PRECISION MEASUREMENT OF THE β-ENERGY SPECTRUM IN 6HE DECAY

By

Xueying Huyan

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Physics – Doctor of Philosophy

2019

ABSTRACT

PRECISION MEASUREMENT OF THE β-ENERGY SPECTRUM IN 6HE DECAY

By

Xueying Huyan

Precision measurements of the β-energy spectrum in nuclear and neutron decays have a high
potential to find possible signatures of new physics beyond the standard electroweak model. Such
signatures would produce a distortion to the β-energy spectrum relative to the Standard Model
prediction. In Gamow-Teller transitions, these distortions would indicate the presence of the exotic
tensor type interactions. An attractive candidate for this study is 6He because the simplicity of
its decay and its other nuclear properties allow an accurate theoretical description of the spectral
shape.

At the National Superconducting Cyclotron Laboratory, we measured the β-energy spectrum
in 6He decay using a calorimetric technique. The radioactive ions were implanted into the active
volume of a detector, eliminating the critical instrumental effect related to the backscattering of β
particles. The goals of the experiment are to provide the first measurement of the weak magnetism
form factor (bWM) in 6He decay and the first direct measurement of the Fierz interference term
(bGT) in Gamow-Teller decay.

This work summarizes the efforts on this experiment, focusing on the data analysis of the main
systematic effects and the projected sensitivities to the observables. The estimates of bWM and bGT

are bWM =? ± 4.9stat ± 11sys and bGT =(cid:2)? ± 3.1stat ± 4.7sys(cid:3) × 10−3, where the central values of

bWM and bGT will be determined in the future analysis. The predicted uncertainty of bGT satisfies
the precision goal of O(10−3), a level which allows new constraints on tensor couplings beyond the
Standard Model.

Copyright by
XUEYING HUYAN
2019

ACKNOWLEDGMENTS

I want to express my highest appreciation to my thesis advisor, Oscar Naviliat-Cuncic. You have
trained me to become a scientist with both high requirement and enormous support. You have
continuously motivated me to come up with creative solutions when the research gets challenging.
This uphill journey has been a rewarding experience for me. And thank you for the great patience
teaching me how to communicate science.

I want to offer my special thanks to my Ph.D. guidance committee members. Vladimir Zelevin-
sky, you are one of the first persons who introduced me the uniqueness of the low-energy high-
precision experiments and you have a special appreciation for my research. You have been a mentor
to me since I took your quantum mechanics class. Wayne Repko, you have provided invaluable
insight into the theoretical part of my research, particularly the radiative correction calculations.
Thank you for giving me a 4.0 in your enjoyable quantum physics class, without which I would
have been kicked out of graduate school after the first year. Kei Minamisono, thank you for the con-
structive suggestions about my data analysis, for challenging me to think harder, and for reminding
me of the big picture of my research. Stuart Tessmer, thank you for your constant encouragement
and support when I was lost. It was great to know that you were backing me up.

My thesis experiment has been a team effort of at least sixteen people at NSCL. I want to
acknowledge the contributions of Shumpei Noji, Alexandra Gade, Anna Simon, Sean Liddick,
Daniel Bazin, Dirk Weisshaar, Mike Bowry, Walter Buhro, Vincent Bader and Saul Beceiro-Novo
during the experiment.
I am particularly grateful to Max Hughes, Stan Paulauskas, Elisabeth
George, and Paul Voytas for our many discussions during the data analysis.

I want to thank Scott Pratt. You provided me the essential support during my difficult transition
from a non-physics major to physics in my first year. I did not know if I could survive graduate
school at that time.

Thanks to my friends in graduate school who shared our vulnerabilities and optimism. It has

been a great experience growing up with your company.

iv

I appreciate my parents for the persistent strong support for my education. Thank you for

understanding my many decisions that broke the traditions. I hope I have made you proud.

My special thanks to my spouse, Sam Marinelli. Thank you for believing me more than I believe
in myself. You have been encouraging me to explore my full potential. You have empowered me.

v

TABLE OF CONTENTS

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1.1 Motivation .

1.2 Observables .

LIST OF TABLES .
LIST OF FIGURES .
CHAPTER 1

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INTRODUCTION .
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1
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1
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2
1.1.1
Phenomenology of Semileptonic Weak Processes . . . . . . . . . . . . . .
5
1.1.2 Current Limits on the Scaler and Tensor Couplings . . . . . . . . . . . . .
6
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6
1.2.1 Correlation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.2.2 The Fierz Interference Term . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Current Status of the Fierz Interference Term . . . . . . . . . . . . . . . . 10
. . . . . 11
1.3 The Shape of the β-energy Spectrum and the Test of the CVC Hypothesis
1.4 The Candidate: 6He .
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1.5 Existing Measurements of β-energy Spectra . . . . . . . . . . . . . . . . . . . . . 15
1.5.1 A Brief History of the Measurements of the β-energy Spectra in 6He Decay 15
1.5.2 More Recent Measurements of the β-energy Spectra
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1.5.3 What is Next? . .

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2.1 Fermi Function .
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2.2 Screening Correction .

CHAPTER 2 THEORETICAL CORRECTIONS TO THE β-ENERGY SPECTRUM . . . 23
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2.2.1 Rose’s Formula .
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2.2.2 Bühring’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3
Simplified Bühring’s Formula . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.4 Weak-Screening Approximation of Bühring’s Formula . . . . . . . . . . . 27
2.3 The Finite Size Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Electrostatic Finite Size Correction . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Convolution Finite Size Correction . . . . . . . . . . . . . . . . . . . . . . 29
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
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2.5.1 Distinguishable e− and γ . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Indistinguishable e− and γ . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.2
2.6 Summary of the Corrections to the β-energy Spectrum in 6He Decay . . . . . . . . 41
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.7 Theoretical Uncertainty .
2.7.1 Calculations of the Endpoint Energy and the Nuclear Radius . . . . . . . . 42
2.7.2 Estimation of the Theoretical Uncertainty . . . . . . . . . . . . . . . . . . 43

2.4 Finite Mass Correction .
2.5 Radiative Correction .
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CHAPTER 3 DESCRIPTION OF THE EXPERIMENT . . . . . . . . . . . . . . . . . . . 46
3.1 General Instrumental Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
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3.2 Calorimetric Technique .

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3.3 Beam Conditions

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3.3.1 Beam Facility .
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3.3.2 The Beam Geometry and the Calculated Range . . . . . . . . . . . . . . . 48
3.3.3 Measurement of the Beam Range . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.4 Beam Purity .
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3.3.5 Beam Cycles
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3.4 Detectors
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3.5 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
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3.6 Data Collections .
. 55
3.6.1 Regular Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 55
3.6.2 Calibration Runs
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3.6.3 Background Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 56
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3.6.4 Gain Drift Test
. 56
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3.7 Sample Data .

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5.1 Simulation Conditions

CHAPTER 4 BIG PICTURE OF THE ANALYSIS . . . . . . . . . . . . . . . . . . . . . 58
CHAPTER 5 DETECTOR SIMULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 60
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.1
Particle Source and Detector Configurations . . . . . . . . . . . . . . . . . 61
5.1.2 Geant4 Physics Constructors . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.3 Geant4 Parameters
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5.1.4 Visualization .
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5.2.1
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5.2.2 Absorption Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3 Detector Response .
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5.4 Continuous β-energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4.1
5.4.2 Absorption Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4.3 Comparison of the Simulated Spectra . . . . . . . . . . . . . . . . . . . . 78

5.2 Discrete Monoenergetic Electrons

Photon Yield .

Photon Yield .

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Introduction .

CHAPTER 6 ENERGY CALIBRATION . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.1
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6.2 Sensitivity to the Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 82
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Sensitivity to the Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Sensitivity to the Offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Sensitivity to the Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . 85
Sensitivity to the Energy Resolution . . . . . . . . . . . . . . . . . . . . . 87
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Summary .
6.3 Determination of the Gain with Auto-Calibration . . . . . . . . . . . . . . . . . . 88
Fitting the Spectrum with the Gain as a Free Parameter . . . . . . . . . . . 89
Statistical Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2.1 General Procedure
6.2.2
6.2.3
6.2.4
6.2.5
6.2.6

6.3.1
6.3.2
6.3.3 Correlations .

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6.4 Determination of the Offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.5 Determination of the Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.6 Determination of the Energy Resolution . . . . . . . . . . . . . . . . . . . . . . . 98

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7.1

CHAPTER 7 RATE EFFECT .
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Introduction .
7.1.1 On-Line Measurement
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7.1.2 Off-Line Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
. . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.1.3 Origin of the Rate Effect
7.2 The Impact of the Rate on the Gain . . . . . . . . . . . . . . . . . . . . . . . . . . 105
. . . . . . . . . . . . . . . . . . . . . . . . . 105
7.3 The Impact of the Rate on the Offset
7.3.1 Rate Correction to the Offset
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7.3.2 Uncertainty of the Offset
7.4 The Impact of the Rate on the Nonlinearity . . . . . . . . . . . . . . . . . . . . . . 108
7.5 The Impact of the Rate on σ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.5.1 Rate Correction to the Width . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.5.2 σ0 in the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.5.3 Uncertainty of the σ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
CHAPTER 8 FITTING THE β-ENERGY SPECTRUM . . . . . . . . . . . . . . . . . . . 115
. 115
8.1.1 The Expression of the Fit Function . . . . . . . . . . . . . . . . . . . . . . 115
8.1.2 Constructing a Function from a Histogram . . . . . . . . . . . . . . . . . . 116
. 123
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8.3.1 Number of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.3.2 Number of Bins .
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. 130
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8.4 Determined Fit Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5 Pile-Up Contribution to the Fit Function . . . . . . . . . . . . . . . . . . . . . .
8.6 Complete Fit Function .

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2 and Log-Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.2.1
8.2.2 Minimizer .
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8.3 Fit Condition . .

χ

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8.2 Fit Procedure .

8.1 Fit Function .

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9.1 Background in the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 9 BACKGROUND STUDY . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
. 133
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9.1.1 Background Component
9.1.2 Background Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.2 Corrections to the Background: Monte Carlo . . . . . . . . . . . . . . . . . . . . . 138
9.2.1 Background Gain Correction . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.2.2 Background Gain-and-Sigma Correction . . . . . . . . . . . . . . . . . . . 140
. . . . . . . . . . . . . . . . . . . . . 142
9.3.1 Corrections to the β + γ Spectrum . . . . . . . . . . . . . . . . . . . . . . 142
9.3.2 Corrections to the β Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 145
9.4 The Impact of the Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

9.3 Corrections to the Background: Experiment

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CHAPTER 10 SENSITIVITIES TO PHYSICS AND BEYOND . . . . . . . . . . . . . . . 155
. 155
. 156

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10.1 Sensitivities .
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10.2 Summary and Outlook .

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APPENDIX .
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APPENDIX DERIVATIONS IN CHAPTER 2 . . . . . . . . . . . . . . . . . . . . . . . 158
. 158
A.1 Derivation of Eq. 2.5 .
A.2 Table of Parameters in the Electrostatic Finite Size Correction . . . . . . . . . . . 158
2 Radiative Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
A.3 Order Zα
A.3.1 Uniform Charge Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 159
A.3.2 Modified Gaussian Charge Distribution . . . . . . . . . . . . . . . . . . . 159
. 160
. 161

A.4 Derivation from Eq. 2.58 to Eq. 2.62 . . . . . . . . . . . . . . . . . . . . . . . .
A.5 Derivation from Eq. 2.62 to Eq. 2.69 . . . . . . . . . . . . . . . . . . . . . . . .

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BIBLIOGRAPHY .

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LIST OF TABLES

Table 1-1: Current most precise values of the hadronic form factors determined with

lattice QCD calculations from Ref. [1].

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5

Table 2-1: The screening corrections for 6He decay for p ≤ 0.3. The values shown here

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are [S(Z, W) − 1]. .

Table 2-2: Sensitivities of bWM and bGT to the uncertainties of the 6Li radius and the

endpoint energy.

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Table 3-1: Experimental conditions for the CsI(Na) and the NaI(Tl) detectors during the

regular measurements.

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Table 5-1: Physics constructors in Geant4 applied in the simulations.

. . . . . . . . . . . . 63

Table 5-2: Default values of the range factor parameter and the geometry factor parameter

in each physics constructor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Table 5-3: Photon yields (in %) produced by a β-energy spectrum generated following
Eq. 5.11. In each simulation, the number of primary events is 2×107. The
results are obtained with four physics constructors.

. . . . . . . . . . . . . . . . 77

Table 5-4: Absorption fractions (in %) produced by a β-energy spectrum generated fol-
lowing Eq. 5.11. In each simulation, the number of primary events is 2×107.
The results are obtained with four physics constructors. . . . . . . . . . . . . . . 77
Table 5-5: Slopes (in 10−8/keV) extracted from the fits to the ratio spectra . . . . . . . . . . 79

Table 6-1: Sensitivities of bWM and bGT to the energy-calibration parameters . . . . . . . . 88
Table 6-2: The fit results of the 137Cs and 60Co γ peaks. The energy values in the first

column are adapted from the NuDat2.7 database [2].

. . . . . . . . . . . . . . . 96

Table 9-1: List of background conditions applied when fitting the observables.

. . . . . . . 149

Table 10-1: Systematic effects considered and the corresponding uncertainties in the error

budget.

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Table A-1: Coefficients for the parameterization of L0(Z, W) for electrons copied from

Ref. [3]. .

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LIST OF FIGURES

Figure 1-1: The current limits of scalar (horizontal) and tensor (vertical) couplings ex-
tracted from the fit to F t values, average neutron lifetime, average neutron β
asymmetry ˜An, and the λAB parameter. The contours around the minimum
correspond to 68%, 95%, and 99% confidence intervals. Figure taken from
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Ref. [1].

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Figure 1-2: A pictorial representation of the most common correlation coefficients, the
two-fold (edges) and the three-fold (faces) correlations between the kinematic
vectors (vertices). Figure taken from Ref. [4].

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6

8

Figure 1-3: Left panel: the decay scheme of the 6He nuclear β decay. Right panel: the
simplified level scheme and transition in the mass A = 6 isospin triplet taken
from Ref. [5].

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. 13

Figure 1-4: The 1σ statistical uncertainties of the Fierz interference term obtained by
fitting the simulated β-energy spectra with different values of the endpoint
energy. Figure taken from Ref. [6].

. . . . . . . . . . . . . . . . . . . . . . . . 14

Figure 1-5: The β-energy spectrum in the decay of a helium isotope measured with a
cloud chamber. The isotope identify was believed to be 6He. Figure taken
from Ref. [7].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

. .

.

Figure 1-6: Kurie plot of the β-energy spectrum in 6He decay. The horizontal axis is the

kinetic energy of β particles in the natural unit (mc2). Figure taken from Ref. [8]. 16

Figure 1-7: Kurie plot of the β-energy spectrum in 6He decay. Figure taken from Ref. [9].

. 16

Figure 1-8: Left panel: a schematic sketch of the apparatus used to determine the electron-
recoil ion correlation in the decay of 6He. The beta-active source volume and
the scintillation spectrometer are located in the common vacuum of the “bell-
the 6He β-energy spectrum measured with the stilbene
jar”. Right panel:
scintillation spectrometer. Figures taken from Ref. [10]. . . . . . . . . . . . .

. 17

Figure 1-9: A scale drawing of the thin lens spectrometer baffle system, the source volume,

and the Geiger counter. Figure taken from Ref. [11]. . . . . . . . . . . . . . . . 18

Figure 1-10: Kurie plot of the β-energy spectrum in 6He decay. Figure taken from Ref. [11].

19

xi

Figure 1-11: Left panel: the layout of the β spectrometer used to measure the kinetic energy
of β particles. It consists of a high-purity germanium detector mounted on the
axis in the bore of a superconducting solenoid. Right panel: the theoretical
statistical distribution (solid line) and the corresponding response from the
detector (dashed line). Figures taken from Ref. [12]. . . . . . . . . . . . . . . . 20

Figure 1-12: Left panel: measured (black) and simulated (red) energy spectra deposited in
the Si(Li) detector with a spectrometer current corresponding to a positron
the β-energy spectrum in 66Ga
kinetic energy of 3.25 MeV. Right panel:
decay. Figures taken from Ref. [13].

. . . . . . . . . . . . . . . . . . . . . . . 20

Figure 1-13: Left panel: Si(Li) spectra measured with the spectrometer current setting at
the β-momentum spectrum in 14O decay
8.8 A and 11.0 A. Right panel:
including the measured spectrum (open circles) and the theoretical spectrum
(solid line). Figures taken from Ref. [14].

. . . . . . . . . . . . . . . . . . . . 21

Figure 2-1: Electrostatic finite size corrections. The dashed lines and the solid line are
calculated with Eq. 2.19 and Eq. 2.20, respectively. The colors indicate the
radius used in the calculations: blue corresponds to R1 and black corresponds
to R3. The results are compared with the numerical values (solid red) from
Ref. [15].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

.

.

.

.

.

.

.

Figure 2-2: Feynman diagrams for the one-photon exchange processes associated with the
order α outer radiative correction. The top row shows the processes involving
virtual photon exchange, and the bottom row shows the processes involving
real photon emission. Figure taken from Ref. [16].

. . . . . . . . . . . . . . . 33

Figure 2-3: The energy spectrum of the inner bremsstrahlung photons emitted in 6He β

decay. The vertical axis has an arbitrary unit. . . . . . . . . . . . . . . . . . . . 37

Figure 2-4: Radiative corrections for 6He decay when the β particles and photons are
distinguishable (blue) calculated with Eq. 2.34 and distinguishable (red)
calculated with Eq. 2.69.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Figure 2-5: Calculated theoretical corrections to the β-energy spectrum in 6He decay.

The dashed line shows the expected spectral function.

. . . . . . . . . . . . . . 41

Figure 3-1: A schematic layout of the beam facility at National Superconducting Cyclotron

Laboratory. .

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 3-2: A cartoon of the top view of the setup. The secondary 6He beam (red)
of 76 MeV per nucleon passes through a degrader (gray). The beam was
decelerated by the degrader down to 46 MeV per nucleon and was implanted
into a CsI(Na) or NaI(Tl) detector (yellow). . . . . . . . . . . . . . . . . . . . . 48

xii

Figure 3-3: Range distributions of the implanted 6He in CsI (left) and NaI (right) calcu-
lated with the LISE++ code. The angle of the degrader with respect to the
beamline is 0°. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

.

.

.

.

Figure 3-4: Range distributions of the implanted 6He in CsI while the angle of the degrader

with respect to the beamline is at 0°(blue) and 42°(red).

. . . . . . . . . . . . . 50

Figure 3-5: Blue: the ratio of the transmitted to the incident 6He ions versus the effective
thickness of the degrader. Red: the corresponding range distribution of the
6He ions. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

. .

. .

.

Figure 3-6: The range straggling curve of the 6He beam from calculations using LISE++

(blue) and measurements (red).

. . . . . . . . . . . . . . . . . . . . . . . . . . 51

Figure 3-7: Energy of the 6He beam measured in the CsI (Na) detector without a degrader
in linear (left panel) and log (right panel) scales. The first dashed line in the
right panel indicates the expected energy of the 8Li contaminant if it was
present; the short peak at 4000 chan marked by the second dashed line and
the data in the energy range of 2400-4000 chan are due to the pile-up in the
DAQ system. . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

. .

Figure 3-8: The setup of the NaI(Tl) detector on the beamline. The 6He ions exiting
from a beam pipe through the 75 µm thick Zr vacuum window (left) were
implanted into the NaI(Tl) detector (right) aligned along the beamline.

. . . . . 53

Figure 3-9: The setup of the CsI(Na) detector on the beamline. The 6He ions exit-
ing from the beam pipe through the 75 µm thick Zr vacuum window (left)
were implanted into the CsI(Na) detector (right) aligned perpendicular to the
beamline. The CsI(Na) detector is surrounded by lead bricks (green). . . . . .

. 54

Figure 3-10: The setup of the OR120 Ge semiconductor detector mounted next to the beamline. 54

Figure 3-11: Left panel: a two-dimensional histogram of the registered time versus de-
posited energy during the beam-off time intervals collected in an one-hour
run with the CsI(Na) detector. Right panel: a background-subtracted β-
energy spectrum measured in the CsI(Na) detector during the beam off period.

. 57

Figure 4-1: Overview of the data analysis steps in this experiment. . . . . . . . . . . . . . . 58

Figure 4-2: The shapes of the energy spectra used in the fit function. . . . . . . . . . . . . . 59

Figure 5-1: Left panel: geometry of the CsI(Na) detector (gray) and position of the
electron source (red) used in the simulations. Dimensions are in cm. Right
panel: geometry of the NaI(Tl) detector (gray) and position of the electron
source (red) used in the simulations. (Published in Ref. [17].) . . . . . . . . .

. 62

xiii

Figure 5-2: Visualization of a Geant4 simulation with the CsI detector. 103 monoener-
getic electrons of 1 MeV were generated. Left: side view of the detector;
Middle: front view; Right: top view. The green lines indicates that a fraction
of the photons escape from the detector. (Published in Ref. [17].) . . . . . . . . 65

Figure 5-3: The photon energy spectrum produced by 2×107 monoenergetic primary
electrons of 3.5 MeV. The physics constructor is Standard. (Published in Ref.
[17].) .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

. 65

Figure 5-4: Photon yields obtained from simulations with the CsI (left panel) and the NaI
(right panel) detectors as a function of the initial kinetic energy of the primary
electron. The simulations were performed with the Standard constructor. The
open, light gray, and dark gray markers correspond to Geant4 simulations;
the black marker corresponds to the values from the NIST ESTAR program.
Details are explained in the text. (Published in Ref. [17].) . . . . . . . . . . . . 67

Figure 5-5: Normalized photon yields obtained from simulations with the CsI (left) and
the NaI (right) detectors as a function of the initial kinetic energy of the
primary electron source. The results are extracted from the simulations with
the Penelope (black markers), the Standard (gray markers), and the Livermore
(open markers) physics constructors by normalizing the photon yields relative
to that obtained with Option4. (Published in Ref. [17].)

. . . . . . . . . . . . . 69

Figure 5-6: Absorption fractions obtained from simulations with the CsI (squares) and
the NaI (circles) detectors as a function of the initial kinetic energy of the
primary electron source. The simulations were performed with the Standard
constructor. (Published in Ref. [17].) . . . . . . . . . . . . . . . . . . . . . . . 70

Figure 5-7: Normalized absorption fraction obtained from simulations with the CsI (left)
and the NaI (right) detectors as a function of the initial kinetic energy of the
primary electron source. The results are extracted from the simulations with
the Penelope (black markers), the Standard (gray markers), and the Livermore
(open markers) physics constructors by normalizing the absorption fractions
relative to that obtained with Option4. (Published in Ref. [17].) . . . . . . . . . 71

Figure 5-8: Upper panel:

the absorbed energy spectrum produced by a monoenergetic
primary electron source of 3.5 MeV in the CsI detector. The number of
primary events is 2 × 107 and the physics constructor is Standard. Lower
panel: the response function at 3.5 MeV obtained by convolving the upper
panel with a Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Figure 5-9: The β-energy spectrum in 6He decay. Blue: energy spectrum of the primary
electron source with 109 events; Red: absorbed energy spectrum in the
CsI detector. The simulation was performed with the Standard constructor.
(Published in Ref. [17].) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

xiv

Figure 5-10: Numerical derivatives of the ratio between the absorbed and the generated
β-energy spectra for the NaI (circles) and CsI (squares) detectors. (Published
in Ref. [17].) . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

. .

Figure 5-11: The photon energy spectrum produced by a β-energy spectrum generated
following Eq. 5.11. The number of primary events is 2×107 and the physics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
constructor is Standard.

. 76

Figure 5-12: Upper panel: the ratio between the absorbed energy spectra simulated with
the Standard and the Option4 constructors. The red line is a linear fit to the
ratio spectrum. Lower panel: the residuals of the fit.

. . . . . . . . . . . . . . . 79

Figure 6-1: Left panel: analytic β-energy spectrum obtained only from the phase space
factor. The theoretical spectrum (red) is obtained by stretching the experi-
mental spectrum (blue) along the horizontal axis by a factor of 1.1. Middle
panel: the ratio between a theoretical spectrum (red) with a stretch factor of
1.001 and the experimental spectrum (blue). Right panel: the slope of the
ratio shown in the middle panel weighted with the square root of the phase
space factor.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

. .

. .

.

Figure 6-2: Left panel:

the average bWM extracted from 200 Monte Carlo simulations
and fits as a function of the relative gain variation. The dashed line indicates
the input value of bWM in the simulations. Right panel: The average bGT
extracted from 200 Monte Carlo simulations and fits as a function of the
relative gain variation. The dashed line indicates the input value of bGT in
the simulations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

.

.

.

Figure 6-3: Left panel: the average bWM extracted from 200 Monte Carlo simulations and
fits as a function of the offset used in the fit. The dashed line indicates the
input value of bWM in the simulations. Right panel: the average bGT extracted
from 200 Monte Carlo simulations and fits as a function of the offset used in
the fit. The dashed line indicates the input value of bGT in the simulations.
Figure 6-4: Left panel: the average bWM extracted from 200 Monte Carlo simulations and
fits as a function of the input nonlinearity in the simulations. Right panel:
the average bGT extracted from 200 Monte Carlo simulations and fits as a
function of the input nonlinearity in the simulations.

. . . . . . . . . . . . . . . 86

. . . 86

Figure 6-5: Left panel:

the average bWM extracted from 200 Monte Carlo simulations
and fits as a function of the width parameter in the simulations. Right panel:
the average bGT extracted from 200 Monte Carlo simulations and fits as a
function of the width parameter in the simulations. . . . . . . . . . . . . . . . . 87

xv

Figure 6-6: The gain extracted from fits (upper panel) and the relative gain variation
multiplied by 1000 (lower panel) obtained from the SM fits as a function of
input values of the gain used in the simulations.

