up (circles) is obtained by taking the
ratio pi /(ni +pi ) for the data in Fig. 3.3. A fit to these data (solid line) is obtained
from the analytic expression for the probability of pile-up of Eqn. 3.16 when the rate
is due to a single exponential decay plus a constant background.
The fit parameters in this function are the dimensionless rate at t = 0 for the decay component a1 = Aτp , the half-life a2 = T1/2 , and the dimensionless background
rate a3 = Bτp . In order to correct for pile-up an unweighted fit was performed on
the simulated probability of pile-up data in Fig. 3.4 and the best-fit parameters of,
Aτp = 0.331 ± 0.008, Bτp = 0.005 ± 0.001, and T1/2 = 1.01 ± 0.03 s, are in excellent
agreement with the input values. It should be emphasized that Fig. 3.4 illustrates a
fit to the bin-by-bin probability of pile-up and not the activity curve. The fact that
the deduced half-life agrees with the input value is a further vindication of the derived
analytic expressions for pile-up. Although impractical due to the lack of statistics in
72
7000
Not-piled-up and γ-ray gated raw data gi
Dead-time and pile-up corrected data
Single decay exponential best fit
Counts recorded in γ gate
6000
5000
Best-fit parameters:
4 -1
A
4000
= (6.97 ± 0.05) x 10 s
T1/2 = 1.003 ± 0.004 s
3000
2
χ /ν = 1.12
2000
1000
0
0
1
2
3
4
5
6
7
8
Time (s)
Figure 3.5: Following the dead-time and pile-up corrections (circles) to the not piledup events in the γ-ray gate gi (squares) the deduced half-life T1/2 = 1.003 ± 0.004 s
agrees with the value T1/2 = 1.002 ± 0.003 s deduced from the mean decay time in
this simulation.
the probability of pile-up spectrum, one could in principle extract the nuclear half-life
from a fit to the probability of pile-up rather than the primary activity decay curve
by making use of the analytic expressions for pile-up derived above.
With the fit to the probability of pile-up data one can now quantify the correction that needs to be applied to the not piled-up and γ-ray gated decay data gi so
that the half-life can be deduced. The fitting of γ-ray-gated decay-curve data following corrections for dead-time and pulse pile-up effects is presented in detail in
Appendix A. From the simulated data plotted in Figs. 3.3 and 3.4 the dead-time and
pile-up corrected γ-ray-gated decay curve is plotted in Fig. 3.5. A fit to these corrected data using a single exponential decay yields a half-life T1/2 = 1.003 ± 0.004 s
73
1.12
1.11
1.10
1.09
1.08
1.07
1.06
1.08
1.09
1.05
1.10
12
1.04
10
1.03
8
1.02
6
1.01
4
1.00
2
0
Measured Half Life (s)
Frequency
Frequency
16
14
12
10
8
6
4
2
0
14
0.99
0.99
1.00
1.01
0.990
Corrected T1/2 (s)
1.000
1.010
Simulated T1/2 (s)
Figure 3.6: Simulation of 25 runs with and without the pile-up correction applied.
The weighted average without the correction applied is T1/2 = 1.0902 ± 0.0009 s,
a result that is 100 standard deviations larger than the pile-up corrected value
T1/2 = 1.0002 ± 0.0008 s and simulated value T1/2 = 1.0001 ± 0.0006 s. The solid
line indicates perfect agreement between the simulation and best-fit results.
which is an order of magnitude more precise than the value obtained from a fit to
the probability of pile-up data and agrees with the true value T1/2 = 1.002 ± 0.003 s
deduced from the mean of the decay times of the simulated events. Ignoring the
pile-up correction and correcting only for dead-time effects, the deduced half-life,
T1/2 = 1.199 ± 0.005 s, would be nearly 40 standard deviations too large, reflecting
the rate-dependent loss of events due to pile-up. A statistical distribution of the
results from 25 simulated decay curves is presented in Fig. 3.6. Histograms representing the uncorrected (top) and pile-up corrected (bottom) fit results are shown
74
in the left panel of Fig. 3.6. Scatter plots showing the simulated versus best-fit results are shown in the right panel with the solid line representing perfect agreement
between the simulated and best-fit values. The weighted average of all 25 half-lives
uncorrected for pile-up yielded T1/2 = 1.0902 ± 0.0009 s, which is 100 standard deviations from the input value, while the average of the 25 pile-up corrected values was
T1/2 = 1.0002 ± 0.0008 s. The pile-up corrected result is in perfect agreement with
the true result of T1/2 = 1.0001 ± 0.0006 s deduced from the mean decay times of
the simulated events and is a further vindication of both the analytic description of
detector pulse pile-up and the pile-up correction methodology.
3.3
Rate-Independent Tests
As a further test of the Monte-Carlo simulation and the analytic expressions derived
for detector pulse pile-up, a series of tests using rate-independent variables including
the number of detectors, the solid angle coverage of each detector, and the γ-ray
decay multiplicity were performed. Any rate-independent effects can only result in
an overall multiplicative factor being applied to the measured activity and therefore
will not affect the slope of the decay curve and hence the deduced half-life.
3.3.1
Detector Solid Angle
In terms of the probability of pile up Eqn. 3.11 any increase in the detector solid angle
will, for the same source activity, lead to an increase in the absolute detector rate x
which will subsequently lead to an increase in the probability of detector pulse pileup. However, because the parameters that are used to fit the probability of pile-up
are free, they will adjust correspondingly to this increased probability of pile-up. If
75
Corrected Half-life (s) Probability of Pile-up
0.8
0.7
0.6
0.5
Linear Approximation: P = 0.68Ω
-1.36Ω
P = 1-e
(1+0.68Ω)
0.4
0.3
0.2
0.1
0.0
1.02
1.01
1.00
0.99
0.98
0.97
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Fractional Solid Angle Ω
0.8
0.9
1.0
Figure 3.7: Pile-up probability versus the detector solid angle (top) showing the lowrate linear approximation P = 0.68Ω (dashed line) and the correct analytic expression
of Eqn. 3.11 that is valid for all rates (solid line). Once corrected for the detector
pile-up, the half-life deduced is consistent with the simulated value T1/2 = 1.0 s and
is independent of the detector solid angle.
one considers the low-rate limit where x < 1 then the exponential in the probability
of pile-up can be expanded to obtain,
P = 1 − e−2x (1 + x),
4
2
≈ 1 − (1 − 2x + 2x2 − x3 + x4 + . . .)(1 + x),
3
3
2 3
≈ x − x + O(x4 ),
3
(3.17)
an expression that, to first order, is linear in the detector rate x. The simulation described above for a single radioactive decay with a constant background was extended
to include the effects of a variable detector solid angle. Results are shown in Fig. 3.7
76
that used the input parameters A = 6.7× 104 s−1 , T1/2 = 1.0 s, B = 1.0 × 103 s−1 ,
τd = 30 µs, τp = 10 µs, and hence the t = 0 dimensionless rate x = 0.68. The solid angle was varied in the simulation from 10% to 100% in increments of 10%. As shown
in Fig. 3.7 the pile-up probability increases linearly as a function of the fractional
solid angle Ω for the lowest rates. The linear dashed curve was obtained using the
first order approximation of Eqn. 3.17, P = 0.68Ω. Once corrected for dead time and
pile-up effects using the complete analytic description there is no bias introduced to
the deduced half-life. At 100 % solid angle coverage these parameters correspond to
a probability of pile up at t = 0 of ∼ 57 % and thus the above simulation covers
the realm of realistic experimental conditions and rates and confirms the fact that
the pile-up correction methodology described above is independent of the particular
solid-angle subtended by the detector.
3.3.2
Number of Detectors
Improving the overall precision in the deduced half-life requires maximizing the collected statistics. It was shown above that one method to increase the statistics would
be to use a larger detector, however, increasing the size of the detector can eventually lead to pile-up probabilities that exceed ideal experimental operating regimes.
Instead, it is more beneficial to cover the same large solid angle using a large number
of smaller detectors. Confirming that the pile-up correction technique is independent
of the number of detectors is critical before employing HPGe detector arrays such as
the 8π γ-ray spectrometer to deduce high-precision β decay half-lives. The simulation
was modified to include a total aquisition dead time, where a trigger in one detector
would lead to all detectors in the array being dead for the duration of the fixed and
non-extendible dead time interval τd . Detector pulse pile-up can, however, only occur
77
0.7
1 Detector
2 Detectors
3 Detectors
4 Detectors
5 Detectors
10 Detectors
Probability of Pile-up
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
Time (s)
Figure 3.8: Pile-up probabilities versus the number of detectors employed. Each curve
was obtained from the same simulation parameters with a total solid angle coverage
of 100% for all detectors. The probability of pile-up can be greatly reduced using
several smaller detectors, however, in all cases the pile-up correction methodology is
independent of the number of detectors used.
if two or more events occur within the pile-up time interval τp in the same detector.
If two events occurred in different detectors within the pile-up time then the second
event was treated as a dead-time loss while the original trigger was counted as a not
piled-up event and was incremented into the γ-ray gated data gi . Using the same simulation parameters as above for the pile-up time, dead time, and activities at t = 0,
the solid angle was fixed at 100 % and the number of detectors was increased, one
at a time, to explore the variation of the measured amounts of pile-up. Best-fits to
the probability of pile-up data are shown in Fig. 3.8 for a selection of cases between
1 and 10 detectors. A large decrease in the probability of pile up is obtained through
78
the use of a 2 detector system with the same solid angle coverage as a 1 detector
system because the detector rate x in the 2 detector setup is halved with respect to
the 1 detector case. As with the solid angle variation no modification to the above
fit function for the pile-up probability was necessary. It was thus confirmed through
simulation that the method for the correction of detector pulse pile-up derived here
can be applied to half-life measurements employing either a 1 detector system or an
array of N detectors such as the 8π spectrometer. In actual experimental data with
the 8π spectrometer it will be shown, however, that a small correction must be applied
to the dead-time correction due to the fact that with finite coincidence time windows
the entire array is not immediately dead following a trigger in a single detector. This
multi-detector dead-time correction is discussed in detail in Chapter 4, Sec. 4.2.1.
3.3.3
γ-ray Multiplicity
The above analysis has not considered the fact that multiple γ rays can be emitted
following each β decay and only one of these γ rays may be the transition of interest that will be included in the γ-ray gate. Extending the Monte-Carlo simulation
to incorporate γ-ray multiplicity Mγ was performed by creating another index that
tracked the frequency of coincident summing si . For a coincidence sum to occur, two
or more γ rays from the same nuclear β decay must be detected by the same detector. The simulation assumed that the time difference between two events in the same
detector from a coincidence sum is sufficiently close to zero (on the order of ps or fs)
that the electronics cannot distinguish this event from a not piled-up singles event.
When a sum event was recorded in the simulation it was therefore counted as a not
piled-up event ni but did not count in the γ-ray gate gi because the energies of the
two events would be summed by the detector. Note that true coincident summing has
79
Mγ = 2
1.010
10
1.005
8
1.000
6
0.995
4
0.990
2
0
14
Frequency
12
Mγ = 1
1.010
10
1.005
8
1.000
6
0.995
4
0.990
2
0
Corrected T1/2 (s)
Frequency
12
Corrected T1/2 (s)
14
0.98 0.99 1.00 1.01 1.02 0.990
1.000
1.010
Simulated T1/2 (s)
Corrected T1/2 (s)
Figure 3.9: Simulation of 25 runs comparing γ-ray multiplicities Mγ = 1 and
Mγ = 2. Higher multiplicities lead to an increased probability of coincident summing and thus an increased statistical uncertainty in the deduced half-life. Because true coincident summing is a rate-independent effect, the half-lives deduced
T1/2 (Mγ = 2) = 0.9997 ± 0.0012 s, T1/2 (Mγ = 1) = 1.0002 ± 0.0008 s are both in
excellent agreement with the simulated value T1/2 = 1.0 s.
a rate-independent probability because the probability that two γ rays from the same
nuclear decay are detected in the same detector is same for every β decay. Pile-up,
or the probability that two events from different nuclear decays are recorded in the
same detector, is a rate-dependent effect because it is a function of the number of
other decays that take place within a short time interval from the decay of interest.
A comparison between the half lives obtained from a series of 25 runs with γ-ray
multiplicity Mγ = 1 and Mγ = 2 is shown in Fig. 3.9 using a 2 detector system
covering 100% of the solid angle. The same simulation parameters were used for both
80
0.7
Mγ = 1.0
Mγ = 2.0
Mγ = 3.0
Mγ = 4.0
Probability of Pile-up
0.6
0.5
Mγ = 5.0
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
Time (s)
Figure 3.10: Pile-up probabilities versus the γ-ray decay multiplicity for multiplicities
up to Mγ = 5.0. It is clearly shown that the pile-up probabilites are saturating even
though the effective γ-ray rates continue to increase for higher-fold multiplicities due
to the fact that the γ-ray activity is not randomly distributed in time.
data sets as above (A = 6.7× 104 s−1 , T1/2 = 1.0 s, B = 1.0 × 103 s−1 , τp = 10 µs,
τd = 30 µs). The data shown in the top panel corresponds to Mγ = 2 and, when
compared with the data in the bottom panel obtained with Mγ = 1, one can clearly
see the effect of coincident summing which is reflected by a loss of statistics in the γray-gated decay curve data and hence increased statistical fluctuations in the deduced
half-life. The resulting half-lives obtained after dead time and pile-up corrections are
both consistent with the simulated value of T1/2 = 1.0 s which again demonstrates
that the pile-up correction technique derived here accounts for all sources of rateindependent effects such as the γ-ray decay multiplicity. A plot showing the best-fit
81
distributions to the probability of pile-up data for Mγ = 1.0 to 5.0 is presented in
Fig. 3.10.
As the multiplicity is increased there are more γ rays emitted for every β decay
and thus the absolute detector rate and probability of pile-up are increased. There is
a saturation that occurs at the highest multiplicities as the γ activity comes in large
γ-ray bursts (1 β decay yields Mγ γ rays all liberated at the same time) followed by
the same γ-ray free time interval that would have been present in the Mγ = 1 case.
As an example, the probability of pile-up for 103 β decays per second with Mγ = 5 is
not equivalent to 5×103 β decays per second with Mγ = 1 even though the effective
γ-ray activity is 5×103 per second in both cases. The former will always lead to more
coincidence summing but smaller pile-up probabilities while the latter can increase
the probability of pile up to unity as shown in Fig. 3.2.
The analytical description of the probability of pile-up given by Eqns. 3.11 and 3.16
have been shown to be completely general and apply for all rate-independent variables
including the number of detectors, the solid angle coverage of each detector, and γ-ray
decay multiplicity. There are, however, additional rate-dependent effects originating
from experimental sources that must be considered before the half-life can be deduced
at the level of 0.05% precision.
3.4
3.4.1
Rate-Dependent Refinements
Pile-up Time Resolution
One must now consider the implications of the fact that the pile-up circuitry will not
be able to resolve pile-up of two or more events (from separate decays) if they occur
within a very short time interval τr . In general this resolution time will be much less
82
than the total pile-up time τr ≪ τp but expressions will be derived that are valid over
the entire range of 0 ≤ τr ≤ τp .
The probability of an event being post-piled-up (independent of whether it was
also pre-piled-up) is given in Eqn. 3.3. In order to include the time resolution of the
pile-up circuitry one needs to modify Eqn. 3.3 to reflect the fact that events occurring
within τr are not detected as piled-up events. The probability of detecting an event as
′
post-piled-up in the presence of a non zero pile-up time resolution (denoted Ppost
) can
again be derived from the Poisson probability distribution that was given in Eqn. 3.1.
For post-pile-up, the result can be expressed as,
′
Ppost
= P (0; τp − τr ) = e−R(τp −τr ) ,
(3.18)
′
Ppost
= P (≥ 1; τp − τr ) = 1 − e−R(τp −τr ) .
(3.19)
A schematic representation of this process is shown in Fig. 3.1c. The expressions for
pre-pile-up detection must similarly be refined to include the pile-up time resolution.
In this case there are 2 possibilities that could give rise to pre-pile-up as seen in
Fig. 3.1d. If the trigger that we are considering occurs at time τr ≤ t ≤ τp after the
release of the dead time of the previous trigger then pre-pile-up can occur if 1 or more
events are recorded in the time interval ∆t = τp - t, and cannot be lost to the pile-up
time resolution. This is the same integral as Eqn. 3.5 with the pile-up time resolution
appearing only in the limits,
′(1)
Ppre
=
=
Zτp
P (≥ 1; ∆t = (τp − t))dPT rig (t),
τr
Zτp
(1 − e−R(τp −t) )Re−Rt dt,
τr
−Rτr
= e
− e−Rτp − R(τp − τr )e−Rτp .
(3.20)
It is also possible for pre-pile-up to occur if the next trigger from the end of the dead
83
time of the previous trigger occurs at a time t ≤ τr , but the time difference between
this trigger and 1 or more events in the dead time of the previous trigger must be
recorded in ∆t = τp - τr , if it is to be detected as pre-piled-up. This probability can
be expressed as,
′(2)
Ppre
=
Zτr
P (≥ 1; ∆t = (τp − τr ))dPT rig (t),
=
Zτr
(1 − e−R(τp −τr ) )Re−Rt dt,
0
0
= 1 − eRτr + e−Rτp − e−R(τp −τr ) .
(3.21)
The probability of detecting an event that is pre-piled-up is given by the sum of
Eqns. 3.20 and 3.21 and the probability for obtaining events that are not detected as
pre-piled-up is given by the normalization condition,
′
Ppre
= 1 − e−Rτp eRτr + R(τp − τr ) ,
′
Ppre
= e−Rτp eRτr + R(τp − τr ) .
(3.22)
(3.23)
Defining ǫr as the ratio of the pile-up time resolution to the total pile-up time
(τr = ǫr τp ), the total probability of detecting pile-up can be expressed solely in terms
of this parameter and the dimensionless detector rate x = Rτp ,
′
′
′
′
′
′
Ppost
+ Ppre
Ppost
+ Ppre
Ppost
,
P ′ = Ppre
= 1 − e−(2−ǫr )x [eǫr x + (1 − ǫr )x] .
(3.24)
In the limit ǫr = 0 this expression is equal to the probability of pile-up derived in
Eqn. 3.11. In the limit where the resolution time is equal to the pile-up time ǫr = 1,
all piled-up events would not be recorded as being piled-up and Eqn. 3.24 correctly
yields P ′ = 0.
84
Following the example where the detector rate is due to a single exponential decay
plus a constant background, fitting the probability of pile-up data with a non-zero
pile-up time resolution can now be achieved with the following analytic function that
properly accounts for these losses,
− ln2
t
a
x(t) = a1 e
2
+ a3 ,
P ′ (t) = 1 − e−(2−a4 )x [ea4 x + (1 − a4 )x] .
(3.25)
where a single new parameter a4 = ǫr has been introduced to the previous function
of Eqn. 3.16. This parameter can typically be fixed at the value determined from
the experimental pile-up time spectra. To illustrate, the Monte Carlo simulation was
modified to treat two or more events occurring in the same detector within the pile-up
time resolution τr as a single not piled-up event ni , but did not record this event in
the γ-ray gated spectra gi because the energy of this “not piled-up” event would not
be correct. Pile-up time spectra were added to the simulation by recording the time of
the pile-up relative to the master trigger. In the case of pure post-pile-up (Ppre Ppost )
the trigger is initially not piled-up until a second event arrives afterward and causes
pile-up. The pile-up time for pure post-pile-up can therefore take on any value t where
0 ≤ t ≤ τp . In the case of pre-pile-up (Ppre Ppost and Ppre Ppost) the trigger itself also
generates the pile-up signal which results in pile-up times of t = 0 for all pre-piled-up
events. Although the pile-up time resolution affects both the pre- and post-piled-up
data, its existence is most readily observed in the pile-up time spectrum by the lack
of post-piled-up events with pile-up times occurring between t = 0 and τr . Simulated
pile-up time spectra for a single exponential decay and constant background with a
pile-up time τp = 10.0 µs, is shown in Fig. 3.11 for the case where there is perfect
pile-up detection τr = 0.0 µs compared to an exaggerated case where τr = 8.0 µs.
85
4
10
3
Counts per 5 ns
10
τr = 0.0 µs
τp = 10.0 µs
εr = 0.0
2
10
τr = 8.0 µs
τp = 10.0 µs
εr = 0.8
τr
1
10
0
10
0
2
4
6
8
10
0
Time of recorded pile-up (µs)
2
4
6
8
10
Time of recorded pile-up (µs)
Figure 3.11: Simulated pile-up time τp with respect to the master-trigger time for
2 different values of the pile-up time resolution τr . When τr = 0.0 µs all piled-up
events are properly recorded. A non-zero resolution time such as τr = 8.0 µs leads to
a loss of detected piled-up events both in the pre-pile-up time peak at t = 0 and the
post-pile-up time continuum from t = 0 to t = τr .
It is important to note that the true pile-up probabilities are always given by
Eqn. 3.11, and are independent of τr . As τr is increased, the probability of pileup does not decrease but the ability of the system to detect these piled-up events
deteriorates. A fit to the detected pile-up probability data using the function in
Eqn. 3.25 is required to determine the parameters a1 to a4 , where a4 = ǫr can be fixed
at the independently determined value. The true probability of pile-up, and hence
the correction that needs to be applied to the γ-ray-gated decay data is, however,
that of Eqn. 3.16 which is equivalent to Eqn. 3.25 with a4 = 0.0.
86
8000
Probability of Pile-up
0.6
Counts recorded in γ gate
7000
6000
5000
4000
Simulated data
Best fit:
a4 = 0.8
Corrected fit: a4 = 0.0
0.5
0.4
0.3
0.2
0.1
0.0
0
3000
2000
1000
5 6 7 8 9
Time (s)
Uncorrected decay data
Corrected with a4 = 0.8
Corrected with a4 = 0.0
Best fit (a4 = 0.8): T1/2 = 1.1307 ± 0.0052 s
Best fit (a4 = 0.0): T1/2 = 0.9995 ± 0.0044 s
1
2
3
4
0
0
1
2
3
4
5
6
7
8
9
10
Time (s)
Figure 3.12: Simulated decay-curve data that is uncorrected (squares), dead-time and
pile-up corrected with ǫr = 0.8 (triangles), and dead-time and pile-up corrected with
ǫr = 0.0 (circles). (Inset) Simulated probability of pile-up data (circles) with best-fit
(solid line) and pile-up time-resolution corrected curve (dashed line). The true halflife for this simulated data set was T1/2 = 0.9949 ± 0.0032 s and is only reproduced
after the pile-up time resolution losses are properly accounted for (dashed lines).
A demonstration of this procedure using a simulated data set with a greatly exaggerated ǫr = a4 = 0.8 is shown in Fig. 3.12. An unweighted fit to the observed pile-up
probability data with the pile-up time resolution fixed at a4 = 0.8 yields the best-fit
parameters Aτp = 0.664 ± 0.017, Bτp = 0.006 ± 0.003, and T1/2 = 1.05 ± 0.05 s. The
pile-up correction to be applied to the not piled-up γ-ray-gated decay data gi is the
true probability of pile-up that is obtained using these best-fit parameters but setting
a4 = 0.0. This is shown as the dashed curve in the inset Fig. 3.12 and is always
87
larger than the value of the fit function which demonstrates that the detected pileup is always less than the true amount of pile-up. After correcting the γ-ray-gated
decay-curve data with this probability of pile-up (with a4 = 0.0, the dashed curve
in Fig. 3.12), the half-life determined for this data set was T1/2 = 0.9995 ± 0.0044 s
which agrees with the true value T1/2 = 0.9949 ± 0.0032 s derived from the mean
decay times for this simulated data set. If instead one were to use the best-fit function
to the detected probability of pile-up (with a4 = 0.8, the solid curve in Fig. 3.12) to
correct the decay data the result, T1/2 = 1.1307 ± 0.0052 s, is more than 26 standard
deviations (or 13.6%) from the correct value.
Experimentally, the pile-up time resolution is typically an order of magnitude less
than the pile-up time (ǫr < 0.1). While this is a much smaller effect than the exaggerated example presented above, the correction for these losses is certainly significant
at the 0.05% level and must therefore be accounted for in experimental superallowed
β-decay data.
3.4.2
Trigger-Energy Threshold
Another refinement to the Monte Carlo simulation of detector pulse pile-up is the
existence of a non-zero trigger-energy threshold. In the above derivation of the probability of pile-up (Eqns. 3.7-3.11) it was assumed that all events that made up the
total detector rate x = Rτp could trigger the system. In this limit, the definition of a
trigger is the next event in the time sequence that follows the release of the dead time
of the previous trigger. Experimentally, however, this is not realistic as a constantfraction-discriminator (CFD) is employed to remove noise in the γ-ray spectrum at
low energies or can be increased intentionally to remove real γ-ray trigger events in
order to reduce the overall dead time of the system. All events, regardless of energy
88
and whether or not they produce triggers, are able to cause pile-up. The existence of
a non-zero CFD threshold must therefore be accounted for in the analysis. Following
the example of a single constant rate source with dimensionless detector rate x, and
defining the fraction of this rate that exceeds the CFD threshold α, the detector rates
above x> and below x< the threshold can be written as,
x> = αx,
x< = (1 − α)x.
(3.26)
Note that once the threshold level has been established α is a constant for a given
source and is independent of rate. In deriving the probability of pile-up expressions in
Sec. 3.1 (Eqns. 3.7-3.11) it was assumed that all events exceeded the CFD threshold
and hence α = 1.0. When α < 1.0, one has to consider its effect on the processes of
pre- and post- pile-up.
The probability of post-pile-up given in Eqn. 3.3 is independent of whether the
events are triggers. Only the events that exceed the CFD threshold can be triggers,
however once a trigger has been recorded any event, regardless of energy, can cause
it to be post-piled-up. The probability of post-pile-up given in Eqn. 3.3 is therefore
independent of the CFD threshold.
The probability of pre-pile-up caused by a γ-ray above the CFD threshold is the
same as that presented above and can be obtained from a direct substitution x> = αx
into Eqn. 3.5,
>
= 1 − e−αx (1 + αx).
Ppre
(3.27)
For pre-pile-up caused by a γ ray below the CFD threshold, the pre-piling-up event
need not be within the dead time of the previous trigger because these events cannot
themselves be triggers. The probability of pre-pile-up is therefore the probability of
89
1.0
Probability of pile-up
0.8
0.6
Analytic, α = 1.0
Analytic, α = 0.5
Analytic, α = 0.1
Monte Carlo
0.4
0.2
0.0
0.0
0.5
1.0
1.5
2.0
Dimensionless Detector Rate x = Rτp
Figure 3.13: Plot of the analytic (lines) and Monte Carlo (circles) pile-up probability
curves versus the dimensionless detector rate Rτp when the fraction of events that
exceed the energy threshold α is varied. The analytic curves were obtained from
Eqn. 3.30 with the indicated α value. The α = 1.0 curve corresponds to that of
Fig. 3.2(b).
obtaining an event within a time interval τp before the trigger event and is obtained
directly from the Poisson probability distribution (Eqn. 3.1),
<
Ppre
= 1 − e(1−α)x .
(3.28)
The total probability of pre-pile-up is obtained from combining these probabilities,
>
<
>
<
>
<
Ppre = Ppre
Ppre
+ Ppre
Ppre
+ Ppre
Ppre
,
= 1 − e−x (1 + αx).
(3.29)
In the limit α = 1, the original definition of pre-pile-up given in Eqn. 3.5 is restored
corresponding to the case where all events exceed the CFD energy threshold. In
90
the limit where α approaches 0, a smaller fraction of events have sufficient energy
to exceed the CFD threshold, while all events produce pile-up, and the symmetry
between pre- and post-pile-up is restored (Eqn. 3.3 = Eqn. 3.29).
Following the calculation of Eqn. 3.11, the probabilities of pre- and post-pile-up
can be combined to yield the total probability of pile-up with a non-zero trigger
energy threshold,
P = 1 − e−2x (1 + αx).
(3.30)
This result is also independent of the dead time assuming the condition τd ≥ τp is
satisfied. The probability of pile-up for a single constant-rate source is plotted against
the dimensionless detector rate x = Rτp for α = 1.0, 0.5, and 0.1 in Fig. 3.13. The
results of the Monte Carlo simulation are overlayed and show perfect agreement over
the entire range of detector rates. Note that raising the CFD threshold leads to an
increase in the probability that trigger events are piled-up and this is true for all
rates.
A common practice is to intentionally raise the CFD thresholds in order to decrease
the overall dead time of the system and record more “good” events at higher energy.
The probability of obtaining a trigger event (or an event that is not lost to the
dead time) D can be expressed in terms of the detector rate that exceeds the CFD
threshold, αR, and the length of the dead-time interval τd [90],
D=
1
.
1 + αRτd
(3.31)
Nuclear half-life measurements with HPGe detectors are typically conducted in the
regime where τd ≥ τp and thus one can substitute τd = βτp with β ≥ 1 and rewrite
Eqn. 3.31 in terms of the dimensionless detector rate x = Rτp ,
D=
1
.
1 + αβx
91
(3.32)
Combining Eqn. 3.32 with the probability of obtaining a not piled-up event, given by
P = 1 - P, leads to the probability of obtaining a trigger that is not-piled-up (and is
therefore a “good event”):
PD =
e−2x (1 + αx)
.
1 + αβx
(3.33)
Equations 3.32 and 3.33 are plotted in Fig. 3.14 for three values of the CFD threshold
α = 1.0, 0.5, 0.1 and two values of the dead time β = 2.0, 10.0. Considering only
the dead time (top row, Fig. 3.14) raising the CFD threshold (decreasign α) leads
to a large increase in the probability of obtaining a trigger and therefore a large
decrease in the subsequent dead-time correction to be applied to the experimental
data. Once the increase in the probability of pile-up is taken into account (bottom
row, Fig. 3.14), this analysis demonstrates that most of the events that are recovered
as a result of raising the CFD threshold are more likely to be piled up. Raising the
CFD threshold therefore only leads to a marginal improvement in the probability of
obtaining a good event. Of interest is the limit where τd = τp (β = 1.0) and every event
recovered by increasing the CFD threshold is, by definition, piled-up. In this limit,
which is not far from typical experimental conditions with HPGe detectors (β ∼ 2.0),
the probability of obtaining a good event is independent of the CFD threshold and as
can immediately be seen from Eqn. 3.33, nothing is actually gained from the common
practice of raising the CFD threshold.
The above calculation for a single source can be repeated for the general case
where there are multiple sources each with different fractions of events that exceed
the CFD threshold and each with dimensionless rates x1 (t), x2 (t), . . .,
x(t) = x1 (t) + x2 (t) + . . .
x> (t) = α1 x1 (t) + α2 x2 (t) + . . .
92
(3.34)
τd = 2τp
τd = 10τp
1.0
α = 1.0
α = 0.5
α = 0.1
0.8
0.6
D
0.4
0.2
0.0
0.8
0.6
PD
0.4
0.2
0.0
0.0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
2.0
Dimensionless Detector Rate x=Rτp Dimensionless Detector Rate x=Rτp
Figure 3.14: Analytic functions describing the probability of obtaining a trigger event
D (top row) and the probability of obtaining a not-piled-up trigger event PD (bottom
row) as a function of the dimensionless detector rate x =Rτp . The fraction of events
that exceed the CFD threshold α is shown by the different lines, and the left(right)
columns are for dead times of 2τp (10τp ), respectively. Raising the CFD threshold
(decreasing α) leads to a large increase in the probability of recording a trigger event
(which reduces the dead-time correction) but simultaneously increases the probability
of pile-up (and hence the pile-up correction, Fig. 3.13). The result is only a very
marginal gain in the probability of obtaining a good event for realistic τd /τp ratios.
93
The probability of pile-up in this general case becomes,
P = 1 − e−2x(t) (1 + α1 x1 (t) + α2 x2 (t) + . . .),
= 1 − e−2x(t) (1 + α(t)x(t)),
(3.35)
where the fraction of events α(t) exceeding the CFD threshold becomes time-dependent
due to the different contributions of the different sources as a function of time and
can be written as,
α(t) =
α1 x1 (t) + α2 x2 (t) + . . .
.
x1 (t) + x2 (t) + . . .
(3.36)
While Eqns. 3.35 and 3.36 give the correct general expression to describe the total
probability of pile-up with a non-zero CFD threshold for any number of decay components, in anticipation of the
26
Na half-life analysis presented in Chapter 4 of this
thesis, it should be noted that these equations, among all of those presented in this
chapter, introduce difficulty in analyzing actual experimental data. Considering the
single-exponential plus a constant-background example of Sec. 3.2, for example, the
function applied to fit the probability of pile-up when α = 1.0 had 3 free parameters
(see Eqn. 3.16): a1 the activity of the decaying exponential at t = 0, a2 the halflife of the exponential, and a3 the constant background rate. With a non-zero CFD
threshold, 2 additional parameters must be included to account for the fraction of
events that exceed the threshold from the exponential decay α1 and the background
α2 , respectively. Because any real experiment would aim to minimize the amount of
pile-up, the statistics available in the probability of pile-up spectrum are generally
insufficient to constrain this number of free parameters. In practice (see Chapter 4)
the approximation,
α(t) ≈ α,
94
(3.37)
where α is averaged over the decay curve, introduces negligible error in the pile-up
correction. This can be understood from the fact that the objective is simply to
obtain a good functional description of the probability of pile-up data Pi . In the
limit of infinite statistics, the bin-by-bin Pi curve itself provides an exact description
of the correction to be made to the γ-ray-gated decay data gi , and no functional form
is required at all (until refinements such as the pile-up time resolution discussed in
Sec. 3.4.1 are included). For finite statistics, the fluctuations of Pi must be smoothed
for a best-fit pile-up correction to the gi by fitting to a function. However, any function
that accurately describes the Pi data produces the same pile-up correction. Although
Eqns. 3.35 and 3.36 provide the exact functional form, in practice the parameters
of Eqn. 3.16 (or its generalization, Eqn. 3.25) are more than sufficient to provide an
accurate description of the probability of pile-up data Pi and below is a demonstration
that the approximation made in Eqn. 3.37 leads to negligible error in the extracted
half-life under realistic experimental conditions.
In the presence of a non-zero pile-up time resolution and a pre-set CFD energy
threshold, the following function is therefore adopted to describe the probability of
detector pulse pile-up,
− ln2
t
a
x(t) = a1 e
2
+ a3 ,
P ′ (t) = 1 − e−(2−a4 )x [ea4 x + a5 (1 − a4 )x] ,
(3.38)
where a single new parameter a5 = α has been added to account for the fraction of
events that exceed the CFD threshold. If the parameters a1 to a3 are left free in the fit
to the probability of pile-up data Pi , then the deduced half-life following the pile-up
correction is very insensitive to the CFD parameter a5 over its entire physical range
0 ≤ a5 ≤ 1. This parameter can thus be fixed at an appropriate value estimated from
95
4.0
Probability of pile-up: pi / (ni + pi)
-5
Fit 1- Fit 2 (x 10 )
0.8
0.6
2.0
0.0
-2.0
-4.0
0
5
0.4
Best-fit parameters (Fit 1):
Aτp = a1 = 0.677 ± 0.021
T1/2 = a2 = 1.02 ± 0.03 s
Bτp = a3 = 0.007 ± 0.001
α1 = 0.1 (fixed)
α2 = 1.0 (fixed)
0.2
10
15
Time (s)
20
25
Best-fit parameters (Fit 2):
Aτp = a1 = 0.685 ± 0.029
T1/2 = a2 = 1.02 ± 0.04 s
Bτp = a3 = 0.004 ± 0.002
α = a5 = 0.11 (fixed)
0.0
0
5
10
15
20
25
Time (s)
Figure 3.15: Simulated probability of pile-up data (circles) and best-fit curve when
a non-zero CFD energy threshold is included. The data was fit using 2 separate
functions (Fit 1 - Eqn. 3.35, Fit 2 - Eqn. 3.30). (Inset) The difference between these
two functions differ by a maximum of 5×10−5 which leads to a difference in the
deduced half-life of only 0.0006%.
the γ-ray spectrum.
This procedure is demonstrated in the simulated probability of pile-up data of
Fig. 3.15 that uses the single exponential plus a constant background example with
the input values, A = 6.7× 104 s−1 , T1/2 = 1.0 s, B = 1.0 × 103 s−1 , τp = 10 µs,
and τd = 30 µs. The simulation used the extreme CFD values of α1 = 0.1 for the
exponential decay rate x1 and α2 = 1.0 for the background rate x2 to test for any
possible breakdown of Eqn. 3.37. The simulated probability of pile-up data were then
fit in two distinct ways. The first (Fit 1) employed the exact function of Eqn. 3.35
treating α1 and α2 as known parameters. The second (Fit 2) used the approximation
96
of Eqn. 3.37 and treated the parameter α = 0.11 as a fixed parameter measured from
the ratio of the events that exceeded the CFD threshold to the total number of events
averaged over the entire decay curve. As seen in Fig. 3.15 the fit parameters in the first
and second fit methods are distinct as a result of the approximation made in Eqn. 3.37,
however, the best-fit curves described by each set are not, differing by no more than
5.0×10−5 . The second fit was also performed fixing the value of α at the extreme
values α = 1.0 and α = 0.0 and the difference in the resultant fit curves differed by
no more than 8.0×10−3 . Using each of these probability of pile-up distributions the
deduced half-lives in this example were T1/2 = 0.9966 ± 0.0122 s for both Fit 1 and
Fit 2 with α = 0.11 differing by only 6.2×10−6 s or 0.0006%. Even when the extreme
values of α = 0.0 and α = 1.0 were used, the half-lives differed from the correct
answer by only 1.3×10−5 s (0.001%) and 7.4×10−5 s (0.007%), respectively. The
approximation made in Eqn. 3.37, with α = 0.11 therefore does not bias the deduced
half-life at the level of 0.0006% in this sample simulation where the probability of
pile-up is as large as ∼ 72% at t = 0 and the α1 and α2 thresholds were chosen to
be very different. For experimental data, where the pile-up corrections are typically
an order of magnitude smaller, and the αi are generally all similar and close to α,
negligible bias is introduced as a result of the approximation made in Eqn. 3.37.
3.4.3
Pile-up Detection Energy Threshold
While the effects of a non-zero trigger-energy threshold have been accounted for
in Sec. 3.4.2, the possibility remains to miss detecting piled-up events due to the
existence of a non-zero energy threshold associated with pile-up detection. A trigger
with sufficient energy to exceed the CFD energy threshold could be piled-up by an
event whose energy is below the pile-up energy threshold leading to the trigger being
97
recorded as a good event. The energy of this type of event would lie in the limited
range between Edep and Edep + δE where Edep is the energy deposited in the detector
by the triggering photon and δE is the pile-up detection energy threshold. Although
these events would not be recorded as pile-up, they would be automatically recovered
by the above analysis if the subsequent γ-ray photopeak energy gate were wider than
the pile-up energy threshold δE (which can typically be set around δE ∼ 20 keV).
It is, however, generally undesirable to set such a wide γ-ray energy gate and one
must therefore apply a correction to account for these undetected low-energy pile-up
events.
Low-energy pile-up losses are characterized by the fraction of the detector rate
that exceeds the pile-up energy threshold. This fraction defines an overall efficiency
with which to detect pile-up ǫp . In the absence of time-resolution (Sec. 3.4.1) and
trigger-energy threshold (Sec. 3.4.2) effects, the probability of detecting pile-up can
be expressed using Eqn. 3.11 as,
P ′ = ǫp P = ǫp [1 − e−2x (1 + x)],
(3.39)
where it has been explicitly assumed that the pile-up detection efficiency due to
energy threshold effects is time-independent. This is expected to be an excellent
approximation (far better than that discussed in Sec. 3.4.2 related to the intentional
raising of the CFD threshold) for any real experiment in which the pile-up energy
threshold is set just above the noise level. It can also be independently confirmed
through inspection of the γ-ray energy spectra recorded as a function of time.
The appropriate function that combines the pile-up detection efficiency ǫp = a6
with the pile-up time resolution a4 , and the trigger-energy CFD threshold a5 can be
98
8000
Probability of Pile-up
0.6
Counts recorded in γ gate
7000
6000
5000
4000
0.5
Simulated Data
Best fit:
εp = 1.0
0.4
Corrected fit: εp = 0.2
0.3
0.2
0.1
0.0
0
3000
2000
1
2
3
7
4
5 6
Time (s)
Uncorrected decay data
Corrected with εp = 1.0
8
9
Corrected with εp = 0.2
Best fit (εp = 1.0): T1/2 = 1.1503 ± 0.0051 s
1000
Best fit (εp = 0.2): T1/2 = 1.0049 ± 0.0042 s
0
0
1
2
3
4
5
6
7
8
9
10
Time (s)
Figure 3.16: Simulated decay-curve data that is uncorrected (squares), dead-time and
pile-up corrected with ǫp = 1.0 (triangles), and dead-time and pile-up corrected with
ǫp = 0.2 (circles). (Inset) Simulated probability of pile-up data (circles) with best fit
(solid line) and pile-up efficiency corrected curve (dashed line). The true simulated
half-life for this data set was T1/2 = 1.0009 ± 0.0032 s and is only reproduced after
the pile-up efficiency losses are properly accounted for (dashed lines).
written as,
t
− ln2
a
x(t) = a1 e
2
+ a3 ,
P ′ (t) = a6 (1 − e−(2−a4 )x [ea4 x + a5 (1 − a4 )x]).
(3.40)
This expression was confirmed using simulated data with a very low pile-up detection efficiency ǫp = 0.2. The results are presented in Fig. 3.16 using the single exponential decay plus a constant background example with input values, A = 6.7× 104 s−1 ,
T1/2 = 1.0 s, B = 1.0 × 103 s−1 , τp = 10 µs, and τd = 30 µs. Correcting the detected
probability of pile-up by the known pile-up detection efficiency ǫp = 0.2 (dashed
99
curves) yields a half-life T1/2 = 1.0049 ± 0.0042 s that agrees with the true simulated value of T1/2 = 1.0009 ± 0.0032 s for this data set. If one does not properly
account for the pile-up efficiency (solid curves), the resulting pile-up correction is too
small and the half-life deduced in this example T1/2 = 1.1503 ± 0.0051 s is too large
by ∼ 29σ (14.9%).
It is important to note that in an actual experiment the pile-up detection efficiency ǫp cannot be treated as a free parameter as it is an overall multiplicative factor
to the probability of detected pile-up. One must therefore determine ǫp experimentally
from the γ-ray spectra. The experimental value deduced can, however, be checked for
consistency by increasing the width of the γ-ray photopeak energy gate. If the γ-ray
gate width extends beyond Eγ + δE then all missed piled-up events would already
be included in the gate and thus ǫp = 1.0. If, however, one takes an energy gate that
is more restrictive, then a correction using ǫp < 1.0 is required. This procedure for
providing the necessary consistency check of ǫp will be discussed in more detail in
Chapter 4, where ǫp = 0.995 was deduced for the
3.4.4
26
Na half-life experiment.
Cosmic Rays
When a high-energy cosmic ray strikes the detector a saturation of the spectroscopy
amplifier and the pile-up circuitry can occur. In this case, a single cosmic-ray event
can be thought of as an event that is self piled-up. Although the pile-up circuitry does
generally fire, these events represent a completely different process than the case of
traditional pile-up described above where two or more events must be recorded in the
same detector within a short time interval τp . Given a constant cosmic-ray detector
rate C, and the probability that a cosmic ray fires the pile-up circuitry ǫc , then the
number of cosmic-ray piled-up and not piled-up events observed in each time bin are
100
given by,
pi = ǫc Ctb ,
(3.41)
ni = (1 − ǫc )Ctb ,
(3.42)
where tb is the length of the time bin. In the case of ǫc = 0, cosmic-ray events would
not trigger the pile-up circuitry and would thus be recorded as not piled-up events.
This would simply be a small addition to the constant background rate B that has
already been incorporated into the pile-up correction method. In the limit where
ǫc = 1, all cosmic-ray events cause the pile-up circuitry to register a piled-up event.
Cosmic-ray self pile-up is an independent process from true detector pulse pile-up,
and can therefore be included in the detected probability of pile-up function through
the addition of a single term,
− ln2
t
a
x(t) = a1 e
2
+ a′3 ,
P ′ (t) = a6 (1 − e−(2−a4 )x [ea4 x + a5 (1 − a4 )x]) +
a7
,
x
(3.43)
where a7 = ǫc Cτp is the dimensionless constant associated with the self pile-up rate
of cosmic-ray events. The inclusion of two constant background components (background and cosmic rays) leads to an infinite covariance in the fit parameters a3 and a7
that must therefore be combined into a single parameter a′3 = Bτp + Cτp . This small
increase in the constant background parameter increases the probability of pile-up in
the first term of Eqn. 3.43 and accounts for the possibility that cosmic-ray events, like
normal events, can cause pile-up of other triggers within the pile-up time interval τp .
The second term in Eqn. 3.43 is due exclusively to the self pile-up nature of the cosmic
rays. The denominator in this term represents the total number of triggers ni + pi
which can be written as the total detector rate x because the factor that accounts for
the dead-time correction will cancel in the ratio.
101
Cosmic rays were added to the Monte Carlo simulation under the assumption
that they have the ability to be piled-up (if not already) and can also cause pileup of the single-exponential decay data and constant-background components. The
final assumption in the simulation was that if a not piled-up cosmic ray was recorded
(this is possible when ǫc < 1.0), the energy associated with this event would not
appear in the γ-ray-gated data because these events saturate the ADC. A sample
of a simulated probability of pile-up distribution and corresponding fit using the
function in Eqn. 3.43 is presented in Fig. 3.17. In this simulation the input values
were the same as those above (A = 6.7× 104 s−1 , T1/2 = 1.0 s, B = 1.0 × 103 s−1 ,
τp = 10 µs, τd = 30 µs). A very large cosmic-ray rate and cosmic pile-up efficiency of
C = 5.0 × 102 s−1 and ǫc = 1.0 were introduced to explore the limiting behaviour of
cosmic-ray pile-up. The most striking feature in the probability of pile-up spectrum
is the creation of a minimum, a feature that is indeed observed experimentally (see
Chapter 4). At early times when the detector rate is due primarily to the decaying
exponential component (i.e. x ≈ a1 ) the cosmic-ray term in the probability of pile-up
is approximately
a7
a1
and is negligible (assuming a1 ≫ a7 ) compared to the multiple-
event pile up described by the first term in Eqn. 3.43. At later times when the
exponential has decayed away to negligible levels, the self pile-up cosmic term is
given by
a7
a′3
which dominates over the probability of pile-up from the multiple γ-ray
pile-up term. As demonstrated in Fig. 3.17, this minimum is exactly described by
the function in Eqn. 3.43, when the contributions from both the multiple-event pileup and the cosmic-ray self pile-up terms are included. The minimum will only be
observed if the probability of multiple γ-ray pile-up from the background rate is less
than the contribution of the cosmic-ray self pile-up term to the probability of pileup distribution. Assuming a low background rate, the condition for the cosmic-ray
102
Probability of pile-up: pi / (ni + pi)
Simulated Data
Best-fit result
Cosmic pile-up term
Pile-up of 2 or more events
0.5
0.4
0.3
Best-fit parameters:
Aτp = a1 = 0.668 ± 0.023
0.2
0.1
T1/2
= a2 = 1.01 ± 0.03 s
Bτp
= a3 = 0.009 ± 0.001
Cεcτp = a7 = 0.005 ± 0.001
0.0
0
5
10
20
15
25
Time (s)
Figure 3.17: Simulated probability of pile-up data (circles) and best-fit curve (solid
line, Eqn. 3.43) when saturating cosmic-ray events are included. The contributions
from cosmic-ray self pile-up (dot dashed) must be separated from the multiple γ-ray
pile-up contributions (dashed) as only the latter applies to the γ-ray-gated photopeak
data.
minimum to be observed can be expressed as Bτp <
set above, Bτp = 0.01, Cτp = 0.005, and thus
C
B+C
C
.
B+C
For the simulated data
= 0.33, and the minimum is
well defined. For the experimental data discussed in Chapter 4 it will be shown that
this minimum can be created or removed by simply adding or removing a calibration
source to the array to intentionally alter the background rate B.
With the assumption that the ADC is always saturated by cosmic-ray events, there
would never be cosmic-ray background under the γ-ray photopeak of interest and
thus the cosmic-ray minimum in the probability of pile-up does not apply to the γ-ray
photopeak-gated data gi . A γ ray from the decay of interest can be piled-up and hence
103
Probability of Pile-up
Counts recorded in γ gate
10000
1000
0.6
Simulated data
Best fit:
a7 = 0.005
Corrected fit: a7 = 0.000
0.5
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4 5 6
Time (s)
7
8
9
100
Uncorrected decay data
Corrected with a7 = 0.005
Corrected with a7 = 0.000
Best fit (a7 = 0.005): T1/2 = 1.0448 ± 0.0047 s
Best fit (a7 = 0.000): T1/2 = 0.9955 ± 0.0042 s
10
0
1
2
3
4
5
6
7
8
9
10
Time (s)
Figure 3.18: Simulated decay-curve data that is uncorrected (squares), dead-time and
pile-up corrected with ǫc Cτp = 0.005 (triangles), and dead-time and pile-up corrected
with ǫc Cτp = 0.0 (circles). (Inset) Simulated probability of pile-up data (circles) with
best fit (solid line) and cosmic-ray subtracted corrected curve (dashed line). The
true simulated half-life for this data set was T1/2 = 0.9975 ± 0.0032 s and is only
reproduced after the saturating cosmic-ray events are properly accounted for (dashed
lines). Note that the bias results primarily from an overcorrection applied to the
background which leads to a increase in the deduced half-life.
lost from the gate by a cosmic ray, but this possibility is included in the increase to the
background term a′3 . Because cosmic-ray self pile-up described by the second term in
Eqn. 3.43 does not affect the gated decay data, its contribution to the total probability
of pile-up must be subtracted before a pile-up correction is applied to the γ-ray gated
decay data gi . For the simulated data set shown in Fig. 3.17 the half-life obtained following a pile-up correction using only the multiple-event pile-up term of Eqn. 3.43 and
removing the cosmic-ray self pile-up contribution is T1/2 = 0.9955 ± 0.0042 s, a result
104
that agrees with the true value T1/2 = 0.9975 ± 0.0032 s for this simulated data set.
A correction to the decay-curve data that does not remove the cosmic-ray self pile-up
component yielded T1/2 = 1.0448 ± 0.0047 s a result that is 10 standard deviations
or 4.9% too large. This discrepancy results from the fact that the inclusion of the
cosmic-ray self pile-up term over-corrects the background (35 % probability of pile-up
correction applied for t > 10 s) while providing a smaller correction for times when
there is still significant exponential decay present. This combination leads to a reduction in the corrected slope of the γ-gated decay curve, as demonstrated in Fig. 3.18,
which results in a larger deduced half-life. Removing the cosmic-ray contribution from
the total fit to the probability of pile-up is performed via the same procedure that
was employed in the pile-up time resolution correction in Sec. 3.4.1. The observed
probability of pile-up data must be fit with the total function of Eqn. 3.43, while the
proper correction to be applied to the γ-ray gated decay data is this function with
the parameters a4 = a7 = 0. Fits to simulated data that varied the cosmic-ray event
rate and cosmic pile-up efficiency confirmed that this procedure correctly accounts for
cosmic-ray saturation events and does not introduce any bias to the deduced half-life
well below the level of 0.05%.
3.5
Summary
The Monte Carlo simulation discussed in this chapter has confirmed the pile-up equations derived in Sec. 3.1 of this work and has been invaluable in providing insight to
the refinements to the pile-up method outlined in Sec. 3.4. It has been shown that
pile-up time resolution, trigger and pile-up energy thresholds, and cosmic-ray self pileup must be fully understood in order to obtain an unbiased estimate of the half-life
105
at the level of precision (0.05%) required for the superallowed Fermi β decay program. Although this level of precision has been enjoyed in β-counting experiments
for decades, the γ-ray photopeak counting method can be superior to β counting in
situations where the signal-to-background levels are low and hence decay selectivity
must be relied upon to cleanly extract the decay of interest. The technique presented
here is not limited to γ-ray counting following a superallowed Fermi β decay but is
widely applicable to all half-life determinations that utilize a γ-ray photopeak energy
to select the decays of a particular isotope. As discussed in Chapter 4, this newly developed method was tested using the β decay of 26 Na to 26 Mg in an experiment that,
for the first time, was able to deduce a β decay half-life to 0.05% via the technique
of γ-ray photopeak counting.
106
Chapter 4
Half-life of 26Na
The methodology described in Chapter 3 for performing the pile-up correction and
fitting the subsequent dead-time and pile-up corrected decay data to determine β decay half-lives was tested with experimental data from the 8π γ-ray spectrometer using
radioactive beams of
26
Na. While not a superallowed decay,
an attractive test case of the methodology because beams of
26
Na β − decay provided
26
Na were readily avail-
able from the existing ISAC surface ionization source, the daughter
26
Mg is stable,
and 100% of β decays are followed by γ-ray emission. Although 84 γ rays from
20 excited states in
26
Mg are known to follow
26
Na β − decay [88], nearly 99% of all
β decays yield the 1809 keV γ-ray from the first excited state to the ground state in
the daughter
26
Mg. A simplified
26
Na β decay level scheme showing the dominant
γ-ray transition at 1809 keV in 26 Mg is shown in Fig. 4.1. In addition, the half-life of
26
Na, T1/2 = 1.07128 ± 0.00025 s [88], was recently determined through a β counting
experiment to a level of precision (0.023%) in excess of the 0.05% goal of this work.