. . . . . . . . . . . . . . . . . 90

Figure 6-7: Left panel: values of bWM extracted from the SM fits with the gain as a free
parameter. The dotted line indicate the input value used in the Monte Carlo.
Right panel: values of bGT extracted from the BSM fits with the gain as a
free parameter. The dotted line indicate the input values used in the Monte Carlo. 90

Figure 6-8: The distributions of the statistical uncertainties of the gain extracted from
the SM fits (left panel) and the BSM fits (right panel) to a sample of 104
simulated spectra.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. 91

Figure 6-9: The distributions of the values of bWM (left panel) and bGT (right panel)
extracted from fits to a sample of 104 spectra, with the gain in the fits as a
free parameter (blue) and fixed to the input value (red).

. . . . . . . . . . . . . 92

Figure 6-10: The scatter plots of the values of br (left panel) and bGT (right panel) relative

to Ar. The results are extracted from fits to a sample of 104 spectra. . . . . . . . 92

Figure 6-11: Left panel:

the γ-energy spectrum of the 137Cs source measured with the
CsI(Na) detector at a rate of 8170.9 ± 5.1 (cps). Right panel: the γ-energy
spectrum of the 60Co source measured with the CsI(Na) detector at a rate of
14153.4 ± 6.3 (cps).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

.

Figure 6-12: The γ-energy spectrum of the 137Cs source measured with the CsI(Na) de-
tector (left panel of Fig. 6-11) zoomed in near the peak. The light and blue
bands are examples of two different energy ranges for the fit.

. . . . . . . . . . 94

Figure 6-13: The centroid (left panel) and width (right panel) of the peak in the 137Cs γ
spectrum (Fig. 6-12 or the left panel of Fig. 6-11) extracted from Gaussian
fits as a function of the band width.

. . . . . . . . . . . . . . . . . . . . . . . . 95

Figure 6-14: The centroid (left panel) and width (right panel) of the left peak in the 60Co
γ spectrum (the right panel of Fig. 6-11) extracted from Gaussian fits as a
function of the band width.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Figure 6-15: The centroid (left panel) and width (right panel) of the right peak in the 60Co
γ spectrum (the right panel of Fig. 6-11) extracted from Gaussian fits as a
function of the band width.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Figure 6-16: Upper panel: the centroid of the γ peaks from 137Cs and 60Co as a function
of the peak energy. The red line is a linear fit to the graph. Lower panel: the
residuals of the fit. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

. 97

xvi

Figure 6-17: Upper panel: the centroid of the γ peaks from 137Cs and 60Co as a function
of the peak energy. The red line is a quadratic fit to the graph. Lower panel:
the residuals of the fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Figure 6-18: Upper panel: the width of the γ peaks from 137Cs and 60Co as a function of
the peak energy. The red line is the result of a fit by Eq. 6.3. Lower panel:
the residuals of the fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Figure 7-1: Two-dimensional histogram of the time versus the deposited energy of 6He
β particles and 137Cs γ rays measured with the CsI(Na) detector recorded
during the beam-off intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Figure 7-2: The measured energy spectrum of β particles in 6He decay and γ rays emitted
by 137Cs projected from four time windows in Fig. 7-1 . The uncertainties
associated with the fitted centroid and width are statistical. . . . . . . . . . . . . 102

Figure 7-3: The centroid (left panel) and the width (right panel) of the 137Cs γ peak as a

function of the average rate in a time window from the on-line measurement. .

. 103

Figure 7-4: The centroid (left panel) and the width (right panel) of the 137Cs γ peak as a
function of the average rate in a time window from the off-line measurement

. . 104

Figure 7-5: The remaining scintillation yield as a fraction of the initial primary scintil-
lation measured at different times after the deposition of a 1.5(±0.2) MeV γ
source in the CsI(Na) and the CsI(Tl) detectors. Figure taken from Ref. [18]. . . 104

Figure 7-6: Upper panel: the corrected centroid of the γ peaks from 137Cs and 60Co as a
function of the peak energy. The red line is a linear fit to the graph. Lower
panel: the residuals of the fit.

. . . . . . . . . . . . . . . . . . . . . . . . . .

. 107

Figure 7-7: The distribution of the reconstructed energy of the 137Cs γ peak extracted

from the on-line measurement with the offset value taken from Eq. 7.6. . . . . . 108

Figure 7-8: Upper panel: the corrected centroid of the γ peaks from 137Cs and 60Co as
a function of the peak energy. The red line is a quadratic fit to the graph.
Lower panel: the residuals of the fit.

. . . . . . . . . . . . . . . . . . . . . . . 109

Figure 7-9: Upper panel: the corrected Gaussian width (keV) of the peaks in the 137Cs
and the 60Co γ spectra as a function of peak energy. The red line is the result
of a fit by Eq. 6.3. Lower panel: the residuals of the fit.

. . . . . . . . . . . . . 111

Figure 7-10: Upper panel:

the uncorrected Gaussian width (keV) of the peaks in 40K,
214Pb, 208Tl and 60Co γ spectra as a function of the peak energy. The red
line is the result of a fit by Eq. 6.3. Lower panel: the residuals of the fit.

. . . . 112

xvii

Figure 8-1: Left panel: values of bWM extracted from SM fits using GetBinContent.
The dotted line indicates the input value used in the Monte Carlo. Right
panel: values of Ar calculated with A extracted from the same fits.

. . . . . . . 117

Figure 8-2: Left panel: values of bGT extracted from BSM fits using GetBinContent.
The dotted line indicates the input value used in the Monte Carlo. Right
panel: values of Ar calculated with A extracted from the same fits.

. . . . . . . 118

Figure 8-3: A typical SM fit to a β-energy spectrum generated by Monte Carlo. The fit
c (E).
uses the number of events in the bin corresponding to E to estimate f 0
The spectrum is zoomed in near the top to show the details of the structure. . .

. 118

Figure 8-4: Left panel: values of bWM extracted from SM fits using Interpolate. The
dotted line indicates the input value used in the Monte Carlo. Right panel:
distribution of the statistical errors of bWM from the same fits. . . . . . . . . . . 119

Figure 8-5: Left panel: values of Ar extracted from SM fits using Interpolate. Right

panel: distribution of the statistical errors of A from the same fits. . . . . . . . . 119

Figure 8-6: Left panel: values of bGT extracted from BSM fits using Interpolate. The
dotted line indicates the input value used in the Monte Carlo. Right panel:
distribution of the statistical errors of bGT from the same fits.

. . . . . . . . . . 120

Figure 8-7: Left panel: values of Ar extracted from BSM fits using Interpolate. Right

panel: distribution of the statistical errors of A from the same fits. . . . . . . . . 120

Figure 8-8: Left panel: values of bWM extracted from SM fits using Tspline3. The
dotted line indicates the input value used in the Monte Carlo. Right panel:
distribution of the statistical errors of bWM from the same fits. . . . . . . . . . . 121

Figure 8-9: Left panel: values of Ar extracted from SM fits using Tspline3. Right

panel: distribution of the statistical errors of A from the same fits. . . . . . . . . 121

Figure 8-10: Left panel: values of bGT extracted from BSM fits using Tspline3. The
dotted line indicates the input value used in the Monte Carlo. Right panel:
distribution of the statistical errors of bGT from the same fits.

. . . . . . . . . . 122

Figure 8-11: Left panel: values of Ar extracted from BSM fits using Tspline3. Right

panel: distribution of the statistical errors of A from the same fits. . . . . . . . . 122

Figure 8-12: A typical SM fit to a β-energy spectrum generated by Monte Carlo. The fit
c (E). The spectrum is zoomed

uses a spline interpolation at E to estimate f 0
in near the top to show the details of the structure.

. . . . . . . . . . . . . . . . 123

xviii

Figure 8-13: Results from SM fits with the χ

2 method. Left panel:

the fraction of the
converged fits as a function of the number of events in each Monte Carlo
histogram. Middle panel: the fraction of fits with p-values above 0.05 as a
function of the number of events. Right panel: the bias as a function of the
number of events.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

.

Figure 8-14: Results from BSM fits with χ

2 method. Left panel:

the fraction of the
converged fits as a function of the number of events in each Monte Carlo
histogram. Middle panel: the fraction of fits with p-values above 0.05 as a
function of the number of events. Right panel: the bias as a function of the
number of events.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

.

Figure 8-15: Results from SM fits with the log-likelihood method. Left panel: the fraction
of the converged fits as a function of the number of events in each Monte
Carlo histogram. Middle panel: the fraction of fits with p-values above 0.05
as a function of the number of events. Right panel: the bias as a function of
the number of events.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Figure 8-16: Results from BSM fits with log-likelihood method. Left panel: the fraction
of the converged fits as a function of the number of events in each Monte
Carlo histogram. Middle panel: the fraction of fits with p-values above 0.05
as a function of the number of events. Right panel: the bias as a function of
the number of events.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Figure 8-17: Left panel: the fraction of convergent SM fits with χ

2 as a function of the
function spectrum rebin factor (RF) and the Monte Carlo histogram rebin
factor (RMC). Right panel: the fraction of the fits with p-values over 0.05
as a function of the function spectrum rebin factor (RF) and the Monte Carlo
histogram rebin factor (RMC). . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Figure 8-18: Left panel: the bias of bWM extracted from SM fits with χ

2 as a function of
the function spectrum rebin factor (RF) and the Monte Carlo histogram rebin
factor (RMC). Right panel: zoomed view of the left panel. . . . . . . . . . . . . 128

Figure 8-19: Left panel: the fraction of convergent SM fits with log-likelihood as a function
of the function spectrum rebin factor (RF) and the Monte Carlo histogram
rebin factor (RMC). Right panel: the fraction of the fits with p-values over
0.05 as a function of the function spectrum rebin factor (RF) and the Monte
Carlo histogram rebin factor (RMC).

. . . . . . . . . . . . . . . . . . . . . . . 129

Figure 8-20: Left panel: the bias of bWM extracted from SM fits with log-likelihood as
a function of the function spectrum rebin factor (RF) and the Monte Carlo
histogram rebin factor (RMC). Right panel: zoomed view of the left panel.

. . . 129

xix

Figure 8-21: A β-energy spectrum in 6He decay measured in the CsI(Na) detector (the

right panel of Fig. 3-11) fitted by Eq. 8.4.

. . . . . . . . . . . . . . . . . . .

. 131

Figure 9-1: The 6He decay spectrum recorded during the beam-off interval. The data is
collected during a single one-hour run. The red and blue dashed lines indicate
the levels of the beam-induced background and the ambient background.

. . . . 134

Figure 9-2: The energy spectrum of the ambient background measured with the CsI(Na)

detector with peak identifications. . . . . . . . . . . . . . . . . . . . . . . . .

. 134

Figure 9-3: The energy spectrum of the beam-induced background measured with the

CsI(Na) detector . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Figure 9-4: Left panel: an example of selected S+B and background time windows. The
time interval of 3.43-3.61 s (left vertical band) is selected as the S+B window,
and time interval of 9.43-9.61 s (right vertical band) is selected as the back-
ground window. Right panel: the energy spectra measured with the CsI(Na)
detector during 3.43-3.61 s (black) and 9.43-9.61 s (blue) corresponding to
the two time windows marked in the left panel. . . . . . . . . . . . . . . . . . . 136

Figure 9-5: The background-subtracted spectrum for the β spectra with an extra 137Cs γ

peak, where the S+B spectra were projected from four time windows. . . . . . . 137

Figure 9-6: Upper panel: the S+B (black) and background (blue) spectra generated with
Monte Carlo. The S+B spectrum has a higher gain and a higher sigma than the
background spectrum. Lower panel: the spectrum after background subtraction. 138

Figure 9-7: Left panel: zoomed-in view of the S+B (black), uncorrected background
(blue), and gain-corrected background (red) spectra generated with Monte
Carlo. Right panel: the full view of S+B (black) and gain-corrected back-
ground (red) spectra in the upper panel, and the new background-subtracted
spectrum in the lower panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 9-8: Left panel: zoomed-in view of the S+B (black), uncorrected background
(blue), and gain-and-sigma corrected background (red) spectra generated
with Monte Carlo. Right panel: the full view of S+B (black) and gain-and-
sigma corrected background (red) spectra in the upper panel, and the new
background-subtracted spectrum in the lower panel.

. . . . . . . . . . . . . . . 141

Figure 9-9: The S+B (black) and background (blue) spectra for the β spectra with an extra

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

137Cs γ peak.

Figure 9-10: Zoomed-in view of the background-subtracted spectra for the β spectra with

an extra 137Cs γ peak (Fig. 9-5).

. . . . . . . . . . . . . . . . . . . . . . . .

. 144

xx

Figure 9-11: The S+B (black), background (blue), and gain-corrected background (red)

spectra for the β spectrum with an extra 137Cs γ peak. . . . . . . . . . . . . . . 145

Figure 9-12: Zoomed-in view of the new background-subtracted spectra for the β spectra
with an extra 137Cs γ peak after gain corrections to the background spectra.

.

. 146

Figure 9-13: The S+B (black), background (blue), and gain-and-sigma corrected back-

ground (red) spectra for the β spectra with an extra 137Cs γ peak.

. . . . . . . . 147

Figure 9-14: Zoomed-in view of the new background-subtracted spectra for the β spectra
with an extra 137Cs γ peak after gain-and-sigma corrections to the background
spectra. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

.

.

.

.

.

.

.

.

Figure 9-15: The S+B (black) and background (blue) spectra for 6He β decay.
Figure 9-16: Zoomed-in view of the background-subtracted β-energy spectra in 6He decay.

. . . . . . . . 149

. 150

Figure 9-17: The S+B (black), background (blue), and gain-corrected background (red)

spectra for 6He β decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Figure 9-18: Zoomed-in view of the new background-subtracted β-energy spectra in 6He

decay after gain corrections to the background. . . . . . . . . . . . . . . . . . . 152

Figure 9-19: The S+B (black), background (blue), and gain-and-sigma corrected back-

ground (red) spectra for 6He β decay. . . . . . . . . . . . . . . . . . . . . . . . 153

Figure 9-20: Zoomed-in view of the new background-subtracted β-energy spectra in 6He

decay after gain-and-sigma corrections to the background. . . . . . . . . . . . . 154

xxi

CHAPTER 1

INTRODUCTION

This chapter provides the background for the precision measurement of the β-energy spectrum in
6He decay. We will start with the physics motivations and the precision goals for the new physics.
We will provide a list of relevant physics observables. It will be shown that one of the observables,
the Fierz interference term (bGT ), could manifest itself through the shape of the β-energy spectrum.
We will explain why the nuclear β decay of 6He is an attractive candidate. In addition, we will
provide a brief history of the 6He β-spectrum measurement. A few recent measurements of the
β-energy spectra in other decay systems will be reviewed, focusing on the common instrumental
challenges.

1.1 Motivation

The Standard Model (SM) of particle physics describes the building blocks of nature and their
interactions. It has been an enormous success since its discovery, particularly with the encouraging
discovery of the Higgs boson. However, today, it is believed that the SM is not the ultimate
description of nature. In the past few decades, a pandora box of puzzles has been opened, such as:

• The SM includes 18 free parameters that are not predicted by theory and are solely determined

by experiments.

• The SM does not include gravity. The unification of gravity with the rest of the fundamental

interactions requires the quantization of gravity.

• The SM does not explain the origin of matter-antimatter asymmetry. The CP violations
observed in hadronic sectors so far are insufficient to describe the amount of matter in the
universe. Extensive searches for new CP violations are taking place.

• The SM suggests that the masses of neutrinos are zero. However, the discovery of the

1

neutrino oscillation indicates non-zero neutrino masses. Besides, the exact masses of the
three neutrino flavors and the neutrino mass hierarchy remain unknown.

• The total mass and energy of the universe include roughly 4.9% of the ordinary matter
described by the SM, 26.8% dark matter, and 68.3% dark energy. The natures of dark matter
and dark energy remain to be cosmological puzzles.

The above open questions motivate physicists to search for a new theoretical framework that
involves new physics beyond the SM. Currently, there exist two frontiers to search for new physics:
the energy frontier, which directly searches for new particles with particle colliders; and the
precision frontier, which indirectly searches for signatures of new physics that manifest in low
and intermediate energy processes. Such signatures would result in small deviations from the SM
predictions and are accessible through high-precision measurements. This work belongs to the
latter category.

1.1.1 Phenomenology of Semileptonic Weak Processes

Historically, the neutron and nuclear β decays have played an essential role in the development
of the electroweak interactions in the SM. For example, the fact that the β-energy spectra are
continuous, rather than discrete, has inspired Wolfgang Pauli to propose a new particle, neutrino,
as a “desperate remedy” to save the law of energy conservation. β decays have also lead to the
discovery of parity violation by Chien Shiung Wu and collaborators through a measurement of the
asymmetry in β emission by polarized 60Co nuclei. In addition, soon after the discovery of parity
violation, β decays have played a prominent role in shaping the V − A structure of weak interactions,
which eventually became part of the SM framework.

Today, there is strong motivation to search for the deviation from the V − A structure of weak
interactions. Despite that the V − A theory has been consistent with data, the existence of other
interactions, i.e., the “exotic” interactions beyond the SM, have not yet been excluded. These
“exotic” interactions would manifest themselves through the scalar or tensor terms in semi-leptonic

2

weak processes. These processes can be studied with an Effective Field Theory (EFT) approach,
which makes a connection between the observables in low-energy and high-energy processes and
allows a comparison between the sensitivities of these observables to new physics. The general
expression of the low-energy (O(1 GeV)) effective Lagrangian for semi-leptonic transitions is [19]

Le f f = − G(0)
F Vud√
2

[(1 + 
L) ¯eγµ(1 − γ5)νl · ¯uγ µ(1 − γ5)d + ˜
L ¯eγµ(1 + γ5)νl · ¯uγ µ(1 − γ5)d

+ 
R ¯eγµ(1 − γ5)νl · ¯uγ µ(1 + γ5)d + ˜
R ¯eγµ(1 + γ5)νl · ¯uγ µ(1 + γ5)d
+ 
S ¯e(1 − γ5)νe · ¯ud + ˜
S ¯e(1 + γ5)νe · ¯ud
− 
P ¯e(1 − γ5)νe · ¯µγ5d − ˜
P ¯e(1 + γ5)νe · ¯µγ5d
+ 
T ¯eσµν(1 − γ5)νe · ¯uσ µν(1 − γ5)d + ˜
T ¯eσµν(1 + γ5)νe · ¯uσ µν(1 + γ5)d] + h.c., (1.1)

F

where G(0)
is the Fermi constant, Vud is the Cabibbo-Kobayashi-Maskawa (CKM) matrix el-
ement describing the weak mixing of the up and down quarks.
l represents the weak flavor.

i(i = V, A, S, T, P) stands for quark-level couplings for vector, axial-vector, scalar, tensor, and
pseudoscalar type interactions. With the focus on the linear order of the couplings, we neglect the
terms related to right-handed neutrinos by setting ˜
L,R,S,P,T = 0, which are of the second order.
In addition, we drop the pseudoscalar terms. As the pseudoscalar bilinear is O(1/M), where M is
nucleon mass, the pseudoscalar contributions are strongly suppressed in the nonrelativistic limit.
Thus the linear-order low-energy effective Lagrangian for neutron and nuclear β decay has the form
[20]

Le f f = − G(0)
F Vud√
2

(1 + 
L + 
R)(cid:8) ¯eγµ(1 − γ5)νe · ¯uγ µ[1 − (1 − 2
R)γ5]d
+ 
S ¯e(1 − γ5)νe · ¯ud + 
T ¯eσµν(1 − γ5)νe · ¯uσ µν(1 − γ5)d(cid:9) + h.c.,

(1.2)

where the factor (1 + 
L + 
R) affects the overall normalization and can be absorbed into Vud. The
couplings related to new physics are:

• The right-handed coupling, 
R, which can be extracted by measuring (1 − 2
R)gA/gV in

3

neutron decays and determining gA/gV from lattice QCD calculations, where gV and gA are
vector and axial form factors.

• The quark-level scalar and tensor couplings, 
S and 
T. They are accessible in both low-
energy precision measurement experiments and high-energy collider experiments. We will
discuss the current limits on 
S and 
T from low-energy and high-energy experiments and
compare their sensitivities in Sec. 1.2.3.

In order to make connections with the nuclear and neutron β decay observables, one could match
the quark-level effective Lagrangian (Eq. 1.2) with a nucleon-level effective Lagrangian obtained
by Lee and Yang [21, 22], who introduced the Lee-Yang effective couplings Ci, C(cid:48)
i(i ∈ V, A, S, T),
where the primed coefficients correspond to parity non-conserving interactions. Once again, we
neglect the right-handed neutrinos, i.e., setting ˜
L,R,S,T = 0, the Lee-Yang effective couplings are
related to the quark-level couplings in the following expressions [19]

CV = gV(1 + 
L + 
R + ˜
L + ˜
R)
C A = −gA(1 + 
L − 
R − ˜
L + ˜
R)
CS = gS(
S + ˜
S)
CP = gP(
P − ˜
P)
CT = 4gT(
T + ˜
T)

C

C

(cid:48)
V = gV(1 + 
L + 
R − ˜
L − ˜
R)
(cid:48)
A = −gA(1 + 
L − 
R + ˜
L − ˜
R)
(cid:48)
S = gS(
S − ˜
S)
C
(cid:48)
P = gP(
P + ˜
P)
(cid:48)
T = 4gT(
T − ˜
T),

C

C

(1.3)

(1.4)

(1.5)

(1.6)

(1.7)

where Ci = GF√
differences, Ci ± C(cid:48)

i

2 VudCi. It is useful to express the effective couplings in terms of their sums and

[1]:

CV + C

C A + C

(cid:48)
V = 2gV(1 + 
L + 
R),
(cid:48)
A = −2gA(1 + 
L − 
R),
(cid:48)
S = 2gS
S,
CS + C
(cid:48)
P = 2gP
P,
(cid:48)
T = 8gT 
T,

CT + C

CP + C

(cid:48)
V = 2gV( ˜
L + ˜
R),
CV − C
(cid:48)
A = 2gA( ˜
L − ˜
R),
C A − C
(cid:48)
CS − C
S = 2gS ˜
S,
(cid:48)
CP − C
P = −2gP ˜
P,
(cid:48)
CT − C
T = 8gT ˜
T,

(1.8)

(1.9)

(1.10)

(1.11)

(1.12)

4

where the sums and differences are sensitive to left-handed and right-handed neutrinos, respectively.
The motivation for separating the contributions of the left-handed and right-handed neutrinos is
that the constraints on the former is much more accessible experimentally. The constants gi(i ∈
V, A, S, P, T) are hadronic form factors, among which gV = 1 is predicted by the Conserved Vector
Current (CVC) hypothesis, and the others are precisely determined with lattice QCD calculations.
The current most precise values are summarized in Table. 1-1 [1]:

Table 1-1: Current most precise values of the hadronic form factors determined with lattice QCD
calculations from Ref. [1].

Value
Reference

gA
1.278(21)(26)
[23]

gS
1.02(11)
[24]

gP
349(9)
[24]

gT
0.987(51)(20)
[25]

1.1.2 Current Limits on the Scaler and Tensor Couplings

The current strongest constraints on the exotic couplings are provided in Ref.
[1], where the
exotic couplings are extracted by fitting 17 experimental input data, including the F t values from
0+ → 0+ transitions and selected neutron data. These fits assume that the right-handed neutrinos
are absent, i.e., C(cid:48)
A)/(CV + C(cid:48)
V),
i
CS/CV = (CS + C(cid:48)
A). The results are shown in Fig.
1-1, where the extracted exotic couplings are

= Ci, and that all couplings are real, i.e., CA/CV = (CA + C(cid:48)
S)/(CV + C(cid:48)

V), CT/CA = (CT + C(cid:48)

T)/(CA + C(cid:48)

CS/CV = 0.0014 ± 0.0012,
CT/CA = 0.0020 ± 0.0022,

(1.13)

(1.14)

and the contours around the minimum correspond to 68%, 95%, and 99% confidence intervals,
respectively.

The most stringent constraint on the scalar couplings is placed by F t values from 0+ → 0+
pure Fermi transitions. The constraint on the tensor coupling, on the other hand, is a factor of two
weaker than that on the scalar coupling, as shown in Fig. 1-1. The weak constraint on the tensor
coupling motivates a new generation of experiments aiming at stronger constraints.

5

Figure 1-1: The current limits of scalar (horizontal) and tensor (vertical) couplings extracted from
the fit to F t values, average neutron lifetime, average neutron β asymmetry ˜An, and the λAB
parameter. The contours around the minimum correspond to 68%, 95%, and 99% confidence
intervals. Figure taken from Ref. [1].

1.2 Observables

1.2.1 Correlation Coefficients

The observables of these low-energy experiments searching for exotic couplings are the correlation
coefficients, proposed by Jackson, Treiman and Wyld [26, 27] for allowed β decays. The correlation
coefficients relate the effective couplings, Ci and C(cid:48)
, to the angular and energy distributions of β
particles. Using the effective Lagrangian (Eq. 1.2), and neglecting the recoil and electromagnetic
corrections, the decay-rate distributions for allowed β decays are derived in Refs [26, 27]. For
polarized nuclei, the distribution in electron energy (Ee), electron direction (Ωe), and neutrino
direction (Ων) is

i

6

(cid:21)(cid:34)

(cid:35)

+ b

me
Ee

1 + a

F(±Z, Ee)
(2π)5
× ξ

(cid:40)
(cid:20)pe · pν
(cid:20)

+ c
(cid:104)J(cid:105)
J

A

+

w((cid:104)J(cid:105)|Ee, Ωe, Ων)dEedΩedΩν =

peEe(E0 − Ee)2dEedΩedΩν
pe · pν
EeEν
− (pe · j)(pν · j)
3EeEν
pe
·
Ee
where pe is the momentum of the β particle, pν
is the momentum of the electron neutrino, E0 is
the endpoint energy of the decay, (cid:104)J(cid:105) is the expectation value of the spin of the initial nuclear state,
J is the unit vector in the direction of (cid:104)J(cid:105), and F(±Z, Ee) is the Fermi function introduced in Sec.
2.1.