In this experiment, 30 µA of 500 MeV protons from the TRIUMF main cyclotron
impinged a 21.8 g/cm2 Ta target. The mass-separated 26 Na radioactive ion beam was
delivered to the 8π spectrometer (see Chapter 2 Sec. 2.2) at a rate of ∼ 106 ions/s.
107
26
Na
1.07128 0.00025 s
3
b
12.20 %
19 Excited States 2.9 - 7.8 MeV
87.80 %
}
1809
2+
~ 99 %
0+
26
Mg
Figure 4.1: Simplifed 26 Na β − decay scheme to the stable daughter 26 Mg. A total of
84 γ rays from 20 excited states [88] are known to follow 26 Na β decay, however, 99%
of all β decays yield the 1809 keV γ-ray.
A tape cycling time of 1-1-30-1 was used which corresponds to counting background
activity for 1.0 s before turning the beam on for 1.0 s to build up a sample of
on the tape. The beam was then turned off and the decay of
26
26
Na
Na was counted for
30.0 s (∼ 30 half-lives). This was followed by a 1.0 s delay while the tape was moved,
bringing a fresh section of tape into the array while simultaneously removing any
long-lived contaminant activities that may have been present in the beam. A “cycle”
therefore lasted ≈ 33 s in duration and recorded the grow-in and decay of a single
sample of 26 Na. Approximately 200 cycles were recorded under the same experimental
108
7
1808.7
2510.5
2524.1
2541.6
Single Escape (1808.7)
+ -
511.0, e e
Counts per 1 keV
5
10
1128.9
6
10
Double Escape (1808.7)
10
4
10
3
10
2
10
1
10
300
600
900
1200
1500
1800
2100
2400
2100
2400
2700
3000
Eγ (keV)
7
1808.7
2598.5
+ -
511.0, e e
Counts per 1 keV
5
10
1238.3
6
10
846.7
10
4
10
3
10
2
10
1
10
300
600
900
1200
1500
1800
2700
3000
Eγ (keV)
Figure 4.2: (Upper panel) Not piled-up γ-ray singles spectra from all 20 detectors
in the 8π spectrometer following the β decay of 26 Na. (Lower panel) To purposely
increase the amount of pile-up in this experiment the event rate was increased by
introducing a 56 Co source into the array for 8 of the 14 experimental runs.
109
Table 4.1: Run-by-run summary of experimental conditions employed in the 26 Na
half-life measurement. The variable dead-time setting had an average value of 25.0 µs
and the 56 Co source was either present “y” or not present “n” in the array. Following
Run 5 the CFD trigger-energy thresholds were increased to reduce the fraction of
dead events.
Run
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Shaping Time
(µs)
6.0
6.0
6.0
6.0
6.0
6.0
2.0
2.0
2.0
2.0
1.0
1.0
1.0
1.0
Dead Time
(µs)
64.0
40.0
40.0
40.0
Variable
Variable
Variable
40.0
40.0
27.0
Variable
Variable
40.0
27.0
CFD Threshold
low
low
low
low
low
high
high
high
high
high
high
high
high
high
56
Co
n
y
y
y
n
n
n
y
y
n
y
y
n
y
Cycles
328
131
133
70
326
259
274
244
129
309
225
60
280
290
conditions which constituted a single “run”. Between runs, experimental conditions
were modified including varying the amplifier shaping times between 1.0, 2.0, and
6.0 µs, the dead-time lengths between variable (measured event-by-event), 27, 40,
and 64 µs, and the HPGe CFD thresholds from a “low” to a “high” setting. In
addition, a
56
Co source was periodically introduced into the array throughout the
experiment to purposely increase the background rate and hence the probability of
pile-up. A total of 14 runs (8 with the
56
Co source) consisting of 3058 cycles, were
obtained under a variety of experimental conditions. A run-by-run summary of the
experimental conditions used throughout the experiment is presented in Table 4.1.
For each cycle, the maximum full-array trigger rate at t = 0 was ∼ 10 kHz with
110
maximum dead-time and pile-up probabilities in the ranges of 25-60% and 1-7%,
respectively. The maximum 60% dead fraction occurred in only a single run (run 1)
where the longest non-extendible dead-time of 64 µs was used. Pile-up probabilities
of 7% occurred in only the first 6 runs (runs 1-6) when the amplifier shaping times
were 6 µs and decreased to 1% when these shaping times were reduced to 1 and 2 µs
in the remaining 8 runs (runs 7-14) (see below Fig. 4.7).
Linear calibration of the HPGe energies were performed on a run-by-run basis
using the 846.8- and 3253.4-keV γ-rays from the
56
Co source that was randomly
inserted into the array throughout the experiment. Germanium γ-ray singles spectra
obtained from the sum of all 20 HPGe detectors for runs with and without the
56
Co
source are presented in Fig. 4.2.
4.1
Compton Suppression
If a single γ ray enters a single HPGe detector, deposits a fraction of its energy in
the crystal, and scatters out, this process is rate independent and would therefore
not bias the measurement of the nuclear half-life. Suppression of these events using
the BGO that surrounds the HPGe detectors would therefore improve the peak-tobackground. With Compton-suppression there is, however, the possibility to “false
veto”, that is, the first γ ray deposits its energy in the HPGe crystal and a random
coincidence strikes the suppression shield within the coincidence time window leading
to the rejection of a good event. Because the probability to false-veto good events
is rate dependent it would bias a half-life measurement. In the 8π spectrometer this
process is minimized by the addition of the tungsten heavy-metal collimators that
prevent γ rays from directly striking the BGO suppression shields. Nonetheless, in
111
order to eliminate the possibility of false suppression the experiment was run in an
“unsuppressed” mode where no Compton-suppression was implemented in hardware.
The BGO time information was recorded such that Compton-suppression could be
implemented in software in order to fully explore false-veto effects. It was determined
that the 26 Na half-life obtained when the software suppression was applied was indeed
biased to a larger value due to the false-veto effect (see Sec. 4.4.1). Unless otherwise
stated all results are presented using the unsuppressed data.
4.2
Dead-Time Corrections
In many high-precision half-life determinations (see Ref. [88] for example) constant
and non-extendible dead-times per-event are chosen to be much longer than the series
dead times from the individual electronic modules in the system. One then relies on
a calibration, such as the source-plus-pulser method [89] (see Chapter 6), to measure
the average dead time per-event before the corrections for dead-time losses can be
applied to the experimental data. With the 8π spectrometer, rather than calibrating
the dead time periodically throughout the experiment these dead times are measured
on an event-by-event basis using the γ-ray singles trigger time stamping information
provided by the ULM. Calculation of the dead-time for each event does not rely on an
average calculation of the dead time and, while a non-extendible dead time τd follows
every trigger event, one is not limited to using fixed and constant dead times. The
total dead-time fraction in the ith time bin Di was calculated from the event-by-event
ULM information and stored in cycle number versus bin number matrices with time
bins of tb = 0.1 s for each experimental run. For processing and read out of signals
from HPGe detectors the dead times are typically quite long (25 to 64 µs used in this
112
work) due to the fact that the pulse shaping time to obtain good energy resolution is
much longer than in β-counting experiments. However, because the dead-time is nonextendible the method for the cycle-by-cycle correction of dead-time losses is identical
to the procedures adopted in β counting [88, 90] and presented in Appendix A of this
thesis.
In the
26
Na experiment, 5 runs were obtained with non-extendible dead times in
a “variable” mode, where the event-by-event dead-time was decided by the system
itself and was based upon the total time required by the aquisition to process and read
out each event. The average value of the dead time in this mode was ∼ 25 µs. The
remaining 9 runs in this experiment were obtained using fixed dead-times per event
using the arbitrary values of 27, 40, and 64 µs. Although these runs were obtained using fixed dead times, the nominal values were not used explicitly as the cycle-by-cycle
dead-time corrections were again calculated using the event-by-event times recorded
by the ULM. It was confirmed through the source-plus-pulser technique [89] that the
measured dead times were in excellent agreement with those that were calculated
using the event-by-event time stamping information. Maximum dead-time fractions
in the
26
Na experiment were typically in the range Di = 25-60% at t = 0 when the
beam was turned off.
4.2.1
Dead-Time Corrections for Multi-Detector Arrays
For multi-detector arrays with N total detectors a γ-ray trigger event recorded in
the ith detector is followed by a non-extendible dead time τd . A subsequent γ-ray
interaction in the same detector within τd will not be recorded as a trigger due to
the fixed dead time and may result in pile-up of the trigger itself (discussed in detail
in Chapter 3). The remaining N−1 detectors remain live for a fraction of τd and,
113
8
10
∆t = 64 ns
Trigger Events
7
10
6
Counts per 4 ns
10
5
10
4
10
3
10
2
10
Random Coincidences
1
10
0
10-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Time of recorded HPGe event (µs)
Figure 4.3: Sum HPGe time spectrum from all 20 detectors in the 8π spectrometer
for a single run with the 56 Co source in the array. The narrow 64.0 ns gate on the
prompt time peak removes most of the random coincidences that were recorded in
the remaining 19 detectors. A correction for these remaining random coincidences
underneath the prompt time peak is performed in Sec. 4.2.
although they do not lead to triggers, these detectors are able to process “coincident”
γ-ray events. The possibility therefore exists to accept random coincidences in the
remaining detectors which would systematically bias the deduced half-life to a smaller
value because the dead-time is assumed to be applied to all N detectors. One can
greatly reduce these random coincidences by applying a narrow coincidence time gate
∆t to the HPGe detector times. An example using the sum of the HPGe time spectra
from the 20 detectors of the 8π spectrometer is shown in Fig. 4.3. Even with a
narrow gate on the HPGe times (64.0 ns in Fig. 4.3) the calculated event-by-event
dead time τd based on the ULM times is still too large because of the additional
114
random coincidences that are accepted in the background underneath the prompt
time peak. This effect can be accounted for by using a corrected dead-time per-event
τd′ after the application of the prompt HPGe time gate given by:
τd′ = τd −
N −1
∆t.
N
(4.1)
For the example shown in Fig. 4.3 a non-extendible dead time τd = 40.0 µs was
employed and the width of the HPGe time gate was ∆t = 64.0 ns. For the 20
detectors of the 8π, the correction to the dead time for this run is therefore 0.15%.
Although this is a small correction to the overall dead-time correction, it certainly
cannot be neglected in experiments with multi-detector arrays if the half-life is to be
deduced to the level of 0.05% necessary in high-precision work.
4.3
Pile-up Probability Analysis
Events in the ith time bin that were not piled-up ni were taken to be all trigger events
(events satisfying the prompt HPGe time gate as discussed in Sec. 4.2) in all detectors
that did not have a pile-up time present in the TDC of the corresponding detector. A
subset of the not piled-up data gi was generated consisting of all of the not piled-up
events that simultaneously satisfied the γ-ray energy gate applied to the 1809-keV
transition in the daughter
26
Mg. Events that were piled-up were stored in a third
data set pi and included all triggers that had at least one entry in the pile-up TDC
of the same detector that the γ ray was registered in. These three data sets were
separately sorted into cycle number versus bin number matrices using bin times of
tb = 0.1 s.
With the 56 Co source in the array a high-statistics pile-up time spectrum obtained
from the sum of all 20 detectors is shown in Fig. 4.4. For each run, the pile-up time
115
5
10
τr = 0.5 µs
4
τp = 17.0 µs
Counts per 4 ns
10
εr = τr/τp = 0.03
3
10
2
10
1
10
0
10
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
Time of recorded pile-up (µs)
Figure 4.4: Sum pile-up time spectra from all 20 detectors in the 8π spectrometer for
a single run with the 56 Co source in the array. The pile-up time resolution is clearly
visible as the loss of counts between the pre-pile-up peak at t = 0.0 µs and ≈ 0.5 µs
of the post-pile-up continuum.
spectra were used to determine the size of the pile-up time resolution correction to
be applied to the observed piled-up data (see Sec. 3.4.1). In the example shown,
the amplifier shaping times were set to the maximum value of 6.0 µs to purposely
increase pile-up, and post-piled-up events were recorded out to ∼ 17.0 µs. Note that
not all pile-up signals at such long times actually prevent the determination of the
photopeak energy of the γ ray which may have already been converted in the peak
sensing ADC’s. To be cautious, all events with pile-up hits out to the full range of
17.0 µs were removed and subsequently corrected for by the procedures outlined in
Chapter 3 of this thesis. This was also true of the few events that were recorded with
116
negative pile-up times as seen in Fig. 4.4. These events are pile-up signals that came
after the TDC’s were cleared from the previous event but before the master trigger of
the current event. These events occurred at times that are larger than the dead time of
the previous trigger and it was assumed that the energy determination of the previous
trigger was therefore not affected by these events. These events do, however, treat
the current trigger as piled-up and the analysis removes these photopeaks and then
corrects for them. In the example of Fig. 4.4 the pile-up time resolution of τr = 0.5 µs
yields ǫr =
0.5
17.0
≈ 0.03. When the amplifier shaping times were reduced to the more
reasonable values of 2.0 µs and 1.0 µs, the pile-up times extended in each case to
≈ 7.0 µs yielding ǫr ≈ 0.07 which is an order of magnitude smaller than the Monte
Carlo simulation example of ǫr = 0.8 that was presented in Sec. 3.4.1 of this thesis.
The data were re-sorted with the piled-up events occurring within the resolution
0.5 µs after the pre-pile-up peak being removed from the probability of pile-up data
and inserted into the not piled-up data to match the pile-up resolution correction of
Eqn. 3.25 that assumes no piled-up events can be recorded in this window.
The use of an average pile-up probability for each run as opposed to fitting the
probability of pile-up on a cycle-by-cycle basis was necessitated due to the lack of
statistics in the probability of pile-up spectra for each cycle. For each run a single
probability of pile-up distribution was generated according to,
Pi =
N
P
pi,j
j=1
N
P
,
(4.2)
(ni,j + pi,j )
j=1
where the index j refers to the cycle number, i indicates the particular time bin and
the quantity N is the total number of cycles in the run. In the limit that there
is no time dependence in the beam rate, an average pile-up correction for each run
is equivalent to a cycle-by-cycle pile-up correction. A plot of the total number of
117
4000
3800
3600
Counts per cycle
3400
3200
3000
2800
2600
2400
2200
2000
1800
3200-3800 counts/cycle acceptance window
1600
0
40
80
120
160
200
240
280
320
Cycle Number
Figure 4.5: Number of 1809-keV photopeaks recorded per-cycle for a single run (run 1,
no 56 Co source). All cycles that fell above or below the indicated acceptance window
were rejected from the analysis.
triggers satisfying the 1809-keV γ-ray energy gate in each cycle for a single run is
presented in Fig. 4.5 and demonstrates that the beam rate did not have significant
time dependence over the course of each experimental run lasting ≈ 2 hours. To
better control constant beam delivery, a window was set for each run that accepted
or rejected cycles whose total counts fell above and below the prescribed values. In
this way cycles where the beam had been turned off (cycles 11-15 in Fig. 4.5 for
example) were removed from the analysis and would not skew the average probability
of pile-up correction.
Plots of the average detected probability of pile-up are shown in Fig. 4.6 for cases
in which the 56 Co was inserted into and removed from the 8π spectrometer. The data
118
shown in Fig. 4.6 were obtained using 131 cycles with the
56
Co source and 328 cycles
without. For the runs without the 56 Co source, the cosmic-ray minimum in the pile-up
probability distribution is clearly visible. When the
56
Co was inserted into the array
the cosmic-ray minimum is removed as described in Sec. 3.4.4. For each of the 14 runs
obtained in this experiment, a probability of pile-up distribution was generated and
fit using the function in Eqn. 3.43 with the pile-up time resolution parameter a4 = ǫr
fixed at the value determined from the corresponding pile-up time spectrum, the above
CFD trigger-energy threshold fraction a5 = α and pile-up energy threshold fraction
a6 = ǫp fixed at the values determined from the γ-ray spectra, and the cosmic-ray self
pile-up term a7 = ǫc Cτp treated as a free parameter. In the runs without the
56
Co
source the depth and shape of the cosmic-ray minimum were sufficient to constrain
the a7 cosmic-ray self-pile-up parameter. When the 56 Co was inserted and the cosmicray minimum removed, this parameter was fixed at the value determined from the
runs without the
56
Co source because of the resulting large covariance between the
background and cosmic-ray parameters.
Both of the probability of pile-up spectra shown in Fig. 4.6 were obtained at the
unusually large shaping time of 6.0 µs and these data therefore represent the runs for
which the probability of pile-up was greatest. In the case of the
56
Co run in Fig. 4.6
(lower panel) the probability of pile-up reached a maximum of ≈ 7.0% in the earliest
time bins. The maximum correction that was applied to the γ-ray gated decay-curve
data following the corrections for the pile-up time resolution, removal of the cosmic
minimum, and the CFD and pile-up TDC energy thresholds (see Sec. 3.4) is plotted
versus run number in Fig. 4.7. Reduction of the amplifier shaping times from 6.0 µs
to 2.0 µs represents the largest decrease in the probability of pile-up and did not
result in a significant reduction in the energy resolution of the HPGe detectors. The
119
0.07
Probability of pile-up: pi / (ni + pi)
26
56
Na data, no Co source
Best-fit result
Cosmic pile-up term
Pile-up of 2 or more events
0.06
0.05
0.04
Best-fit parameters:
Aτp = a1 = 0.136 ± 0.005
T1/2 = a2 = 1.01 ± 0.02 s
−4
Bτp = a3 = (2.8 ± 0.3)×10
εr
= a4 = 0.03 (fixed)
α
= a5 = 0.957 (fixed)
εp
= a6 = 0.995 (fixed)
−5
Cεcτp = a7 = (1.2 ± 0.1)×10
0.03
0.02
0.01
0.00
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
Time (s)
0.08
26
0.07
Probability of pile-up: pi / (ni + pi)
56
Na data, Co source
Best-fit result
Cosmic pile-up term
Pile-up of 2 or more events
0.06
Best-fit parameters:
Aτp = a1 = 0.142 ± 0.002
T1/2 = a2 = 1.01 ± 0.03 s
−3
Bτp = a3 = (11.5 ± 0.5)×10
εr
= a4 = 0.03 (fixed)
α
= a5 = 0.919 (fixed)
εp
= a6 = 0.995 (fixed)
−5
Cεcτp = a7 = 1.2×10 (fixed)
0.05
0.04
0.03
0.02
0.01
0.00
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
Time (s)
Figure 4.6: (Upper panel) A typical run at 6.0 µs shaping where the 56 Co source was
not in the array and the contribution from the cosmic rays dominates the observed
probability of pile-up spectrum at late times. (Lower panel) A typical run at 6.0 µs
shaping with the 56 Co source installed inside the array. The cosmic rays are dominated
by the 56 Co background and the cosmic minimum is not observed.
120
0.09
CFD = Low
Maximum probability of pile-up
0.08
CFD = High
Shape = 6.0 µs
Shape = 2.0 µs
Shape = 1.0 µs
0.07
0.06
0.05
0.04
0.03
56
Co source removed
Co source inserted
CFD level change
Shape time change
0.02
56
0.01
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Run Number
Figure 4.7: Run-by-run plot of the maximum probability of pile-up taken at the start
of the decay for each run (t = beam off) in the 26 Na experiment. The largest decrease
in the probability of pile-up is due to the decrease in the shaping time from 6.0 µs to
2.0 µs. The influence of the 56 Co source is visible as a small increase in the probability
of pile-up for each shaping time subset of data.
effect of the
56
Co source can be seen in Fig. 4.7 as a small increase to the observed
probability of pile up when the source was inserted into the 8π array. A comparison of
Runs 5 and 6 in Fig. 4.7 obtained before and after the raising of the CFD thresholds
to ≈ 500 keV with all other electronic settings equal, demonstrates that at the level
of statistical uncertainty in the pile-up probability distributions, this experiment was
insensitive to the expected slight increase in the probability of pile-up discussed in
Sec. 3.4.2.
121
4.4
Half-life Analysis
Following the data pre-selection criteria and the run-by-run fits to the average probability of pile-up data, the pile-up and dead-time corrections were applied to the
not piled-up and unsuppressed data gated on the 1809-keV photopeak cycle-by-cycle
using the procedure described above and in Appendix A. These decay-curve data
were summed for each run and then fit to a function containing the exponential decays of
26
Na and
26m
Al plus a free constant background parameter. The
small contamination that was present at the level of ∼ 10−3 (relative to
A = 26 beam. While
26m
26m
26
Al was a
Na) in the
Al β + decay (T1/2 = 6.3450 ± 0.0019 s [1]) does not yield
γ-ray radiation, it would be possible for the 1809-keV γ-ray gate to be contaminated
by bremsstrahlung or in-flight positron annihilation from this decay. Taking a gate
on the background just above the 1809-keV photopeak for the runs without the
source the
26m
56
Co
Al contamination inside the 1809-keV gate was deduced to be only
3.8 ± 3.6 ppm at the time the beam was turned off. In the fits to the 1809-keV γ-ray
gated decay data, the
of the
26
26m
Al bremsstrahlung intensity was therefore fixed at 3.8 ppm
Na intensity at t = 0 and its half-life was fixed at 6.3450 s. Uncertainties
associated with these values are considered below in Sec. 4.4.2. A powerful demonstration of the improved peak-to-background that is possible using the technique of
γ-ray photopeak counting is shown in Fig. 4.8 where the
26m
Al contaminant in the
resulting decay spectrum is reduced by nearly two orders of magnitude over a previous
β counting experiment [88].
A plot of a typical dead-time and pile-up corrected decay curve showing the deduced half-life of
26
Na with its statistical uncertainty T1/2 = 1.0706 ± 0.0012 s is
presented in Fig. 4.9 for a single run (Run 11) comprised of 225 cycles. The pile-up
122
6
10
5
10
β counting
26
4
T1/2( Na) = 1.0716(7) s
2
χ /ν = 1.06
Counts
10
3
10
26m
26
Al/ Na (t = 0) ≈ 0.3 %
γ-ray counting
2
10
26
T1/2( Na) = 1.0706(12) s
2
χ /ν = 1.08
1
10
26m
26
Al/ Na (t = 0) ≈ 0.004 %
0
10 0
5
10
15
20
25
Time (s)
Figure 4.8: Comparison of the 26 Na decay curves obtained from single runs via a
direct β counting half-life determination at GPS and a γ-ray photopeak counting
determination at the 8π spectrometer following a γ-ray gate on the 1809 keV transition in the daughter 26 Mg. Isobaric contamination from 26m Al (T1/2 = 6.345 s) is
suppressed in the γ-ray counting experiment by nearly two orders of magnitude.
correction applied to these data was described using the central values of the fit parameters Aτp , Bτp , and T1/2 following the fit to the probability of pile-up spectra.
The uncertainties associated with the pile-up fit parameters were explored on a runby-run basis by varying these parameters within their ± 1σ limits and observing the
effect each had independently on the deduced half-life of
26
Na. The change in the
measured half-life was then used to assign an overall uncertainty due to the pile-up
correction for each run. The statistical uncertainty was then combined in quadrature
with the uncertainty deduced from the pile-up fitting procedure. A plot of the half-life
obtained for each run versus the corresponding run number is given in Fig. 4.10 and
123
5
Counts per 1 keV
Counts in 1809-keV γ-ray Gate
10
4
10
3
10
5
10
4
10
3
10
1790
1800
2
10
1
1810
Eγ (keV)
1820
22
26
1830
26
10
T1/2( Na) = 1.0706 (12) s
2
χ /ν = 1.08
0
10
0
2
4
6
8
10
12
14
16
18
20
24
28
30
32
Time (s)
Figure 4.9: Typical pile-up and dead-time corrected decay curve obtained from a
single run (run 14, no 56 Co source, 290 cycles) following a gate on the 1809-keV
transition in 26 Mg (inset). The decay spans nearly 15 half-lives before the background
is reached demonstrating the power of the γ-ray counting technique.
includes the small increases in the uncertainties resulting from the pile-up correction
(typically < 5% of the statistical uncertainty). The half-life of
26
Na deduced for each
run is presented in Fig. 4.10 and Table 4.2. The overall half-life of
26
Na, obtained
from a weighted average of these 14 run-by-run values, is T1/2 = 1.07167 ± 0.00042 s,
a result that agrees with the value T1/2 = 1.07128 ± 0.00025 s obtained in a highprecision β-counting experiment [88]. Treating each of the 14 runs as independent
measurements of the 26 Na half-life results in a reduced χ2 value of 1.70. According to
the method of the Particle Data Group [4], when the reduced χ2 value exceeds unity,
the square root of this value must be included as an estimate of the systematic
124
1.082
26
T1/2( Na) = 1.07167 ± 0.00042 s
1.080
2
χ /ν = 1.70
1.078
Half-life (s)
1.076
1.074
1.072
1.070
1.068
1.066
1.064
1.062
1
2
3
4
5
7
6
8
9
10
11
12
13
14
Run Number
Figure 4.10: Half-life of 26 Na with statistical errors versus run number for all 14 runs
in this analysis. A weighted average of T1/2 = 1.07167 ± 0.00042 s is deduced from
these data where the uncertainty is statistical and does not reflect any systematic
effects.
Table 4.2: Half-life of 26 Na deduced on a run-by-run basis with statistical errors and
resulting reduced χ2 values for all 14 runs in this analysis. A weighted average of
T1/2 = 1.07167 ± 0.00042 s is deduced from these data where the uncertainty is
statistical and does not reflect any systematic effects.
Run
Number
1
2
3
4
5
6
7
stat
T1/2 σT1/2
(s)
1.07067(132)
1.07571(223)
1.07707(240)
1.07179(311)
1.07175(129)
1.07345(159)
1.07034(130)
2
χ /ν
1.02
1.09
0.75
0.94
0.97
1.01
1.01
Run
Number
8
9
10
11
12
13
14
125
stat
T1/2 σT1/2
(s)
1.07015(152)
1.06879(200)
1.07467(138)
1.07243(149)
1.06826(285)
1.07059(127)
1.07056(132)
χ2 /ν
1.10
0.87
1.00
1.15
0.99
0.98
1.08
uncertainty. The half-life of 26 Na is therefore T1/2 = 1.07167(42)(35) s where the first
uncertainty is statistical and the second systematic.
4.4.1
Tests of the Pile-up Method
The half-life of
26
Na and its statistical uncertainty, T1/2 = 1.07167 ± 0.00042 s,
were obtained from an “optimal analysis” method (hereafter denoted method 6) that
performed the full pile-up correction including all of the refinements to the pile-up
technique in order to properly account for the effects of the pile-up time resolution,
the influence of cosmic rays, and the CFD and pile-up detection energy thresholds.
This was compared with several other correction methods that deliberately ignored
some of these effects in order to explore the size and accuracy of each of the corrections involved. For this comparison the half-life of
26
Na was extracted from the
same data set in 5 additional ways. The first method (method 1) ignored the pileup correction completely and only applied a dead-time correction to the 1809-keV
photopeak-gated decay data. The second method (method 2) performed the pile-up
correction using the bin-by-bin probability of pile-up data. This method therefore did
not fit the probability of pile-up data to the analytic function of Eqn. 3.43, which is
equivalent to not performing the pile-up time resolution, cosmic-minimum removal,
and energy threshold corrections discussed in Sec. 3.4. A third method (method 3)
fit the probability of pile-up data to the correct analytic function, properly removed
the cosmic minimum, and accounted for the CFD and pile-up energy thresholds, but
did not make the correction for the pile-up time resolution discussed in Sec. 3.4.1.
In the optimal analysis of method 6 a pile-up efficiency was determined from each
of the 14 γ-ray spectra and the average of these results ǫp = 0.995 was used to correct each run for low-energy pile-up energy threshold losses. In the next method
126
Table 4.3: Comparison of the half-life of 26 Na obtained as each correction to the
pile-up method was applied to the raw γ-ray gated data. All methods listed here
include the correction for dead-time losses and are compared to the complete pile-up
correction of method 6 both in terms of percentage differences (%) and statistical
standard deviations (σ) with a + indicating that the value obtained is greater than
the value obtained from the complete correction. A total correction for detector pulse
pile-up of 0.940% or 27.2σ was applied in this work in which the half-life of 26 Na was
ultimately determined to ± 0.05% precision (see Sec. 4.4.2).
Comparison
# Description
T1/2 σTstat
1/2
to method 6
1
2
3
4
5
6
7
8
No pile-up correction
Bin-by-bin pile-up correction
Analytic, no τr correction
Analytic, no ǫp correction
Analytic, no α correction
Analytic, complete
Analytic, complete (suppressed)
Analytic, complete (no ∆t , Sec.4.2)
(s)
(%)
(σ)
1.08174(37)
1.07389(37)
1.07205(42)
1.07171(42)
1.07167(42)
1.07167(42)
1.07228(42)
1.07131(42)
+0.940
+0.207
+0.035
+0.004
< +6×10−4
—
+0.057
-0.034
+27.2
+6.0
+0.9
+0.1
< +1×10−2
—
+1.5
-0.9
considered (method 4) this correction was ignored by assuming a pile-up efficiency
of unity, ǫp = 1.0. The final method considered (method 5) did not use the average CFD trigger-energy threshold α = a5 deduced from the γ-ray spectra for each
run, but instead ignored this correction and used α = 1.0 for both the low and high
CFD data. In all of these methods the event-by-event dead-time corrections were
performed so the differences between them are due solely to the differences in the
pile-up corrections that were applied.
The half-life of
26
Na deduced from method 1, the weighted average of the 14
experimental runs when no corrections for detector pulse pile-up were applied, was
T1/2 = 1.08174 ± 0.00037 s, a result that is 28.3σ (statistical standard deviations)
larger than the high-precision β counting result of Ref. [88], T1/2 = 1.07128 ± 0.00025 s.
The uncertainty of ± 0.00037 s is the statistical uncertainty from the fit procedure
127
and was not increased by any pile-up uncertainties because no corrections for pileup were applied. Using the bin-by-bin pile-up correction of method 2 the result,
T1/2 = 1.07389 ± 0.00037 s is 7.1σ larger than the β counting result. The fact that
this half-life is greater than the β result was also observed in the Monte Carlo simulation (Sec.3.4.4) and is due to the fact that cosmic-ray self pile-up does not apply
to the γ-ray photopeak gated decay data. In the analysis of method 3, ignoring the
pile-up time resolution correction led to a result T1/2 = 1.07205 ± 0.00042 s that is
1.8σ larger the β counting value. The analysis of method 4 that assumed a pile-up
efficiency of 100% led to the value T1/2 = 1.07171 ± 0.00042 s, a result that agrees
with the β-counting result at ∼ 1.0σ and is only 0.1σ larger than the optimal analysis value of method 6 that used the average value ǫp = 0.995. In the final analysis
considered (method 5) the assumption of CFD trigger-energy threshold of α = 1.0
led to no measurable change in the half-life of
26
Na deduced as the value obtained
T1/2 = 1.07167 ± 0.00042 s was identical to the optimal analysis of method 6. These
results are summarized in Table 4.3.
In Sec. 4.1 it was discussed that a potential systematic bias associated with ratedependent false-veto effects when Compton-suppression is implemented may occur.
All of the data and analysis have therefore been performed using the unsuppressed
data. However, one can test for the presence, and relative size, of this systematic by
determining the half-life of
26
Na using the suppressed data. The not-piled-up and
suppressed data nsi were generated as all events in every detector that i) satisfied the
prompt HPGe time gate, ii) did not have a corresponding time in the pile-up TDC
of that detector, and iii) did not have a corresponding time in the BGO TDC of
that detector. The not-piled-up, suppressed, and gated data gis satisfied all of these
conditions and, in addition, satisfied the γ-ray photopeak energy gate at 1809 keV.
128
The piled-up and suppressed data psi were all pile-up events that did not have an
associated BGO time in the TDC of that detector. These data sets were then analyzed using the identical method to that described above for the unsuppressed data.
Using the complete pile-up correction of method 6 on the suppressed data (denoted
method 7) the half-life of
26
s
Na deduced was T1/2
= 1.07228 ± 0.00042 s. This result
is 0.07% larger than the value deduced from the unsuppressed data and illustrates
that even with the tungsten heavy-metal collimators installed in the 8π spectrometer,
the systematic bias associated with false-veto effects is significant in high-precision
half-life measurements. This result is listed, for comparison, in Table 4.3 where this
systematic bias is larger than nearly all of the other refinements that employed the
analytic fit to the probability of pile-up data (methods 3-6). One should therefore not
apply Compton-suppression in high-precision half-life measurements with HPGe detectors due to the relatively large systematic bias associated with the rate-dependent
losses of good events due to false-suppression effects.
A final method (method 8) to correct the data employed the full pile-up correction
from method 6 on the unsuppressed γ-ray gated decay data but ignored the correction
to the dead time for remaining random coincidences underneath the prompt HPGe
time peak. This correction was discussed in Sec. 4.2.1 and is only applicable in
experiments with multi-detector arrays. Neglecting this correction led to a decrease
in the 26 Na half-life and results from the fact that the calculation of the dead time on
an event-by-event basis is too large because of these additional random coincidences.
When this correction was removed from the analysis, the resulting half-life of
26
Na
was T1/2 = 1.07131 ± 0.00042 s. In this experiment, where the dead fractions were
25-60% at the start of each cycle, this correction therefore increased the half-life by
0.03% or 0.85σ. This value is also listed, for comparison, in Table 4.3.
129
From the raw data obtained in this experiment, the complete pile-up correction
therefore reduced the uncorrected half-life T1/2 = 1.08174 ± 0.00037 s by 27.2σ or
0.94%. Using the bin-by-bin pile-up correction of method 2 the half-life obtained was
6.0σ (0.21%) larger than the optimal value of method 6 and is due to the influence of
saturating cosmic-ray events that can trigger the pile-up circuitry. While the pile-up
time resolution parameter ǫr in this experiment never exceeded 0.07, this correction
still led to a decrease in the measured half-life by 0.9σ (0.04%) and is therefore significant in high-precision half-life measurements. The CFD and pile-up energy thresholds
were much less significant in this work with only the latter leading to a measurable
decrease of 0.1σ (0.004%) in the deduced half-life of 26 Na. Once the complete pile-up
correction was applied the half-life of
26
Na obtained, T1/2 = 1.07167 ± 0.00042 s,
agrees with T1/2 = 1.07128 ± 0.00025 s obtained from the β counting experiment [88]
at the 1σ (statistical) level.
As a further test for potential rate dependence in the final corrected data, channels
were systematically removed from the beginning of each decay curve. A plot of
the deduced half-life versus the number of channels removed is given in Fig. 4.11
for methods 1,2, and 6 and includes only the statistical uncertainties. For clarity,
methods 3,4, and 5 are not displayed in Fig. 4.11 as these are sufficiently similar to
method 6. The systematic increase in the uncertainty on the deduced half-life results
from the fact that the channels with the highest statistics are being removed. It should
also be stressed that these data are not randomly scattered about the mean because
these are not independent but rather highly correlated data as each point contains
all of the data to its right. Ignoring the pile-up correction completely (method 1)
clearly shows the rate-dependent losses resulting from detector pulse pile-up as the
deduced half-life systematically decreases as the high-rate data are removed. Perhaps
130
1.084
Method 1: No Pile Up Correction
Method 2: Bin-by-bin Correction
Method 6: Complete Pile Up Correction
Half Life via β counting: 1.07128(25) s
1.082
1.080
Half-life (s)
1.078
1.076
1.074
1.072
1.070
1.068
1.066
0
5
10
15
20
25
30
35
40
45
50
Number of leading channels removed (1 chan = 0.1 s)
Figure 4.11: Deduced half-life of 26 Na versus the number of leading channels removed.
These data contain only the statistical uncertainties. When 50 channels are reached,
nearly 5 half-lives and hence 97% of our data has been removed. The high-precision
result obtained in a previous β-counting experiment [88] is overlayed for comparison.
more remarkable is the fact that the bin-by-bin pile-up correction (method 2) which
does not remove the cosmic-ray minimum shows no evidence of rate dependence.
The cosmic-ray activity has, however, increased the deduced half-life by more than
6 statistical standard deviations (0.2%). This indicates that while removing leadingchannel plots are suggestive they are by no means conclusive. The data following
the complete pile-up correction (method 6) do appear to show a small residual rate
dependence, however, the half-life deduced for all numbers of channels removed (out
to 5
26
Na half-lives or 97% of the data being removed) is always within ∼ 2σ of the
high-precision β counting result when considering only the statistical uncertainties.
131
7
Counts per 1 keV
10
6
(1)
(a)
(b)
(c)
(d)
(e)
10
5
10
4
Half-life (s)
10
1800
1810
1820
1830
Energy (keV)
1840
1850
1860
0
10
20
30
Gate width (keV)
40
50
60
1.0724
1.0722
1.0720
1.0718
1.0716
1.0714
1.0712
1.0710
Figure 4.12: (Upper panel) The width of the γ-ray gate on the entire data set was
adjusted to explore the effects of undetected low-energy pile-up events. (Lower panel)
A small decrease of 0.00004 s in the half-life of 26 Na was observed when the gate width
was increased from 20- to 30-keV ([1,a] to [1,b]) and the pile-up efficiency was assumed
to be unity. This decrease is consistent with the value of ǫp = 0.995 deduced from
the γ-ray spectra. Increasing the gate width further to as much as 60 keV ([1,e])
demonstrates no change associated with increasing the gate width.
While a completely flat channel-removal plot would be more satisfying, (although, as
noted above, potentially misleading) the trend observed in Fig. 4.11 for the complete
pile-up correction does not provide statistically conclusive evidence of any systematic
bias in the analysis procedure. Additional sources of systematic uncertainty will be
discussed below in Sec. 4.4.2.
A final test of the pile-up method was performed which aimed to quantify the pileup efficiency parameter a6 = ǫp whose average was deduced from the γ-ray spectra
132
to be ǫp = 0.995. This value was included as a fixed parameter in the pile-up fitting
procedure and is included in the optimal analysis of method 6 that yielded the overall
half-life of
26
Na, T1/2 = 1.07167 ± 0.00042 s. Because this parameter accounts for
missed pile-up resulting from a ∼ 20 keV pile-up detection threshold, a consistency
check was performed that set this parameter to unity and attempted to recover these
lost events by systematically increasing the width of the γ-ray energy gate on the
1809-keV photopeak. This analysis is presented in Fig. 4.12 where the energy gate
was increased from the optimal gate width of 20 keV (1800 keV to 1820 keV, position
[1,a] in Fig. 4.12) up to a maximum of 60 keV (1800 keV to 1860 keV, position [1,e] in
Fig. 4.12) in increments of 10 keV. As shown in Fig. 4.12 (lower panel) increasing the
gate width does not significantly alter the deduced half-life of
26
Na and thus events
are not being lost from the 1809-keV γ-ray gate from low-energy photon pile-up at
the level of the statistics in the experiment. The largest change in the
occurs from the gate width of 20- to 30-keV where the
26
26
Na half-life
Na half-life decreases from
1.07171 ± 0.00042 s to 1.07167 ± 0.00042 s, respectively. This change is perfectly
consistent with the calculated value of the overall pile-up efficiency ǫp = 0.995 from
the γ-ray spectra that was already applied in the optimal analysis of method 6 and
yielded T1/2 = 1.07167 ± 0.00042 s with the 20 keV wide energy gate.
4.4.2
Systematic Uncertainties
With the pile-up correction method established, tests of systematic uncertainties in
the result can now be made. Treating each of the 14 runs in Fig. 4.10 as independent measurements of the
26
Na half-life a reduced χ2 -value of 1.70 was obtained. For
13 degrees of freedom the probability of obtaining a reduced χ2 that is this large is
less than 6% and therefore likely indicates additional systematic uncertainties in the
133
method that have not yet been taken into account. Adopting the method of the Particle Data Group [4], this led to the overall half-life of 26 Na, T1/2 = 1.07167(42)(35) s,
where the first uncertainty is statistical and the second systematic.
The possibility of a small contamination due to bremsstrahlung and in-flight
positron annihilation from
26m
Al in the background under the 1809-keV γ-ray gate
was also considered and, as discussed above, was measured to be 3.8 ± 3.6 ppm of the
26
Na intensity at t = 0. The intensity of the 26m Al decay was treated as a fixed param-
eter in the optimal analysis using the mean value of 3.8 ppm, however, an identical
analysis to that above with this parameter fixed at its 1σ limits of 0.2 and 7.4 ppm
was also performed. Varying the
26m
Al contamination between these limits led to
values of 1.07170 ± 0.00042 s and 1.07164 ± 0.00041 s being deduced for the half-life
of
26
Na for the −1σ and +1σ limits, respectively. Similarly the half-life of
26m
Al was
varied between its ±1σ limits which led to no measurable change in the 26 Na half-life.
An additional systematic uncertainty of ± 0.00003 s from the contamination of 26m Al
was assigned in this experiment.
To further explore possible systematic effects, the half-lives deduced on a runby-run basis were combined and averaged according to common electronic settings
that were adjusted between runs. A summary of the settings used for each run was
previously presented in Table 4.1. Treating each setting as an independent measurement of the 26 Na half-life yields the various combinations plotted in Fig. 4.13 with the
half-life obtained from the weighted average of all 14 runs T1/2 = 1.07167 ± 0.00042 s
overlayed for comparison. The average half-life deduced from the 8 runs with the
56
Co source inside the array was T1/2 = 1.0715 ± 0.0007 s while the 6 runs obtained
without the
56
Co source yielded T1/2 = 1.0718 ± 0.0006 s. If the pile-up correction
134
1.076
1.075
1.074
Deduced half-life overlayed for comparison:
26
T1/2( Na) = 1.07167 ± 0.00042 s
1.073
Half-life (s)
1.072
1.071
1.070
1.069
1.068
1.067
1.066
1.065
2
χ /ν = 1.56
2
χ /ν = 0.47
2
χ /ν = 1.33
2
χ /ν = 0.07
All data
Shaping Time: 6.0 µs, 2.0µs, 1.0µs
Dead Time: 64 µs, 40 µs, 27 µs, Variable
CFD Setting: Low, High
56
Co Source: In, Out
1.064
Figure 4.13: Half-life of 26 Na with statistical errors deduced from an average of all runs
at each common adjustable setting. The half-life deduced from a weighted average of
the 14 runs is overlayed for comparison. Treating each group of common settings as an
independent measurement of the 26 Na half-life the reduced χ2 values were computed
and are indicated for each setting.
was not properly taken into account, the addition of the
56
Co source would have re-
sulted in a greater loss of 1809-keV photopeaks that would lead to a larger half-life
of
26
Na. This analysis yields results that are in agreement with each other. Treating
these as 2 independent measurements of the 26 Na half-life yields a reduced χ2 of only
0.07. Similar analysis of the dead times (4 independent measurements, χ2 /ν = 0.47)
and the CFD settings (2 independent measurements, χ2 /ν = 1.33) yields results that
are consistent. From a comparison of the amplifier shaping times, however, a reduced χ2 of 1.56 is obtained from these 3 independent measurements. Using the
method of the Particle Data Group [4], an estimate of the systematic uncertainty of
135
± 0.00032 s is obtained. This value is smaller than the systematic uncertainty of
± 0.00035 s deduced from the run-by-run comparison and, to be cautious, the larger
value of ± 0.00035 s is retained as the estimate of the overall systematic uncertainty
associated with the pile-up correction.
The half-life of 26 Na deduced in this work is therefore T1/2 = 1.07167(42)(35)(3) s,
where the first uncertainty is statistical and includes an increase of ∼ 5% due to
the uncertainties on the best-fit parameters from fits to the probability of pile-up
data, the second reflects a systematic uncertainty due to the reduced χ2 value of
1.70 from the run-by-run analysis, and the third reflects a small isobaric contamination from the decay of
26m
Al that was present in the beam. Combining these
uncertainties in quadrature yields T1/2 = 1.07167 ± 0.00055 s, a result that is precise
to 0.051% and agrees with the previous high-precision β counting determination of
T1/2 = 1.07128 ± 0.00025 s [88]. This work therefore demonstrates the feasibility of
using HPGe γ-ray detector arrays to measure half-lives to precisions of 0.05% (the
required level of precision for superallowed Fermi β decay studies). In this particular
experiment, the overall precision was ultimately limited by the estimated systematic
uncertainty of ± 0.00035 s that was nearly equal in size to the statistical uncertainty
in the measurement. It should be emphasized that attempts to intentionally magnify
the pile-up effects were made in this experiment which led to an overall pile-up correction that shifted the half-life by 27.2σ. Assuming that the systematic uncertainty
from the run-by-run analysis is due entirely to remaining systematics associated with
the pile-up methodology presented here one can express this as an overall uncertainty
in the method. Noting that a correction of 27.2σ yielded a result that agrees within
1σ of the high-precision β counting value an estimate of any remaining uncertainty
136
in our method is therefore only 3.7 % of the pile-up correction itself. Although intentionally increasing the probability of pile-up proved invaluable in understanding the
process of detector pulse pile-up, in particular the cosmic-ray minimum, this would,
of course, not be repeated in any true half-life experiments.
4.5
Summary
Methodology for the correction of detector pulse pile-up affected γ-ray-gated β decaycurve data has been presented and was demonstrated to be independent of several
parameters such as the number of detectors employed, the solid angle coverage of
each detector, and the γ-ray decay multiplicity. With radioactive beams of
delivered to the 8π spectrometer, the half-life of
26
26
Na
Na was determined following
a γ-ray gate on the 1809-keV transition in the daughter
26
Mg and was found to
be T1/2 = 1.07167 ± 0.00055 s, a result that agrees with the high-precision result
T1/2 = 1.07128 ± 0.00025 s obtained from a previous β-counting experiment [88].
The pile-up correction methodology presented here demonstrates the feasibility of using HPGe detectors to determine β decay half-lives at the level of ± 0.05% necessary
for the superallowed Fermi β decay program. Although this level of precision has
been enjoyed in β-counting experiments for decades, the γ-ray photopeak counting
method can be superior to β counting in situations where the signal-to-background
levels are low and decay selectivity must be relied upon to cleanly extract the decay
of interest.
Additional rate-dependent corrections that were not accounted for in this analysis
are assumed to be contained within a systematic uncertainty of ± 3.7% of the pileup correction itself. With lower rates, lack of intentional
137
56
Co background sources,
and typical shaping times used in real experiments, the pile-up corrections are much
smaller than in the current work and a 3.7% systematic uncertainty in the pile-up
correction is expected to be entirely negligible even for the demanding cases of superallowed Fermi β decay studies. In the following chapter, this will be demonstrated
using the half-life of the superallowed β + emitter
18
Ne as a demonstration of the
pile-up correction methodology under more typical experimental conditions.
138
Chapter 5
Half-life of 18Ne
Combining the pile-up correction technique discussed in Chapter 3 with the successful
and necessary calibration of the 8π spectrometer described in Chapter 4, measurements of the half-lives of several superallowed Fermi β decays can now be achieved to
the necessary level of precision (0.05%) using the method of γ-ray photopeak counting. The first superallowed half-life measurement with the 8π spectrometer was for
the β + emitter
18
Ne [79]. This particular decay is an attractive case to study because
of the relatively large isospin symmetry breaking correction δC = 0.620(32)% [23]
obtained from the Woods-Saxon model of Towner and Hardy (see Fig. 1.3) while
there has not been a calculation of δC for this decay using the Hartree-Fock model
of Ormand and Brown. The decay of
18
Ne, a Tz = −1 emitter, is complicated by
the fact that the superallowed branch does not feed the ground state of the daughter
nucleus as it does for all Tz = 0 decays. For
18
Ne decay the superallowed branch is
only 7.70 ± 0.21 % [91, 92, 93] and populates the 1042 keV level in the daughter
18
F. A
18
F level scheme populated via
the significant portion of
18
18
Ne β + decay is presented in Fig. 5.1. While
Ne β decays that populate excited states in 18 F facilitates
139
18
Ne
ft = 2922 80 s
0+ 1672.1 4.6 ms
+
b
1701
1+
0+
0
1081
659
0.188 %
0.0021 %
7.7 0.2 %
1042
1+
92.11 %
109.8 min
+
b
18
F
Figure 5.1: Decay scheme for 18 Ne β + decay to 18 F. Because 7.7% of the 18 Ne β
decays yield a 1042 keV γ ray measurements of the 18 Ne half-life can be performed
using the technique of γ-ray photopeak counting.
a half-life measurement via the γ-ray photopeak counting technique, this large nonsuperallowed ground state branch of 92.1% hinders high-precision measurements of
the superallowed β decay branching ratio. For this reason the
18
Ne f t value is not
one of the thirteen most precisely determined. Nonetheless, with the new technique
described in Chapters 3 and 4 of this thesis, a high-precision half-life measurement
for
18
Ne is now possible and, when combined with the Q value for this decay that
has already been determined to the level of 0.018% precision [1], will provide the
necessary motivation to improve the existing measurement of the
branching ratio.
140
18
Ne superallowed
Prior to this work the world average
18
Ne half-life T1/2 = 1.672 ± 0.008 s [80,
91] was limited to a precision of 0.5% (10 times too large for superallowed decay
studies) and was dominated by two measurements [93, 94] that do not agree within
experimental uncertainty. Both of these measurements were performed in 1975, where
the first, T1/2 = 1.687 ± 0.009 s [93], was obtained by counting 1042 keV γ-ray
photopeaks using a 40 cm3 Ge(Li) detector and the second, T1/2 = 1.669 ± 0.004 s [94],
employed a fast scintillator to detect the 18 Ne decay positrons directly. These results
differ by 0.018 s which is two times larger than the largest quoted uncertainty. As
discussed in Chapters 3 and 4 of this thesis the systematic loss of photopeaks due to
detector pulse pile-up can bias γ-ray photopeak counting half-life measurements to
larger values which could explain the discrepancy observed between these two values.
For this reason a new measurement of the
18
Ne half-life using the newly developed
γ-ray photopeak counting technique was performed using the 8π γ-ray spectrometer
at the TRIUMF-ISAC facility.
Radioactive beams of
18
Ne were produced in collisions of 500-MeV protons from
the TRIUMF main cyclotron (see Sec. 2.1), with intensities of up to 30 µA, on a SiC
target. Samples of
18
Ne were extracted using ISAC’s 2.45 GHz electron-cyclotron-
resonance ion source [70] operating in its first experimental run. Following mass
separation, a beam of approximately 4×105
18
Ne ions/s was delivered to the 8π spec-
trometer for approximately 90 hours. The 30-keV beam was collected using the
mylar-backed aluminum tape system (Sec. 2.6), moving through the mutual center of
the 8π spectrometer (Sec. 2.2) and SCEPTAR (Sec. 2.3). Each cycle lasted 49 seconds
in duration and was comprised of cycling times of 1-7-40-1, corresponding to counting
background for 1 s, implanting the
18
Ne beam for 7 s (∼ 4 half-lives), counting the
decay of the sample for 40 s (∼ 24 half-lives), followed by a tape-move time of 1 s. In
141
Table 5.1: Run-by-run summary of experimental conditions employed in the 18 Ne halflife measurement. The variable dead-time setting had an average value of 25.0 µs.
Following Run 5 the CFD trigger-energy thresholds were increased to reduce the
fraction of dead events.
Run
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
the
18
Shaping Time
(µs)
1.0
1.0
1.0
1.0
0.5
0.5
0.5
2.0
2.0
2.0
2.0
1.0
1.0
1.0
0.5
Dead Time
(µs)
Variable
Variable
27.0
40.0
Variable
27.0
40.0
Variable
40.0
27.0
Variable
Variable
27.0
40.0
40.0
CFD Threshold
Cycles
low
low
low
low
low
low
low
low
high
high
high
high
high
high
high
427
443
215
256
274
285
298
217
214
266
216
267
296
375
205
Ne half-life experiment, SCEPTAR was used solely for determining the levels
of isobaric contaminants in the A = 18 beam of
addition to the
Neither
17
18
F nor
18
F (T1/2 = 109.77(5) min [80]), in
Ne daughter activity, and molecular H17 F (T1/2 = 64.49(16) s [95]).
18
F β + decay gives rise to γ-ray radiation, however, as discussed in
Chapter 4 for the case of 26m Al, these decays could in principle produce a small background inside the 1042 keV γ-ray gate due to bremsstrahlung and in-flight positron
annihilation processes.
In order to test for the presence of systematic effects the non-extendible deadtimes, amplifier shaping times, and the constant-fraction-discriminator (CFD) thresholds were varied throughout the experiment. Data were collected with combinations
142
4
10
10
20000
10000
0
1030
1040
1050
1701
10
5
1081
Counts
10
30000
1042
6
659
10
7
511
10
3
2
10
1
0
1000
500
2000
1500
3000
2500
Energy (keV)
10
60000
1081
30000
4
10
1042
5
659
Counts
10
10
90000
6
511
10
7
0
1030
1040
1050
1701
10
3
2
10
1
0
10 0
500
1000
1500
2000
2500
3000
Energy (keV)
Figure 5.2: Singles spectra of γ rays following the β decay of 18 Ne with (upper panel)
low CFD thresholds and 0.5 µs HPGe shaping times and (lower panel) high CFD
thresholds and 2.0 µs shaping times. Transitions between states in 18 F are labeled
with their energy in keV. The insets show the region around the 1042-keV gating
transition on a linear scale and illustrates the change in energy resolution between
0.5 µs and 2.0 µs shaping time.
143
of three dead-time settings (variable, fixed 27 µs and fixed 40 µs), three HPGe spectroscopy amplifier shaping times (0.5 µs, 1.0 µs and 2.0 µs), and “low” and “high”
CFD thresholds. Note that these settings are considered to span the “optimal” ranges
for these types of experiments and were not adjusted to purposely increase the probability of detector pulse pile-up as was done in the
26
Na calibration experiment of
Chapter 4. A summary of the settings employed for each run is presented in Table 5.1.