J(J + 1) − 3(cid:104)J · j(cid:105)2

EeEν
pν
Eν

pe × pν
EeEν

J(2J − 1)

(cid:21)(cid:41)

+ B

+ D

,

(1.15)

The correlation coefficients, ξ, a, b, c, A, B, and D, depend on different combinations of the
sixteen Lee-Yang effective couplings, C and C(cid:48), and are accessible experimentally. Specifically,
a is the β-neutrino asymmetry, b is the Fierz interference term, A is the β asymmetry, B is the
neutrino asymmetry, and D is a triple correlation coefficient. The c coefficient vanishes if the initial
nucleus is nonoriented (J(J + 1) = 3(cid:104)J · j(cid:105)2).

In addition, for unpolarized nuclei, the distribution function in electron and neutrino directions,
electron energy, and electron polarization for allowed β decay depend on correlation coefficients
G, H, K, L. For polarized nuclei, the distribution function in electron energy, electron direction, and
electron polarization depend on coefficients A, J, N, Q, R. Details of their origins and dependences
on the effective couplings can be found in Refs. [26, 27]. The most common correlation coefficients
are summarized with pictorial representation in Fig. 1-2.

It is important to identify the correlation coefficients that have linear dependences on the
exotic couplings, which makes them most sensitive to new physics. Among them, the coefficients
that have been accessed experimentally are: b, B, N, Q, R and D [28]. Both the scalar and
tensor couplings contribute linearly to the Fierz interference term b and the neutrino asymmetry
correlation coefficient B. The measurement of B is challenging because it involves the neutrino

7

Figure 1-2: A pictorial representation of the most common correlation coefficients, the two-fold
(edges) and the three-fold (faces) correlations between the kinematic vectors (vertices). Figure
taken from Ref. [4].

It is also difficult to access the polarization-spin
momentum and the nuclear polarization [1].
correlation N and the longitudinal polarization from polarized nuclei Q, as the analysis of the spin
of β particles emitted from polarized neutrons or nuclei is required [28].

1.2.2 The Fierz Interference Term

The correlation coefficients that are most accessible in β decays of unpolarized nuclei are the β-
neutrino asymmetry a and the Fierz interference term b. They are related to the Lee-Yang effective
couplings in the following expressions [1]
V|2 − |CS|2 − |C(cid:48)
(cid:17)
CVC∗
S + C(cid:48)
V|2 + |CS|2 + |C(cid:48)

(cid:16)|CA|2 + |C(cid:48)
S|2(cid:17) − |MGT|2
(cid:1)(cid:105)
+ |MGT|2(cid:0)CAC∗
+ |MGT|2(cid:16)|CA|2 + |C(cid:48)
S|2(cid:17)

T|2(cid:17)
T|2(cid:17)

aξ = |MF|2(cid:16)|CV|2 + |C(cid:48)
(cid:104)|MF|2(cid:16)
ξ = |MF|2(cid:16)|CV|2 + |C(cid:48)

bξ = ±2γRe

A|2 + |CT|2 + |C(cid:48)

A|2 − |CT|2 − |C(cid:48)

T + C(cid:48)

AC(cid:48)∗

T

VC(cid:48)∗

S

3

,

(1.16)

(1.17)

, (1.18)

2Z2 and α is the fine structure constant. As the SM predicts
where ξ is an overall factor, γ =
zero values of the scalar and tensor couplings, the Fierz interference term is predicted to be zero

√
1 − α

,

8

by the SM. Any deviation of the measured Fierz interference term from zero would indicate the
presence of new physics.

It is worth noticing that the β-neutrino asymmetry a depends quadratically on the scalar and
tensor coupling constants, while the Fierz interference term b depends linearly on these couplings.
Therefore, the measurement of a requires higher precision than b to have comparable constraints to
the exotic couplings. The linear dependence of the Fierz interference term on the exotic couplings
allows high sensitivity to new physics.

The Fierz interference term contributes to almost all other correlation coefficients. For most
of the experiments determining the correlation coefficients, instead of directly measuring the
correlation coefficient themselves, the actual measured quantity includes a contribution from the
Fierz interference term [29]:

X

˜X =

1 + b(cid:104)m/Ee(cid:105),

(1.19)
where X = a, A, B, etc., which stands for a correlation coefficient, and (cid:104)m/Ee(cid:105) stands for the
weighted average over the observed part of the β-energy spectrum. The sensitivities of these
correlation coefficients are affected by the contribution of the Fierz interference term. The Fierz
interference term, on the other hand, can be accessible experimentally through not only indirect
measurements of the other correlation coefficients but also a direct measurement of b.

The expression of b can be simplified for pure Fermi and pure Gamow-Teller transition:

(cid:32)CS + C(cid:48)
(cid:32)CT + C(cid:48)

CV

(cid:33)
(cid:33)

CA

bF = ±γRe

bGT = ±γRe

S

T

,

,

(1.20)

(1.21)

where bF and bGT stand for the Fierz interference terms in Fermi and Gamow-Teller transitions,
respectively. In this case, bF and bGT are independent of the nuclear matrix elements, |MF| and
|MGT|. This is a great advantage because it circumvents a dedicated evaluation of |MF| or |MGT|.
In summary, the discussion above shows that the Fierz interference term has unique advantages
to place a stringent constraint on exotic couplings.
In this work, the observable of choice is,
therefore, the Fierz interference term in Gamow-Teller transitions, bGT, which is sensitive to tensor

9

couplings. Remarkably, in most of the past experiments aiming at the exotic coupling searches, little
attention had been paid to the Fierz interference term. There have been no direct measurements of
the Fierz interference term in Gamow-Teller decays in literature. The long-term goal of this work
is to provide the first direct measurement of the Fierz interference term in Gamow-Teller decay.

1.2.3 Current Status of the Fierz Interference Term

The Fierz interference term in Fermi transitions, bF, has been extracted by Hardy and Towner
through years of analysis of the F t values in superallowed 0+ → 0+ Fermi transitions. The latest
value is [30]

bF = −0.0028 ± 0.0026,

(1.22)

providing a strong constraint on the scalar couplings.

The first and only direct extracted Fierz interference term in neutron decay (bn) has be obtained
by the UCNA Collaboration, finding [31] bn = 0.067 ± 0.005stat
, which is consistent
with zero, as predicted by the SM. While the statistical uncertainty is on the order of 10−3, the
systematic uncertainty, dominated by the energy calibration, is too high for a stringent constraint
on the exotic couplings. Significant improvement in the characterization of the energy response
of the detection system is required. One ongoing effort aiming to improve the precision of bn in
neutron decay is the Nab experiment [32]. Their precision goal is δbn (cid:39) 3 × 10−3.

+0.090
−0.061sys

For the measurement of the Fierz interference term in nuclear Gamow-Teller decays (bGT), sev-
eral experiments are ongoing or planned. At the National Superconducting Cyclotron Laboratory,
the β-energy spectra of 6He and 20F have been measured with scintillating detectors [33]. The
ongoing data analyses aim at a O(10−3) precision of bGT. The β-energy spectrum in 45Ca decay
has been measured by the UCNA-Nab-Leuven collaboration with a 4π magnetic spectrometer.
The ongoing data also aim at a O(10−3) sensitivity to bGT. Another experiment dedicated to the
measurement of the β-energy spectrum uses the miniBETA spectrometer [34], which incorporates
a low-pressure multiwire drift chamber and is currently being commissioned.

10

The current benchmark uncertainty on the Fierz interference term is provided in Ref.

[1],
where the uncertainty is derived from the exotic couplings extracted from a fit assuming no right-
handed neutrinos. Twenty-eight experimental input data were incorporated in the fit, including
the F t values from 0+ → 0+ transitions and all neutron and nuclear data. The resulting absolute
uncertainties for neutron and Gamow-Teller decays are respectively

δbn = 3.2 × 10−3
δbGT = 3.9 × 10−3

,

.

(1.23)

(1.24)

These uncertainties provide a precision level any new measurement needs to reach to improve the
current constraint on the tensor couplings. The above result for bGT is extracted from indirect
measurements.

The sensitivities to new physics in low-energy experiments can be compared with that in high-
energy experiments with the EFT approach. The analysis of the data taken at the LHC in the SMEFT
framework has provided a constraint on the quark-level tensor couplings. The 90% confidence level
bound on 
T is |
T| (cid:46) 0.6 × 10−3. The corresponding 1σ uncertainty is 0.37 × 10−3. This
uncertainty can be converted to the uncertainty of bGT, according to Eq. 1.12:

GT (cid:46) 2.9 × 10−3
bLHC

.

(1.25)

Equation 1.25 provides a precision level any new low-energy measurement has to reach to have a
comparable level of constraint with the LHC searches.

1.3 The Shape of the β-energy Spectrum and the Test of the CVC Hypothesis

The Fierz interference term can be extracted by precision measurements of the energy spectrum
of β particles in a neutron or nuclear β decay. A non-zero Fierz interference term would induce
a distortion to the β-energy spectrum. For allowed decays, the shape of the β-energy spectrum is
described by

N(W)dW = pW(W0 − W)2

η(W)dW

11

(cid:18)

(cid:19)

1 +

bGT
W

f1 (E, bWM) ,

(1.26)

where p is the momentum of β particle, W is the electron total energy in the unit of the rest energy
of the electron, mc2 (natural unit), where m is the electron mass. W0 is the maximum electron
total energy allowed by kinematics and is referred to as the endpoint energy. pW(W0 − W)2dW is
the dominating statistical factor that is referred to as the phase space factor. η(W) is the product
of several theoretical corrections that are discussed in details in Chapter 2. (1 + bGT/W) is the
distortion to the β-energy spectrum by the possible presence of the Fierz interference term bGT.
f1(E, bWM) is the spectral function and it is associated with the weak magnetism form
Finally,
In
the Gamow-Teller decay of 6He [35], with the inclusion of the recoil to first order, assuming the
absence of the second class currents and the induced tensor produced by the second class currents
[36][37], the spectral function has the expression:

factor, bWM, where E is the kinetic energy of β particle in the SI unit (cid:16)

E = (W − 1)mc2(cid:17).

f1(E, bWM) = 1 + C0 + C1W +

C−1
W ,

where

(cid:19)

(cid:18)

W = E + mc2
,
(cid:18)
C0 = −2W0mc2
3M
(cid:18)
5 +
(cid:118)(cid:116)6ΓM1M2
C−1 = −2mc2
3M

2
3M

C1 =

bWM =

αE3

γ

(cid:19)

1 +
2bWM

c
bWM
1 +

(cid:19)

bWM

= −1.234(14)%,
c
= 0.650(7)%/MeV[5],

= −0.0802(9)%MeV[5],

c
= 68.22(79).

(1.27)

(1.28)

(1.29)

(1.30)

(1.31)

(1.32)

Here, ΓM1 is the M1 γ decay width of an analog state within an isospin triplet [38], Eγ is the
energy of the γ transition [39], M is the nuclear mass of the daughter nucleus 6Li [40], and
c = gAMGT = 2.7491 is the dominant Gamow-Teller strength [41].

The above expression of the weak magnetism form factor, bWM, is predicted in terms of the M1
electromagnetic decay width of an analog state in an isospin triplet. This prediction is based on
the CVC hypothesis, which stands for Conservation of the Vector Current that is analogous to the

12

conservation of electric charge, and it relates the matrix elements of the charged weak polar vector
current with the corresponding electromagnetic amplitudes. The CVC hypothesis can be tested by
measuring the β-energy spectrum and extracting the value of bWM.

While bWM is present in all C0, C1 and C−1 coefficients, its most significant impact on the
shape of β-energy spectrum is through the C1 coefficient, which mostly dominates the slope of the
spectral function. Most of the existing tests of the CVC hypothesis [42, 43, 44, 12] were performed
by measuring the slopes of the spectral functions and comparing them with the slopes predicted by
the CVC hypothesis, rather than directly measuring the values of bWM.
1.4 The Candidate: 6He

The candidate selected for this study is the nuclear β decay of 6He. The decay scheme is shown
in the left panel of Fig. 1-3, where a 6He decays into a 6Li in an allowed pure Gamow-Teller
transition. The right panel of Fig. 1-3 shows the simplified level scheme in the mass A = 6 isospin
triplet, where the 6He ground state forms an isospin triplet with the 0+ state of 6Li at 3563 keV and
the unbound ground state of 6Be. The CVC value of bWM in 6He decay can be calculated with the
width and energy of the M1 transition in 6Li.

Figure 1-3: Left panel: the decay scheme of the 6He nuclear β decay. Right panel: the simplified
level scheme and transition in the mass A = 6 isospin triplet taken from Ref. [5].

The 6He decay is an attractive candidate for several advantages listed below:

• This transition is sensitive to tensor couplings because the nuclear β decay of 6He is a

13

100% Gamow-Teller decay. Since the Fierz interference term is proportional to tensor
couplings in Gamow-Teller decays, this decay process is sensitive to tensor couplings through
a measurement of the Fierz interference term.

• The endpoint of 6He decay is about 3.5 MeV, which results in high kinematic sensitivity to
the Fierz interference term. Reference [6] has studied with Monte Carlo the impact of the
endpoint on the statistical uncertainty of b. The results are shown in Fig. 1-4, where the 1σ
statistical uncertainties of b are extracted by fitting the β-energy spectrum corresponding to
each endpoint. It is worth noticing that the β decays of neutron and 6He have high statistical
sensitivities, and they are comparable.

Figure 1-4: The 1σ statistical uncertainties of the Fierz interference term obtained by fitting the
simulated β-energy spectra with different values of the endpoint energy. Figure taken from Ref.
[6].

• There is no subsequent γ ray emitted in the decay of 6He, which is a great advantage for a clean
measurement of β-energy spectrum. In addition, Chap. 3 has shown that the measurements
are performed in cycles, and each cycle consists of a beam implantation time window and a
decay time window. The β decay of 6He has a half-life of about 806 ms [45], which makes
it convenient to manipulate the beam cycles and to perform decay measurements.

14

There have been no measurements of the weak magnetism form factor in 6He decay reported
in literature. The short-term goal of this work is to provide the first measurement of bWM in 6He
decay as a test of CVC, before reaching the long-term goal of the first direct measurement of bGT
in Gamow-Teller decay.

1.5 Existing Measurements of β-energy Spectra

1.5.1 A Brief History of the Measurements of the β-energy Spectra in 6He Decay

T. Bjerge et al. [7] have reported the earliest measurement of the β-energy spectrum in the decay
of a helium isotope in the 1930s. The kinetic energy of β particles was measured with a cloud
chamber. The obtained β-energy spectrum is shown in Fig. 1-5, from which the endpoint energy
of this decay was determined to be 3.7 ± 0.5 MeV. The authors concluded that the most reasonable
assumption of the isotope identity was 6He.

Figure 1-5: The β-energy spectrum in the decay of a helium isotope measured with a cloud
chamber. The isotope identify was believed to be 6He. Figure taken from Ref. [7].

V. Perez-Mendez et al.

[8] have reported the first dedicated measurement of the β-energy
spectrum in 6He decay. The energy of β particles was measured with a semicircular focusing
magnetic spectrometer. The obtained Kurie plot of the β-energy spectrum is shown in Fig. 1-6,
where the tail near the endpoint is because the energy resolution is ∼6%.

15

Figure 1-6: Kurie plot of the β-energy spectrum in 6He decay. The horizontal axis is the kinetic
energy of β particles in the natural unit (mc2). Figure taken from Ref. [8].

C.S. Wu et al. [9] have remeasured the β-energy spectrum in 6He decay with the Columbia
magnetic solenoidal spectrometer. The Kurie plot shown in Fig. 1-7 is linear in the energy range
of 0.6 MeV to the endpoint. The determined endpoint energy is 3.50 ± 0.05 MeV.

Figure 1-7: Kurie plot of the β-energy spectrum in 6He decay. Figure taken from Ref. [9].

B.M. Rustad et al. [10] have performed a measurement of the β-neutrino angular correlation in
6He decay in the 1950s. As a part of this experiment, a β-energy spectrum in 6He was measured to
determine whether an impurity with a half-life close to that of 6He but of different endpoint energy
was present. A schematic sketch of the experimental setup is shown in the left panel of Fig. 1-8,
where the 6He β-active volume and the scintillation spectrometer are located in a common vacuum

16

volume. The scintillation spectrometer measures the energy and direction of the β particles. The
energy distribution of the β particles striking the stilbene plastic is shown in the right panel of
Fig. 1-8. This experimental setup, with the decay source next to the detector, faced instrumental
difficulties due to the backscattering of β particles from the stilbene plastic that greatly distorted
the β-energy spectrum.

Figure 1-8: Left panel: a schematic sketch of the apparatus used to determine the electron-recoil
ion correlation in the decay of 6He. The beta-active source volume and the scintillation
spectrometer are located in the common vacuum of the “bell-jar”. Right panel: the 6He β-energy
spectrum measured with the stilbene scintillation spectrometer. Figures taken from Ref. [10].

It is interesting to notice that the motivation of this experiment was to determine the nature of
weak interactions. The result had led the authors to believe that weak interaction was dominated by
tensor couplings, which is different from what we know today: the contribution of tensor couplings
is minimal if non-zero, and is precisely what this work is searching.

A. Schwarzschild [11] has reported in his Ph.D. thesis a dedicated precise measurement of the
shape of β-energy spectrum in 6He decay to provide limits on the axial vector and pseudoscalar
couplings in β decay. The energy of β particles was measured with a magnetic spectrometer. Figure
1-9 shows a scale drawing of the spectrometer baffle system, the source volume, and the Geiger
counter.

The greatest contributions to the systematics in this experiment came from the scattering of β
particles. First of all, the scattering of the electrons in the source with a finite thickness caused

17

Figure 1-9: A scale drawing of the thin lens spectrometer baffle system, the source volume, and
the Geiger counter. Figure taken from Ref. [11].

the energy loss of electrons, distorting the measured β-energy spectrum. Secondly, the scattering
of the electrons in the counter window affects the shape of β-energy spectrum, as the thickness
of the counter window also reduces the energy of β particles. Lastly, the most severe distortion
to the β spectrum came from the scattering of electrons from the baffle system. This effect is far
more difficult to investigate than the previous two, and the authors attempted to set an upper limit
on the distortion to the β spectrum in the low energy region. Other effects taken into account in
the analysis are the inconsistency of the activity of 6He beam delivered to the spectrometer source
volume, the background of the counter, the corrections for the energy resolution, and the corrections
for the 23Ne contaminant. Figure 1-10 shows the resulting Kurie plot of the 6He beta spectrum,
where the deviations in the low energy range are due to the scattering processes described above,
dominated by the scattering of electrons from the baffle system.

1.5.2 More Recent Measurements of the β-energy Spectra

20F

The most recent measurement of the β-energy spectrum in 20F decay was performed by D. W.
Hetherington et al.[12]. The extracted slope of the spectral function is in good agreement with

18

Figure 1-10: Kurie plot of the β-energy spectrum in 6He decay. Figure taken from Ref. [11].

the prediction by CVC. In this measurement, the kinetic energy of β particles was measured with
a high-purity germanium detector in a high-field superconducting solenoid. The layout of the
spectrometer is shown in the left panel of Fig. 1-11, where the decay source is placed outside the
active volume of the detector. Due to the backscattering and the bremsstrahlung production, the
corresponding response from the detector (dashed line) shown in the right panel of Fig. 1-11 is
greatly distorted relative to the theoretical spectrum (solid line).

66Ga and 14O

G. W. Severin et al.[13] have measured the β-energy spectrum in 66Ga decay. The measurement
was performed with a Wisconsin superconducting beta spectrometer [46], a Wu-type magnetic
spectrometer similar to that described in Ref. [47]. The β particles passing through the annular
slits are detected with a Si(Li) detector. The β-energy spectrum was obtained by recording the
number of the detected β particles at several spectrometer currents in sequence. The left panel of
Fig. 1-12 shows the energy spectrum deposited in the Si(Li) detector at a spectrometer current
setting corresponding to a positron kinetic energy of 3.25 MeV. The long tail below 1.5 MeV

19

Figure 1-11: Left panel: the layout of the β spectrometer used to measure the kinetic energy of β
particles. It consists of a high-purity germanium detector mounted on the axis in the bore of a
superconducting solenoid. Right panel: the theoretical statistical distribution (solid line) and the
corresponding response from the detector (dashed line). Figures taken from Ref. [12].

originates from the energy loss due to the backscattering of positrons and the bremsstrahlung
process. The resulting Kurie plot of the β-energy spectrum is shown in the right panel of Fig. 1-12.

Figure 1-12: Left panel: measured (black) and simulated (red) energy spectra deposited in the
Si(Li) detector with a spectrometer current corresponding to a positron kinetic energy of 3.25
MeV. Right panel: the β-energy spectrum in 66Ga decay. Figures taken from Ref. [13].

20

With the same apparatus, E. A. George et al.[44] have measured the β-momentum spectrum in
14O decay. The left panel of Fig. 1-13 shows the positron energy spectra deposited in the Si(Li)
detector with two spectrometer current settings. Similar to the left panel of Fig. 1-12, the tails
below the primary peak and above a channel number of 120 originates primarily from the energy
loss due to the backscattering of positrons out of the detector. The resulting β-momentum spectrum
is shown in the right panel of Fig. 1-13. In this experiment, the previously measured 66Ga spectra
[13] were used as model spectra in the analysis. It is shown in Refs. [44, 14] that the backscattering
events in the 66Ga spectra cannot be accurately described by Monte Carlo simulations. For all
spectrometer currents, the simulations overpredict the backscattering events [44]; in the current
range of 4-8 A, the simulations overpredict by 7% [14].

Figure 1-13: Left panel: Si(Li) spectra measured with the spectrometer current setting at 8.8 A
and 11.0 A. Right panel: the β-momentum spectrum in 14O decay including the measured
spectrum (open circles) and the theoretical spectrum (solid line). Figures taken from Ref. [14].

21

1.5.3 What is Next?

It is clear now that among the experiments measuring the shapes of β-energy spectra, the most
common and most severe instrumental effect is the backscattering of β particles from the active
volume of the detector. For the new generation of experiments measuring the β-energy spectra,
higher precisions are required to access any signatures of new physics. The new precision goals
require a new experimental technique capable of eliminating the effect of backscattering.

22

CHAPTER 2

THEORETICAL CORRECTIONS TO THE β-ENERGY SPECTRUM

In Chap. 1, the full theoretical description of the allowed β-energy spectrum in 6He decay was
introduced in Eq. 1.26, where the factor η(W), the product of several theoretical corrections, was not
discussed. The corrections include: the Fermi function F(Z, W), the screening correction S(Z, W),
the electrostatic finite size correction L0(Z, W), the convolution finite size correction C A(Z, W, W0),
the finite mass correction RN(W, W0), and the radiative correction δR(Z, W, W0) or δ(E). This
chapter discusses the origin of each theoretical correction and calculates each correction for 6He
decay. Since the full theoretical description of the β spectrum has an uncertainty, the uncertainties of
the observables associated with the theoretical description (theoretical uncertainties) are estimated.
Equation 1.26 is derived including the recoil to first order, assuming the absence of second class
currents, and neglecting the induced tensor form factor in 6He decay. To describe the transition to
first order in recoil, it requires only three form factors [48, 36, 37], i.e., the dominant Gamow-Teller
strength (c), the weak magnetism form factor (bWM), and the induced tensor form factor (d), where
d is the sum of a first class part (dI) and a second class part (dII). The first class part is neglected
here, but its impact will be evaluated in the complete analysis. The second class part is absent in
6He decay because a preliminary analysis in Ref. [48] found no evidence of a dII term in 6He eν
correlation measurements.

The theoretical evaluation is the first step towards an accurate description of the β spectrum,
and it lays the foundation for the study of a series of instrumental effects discussed in later chapters.
The study in this chapter applies for β− and Gamow-Teller decays. Throughout the chapter, the
variables are expressed in the natural unit (mc2), with  = m = c = 1.

2.1 Fermi Function

The Fermi function describes the electrostatic interaction between the emitted β particle and
the Coulomb field of the daughter nucleus. It is derived from the solution to the Dirac equation for

23

the electron wave function at a distance R from an assumed infinitely heavy, point-like daughter
nucleus. The Fermi function has the expression [49]

where Z is the atomic number of the daughter nucleus, γ =(cid:112)1 − (αZ)2, y = αZW/p, and Γ is

F(Z, W) = 2(γ + 1)(2pR)2(γ−1)eπy |Γ(γ + iy)|2
[Γ(2γ + 1)]2,

(2.1)

the complex gamma function. For a point charge, the solution to the Dirac equation diverges at the
origin. To circumvent the divergence, a cut-off radius R without physical significance is introduced
[50]. The radius of the daughter nucleus is taken as the value of R and is determined in Sec. 2.7.

2.2 Screening Correction

For a departing electron, the presence of the other electrons in the atom of the daughter bound
state produces a screening of the nuclear charge and effectively reduces the nuclear charge “seen”
by the departing electron. This change of the electromagnetic field distorts the β-energy spectrum
and can be taken into account by a screening correction. There are a few methods to calculate the
screening correction in literature. In this work, four different formulas were considered: Rose’s
formula, Bühring’s formula, simplified Bühring’s formula, and weak-screening approximation of
Bühring’s formula. These formulas require a common input:
the screening potential, which is
derived here.