For each cycle, the maximum full-array trigger rate at the time the beam was
turned off was ∼2.5 kHz with maximum dead-time fractions of 10% and 40% at the
start of the decay curves for the variable (∼25 µs) and 40 µs dead-time settings,
respectively. For all runs, the pile-up corrections were less than 1% at all times
(see Fig. 5.4). Linear calibration of the HPGe energies were performed periodically
between runs using standard
56
Co and
152
Eu sources. Sample γ-ray singles spectra
obtained under different experimental conditions are shown in Fig. 5.2. The upper
panel of Fig. 5.2 was obtained with low CFD thresholds and amplifier shaping times
set to 0.5 µs, while the lower panel shows the corresponding spectrum with high
CFD thresholds and 2.0 µs shaping times, illustrating the improvement in energy
resolution.
5.1
Pile-up Probability Analysis
In the offline analysis, a threshold (unique to each run) was set that rejected any cycles for which the total number of counts fell above or below prescribed values. This
was required in order to define an average pile-up probability for each run using the
method described in detail in Sec. 4.3 of Chapter 4 of this thesis. Plots of the average
144
detected probability of pile-up are shown in Fig. 5.3 for two cases in which low and
high CFD settings were employed. The data shown in Fig. 5.3 were obtained using
427 cycles at the low CFD setting and 205 cycles at the high setting. For the β + decay of
18
Ne (and the β + decay of the daughter
18
F), a significant fraction of the total
γ-ray activity is measured at 511 keV (or less due to Compton scattering processes)
as the liberated β + particles interact with electrons in the surrounding material producing annihilation radiation. When the CFD thresholds were raised to intentionally
decrease the overall dead time of the system, a large portion ≈ 93% of the total activity was removed from the resulting γ-ray spectrum as demonstrated in Fig. 5.2. As
anticipated from the simulations of Chapter 3 and discussed in Sec. 3.4.2, one clearly
observes the expected increase in the probability of pile-up for these recorded events
as a result of raising the CFD thresholds. In the
26
Na experiment it was shown in
Sec. 4.3 (Fig. 4.7) that the expected increase to the probability of pile-up was not
statistically significant. This is due to the fact that 99% of
lowed by an 1809 keV γ-ray transition and
26
26
Na β decays are fol-
Na is a β − decay therefore much less of
the total activity occurs at energies ≤ 511 keV. Increasing the CFD thresholds in the
26
Na experiment led to a much smaller reduction in the overall γ-ray activity (≈ 10%)
and the resulting increase to the probability of pile-up was proportionally smaller. In
addition, the increase to the CFD thresholds were performed in the
26
Na experiment
while the amplifier shaping times were at the unreasonable setting of 6.0µs. For the
18
Ne experiment, one can also clearly observe the increase in the probability of pile-up
due to the raised CFD thresholds in Fig. 5.3 by the increase from 1.0% to 2.5% in the
probability of pile-up at late times dominated by the cosmic-ray self pile-up events.
The maximum correction that was applied to the γ-ray gated decay-curve data
following the corrections for the pile-up time resolution, removal of the cosmic-ray
145
0.045
Best-fit parameters:
Aτp = a1 = 0.036 ± 0.015
T1/2 = a2 = 1.67 ± 0.09 s
−5
Bτp = a3 = (3.4 ± 0.8)×10
εr
= a4 = 0.03 (fixed)
α
= a5 = 0.93 (fixed)
εp
= a6 = 0.995 (fixed)
−7
Cεcτp = a7 = (3.1 ± 0.7)×10
Probability of pile-up: pi / (ni + pi)
0.040
0.035
0.030
0.025
0.020
18
Ne data, low CFD
Best-fit result
Cosmic pile-up term
Pile-up of 2 or more events
0.015
0.010
0.005
0.000
0
5
10
15
20
30
25
35
40
45
Time (s)
0.045
Best-fit parameters:
Aτp = a1 = 0.0117 ± 0.0006
T1/2 = a2 = 1.7 ± 0.1 s
−4
Bτp = a3 = (1.44 ± 0.21)×10
εr
= a4 = 0.03 (fixed)
Probability of pile-up: pi / (ni + pi)
0.040
0.035
0.030
18
Ne data, high CFD
Best-fit result
Cosmic pile-up term
Pile-up of 2 or more events
0.025
0.020
0.015
α
= a5 = 0.131 (fixed)
εp
= a6 = 0.995 (fixed)
−6
Cεcτp = a7 = (3.6 ± 0.5)×10
0.010
0.005
0.000
0
5
10
15
20
25
30
35
40
45
Time (s)
Figure 5.3: Experimental probability of pile-up spectra for the 18 Ne experiment.
(Upper panel) A typical run at 1.0 µs shaping and CFD thresholds at the “low”
setting. (Lower panel) A typical run at 1.0 µs shaping and CFD thresholds at the
“high” setting. At the high threshold setting only ∼ 13% of the total activity has
sufficient energy to trigger the system.
146
0.012
Maximum probability of pile-up
CFD = Low
CFD = High
0.010
0.008
0.006
0.004
0.5 µs shape time
1.0 µs shape time
2.0 µs shape time
0.002
0.000
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Run Number
Figure 5.4: Run-by-run plot of the maximum probability of pile-up taken at the start
of the decay for each run (t = beam off) in the 18 Ne experiment. The largest decrease
in the probability of pile-up is due to the increase in the shaping time corresponds to
the expected increase when the CFD thresholds were increased from the low to the
high setting after Run 8.
contribution, and the CFD and pile-up energy thresholds is plotted versus run number
in Fig. 5.4 where all of the pile-up corrections were less than 1% at the time the beam
was turned off. The large increase in the probability of pile-up was associated with
the intentional raising of the CFD thresholds which occurred after Run 8. The final
run (Run 15) was obtained at the shortest shaping time of 0.5µs and the subsequent
decrease to the overall probability of pile-up is readily observed in Fig. 5.4 compared
to the other runs obtained at 1.0 and 2.0 µs shaping times at the high CFD threshold
setting.
147
5.2
Half-life Analysis
The half-life of 18 Ne was determined by selecting events in which the 1042-keV γ ray,
which follows the superallowed decay of
the 1+ ground state in the daughter
18
18
Ne and connects the analog 0+ state to
F (see Fig. 5.1) was detected. Decay curves
associated with the 1042-keV γ ray were dead-time corrected on a cycle-by-cycle basis
and pile-up corrected based on the run-by-run average probability of pile-up using the
procedures outlined in Chapter 4 of this thesis and explained in detail in Appendix A.
These corrected data were summed for each run and fit to a function containing
the exponential decay of
18
Ne with free intensity and half-life parameters plus a
third free parameter to describe the constant background. Additional exponentials
were added to the fit function to describe the grow-in and decay of the daughter
18
F and the decay H17 F isobaric contaminant and these will be considered in detail
below (see Sec. 5.2.1). A typical grow-in and decay curve from a single run, and the
corresponding fit to the data considering only 18 Ne decay with a constant background
is shown in Fig. 5.5. These data include corrections for dead-time and pulse pile-up
effects using the method described in Chapter 4 and Appendix A. The half-life
obtained for each of the fifteen experimental runs is plotted in Fig. 5.6 and listed in
Table 5.2 in addition to the run-by-run reduced χ2 values determined from each of
the fits. The weighted average of these values yield T1/2 = 1.6656 ± 0.0017 s and a
reduced χ2 value of 0.55 with 14 degrees of freedom.
5.2.1
Systematic Uncertainties
As with the
26
Na experiment described in Chapter 4, each of the fifteen runs were
grouped according to their common electronic settings (see Table 5.1 for a summary)
148
4
10
10
Counts
Counts in 1042 keV γ-ray Gate
10
3
10
5
4
10
3
1025 1030 1035 1040 1045 1050 1055
10
Eγ (keV)
2
18
10
1
0
T1/2( Ne) = 1.6644 (47) s
2
χ /ν = 1.06
5
10
15
20
25
30
35
40
45
50
Time (s)
Figure 5.5: Pile-up and dead-time corrected decay curve obtained from a single run
following a gate on the 1042-keV transition in 18 F (inset).
and are shown in Fig. 5.7. Of the fifteen runs collected in this experiment, 4 were
obtained with a shaping time of 0.5 µs, 7 were at 1.0 µs, and 4 were at 2.0 µs. Treating
these three settings as three independent measurements of the
18
Ne half-life (with 2
degrees of freedom) yields an average of T1/2 = 1.6656 ± 0.0017 s, and a reduced
χ2 value of 1.30. A similar analyis was performed using the 3 dead-time values of
variable, 27, and 40 µs (2 d.o.f, χ2 /ν = 0.04), and the 2 CFD threshold settings of low
and high (1 d.o.f, χ2 /ν = 0.05). Following the method of the Particle Data Group [4]
the largest reduced χ2 value of 1.30 is retained as an estimate for any unidentified
systematic effects and the statistical uncertainty must therefore be increased by the
square root of this value. This yields a systematic uncertainty of ± 0.0009 s so that
when added in quadrature with the statistical uncertainty of ± 0.0017 s yields the
149
1.70
18
Ne Half-life (s)
1.69
1.68
1.67
1.66
1.65
18
T1/2( Ne) = 1.6656 ± 0.0017 s
1.64
2
χ /ν = 0.55
1.63
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Run Number
Figure 5.6: Half-life of 18 Ne versus individual run number. The weighted average of
all fifteen runs and its statistical error are T1/2 = 1.6656 ± 0.0017 s.
Table 5.2: Half-life of 18 Ne deduced on a run-by-run basis with statistical errors
and resulting reduced χ2 values for all 15 runs in this analysis. A weighted average
of T1/2 = 1.6656 ± 0.0017 s is deduced from these data where the uncertainty is
statistical and does not reflect any systematic effects.
Run
Number
1
2
3
4
5
6
7
8
T1/2 σTstat
1/2
(s)
1.6555(69)
1.6718(70)
1.6672(95)
1.6656(63)
1.6645(67)
1.6647(74)
1.6670(81)
1.6858(114)
χ2 /ν
1.02
1.19
1.12
0.90
1.05
0.97
0.99
1.04
Run
Number
9
10
11
12
13
14
15
150
T1/2 σTstat
1/2
(s)
1.6698(75)
1.6691(67)
1.6686(77)
1.6637(53)
1.6615(52)
1.6644(47)
1.6658(64)
χ2 /ν
1.03
0.90
0.98
0.90
1.02
1.06
1.03
1.680
2
χ /ν =
1.30
0.04
0.05
18
Ne Half-life (s)
1.675
1.670
1.665
1.660
All Runs (15)
Shape time: 0.5µs (4); 1.0µs (7); 2.0µs (4)
Dead time: variable (6); 27µs (4); 40µs (5)
CFD: low (8); high (7)
1.655
1.650
Figure 5.7: Half-life measurements of 18 Ne sorted by adjustable electronic setting.
The reduced χ2 values for each group are displayed at the the top and the number of
runs that were combined into each particular group are indicated in the () brackets.
total uncertainty of ± 0.0019 s.
The overall pile-up correction in this measurement amounted to a reduction in
the deduced
18
Ne half-life by ∼ 1.4σ from a value T1/2 = 1.6679 ± 0.0016 s that was
obtained when the pile-up correction was not applied to the data. This correction is
small compared with the pile-up corrections that were successfully applied in the 26 Na
experiment described in Chapter 4, in which a 0.9% (equivalent to ∼ 27σ) pile-up correction in a ± 0.05% measurement for 26 Na was confirmed through a comparison with
traditional β counting techniques. From the
26
Na analysis, a systematic uncertainty
in the application of the pile-up correction of 3.7% of its own value was assigned.
For the
18
Ne analysis this amounts to ± 0.05σ or ± 0.00009 s. The half-life of
151
18
Ne
deduced in this work is therefore T1/2 = 1.6656(17)(9)(1) s, where the uncertainties
are statistical, an estimated systematic effect due to the electronic settings, and an
estimated systematic effect resulting from the pile-up correction, respectively.
While neither
18
F or
17
18
F nor
17
F β decay give rise to γ radiation, bremsstrahlung from
F β + particles, inner bremsstrahlung from the electron capture process, and
in-flight annihilation processes may produce a small time-dependent background beneath the 1042-keV gating transition. Because the fractions of the total measured
β activity from SCEPTAR at the start of the decay curve from
18
F and
17
F β de-
cay were only 0.3% and 0.4% respectively [96], it was expected that these processes
would yield negligible contamination at 1042 keV in the γ-ray activity. In order to
provide an estimate of these contaminant levels, only the bremsstrahlung from the β +
particles, the dominant process at 1042 keV was considered. Based on the QEC values of 1655 keV [80] and 2761 keV [95] for
18
F and
17
F respectively, only the latter
could give rise to β + bremsstrahlung contamination at 1042 keV. Assuming that the
relative bremsstrahlung yields of
17
F to
18
Ne (denoted IBS ) at the start of the decay
counting period (t = 8 s) inside the 1042 keV energy gate is proportional to the relative β activities Iβ at t = 8 s measured in the SCEPTAR array one can write the
expression,
IBS (17 F)
Iβ (17 F)
=
B
,
IBS (18 Ne)
Iβ (18 Ne)
(5.1)
where B is a proportionality constant and depends upon the β decay Q values and
the specific material (in this case the delrin plastic that forms and surrounds the
target chamber and covers the faces of the HPGe detectors in the 8π spectrometer)
that is acting to slow the β particles. The ratio of the expected bremsstrahlung
yields for equal numbers of
17
F and
18
Ne β particles in delrin were calculated and,
as shown in Fig. 5.8, provide B = 3.8% at 1042 keV. There are higher-order effects
152
6
Bremsstrahlung Yield (photons/keV/10 β Decays)
10
4
3
18
2
17
10
10
10
10
10
10
F:
QEC = 2761 keV
1
0
-1
-2
10
10
Ne: QEC = 4446 keV
-3
Eβ = 1042 keV
-4
10
-5
-6
10 0
500
1000
2000
1500
2500
3000
3500
Eβ (keV)
Figure 5.8: Calculated bremsstrahlung yield of 17 F and 18 Ne β particles in delrin. For
equal numbers of 18 Ne and 17 F β particles the relative bremsstrahlung yield in delrin
17
BS ( F)
at 1042 keV is IIBS
= B = 3.8%.
(18 Ne)
that were not considered in the calculation such as the summing of a bremsstrahlung
photon with a 511 keV annihilation photon that always accompanies β + decay, in
addition to higher energy bremsstrahlung photons Compton scattering out of the
HPGe crystal and depositing only a fraction of their energy. To account for these
effects and provide a conservative upper-limit, the 17 F and 18 Ne bremsstrahlung yields
of Fig. 5.8 were integrated from 531 keV (1042 − 511) to their respective endpoints
to yield an adopted upper-limit value of B = 10%. Combining this value with the
0.4% relative β activity of
17
F to
18
Ne observed in SCEPTAR [96] yields an upper-
limit of 0.04% relative contamination from
17
F bremsstrahlung compared to
18
bremsstrahlung in the 1042 keV γ-ray gate at the time the beam was turned off.
153
Ne
3
Counts
Counts in 1052 keV γ-ray Gate
10
10
6
1042 keV
10
10
5
4
2
10
Background gate
10
207
Bi:
1064 keV
3
1040 1045 1050 1055 1060 1065 1070
Eγ (keV)
10
1
18
T1/2( Ne) = 1.62 (6) s
18
IBS( Ne) = 1450 ± 224 s
0
5
10
15
-1
20
25
30
35
40
45
50
Time (s)
Figure 5.9: Pile-up and dead-time corrected decay curve obtained from a single run
following a gate directly above the 1042-keV transition in 18 F (inset). The 18 Ne
activity is due primarily to bremsstrahlung from the 18 Ne β + particles.
This value was combined with the observed bremsstrahlung yield of
18
Ne at
1042 keV, IBS (18 Ne), which was obtained from taking a γ-ray gate directly above the
1042-keV photopeak. It was assumed that the smoothly varying bremsstrahlung yield
for
18
Ne inside the 1042 keV photopeak gate and in the background directly above
the 1042 keV γ-ray gate were equal. A sample of the time distribution of events from
just above the 1042 keV photopeak for a single run is presented in Fig. 5.9. The 18 Ne
intensity for this particular run was found to be IBS (18 Ne) = 1450 ± 224 s−1 at the
time the beam was turned off (t = 8 s). From Eqn. 5.1 this yields an upper limit
of only IBS (17 F) ≤ 0.6 counts per second at the start of the decay curve shown in
Fig. 5.5.
154
1.676
1.674
18
Ne Half-life (s)
1.672
1.670
1.668
1.666
1.664
1.662
1.660
1.658
1.656
0
4
8
12
16
20
24
28
32
Number of leading channels removed (1 chan = 0.1 s)
Figure 5.10: Deduced half-life of 18 Ne versus the number of leading channels removed
(1 channel = 100 ms). These data are not randomly scattered about the mean because
they are highly-correlated, with each data point containing all of the data to the right
of it.
The data for each run were re-fit using a function containing the exponential
decays of
18
Ne and
17
F plus a constant background with the
17
F half-life fixed at
T1/2 (17 F) = 64.49 s, and the 17 F intensity fixed at the deduced upper limit for each run.
The half-life of 18 Ne obtained via this procedure, T1/2 = 1.6656 ± 0.0017 s, is identical
to that above where no contamination from 17 F was considered. Bremsstrahlung contamination from
deduced
18
17
F β decay was therefore concluded to have negligible effect on the
Ne half-life. Inner bremsstrahlung and in-flight annihilation are expected
to contribute at even lower levels than the 0.04% deduced from a calculation of the
outer bremsstrahlung. This possibility was nonetheless considered by performing an
155
additional fit to the data that treated the
this analysis the
18
17
F intensity as a free parameter. From
Ne half-life was unchanged and the
17
F intensities deduced were
consistent with zero, but with large uncertainties due to a large covariance between
the 17 F intensity and the constant background rate. For 18 F, outer bremsstrahlung is
not possible at 1042 keV and any contamination could only come from annihilation
in-flight or inner bremsstrahlung processes. The data were therefore fit using a free
18
F intensity (and included the grow-in from 18 Ne decay) and this procedure also had
no effect on the deduced half-life of
18
Ne.
To test for unknown rate-dependent systematic effects, the data from the first
three seconds (∼ 2 half-lives or 75% of the data) after the beam was switched off
were eliminated from the data set channel-by-channel and re-fit using the function
described above that assumed a negligible contribution from
17
F. The results are
plotted in Fig. 5.10 where no evidence for a change in the half-life, and hence the
presence of any additional rate-dependent systematic effects, was detected.
5.2.2
Diffusion and the Half-life of
23
Ne
Another systematic effect in the study of half-lives of implanted noble-gas isotopes
using the techniques described above is associated with their potential diffusion from
the implantation site that would systematically bias the deduced half-life to a smaller
value.
It has been observed that, under certain conditions, approximately 10%
of noble-gas ions implanted into Al [97], diffuse out with diffusion “half-lives” of
< 100 ms. Short-lived diffusion effects for a portion of the
18
Ne atoms in this experi-
ment have already been shown to be less than the statistical error in this experiment
from Fig. 5.10 where a systematic increase in the deduced
18
Ne half-life as channels
are removed from the start of the decay curve was not observed. To test whether
156
10
40
3
2982
10
2542
208
Tl: 2615
4
2076
Bi: 1064
10
5
207
Counts
10
K: 1461
1636
6
511
10
7
440
10
2
10
1
0
1000
500
2000
1500
2500
3000
3500
Energy (keV)
Figure 5.11: Singles spectra of γ rays following the β decay of 23 Ne with low CFD
thresholds and 2.0 µs HPGe shaping times. Transitions between states in 23 Na are
labeled with their energy in keV.
diffusion effects could be present on longer time scales (> 1 s), the half-life of the
longer-lived β − decay of
set-up employed in the
23
18
Ne to
23
Na was determined using the same experimental
Ne measurement. The A = 23 beam was cycled with an
18.6 s implant time and 363.0 s decay time with a total of 20 cycles being collected.
This single run was collected using amplifier shaping times of 2.0 µs, a variable dead
time, and the CFD thresholds were at the low setting. A γ-ray singles spectrum
containing the entire collected data set is presented in Fig. 5.11.
A probability of pile-up spectrum obtained from the average of the 20 cycles
collected is shown in Fig. 5.12. Due to the longer half-life of
of
18
23
Ne compared to that
Ne it takes nearly 340 seconds after the beam is switched off to attain the level
157
Probability of pile-up: pi / (ni + pi)
0.015
Best-fit parameters:
Aτp = a1 = 0.0029 ± 0.0004
T1/2 = a2 = 33.8 ± 1.9 s
−5
Bτp = a3 = (4.3 ± 0.6)×10
εr
= a4 = 0.03 (fixed)
α
= a5 = 0.96 (fixed)
εp
= a6 = 0.995 (fixed)
−7
Cεcτp = a7 = (3.4 ± 0.5)×10
0.010
23
Ne data, low CFD
Best-fit result
Cosmic pile-up term
Pile-up of 2 or more events
0.005
0.000
0
50
100
150
200
250
300
350
Time (s)
Figure 5.12: Experimental probability of pile-up spectra for the 23 Ne experiment.
This run was obtained at 2.0 µs shaping and CFD thresholds at the “low” setting.
23
4
T1/2( Ne) = 37.11 (6) s
2
χ /ν = 0.88
10
3
10
10
Counts
Counts in 440 keV γ-ray Gate
10
2
6
10
10
5
4
10
10
3
1
0
430 435 440 445 450
Eγ (keV)
50
100
150
200
250
300
350
Time (s)
Figure 5.13: Pile-up and dead-time corrected decay curve for 23 Ne obtained from a
single run following a gate on the 440-keV transition in 23 Na (inset).
158
38.5
23
Ne Half-life (s)
38.0
37.5
37.0
36.5
23
36.0
35.5
T1/2( Ne) = 37.11 ± 0.06 s
2
χ /ν = 0.86
0
2
4
6
8
10
12
14
16
18
20
Cycle Number
Figure 5.14: Half-life of
23
Ne deduced on a cycle-by-cycle basis.
Table 5.3: Half-life of 23 Ne deduced on a cycle-by-cycle basis with statistical errors
and resulting reduced χ2 values for all 20 cycles in this analysis. A weighted average
of T1/2 = 37.11 ± 0.06 s is deduced from these data where the uncertainty is statistical
and does not reflect any systematic effects.
Cycle
Number
1
2
3
4
5
6
7
8
9
10
T1/2 σTstat
1/2
(s)
37.056(250)
37.061(251)
37.037(253)
37.238(256)
36.861(255)
37.167(280)
36.983(251)
37.116(256)
37.021(261)
37.065(254)
χ2 /ν
1.00
0.96
0.88
1.10
1.09
0.92
0.95
0.92
1.09
1.17
Cycle
Number
11
12
13
14
15
16
17
18
19
20
159
T1/2 σTstat
1/2
(s)
36.905(256)
37.682(262)
37.163(250)
37.110(255)
37.288(254)
37.039(255)
37.391(253)
37.367(257)
36.492(252)
37.258(256)
χ2 /ν
1.02
1.03
0.95
1.08
1.00
0.94
1.06
1.14
0.96
1.03
of approximately 1.0% for the cosmic-ray probability of self pile-up. Note that for
low CFD settings in the
18
Ne experiment (see Fig. 5.3) the same saturation level of
1.0% was achieved in a proportionally shorter amount of time (∼ 15 seconds). The
largest probability of pile-up applied to the gated γ-ray data for the 23 Ne experiment
was only 0.2% at the time the beam was turned off.
Following a γ-ray energy gate on the 440-keV γ ray in the daughter
23
Na, the
resulting grow-in and decay curve obtained from all 20 cycles is presented in Fig. 5.13.
These data were fit using a two-exponential function with one of the exponentials
having a fixed half-life of T1/2 (23 Mg) = 11.317 ± 0.011 s [98, 99] corresponding to
the
23
Mg isobaric contaminant in the beam. Following corrections for dead-time and
23
pile-up effects, the half-life of
was not affected when the
23
Ne was determined to be T1/2 = 37.11 ± 0.06 s and
Mg half-life was varied between its ± 1σ limits. The
total pile-up correction in the
23
Ne experiment was negligible as the identical result
of T1/2 = 37.11 ± 0.06 s was obtained with no correction for pulse pile-up applied to
the data set.
In order to test each cycle for consistency, the half-life of
23
Ne was determined on
a cycle-by-cycle basis. The average probability of pile-up from all 20 cycles, shown in
Fig. 5.12, was used to correct every cycle individually in the cycle-by-cycle analysis.
The half-life of
23
Ne obtained for each cycle is plotted in Fig. 5.14 and listed in
Table 5.3. Treating each cycle as an independent measurement of the
23
Ne half-life
yields a reduced χ2 value of 0.86 and confirms the consistency among the 20 cycles
obtained in this experiment.
The half-life of
23
Ne obtained in this analysis is a factor of 2 more precise than,
and consistent with, a previous determination of the
23
Ne half-life that obtained
T1/2 = 37.24 ± 0.12 s [100]. Because this previous measurement was performed by
160
37.8
23
37.7
Half-life of Ne overlayed for comparison
37.6
T1/2( Ne) = 37.11 ± 0.06 s
23
23
Ne Half-life (s)
37.5
37.4
37.3
37.2
37.1
37.0
36.9
36.8
36.7
0
30
90
60
120
150
Number of leading channels removed (1 chan = 0.8 s)
Figure 5.15: Deduced half-life of 23 Ne versus the number of leading channels removed
(1 channel = 800 ms). These data are not randomly scattered about the mean because
they are highly-correlated, with each data point containing all of the data to the right
of it.
trapping the 23 Ne noble gas ions in a stainless steel counting cell it was therefore free
of any diffusion effects. Agreement between this prior measurement and the new value
presented here provides confirmation that diffusion of
23
Ne from the aluminum tape
was negligible on the time scale of 10 half-lives (∼ 360 s) studied in this analysis. As a
complementary test, channels were systematically removed from the beginning of the
data set as described above for 18 Ne. As shown in Fig. 5.15, no evidence for a change in
the half-life was observed on time scales of ∼ 100 s thus ruling out a possible diffusion
component on this time scale. With a short-lived diffusion component already ruled
out from the complementary plot for
18
Ne (Fig. 5.10), diffusion at any time scale
161
relevant to analysis of the
18
Ne and
23
Ne half-life experiments was concluded to be
negligible.
5.3
Comparison to Previous Results
With possible diffusion effects deemed negligible, the half-life of
18
Ne deduced in this
work, T1/2 = 1.6656 ± 0.0019 s, was obtained by combining the statistical (0.0017 s),
electronic systematic (0.0009 s), and pile-up systematic (0.0001 s) uncertainties in
quadrature. This value agrees with three of the four previous half-life determinations
and is four times more precise than the previously-accepted world-averaged value
T1/2 = 1.672 ± 0.008 s [80, 91] comprised of two measurements [93, 94]. A comparison
of this work to all previous measurements of the
18
Ne half-life [93, 94, 101, 102] is
presented in Fig. 5.16 and listed in Table 5.4.