One of the simplest models describing a screened Coulomb potential is the Hulthén potential

[51]:

Its behavior near the origin is [52]

V(r) = − αZ βs

eβsr − 1 .

V(r) = − αZ
r

1
2 αZ βs + O(r),

+

(2.2)

(2.3)

where the term 1
2 αZ βs is a shift of the potential near the nucleus due to the screening. This shift of
potential is sometimes also referred to as the screening potential, represented by VS, and is negative
for β− decays [51]:

VS = −1

2 αZ βs, βs = 2C(| ˜Z|)α| ˜Z|1/3

.

(2.4)

24

Here, ˜Z = 2 is the atomic number of the mother nucleus 6He, C(| ˜Z|) is a slowly increasing function
of ˜Z. The value of C(| ˜Z|) is determined following the approach in Ref. [51]. Equating the Hartree
potential in Ref. [53] with the three Yukawa potentials in Ref. [54] gives

C(| ˜Z|) = 1.13


aj bj,

j

(2.5)

which is derived in Sec. A.1 in the Appendix. aj and bj are coefficients corresponding to the
mother nucleus and their values for helium are adapted from Table I in Ref. [53]. This yields the
value of C(| ˜Z|) for helium:

(2.6)
The screening potential is then determined by substituting the values of C(2) and | ˜Z| into Eq. 2.4.

C(2) (cid:39) 1.386.

2.2.1 Rose’s Formula

The earliest work on the screening correction was reported by Rose [49]. The correction is
determined following a approximate WKB procedure [55] and has the form

(cid:19)2γ−1

(cid:18) ˜pR

p

eπ( ˜yR−y)(cid:12)(cid:12)(cid:12)(cid:12) Γ(γ + i ˜yR)

Γ(γ + iy)

(cid:12)(cid:12)(cid:12)(cid:12)2

,

S(Z, W) =

˜W
˜pR
pW

F(Z, ˜W)
F(Z, W) =

˜W
W

=

where

(cid:112) ˜W2 − 1,

˜W = W + VS,
˜pR =
˜yR =

αZ ˜W
˜pR

.

(2.7)

(2.8)

(2.9)

(2.10)

2.2.2 Bühring’s Formula

The accurate calculation of the screening correction requires solving the Dirac equation in the
superposition field of the nucleus and the extended electron shells or an acceptable surrogate for the
two together [56]. Alternatively, one may solve the Klein-Gordon equation in the Hulthén potential,
which is a useful surrogate for the superimposed fields [56]. Bühring has studied the Hulthén

25

potential and derived an approximate but close evaluation of the wave function normalization
factor of the Dirac equation for a semi-realistic case [56]. The resulting expression for the screening
correction is [3, 51]

βs

Γ(γ + iy)

(cid:19)2(1−γ)(cid:12)(cid:12)(cid:12)(cid:12) Γ(γ + i ˜y)
(cid:18)2p
(cid:19)(cid:18) βs
(cid:19)2
(cid:18) ˜W + γ
(cid:19)1/2(cid:21)−2
(cid:19)2(cid:20)
(cid:18) βs
(cid:3)1/2

˜W
W − 1
˜W

1
8

p

p

,

S(Z, W) =X

˜W
W

e−πy

1
4

×

(cid:19)2(cid:21)−1
(cid:18) βs
(cid:26)
(cid:18)
2(cid:20)
(cid:2)p2 − 2αZ ˜W βs

1 − αZ βs
W + 1

1 +

p

1 +

,

Γ(1 + 2ip/βs)

(cid:12)(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)(cid:12)2(cid:12)(cid:12)(cid:12)(cid:12) Γ(γ + 2i ˜p/βs)
(cid:18) βs

8(1 − γ) 1

1 − 1

p

γ

(cid:19)2(cid:21)(cid:27)

,

where

X =

˜p =

˜y =

(cid:20)

1 +

1
2 γ

+

1
2

1
2 p +
αZ ˜W
˜p .

(2.11)

(2.12)

(2.13)

(2.14)

(2.15)

Bühring’s formula is a significant improvement compared with Rose’s formula [51].
It can be
shown that, in a certain limit, the latter can be derived from the former. Note that ˜p and ˜pR differ
only in higher-order terms:

(cid:2)p2 − 2αZ ˜W βs

(cid:3)1/2

1
2 p +

1
2

˜p =
= ( ˜W2 − 1)1/2 + O(β
= ˜pR + O(β

s).
2

s)
2

As βS → 0, ˜p → ˜pR.

2.2.3 Simplified Bühring’s Formula

In Eq. 2.12 of Bühring’s formula, the factor X can be expressed as

X = 1 + O(β

s).
2

(2.16)

As βS → 0, X → 1. Therefore the Bühring’s formula with X = 1 can be considered a good
approximation to the Bühring’s formula with the X calculated with Eq. 2.12. This gives the

26

simplified Bühring’s formula:
˜W
W

S(Z, W) =

e−πy

(cid:18)2p

βs

(cid:19)2(1−γ)(cid:12)(cid:12)(cid:12)(cid:12) Γ(γ + i ˜y)

Γ(γ + iy)

(cid:12)(cid:12)(cid:12)(cid:12)2(cid:12)(cid:12)(cid:12)(cid:12) Γ(γ + 2i ˜p/βs)

Γ(1 + 2ip/βs)

(cid:12)(cid:12)(cid:12)(cid:12)2

.

(2.17)

2.2.4 Weak-Screening Approximation of Bühring’s Formula

In Eq. 2.11 of Bühring’s formula, if one performs an asymptotic expansion in power of βs and omits
the terms of order β

2
s , the weak-screening approximation of Bühring’s formula can be obtained:

.

(2.18)

(cid:18) ˜p

p

S(Z, W) =

˜W
W

(cid:19)2γ−1

eπ( ˜y−y)(cid:12)(cid:12)(cid:12)(cid:12) Γ(γ + i ˜y)

Γ(γ + iy)

(cid:12)(cid:12)(cid:12)(cid:12)2

This is essentially the Rose’s formula apart from the difference between ˜p and ˜pR, which is also
responsible to the difference between ˜y and ˜yR. Once again, if β → 0, ˜p → ˜pR (Eq. 2.15), then
Eq. 2.18 becomes Rose’s formula. Therefore this result is also referred to as Rose† in Ref. [56].
Table 2-1 shows the calculated screening corrections in 6He decay with the above four formulas.
The Bühring’s formula, the simplified Bühring’s formula, and the weak-screening approximation
of Bühring’s formula give the same results at the level of 10−5. The relative difference between
the correction with Rose’s formula and the rest is smaller than 2% for p < 0.2. For p > 0.2, the
four formulas give identical results at the level of 10−5. In the analysis in this work, the Bühring’s
formula is applied.
Table 2-1: The screening corrections for 6He decay for p ≤ 0.3. The values shown here are
[S(Z, W) − 1].

Weak-screening
Approximation

to Bühring

-0.01372
-0.00529
-0.00285

Rose
-0.01352
-0.00528
-0.00285

p
0.1
0.2
0.3

Bühring
-0.01372
-0.00529
-0.00285

Simplified
Bühring
-0.01372
-0.00529
-0.00285

27

2.3 The Finite Size Corrections

2.3.1 Electrostatic Finite Size Correction

As mentioned in Sec. 2.1, the Fermi function assumes an infinitely massive, point-like daughter
nucleus. In reality, the daughter nucleus has a finite charge distribution, and its deviation from the
point charge distribution affects the solutions to the Dirac equation. Solving the Dirac equation in
the field of the actual charge distribution of the daughter nucleus introduces an electrostatic finite
size correction L0(Z, W) to the Fermi function.

The values of L0(Z, W) for Z < 30 were tabulated by H. Behrens and J. Jänecke [15] as a result
of numerical computations assuming the daughter nucleus is an uniformly charged sphere of radius
R. Based on these values, Wilkinson [57] performed an empirical fit and obtained an analytic
expression of L0(Z, W) that applies to Z < 40:

L0(Z, W) = 1 + 0.206(αZ)2 − 1.039W RαZ − 0.0058(pR)2 − 0.54 RαZ
W .

(2.19)

With the same tabulation as Ref. [15], Wilkinson improved the fit and extended the domain to
Z ≤ 60 [3]. The updated expression of L0(Z, W) for electron is
41 − 26γ
13
15(2γ − 1) − αZ Rγ
60(αZ)2 − W RαZ
an(W R)n + 0.41(R − 0.0164)(αZ)4.5

30W(2γ − 1) + a−1

L0(Z, W) = 1 +

17 − 2γ

(2.20)

R
W

.

5


n=0

+

6


x=1

In the above expression, the a-values are given by the parameterization

a =

bx(αZ)x,

(2.21)

where the bx-values are given in Table A-1 in Sec. A.2 of the Appendix and are different for each
an.

Figure 2-1 shows the calculated L0(Z, W) applying the analytic expressions described above.
The dashed blue line is obtained with Eq. 2.19, which has a small deviation from the numerical
values of Ref. [15] (solid red line). The solid blue line is obtained with the improved analytic
expression, Eq. 2.20, resulting in a better agreement with the values in Ref. [15].

28

In the above calculations with Eqs. 2.19 and 2.20, the same nuclear radius was applied as the
radius used by H. Behrens and J. Jänecke in Ref. [15]: R1 = 0.42587A1/3. These calculations only
serve the purpose of comparing the results with the values in Ref. [15]. It is shown in Sec. 2.7
that there are a few options for the nuclear radius in literature. In the actual analysis in this work,
a different radius, R3, described in Sec. 2.7, is used. With R3, the calculated L0(Z, W) values are
shown in Fig. 2-1, where the dashed black line is based on Eq. 2.19 and the solid black line is
based on Eq. 2.20. The calculations with R3 give lower values of L0(Z, W) than those with R1.

Figure 2-1: Electrostatic finite size corrections. The dashed lines and the solid line are calculated
with Eq. 2.19 and Eq. 2.20, respectively. The colors indicate the radius used in the calculations:
blue corresponds to R1 and black corresponds to R3. The results are compared with the numerical
values (solid red) from Ref. [15].

2.3.2 Convolution Finite Size Correction

Another effect related to the finite nuclear size is the convolution of the leptonic and nucleonic
wave functions through the nuclear volume. For Gamow-Teller decays it has the form [3]

C A(Z, W, W0) = 1 + C A0 + C A1 W + C A2 W2

,

(2.22)

29

where

630(αZ)2 − (W0R)2
C A0 = −233
5
C A1 = −21
4
9W0R2
35 RαZ +
C A2 = −4
9 R2

.

,

2
35W0RαZ,

+

(2.23)

(2.24)

(2.25)

Here the superscript A indicates the axial vector nature of the Gamow-Teller transition.

2.4 Finite Mass Correction

In the Fermi function, two assumptions about the mass are made: the neutrino is considered
massless and the daughter nucleus is considered infinitely massive. The effect of the non-zero
neutrino mass on the β spectrum can be safely neglected with a neutrino mass below 30 eV [3]. The
effect related to the finite nuclear mass, however, requires a recoil correction. The correction for
Gamow-Teller decays can be included by multiplying the phase space factor by a factor RN(W, W0)
provided in Ref. [3]:

RN(W, W0) =1 + r A0 +

r A1
W

+ r A2 W + r A3 W2
,

where

0

+

− W2
r A0 = − 2W0
6M2 − 77
18M2,
3M
r A1 = − 2
7W0
9M2,
3M
− 28W0
10
r A2 =
9M2 ,
3M
88
9M2 .

r A3 =

(2.26)

(2.27)

(2.28)

(2.29)

(2.30)

Here, M is the nuclear mass of 6Li in the natural unit [40]. Once again, the superscript A corresponds
to the axial vector nature of the Gamow-Teller transition.

2.5 Radiative Correction

During β decay, the virtual and real photons can be emitted and change the β-decay kinematics.
The exchange of the virtual photons and the emission of the real photons can be taken into account

30

by multiplying the phase space factor by a radiative correction, R(Z, W, W0). It can be split into
two parts [58]: the “inner radiative correction” which depends on the special form of the weak and
strong interactions theory, and the “outer radiative correction” which is independent of the details
of strong and weak interactions. The complete radiative correction has the form [16]

(cid:16)1 + ∆

V/A
R

(cid:17) [1 + δR(Z, W, W0)],

where (cid:16)1 + ∆

V/A
R

respectively.

R(Z, W, W0) =

(cid:17) and [1 + δR(Z, W, W0)] represent the inner and outer radiative corrections,

(2.31)

The inner radiative correction is energy independent and nucleus independent. It affects the
integrated form of the β-energy spectrum and is crucial for the analysis of the F t values. Because of
the energy independence, the inner radiative correction manifests itself as a normalization constant
in the differential form of the β-energy spectrum. Therefore it is irrelevant for the scope of this
work.

The outer radiative correction depends on the energy of the outgoing β particles and distorts the
shape of the β-energy spectrum. It includes processes involving virtual and real photons. Details
about these processes are described below. Note that the real photon emitted during a decay is also
referred to as the inner bremsstrahlung radiation, which is to be distinguished from both the inner
radiative correction and the outer bremsstrahlung photon emitted as the β particle slows down in
the detector.

This section focuses on the outer radiative correction.

In the rest of the section, the term
“radiative correction” refers to only the outer radiative correction. The radiative correction are
discussed in two categories of experimental conditions: 1) the electrons and photons are completely
distinguishable so that the energy measured in the detector is the energy of β particles only; 2)
electrons and photons are completely indistinguishable so that the measured energy is the sum of
electron and photon energy.

31

2.5.1 Distinguishable e− and γ

The radiative correction δR includes the corrections of different orders of the fine-structure constant
α. The first three orders are the most frequently discussed in literature:

δR(Z, W, W0) = δ1(W, W0) + δ2(Z, W) + δ3(Z, W),

where δ1(W, W0), δ2(Z, W), and δ3(Z, W) are the radiative corrections of the orders α, Zα
Z2

3, respectively.
α
The order α radiative correction is on the order of a few percent; it is one of the two dominant
3 corrections have
corrections (the other one is the Fermi function). The order Zα
a remarkable influence on the F t value and are of small but non-negligible importance for the
differential form of the spectrum.

2 and Z2

α

(2.32)

2, and

Order α

The first order of the radiative correction is associated with the one-photon exchange processes,
shown in Fig. 2-2. The pioneering work on the order α correction was published by Sirlin [59]
in 1967 and it is still being applied in the precision electroweak studies as the default radiative
correction:

δ1(W, W0) =

gα(W, W0),

(cid:18) 2βe

(cid:19)

(cid:34)

L

4
βe
2(1 + β

1 + βe
e) +
2

where
gα(W, W0) =3 ln mp − 3
4 +
tanh−1
βe
ln(1 − t)

Þ x

βe

+

L(x) =

0

t

dt.

α
2π

(cid:32)tanh−1

+ 4

(W0 − W)2

6W2

βe

− 1

βe
− 4 tanh−1

(cid:33)(cid:20)W0 − W
(cid:35)

3W

βe

,

(2.33)

(cid:21)

− 3
2 + ln[2(W0 − W)]

(2.34)

(2.35)

Here, mp is the proton mass, βe = p/W, and L(x) is the Spence function or Dilogarithm. The
ln[2(W0 − W)] term leads to a divergence of Eq. 2.34 as W approaches the endpoint energy W0.
This behavior can be eliminated [60] by replacing t(βe) ln(W0 − W) by (W0 − W)t(βe) − 1, where

32

t(βe) = 2α/π(cid:2)(tanh−1

βe)/βe − 1(cid:3) . Note that t(βe) ln(W0 − W) is just the leading term of the

expansion of (W0 − W)t(βe) − 1.

Figure 2-2: Feynman diagrams for the one-photon exchange processes associated with the order α
outer radiative correction. The top row shows the processes involving virtual photon exchange, and
the bottom row shows the processes involving real photon emission. Figure taken from Ref. [16].

Order Zα

2

The order Zα

2 of the radiative correction has the form [61]

δ2(Z, W) = αZ2

It is convenient to split ∆1 into two parts:

4


i=1

∆i(W).

1(W) + ∆F1 ,
0
∆1 = ∆

33

(2.36)

(2.37)

0
where ∆
1 corresponds to a point nucleus and is energy dependent, and ∆F1 is energy indepen-
dent but depends on the charge distribution of the daughter nucleus. In the extreme relativistic
1(W) + ∆4(W) can be expressed analyticly:
0
approximation, ∆

0

1(W) + ∆4(W) = ln mp − 5
3

∆

ln(2W) +

43
18 .

(2.38)

The terms ∆F1 , ∆2 and ∆3 are finite nuclear size effects and depend on the charge distribution of the
daughter nucleus. The analytic forms of the three terms are available in literature for three types of
charge distributions: uniform distribution [61], modified Gaussian distribution [61], and Yukawa
distribution [62]. For the first two charge distributions, the expressions of ∆F1 , ∆2 and ∆3 are shown
in Sec. A.3 of the Appendix. For the Yukawa charge distribution,

(cid:18)

(cid:19)

π
4

(cid:18)

2
Λ
mp
π
1 +

1 +

Λ
mp

,

∆F1 =

ln Λ
mp
π
ln Λ
Λ
Λ
M
mp
mp
−1 + π
gA(1 + κV) Λ
mp

(cid:21)
where 1 + κV = 4.7059, Λ is related to the radius of the daughter nucleus: Λ ≡ √

ln Λ
mp

π
8

Λ
mp

− 2

1 +

1
π
1
π

Λ
mp

∆3 =

∆2 =

(cid:18)

(cid:19)

.

− 2

(cid:19)

,

(cid:20)

10/R.

(2.39)

(2.40)

(2.41)

(2.42)

Order Z2

α

3

The order Z2

α

(cid:21)

3 radiative correction has the expression [61, 63, 64]:

3(cid:20)

(cid:18) Γ

(cid:19)

mp

δ3 = Z2

α

a ln

+ b f(W) + dg(W) + h ln(2W0)

34

where,

(cid:32)

(cid:33)(cid:30)

π

2

(cid:32)11
3 − 3

π,

2
2
4 − γE − π
6

(cid:33)

,

a =

b =

4
3π
4
3π

,

d =
h = −0.649,

(2.43)

(2.44)

(2.45)

(2.46)

(2.47)

(2.48)

5
3

ln(2RW).

f(W) = ln(2W) − 5
6,
g(W) =

(cid:2) ln2 R − ln2(2W)(cid:3) +

1
2

Here, γE is the Euler–Mascheroni constant.

The above radiative correction, δR(Z, W, W0), is based on the assumption that the β spectrometer
can separate photons from the β particles so that no energy of photons is detected. The validity of
this assumption and its impact on the radiative correction need to be discussed for the experimental
setup in this work shown in Chap. 3.

2.5.2

Indistinguishable e− and γ

As the radiative correction involves the emission of real photons (i.e., the inner bremsstrahlung
radiation), it is possible for the real photons to be detected along with the β particles in a measure-
ment. The direct consequence will be that the measured energy spectrum would no longer be a
clean β-energy spectrum, but rather a (β + γ)-energy spectrum. Whether the real photons can be
detected and if so, the fraction of the detected photon energy, are decided by the experimental tech-
nique. Therefore the radiative correction for indistinguishable e− and γ requires to be approached
differently from the previous theoretical corrections, which are independent of the experimental
setup.

The inner bremsstrahlung radiation is usually treated as a two-step process. First the β particle
e(cid:48) of energy W(cid:48) is emitted in a virtual intermediate state, then it goes to its final state e− of energy

35

W accompanied by a photon of energy k:

n → p + ¯νe + e(cid:48) → p + ¯νe + e− + γ.

(2.49)

For allowed β transitions, the theory of inner bremsstrahlung was suggested by Knipp and
Uhlenbeck [65] and independently by Bloch [66], referred to as the KUB theory. According to the
KUB theory, the probability for a β particle of initial energy W(cid:48) to release a photon of energy k is
given by

Φ(W(cid:48)
, k) =
√
where W(cid:48) = W + k and p(cid:48) =
W(cid:48)2 − 1. The inner bremsstrahlung spectrum is obtained by
integrating the product of Φ(W(cid:48), k) and the phase space factor over the energy range of k + 1 to W0:

ln(W + p) − 2

αp
πkp(cid:48)

W(cid:48)p

(2.50)

(cid:35)

,

(cid:34)

W2 + W(cid:48)2

Þ W0

k+1

S(k) =

pW(cid:48)(W(cid:48) − W0)2dW(cid:48)Φ(W(cid:48)

, k).

(2.51)

With Eqs. (2.50) and (2.51), the inner bremsstrahlung spectrum emitted in 6He β decay is calculated
and shown in Fig. 2-3. The majority of the photons are in the low energy part of the spectrum.
These photons can in principle be detected in a calorimetric detector used in this work (see Chap.
3). For an efficient calorimeter, it is reasonable to assume that all bremsstrahlung photons fully
deposit their energy in the detector. Therefore for this work, the radiative correction should take
into account the indistinguishability of β particles and photons and the resulting energy summing
effect.

For a decay event taking place in a calorimeter, the total energy recorded by the detector is
represented by E, and E = k + W if an inner bremsstrahlung photon is emitted. The maximum
photon energy kmax is then equal to kmax = E − 1, where this 1 is the electron rest mass. To first
order, the radiative correction can be written as

α
π

g(E),

δ(E) = 1 +
g(E) = gv(E) + gb(E),

(2.52)

(2.53)

where gv(E) and gb(E) stand for the corrections associated with the virtual and real photon
processes, respectively.

36

Figure 2-3: The energy spectrum of the inner bremsstrahlung photons emitted in 6He β decay.
The vertical axis has an arbitrary unit.

In the low photon-energy regime, i.e., k (cid:28) mp, the correction associated with the virtual photon

is [67]

(cid:18) µ
(cid:19)
(cid:18)1 + β

mp

1 − β

(cid:19)

(cid:18) µ

(cid:19)
+ A − 3
8,
MA
ln mp − 1
3
2

(cid:20)1

2

β

ln

(cid:21)

(cid:19)

A =

3
2

ln

+ 3Q ln

(cid:20) 1

1
2 β ln

gv(
) =

(cid:18)1 + β

(cid:18) 2β
where β =(cid:112)E2 − 1(cid:14)E, λ is an artificial “photon mass”. It will be shown that ln λ is canceled out

(cid:18)1 + β

− 1 + 2 ln λ

1 + β
(2.55)

(cid:19)(cid:21)2

1 − β

1 − β

− 1

+

ln

2β

+

1
β

L

(2.54)

(cid:19)

,

when calculating the sum of gv(E) and gb(E). Q is the average charge of the quarks involved in the
transition:

(2.56)
MA = 1260 MeV [68], and µ ∼ 1 GeV is the energy scale under which the EFT is valid. As
ln(µ/mp) ∼ 0 and Q ln(µ/MA) ∼ 0, gv(E) is calculated to be

Q =

1
2(Qu + Qd) =

1
6 .

gv(E) = A − 3
8

,

(2.57)

which is consistent with the expression of gv(E) in Ref. [69].

The correction associated with the real photons, gb(E), has been derived by Gardner et al. [69],
who introduced a quantity ∆E that is referred to as “energy resolution”, such that for k (cid:54) ∆E,

37

01234310·Photon energy k (keV)4-102-101210410gb(E) =3 − 2
− 1

β
− 2

+ 2 ln

where

(cid:33)

λ

+

+

β

+

1 + β

β

1
β

I1
k

β

1
β

L

I0
k

+

1
2E2

tanh−1
β

(cid:33)
(cid:32)
(cid:19)
(cid:18) 2β
(cid:19)
(cid:18)2kmax
1 − tanh−1
ln
β
(cid:32)E − kmax −(cid:112)(E − kmax)2 − 1
(cid:112)(E − kmax)2 − 1
k − 1
(tanh−1
I−1
β)2 +
E β
(cid:112)E2 − 1
E −(cid:112)E2 − 1
(cid:35)
(cid:34)E2 − 1 − Ekmax +(cid:112)E2 − 1 ·(cid:112)(E − kmax)2 − 1
2
ln
β
(cid:18)1 + βe
Þ kmax
(cid:18)1 + βe
Þ kmax
(cid:112)(E − k)2 − 1
(cid:19)

· 1 − β
(cid:19)
1 + β

2(E2 − 1)

, n = 0, 1,

1 − βe

1 − βe

dkkn ln

E − k

0
p
W

I−1
k

=

dk

ln

βe =

=

(cid:19)

,

In
k

=

1
k

β − 1

β +

− 2 ln[2(E + 1)] +
3
tanh−1
β

β +

1
β

3
2

(cid:19)
8 − 2
ln mp +
E2 − 2I0
E

I1

β

,

2I−1 +

(cid:18)

0

.

dE

(E − E) ln E + p
E(1 + β),
dE(E − E)n ln(E + p), n = 1, 0.

(2.58)

(2.59)

(2.60)

(2.61)

β)2

(2.62)

(2.63)

(2.64)

E = W + k is recorded. Under the previous assumption that all the bremsstrahlung photons deposit
all the energy in the detector, ∆E = kmax = E − 1, the real photon radiative correction gb(E) can
be expressed as [69]

Summing up gv(E) and gb(E) above gives the total radiative correction g(E):
13

g(E) =2 ln[2(E − 1)]

(tanh−1

+

2
β

L

where

(cid:18) 1
(cid:19)

(cid:18) 2β

1 + β

β

tanh−1
+ β tanh−1

Þ E
Þ E

1

1

I−1 =

In =

Equation 2.62 is equivalent to the Eq. (19) obtained by Vogel in Ref. [70]. The derivation of Eq.
2.62 from Eq. 2.58 is shown in Sec. A.4 of the Appendix.