Compared to the two measurements that defined the previously adopted world
average, the result presented here T1/2 = 1.6656 ± 0.0019 s agrees with the direct
β counting measurement of Ref. [94], T1/2 = 1.669 ± 0.004 s, but does not agree with
T1/2 = 1.687 ± 0.009 s [93] at the level of 2.4σ which was obtained using an older
γ-ray photopeak counting technique. While the reason for this discrepancy is not
immediately clear, one possibility could be an incomplete treatment of detector pulse
pile-up effects in Ref. [93] which led to the larger quoted half-life. At the present level
of precision, however, one cannot exclude the measurement of Ref. [93] as anything
more than a statistical fluctuation. The new world average, obtained from all five
measurements, is therefore T1/2 = 1.6670 ± 0.0017 s with a reduced χ2 value of 1.52.
Adopting the procedure of the Particle Data Group [4] the uncertainty in the half-life
of
18
Ne must therefore be increased by the square root of 1.52.
162
18
T1/2( Ne) = 1.6670 ± 0.0017 s
1.73
2
χ /ν = 1.52
1.72
18
Present Work
Ne Half-life (s)
1.71
1.70
1.69
1.68
1.67
1.66
1.65
1.64
[Asl70]
[Alb70]
[Har75] [Alb75]
8π
Figure 5.16: Comparison of 18 Ne half-life measurements. The new world average
obtained from a weighted average of these five results, T1/2 = 1.6670 ± 0.0017 s, is
overlayed for comparison. The previous half-life measurements were obtained from
references [Asl70] ([101]), [Alb70] ([102]), [Har75] ([93]), and [Alb75] ([94]), respectively.
Table 5.4: Summary of all high-precision 18 Ne half-life measurements. The new world
average of T1/2 = 1.6670 ± 0.0021 s with a reduced χ2 value of 1.52 is obtained from
a weighted average of these 5 measurements.
Reference
present work
D.E. Alburger et al. [94]
J.C. Hardy et al. [93]
D.E. Alburger [102]
E. Aslanides et al. [101]
Year
2007
1975
1975
1970
1970
World Average (χ2 /ν = 1.52)
163
T1/2 (s)
1.6656
1.669
1.687
1.670
1.690
σ (s)
0.0019
0.004
0.009
0.020
0.040
1.6670
0.0017
This procedure results in the value T1/2 = 1.6670 ± 0.0021 s and establishes the
18
Ne half-life to 0.13% precision. This value is overlayed, for comparison, with the
experimental values in Fig. 5.16.
Measurement of the
18
Ne half-life to a precision of 0.11% presented in this thesis,
while 4 times more precise than the previous world average, was limited entirely by
statistics and could be improved by repeating the experiment with a higher intensity
18
Ne beam from the ISAC FEBIAD ion source [70] or an improved ECR ion source
design.
5.4
Present Status of the
Combining the
18
18
Ne f t and Ft values
Ne half-life T1/2 = 1.6670(21) s with the accepted values for the
superallowed β branching ratio BR = 7.70(21) % [1], and the statistical rate function
f = 134.48(15) [24] yields,
f t(18 Ne) = 2912(3)f (4)T1/2 (78)BR s
= 2912(78) s,
(5.2)
for the f t value for the superallowed decay of 18 Ne. This result is nearly two orders of
magnitude less precise than the best measured cases (see Fig. 1.2 of Chapter 1) and
is entirely limited by the measurements of the superallowed branching ratio. While
modest improvements to the
18
Ne branching ratio could be achieved, a measurement
of this quantity to 0.05% precision required for the superallowed program seems unlikely in the immediate future. A meaningful test of the isospin symmetry breaking
calculations of Towner and Hardy therefore cannot be achieved using the present 18 Ne
f t value.
164
Although the f t value for 18 Ne is too imprecise to test calculations of isospin symmetry breaking one can instead test the experimental branching ratio directly using
the world average F t value from the thirteen most precisely determined superallowed
decays. Using the Towner and Hardy calculated value for isospin symmetry breaking
in 18 Ne, δC = 0.620(32)% [23], the radiative correction δR = 1.219(37)%, the statistical rate function f = 134.48(15) [24], the world average half-life T1/2 = 1.6670(21) s
and the average value F t = 3074.0(8) s from Towner and Hardy’s calculations of δC ,
rearranging Eqn. 1.45 yields,
BR(18 Ne) =
f T1/2
(1 + δR )(1 − δC )
Ft
= 7.336(13) %.
(5.3)
This result is nearly 2σ lower and more than an order of magnitude more precise than
the experimental value BR = 7.70(21) % [1], and, assuming the validity of the CVC
hypothesis, can be used to provide a benchmark for future measurements of the
18
Ne
branching ratio.
5.5
Summary
The half-life of the superallowed-Fermi β + emitter
18
Ne was determined via a newly
developed γ-ray photopeak counting technique to be T1/2 = 1.6656 ± 0.0019 s, representing an improvement in precision by a factor of four over the previously adopted
world average. This measurement is the first high-precision superallowed half-life
determined via γ-ray photopeak counting with the 8π spectrometer and includes
a 1.4σ correction for detector pulse pile-up effects. Although
18
Ne is an attrac-
tive case to study because of the large isospin symmetry breaking correction of
δC = 0.620(32)% [23] predicted for this decay, the present experimental f t value
165
f t = 2912(78) s is limited by the superallowed branching ratio and remains too
imprecise to test this theoretical prediction. In order to provide this test, an improvement to the 18 Ne branching ratio precision by a factor of 25 or more would have
to be realized which seems unlikely in the immediate future.
Stringent tests of the theoretical models for isospin symmetry breaking can instead
be performed using cases where all three quantities (β branching ratio, Q value, and
β decay half-life) are all amenable to precision measurements. Recent measurements
of the superallowed β branching ratio to the level of 0.01% [10] and the statistical
rate function to 0.03% [6] have been performed for the superallowed β + emitter 62 Ga.
The
62
Ga half-life prior this thesis work was dominated by a single measurement
T1/2 = 116.19 ± 0.04 ms [103], and was the least precisely known of the three quantities. Furthermore, as the A ≥ 62 decays have the largest predicted corrections for
isospin symmetry breaking effects (∼1.4% for 62 Ga), a new measurement of 62 Ga halflife was considered necessary to contribute to the stringent tests of isospin symmetry
breaking in this mass region. This new measurement is described in Chapter 6 of this
thesis.
166
Chapter 6
Half-life of 62Ga
High-precision measurements of the f t values for superallowed transitions in the
A ≥ 62 region can provide a rigorous test of the theoretical calculations of isospin
symmetry breaking because these decays have large predicted corrections (> 1%)
and show greater model dependency than in the lighter decays. The superallowed
f t value for
62
Ga, in particular, is presently the most precisely determined in the
A ≥ 62 region. Its superallowed branching ratio has recently been deduced to highprecision, BR = 99.859(8)% [41], using the excited 2+ states in the daughter
62
Zn
as collectors for the γ-decay flux from the weak and unobserved β decay branches
to high-lying 1+ states within the Q-value window. With this result, combined with
a recent high-precision Q-value measurement [6], the f t value for
62
Ga prior to the
present work, f t = 3075.6(14) s, was limited by the overall precision in the half-life.
Although six previous measurements of the
62
Ga half-life have been performed with
precisions to better than 0.3%, the previous world average, T1/2 = 116.17 ± 0.04 ms,
was dominated by a single measurement T1/2 = 116.19 ± 0.04 ms [103] that was 4
times more precise than any of the other five measurements. In this chapter, the
results of a new
62
Ga half-life determination that is 1.6 times more precise and 2.3σ
167
lower than that of Ref. [103] will be presented. When combined with all previous
half-life measurements this work leads to a 0.9σ decrease and a 20% improvement
in the overall precision of the
62
Ga f t value. This new
62
Ga f t value now rivals the
precision of the best measured superallowed decays and leads to an improved test
of the theoretical calculations for isospin symmetry breaking corrections in A ≥ 62
nuclei.
The superallowed β + decay of
62
Ga to the daughter
ground-state transition occurring in 99.859(8)% of
62
62
Zn is a ground-state-to-
Ga β decays. This decay is
therefore not suited to a half-life determination via γ-ray photopeak counting and
one must therefore rely on the direct β counting technique. Because β + decay results in a 3-body final state the β + particles are emitted with a broad spectrum of
energies ranging from zero up to the β decay Q value. Unlike γ-ray counting, one
therefore cannot use the β energies to discriminate one decay from another. This is
an advantage in the sense that pile-up does not effect these types of measurements.
However, this can also be a disadvantage because the peak-to-background can be
relatively poor in β counting experiments as contaminant and daughter activities
cannot be discriminated against and suppressed. The technique of direct β counting
for half-life determinations is simpler because one does not require a shaped pulse to
extract energies and thus dead-time corrections are typically much smaller than in
γ-ray counting measurements. Any pulse that exceeds a set threshold is counted as
a single β event and if this pulse happens to be larger or malformed, due to a second
β particle arriving within a short time-interval τp afterwards, the first event is not lost
because no energy gate is taken and the second event (which is lost) will be included
in the correction for dead-time losses.
168
The 62 Ga half-life experiment was performed at the ISAC facility using a radioactive beam of ∼ 8000
62
Ga ions/s produced following the bombardment of a ZrC
production target (14.78 g/cm2 Zr) by 35µA of 500 MeV protons from the TRIUMF
main cyclotron. Spallation reaction products diffused from the ZrC target surface and
were ionized using the TRIUMF Resonant Ionization Laser Ion Source (TRILIS) [71]
which was tuned to selectively ionize Ga isotopes. Compared to the
62
Ga beams
produced with TRILIS in an earlier experiment [10], the present beam intensity of
∼ 8000
62
Ga ions/s in the experiment described here was 2 times larger, while the
level of isobaric contamination, specifically
62
Cu/62 Ga = 0.8(2), was lowered by more
than an order of magnitude by using a bare Ta transfer tube without a Re foil to
suppress surface ionization. Following laser ionization, mass separated
62
Ga was ex-
tracted as a 1+ ion beam and delivered to the fast-tape-transport system and 4π gas
proportional counter in the ISAC experimental hall (see Sec. 2.5).
The low-energy (30 keV) beam was implanted, under vacuum, into a 25 mm wide
aluminized mylar tape for ∼ 0.5 s or approximately 4.5
62
the collection, the beam was turned off and the sample of
Ga half-lives. Following
62
Ga was moved rapidly
(36 cm in 130 ms) out of vacuum and into the 4π proportional counter. The gas
counter was operated in the plateau region, as determined by a
90
Sr source, which
corresponded to voltages between 2600 and 2850 V. The β particles from the decay
of the sample were multi-scaled using 2 independent CAMAC multi-channel scaler
modules (MCS’s) into 250 bins of adjustable bin-time widths. The bin times were
varied on a run-by-run basis between 8, 10, and 12 ms in this experiment. The
decay collection time therefore spanned 2.0-3.0 s or 17-26 62 Ga half-lives. A Stanford
Research Systems 1 MHz ± 1 Hz precision laboratory clock scaled to 100 kHz was
used to provide a time standard for the experiment. The clock was calibrated after the
169
Table 6.1: Single cycle timing sequence as determined by a 100 kHz oscillator for a
3.0 s counting cycle. These values were kept constant during the 62 Ga experiment
with the exception of the counting time which was altered on a run-by-run basis
betweeen 2.0, 2.5, and 3.0 seconds.
Time (s)
0.00001
0.53250
0.53750
0.53750
0.55150
0.60000
0.61000
0.62000
0.62050
0.66200
3.62200
3.62200
Process
Beam
Beam
Upstream brake
Downstream brake
Tape movement
Tape movement
Upstream brake
Downstream brake
Multi-channel scalars
Counting
Counting
Cycle
Status
on
off
released
released
begins
ends
engaged
engaged
awakened
starts
finishes
complete
experiment and yielded 99.999397 kHz. A summary of the timing for a single cycle
is given in Table 6.1. Non-extendible and fixed dead-times per-event of τ1 ≈ 3 µs
and τ2 ≈ 4 µs were applied to each of the multi-channel scalars. These dead times
were chosen to be much longer than the series dead times in the system and were
measured to be τ1 = 2.9489 ± 0.0079 µs and τ2 = 3.9671 ± 0.0079 µs using the sourceplus-pulser technique [89]. One dead time was applied to each of the MCS’s and
they were interchanged throughout the experiment to investigate possible systematic
effects. To further explore potential systematics associated with the electronics, the
detector operating voltage, lower-level-discriminator threshold, and dwell times were
also altered on a run-by-run basis. A table summarizing the parameters for each run
is given in Table 6.2.
A total of 83445 cycles were collected in this experiment. In the offline analysis,
170
Table 6.2: Run-by-run summary of the electronic settings used in the 62 Ga experiment. The column titles of “Min” and “Good” refer to the minimum cycle threshold
applied to each run and the total number of good cycles that remained after the
pre-selection criteria were applied, respectively. The “TAPE” column is whether the
sample was collected at the begining (B), middle (M), or end (E) of the tape.
Run
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Voltage Disc.
(V)
(mV)
2700
100
2600
100
2800
100
2800
125
2650
125
2750
125
2600
125
2800
100
2700
100
2600
100
2600
75
2750
75
2650
75
2800
75
2800
75
2700
75
2600
75
2600
100
2750
100
2650
100
2850
100
2850
125
2650
125
2650
50
2650
50
2700
50
2750
50
2850
50
Dwell
(s)
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.10
0.08
0.08
0.08
0.08
τdOLD
(µs)
4
4
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
171
τdNEW
(µs)
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
TAPE
Min
Cycles
M
E
B
M
M
E
B
E
B
M
M
E
B
E
B
E
B
E
B
E
B
E
B
E
B
E
B
E
420
500
490
480
440
430
580
490
420
570
400
400
400
350
400
450
470
450
480
450
470
440
420
420
360
290
300
280
1469
1092
1508
1304
1434
1369
1231
1553
1480
894
1459
1313
1479
1068
1358
1310
1387
1163
1538
1011
1451
1639
1668
1092
1256
1147
1564
1426
Table 6.3: Run-by-run summary of the electronic settings used in the 62 Ga experiment. The column titles of “Min” and “Good” refer to the minimum cycle threshold
applied to each run and the total number of good cycles that remained after the
pre-selection criteria were applied, respectively. The “TAPE” column is whether the
sample was collected at the begining (B), middle (M), or end (E) of the tape.
Run
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
Voltage Disc.
(V)
(mV)
2800
125
2700
125
2750
125
2850
125
2850
75
2650
75
2750
75
2800
75
2800
75
2800
50
2700
50
2850
50
2850
50
2700
50
2700
100
2600
100
2600
100
2600
125
2600
125
2600
75
2750
75
2650
50
2800
50
2800
125
2700
50
2700
100
2600
50
2850
50
Dwell
(s)
0.08
0.08
0.08
0.08
0.08
0.08
0.08
0.08
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
τdOLD
(µs)
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
172
τdNEW
(µs)
4
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
TAPE
Min
Cycles
B
E
B
E
B
E
B
E
B
E
B
E
B
E
B
E
B
M
E
B
E
B
E
B
E
B
E
B
240
240
260
340
280
280
200
290
360
350
350
410
340
380
430
430
370
470
380
450
500
500
480
630
430
420
430
430
1703
1490
1580
1602
1379
1558
1465
448
1475
1531
1575
1472
1602
1291
1184
519
1357
938
456
1340
1213
1218
1243
1284
1402
1350
1069
1379
430-950 gas counts/cycle acceptance window
Counts per Cycle
1000
750
500
250
0
0
200
400
800
600
Cycle Number
1000
1200
1400
Counts in Gas Counter / Counts in Scintillator
2.25
1.15-1.85 gas counter/scintillator acceptance window
2.00
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.00
0
200
400
800
600
Cycle Number
1000
1200
1400
Figure 6.1: Cycle-by-cycle plots of (upper panel) the total number of counts per cycle
in the gas counter for MCS1 and (lower panel) the ratio of the of the number of counts
recorded in the gas counter to that of a scintillator located at the beam implantation
site. The cycle selection thresholds are indicated. In this run, no additional cycles
were removed due to the gas counter to scintillator ratio cut.
173
a threshold (unique to each run) was set that rejected any cycles for which the total
number of counts fell below a prescribed value. This rejection criterion removed a
total of 6417 cycles which were all cycles in which the primary proton beam had
tripped off. A second criterion used the ratio of the number of counts recorded by
the 4π gas counter to that of a scintillator located at the beam implantation site in
order to reject 3242 cycles where the 62 Ga sample was not accurately centered within
the gas counter. Sample spectra illustrating these pre-selection criteria are shown in
Fig. 6.1. Note that the cycles in the lower panel of Fig. 6.1 where the ratio is zero
are those cycles that were already rejected based on the first criterion. No new cycles
were removed for this run based on the gas counter to scintillator ratio. The 9659
cycles (11.6% of the total number of cycles collected) rejected by these criteria were
among the poorest statistically and contained only 5.9% of the raw data.
6.1
Half-life Analysis
Following the pre-selection criteria described above, a total of 73786 cycles remained,
divided amongst 56 experimental runs of approximately equal duration. Each cycle
was dead-time corrected using the measured dead times of τ1 = 2.9489 ± 0.0079 µs
and τ2 = 3.9671 ± 0.0079 µs and the procedure of Refs. [88, 90] that is described in
Appendix A. The maximum detector rate at the beginning of the counting period was
approximately 4000 counts per second which corresponded to a maximum dead-time
correction of ∼ 3% at t = 0.
In order to test for the presence of small isobaric contaminants in the massseparated A = 62 beam, a β-γ coincidence spectrum was obtained using a single
HPGe detector placed next to the gas counter (see Fig. 2.7). Because the beam-on
174
time of 0.5 s was not optimal to detect longer lived radioactive species, very little information regarding low-level beam contamination could be extracted from the
spectrum shown in the upper panel of Fig. 6.2. Instead the
62
Ga beam was delivered
to the 8π spectrometer and a second β-γ coincidence spectrum was obtained [41, 104]
using the 20 HPGe detectors of the 8π spectrometer and the 20 plastic scintillators
of SCEPTAR. The beam-on time was 30 s (as opposed to 0.5 s in the half-life measurement) which enhanced the relative activities of the longer-lived isobars. From
the β-γ coincidence spectrum obtained with the 8π, shown in the lower panel of
Fig. 6.2, isobaric contamination from
(T1/2 = 1.50(4) min [106]), and
62m
62
Cu (T1/2 = 9.67(3) min [105, 106]),
62g
Co
Co (T1/2 = 13.91(5) min [106]) was observed.
62
Relative yields, with respect to
Ga, of these isobars were calculated using the
known β branching ratios [106], a relative HPGe efficiency calibration using standard
56
Co, 133 Ba and 152 Eu sources [41, 104], and the spectral areas of the fitted photopeaks.
The β-γ coincidence spectrum at the 8π spectrometer was obtained during a beam-on
time of ton ≈ 30 s and the activities of all species are expected to grow-in according
to,
Ai (t) = Ri (1 − eλi t ),
(6.1)
where A(t) is the activity, R is the beam rate (or yield), λ is the decay constant, and
the index i represents the particular contaminant
62g
Co,
62m
Co,
62
Cu,
62
Mn, or
62
Fe
being considered. The number of decays of each species Di that occur during the
beam on period ton can be obtained from the integral of the activity over the 30 s
beam-on time,
Di =
Z
0
ton
Ri (1 − eλi t )dt,
Ri =
λi ton + e−λi ton − 1 .
λi
175
(6.2)
10
6
400
511
10
62
+
954 keV Ga β decay
40
300
5
1460 keV ( K)
Counts per 1 keV
200
10
4
100
60
10
10
1173, 1332 keV ( Co)
3
954
1000
1200
1400
2
10
1
0
10 0
1000
2000
3000
4000
Eγ (keV)
10
10
7
10
6
4
511
Counts per 1 keV
10
1388 keV ( Ga)
3
5
10
4
2
62m
1163 keV (
954
10
10
40
1460 keV ( K)
62
10
10
62
954 keV ( Ga)
1000
3
1388
62
Co), 1173 keV ( Cu,
1200
Co)
1400
2227
2
10
62g
3494
1
0
10 0
1000
2000
3000
4000
Eγ (keV)
Figure 6.2: Summed γ-ray spectra from (upper panel) the single HPGe detector in
coincidence with the gas counter at GPS, and (lower panel) the 20 HPGe detectors
of the 8π spectrometer when detected in coincidence with the SCEPTAR array. (Insets) Isobaric contamination from the decays of 62 Cu, 62g Co, and 62m Co was directly
observed at the 8π (but not at GPS) while only upper limits were deduced from the
8π spectrum on the possible contributions from 62 Mn and 62 Fe.
176
Table 6.4: Isobaric contamination in the A = 62 beam deduced from β-γ coincidences
between the 20 HPGe detectors of the 8π spectrometer and the 20 plastic scintillators
of the SCEPTAR array. The relative yields Rrel and activities Arel in the half-life
measurement at t = 0 are calculated (following a 0.5 s grow-in time and 0.13 s tape
movement) relative to 62 Ga.
Decay
parent
62
Ga
62
Cu
62g
Co
62m
Co
62
Fe
62
Mn
T1/2
Eγ
(keV)
Mi
fiγ
(%)
ǫγrel
(%)
116.17(4) ms
9.67(3) min
1.50(4) min
13.91(5) min
68(2) s
0.88(15) s
954
876
1173
1163
506
877
5596(83)
155(37)
166(78)
185(31)
< 6048
< 30
0.08
0.15
84.6
71.1
100.0
90.0
1.00
1.06
0.87
0.88
1.46
1.06
Rrel
×10−4
S
AGP
rel
×10−4
10000
10000
7900(1900) 11.2(27)
3.0(14)
0.028(13)
35(6)
0.035(6)
< 43
< 0.53
< 0.047
< 0.033
The number of photopeaks that will be observed in the 8π during this time will be
decreased from the total number of decays by the fraction of those β decays that yield
the γ ray of interest fiγ , and further by the absolute photopeak efficiency of the 8π
spectrometer at the particular γ-ray energy ǫγi . The measured number of photopeaks
for each species Mi is therefore given by,
Mi =
Ri fiγ ǫγi λi ton + e−λi ton − 1 .
λi
(6.3)
The quantities Mi , fiγ , ǫγi , and λi for all 6 species relevant to this work are listed in
Table 6.4. Relative yields or beam intensities Ri of each species, with respect to 62 Ga,
can then be deduced from Eqn. 6.3. In order to relate these quantities to the half-life
experiment, the relative activities at t = 0 in the gas counter must be calculated from
the relative rates obtained from Eqn. 6.3 and combined with a beam-on time of 0.5 s
and a tape-move time of 0.13 s via,
S
AGP
(t = 0) = Ri (1 − eλi ton )e−λi tmove .
i
(6.4)
S
The relative yields, Rrel , and activities (at t = 0), AGP
rel , in the half-life experiment
177
10
5
Residual (yi-yfit)/σi
Counts per 10 ms
10
4
3
2
1
0
-1
-2
-3
µ = -0.005(64), σ = 1.006
0.0
10
0.5
3
1.0
1.5
Time (s)
2.0
2
χ /ν = 1.02
10
2
0.0
62
T1/2( Ga) = 116.06 (15) ms
0.5
1.0
1.5
2.0
2.5
Time (s)
Figure 6.3: Typical dead-time corrected decay curve from a single 62 Ga run (run 23)
summed over 1668 cycles. (Inset) The residuals; (yi -yf it )/σi , while not used directly
in the Poisson maximum likelihood fit, remain a measure of the goodness of fit and
yield a mean of µ = -0.005(64) and standard deviation of σ = 1.006, values that are
consistent with the expectation of a normal distribution.
are also summarized in Table 6.4. The 8π yield experiment was performed immediately after the half-life experiment, and the measured ratios are expected to differ
by much less than 20% (the most precisely determined ratio in Table 6.4) in these
subsequent measurements. Although γ-ray photopeaks expected from the decays of
62
Mn (T1/2 = 0.88(15) s [106, 107]) and
62
Fe (T1/2 = 68(2) s [106]) were not observed
in the 8π β-γ coincidence spectrum, upper limits were set on their existence using the
+1σ uncertainties following null-area fits to the expected locations of the photopeaks.
The cycle-by-cycle dead-time corrected decay data were summed into a single
decay curve for each experimental run, corrected by the clock calibration, and were fit
178
62
116.8
T1/2( Ga) = 116.100 ± 0.022 ms
2
χ /ν = 0.77
62
Ga Half-life (ms)
116.6
116.4
116.2
116.0
115.8
115.6
115.4
0
10
5
15
20
25
30
35
40
45
50
55
Run Number
Figure 6.4: Half-life of 62 Ga versus the experimental run number. The weighted
average of all 56 runs and its statistical uncertainty T1/2 = 116.100 ± 0.022 ms are
displayed as horizontal solid and dotted lines, respectively.
to a function that contained 4 exponentials (62 Ga, 62 Cu, 62g Co, 62m Co) plus a constant
background. The t = 0 relative activities and half-lives for each of the contaminants
were fixed at the central values listed in Table 6.4. The fit function therefore contained
only 3 free parameters: (i) the activity of 62 Ga at t = 0, (ii) the half-life of
62
Ga, and
(iii) a constant background rate. This procedure is defined as the “best-fit” result for
extracting the half-life of
62
Ga. In Sec. 6.1.1 below, the effects of the uncertainties
on the fixed parameters are considered and are tested for consistency in the fitting
procedure by using additional permutations of the fit function including the upper
limits of
62
Mn and
62
Fe isobaric contamination. A sample dead-time corrected decay
curve, resulting fit, and corresponding residuals; (yi -yf it )/σi , from a single run (Run
179
23, 1668 cycles) is presented in Fig. 6.3.
The half-lives of
62
Ga obtained from each of the 56 runs (with statistical uncer-
tainties) are shown in Fig. 4.10. A weighted average of these 56 runs yields the
62
Ga
half-life (and statistical uncertainty) deduced in this work, T1/2 = 116.100 ± 0.022 ms,
with a reduced χ2 value of 0.77.