38

The closed forms of the integrals I0, I1, I−1 are given in Ref. [71]:

I0
E =
I1
E2 =

I−1 =

where

ln

(cid:33)

(cid:32)3

1
1 + β
1 − β
2
2
8 − β
8
1 + β
1
ln
1 − β
2

− β,
1 + β
1 − β
1 +
2

(cid:18)

(cid:20)

ln

ln

− 3
4 β,

1(cid:112)1 − β

2

(cid:18) 2β

1 + β

(cid:19)

,

− 1
4

ln2 1 + β
1 − β

− Li2

x4
k2

(2.65)

(2.66)

(2.67)

(2.68)

(cid:19)(cid:21)
∞


k=1

Li2(x) =

is the polylogarithm of the second order. Li2(x) can be substituted by the Spence function −L(x)
as Li2(x) is the approximate form of −L(x) when |x| (cid:54) 1. In addition, one can substitute Eqs.
2.65-2.67 into Eq. 2.62. This results in

(cid:26) 2

β

(cid:21)

(cid:20) 2β

1 + β

ln

g(E) = ln

1 + β
1 − β

+

3
8 β +

7
8β

− 2 ln

(cid:27)

(cid:19)

(cid:18) 4β

2
1 − β

2

(cid:19)

(cid:18) 2β

1 + β

+

L

4
β

23
8 +

3
2

+

ln mp,

(2.69)

which is equivalent to Eq. (25) in Ref. [71]. The derivation of Eq. 2.69 from Eq. 2.62 is shown in
Sec. A.5 of the Appendix.

Equation (2.69) is the final expression of the radiative correction for measurements in which
the β particles and photons are completely indistinguishable. It is equivalent to Eq. 2.62 which is
identical to Eq. (19) in Ref. [70] as well as Eq. 2.58 which is derived from Eq. (9 - 13) in Ref. [69]
with ∆E = kmax = E − 1. With Eq. 2.69, the radiative correction for indistinguishable β particles
and photons in 6He decay is calculated and shown in Fig. 2-4. The size of the correction is on the
order of 10−2.

For comparison, Fig. 2-4 also includes the radiative correction for distinguishable β particles
and photons calculated with Eq. 2.34, which is of the same order as the indistinguishable correction
for the most relevant part of the spectrum. However, the slopes of the two corrections differ by
a factor of 10. This difference is because in the distinguishable case, the real photons emitted
during the decays reduce the measured energy of β particles in the detector, distorting the β-energy

39

spectrum by introducing a negative slope; while in the indistinguishable case, the emitted inner
bremsstrahlung photons are collected back in the detector volume, which essentially cancels the
effect of the possible energy loss due to the real photon emission. The remaining slope is mostly
due to the change of β decay kinematics by the virtual photon exchange.

Figure 2-4: Radiative corrections for 6He decay when the β particles and photons are
distinguishable (blue) calculated with Eq. 2.34 and distinguishable (red) calculated with Eq. 2.69.

The calculation of the radiative correction for indistinguishable β particles and photons with
Eq. 2.69 is based on the assumption that all the inner bremsstrahlung photons would fully stop in
the detector. In reality, it is possible for the photons populating the high energy part of Fig. 2-3 to
escape from the detector. Therefore the exact radiative correction for this work is Eq. 2.69 with an
additional correction taking into account the escape of the inner bremsstrahlung photons from the
detector. As the probability for a photon to escape depends on the photon energy and the geometry
and material of the detector, the size of the correction needs to be determined with a detector
simulation, which is part of the undergoing effort. The exact radiative correction is expected to be

40

0123Electron kinetic energy (MeV)6-4-2-02Correction - 1 (%)DistinguishableIndistinguishablebetween the two extreme cases represented by the two corrections in Fig. 2-4.

2.6 Summary of the Corrections to the β-energy Spectrum in 6He Decay

Equipped with the studies in the previous sections, the size of each correction is calculated for
6He decay. The results are shown in Fig. 2-5. The dominating correction is the Fermi function
and the sizes of the rest of the corrections are all below 2%. The complete theoretical β-energy
spectrum described by Eq. 1.26 is now determined.

Figure 2-5: Calculated theoretical corrections to the β-energy spectrum in 6He decay. The dashed
line shows the expected spectral function.

2.7 Theoretical Uncertainty

The theoretical β-energy spectrum described by Eq. 1.26 depends on a few inputs determined
from experiments. The experimental uncertainties of these inputs introduce an uncertainty to the
theoretical spectrum. The dominant contributions to the theoretical β-energy spectrum are from the

41

0.00.51.01.52.02.53.03.5Electron kinetic energy (MeV)4-2-02468101214Correction - 1 (%)Fermi functionScreeningElectrostatic finite sizeConvolution finite sizeFinite massRadiative - indistinguishableSpectral function(cid:17)

∆

(cid:16)1 + 1
2M∆
1 + ∆2M
1
2(M(cid:48) + M),

M =
∆ = M(cid:48) − M,

W0 =

,

(2.70)

(2.71)

6Li radius R and the endpoint energy W0. Specifically, R enters the Fermi function, the screening
2
correction, the electrostatic finite size correction, the convolution finite size correction, and the Zα
and Z3
3 order radiative corrections; W0 enters the α order radiative correction, the finite mass
correction, the convolution finite size correction, and most importantly, the phase space factor.

α

2.7.1 Calculations of the Endpoint Energy and the Nuclear Radius

The calculation of the endpoint energy is performed following the procedure described in Ref. [72],
which includes the recoil effect of the daughter nucleus:

(2.72)
where M(cid:48) is the nuclear mass of the parent nucleus. The atomic masses used to calculate M(cid:48) and
M are taken from Ref. [73] for 6He and Ref. [40] for 6Li. The calculated W0 for 6He decay has
the value of

W0 = 7.8568 ± 0.0001

(2.73)

in the natural unit, where the uncertainty is from the uncertainties of the atomic masses of 6He and
6Li.

Among the different expressions of the nuclear radius available in literature, the most commonly

used is

(2.74)
where r0 = 0.42587 is a constant fitted from electron scattering experimental data [15]. For 6Li,
R1 has the value of

,

R1 = r0A1/3

R1 = 5.7 × 10−3

in natural unit. Another widely used description of the radius is the Elton formula [74]:

R2 = 0.0029A1/3 + 0.0063A−1/3 − 0.017A−1

.

(2.75)

(2.76)

42

For 6Li, R2 has the value of

R2 = 5.9 × 10−3

(2.77)

in natural unit. Note that both Eq. 2.74 and Eq. 2.76 give general descriptions of the nuclear radius
as a function of the mass number A, covering a wide range of isotopes. The calculated radius with
Eq. 2.74 and Eq. 2.76 increases as a function of A. However, G. Ewald et al. [75] have measured
the nuclear charge radius of Li isotopes and found that the rms charge radius of 6,7,8,9Li decreases
as the neutron number increases. Such details are not taken into account by Eq. 2.74 or Eq. 2.76.
In this work, the description of the nuclear radius as a function of A is not relevant, rather, this
work only needs one accurate radius of 6Li. The measured rms nuclear charge radius of 6Li is [76]

The corresponding radius of the uniformly charged sphere converted to natural unit is [3]

(2.78)

(2.79)

(2.80)

(cid:10)r2(cid:11)1/2
(cid:114)5

R3 =

3

= 2.589 ± 0.039 fm.

(cid:10)r2(cid:11)1/2 × 2.5896 × 10−3

.

In the following text, R is used to represent R3 and it has the value of

in natural unit, where the uncertainty is from(cid:10)r2(cid:11)1/2.

R = (8.66 ± 0.13) × 10−3

2.7.2 Estimation of the Theoretical Uncertainty

The theoretical uncertainties, i.e., the impacts of the uncertainties of the theoretical β-energy
spectrum on the observables, are estimated here. This was done by estimating the sensitivities of
bWM and bGT to the uncertainties of the 6Li radius and the endpoint energy. For illustration, the
sensitivity of bWM to the radius is determined with the following procedure:

1. Calculate the theoretical β-energy spectrum with the radius and endpoint energy equal to the
central values of R in Eq. 2.80 and W0 in Eq. 2.73; and with bWM = 68.22, bGT = 0. Two
parameters associated with the calibration of the detector are introduced in the spectrum: an
offset and a gain. Details of the offset and the gain are discussed in Chapter 6.

43

2. Construct a fit function with the radius equal to R + δR and the endpoint energy equal to W0,
and with bWM and bGT being parameters in the fit function. The offset is fixed at the initial
value and the gain is a free parameter.

3. The spectrum calculated in Step 1 is fitted by the function constructed in Step 2 using the χ
method with the “SM” fit, i.e., bGT was fixed to 0 and bWM was left as a free parameter along
with the overall normalization and the gain. The offset is fixed at the initial value. The fitted
bWM is labeled as bWM(R + δR, W0).

2

4. Set the radius in Step 2 at R − δR and repeat Step 3. The fitted bWM is labeled as bWM(R −

δR, W0).

5. The uncertainty of bWM arising from δR can be estimated by

δbWM =

|bWM(R + δR, W0) − bWM(R − δR, W0)|

2

.

(2.81)

Similar to the sensitivity of bWM to the radius, the sensitivity of bWM to the endpoint energy
is determined with the endpoint energy in Step 2 and Step 4 fixed at W0 + δW0 and W0 − δW0,
respectively, and with the radius fixed at the central value. In addition, similar to the sensitivities of
bWM, the sensitivities of bGT can be determined by performing "BSM" fits, i.e., fix bWM to 68.22
and leave bGT as a free parameter along with the normalization and the gain.

The results from the above procedure are summarized in the first two rows of Table 2-2, where
δW0 and δR have comparable impacts on bWM, whereas for bGT, δW0 has a larger impact than δR.
The total uncertainties of bWM and bGT are shown in the third row of the table. The theoretical
uncertainty of bWM is a factor of 10 smaller than the uncertainty of the CVC value of bWM (0.79,
see Eq. 1.32). The theoretical uncertainty of bGT is a factor of 100 smaller than the precision goal
(O(10−3)).

In the actual analysis, the theoretical β-energy spectrum is convolved with the detector response,
including the detector simulations with Geant4 and the subsequent convolution with the energy
resolution, and then fitted to the experimental data. The actual theoretical uncertainties will be

44

Table 2-2: Sensitivities of bWM and bGT to the uncertainties of the 6Li radius and the endpoint
energy.

δbWM (10−2)

5.9
-5.7
8.2

δbGT (10−5)

-1.8
2.4
3.0

δR
δW0
total

determined by the full analysis. Therefore the values estimated in Table 2-2 are the upper limits of
the theoretical uncertainties of bWM and bGT.

45

CHAPTER 3

DESCRIPTION OF THE EXPERIMENT

Precision measurement of the β-energy spectrum requires the minimum distortion to the measured
spectrum from the true spectrum. The measured spectrum could be distorted by any instrumental
effect that prevents the β particles from depositing all the energy in the detector. Several instrumental
challenges are discussed at the beginning of this chapter, followed by a new instrumental technique
proposed to avoiding these instrumental effects. An experiment performed using this new technique
is described. Sample data from a typical experimental run are provided.

3.1 General Instrumental Challenges

Two of the common instruments to study the β-energy spectrum are a magnetic spectrometer
[77] and a detecting system with the β emitter located outside the active volume of the detector
[12]. In such measurements, some instrumental effects have great impacts on the detector response
and distort the measured β-energy spectrum [12]. These effects are described below.

• The backscattering occurs when a β particle enters the surface of a material and undergoes
sufficient deflections so that it reemerges from the surface through which it entered [78]. It
is most pronounced for electrons with low incident energy and absorbers with high atomic
number. The backscattered electrons do not deposit all the energy in the medium and
therefore can have a significant effect on the detector response, resulting in a big distortion
to the measured spectrum [12]. Such a distortion makes the backscattering one of the most
severe instrumental effects in the detection of β particles.

• The outscattering occurs when an electron leaves the edge of the detector after multiple
Coulomb scatterings. The electron carries away energy when it leaves the detector volume,
resulting in partial energy deposition.

• The dead layer is a surface layer on the active volume that the incident electrons must pass

46

before reaching the active volume. This is a disadvantage because a portion of the electron
energy is lost before reaching the active volume.

3.2 Calorimetric Technique

The instrumental effects discussed above can be eliminated by confining the β decay source
inside the active volume of the detector. The 6He decays inside of the detector, emitting β particles.
The detector encloses the whole 4π solid angle of the decay products and absorbs the energy of β
particles. The 6He decay source can be sent into the detector by implanting a 6He ion beam into
the detector. If the beam is implanted deep enough so that the decay source is fully enclosed in the
detector volume, no β particles would be able to escape.

3.3 Beam Conditions

3.3.1 Beam Facility

The 6He beam was produced at the Coupled Cyclotron Facility shown in Fig. 3-1 located at
National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University (MSU). A
beam of 18O+ stable ions from the electron cyclotron resonance ion source was injected into the
K500 cyclotron and accelerated to an energy sufficient to strip most of the remaining electrons
in a foil positioned near the center of the K1200 cyclotron [79]. The highly stripped ions were
then accelerated to an energy of 120 MeV per nucleon. The primary beam impinged on a 2.54
g/cm2 thick Be target, where 6He was produced from fragmentation reactions. The fragments were
separated by the A1900 projectile fragment separator [80][79], extracting a 6He secondary beam at
the energy of 76 MeV per nucleon, which was directed to the experimental area. In the experimental
area, the 6He ions exited from the beam pipe through a 75 µm thick Zr vacuum window and were
implanted into a CsI(Na) or a NaI(Tl) detector. Details of the detectors are discussed in Sec. 3.4.
It is important to control the implantation of the 6He beam into the detector such that the
detector volume fully encloses the range of the β particles from the decays (6 mm in CsI). We have
placed an Al degrader 1 m upstream from the detector, as shown in Fig. 3-2. The degrader can be

47

Figure 3-1: A schematic layout of the beam facility at National Superconducting Cyclotron
Laboratory.

rotated along the vertical axis. The angle of the rotation determines the effective thickness passed
through by the beam to control the beam energy. During regular measurements, the energy of the
implanted beam was 46 MeV per nucleon corresponding to a degrader angle of 0°, i.e., when the
effective thickness of the degrader was the lowest and the implantation depth was the highest.

Figure 3-2: A cartoon of the top view of the setup. The secondary 6He beam (red) of 76 MeV per
nucleon passes through a degrader (gray). The beam was decelerated by the degrader down to 46
MeV per nucleon and was implanted into a CsI(Na) or NaI(Tl) detector (yellow).

3.3.2 The Beam Geometry and the Calculated Range

The transverse profile of the implanted beam in the detector was measured from an image taken by
a digital night vision camera which has high photosensitivity in low-light environments. The beam

48

was cut by the A1900 focal plane slits that were set to 2×1.5 mm2. The beam is magnified during
the transport and produced a rectangular image of 4.5×4.6 mm2 recorded by the camera.

The longitudinal profile of the implanted beam in the detector was determined with beam
transport calculations using the LISE++ code [81]. Figure 3-3 shows the range distributions of the
implanted beam in the CsI and the NaI detectors while the angle of the degrader with respect to
the beamline is 0°. The range, defined as the distance from the surface where the beam enters, has
a mean value of 1.23 cm for CsI and 1.48 cm for NaI. Figure 3-4 shows the range distributions in
CsI while the angle of the degrader with respect to the beamline is at 0°and 42°. The smaller angle
of the degrader gives a lower effective thickness along the beamline, thus a higher energy of the
implantation beam, resulting in a larger range. The widths of the distributions resulting from the
range straggling are discussed in the next subsection. For any case shown in Figs. 3-3 and 3-4, the
deposited 6He ions are fully enclosed by the detector.

Figure 3-3: Range distributions of the implanted 6He in CsI (left) and NaI (right) calculated with
the LISE++ code. The angle of the degrader with respect to the beamline is 0°.

3.3.3 Measurement of the Beam Range

A measurement of the beam range was performed during the experiment. The range distribution of
the 6He ions can be determined by passing the beam through different thicknesses of the degrader
and measuring the ratio of the transmitted to the incident ions. We measured the counts in the

49

11.011.512.012.513.013.514.0He (mm)6Range of 0.00.10.20.30.40.50.60.70.80.9a.u.14.014.515.015.516.0He (mm)6Range of 0.000.050.100.150.200.250.300.350.400.45a.u.Figure 3-4: Range distributions of the implanted 6He in CsI while the angle of the degrader with
respect to the beamline is at 0°(blue) and 42°(red).

CsI(Na) detector while the degrader was at fourteen different angles with respect to the beamline
and divided each count by the count measured without the degrader. This ratio, or transmission,
is shown as the range number-distance curve [82] in Fig. 3-5. From 0 mm to 30 mm, the ratio
decreases slowly because of the increasing nuclear reactions between the beam and the degrader.
In the thickness range of 30-35 mm, the ratio suddenly drops to zero over a spread of thickness
rather than a single value of thickness because when the implantation beam enters the detector, the
energy loss is not continuous but statistical in nature. The absolute value of the derivative of the
ratio, known as the range straggling curve, is shown in Fig. 3-5, where the range of the 6He ions is
then a statistical distribution centered about a mean value rather than being a constant [82].

The measured range straggling curve can be compared with the beam transport calculations by
LISE++. The results are shown in Fig. 3-6, where the red curve is from measurements and the
blue curve is from LISE ++ calculations. The spread of the range distribution is higher from the
calculations than from the experiments. Similar overestimates have been seen in Ref. [83], where
the range straggling has been determined from experiments and TRIM95 [84] calculations. The
Table 1 in Ref. [83] provided the range straggling values for helium isotopes in three scintillators:
GsO(Ce), CsI(Tl), and BGO. The TRIM95 code systematically gives larger range straggling values

50

567891011121314He (mm)6Range of 0.00.10.20.30.40.50.60.70.80.9a.u.0 Degree42 DegreesFigure 3-5: Blue: the ratio of the transmitted to the incident 6He ions versus the effective
thickness of the degrader. Red: the corresponding range distribution of the 6He ions.

than the experiments. Specifically, the range straggling of 6He in CsI(Tl) at 258 MeV determined
from experiments and TRIM95 are 2.2% and 4.0%, respectively.

Figure 3-6: The range straggling curve of the 6He beam from calculations using LISE++ (blue)
and measurements (red).

3.3.4 Beam Purity

The beam purity is determined by measuring beam energy without a degrader shown in Fig. 3-7.
The left panel is the beam energy measured in the CsI (Na) detector in channel (chan), which is an
energy unit related to the data acquisition (DAQ) system. The HV of the detector was at +300 V.

51

05101520253035400.00.20.40.60.81.00510152025303540Degrader thickness (mm)0.00.20.40.60.81.0Transmission00.20.40.60.81a.u.Range Number-Distance CurveRange Straggling Curve2426283032343638Degrader thickness (mm)0.000.050.100.150.200.250.300.35a.u.LISE++ExperimentThe dominant feature is the main peak with a narrow width. Details of the energy distribution are
shown in the right panel in the log scale. Specifically, the tail in the energy range between 0-2000
chan is due to the beam-induced reactions in the detector; the short peak at 4000 chan and the data
in the energy range of 2400-4000 chan are due to the pile-up in the DAQ system. One possible
contaminant could be 8Li with a half-life of (838.4 ± 0.36) ms [85], which is very close to the
half-life of 6He [45]. If 8Li was present, its energy is expected to be the energy of 6He times 8/6,
corresponding to a channel number roughly 4/3 of the channel number of 6He. A dashed line in
the plot marks the expected channel number of 8Li if it was present, where no visible traces of 8Li
are seen. From this spectrum, the level of contaminants was deduced to be lower than 2 × 10−5.

Figure 3-7: Energy of the 6He beam measured in the CsI (Na) detector without a degrader in
linear (left panel) and log (right panel) scales. The first dashed line in the right panel indicates the
expected energy of the 8Li contaminant if it was present; the short peak at 4000 chan marked by
the second dashed line and the data in the energy range of 2400-4000 chan are due to the pile-up
in the DAQ system.

3.3.5 Beam Cycles

During the measurement, the beam was chopped periodically by dephasing the radio-frequency
signal of one of the cyclotrons. This forms implantation and decay cycles. Each cycle consists of a
2.5 s implantation interval with the beam on, referred to as the beam-on interval, and a subsequent
decay/measurement period with the beam off, referred to as the beam-off interval. The beam-off

52

0.00.51.01.52.02.53.03.54.04.55.0310·Energy (chan)0123456310·Counts/bin0.00.51.01.52.02.53.03.54.04.55.0310·Energy (chan)110210310410Counts/binLi8energy of expected pile-upperiod is 10 s for the CsI(Na) detector and 15 s for the NaI(Tl) detector. The long beam-off intervals
were chosen for background estimations discussed in Chap. 9.

3.4 Detectors

As mentioned earlier, the 6He beam was implanted into two types of scintillating detectors, a
CsI(Na) detector, and a NaI(Tl) detector. The CsI(Na) detector is a 9.8 × 5 × 5 cm3 rectangular
cuboid scintillator from the CAESAR array [86] manufactured by ScintiTech. The NaI(Tl) detector
is a Ø7.6× 7.6 cm2 cylindrical scintillator, where the scintillation crystal is manufactured by Saint-
Gobain (Bicron). Figure 3-8 shows the setup of the NaI(Tl) detector on the beamline. The setup for
the CsI(Na) detector is shown in Fig. 3-9, where the CsI(Na) detector was surrounded by a shield
constructed with lead bricks to reduce the ambient background during the regular measurement
runs. In addition, an OR120 Ge semiconductor detector shown in Fig. 3-10 was placed next to the
beamline to measure the γ rays induced by the beam in the detectors.

Figure 3-8: The setup of the NaI(Tl) detector on the beamline. The 6He ions exiting from a beam
pipe through the 75 µm thick Zr vacuum window (left) were implanted into the NaI(Tl) detector
(right) aligned along the beamline.

3.5 Data Acquisition System

The DAQ system of this experiment was based on the implementation of the Digital Data Acqui-
sition System (DDAS) developed at the NSCL [87]. Two Pixie-16 digitizing modules manufactured

53

Figure 3-9: The setup of the CsI(Na) detector on the beamline. The 6He ions exiting from the
beam pipe through the 75 µm thick Zr vacuum window (left) were implanted into the CsI(Na)
detector (right) aligned perpendicular to the beamline. The CsI(Na) detector is surrounded by lead
bricks (green).

Figure 3-10: The setup of the OR120 Ge semiconductor detector mounted next to the beamline.

by XIA LLC were used. Each Pixie-16 module is capable of digitizing signals at a rate of 100
MHz. The signals from both the CsI(Na) and the NaI(Tl) detectors were sent to the first module,
and the signal from the Ge detector was sent to the second module. Both modules received three
additional signals associated with the beam-on start, the beam-off start, and a 100 Hz pulser. The
digitizers provided the information of the time stamp and the energy of each signal sent to each
channel in the two modules, where the energy was converted using a trapezoidal filter [87].

For each incoming signal, the waveform was digitized by a Pixie-16 module. Once the waveform

54

was above a certain threshold, it was processed by the Field-Programmable Gate Arrays (FPGAs)
in the Pixie-16 system, and the information was recorded. The FPGAs provide an estimate of the
energy of each signal, during which no other triggers were accepted. If multiple signals took place
too close in time to each other, their waveforms would overlap, and the overlapping signals would
be reported as one signal. The energy of the first signal would, therefore, be overestimated, and
the reported value would be between the energy of the first signal and the sum of the overlapping
signals. This is the origin of the pile-up in the DAQ system mentioned in Fig. 3-7. The contribution
of the pile-up has a great impact on the measured β-energy spectrum and is discussed in the data
analysis in Chaps. 4 and 8.

3.6 Data Collections

3.6.1 Regular Measurements

The regular measurements consisted of 22 runs with the CsI(Na) detector and 30 runs with the
NaI(Tl) detector. Most of the runs were about one hour long. The experimental conditions for the
two detectors are summarized in Table 3-1, including the high voltage, the primary beam intensity,
the lengths of beam-on and beam-off time intervals, and the number of collected implantation-and-
decay cycles.

Table 3-1: Experimental conditions for the CsI(Na) and the NaI(Tl) detectors during the regular
measurements.
Detector HV (V) Beam intensity (µA)
CsI(Na)
NaI(Tl)

Implantation time (s) Decay time (s) Collected cycles

5760
4114

+860
+1000

1.6±0.1
1.2±0.1

2.5
2.5

10
15

3.6.2 Calibration Runs

Detector calibration measurements with the 60Co and the 137Cs γ sources were performed before,
during, and after the experiment. Most of the measurements lasted for a few minutes. The voltage
settings were the same as in Table 3-1.

55

3.6.3 Background Measurements

This experiment involves two types of background: the ambient background and the beam-induced
background. The ambient background was measured before and after the beam time, with and
without the lead shield around the CsI(Na) detector. It was observed that the ambient background
event rate is reduced by 13% with the presence of the lead shield. Therefore for all regular runs
collecting β decay events with the CsI(Na) detector, the detector was placed in the lead shield. The
beam-induced background was measured in four runs between the regular runs with the CsI(Na)
detector. The obtained energy spectra have both the signal (β decay) events and the background
events. These measurements provide valuable inputs to the background analysis in Chap. 9.

3.6.4 Gain Drift Test

To test the drifts of the system gain, a dedicated experimental run was performed with a 137Cs γ
source placed next to the CsI(Na) detector during the beam implantation. The energy spectrum of
this run is then a superposition of the β-energy spectrum from 6He and the γ-energy spectrum from
the 137Cs source. The shift of the full-energy peak in the γ spectrum would indicate a possible
gain drift during the 6He decay. The results are discussed in Chap. 7.

3.7 Sample Data

The sample data collected in a typical run with the CsI(Na) detector is shown in Fig. 3-11.
The left panel shows a two-dimensional histogram of the registered time versus deposited energy
during the beam-off time intervals (2.5-12.5 s in each beam cycle) collected in an one-hour run.
The projection of the two-dimensional histogram on the energy axis is a β-energy spectrum, which
is shown in the right panel of Fig. 3-11 after subtracting events in the 9.43-9.61 s interval from
events in the 3.43-3.61 s interval, see discussions about the background subtraction in Chap. 9. The
structure above 5200 chan are due to the pile-up in the DAQ system. The analysis of the measured
β-energy spectrum is the focus of the rest of the thesis.

56

Figure 3-11: Left panel: a two-dimensional histogram of the registered time versus deposited
energy during the beam-off time intervals collected in an one-hour run with the CsI(Na) detector.
Right panel: a background-subtracted β-energy spectrum measured in the CsI(Na) detector during
the beam off period.

57

051015310·Electron kinetic energy (chan)110210310410Counts/binCHAPTER 4

BIG PICTURE OF THE ANALYSIS

The rest of the thesis focuses on analyzing the shape of the measured β-energy spectrum in 6He
decay. As the effect of the CVC value of bWM on the spectrum is small, and the effect of the
SM value of bGT on the spectrum is zero, dedicated study of each systematic effect is performed
focusing on its impact on the shape of the spectrum. An overview of the data analysis is illustrated
in Fig. 4-1, where the top row focuses on building an accurate and precise function describing the
measured β-energy spectrum, and the bottom row focuses on the experimental data; the function is
then fitted to the data to extract the observables.

Figure 4-1: Overview of the data analysis steps in this experiment.

To analyze the measured β-energy spectra (Chap. 3), an accurate theoretical description of
the β-energy spectrum, which is a product of the phase space factor (Chap. 1) and the theoretical

58

Phase SpaceDetector SimulationsDetector ResolutionAuto-Calibration
&FitPile-UpExperimental
DataBackground SubtractionbWMTheoretical Correctionscorrections (Chap 2), is calculated and taken as the input for the detector simulations with the
Geant4 Monte Carlo toolkit (Chap. 5) to take into account part of the detector response [17]. The
output of the detector simulation, the absorbed spectrum shown in Fig. 4-2 in red, is convolved with
a Gaussian to take into account the detector energy resolution, resulting in the convolved spectrum
in green, which is used to calculate a pile-up spectrum in black. A superposition of the convolved
spectrum and the pile-up spectrum expressed in channel (chan) after the energy calibration (Chaps.
6 and 7), multiplied by the spectral function and (1 + bGT/W) in Eq. 1.26, was used as a fit
function that describes the shape of the measured β-energy spectrum. The fit function has four free
parameters: an observable (bWM or bGT), a calibration parameter (gain), a normalization factor for
the convolved spectrum, and a normalization factor for the pile-up spectrum.

Figure 4-2: The shapes of the energy spectra used in the fit function.

The measured spectra include both the β-decay events and the background events. The spectra
after background subtractions (Chap. 9) are fitted by the fit function to extract the observables with
a single-step auto-calibration and fit method [88] (Chap. 6). The observables can be extracted
without bias in the fits (Chap. 8).

The analysis is currently entering the final stage and is expected to provide the first measure-
ment of the weak magnetism form factor in 6He decay, the first direct measurement of the Fierz
interference term in Gamow-Teller decay, and a new constraint on the tensor interactions beyond
the SM.

59

CHAPTER 5

DETECTOR SIMULATIONS

This chapter simulates the responses of the CsI(Na) and NaI(Tl) detectors to the β-energy spectrum.
Two approaches were taken: 1) simulating the detector responses to monoenergetic electrons and
parameterizing the response functions as a function of energy; 2) directly simulating the energy
spectra absorbed in the detectors using the theoretical spectrum as a continuous input source.
For both approaches, the photon production and the energy absorption fraction were studied. In
addition, the systematic uncertainties associated with the detector simulations were estimated.

The response function of the detector is defined as the energy spectrum observed in the detector
as a response to a given monoenergetic radiation. After eliminating the backscattering effect in
Chap. 3, there is another instrumental effect that prevents the electrons from fully depositing the
energy in the detectors. Some electrons in the detectors may emit bremsstrahlung photons as they
decelerate, and these photons may escape from the detector and carry away energy. This effect
produces a low energy tail in a response function, distorting the observed β energy spectrum in
the detectors. As the probability for a photon to escape depends on the photon energy and the
geometry and material of the detector, this effect needs to be studied with detector simulations.
The simulations were performed using the Geant4 toolkit [89].

In practice, because of the statistical nature of the interaction between the electrons and the
detector, the energy deposited by a monoenergetic source is dominated by a Gaussian distribution
instead of a Delta function. The standard deviation of the Gaussian is associated with the energy
resolution of the detector. The finite detector resolution is taken into account by convolving the
absorbed energy spectrum simulated with Geant4 with the Gaussian distribution. A parametrization
of the response function consisting of a Gaussian peak and a low energy tail is provided, but will
not be used in the analysis. The second approach mentioned above was chosen when analyzing the
β-energy spectrum.

60

5.1 Simulation Conditions

Electrons lose energy in the detectors by atomic collisions and bremsstrahlung radiation. Col-
lisions include the ionization and excitation processes which can produce Auger electrons and
X-rays. These processes are incorporated in the Geant4 toolkit. In this work, the simulations were
performed with the version 10.3.0 of Geant4. The majority of the work in this chapter has been
published in Ref. [17].

5.1.1 Particle Source and Detector Configurations

The geometries of the electron sources and the detectors used in the simulations are shown in Fig.
5-1. They correspond to the actual experimental conditions described in Chapter 3. Specifically,

• The geometries of the detectors are a 9.8 × 5 × 5 cm3 rectangular cuboid for the CsI(Na)
detector and a Ø7.6 × 7.6 cm2 cylinder for the NaI(Tl) detector.

• The geometry of the electron sources are described by a rectangular cuboid with the dimen-
sions of 4.6 × 4.5 × 1.5 mm3 in the CsI(Na) detector and 4.6 × 4.5 × 1.7 mm3 in the NaI(Tl)
detector.

• In the CsI(Na) detector, the center of the electron source is located 2.44 cm from the top
and 1.23 cm from the front face. In the NaI(Tl) detector, the center of the source is on the
cylindrical axis and is 1.48 cm from the front face.

The absorber materials implemented in the simulations are CsI for the CsI(Na) detector and NaI
for the NaI(Tl) detector. The CsI and NaI materials were constructed with the predefined materials
in the Geant4 NIST database: G4_CESIUM_IODIDE and G4_SODIUM_IODIDE.

The primary particle source was defined with the G4GeneralParticleSource (GPS) class in
Geant4 toolkit. GPS allows the specifications of the spectral, spatial, and angular distribution of
the primary particles. In this work, electrons with an isotropic angular distribution were generated
and uniformly distributed within the volumes of the sources inside the detectors shown in Fig. 5-1.

61

Two types of energy distributions were implemented in this study: a list of monoenergetic primary
electron sources in the energy range of 0-4 MeV and a continuous theoretical β-energy spectrum
in 6He decay including the phase space factor and the theoretical corrections.

Figure 5-1: Left panel: geometry of the CsI(Na) detector (gray) and position of the electron
source (red) used in the simulations. Dimensions are in cm. Right panel: geometry of the NaI(Tl)
detector (gray) and position of the electron source (red) used in the simulations. (Published in
Ref. [17].)

5.1.2 Geant4 Physics Constructors

The physical processes of interest are electromagnetic and are taken into account by the electro-
magnetic physics constructors in Geant4, where the electromagnetic physics lists are implemented.
The constructors can provide different CPU performances, and some constructors are adapted to
low-energy processes.
In addition, these physics constructors may differ in the production and
absorption of photons, resulting in different response functions. To study the possible differences
among the simulations, this work has included the constructors in Table 5-1.

The Livermore (liv), Penelope (pen) and Option4 (opt4) constructors are optimized for low-
energy processes, among which the Option4 constructor is considered “using the most accurate
standard and low-energy models” according to an on-line documentation [90]. The Standard

62

Table 5-1: Physics constructors in Geant4 applied in the simulations.

Physics constructors
PhysListEmStandard
G4EmLivermorePhysics
G4EmPenelopePhysics
G4EmStandardPhysics_option4

Denomination

Standard
Livermore
Penelope
Option4

constructor is not specially optimized for low-energy processes and is included here to compare its
results with the other constructors’. In this work, the Standard constructor is taken from Ref. [91].
For each physics constructor, the models describing the electromagnetic processes are dis-
cussed in the Geant4 Physics Reference Manual [92]. In particular, the bremsstrahlung process in
the Standard, Livermore and Penelope constructors is described by G4SeltzerBergerModel,
G4LivermoreBremsstrahlungModel and G4PenelopeBremsstrahlungModel, respectively.
An inspection of the source code indicates that Option4 uses G4SeltzerBergerModel for the
electron energy range considered in this work. Note that these constructors are not mutually
independent, as the Option4 constructor is using the same bremsstrahlung model as Standard in
this work. In addition, Livermore and Penelope use common data libraries such as the Evaluated
Electron Data Library [93], according to the Physics Reference Manual [92].

5.1.3 Geant4 Parameters

Some parameters control the physical processes in each physics constructor. The default parameters
in each physics constructor have been adapted in this work. Specifically, the range factor parameter,
FR, and the geometry factor parameter, FG, affect the multiple scattering process [94]. Their default
values in each physics constructor are summarized in Table 5-2.

Table 5-2: Default values of the range factor parameter and the geometry factor parameter in each
physics constructor.

FR
Standard
0.04
Livermore 0.02
0.02
Penelope
Option4
0.02

FG
2.5
2.5
2.5
2.5

63

The event tracking is controlled by a parameter referred to as the range cut, or “cut for sec-
ondaries”.
It is worth clarifying that the only kind of primary particle in the simulation is the
primary electron source. All the photons and electrons produced by the primary electrons are
considered secondary particles. The range cut controls the creation of secondary particles in the
following way: if a primary particle has enough energy to produce a secondary particle offspring
that can travel over a distance greater or equal to the range cut in a given material, the secondary
particle is created. Otherwise, the energy of the primary particle is absorbed locally. In this work,
unless stated otherwise, the range cut values in the CsI and NaI detectors are 5.0 µm and 5.7 µm,
respectively. These values correspond to an energy of 20 keV for the primary electrons.

5.1.4 Visualization

The simulation process can be visualized with the OpenGL event display. Figure 5-2 illustrates a
simulation with the CsI detector, where 103 monoenergetic electrons of 1 MeV were generated. The
trajectories of the particles are color-coded with red representing electrons and green representing
photons. All the electrons stop inside the detector while a small fraction of photons escapes. This
simulation was performed applying the Standard physics constructor.

5.2 Discrete Monoenergetic Electrons

In this section, the production and the absorption of photons for the monoenergetic primary elec-
tron sources are studied. For illustration, Fig. 5-3 shows the photon spectrum produced by 2×107
electrons of 3.5 MeV in the CsI detector. The dominant contributions are from bremsstrahlung
photons, a majority of which populate the low energy range of the spectrum. The peak at 0.511
MeV is due to positron annihilations following pair productions by the bremsstrahlung photons
above 1.022 MeV. In addition, characteristic X-ray lines are produced under 50 keV, with details
shown in the insert to Fig. 5-3. The X-ray lines correspond to the Kα1, Kα2, Kβ1, and K-edge
lines for I and Cs and the L-lines at lower energy. This simulation was performed with the Standard
constructor, and similar simulations with the other three constructors were also performed. The

64

Figure 5-2: Visualization of a Geant4 simulation with the CsI detector. 103 monoenergetic
electrons of 1 MeV were generated. Left: side view of the detector; Middle: front view; Right:
top view. The green lines indicates that a fraction of the photons escape from the detector.
(Published in Ref. [17].)

intensity (I) of the X-ray peaks produced by the constructors show a relative difference of up to
20%, with Ipen > Istd > Iliv > Iopt4.

Figure 5-3: The photon energy spectrum produced by 2×107 monoenergetic primary electrons of
3.5 MeV. The physics constructor is Standard. (Published in Ref. [17].)

65

The above results show that at least the photons produced by the atomic processes vary when
applying different physics constructors. As a result, the energy absorption may also be different.
The differences in the photon production and the energy absorption associated with different physics
constructors were studied with detector simulations with the monoenergetic electrons. For both the
CsI and the NaI detectors, simulations with each physics constructor were performed by varying
the energy of the primary electron source in the range of 0.5-4 MeV in steps of 0.5 MeV. In each
simulation, 2×107 primary events were generated, and the information about the photon production
and the energy absorption was recorded. The rest of this section discusses the simulation results
focusing on the photon production and the energy absorption characterized by the photon yield and
the absorption fraction, respectively.

5.2.1 Photon Yield

The photon yield, Y, is defined as the fraction of the initial kinetic energy of the primary electrons
converted into photons. Figure 5-4 shows the photon yields in the CsI and the NaI detectors
obtained with the Standard constructor as a function of the initial energy of the primary electron
source. As the photon can come from different production mechanisms, the sources of photon
production were studied quantitatively with three simulation conditions:

1. All photon production processes are switched on, corresponding to the open markers (squares

and circles) in Fig. 5-4.

2. The atomic deexcitations, including the fluorescence photons, the Auger electrons, and the
Particle Induced X-ray Emission, are switched off. The results are represented by the light
gray markers in Fig. 5-4.

3. Only the bremsstrahlung photons from the primary electrons are collected. This requires
that, on the one hand, the photon threshold should be reasonably low to include most of the
photons produced by the primary electrons; on the other hand, besides the photon production
by the primary electrons, all other photon production processes should be switched off.

66

The former can be done by setting the range cuts for photons at 3.5 µm in the CsI and
5 µm in the NaI, respectively. The latter requires that in addition to disabling the atomic
deexcitation processes, the interactions of the photons and the secondary electrons with the
detectors should be switched off. The interaction of photons with the detectors is disabled by
switching off the photoelectric effect, the Compton scattering, and the pair production. The
interaction of the secondary electrons with the detectors is disabled by switching off multiple
scattering. In addition, by setting the range cut for electrons at 1 km which is considered
“infinity” in Geant4, any photon production process by the secondary electrons is disabled.

Figure 5-4: Photon yields obtained from simulations with the CsI (left panel) and the NaI (right
panel) detectors as a function of the initial kinetic energy of the primary electron. The simulations
were performed with the Standard constructor. The open, light gray, and dark gray markers
correspond to Geant4 simulations; the black marker corresponds to the values from the NIST
ESTAR program. Details are explained in the text. (Published in Ref. [17].)

The conditions above allow one to deduce the relative contributions of the main photon produc-
tion mechanisms. It is shown by the dark gray markers in Fig. 5-4 that the dominant contribution
is from the bremsstrahlung photons produced by the primary electrons. The second greatest contri-
bution is from the atomic deexcitations. The smallest contribution is from the photons produced by
the secondary electrons and photons. The above simulation results are compared with the radiation
yield (black markers) from the NIST ESTAR program, which is defined as the average fraction of
the initial kinetic energy of an electron that is converted to bremsstrahlung energy as the particle

67

slows down to rest calculated in the continuous-slowing-down approximation. Between the ESTAR
values and the dark gray marker values, there is a small but visible difference. The origin of this
difference is unclear but can be attributed to the details in the calculations of the radiation yield,
including the cutoff in energy integration, or the difference between the continuous-slowing-down
approximation used in ESTAR and the discrete steps used in Geant4. In practice, and in the rest
of the chapter, all photon contributions are included because once a photon is produced, it has a
chance to escape from the detector.

Similarly, the photon yields can be obtained from simulations with the three other constructors.
The results are indistinguishable from those obtained with the Standard constructor on the scale of
Fig. 5-4. The differences can be made visible by normalizing the results obtained with the Standard,
Penelope, and Livermore constructors to those with the Option4 constructor. The normalized yields
are defined as

rX = YX/Yopt4 − 1,

(5.1)

where X = std, pen, liv. The results are shown in Fig. 5-5, where it is observed that the normalized
yields obtained with the Standard and Penelope constructors show a larger energy dependence than
that obtained with Livermore. The Livermore and Option4 constructors produce the closest results
in the photon yield and the energy dependence of the normalized yield.

From the above result, one can estimate the uncertainty of the photon yield of the β-energy
spectrum in 6He decay shown in Fig. 5-9, which has a mean energy of 1.6 MeV. From Fig. 5-5,
the normalized yields at 1.6 MeV calculated with the four constructors show maximum spreads of
about 5% for the CsI detector and 4% for the NaI detector. These spreads are taken as an estimate
of the relative uncertainties applying to the photon yields of 8% for the CsI and 6% for the NaI
(Fig. 5-4). Therefore one can expect the absolute uncertainty of the photon yield from the β-energy
spectrum in 6He decay to be on the order of 10−3.

The differences in the photon production obtained with different physics constructors do not
imply that the difference in the energy absorption will have the same magnitude. The former is
uniquely determined by the energy of the primary electron source and the detector material. The

68

Figure 5-5: Normalized photon yields obtained from simulations with the CsI (left) and the NaI
(right) detectors as a function of the initial kinetic energy of the primary electron source. The
results are extracted from the simulations with the Penelope (black markers), the Standard (gray
markers), and the Livermore (open markers) physics constructors by normalizing the photon
yields relative to that obtained with Option4. (Published in Ref. [17].)

latter is also affected by the detector geometry. If the detectors were infinitely large, regardless of
the differences in the photon production, all photons and thus all the energy would be absorbed.

5.2.2 Absorption Fraction

From the simulations described at the beginning of Sec. 5.2, one can extract the absorption
fraction, f , defined as the fraction of the initial kinetic energy of the primary electrons absorbed in
the detector. Figure 5-6 shows the absorption fractions in the CsI and the NaI detectors obtained
with the Standard constructor as a function of the initial energy of the primary electron source. It
is worth emphasizing that the geometries of the two detectors are different and would affect the
results.

With the results in Fig. 5-6, one can estimate the average size of the correction to the β-energy
spectrum in 6He decay due to the photons escaping from the detector. Taking again a mean energy
of 1.6 MeV shown in Figure 5-9, the average correction is the average fraction of energy lost by
the detector, which is about 1%-2% according to Fig. 5-6. Since the absorption fraction is energy-
dependent, the correction to the β-energy spectrum is energy-dependent as well. As the electron
energy increases, the absorption fraction decreases, and the size of the correction increases.

69

Figure 5-6: Absorption fractions obtained from simulations with the CsI (squares) and the NaI
(circles) detectors as a function of the initial kinetic energy of the primary electron source. The
simulations were performed with the Standard constructor. (Published in Ref. [17].)

The absorption fractions from simulations with the three other constructors can be obtained
similarly. The results are indistinguishable from those obtained with the Standard constructor
on the scale of Fig. 5-6. Once again, the differences can be made visible by normalizing the
results obtained with the Standard, Penelope, and Livermore to those obtained with the Option4
constructor. The normalized absorption fraction is defined as

sX = fX/ fopt4 − 1.

(5.2)

The results are shown in Fig. 5-7, where the error bars are derived from the statistical uncertainties
of the mean absorbed energy per event. Unlike the normalized photon yields shown in Fig. 5-5, the
normalized absorption fractions do not show a common trend as a function of the electron energy.
As the electron energy increases, the normalized absorption fraction obtained with the Penelope
constructor increases, whereas the result obtained with the Livermore constructor decreases. These
two trends result in a spread that increases as a function of electron energy.
In addition, the
difference in the results obtained with the Livermore and the Option4 constructors is larger than
that obtained with the Penelope and the Option4 constrictors, which is possibly because the pair
production model in Option4 (G4PenelopeGammaConversionModel) is taken from the Penelope
constructor.

With the above result, one can estimate the uncertainty of the absorption fraction for the β-

70

Figure 5-7: Normalized absorption fraction obtained from simulations with the CsI (left) and the
NaI (right) detectors as a function of the initial kinetic energy of the primary electron source. The
results are extracted from the simulations with the Penelope (black markers), the Standard (gray
markers), and the Livermore (open markers) physics constructors by normalizing the absorption
fractions relative to that obtained with Option4. (Published in Ref. [17].)

energy spectrum in 6He decay shown in Fig. 5-9. Taking again a mean energy of 1.6 MeV, the
normalized absorption fractions calculated with the four constructors show maximum spreads of
about 2×10−4 for the CsI detector and 1.5×10−4 for the NaI detector. These spreads are considered
the relative uncertainties that apply to the absorption fractions that are close to 1. Therefore one
can expect the absolute uncertainties of the absorption fraction from the β-energy spectrum in 6He
decay to be on the order of 10−4 for both detectors. These uncertainties are an order of magnitude
lower than those of the photon yields. Such small differences in the absorption fractions obtained
with different constructors are difficult to benchmark experimentally.

5.3 Detector Response

The absorbed energy spectra produced by the monoenergetic electron sources are obtained
from the simulations. For a 3.5 MeV electron source, the spectrum is shown in the upper panel of
Fig. 5-8. The number of primary events is 2 × 107, and the physics constructor is Standard. The
spectrum is dominated by a full-energy peak at 3500 keV followed by a continuous electron-energy
spectrum below 3500 keV due to the escape of bremsstrahlung photons. The two peaks of energy
0.511 MeV and 1.022 MeV below the full energy peak are the “single escape peak” and the “double

71

escape peak” originated from the annihilations following the pair productions of the bremsstrahlung
photons above 1.022 MeV.

Figure 5-8: Upper panel: the absorbed energy spectrum produced by a monoenergetic primary
electron source of 3.5 MeV in the CsI detector. The number of primary events is 2 × 107 and the
physics constructor is Standard. Lower panel: the response function at 3.5 MeV obtained by
convolving the upper panel with a Gaussian.

To take into account the energy resolution, the above absorbed spectrum is convolved with a
Gaussian, resulting in a convolved spectrum describing the response function. Considering that
the energy resolution is energy-dependent, the convolution is performed in a bin-by-bin case. The
events in each bin are redistributed following a Gaussian distribution with the following mean (µ)
and standard deviation (σE)

µ = E,

σE = σ0

√

E,

(5.3)

(5.4)

where E is the kinetic energy in keV at the center of a bin, and the coefficient σ0 is determined with
γ sources (Chaps. 6 and 7). In this example, σ0 = 1.044 keV1/2. For each bin, the redistributed

72

0.00.51.01.52.02.53.03.54.0310· [keV]110210310410510610710counts/bin0.00.51.01.52.02.53.03.54.0310·Energy (keV)110210310410counts/binevents are filled in a new histogram, resulting in the convolved spectrum shown in the lower panel
of Fig. 5-8.

The response function described by the shape of the convolved spectrum can be expressed as

fr(E; E0, P0, ..., P7) = tr(E; P0, ..., P7) + P8e

−1
2

P9

.

(5.5)

(cid:18) E−E0

(cid:19)2

Here, E0 is fixed at the initial energy of the monoenergetic electron in keV. The second term in
Eq. 5.5 corresponds to the main Gaussian peak and tr(E; P0, ..., P7) describes the shape of the
asymmetric tail and has the form

tr(E; P0, ..., P7) = P7 ˜L(E; P0, ..., P6) + (1 − P7) ˜G(E; P0, ..., P6),

(5.6)

˜L and ˜G are the modified Lorentzian distribution and the modified Gaussian distribution,

where,
respectively:

(cid:104)

P0
E−P1

(cid:20)

˜γ(E;P1,...,P6)
E−P1
−1
2

˜γ(E;P1,...,P6)

(cid:105)2,
(cid:21)2

,

˜L(E; P0, ..., P6) =

1 +

˜G(E; P0, ..., P6) = P0e
˜γ(E; P1, ..., P6) =

P2

1 + P3eP4(E−P1) + P5eP6(E−P1) .

(5.7)

(5.8)

(5.9)

With the ten free parameters, P0, ..., P9, and one fixed parameter, E0, one can use Eq. 5.5 to
describe a spectral shape shown in the lower panel of Fig. 5-8, where the red curve shows a best
fit that determines a set of values for the ten free parameters. Similarly, new sets of values for the
ten parameters can be obtained by fitting the convolved spectra produced by other monoenergetic
electron sources. Each parameter as a function of the electron source energy is fitted by a linear
function. In this way, the response function of a monoenergetic electron source with any energy in
the range of 0-4 MeV can be determined.

For the β-energy spectrum, the detector response can be expressed as a linear combination of a

list of response functions

Fr(E) =

Emax


E=Emin

fr(E; E0, P0, ..., P7)wE,

(5.10)

73

where wE is the statistical weight at the energy E. The number of functions in the sum is determined
by the energy step size.

This seemingly direct approach to determine the response of the detector to a β-energy spectrum
is long and tedious. Also, the expression of the response function, Eq. 5.5, has no physical
significance: it only serves the purpose of describing the shape of the convolved spectra. Instead
of simulating the absorbed energy spectra of monoenergetic electrons, the β-energy spectrum can
be directly used as a continuous primary particle source in Geant4. The GPS class in Geant4 is
capable of defining a primary particle source with continuous energy distribution. The simulation
with the continuous source provides the absorbed β-energy spectrum, which is convolved with the
Gaussian, producing a convolved β-energy spectrum that represents the response of the detector to
a β-energy spectrum. This approach includes only two steps: the simulation with Geant4 and the
convolution with Gaussian.

5.4 Continuous β-energy Spectrum

A β-energy spectrum with 109 events was generated following Eq. 5.11, which is Eq. 1.26
without the (1 + bGT/W) and the f1 (E, bWM) factors associated with the observables. These two
factors will be included in the spectrum when fitting the measured β-energy spectrum, with bWM
and bGT being free parameters in the fits, which is discussed in Chap. 8. This generated spectrum
was used as a primary electron source for the simulation, shown in Fig. 5-9 in blue. From a
simulation with the CsI detector, the absorbed β-energy spectrum obtained, as shown in Fig. 5-9
in red, where a clear distortion of the spectrum is observed. The difference between the generated
and the absorbed spectra originates from the energy loss due to the escapes of the bremsstrahlung
photons from the detector. Such a distortion is smaller in the NaI detector.