6.1.1
Systematic Uncertainties
Two separate and independent multi-channel scalers were used to bin the decay data,
with each MCS receiving a different fixed and non-extendible dead time of either
3 or 4 µs. These dead times were periodically swapped throughout the experiment. The half-lives of
62
Ga obtained using each of the MCS data streams were
MCS2
MCS1
= 116.099 ± 0.022 ms. Because the two
= 116.101 ± 0.022 ms and T1/2
T1/2
scalers independently bin the same decay data, these are not independent measurements of the
62
Ga half-life but instead provide an important consistency check of
the dead-time corrections. Because these two values are consistent, the unweighted
average, T1/2 = 116.100 ± 0.022 ms, is taken as the best measure of the
62
Ga half-
life. The data shown in Fig. 6.4 already include the unweighted average, for each
run, of the 2 MCS data streams. As demonstrated in Fig. 6.5, separation of the
data set into the two dead-time values of 3 and 4 µs yielded identical results of
T1/2 = 116.100 ± 0.022 ms for the
62
Ga half-life.
In order to test for further potential systematic uncertainties, several electronic
settings were modified throughout the experiment on a run-by-run basis. These
modifications included altering the detector voltage within the plateau region (26002850 V), changing the MCS bin times (and hence the decay time length) between
180
116.3
62
T1/2( Ga) = 116.100 ± 0.022 ms
62
Ga Half-life (ms)
116.2
116.1
116.0
115.9
115.8
2
χ /ν = 0.45
0.25
0.91
1.29
Entire Data Set
MCS 1; MCS 2
3µs MCS 1; 4µs MCS 1; 3µs MCS 2; 4µs MCS 2
all 3µs; all 4µs
Discriminator (mV) 0.50; 0.75; 1.00; 1.25
Bin Time (ms) 8; 10; 12
Voltage (V) 2600; 2650; 2700; 2750; 2800; 2850
Start of tape spool ; Middle; End
115.7
Figure 6.5: Half-life measurements of 62 Ga sorted by adjustable electronic and experimental settings. All reduced χ2 values for the independent groups are less than
unity with the exception of the tape position value χ2 /ν = 1.29 which is used as an
estimate of possible sources of systematic uncertainty in this analysis.
8 ms/bin (2.0 s decay) and 12 ms/bin (3.0 s decay), and adjusting the lower-leveldiscriminator threshold between 50 and 125 mV. A summary of the
62
Ga half-life
obtained at each of the adjustable settings considered is shown in Fig. 6.5. Of the 56
runs collected in this experiment 15 were obtained using a lower-level-discriminator
setting of 50 mV, 14 were at 75 mV, 14 at 100 mV, and 13 at 125 mV. Because this
group of 4 settings contain all of the experimental data, the weighted average of these
4 groups is the total average, T1/2 = 116.100 ± 0.022 ms. Treating these 4 settings
as 4 independent measurements of the
62
Ga half-life (with 3 degrees of freedom) a
reduced χ2 value of 0.45 is obtained. According to the method of the Particle Data
181
Group [4] a reduced χ2 value that is less than unity indicates that the
62
Ga half-life
obtained is consistent with there being no systematic uncertainty associated with the
4 discriminator settings. A similar analyis was performed using the 3 bin time values
of 8, 10, and 12 ms (2 d.o.f, χ2 /ν = 0.25), and the 6 gas-counter voltage settings of
2600, 2650, 2700, 2750, 2800, and 2850 V (5 d.o.f, χ2 /ν = 0.91) indicating that these
groups are also consistent with there being no systematic uncertainties pertaining to
these settings.
Although not an electronic grouping, a fourth setting was considered that grouped
the data according to the beam implantation position within the tape spool. The
aluminized mylar tape at the β counting station is not a continuous loop but is
collected on a spool. Due to its finite length, it was necessary to rewind the tape
after every second or third run which permitted a grouping of the data based on
whether the run was obtained before or immediately after a tape rewind. Of the 56
runs collected in this experiment, 25 were obtained after a rewind (at the beginning of
the tape spool), 25 were obtained before a rewind (at the end of the tape spool), and
6 were obtained in the middle of the tape when 3 experimental runs were collected
between tape rewinds. The reduced χ2 value obtained from these 3 settings (with
2 degrees of freedom) is χ2 /ν = 1.29. Although it is unclear as to how there could
be any potential systematic effect associated with the tape spool location, to be
cautious, the method of the Particle Data Group [4] is used to inflate the statistical
uncertainty of 0.022 ms by the square root of the reduced χ2 value which leads to an
overall uncertainty of 0.025 ms. Assuming the statistical and systematic uncertainties
are independent quantities and can be combined in quadrature to obtain the overall
uncertainty, the half-life of 62 Ga deduced in this work is T1/2 = 116.100 ± 0.022(stat.)
± 0.012(sys.) ms.
182
116.25
62
T1/2( Ga) = 116.100 ± 0.0022 ms
62
Ga Half-life (ms)
116.20
116.15
116.10
116.05
116.00
115.95
0
10
20
30
Number of Leading Channels Removed
Figure 6.6: Deduced half-life of 62 Ga as a function of the number of leading channels
removed from the analysis with the half-life and statistical uncertainty (at 0 channels
removed) overlayed for comparison. The 62 Ga half-life obtained in this work remains
constant even after 3 half-lives, or 88% of the data, have been removed. These data
are not randomly scattered about the mean because they are highly-correlated, with
each data point containing all of the data to the right of it.
In order to test for any residual rate dependence in this result, leading channels
were removed from the data set in increments of 2 channels (∼ 20 ms) to a maximum
of 36 channels or ∼ 3 62 Ga half-lives. The result of this analysis is presented in Fig. 6.6
and demonstrates that the half-life of
62
Ga deduced in this work is consistent even
when 3 half-lives, or 88% of the data, has been removed from the analysis. Because
the beam intensity from TRILIS fluctuated between 4000 to 8000
62
Ga ions/s, a
complementary test for residual rate dependencies was performed by plotting the
cycle-averaged detector rate at the start of the decay curve versus the half-life obtained
183
116.8
116.4
62
3
Ga half-life×10 (s)
116.6
−8 2
Slope = (-1.7 ± 3.0)×10 s
T1/2 = 116.100 ± 0.022 ms
116.2
116.0
115.8
115.6
115.4
2000
3000
4000
5000
Detector rate at t = 0 (counts/s)
Figure 6.7: Half-life of 62 Ga versus the detector rate at t = 0 for the 56 runs analyzed
in this work. The best fit result (solid line) T1/2 = 116.100(22) ms is obtained by
fitting these data to a constant. A weighted linear regression of these data (dashed
line) yields a slope of (-1.7 ± 3.0)×10−8 s2 and is consistent with there being no rate
dependence in the deduced half-life of 62 Ga.
for each of the 56 runs as shown in Fig. 6.7. A weighted linear regression applied to
these data resulted in a slope of (-1.7 ± 3.0)×10−8 s2 and confirms the expectation
that the deduced half-life is rate independent.
Tests of the fit function were investigated by refitting the decay-curve data to a
variety of functions that either included or removed the contributions of the various
isobaric contaminants. The largest contamination in this experiment came from the
decay of
62
Cu which had a relative activity at t = 0 of only 11.2(27)×10−4 (see
Table 6.4) and a half-life (∼ 10 min) that was very long on the data collection time
scale of 2.0-3.0 s used in this experiment. The data were refit using only a single
184
exponential (62 Ga decay) plus a free constant background under the assumption that
the
62
Cu activity could be approximated by a constant background. The result of
this analysis yielded T1/2 = 116.101 ± 0.022 ms and is in excellent agreement with
the best-fit result T1/2 = 116.100 ± 0.022 ms that used 4 exponentials plus a constant
background. This analysis demonstrates the independence of the final result on the
low levels of isobaric contamination and any small variation in this ratio between the
half-life measurement and the γ-ray experiment with the 8π spectrometer that was
used to fix the relative contaminant activities. The data were also refit under the
assumption that all of the background was due to the decay of
T1/2 = 116.097 ± 0.022 ms was obtained for the
62
62
Cu and the value
Ga half-life, which is again in
excellent agreement with the best-fit result.
In order to test for additional sources of systematic uncertainty associated with
possible isobaric contamination from
62
Mn and
62
Fe decay, these exponential decays
were added to the fit function with their half-lives and intensities fixed at the values
listed in Table 6.4. The fit function in each case consisted of 5 exponential decays plus
a constant background (3 free parameters) and yielded T1/2 = 116.098 ± 0.022 ms
for the half-life of
62
Ga when the
62
Mn upper limit was included in the fit func-
tion and T1/2 = 116.099 ± 0.022 ms when including only
62
62
Fe. Because
62
Mn and
Fe decay were not directly observed in the β-γ coincidence spectrum, the best-fit
value for the half-life of
62
Ga is not altered as a result of this analysis but rather
the differences between the results when including these possible contaminants and
the best-fit answer (∆T = 1.7 × 10−3 ms for
62
Mn) is retained as a measurement of
an unaccounted for systematic uncertainty in the best-fit result. A recent measurement of the
62
Mn half-life T1/2 = 0.67(5) s [108] is significantly lower than the value
185
Table 6.5: Differences |∆T | between the best-fit 62 Ga half-life using parameters fixed
at their central values and the result obtained when these parameters were fixed at
their ±1σ uncertainties. Treating each of these parameters independently, a total
estimate of the systematic uncertainty is obtained from the quadrature sum.
Fixed Parameter
Value
Intensity I(62 Cu/62 Ga)
Half-life T1/2 (62 Cu)
Intensity I(62g Co/62 Ga)
Half-life T1/2 (62g Co)
Intensity I(62m Co/62 Ga)
Half-life T1/2 (62m Co)
Include I(62 Mn/62 Ga)
Half-life T1/2 (62 Mn)
Alternate T1/2 (62 Mn)
Include I(62 Fe/62 Ga)
Half-life T1/2 (62 Fe)
1.12(27)×10−3
9.67(3) min
2.8(13)×10−6
1.50(4) min
3.5(6)×10−6
13.91(5) min
3.3×10−6
0.88(15) s
0.67(5) s
5.3×10−5
68(2) s
2.9489(79) µs
3.9671(79) µs
Measured dead times
Total
|∆T | (ms)
±
±
±
±
±
±
±
±
±
±
±
5.8×10−4
6.7×10−6
1.8×10−5
1.0×10−6
9.7×10−7
2.6×10−8
1.7×10−3
3.6×10−4
1.4×10−4
9.7×10−4
9.4×10−4
± 7.7×10−4
0.0024 ms
T1/2 = 0.88(15) s [107, 106]. If instead the most recent value is used, the
62
Ga half-
life deduced is T1/2 = 116.098 ± 0.022 ms, and differs from the value above by only
∆T = 1.4 × 10−4 ms. This difference, which arises from a systematic uncertainty in
the 62 Mn half-life, is also included in the total estimate of the systematic uncertainty
in this work.
A similar procedure was adopted to account for additional sources of systematic
uncertainty associated with fixing specific parameters in the analysis at their central values. For all of the parameters that were fixed in order to arrive at the best-fit
answer the above analysis was repeated for each parameter fixed at ± 1σ from its central value. The differences between the best-fit result of T1/2 = 116.100 ± 0.022 ms
and the half-life obtained using the ± 1σ values are summarized in Table 6.5 for
186
all fixed parameters. A total systematic uncertainty associated with fixing parameters is 0.0024 ms and is obtained from the quadrature sum of the ∆T column in
Table 6.5. This uncertainty is negligible when combined in quadrature with the statistical uncertainty of 0.022 ms. The half-life of 62 Ga deduced in this work is therefore
T1/2 = 116.100 ± 0.022(stat.) ± 0.012(sys.) ms.
6.2
Comparison to Previous Results
The half-life of
62
Ga, (adding the statistical and systematic uncertainties in quadra-
ture), T1/2 = 116.100 ± 0.025 ms is precise to the level of 0.022% and represents the
most precise measurement of any superallowed half-life to date. Compared to previous measurements of the
62
Ga half-life (Table 6.6, Fig. 6.8), this result is a factor of
1.6 times more precise than Ref. [103] and is 7 times more precise than any of the
other five previous determinations [109, 110, 111, 112, 37]. Taking a weighted average
of all seven half-life measurements, shown in Fig. 6.8 and Table 6.6, yields the world
average T1/2 = 116.121 ± 0.021 ms. This value is 0.04% or 2.3σ lower than the previous world average [1] and reflects the fact that the value deduced in Ref. [103] does
not agree with the measurement presented here at the level of 0.08% or 2.3σ. While
the reason for the discrepancy between the two highest precision measurements is not
understood, the reduced χ2 value obtained from the full set of seven
62
Ga half-life
measurements is 1.006. Treated as a group, these seven measurements are therefore
a consistent set and the uncertainty on the world average need not be increased according to the method of the Particle Data Group. As a result of the high-precision
half-life determination presented in this work, the average 62 Ga half-life has thus been
decreased by 2.3σ to T1/2 = 116.121 ± 0.021 ms and its overall precision of 0.018%
187
116.8
62
T1/2( Ga) = 116.121 ± 0.021 ms
2
χ /ν = 1.006
116.4
116.2
115.8
115.6
[Alb78] [Dav79] [Hym03] [Bla04][Can04] [Hyl05]
Present work
116.0
62
Ga Half-life (ms)
116.6
115.4
Figure 6.8: Comparison of all high-precision 62 Ga half-life measurements. The new
world average of T1/2 = 116.121 ± 0.0021 ms with a reduced χ2 value of 1.006 is
obtained from a weighted average of these 7 measurements and is overlayed for comparison.
Table 6.6: Summary of all high-precision 62 Ga half-life measurements. The new world
average of T1/2 = 116.121 ± 0.0021 s with a reduced χ2 value of 1.006 is obtained
from a weighted average of these 7 measurements.
Reference
present work
B. Hyland et al. [37]
G. Canchel et al. [112]
B. Blank et al. [103]
B.C. Hyman et al. [111]
C.N. Davids et al. [110]
D.E. Alburger [109]
Year
2007
2005
2005
2004
2003
1979
1978
World Average (χ2 /ν = 1.006)
188
T1/2 (ms)
116.100
116.01
116.09
116.19
115.84
116.34
115.95
σ (ms)
0.025
0.19
0.17
0.04
0.25
0.35
0.30
116.121
0.021
has been improved upon by nearly a factor of 2. The half-life of 62 Ga is now the most
precisely determined for any of the superallowed emitters.
6.3
62
Present Status of the
Ga f t and Ft values
With the significant decrease of 2.3σ (0.04%) in the average 62 Ga half-life, the experimental f t value for this decay, which was previously known to 0.05% and was limited
by the uncertainty in the half-life, is also significantly impacted. Combining the world
averaged 62 Ga half-life, T1/2 = 116.121(21) ms, with the superallowed branching ratio
BR = 99.859(8)% [41], the calculated electron conversion fraction PEC = 0.137% [1],
and the statistical rate function f = 26401.6(83) [6] one obtains,
f t(62 Ga) = 3074.3(2)BR (5)T1/2 (10)f s
= 3074.3(11) s,
for the
62
(6.5)
Ga f t value. This result is precise to 0.04% and is now limited by the
precision in the statistical rate function f (or Q value). As a result of the half-life
measurement presented in this work, the
62
Ga f t value has been decreased by 0.9σ
and its overall precision has been improved by more than 20% compared with its
previous value [10]. With this 20% improvement, the
62
Ga f t value is now one of the
most precisely determined for any of the superallowed decays.
Using the correction terms δR′ = 1.459(87)% [24], δN S = −0.036(20)% [24] and
δC = 1.38(16)% [23], the corrected f t value for
62
Ga can be expressed as,
F t(62 Ga) = 3075.0(6)δNS (11)f t (26)δR′ (50)δC s
= 3075.0(58) s.
(6.6)
This result has been reduced by 0.9 s compared to its previous value [10] due to the
189
half-life measurement presented here. It is in excellent agreement with the world
average F t = 3075.0(12) s calculated in Chapter 1 of this thesis (Eqn. 1.48) but
is entirely limited in precision by the theoretical corrections for isospin symmetry
breaking δC and radiative effects δR′ .
A test of the δC corrections can be performed by calculating the isospin symmetry
breaking correction that is required to satisfy the CVC hypothesis. Using the value of
F t(TH) = 3074.0(8) s obtained in Chapter 1 (Eqn. 1.46) using the isospin symmetry
breaking corrections of Towner and Hardy, the deduced value of δC can be obtained
from Eqn. 1.45 and expressed as,
δC (62 Ga) = 1 −
F t(TH)
,
f t(1 + δR )
= 1.41(2)δNS (3)F t (4)f t (8)δR′ %,
= 1.41(10) %.
(6.7)
This result is in excellent agreement with the Woods-Saxon model calculations of
Towner and Hardy that predict δC = 1.38(16)% [23]. A similar test of the selfconsistent Hartree-Fock calculations was performed using the world average F t value
F t(OB) = 3075.8(8) s obtained with the δC corrections of Ormand and Brown. The
result,
δC (62 Ga) = 1.35(2)δNS (3)F t (4)f t (8)δR′ %,
= 1.35(10) %,
(6.8)
also agrees with the calculated range of values δC = 1.26-1.32% [34].
The value for δC deduced under the assumption that CVC is satisfied, is 1.6 times
more precise than the theoretical values and may be used to further constrain the
theoretical calculations of isospin symmetry breaking in A ≥ 62 superallowed β decays. It should be stressed, however, that the deduced value of δC for
190
62
Ga is now
entirely limited by the uncertainty in the δR′ calculation for this high-Z superallowed
emitter. A significant sharpening of this test of the isospin symmetry breaking corrections could thus be achieved with a reduction in the uncertainties of δR′ for the heavy
superallowed emitters by extending the radiative corrections to higher order [27].
6.4
Summary
The half-life of the superallowed β + emitter 62 Ga has been deduced using a 4π proportional counter and fast-tape-transport system. The result, T1/2 = 116.100 ± 0.025 ms,
is the most precise measurement of any superallowed half-life to date and leads to a
2.3σ decrease in the
62
Ga world average half-life. The new world average, obtained
from a weighted average of all seven measurements to date yields T1/2 = 116.121 ± 0.021 ms
with a reduced χ2 value of 1.006. Combining the average
measurements of the
62
62
Ga half-life with recent
Ga decay Q value and the superallowed β branching ratio
yields f t = 3074.3(11) s for
value [10]. The
62
62
Ga, which has been reduced by 0.9σ from its previous
Ga f t value is now known to ± 1.1 s or 0.04%, and rivals the
precision of the best known superallowed f t values for A < 62. This high-precision
superallowed f t value provides a new benchmark for tests of isospin symmetry breaking calculations for the A ≥ 62 superallowed β decays. Improved precision of the
theoretical radiative corrections for the high-Z superallowed emitters will, however,
be required to improve this test. The precision of the experimental
62
Ga f t value
is now limited by the precision of the statistical rate function f that results from a
single high-precision Q-value measurement [6] and an independent confirmation of
this result would also be highly desirable.
In Chapter 4 of this thesis, the
26
Na calibration experiment was presented which
191
allows, for the first time, a method with which to deduce β decay half-lives via γray photopeak counting to the level of precision (0.05% or better) required by the
superallowed Fermi β decay program. This technique was then applied to deduce
the half-life of
18
Ne, a Tz = -1 emitter with a relatively large isospin symmetry
breaking correction, in Chapter 5 to the level of 0.11% precision. In this Chapter
the half-life of
62
Ga was determined through the previously established technique of
direct β counting and consequences on isospin symmetry breaking in the A ≥ 62
mass region were discussed. It has been demonstrated throughout this work that
the specific method employed (either direct β or γ-ray counting) depends on the
specific decay being investigated. In the case of
62
Ga, for example, only 0.1% of all
β decays produce γ-ray activity which would statistically hinder a γ-ray counting
measurement. In the final chapter of this thesis several additional applications of
both of these techniques to the superallowed half-life measurement program will be
presented.
192
Chapter 7
Conclusion and Future Work
With the introduction of a novel γ-ray photopeak counting technique in this thesis
to deduce, to high-precision, β decay half-lives, there are many attractive examples
of superallowed decays for which such a technique would be beneficial. In Chapter 5
it was shown that a new measurement of the
18
Ne half-life, the first measurement of
a superallowed half-life using this technique, may have uncovered the reason behind
the disagreement between two previous measurements [93, 94] of this quantity. In
addition to
18
Ne, there are several additional examples of superallowed decays for
which the use of such a technique could significantly improve upon and complement
previous measurements. The superallowed decays of
10
C and
14
O are two specific
examples where previous direct β counting and γ-ray photopeak measurements have
led to systematically different results. A third example is the decay of the Tz = −1
superallowed emitter
34
Ar whose daughter,
34
Cl, has a half-life that is approximately
twice that of 34 Ar leading to a large covariance between fitted half-lives in the resulting
β spectrum. All of these examples will be discussed in detail below and will be used
to demonstrate the need for measurements of all three of these half-lives using the
newly developed technique of γ-ray photopeak counting. First, however, an additional
193
motivation for the study of
10
C and
14
O, the lightest superallowed emitters, will be
considered and is based on constraining the existence of scalar currents in the weak
interaction.
7.1
Scalar Interactions in Superallowed Decay
The existence of a fundamental scalar current, or any scalar current induced by the
vector part of the weak interaction, would lead to corrected F t values that would
no longer be constant for all superallowed decays. A scalar interaction would alter
the calculation of the statistical rate function f by including into the Fermi function
F (Z, W ) (see Eqn. 1.31) a term that is proportional to the inverse of the average decay
energy W . This correction function can be approximated by the expression [1, 49],
C(Z, W ) ≈ |MF |
2
CV2
bF γ
1+
+··· ,
W
(7.1)
where |MF |2 = 2 is the Fermi matrix element for superallowed decays between isospin
1
T = 1 states, CV is the normalized vector coupling constant, and γ = (1-(αZ)2) 2 .
The quantity bF is called the Fierz interference term and is given by [1],
CS ,
bF = ±2µ1 CV (7.2)
where ±µ1 is a β-decay Coulomb function for β ∓ decays that characterize the wave
functions of the electrons and is of order unity, and
CS
CV
is the ratio of the coupling
constants for the scalar and vector components in the β decay process.
The expression for the corrected F t values for superallowed Fermi β decays would
thus be modified in the presence of a non-zero scalar current yielding,
Ft ≈
2GV
2
1
K
,
V 1+b
(1 + ∆R )
194
(7.3)
3110
Ft
3105
= 3074.0 ± 0.8 s
2
χ /ν = 0.86
3100
74
Rb
3095
Ft (s)
22
3090
3085
10
3080
Mg
46
C
14
34
O
Ar
42
V
62
Sc
54
3075
3070
26m
Al
34
Cl
38m
K
3065
3060
0
10
Ga
50
Co
Mn
20
30
40
Z of daughter
Figure 7.1: Current F t values for the thirteen most precisely determined superallowed decays and weighted average F t with corresponding statistical uncertainty.
The curved-dashed lines represent the approximate loci the F t values would follow if
one includes a scalar interaction that contributes at the level of | CCVS | = ± 0.2%.
where b =
− 2µW1 γ
CS CV for β + decays. Constraints on the existence of scalar currents
were performed in the previous superallowed survey [1] by fitting the F t values shown
in Fig. 7.1 to a version of Eqn. 7.3 that included higher-order corrections. The
limit, deduced from the superallowed decays, on the existence of scalar interactions
is presently [1],
CS CV ≤ 0.0013.
(7.4)
The approximate curvature the F t values would follow with the relative scalar interaction | CCVS | = ± 0.002 is overlayed, for comparison, in Fig. 7.1 and demonstrates
the importance of the F t values, from the superallowed decays of
195
10
C and
14
O in
particular, for tightening the above limit on the existence of scalar currents in the
weak interaction.
7.1.1
Half-life of
Prior to the
62
14
O
Ga half-life measurement discussed in Chapter 6 of this thesis, the
most precisely determined superallowed β decay half-life was that of
14
O whose av-
erage, T1/2 = 70.620 ± 0.014 s, is comprised of eight measurements and is precise
to ± 0.019%. The β decay of
14
O is an ideal candidate for the γ-ray photopeak
counting method as 99.4% of all decays yield a 2.3 MeV γ-ray in the daughter
14
N
(see Fig. 7.2). Of the eight half-life measurements, six were deduced by the method
of γ-ray photopeak counting, while the remaining two (and the two most recent determinations) employed the method of direct β counting. In one of these β counting
measurements, the authors had attempted to use a HPGe detector to count 2.3 MeV
photopeaks. However, because a suitable method for performing a pile-up correction
to the level of precision required could not be found, the “use of a germanium detector
was abandoned” [86].
All eight
14
O half-life measurements separated by counting method are presented
in Fig. 7.3. Combining the six γ-ray photopeak counting measurements yields an
average value of T1/2 (γ) = 70.598 ± 0.017 s, while the two β counting measurements
yield T1/2 (β) = 70.648 ± 0.019 s. These results disagree with each other at the level
of 0.07% or 2.6σ. Treating these two values as independent measurements of the
14
O
half-life, the reduced χ2 value is ∼ 4.0 and the Particle Data Group method would
√
imply that the uncertainty in adopted half-life must be increased by a factor of 4 = 2.
If one instead assumes that the average of one of these two methods is correct, the
accepted
14
O half-life would either increase or decrease (depending on the adopted
196
14
O
ft = 3043.3 1.9 s
0+ 70620 14 ms
+
b
1635
1+
0+
3948
2313
0.0545 %
99.334 %
1+
0.611 %
Stable
14
N
Figure 7.2: Decay scheme for 14 O β + decay to 14 N. Because 99.4% of all 14 O β decays
yield a 2.3 MeV γ ray, measurements of the 14 O half-life can be efficiently performed
using the technique of γ-ray photopeak counting.
197
70.9
14
O Half-life (s)
70.8
T1/2(γ) = 70.598 ± 0.017 s
T1/2(β) = 70.648 ± 0.019 s
70.7
70.6
70.5
70.4
70.3
70.2
2
χ /ν = 0.77
γ-ray Counting
2
χ /ν = 0.97
β Counting
Figure 7.3: Previous 14 O half-life measurements separated by counting method.
The left plot contains six γ-ray measurements and an overall weighted average
T1/2 (γ) = 70.598 ± 0.017 s is obtained from Refs. [83, 84, 85, 115, 113, 114]. The right
plot contains the two most recent measurements that employed direct β counting and
yield the average T1/2 (β) = 70.648 ± 0.019 s from Refs. [35, 86]. These two methods
give half-lives that disagree at the level of 2.6σ or 0.07%.
method) by 0.022 s or 1.6σ from the present total average. This would have important
implications for precision tests of the Standard Model because the possibility exists
that one of the most precisely known F t values, F t(14 O,TH) = 3072.0(32) s, could be
shifted by more than 0.5σ. Shown in Fig. 7.4 is a plot of the present world F t values.
The insets show the same plot under the assumption that one of these two counting
techniques are correct. The approximate loci of the F t values with a scalar interaction
of ± 0.02% are overlayed for comparison.