N(W)dW = pW(W0 − W)2

η(W)dW .

(5.11)

The size of the distortion can be characterized by a spectral slope. The absorbed spectrum
(red) is divided by the generated spectrum (blue), the derivate of the resulting ratio is the slope
introduced by the distortion. Figure 5-10 shows the slope calculated from the simulations with the

74

Figure 5-9: The β-energy spectrum in 6He decay. Blue: energy spectrum of the primary electron
source with 109 events; Red: absorbed energy spectrum in the CsI detector. The simulation was
performed with the Standard constructor. (Published in Ref. [17].)

CsI (squares) and the NaI (circles) detectors as a function of the β particles energy. The slopes
vary by a factor of 4 and change the sign at 0.9 MeV. Near the endpoint energy, the slopes are
the largest, but the weights are much smaller than the weight at the mean energy of the β-energy
spectrum. The mean values of the slopes derived from Fig. 5-10 taking into account the weights
are -5.4%/MeV for the CsI detector and -4.8%/MeV for the NaI detector. These two slopes provide
estimates of the corrections for the effect of the photon escape applied to the generated spectrum.

Figure 5-10: Numerical derivatives of the ratio between the absorbed and the generated β-energy
spectra for the NaI (circles) and CsI (squares) detectors. (Published in Ref. [17].)

Section 5.2 has shown the differences in the photon production and the energy absorption

75

0.51.01.52.02.53.0Electron kinetic energy (MeV)20-15-10-5-0510Slope (%/MeV)2 7.6 cm· 7.6 ONaI - 3 5 cm· 5 ·CsI - 9.8 among the physics constructors for monoenergetic electrons and estimated the mean effects on
the β-energy spectrum in 6He decay. This effect can be more accurately calculated by performing
Geant4 with the continuous β-energy spectrum as the primary electron source. For both the CsI and
the NaI detectors, a simulation with each physics constructor was performed. In each simulation,
2×107 primary events were generated following Eq. 5.11, and the information about the photon
production and the energy absorption was recorded.

5.4.1 Photon Yield

The photon yield of the continuous β-energy spectrum can be calculated from the simulated photon
energy spectrum. For illustration, Fig. 5-11 shows the photon energy spectrum obtained from a
simulation with 2 × 107 primary electrons with the Standard constructor. It shows similar features
to the photon spectrum produced by the monoenergetic electrons shown in Fig. 5-3. Both spectra
have a dominant continuous bremsstrahlung distribution, a 511 keV peak, and some characteristic
X-ray lines. The photon yield can be calculated by dividing the total energy of photons in Fig. 5-11
by the total energy of the primary electrons.

Figure 5-11: The photon energy spectrum produced by a β-energy spectrum generated following
Eq. 5.11. The number of primary events is 2×107 and the physics constructor is Standard.

The photon yields calculated from the simulations with the four physics constructors are sum-
marized in Table 5-3. The largest value of the photon yield is produced by the Penelope constructor,

76

followed by the Standard, the Livermore and the Option4 constructors, showing a similar trend to
the simulations with the monoenergetic source (Figs. 5-4 and 5-5). The largest differences in the
photon yields calculated with the four constructors are 3.4×10−3 for the CsI and 2.2×10−3 for the
NaI, as expected from the estimates at 1.6 MeV in Sec. 5.2.1.

Table 5-3: Photon yields (in %) produced by a β-energy spectrum generated following Eq. 5.11.
In each simulation, the number of primary events is 2×107. The results are obtained with four
physics constructors.

pen
8.30
6.75

std
8.16
6.68

liv
8.07
6.60

opt4
7.96
6.53

CsI
NaI

5.4.2 Absorption Fraction

The absorption fraction of the continuous β-energy spectrum can be calculated from the simulated
β-energy spectrum. The total energy in the absorbed spectrum divided by the total energy of the
primary electrons gives the absorption fraction. The results calculated from the simulations with
the four physics constructors are summarized in Table 5-4. The largest value of the absorption
fraction is produced by the Penelope constructor, followed by the Option4, the Standard, and the
Livermore constructors, showing a similar trend to the simulations with the monoenergetic source
(Figs. 5-6 and 5-7). The largest differences in the absorption fractions calculated with the four
constructors are 3×10−4 for the CsI and 2×10−4 for the NaI, as expected from the estimate at 1.6
MeV in Sec. 5.2.2. As previously indicated, these differences are an order of magnitude smaller
than the differences among the photon yields.

Table 5-4: Absorption fractions (in %) produced by a β-energy spectrum generated following Eq.
5.11. In each simulation, the number of primary events is 2×107. The results are obtained with
four physics constructors.

pen
98.15
98.39

std
98.14
98.39

opt4
liv
98.12
98.14
98.37 98.39

CsI
NaI

77

5.4.3 Comparison of the Simulated Spectra

The absorbed energy spectra obtained from the above simulations are similar to the red spectrum
shown in Fig. 5-9 but with 2 × 107 events each. The spectra obtained with different physics
constructors are indistinguishable from each other at those scales. As indicated by the absorption
fractions in Table 5-4, the differences are expected to be small. Therefore simulations with high
statistics are required to quantify the differences.
In addition to the red spectrum in Fig. 5-9
simulated with the Standard constructor, the absorbed β-energy spectra were simulated with the
three other physics constructors for both the CsI and the NaI detectors. In each simulation, 109
primary events were generated. The outcomes of these simulations are four spectra with high
statistics for each detector.

If one takes the spectrum obtained with Option4 as a reference, and divide the spectra simulated
with the other three constructors by the one simulated with Option4, the slopes of the resulting ratio
spectra are taken as estimates of the uncertainties of the absorbed spectra related to the differences
among the constructors. The slopes are extracted by fitting the ratio spectra by a linear function.

For illustration, the upper panel of Fig. 5-12 shows the ratio between the absorbed spectra
simulated with the Standard and the Option4 constructors. The ratio spectrum was fitted by a linear
2 minimization method in the energy range of 100-3000 keV. The lower panel
function with the χ
of the figure shows the residuals of the fit defined as

Residual =

ybin − yfit

σbin

,

(5.12)

where, ybin and σbin are the content and error of a bin, respectively, yfit is the value of the fitted
function at the bin. Table 5-5 summarizes the slopes extracted from the fits to the ratio spectra1,
where the uncertainties are statistical uncertainties calculated with the minimization algorithm.

1In Ref. [17], other values of the slopes were provided. This is because the generated spectra
in Ref. [17] and in this work are different. The former includes a radiative correction assuming the
electrons and the photons are distinguishable in the detector, whereas the latter includes a radiative
correction for electrons and the photons that are indistinguishable in the detector. Details of this
distinction are discussed in Sec. 2.5.

78

Figure 5-12: Upper panel: the ratio between the absorbed energy spectra simulated with the
Standard and the Option4 constructors. The red line is a linear fit to the ratio spectrum. Lower
panel: the residuals of the fit.

Table 5-5: Slopes (in 10−8/keV) extracted from the fits to the ratio spectra

pen

20.0 ± 6.7
6.8 ±6.7

std

-7.1±6.7
-5.3±6.7

liv

-65.1±6.7
-44.2±6.7

CsI
NaI

In Table 5-5, both the signs and the absolute values of the slopes are consistent with the trend
and amplitude of the absorption fractions for monoenergetic electrons shown in Fig. 5-7. Among
the slopes calculated with the four constructors, the largest spreads are ±4.2(1) × 10−7/keV for the
CsI and ±2.6(1) × 10−7/keV for the NaI. These values are used to calculate the upper limits of
the uncertainties of the absorbed spectra. As the mean corrections to the spectra due to the effect
of photon escape were estimated to be -5.4%/MeV for CsI and -4.8%/MeV for NaI, the slopes
introduced by the difference among physics constructors represent relative uncertainties of 0.78%
and 0.54% for the CsI and the NaI detectors, respectively.

The largest spreads of the slopes for the two detectors are also used to derive the uncertainties

of the observables based on the spectral function (Eq. 1.27). This results in

δbWM = 4.41,

δbGT = 1.76 × 10−3

(5.13)

79

for the CsI detector, and

δbWM = 2.73,

δbGT = 1.09 × 10−3

(5.14)

for the NaI detector. These are the uncertainties of bWM and bGT associated with the Geant4
simulations.

80

CHAPTER 6

ENERGY CALIBRATION

6.1 Introduction

A crucial part of the analysis of the β-energy spectrum is the energy calibration. Its significance
can be seen in a measurement of the Fierz interference term in free-neutron β decay [31], where the
systematic uncertainty of bn, dominated by the absolute energy calibration and the linearity of the
β spectroscopy energy response, is 6.1%. The measured bn with such a high systematic uncertainty
provides no sensitivity to the exotic couplings.

In this work, a β particle of energy E in a detector produces a signal, which undergoes the
digitizing process, resulting in a measured kinetic energy expressed in channels (chan), which is
also referred to as the “ADC units” or the “digitizer units”. Assuming the detector chain has a linear
energy response, the relation between the kinetic energy and the channel value is

CH(E) = A · E + B chan,

(6.1)

where A is the system gain in chan/keV, B is the pedestal or offset in chan.

In practice, the energy response of the detector chain could have a small deviation from the
perfect linear response described in Eq. 6.1. This deviation is described by introducing an additional
quadratic term:

CH(E) = P · E2 + A · E + B chan,

(6.2)
where the coefficient P describes the nonlinearity of the detector chain and its unit is chan/keV2.
Another important contribution to the energy calibration is a Gaussian standard deviation (σE)
associated with the detector energy resolution. As mentioned in Chap. 5, the energy resolution
is taken into account by convolving the absorbed energy spectrum simulated using Geant4 with a
Gaussian distribution that has an energy-dependent standard deviation:

√

E,

σE = σ0

81

(6.3)

where σ0, referred to here as the “width parameter”, is determined by fitting the full-energy peaks
of several of γ sources.

In summary, the energy calibration in this work requires to determine four parameters: gain
(A), offset (B), nonlinearity (P), and the width parameter (σ0). The determinations of the energy-
calibration parameters have significant impacts on the observables. The sensitivities of the observ-
ables to the energy-calibration parameters are studied, which is followed by initial determinations
of the calibration parameters. For the gain, an auto-calibration method is described. For the offset,
the nonlinearity and the width parameter, initial estimates are provided in this chapter, which are
updated in Chap. 7 after taking into account an additional systematic effect related to the event
rate.

6.2 Sensitivity to the Energy Calibration

The two observables, bWM and bGT, manifest themselves in the shape of the β-energy spectrum
through a positive and a negative slope, respectively. Any inaccuracy of the energy-calibration
parameters would cause a wrong description of the measured β-energy spectrum, introducing
systematic uncertainties of bWM and bGT.

Figure 6-1 illustrates how an incorrect gain produces such a systematic effect and how similar
arguments hold for other energy-calibration parameters. The left panel shows a hypothetical
experimental spectrum (blue) and a theoretical spectrum (red) to be adjusted to the experimental
spectrum. The experimental spectrum is the phase space factor. To make the effect visible,
the theoretical spectrum is obtained by stretching the experimental spectrum (blue) along the
horizontal axis by a factor of 1.1. If we consider the gain of the experimental spectrum to be 1,
then the theoretical spectrum has a gain of 1.1, overestimating the gain by 10%. The middle panel
shows the ratio between a similar theoretical spectrum with a gain of 1.001 and the experimental
spectrum with a gain of 1. The systematically wrong description of the experimental spectrum by
the theoretical function produced a slope which increases towards the endpoint energy. The right
panel shows the derivative of the ratio shown in the middle panel weighted with the square root of

82

the phase space factor. The average slope in the energy range of 1-2 MeV is about 0.25%/MeV,
which is about 40% of the value of C1, the dominant slope of the spectral function. Therefore a
0.1% systematic uncertainty of the gain produces a large spectral slope that has significantly effects
on the correct extractions of the observables.

Figure 6-1: Left panel: analytic β-energy spectrum obtained only from the phase space factor.
The theoretical spectrum (red) is obtained by stretching the experimental spectrum (blue) along
the horizontal axis by a factor of 1.1. Middle panel: the ratio between a theoretical spectrum (red)
with a stretch factor of 1.001 and the experimental spectrum (blue). Right panel: the slope of the
ratio shown in the middle panel weighted with the square root of the phase space factor.

6.2.1 General Procedure

The sensitivities of bWM and bGT to the energy-calibration parameters can be studied with Monte
Carlo simulations. As the shape of β-energy spectrum is dominated by the phase space factor, one
can neglect the theoretical corrections and the detector response for the moment, and estimate the
impacts of the energy-calibration parameters on the observables with the following function as an
approximation to the β-energy spectrum:

f(CH; N0, A, b) = pW(W0 − W)2 × f1(E, bWM) ×

E =

CH − B

A

,

where the spectrum is expressed in chan.

83

(cid:18)

(cid:19)

1 +

bGT
E

,

(6.4)

(6.5)

01234051015a.u.0123411.021.04Ratio012340.5−00.511.5 (%/MeV)WSElectron kinetic energy (MeV)WM = 68.22 and bGT = bSM

A Monte Carlo sample consisting of 200 β-energy spectra were generated with the SM values of
GT = 0. Each spectrum contains 106 events
the observables, i.e., bWM = bSM
distributed among 256 bins. The spectra were fitted by Eq. 6.4 in a channel range corresponding
to the energy range of 300-3200 keV. Depending on the observable of interest, the fit can be either
a “SM” fit with bWM as a free parameter and bGT fixed at bSM
, or a “BSM” fit with bGT as a free
GT
parameter and bWM fixed at bSM
. In both fits, there are two additional free parameters: the gain
WM
and an overall normalization factor.

The above is the general procedure to study the impacts of each energy-calibration parameter on
the physics observables. Depending on the effect of interest, the procedure differs when generating
or fitting the spectra.

6.2.2 Sensitivity to the Gain

The sensitivity to the gain was studied by generating β-energy spectra following Eq. 6.4 with
A = AMC = 1.5 chan/keV and B = BMC = 20 chan. The sample was fitted by Eq. 6.4 with the
gain fixed in the range of 1.495-1.505 chan/keV in steps of 0.001 chan/keV and the offset fixed at
B = BMC chan. For each fixed gain, an average bWM was extracted from the 200 SM fits, and an
average bGT was extracted from the 200 BSM fits. Figure 6-2 shows the results as a function of the
relative gain variation defined as

Ar =

A

AMC

− 1,

(6.6)

where A is the gain in the fit function. The extracted sensitivities of bWM and bGT to a 0.1%
variation of Ar are -32 and 1.4 × 10−2, respectively. This variation of bWM produces a slope of
-0.29%/MeV for C1, which is comparable to the estimates made in Sec. 6.2 with the analytic
spectra.

It is worth pointing out that it is the relative gain variation rather than the absolute gain variation
that is relevant to characterize the sensitivities of bWM and bGT. The Monte Carlo simulations
with a different AMC but the same range of values of Ar result in the same sensitivities of bWM and
bGT to Ar.

84

Figure 6-2: Left panel: the average bWM extracted from 200 Monte Carlo simulations and fits as a
function of the relative gain variation. The dashed line indicates the input value of bWM in the
simulations. Right panel: The average bGT extracted from 200 Monte Carlo simulations and fits
as a function of the relative gain variation. The dashed line indicates the input value of bGT in the
simulations.

6.2.3 Sensitivity to the Offset

The same Monte Carlo sample as above was used to study the sensitivity to the offset. These spectra
were fitted by Eq. 6.4 with the gain fixed at AMC and the offset fixed in the range of 15-25 chan and
in steps of 1 chan. For each fixed offset, an average bWM was extracted from the 200 SM fits, and
an average bGT was extracted from the 200 BSM fits. Figure 6-3 shows the results as a function of
the offset used in the fits. The extracted sensitivities of bWM and bGT to a 1-chan variation of B are
-12 and 5.7 × 10−2, respectively. It is worth pointing out that it is the absolute offset variation that
is relevant to characterize the sensitivities of bWM and bGT.

6.2.4 Sensitivity to the Nonlinearity

The sensitivity to the nonlinearity was also studied by generating β-energy spectra following Eq.
6.4 but with a quadratic response shown in Eq. 6.2, with A = AMC = 1.5 chan/keV, B = BMC = 20
chan, and P fixed in the range of 2 × 10−7 − 1 × 10−6chan/keV2 in steps of 1 × 10−7chan/keV2.
For each value of P, a sample of 200 spectra was generated. Each sample was fitted assuming a
linear response shown in Eq. 6.1 with the gain and the offset fixed at A = AMC and B = BMC,

85

4-3-2-1-012343-10·rA0100WMb4-3-2-1-012343-10·rA5-05 (%)GTbFigure 6-3: Left panel: the average bWM extracted from 200 Monte Carlo simulations and fits as a
function of the offset used in the fit. The dashed line indicates the input value of bWM in the
simulations. Right panel: the average bGT extracted from 200 Monte Carlo simulations and fits as
a function of the offset used in the fit. The dashed line indicates the input value of bGT in the
simulations.

respectively, and an average bWM was extracted from the 200 SM fits and an average bGT was
extracted from the 200 BSM fits. Figure 6-4 shows the results as a function of P. The extracted
sensitivities of bWM and bGT to a 1 × 10−7chan/keV2 variation of P are 3.5 and −1.3 × 10−3,
respectively.

Figure 6-4: Left panel: the average bWM extracted from 200 Monte Carlo simulations and fits as a
function of the input nonlinearity in the simulations. Right panel: the average bGT extracted from
200 Monte Carlo simulations and fits as a function of the input nonlinearity in the simulations.

86

14161820222426 (chan)B50100WMb14161820222426 (chan)B2-02 (%)GTb10-8-6-4-2-0)2 (chan/keV7 10· P3040506070WMb10-8-6-4-2-0)2 (chan/keV7 10· P0.5-0.00.51.01.5 (%))GTb6.2.5 Sensitivity to the Energy Resolution

The sensitivity to the width parameter σ0 which is associated with the energy resolution was studied
with Monte Carlo simulations. A β-energy spectrum following the phase space factor was generated
with Monte Carlo. The spectrum is convolved with a Gaussian distribution with a varying width
parameter in the range of 1-1.25 in steps of 0.025, producing ten convolved spectra. For each value
of σ0, a sample of 200 spectra was generated following the distribution that is a product of the

corresponding convolved spectrum and f1(E, bWM) ×(cid:16)1 + bGT

(cid:17), with A = AMC = 1.5 chan/keV

E

and B = BMC = 20 chan. Each sample was fitted by Eq. 6.4 with the gain and the offset fixed at
A = AMC and B = BMC, respectively, and an average bWM was extracted from the 200 SM fits and
an average bGT was extracted from the 200 BSM fits. Figure 6-5 shows the results as a function of
σ0. The extracted sensitivities of bWM and bGT to a 1% variation of σ0 are -0.24 and 2.0 × 10−4,
respectively.

Figure 6-5: Left panel: the average bWM extracted from 200 Monte Carlo simulations and fits as a
function of the width parameter in the simulations. Right panel: the average bGT extracted from
200 Monte Carlo simulations and fits as a function of the width parameter in the simulations.

6.2.6 Summary

The sensitivities of bWM and bGT to the energy-calibration parameters are summarized in Table
6-1, which is used to evaluate the systematic uncertainties of the observables associated with the

87

1.001.051.101.151.201.250s506070WMb1.001.051.101.151.201.250s0.00.51.0 (%)GTbenergy calibration.

Table 6-1: Sensitivities of bWM and bGT to the energy-calibration parameters

10−3 variation of Ar
1-chan variation of B
10−7chan/keV2 variation of P
10−2 variation of σ0

∆bWM ∆bGT × 103
-32
-12
3.88
-0.8

14
5.7
-1.4
0.4

6.3 Determination of the Gain with Auto-Calibration

From the sensitivity studies, it is clear that the energy calibration has a significant impact on
the systematic uncertainties of bWM and bGT. To meet the precision goal of O(10−3) for bGT, the
systematic uncertainty of the gain needs to be at the level of 10−4 chan/keV or lower, which is
extremely challenging to reach. Even if the gain could be determined with this accuracy, it is still
crucial to be able to monitor the gain during the measurement of the β-energy spectrum with the
same level of accuracy. A common way to calibrate scintillating detectors such as CsI(Na) and
NaI(Tl) is to use external γ sources. Such a procedure can be problematic to determine the gain
in this experiment. First of all, the CsI(Na) and the NaI(Tl) detector crystals respond differently
to γ and electrons; the former produces more photons than the latter, resulting in a higher gain.
Secondly, the gain determined from an external calibration source depends on the location of the
source relative to the detector, and it is impossible to place the γ source at the identical place
with the β particles emitting inside the calorimeter. Thirdly, a rate-dependent gain drift has been
observed, as discussed in Chap. 7.

All the above effects producing gain shifts can be eliminated by an auto-calibration technique,
which determines the gain and an observable (here either bWM or bGT) simultaneously by fitting the
β-energy spectrum. It is, however, possible that this procedure would induce a correlation between
the gain and the observable. The potential correlation effect, as well as the performance of the
auto-calibration technique, are studied in this section.

88

6.3.1 Fitting the Spectrum with the Gain as a Free Parameter

WM = 68.22, bGT = bSM

A similar general procedure to that in Sec. 6.2.1 was followed. A Monte Carlo sample consisting of
200 β-energy spectra was generated following Eq. 6.4, with bWM = bSM
GT = 0,
and B = BMC = 20. Each spectrum contains 106 events distributed among 256 bins. Here instead
of having a fixed value of AMC when generating the spectra, a randomly generated AMC in the range
of 1.5-1.8 chan/keV is applied to each spectrum. This range of gain covers a 20% relative variation
of the gain. The spectra were fitted by a Eq. 6.1 but with an additional normalization parameter.
These fits were performed in a channel range corresponding to an energy range of 300-3200 keV
and with the offset fixed at 20 chan. Three free parameters are included in each fit: a normalization
factor, the gain (A), and an observable (bWM or bGT ).

Figure 6-6 shows the values of the gain extracted from the SM fits. The results from the BSM
fits are similar, but the statistical uncertainties of the gain are about 30% smaller than those from
the SMs fit. The upper panel shows the gain extracted from the fits as a function of the random
input gain in the Monte Carlo. The lower panel shows the relative gain variation defined in Eq.
6.6. It is worth pointing out that the fluctuations of the values of Ar are not purely statistical, as
discussed in Sec. 6.3.3.

From the same fits, the values of the observables are extracted, as shown in Fig. 6-7, where
the dotted lines indicate the input values in the Monte Carlo. Remarkably, the fitted values of the
observables fluctuate around the input values without showing any correlation with the gain. The
absence of such a correlation is because the dominant contribution of the fit function is the phase
space factor, which is the decisive factor that determines the gain in the fit.

6.3.2 Statistical Uncertainties

As the auto-calibration technique extracts the observables and the gain from the same data, it is
interesting to study the statistical impacts of the technique. This study requires simulations with
a larger sample size. Following the same procedure as that in Sec. 6.3.1, the simulations and fits
were performed with 104 spectra in a sample. The distributions of the statistical uncertainties of

89

Figure 6-6: The gain extracted from fits (upper panel) and the relative gain variation multiplied by
1000 (lower panel) obtained from the SM fits as a function of input values of the gain used in the
simulations.

Figure 6-7: Left panel: values of bWM extracted from the SM fits with the gain as a free
parameter. The dotted line indicate the input value used in the Monte Carlo. Right panel: values
of bGT extracted from the BSM fits with the gain as a free parameter. The dotted line indicate the
input values used in the Monte Carlo.

90

1.51.61.71.8 (chan/keV)MCA2-023-10·3 10· rA1.51.61.71.81.51.61.71.8 (chan/keV)A1.51.61.71.8 (chan/keV)MCA0100200WMb1.51.61.71.8 (chan/keV)MCA5-05 (%)GTbthe gain are shown in Fig. 6-8, where the mean statistical uncertainties are 1.2× 10−3 chan/keV for
the SM fits and 0.9 × 10−3 for the BSM fits.

Figure 6-8: The distributions of the statistical uncertainties of the gain extracted from the SM fits
(left panel) and the BSM fits (right panel) to a sample of 104 simulated spectra.

The left panel of Fig. 6-9 shows the distribution of the bWM values extracted from the SM fits
to the sample. The blue histogram shows the distribution of bWM extracted from the fits with the
gain as a free parameter and has a standard deviation of 28.51. If we assume that the gain is exactly
known and is equal to the input values in the simulations, the gain can be fixed at AMC during
the fit. The resulting distribution of bWM extracted fits with fixed values of gain is shown by the
red histogram with a standard deviation of 16.74. Therefore leaving the gain as a free parameter
increases the statistical uncertainty of the mean bWM by a factor 1.7. The right panel of Fig. 6-9
shows the similar results extracted from the BSM fits to the sample. The standard deviations of
bGT with the free and the fixed gains are 1.25% and 0.98%, respectively. Leaving the gain as a free
parameter increases the statistical uncertainty of bGT by a factor 1.28.