198
3120
3085
3115
3080
3110
3075
3105
74
3070
3100
Ft (s)
b)
a)
10
3065
3095
22
3090
14
C
10
O
C
14
Rb
O
Mg
46
3085
34
3080
42
Ar
Sc
62
V
54
Ga
Co
3075
3070
26m
3065
3060
10
0
14
C
Al
O
10
34
38m
Cl
K
50
Mn
20
Ft
= 3074.0 ± 0.8 s
2
χ /ν = 0.86
30
40
Z of daughter
Figure 7.4: Systematic effect between γ-ray and β counting half-life determinations
on the 10 C and 14 O F t values. A scalar interaction of |CS /CV | = ± 0.2% is overlayed
for comparison. The insets show the same plot for 10 C and 14 O under the assumption
that (a) only the γ-ray counting half-life determinations are accepted (b) only the
β counting measurements are accepted.
A comparison between two new measurements of the 14 O half-life using both γ-ray
and direct β counting techniques to overall precisions of < 0.05% in each experiment,
would provide the necessary tests of the systematic differences observed between
these techniques. With the γ-ray pile-up correction technique developed in Chapter 3
of this thesis, there is now the capability to perform the γ-ray measurement with
confidence that the systematic effect due to γ-ray pulse pile-up is not biasing the
deduced half-life. While these two new half-life determinations may not significantly
improve the precision of the existing world data, they would have the potential to
identify the systematic bias present in one of these measurement techniques, enabling
199
the adoption of a consistent set of data for the
14
O half-life and corresponding F t
value.
7.1.2
Half-life of
10
C
Measurement of the half-life of
ing as 99.6% of all
ter
10
10
10
C is also equally suited for γ-ray photopeak count-
C β decays yield a 718 keV γ-ray transition in the daugh-
B (see Fig. 7.5). There are presently only two half-life measurements of
10
C
(T1/2 = 19.282 ± 0.020 s [115], T1/2 = 19.295 ± 0.015 s [116]), both of which were
determined through the technique of γ-ray photopeak counting. The accepted world
average of the
10
C half-life from these two measurements is T1/2 = 19.290 ± 0.012 s
and is precise to 0.06%. The author of the recent
14
O half-life article whose “inabil-
ity to find a justifiable and trustworthy method of analyzing the half-life data” [86]
is also responsible, in an older work [116], for the most precise
10
C γ-ray counting
half-life measurement that currently dominates the world average. Although the systematic effect of detector pulse pile-up was discussed in their work on the 10 C half-life,
this systematic could only be minimized by removing the high-rate data. Even after
this rejection procedure, the individual reduced χ2 values from the fits to the data
were anomalously large which led the authors to proclaim that the “final values are
somewhat worrying” [116].
A new measurement of the 10 C half-life via direct β counting has led to the preliminary value T1/2 = 19.312 ± 0.004 s [117]. This result is 1.8σ larger than the average
of the γ-ray measurements and suggests that the same systematic observed in the
14
O data is also present in the
10
10
C half-life result is correct, then the corrected F t value for this decay would in-
C data. If the assumption that the new β counting
crease by 0.7σ as demonstrated in Fig. 7.4b. Performing two new high-precision
200
10
C
10
C
ft = 3039.5 4.7 s
0+ 19290 12 ms
+
b
0+
1+
1022
1740
718
1.465 %
98.53 %
3+
Stable
10
B
Figure 7.5: Decay scheme for 10 C β + decay to 10 B. Because 99.6% of all 10 C β decays
yield a 718 keV γ ray, measurements of the 10 C half-life can be efficiently performed
using the technique of γ-ray photopeak counting.
201
half-life measurements to precisions of < 0.05% each, would improve the precision
of the existing world average, and perhaps more importantly, probe the accuracy of
these previous results by testing for the presence of systematic differences between
the techniques of γ-ray and β counting. As in the case of
14
O, the γ-ray pile-up
technique developed in Chapter 3 of this thesis now permits the γ-ray component of
the
10
C half-life measurements to be made with the confidence that pile-up correc-
tions are not a source of systematic bias. Significant shifts in the F t values for
and
14
10
C
O resulting from the rejection of results from one of the previous techniques
would both be in the same direction (dependent on which method is adopted) and
would therefore have important implications on the present limits on scalar current
contributions to the weak interaction (see Fig. 7.4).
7.2
Half-life of
34
Ar
Decay of the Tz = −1 superallowed β + emitter
34
Ar can also be used to constrain
isospin symmetry breaking corrections. The calculated values of δC = 0.64(4)% [23]
via the Woods-Saxon model and δC = 0.39(9)% [29] obtained with the self-consistent
Hartree-Fock calculation represents the largest difference between these calculations
for A < 62. Measurement of the half-life of this Tz = −1 superallowed β + emitter
also represents a considerable experimental challenge because the daughter,
a Tz = 0 superallowed emitter whose half-life is nearly twice that of
34
34
Cl, is
Ar. In general,
one can express the time-dependent activities of the parent, with decay constant λP ,
and the daughter, with decay constant λD , as,
AP (t) = λP No,P e−λP t ,
λP λD −λP t
λP
AD (t) = No,P
e
+ λD No,D − No,P
e−λD t ,
λD − λP
λD − λP
202
(7.5)
where No,P and No,D are the initial number of parent and daughter nuclei (at t = 0).
The total activity observed in a β counting experiment will be the sum of these two
activities which can be expressed as,
2λD − λP
λP
−λP t
A(t) = λP No,P
e
+ λD No,D − No,P
e−λD t .
λD − λP
λD − λP
(7.6)
In the special case where the decay constant of the parent is exactly twice that of the
daughter, λP = 2λD , the first term cancels and the total activity will appear exponentially distributed in time with a half-life corresponding to that of the daughter.
For
34
Ar, T1/2 (34 Ar) = 843.8(4) ms [12], and
34
Cl, T1/2 (34 Cl) = 1526.79(33) ms [1],
the ratio of the half-lives is 1.81 which is sufficiently close to this factor of 2 and leads
to a partial cancellation of the first term. A simulation of
34
Ar decay is shown in
Fig. 7.6 with the initial constraint No,D = 0. It is clear that the total activity closely
resembles a single exponential with the
the
34
34
Cl half-life leaving very little sensitivity to
Ar half-life.
The fact that the first term of Eqn. 7.6 is not exactly cancelled in the case of 34 Ar
decay to 34 Cl (which is also represented by the slight curvature near t = 0 in Fig. 7.6)
means that direct β counting techniques to deduce the 34 Ar half-life are possible, but
pose significant challenges which are compounded further if No,D is non-zero. Recently
the
34
Ar half-life was determined to high-precision, T1/2 = 843.8 ± 0.04 ms [12]
and relied on the slight deviation of the total decay curve (similar to Fig. 7.6) from
that of a single pure exponential with the
constraint on the number of
34
34
Cl half-life, and used an independent
Cl nuclei obtained from event-by-event counting of
the implanted ions. With the extreme challenges of this β counting technique, an
independent confirmation of this single high-precision half-life measurement for
would certainly be desirable. A measurement of the half-life of
34
34
Ar
Ar could therefore
benefit from the technique of γ-ray photopeak counting as a γ-ray energy gate on a
203
10
5
Total β Activity
Counts
10
4
10
3
34
10
Cl Activity
2
34
10
Ar Activity
1
0
10 0
10
5
15
20
25
30
Time (s)
Figure 7.6: Total simulated β activities for 34 Ar and 34 Cl decay. Due to the fact that
the 34 Cl daughter half-life is nearly twice that of the parent 34 Ar, the resulting total
activity β spectrum resembles a single exponential with the half-life of 34 Cl.
transition in the daughter
34
Cl can, in principle, remove all of the daughter activity
from the resulting decay spectrum. Although this technique is hindered by the fact
that the largest β branch is 2.49% [1] yielding a 666 keV γ-ray (see Fig. 7.7), this loss
of overall statistics relative to β counting could be more than compensated by the
fact that the
34
Cl daughter activity is removed from the resulting γ-ray gated decay
spectrum.
Preliminary investigations of an
34
Ar half-life measurement were carried out on
two separate occasions at ISAC during ion source development. The first test was
performed in May 2005 during the first online test of a 2.45 GHz ISAC-ECR source
with a TiC target. The observed yield of 100 34 Ar ions per second was, however, three
204
34
Ar
ft = 3053 8 s
0+ 843.8 0.4 ms
+
b
1+
3129
1+
1+
1+
2580
666
1.30 %
0.85 %
2.49 %
461
0.91 %
0+
94.45 0.25 %
1526.6 0.4 ms
+
b
34
Cl
Figure 7.7: Decay scheme for 34 Ar β + decay to 34 Ar. The technique of γ-ray photopeak counting can provide a superior method to deduce the 34 Ar half-life as it removes
the contamination from the daughter 34 Cl which β decays with a similar half-life.
205
15000
10000
34
Counts
T1/2( Ar) = 0.78 ± 0.21 s
2
χ /ν = 1.3
5000
0
0
2
4
6
8
10
12
14
16
18
20
22
Time (s)
Figure 7.8: Initial attempt to deduce the 34 Ar via direct β counting with the 20
plastic scintillators of the SCEPTAR array. In this run, comprised of 90 cycles, the
half-life of 34 Ar was determined to be T1/2 = 0.78 ± 0.21 s.
orders of magnitude too low to use the γ-ray photopeak counting technique to improve
upon previous measurements of the 34 Ar half-life. Tests were instead performed using
the SCEPTAR array as a high-precision β counter. Shown in Fig. 7.8 is a grow-in
and decay curve obtained from a single run containing 90 cycles with a tape cycle
time of 1-5-15-1 s. From a fit to these data with two exponentials (including the
grow-in component of
34
Cl) with a fixed
34
Cl half-life and a free background (4 free
parameters) the 34 Ar half-life of T1/2 = 0.78 ± 0.21 s was obtained for this run. Fitting
these data to a single exponential with a free half-life yielded T1/2 = 1.563 ± 0.006 s
a result that is similar to the known
34
Cl half-life of T1/2 (34 Cl) = 1.52679(33) s [1]
and demonstrates the similarity of the total β activity spectrum to that of a single
206
exponential decay with the daughter half-life as discussed above. From a total of 5
hours of collected data an overall precision in the 34 Ar half-life of 50 ms was attained.
This result is more than two orders of magnitude larger than the desired 0.4 ms that
would establish the 34 Ar half-life to 0.05%. In order to achieve this goal with the given
beam of intensity one would require an experiment lasting nine years! It is interesting
to compare this to the expected result if the 34 Cl activity could be completely removed
from the spectrum. With a beam intensity of 100 34 Ar ions per second, a total sample
of 120 34 Ar nuclei is collected in each cycle which provides a statistical uncertainty of
77 ms per cycle. With a cycle lasting 22 s in duration a total precision of 0.4 ms could
therefore be achieved in less than 10 days. Clearly the addition of the unwanted
34
Cl
daughter activity severely hinders the method of direct β counting.
A second test of the
34
Ar half-life measurement was performed in November 2006
using a CaZrO3 target coupled to a FEBIAD ion source in its first commissioning
run. The yield of
34
Ar at the beginning of the experiment peaked at 3×104
34
Ar
per second which was an improvement, by more than two orders of magnitude, over
the previous experiment. A sample γ-ray spectrum is shown in Fig. 7.9 where the
666 keV transition in
34
Cl that follows
34
Ar β + decay is clearly visible. In addition,
γ-ray peaks following the β decay of the isobaric contaminant
were also present. A measurement of the half-life of
34
34m
Cl (T1/2 = 32 min)
Ar via the γ-ray activity
was investigated, however, as shown in Fig. 7.10, a significant contamination was
observed under the 666 keV photopeak due to bremsstrahlung radiation from the
decay of the
34
Cl daughter. Following a gate on the background directly above the
666 keV γ-ray transition, the resulting decay spectrum (upper-left panel in Fig. 7.10)
shows a clear time dependence. Fitting these data to a single exponential with a
free half-life yields T1/2 = 1.5 ± 0.1 s, a result that agrees with the known
207
34
Cl
10
8
20000
34
Counts
10
7
6
511
15000
10000
5000
146
*
10
Counts
10
Ar
666
2128
1987
*
*
5
0
620 640 660 680 700 720
Eγ (keV)
3304
10
10
10
*
4
3
4115
*
2
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Eγ (keV)
Figure 7.9: Sum γ-ray singles spectrum obtained with the 8π spectrometer comprised
of 10 hours of A = 34 beam with γ rays from a contamination of 34m Cl indicated by
(*). (Inset) The 666-keV γ-ray in the daughter 34 Cl that follows the decay of 34 Ar.
half-life T1/2 = 1.52679(33) s [1]. As
34
Cl decay does not give rise to γ-radiation
this decay spectrum must be due solely to bremsstrahlung and in-flight annihilation
processes. Extracting the
34
Ar half-life from the 666 keV γ-ray energy gated data
must therefore take this contamination into account. Fitting these gated data with
a function containing two exponentials (using a fixed
34
Cl half-life and including the
grow-in) and a free background yielded T1/2 = 864 ± 94 ms for the half-life of
34
Ar,
a result that agrees with the high-precision value T1/2 = 843.8 ± 0.4 ms [12]. Note
that the spectral area of the 666 keV photopeak is comprised of a total of 6622(81)
events which, in the limit of no background or bremsstrahlung activities, would yield
a precision of 10 ms for the
34
Ar half-life. The addition of the
208
34
Cl contamination
500
400
5
4
300
3
200
2
100
1
5
10
15
Time (s)
20
650
660
700
Counts in 666 keV γ gate
6
Gate (a) Gate (b)
Gate (b)
34
T1/2( Cl) = 1.5 ± 0.1 s
670
Eγ (keV)
680
690
3
600
Counts per keV (x 10 )
Counts below 666 keV γ gate
7
700
0
600
Gate (a)
500
34
400
T1/2( Ar) = 864 ± 94 ms
300
χ /ν = 0.86
2
200
100
0
0
5
10
15
20
Time (s)
Figure 7.10: (Upper left) Growth and decay curve from 34 Cl bremsstrahlung radiation
obtained from a gate (Gate (b)-upper right) on the background just above the 666keV transition. Following a gate on the 666-keV transition itself (Gate (a)) the growth
and decay curve of 34 Ar is obtained but is also contaminated by 34 Cl bremsstrahlung.
activity to the energy gate therefore degrades the expected precision by an order of
magnitude. As this run was comprised of a total of 337 cycles, with a cycling time
of 1-4-15-1 s, an overall precision of 0.4 ms could not be achieved with this beam
intensity in a reasonable time. A γ-ray counting experiment to deduce the half-life of
34
Ar lasting two weeks in duration at a beam intensity of 3×104 ions per second of
34
Ar would only achieve an overall precision of ∼ 7 ms which is still more than an order
of magnitude larger than desired. Using the same data, a measurement of the
34
Ar
half-life from the β activity detected in the SCEPTAR array was also investigated
despite the difficulties described above and with the added complication of the
209
34m
Cl
10
6
Best Fit
34
Ar
34
Cl
34m
Cl
34
T1/2( Ar) = 0.88 ± 0.05 s
2
Counts
χ /ν = 1.51
10
10
5
Components at beam off
34
34
Ar/ Cl = 0.37
34
34m
Ar/ Cl = 4.25
34
34m
Cl/ Cl = 11.46
4
34
Ar/Total = 0.25
Cl/Total = 0.69
34
0
5
10
20
15
25
Time (s)
Figure 7.11: Typical growth and decay curve obtained from the sum of all 20 SCEPTAR plastic scintillators for a single run. The contributions of 34 Ar, 34g Cl, and 34m Cl
to the composite curve demonstrate that 34 Ar is only a small component of the total β activity.
contaminant. As shown in Fig. 7.11 the half-life of
34
Ar, T1/2 = 880 ± 50 ms, was
obtained from a fit that assumed all of the background was due to the decay of 34m Cl.
Note that the data shown from γ-ray counting (Fig. 7.10) and β counting (Fig. 7.11)
were obtained from the same experimental run. In this case, the precision obtained
in the
34
Ar half-life via β counting, 0.5 s, was nearly a factor of two better than the
0.9 s obtained via γ-ray counting.
These test experiments have provided a first step towards overcoming the challenges of an
34
Ar half-life determination by the γ-ray photopeak counting technique.
It is clear, however, that the present yield of
210
34
Ar at ISAC needs to be improved by
at least an order of magnitude and work to this end is in progress. From the above
analysis, at rates of ∼ 104
34
Ar per second the β counting technique is actually a
slightly more precise method to deduce the
34
Ar half-life. This would no longer be
true if the yield was significantly increased as the precision in the γ-ray result would
be expected to scale inversely with the square-root of the yield, while the β counting
result is not limited by statistics but instead by the covariance between the similar
half-lives of
34
Cl and
34
Ar. At higher rates it is anticipated that a measurement via
the γ-ray technique will result in a more precise value for the
34
Ar half-life than the
method of direct β counting and will provide an essential test of the single highprecision β counting result [12] that presently determines the adopted half-life for
this important Tz = −1 superallowed emitter.
7.3
Conclusion
High-precision measurements of the f t values for superallowed Fermi β decays between 0+ isobaric analogue states provide a rigorous test of the Standard Model
description of electroweak interactions. Owing primarily to their relative insensitivity to nuclear structure effects, which enter only as small corrections at the percent
level, these decays have set strict limits on possible extensions to the Standard Model
that include scalar and right-handed currents [3], have confirmed the conserved vector
current (CVC) hypothesis to 1.3 parts in 104 [2], and have provided the most precise
determination of the CKM matrix element Vud [2, 28]. In Chapter 1 of this thesis it
was shown that the β decays of the free neutron and the pion can alternatively be
used to deduce the CKM matrix element Vud . However, these decays are presently
limited by the experimental determinations of the axial-vector to vector asymmetry
211
parameter λ in the case of neutron decay and an extremely weak branching ratio
∼ 10−8 in the case of pion decay. At the present time superallowed Fermi β decay
thus provides the most sensitive measure of Vud .
In the case of the superallowed decays, measurements of the β decay half-lives
and β branching ratios must be deduced to 0.05% precision or better while Q values must be determined to 0.01% in order to deduce the experimental f t values
at the level of 0.1% necessary for these sensitive tests. To date, nine superallowed
f t values have been determined to better than 0.1% and four more are known to
better than 0.5%. Together they establish the world-average corrected f t value to
F t = 3074.9(8)f t(9)δC s, where the first error is statistical and the second is an
estimated systematic effect resulting from differences between two independent calculations of isospin symmetry breaking. Comparison of the vector coupling constant
GV deduced from the superallowed decays with the Fermi coupling constant from
purely leptonic decay GF yields the up-down element of the CKM matrix element
Vud = 0.97376(12)F t(15)δC (18)∆VR , a result that is currently dominated by the uncertainty in the nucleus-independent radiative correction ∆VR but with recent progress
may soon be dominated by the systematic uncertainty associated with the δC corrections.
The theoretical corrections for isospin symmetry breaking can be scrutinized by
experimental means either by focusing on the Tz = 0 superallowed emitters such as
62
Ga and 74 Rb in the A ≥ 62 region where large δC corrections (> 1%) are predicted,
or through measurements of the Tz = −1 superallowed decays in the 18 ≤ A ≤ 42
region, where the discrepancies between the δC calculations are largest (14 O,
30
18
Ne,
S, 34 Ar, see Fig. 1.3). Regardless of which of these two paths is investigated, achiev-
ing overall precisions in the f t values of better than 0.1% represents a considerable
212
experimental challenge and in many cases new, and sometimes unique, methods must
be developed on a case-by-case basis.
In this thesis, a new method for the measurement of β decay half-lives to the level
of 0.05% using the technique of γ-ray photopeak counting was presented and, for
the first time, provides a quantitative description of detector pulse pile-up that has
previously hindered these types of measurements. While superallowed Fermi β decay
studies provided the main motivation for this work, the non-superallowed β − decay of
26
Na was used to develop the technique because beams of 26 Na were readily available
from existing ISAC surface ionization sources, the daughter 26 Mg is stable, and 100%
of β decays are followed by γ-ray emission. The half-life of
26
Na could therefore
be determined either by direct β counting or γ-ray photopeak counting methods.
Following a careful analysis of cosmic-ray self pile-up, pile-up time resolution losses,
and CFD and pile-up energy threshold effects, the half-life of
26
Na was determined
using this new technique to be T1/2 = 1.07167 ± 0.00055 s, a result that includes
a 1.0% or ∼ 27σ correction for pile-up and agrees with the high-precision result
T1/2 = 1.07128 ± 0.00025 s obtained from a previous β-counting experiment [88].
The pile-up correction methodology presented here demonstrates the feasibility of
using HPGe detectors to determine β-decay half-lives at the level of 0.05% necessary
for the superallowed-Fermi β-decay program.
As a first superallowed β decay half-life measurement with this novel technique,
the half-life of the Tz = −1, β + emitter
of
18
18
Ne was chosen. In this decay 7.70(21)% [1]
Ne β decays result in a 1042 keV γ-ray which connects the analogue 0+ state to
the 1+ ground state in the daughter 18 F. The half-life of 18 Ne deduced in this work, by
measuring the time dependence of these γ-rays, T1/2 = 1.6656 ± 0.0019 s, represents
an improvement in precision, by a factor of four, over the previously adopted world
213
average that was comprised of two measurements [93, 94] that do not agree within
experimental uncertainty. Of these two previous measurements, the result presented
here is in excellent agreement with a previous β counting determination [94] but
does not agree with, and is 2.4σ smaller than, an older γ-ray photopeak counting
determination [93]. The fact that this previous γ-ray counting measurement resulted
in the largest half-life ever measured for
18
Ne may be the result of the systematic
effects of detector pulse pile-up that have been a main focus of this thesis. Because of
the possibility for diffusion of the implanted noble gas ions from the implantation site
to bias the
18
Ne half-life measurement, the half-life of the heavier isotope
23
Ne was
determined to be T1/2 = 37.11 ± 0.06 s, a factor of two improvement over the previous
world average. Through a careful study of the longer-lived
23
Ne decay, no diffusion
effects at long time scales (∼ 100 s) were observed at the level of the statistical
uncertainty and, when combined with the systematic channel removal analysis for
18
Ne which ruled out short time-scale effects, diffusion was deemed negligible in the
18
Ne experiment.
A major experimental achievement in the past two years has been the addition
of the
62
Ga f t value to the above list of superallowed decays representing the first
A ≥ 62 decay determined to this level of precision and providing an important test of
the isospin symmetry breaking corrections in this mass region. Prior to this thesis, the
62
Ga f t was established to the level of 0.05% and was limited by the half-life, which
itself was dominated by a single measurement T1/2 = 116.19 ± 0.04 ms [103]. As only
0.1% of all
62
Ga β decays result in γ-ray activity a new measurement of the
62
Ga
half-life was performed by the method of direct β counting. The half-life deduced
in this work, T1/2 = 116.100 ± 0.025 ms, represents the most precise superallowed
half-life measurement ever reported and establishes the world average
214
62
Ga half-life
to T1/2 = 116.121 ± 0.021 ms, a result that is precise to the level of 0.018%. With this
new measurement of the half-life, the
62
Ga f t value, f t = 3074.3 ± 1.1 s, now rivals
the precision of the best measured superallowed decays in the lighter mass region.
Assuming the CVC hypothesis is satisfied and using the average F t value from the
twelve other high-precision superallowed transitions permits an experimental extraction of the isospin symmetry breaking correction required by
62
Ga decay to conform
to CVC. This procedure results in the value δC (62 Ga) = 1.41(10) %, a result whose
uncertainty is not dominated by experiment, but rather by the theoretical calculation
of the radiative correction δR′ which can, in principle, be improved by extending the
radiative corrections to higher order [27]. It should be emphasized that the overall
precision in the
62
Ga f t value has been improved upon by a factor of 40 in the past
two years alone. Higher-order calculations to improve the radiative corrections for
these high-Z decays were therefore previously unwarranted. The precision in the 62 Ga
f t value only now provides a strong argument for these calculations to be extended.
Superallowed Fermi β decays have, for decades, provided rigorous tests of the
Standard Model and with new and exotic radioactive ion beam facilities such as
ISAC producing beams of unprecedented intensities, these basic and fundamental
studies are being vigorously pursued with renewed interest. Combined with new
experimental techniques, such as γ-ray photopeak counting for high-precision halflife measurements, ultra-high sensitivity to weak β decay branches, and the use of
precision Penning traps for sub-keV mass measurements, these studies will continue
to produce some of the most impressive and novel techniques and results for decades
to come.
215
Appendix A
Dead-time and Pile-up Corrected
Decay-curve Fitting
The fitting of decay-curve data was performed with a Levenberg-Marquardt χ2 minimization algorithm [118] that was modified to include the definition of the χ2 value
that is obtained by a direct application of maximum likelihood to the Poisson probability distribution [119]. This procedure has been shown to introduce negligible bias
in counting experiments with small numbers of counts per bin [119] and has been
universally adopted in the evaluation of superallowed β decay data [1]. A bin-by-bin
weighting factor Wi was previously introduced to increase the variances associated
with performing dead-time corrections [88, 120]. Modification of this procedure to
incorporate a pile-up correction was performed in an analogous way. The χ2 that was
employed in the fitting of pile-up and dead-time affected data was,
2
χ =2
N
X
i=1
Wi
yi
yf it − yi + yiln
yf it
,
(A.1)
where yi is number of counts in the γ-ray gate in the ith time bin following the deadtime and pile-up corrections, yf it is the value of the fit function in that particular bin
216
and N is the total number of bins included in the fit. The weighting factor Wi can
be written as [88],
Wi =
yi
,
σi2
(A.2)
where σi2 is the variance of the γ-ray gated decay data in the ith bin, following corrections for dead-time and pile-up effects. Application of the dead-time correction
was achieved by determining the bin-by-bin fraction of the time that the system was
dead Di . This fraction is given by total number of triggers in that particular bin, the
dead-time per event of the system τd , and the bin time tb [90],
Di = (ni + pi )
τd
.
tb
(A.3)
It is important to note that the total dead time in each bin is related to all trigger
events (ni +pi ), where ni and pi are the total number of not piled-up and piled-up
trigger events, respectively and are not related to the γ-ray-gated data gi that are
being fit.
In an analogous way, introduction of the pile-up correction to this fitting routine
can be achieved by defining the probability of pile-up Pi′ (t) which is not given by the
bin-by-bin data, but rather from an unweighted fit to the piled-up data using the
function (Eqn. 3.43) discussed in Sec. 3.4 of Chapter 3 that accounts for the existence
of cosmic rays, non-zero pile-up time-resolution effects and non-zero CFD and pileup energy thresholds. The dead-time and pile-up corrected number of counts and
associated variances are then given by,
gi
,
(1 − Di )(1 − Pi′ (t))
gi
.
=
[(1 − Di )(1 − Pi′ (t))]2
yi =
(A.4)
σi2
(A.5)
In β-counting experiments, detector pulse pile-up is not relevant and an energy gate
is not used, thus gi and ni are equivalent. Substitution of this limit (P ′ (t) = 0,
217
pi = 0, gi = ni ) into the above expressions restores the fitting procedure to that used
in high-precision β counting experiments [88, 90, 120].
218
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