6.3.3 Correlations

It is worth noticing from the lower panel of Fig. 6-6 and the left panel of Fig. 6-7, where the
results were extracted from the same fits, that the values of bWM have shown an anti-correlation
with the relative gain variation, despite the absence of the correlation between bWM and the gain

91

0.00.51.01.52.02.53-10· (chan/keV)As0.00.51.01.52.0310·counts/bin0.00.51.01.52.02.53-10· (chan/keV)As0123310·counts/binFigure 6-9: The distributions of the values of bWM (left panel) and bGT (right panel) extracted
from fits to a sample of 104 spectra, with the gain in the fits as a free parameter (blue) and fixed to
the input value (red).

value. Similar to Ar, we define a relative variation of bWM :

br =

− 1,

bWM
bSM
WM

(6.7)

where bWM is the extracted value from the fits. The scatter plot of br relative to Ar is shown in the
left panel of Fig. 6-10, where the results are extracted from fits to the sample of 104 spectra and
show an anti-correlation. The scatter plot of bGT relative to Ar is shown in the right panel, where
a correlation is observed.

Figure 6-10: The scatter plots of the values of br (left panel) and bGT (right panel) relative to Ar.
The results are extracted from fits to a sample of 104 spectra.

92

050100150WMb0100200300400counts/binFree gainFixed gain4-2-024 (%)GTb0100200300400counts/binFree gainFixed gain2-023-10·rA1-01rb2-023-10·rA5-05 (%)GTb6.4 Determination of the Offset

The offset was estimated with external γ sources. Here we include two γ-energy spectra of a
137Cs source measured with the CsI(Na) detector at a rate of 8170.8 ± 5.1 (cps), shown in the left
panel of Fig. 6-11, and a 60Co source measured with the same detector at 14153.4 ± 6.3 (cps),
shown in the right panel of Fig. 6-11. The salient features are a full-energy peak in the 137Cs
spectrum and two full-energy peaks in the 60Co spectrum. Since the full-energy peaks have known
values of energy, if the peak centroids in chan can be determined, the offset can then be extracted
from a fit, assuming a linear energy response. Therefore to extract an accurate offset, it is crucial
to determine the peak centroids accurately.

Figure 6-11: Left panel: the γ-energy spectrum of the 137Cs source measured with the CsI(Na)
detector at a rate of 8170.9 ± 5.1 (cps). Right panel: the γ-energy spectrum of the 60Co source
measured with the CsI(Na) detector at a rate of 14153.4 ± 6.3 (cps).

The shape of each full-energy peak in Fig. 6-11 can be described with a Gaussian function,
where the Gaussian mean (µ) and standard deviation (σE) represent the centroid and width of the
peak and can be extracted from a best fit. However, the presence of the Compton events under the
Gaussian peaks affects the fitted value of the centroid. This effect is the smallest when the ratio
between the number of Gaussian events and the number of Compton events is the highest. This
ratio can be varied by changing the energy range in the fit, as shown in Fig. 6-12 for the 137Cs peak
in the left panel of Fig. 6-11, where the light and dark bands represent two energy ranges. Each

93

0.00.51.01.5310·Photon energy (chan)0510310·counts/bin012345310·Photon energy (chan)0246310·counts/binband has a width of N chan, covering the same number of channels (N/2) above and below the
Gaussian centroid determined from a crude estimate. An increase of the band width would reflect
the decrease of the Gaussian to Compton ratio.

Figure 6-12: The γ-energy spectrum of the 137Cs source measured with the CsI(Na) detector (left
panel of Fig. 6-11) zoomed in near the peak. The light and blue bands are examples of two
different energy ranges for the fit.

The 137Cs γ peak shown in the left panel of Fig. 6-11 or 6-12 is fitted by a Gaussian function
over the ranges of different band widths. The band width varied from 0 to two Gaussian standard
deviations determined from a crude estimate. The extracted mean and standard deviation of the
peak as a function of the band width are shown in Figs. 6-13. For both the centroid and the width
of the peak, the fitted values show clear trends of either increasing or decreasing as the band width
increases. As the band width decreases, the fitted results converge to stable values with higher
accuracies because of the higher Gaussian to Compton ratio but lower precisions because of the
fewer number of events included in the fits.

As the band width decreases from two standard deviations to zero, both the centroid and the
width start to converge at a certain band width. This is the optimal band width as a result of a
compromise between the accuracy and the precision. For the 137Cs peak, the optimal band width
is 72 chan. Similar studies have been done for the two peaks in the 60Cs γ spectrum; the results
are shown in Fig. 6-14 and Fig. 6-15 for the left and the right peaks. The optimal band width is 98
chan for the left peak and 168 chan for the right peak.

94

0.91.01.11.21.31.4310·Photon energy (chan)0510310·counts/binFigure 6-13: The centroid (left panel) and width (right panel) of the peak in the 137Cs γ spectrum
(Fig. 6-12 or the left panel of Fig. 6-11) extracted from Gaussian fits as a function of the band
width.

Figure 6-14: The centroid (left panel) and width (right panel) of the left peak in the 60Co γ
spectrum (the right panel of Fig. 6-11) extracted from Gaussian fits as a function of the band
width.

95

050100150Band width (chan)1.1981.1991.2001.201310· (chan)m050100150Band width (chan)354045505560 (chan)s050100150Band width (chan)1.1981.1991.2001.201310· (chan)m050100150Band width (chan)354045505560 (chan)sFigure 6-15: The centroid (left panel) and width (right panel) of the right peak in the 60Co γ
spectrum (the right panel of Fig. 6-11) extracted from Gaussian fits as a function of the band
width.

Table 6-2 summarizes the fit results of the three peaks in 137Cs and the 60Co γ spectra. For
each peak, the fits were performed at the optimal band width determined above. The upper panel
of Fig. 6-16 shows the centroid as a function of the peak energy. A linear fit gives the following
values of gain and offset

A = 1.7860 ± 0.0003 chan/keV, B = 16.67 ± 0.32 chan

(6.8)

Table 6-2: The fit results of the 137Cs and 60Co γ peaks. The energy values in the first column are
adapted from the NuDat2.7 database [2].

µ (chan)

E (keV)
1198.31 ± 0.14
661.657(3)
1173.228(3) 2115.44 ± 0.25
1332.492(4) 2395.58 ± 0.15

σE (chan)
47.35 ± 0.36
68.28 ± 0.67
71.90 ± 0.25

Band width (chan)

72
98
168

Rate (cps)
8170.8 ± 5.1
14153.4 ± 6.3
14153.4 ± 6.3

It is worth emphasizing that the above calibration result is extracted from the 137Cs and 60Co γ
spectra measured at high rates of 8170.78 ± 5.14 (cps) and 14153.4 ± 6.3 (cps). It will be shown
in Chap. 7 that both the fitted centroid and the width of the peak are affected by the event rate.
Therefore rate corrections have to be made to the centroid before determining a final value of the
offset. More details about the rate effect are discussed in Chap. 7, where the γ calibration is
updated after the rate correction.

96

050100150Band width (chan)1.1981.1991.2001.201310· (chan)m050100150Band width (chan)354045505560 (chan)sFigure 6-16: Upper panel: the centroid of the γ peaks from 137Cs and 60Co as a function of the
peak energy. The red line is a linear fit to the graph. Lower panel: the residuals of the fit.

6.5 Determination of the Nonlinearity

The same data with those in Fig. 6.8 are fitted by a quadratic function shown in Eq. 6.2, where
A, B, and P are all free parameters in the fit. The result of the fit is shown in Fig. 6-17, where the
extracted value of the non-linearity is

P = (−5.04 ± 0.34) × 10−5 chan
keV2 .

(6.9)

Similar to the offset, since P is determined based on the values of the peak centroid, the rate
corrections to the centroid have to be included before determining a final value of P. An updated
value of P is provided in Chap. 7.

97

0.60.81.01.2310·Photon energy (chan)1.52.02.5310·Centroid (chan) / ndf 2c 215.5 / 1   B 0.3203– 16.97    A 0.0003056– 1.786  / ndf 2c 215.5 / 1   B 0.3203– 16.97    A 0.0003056– 1.786 0.60.81.01.2310·Photon energy (keV)5-051015ResidualsFigure 6-17: Upper panel: the centroid of the γ peaks from 137Cs and 60Co as a function of the
peak energy. The red line is a quadratic fit to the graph. Lower panel: the residuals of the fit.

6.6 Determination of the Energy Resolution

The fitted peak widths shown in Table 6-2 can be converted to keV using Eq. 6.8. The resulting
width in keV as a function of the peak energy is shown in Fig. 6-18. The graph is fitted by Eq. 6.3,
where σ0 is a free parameter. The extracted value of σ0 is

σ0 = 1.086 ± 0.003(keV)1/2

.

(6.10)

An updated value of σ0 is provided in Chap. 7 after both the fitted peak width and the γ calibration
are updated after the rate correction.

98

0.60.81.01.2310·Photon kinetic energy (chan)1.52.02.5310·Centroid (chan) / ndf 2c15 / 0- 5.418e   B 3.013–27.01 -    A 0.00678– 1.885    P06- 3.433e–05 -5.039e-  / ndf 2c15 / 0- 5.418e   B 3.013–27.01 -    A 0.00678– 1.885    P06- 3.433e–05 -5.039e- 0.60.81.01.2310·Photon energy (keV)60-40-20-09-10·ResidualsFigure 6-18: Upper panel: the width of the γ peaks from 137Cs and 60Co as a function of the peak
energy. The red line is the result of a fit by Eq. 6.3. Lower panel: the residuals of the fit.

99

0.60.81.01.2310·Photon energy (chan)25303540 (keV)Es / ndf 2c 78.08 / 2 0s 0.003327– 1.086  / ndf 2c 78.08 / 2 0s 0.003327– 1.086 0.60.81.01.2310·Photon energy (keV)5-0ResidualsCHAPTER 7

RATE EFFECT

The previous chapter has described the energy-calibration parameters without considering the effect
of the event rate. However, it was found that the event rate has a significant impact on the recorded
spectra. This chapter introduces the size and origin of the rate effect and applies corrections to the
energy-calibration parameters accordingly. The updated values of the parameters as well as their
uncertainties are provided.

7.1 Introduction

The potential rate effect in the energy calibration was monitored with a 137Cs γ source, whose
full-energy peak is described with a Gaussian distribution. The centroid and the width of the peak
were measured as a function of the rate with the same CsI(Na) detector described in Chapter 3.
The measurements were performed in two conditions: 1) with the 6He beam implanting into the
detector at the same time (on-line measurement), and 2) without the implanting 6He beam (off-line
measurement). Since the energy of the γ peak is a constant, any changes in the measured peak
centroid and peak width are due to the changes in the energy calibration.

7.1.1 On-Line Measurement

In one of the runs during the experiment, a 137Cs source was placed next to the CsI(Na) detector
while the detector was being implanted by the 6He beam. The recorded events include both the
β particles from 6He decay and the γ rays emitted by the 137Cs source. Figure 7-1 shows a
two-dimensional histogram of the time versus the deposited energy recorded during the beam-off
intervals. The main features of this two-dimensional histogram are similar to the one shown in the
left panel of Fig. 3-11 . In both cases, the number of events decreases as a function of time due
to the decay of 6He. A noticeable difference between these two two-dimensional histograms is the
presence of a line at about 1000 chan in Fig. 7-1, which is produced by the γ ray from the 137Cs

100

source.

Figure 7-1: Two-dimensional histogram of the time versus the deposited energy of 6He β particles
and 137Cs γ rays measured with the CsI(Na) detector recorded during the beam-off intervals

The effect of the rate on the centroid and the width of the γ peak can be studied with the
energy spectra projected from different time windows in the two-dimensional spectrum. Figure 7-2
shows the energy spectra projected from four time widows in this run. Each time window contains
approximately half a million events.

The energy spectrum corresponding to each time window was fitted by a function consisting of
a Gaussian and a β-energy spectrum. The Gaussian centroid extracted from the fits and the average
event rate in the time window are shown in each panel of Fig. 7-2. It is clear that as the event rate
decreases, both the centroid and the width of the peak decrease as well, indicating the apparent
changes to the gain related to the rate.

To quantify the effect of the rate on the peak centroid and width, the data in Fig.7-1 were divided
into 49 time windows, and from each, a projected energy spectrum was obtained. Similar fits to
the spectra have been performed and the extracted centroid as a function of the average rate in the
time window is shown in the left panel of Fig. 7-3. The peak centroid appears to increase linearly
as a function of the rate. The data were fitted by the function

µ(r) = aµ × r + bµ,

(7.1)

101

Figure 7-2: The measured energy spectrum of β particles in 6He decay and γ rays emitted by
137Cs projected from four time windows in Fig. 7-1 . The uncertainties associated with the fitted
centroid and width are statistical.

where r represents the average rate in a time window, µ(r) represents the centroid of the peak in
chan at rate r, and bµ is the centroid at zero rate. The fitted slope aµ is 2.3 × 10−3 chan/cps.
Similarly, the peak width appears to increase linearly as a function of the rate, as shown in the right
panel of Fig. 7-3. The data were fitted by the function

σ(r) = aσ × r + bσ,

(7.2)

where σ(r) represents the width of the peak in chan at rate r, and bσ is the width at zero rate. The
fitted slope aσ is 4.2 × 10−4 chan/cps.

102

0123456310·Energy (chan)0510310·counts/binTime window: 343.00 - 370.00 (sec)Average rate = 18078 (cps)Centroid = 1002.41 +/- 0.59(chan)Width = 43.99 +/- 0.56(chan)0123456310·Energy (chan)0510310·counts/binTime window: 371.00 - 407.00 (sec)Average rate = 14128 (cps)Centroid = 993.69 +/- 0.45(chan)Width = 42.93 +/- 0.43(chan)0123456310·Energy (chan)051015310·counts/binTime window: 408.00 - 461.00 (sec)Average rate = 10106 (cps)Centroid = 983.85 +/- 0.32(chan)Width = 41.04 +/- 0.30(chan)0123456310·Energy (chan)0102030310·counts/binTime window: 462.00 - 563.00 (sec)Average rate = 5981 (cps)Centroid = 974.79 +/- 0.20(chan)Width = 39.61 +/- 0.18(chan)Figure 7-3: The centroid (left panel) and the width (right panel) of the 137Cs γ peak as a function
of the average rate in a time window from the on-line measurement.

7.1.2 Off-Line Measurements

The energy of the 137Cs γ peak was also measured at different rates without the 6He beam
implanting the CsI(Na) detector. The rate was varied by changing the distance of the 137Cs source
from the CsI(Na) crystal. Five measurements were performed with rates at approximately 2, 5, 10,
20, and 40 kcps. The γ peak in each spectrum was fitted with a Gaussian function. The extracted
centroid and width of the peak as functions of the event rate are plotted in Fig. 7-4. Similar to
the on-line measurement, both the centroid and the width increase linearly with the rate, where the
fitted slopes are aµ is 1.458(2) × 10−3 chan/cps and aσ is 1.27(4) × 10−4 chan/cps, respectively.

7.1.3 Origin of the Rate Effect

A possible explanation of the origin of the observed rate effect is the after-glow or phosphorescence
in the crystal. The remaining scintillation yield as a fraction of the initial primary scintillation was
measured at different times after the deposition of a γ source of 1.5(±0.2) MeV in a CsI(Na) and
a CsI(Tl) detector, as shown in Fig. 7-5 [18]. It is clear that the scintillation is still present 10−2
seconds after the interaction. In the measurement of the β-energy spectrum, the remaining after-
glow scintillation introduces a pile-up effect. As the event rate increases, more scintillation photons
corresponding to the previous events pile up to those corresponding to the current event, causing

103

5101520310·Average rate (cps)0.981.00310·Centroid (chan) / ndf 2c 55.89 / 47  mb 0.2141– 961.6   ma05- 2.9e– 0.002272  / ndf 2c 55.89 / 47  mb 0.2141– 961.6   ma05- 2.9e– 0.002272 5101520310·Average rate (cps)404550Width (chan) / ndf 2c 51.66 / 47 sb 0.1872– 36.99  sa05- 2.66e– 0.0004159  / ndf 2c 51.66 / 47 sb 0.1872– 36.99  sa05- 2.66e– 0.0004159 Figure 7-4: The centroid (left panel) and the width (right panel) of the 137Cs γ peak as a function
of the average rate in a time window from the off-line measurement

an apparent increase of the observed gain, which explains the increase of the γ peak centroid as a
function of rate.

Figure 7-5: The remaining scintillation yield as a fraction of the initial primary scintillation
measured at different times after the deposition of a 1.5(±0.2) MeV γ source in the CsI(Na) and
the CsI(Tl) detectors. Figure taken from Ref. [18].

104

010203040310·Rate (cps)1.581.601.62310·Centroid (chan) / ndf 2c  1449 / 3  mb 0.0408–  1564   ma06- 2.343e– 0.001458  / ndf 2c  1449 / 3  mb 0.0408–  1564   ma06- 2.343e– 0.001458 010203040310·Rate (cps)606264Width (chan) / ndf 2c 40.57 / 3 sb 0.04641– 59.21  sa06- 3.56e– 0.0001268  / ndf 2c 40.57 / 3 sb 0.04641– 59.21  sa06- 3.56e– 0.0001268 7.2 The Impact of the Rate on the Gain

Section 6.3 has discussed the disadvantages of the common calibration method using external γ
sources. One of the disadvantages is the rate-dependent gain drift, which would introduce a serious
consequence in this experiment. In the on-line measurement shown in Fig. 7-3, the centroid of
the γ peak increased by a factor of 3.6% from the lowest rate to the highest, indicating a 3.6%
uncertainty of the gain if the γ source were used to determine the gain. According to Table 6-1,
this variation of the gain introduces a -1152 variation of bWM and a 0.504 variation of bGT. Such
huge systematic uncertainties would make it impossible to extract any useful information of the
observables. Therefore the calibration using external γ sources cannot be applied in this analysis.
In the actual analysis, it is shown in Chap. 9 that each one-hour run is divided into six time
windows to separate events in different rate ranges. For a given time window, the correct gain
corresponds to this particular set of data with the unique amount of after-glow pile-up determined
by the history of the crystal. With the auto-calibration method described in Sec. 6.3, the gain
corresponding to each time window is extracted from the data themselves, eliminating a potential
rate effect on the gain similar to that observed in γ source calibrations.

7.3 The Impact of the Rate on the Offset

7.3.1 Rate Correction to the Offset

The offset was previously estimated in Chap. 6 with the 137Cs and 60Co γ sources. That estimate
did not take into account the rate effect on the peak centroid used in the calibration fit shown in
Fig. 6.8. Assuming the offset is independent of the event rate, the correct offset is extracted by
extrapolating the centroid to zero rate and redoing the fit.

The dependence of the centroid of the 137Cs peak on the rate shown in Eq. 7.1 is generalized

for a peak with any energy E:

(7.3)
where µE(r) is the centroid of a γ peak of energy E measured at a rate r, µE(0) is the expected

µ × r + µE(0),

µE(r) = aE

105

centroid of the same peak at zero rate, and aE
µ

is related to the fitted slope in the left panel Fig. 7-4:

aE
µ

=

E
E0

aµ,

(7.4)

where E0 is the energy of the 137Cs γ peak. Substituting Eq. 7.4 into Eq. 7.3, the expected centroid
of a γ peak of energy E at zero rate has the expression

µE(0) = µE(r) − E
E0

aµ × r.

(7.5)

Using the values of µE(r) and the rate in Table 6-2, and the value of aµ from off-line measurements,
the extrapolated centroid at zero rate as a function of energy is shown in Fig. 7-6. A linear fit gives
a new calibration:

A = 1.7410 ± 0.0003 chan/keV, B = 34.52 ± 0.32 chan.

(7.6)

By comparison, with the peak centroids at non-zero rates shown in Fig. 6-16, the extracted
offset is 16.97± 0.32 chan. Compared with the new offset after the rate corrections to the centroid,
the offset without rate corrections is underestimated by 18 chan. According to Table 6-1, this
variation of the offset introduces a -216 variation of bWM and a 0.103 variation of bGT. Such
high systematic uncertainties would make it impossible to extract any useful information of the
observables. Therefore it is crucial to perform rate corrections to the γ centroid when determining
the offset.

7.3.2 Uncertainty of the Offset

The uncertainty of the offset originates from three sources.

• δB1 = 0.32 is the uncertainty in the calibration fit shown in Fig. 7-6, which is related to the

error of the peak centroid at the non-zero rate shown in Table 6-2.

• δB2 is the uncertainty related to the error of the fitted aµ in Fig. 7-4. By replacing the aµ
in Eq. 7.5 by aµ − δaµ and aµ + δaµ, two sets of corrected centroids are calculated and are
used to determine B. The resulting two offsets are considered the upper and lower limits of

106

Figure 7-6: Upper panel: the corrected centroid of the γ peaks from 137Cs and 60Co as a function
of the peak energy. The red line is a linear fit to the graph. Lower panel: the residuals of the fit.

the offset related to the error of aµ. Their difference divided by two is an estimate of δB2 of
0.03 chan.

• δB3 is the uncertainty related to the rate of the online measurement, which is extracted from
the same time windows with those produced Fig. 7-3. The β and γ energy spectra projected
from each time window is fitted with an offset shown in Eq. 7.6, producing a reconstructed
energy of the 137Cs γ peak. The distribution of the reconstructed γ energy from these time
windows is shown in Fig. 7-7. A Gaussian fit to the distribution gives a mean γ energy of
674.9 ± 0.4 keV, where δE0 = 0.4 is considered the uncertainty of the γ energy associated
with the rate. Similarly, a different mean γ energy is obtained with a different offset (e.g. ,

107

0.60.81.01.2310·Photon energy (chan)1.52.0310·Centroid (chan) / ndf 2c 52.94 / 1   B 0.3203– 34.52    A 0.0003056– 1.741  / ndf 2c 52.94 / 1   B 0.3203– 34.52    A 0.0003056– 1.741 0.60.81.01.2310·Photon energy (keV)05Residualsfrom Eq. 6.8). These results are used to calculate dB/dE0, which, multiplied with δE0 = 0.4,
gives the an estimate of δB3 of 0.7 chan.

Figure 7-7: The distribution of the reconstructed energy of the 137Cs γ peak extracted from the
on-line measurement with the offset value taken from Eq. 7.6.

To summarize, the uncertainty of the offset is

(cid:113)(δB1)2 + (δB2)2 + (δB3)2 = 0.77 chan.

δB =

(7.7)

According to Table 6-1, the projected systematic uncertainties of bWM and bGT are 9.24 and
4 × 10−3, respectively.

7.4 The Impact of the Rate on the Nonlinearity

The non-linearity was previously estimated in Chapter 6 by fitting the centroid of the 137Cs and
the 60Coγ peaks with a quadratic function, which did not take into account the rate effect on the
peak centroid. With the corrected centroid calculated in Sec. 7.3.1, a new fit with the quadratic
function can be performed. The result of the fit is shown in Fig. 7-8, where the extracted value of
the non-linearity is

P = (−2.5 ± 0.3) × 10−5 chan
keV2 .

(7.8)

Compared with the P extracted without the rate corrections shown in Eq. 6.9, the updated P has
reduced by a factor of two.

108

 / ndf 2c 1.766 / 4Constant  2.70– 14.77 Mean      0.4– 674.9 Sigma     0.296– 2.564 660670680690Photon Energy (keV)051015counts/bin / ndf 2c 1.766 / 4Constant  2.70– 14.77 Mean      0.4– 674.9 Sigma     0.296– 2.564 The result in Fig. 7-8 can also be compared with Fig. 7-6, where the same data were fitted by
a linear function instead. It is worth noticing that the residuals in Fig. 7-8 are significantly larger
than those in Fig. 7-6, which indicates that a linear function describes these data better than a
quadratic function. Therefore it is safe to assume a linear response between channel and energy in
the analysis and consider P to be zero.

Figure 7-8: Upper panel: the corrected centroid of the γ peaks from 137Cs and 60Co as a function
of the peak energy. The red line is a quadratic fit to the graph. Lower panel: the residuals of the fit.

109

0.60.81.01.2310·Photon kinetic energy (chan)1.52.0310·Centroid (chan) / ndf 2c09 / 0- 1.401e   B 3.013– 12.73    A 0.00678–  1.79    P06- 3.433e–05 -2.498e-  / ndf 2c09 / 0- 1.401e   B 3.013– 12.73    A 0.00678–  1.79    P06- 3.433e–05 -2.498e- 0.60.81.01.2310·Photon energy (keV)40-20-06-10·Residuals7.5 The Impact of the Rate on σ0

7.5.1 Rate Correction to the Width

In Chaps. 5 and 6, the width of the peak as a function of energy is described:

√

E,

σE = σ0

(7.9)

where σ0 is estimated in Chapter 6 with the 137Cs and the 60Co γ sources, which did not take into
account the rate effect on the peak width.

The dependence of the width of the 137Cs peak on the rate shown in Eq. 7.2 is generalized for

a peak with any energy E:

(7.10)
where σE(r) is the width of a γ peak of energy E measured at a rate r, σE(0) is the expected width
of the same peak at zero rate, and aE
σ

is related to the fitted slope in the right panel Fig. 7-4:

σE(r) = aE

σ × r + σE(0),

(cid:115)

(cid:115)

aE
σ

=

E
E0

aσ,

(7.11)

where E0 is the energy of the 137Cs γ peak. Substituting Eq. 7.11 into Eq. 7.10, the expected
width of a γ peak of energy E at zero rate has the expression

σE(0) = σE(r) −

aσ × r.

E
E0

(7.12)

Using the values of σE(r) and the rate in Table 6-2, and the value of aσ extracted from the
off-line measurements, the extrapolated width at zero rate as a function of the γ energy is calculated
and converted to keV according to the calibration relation shown in Eq. 7.6. The new width in keV
at zero rate as a function of energy in keV is shown in Fig. 7-9. A fit by Eq. 6.3 gives the new
value of σ0:

σ0 = 1.077 ± 0.003(keV)1/2

.

(7.13)

By comparison, with the peak width at non-zero rates shown in Fig. 6-18, the extracted σ0 is
1.086 ± 0.003 keV1/2. Compared with the new value of σ0 after the rate corrections to the width

110

Figure 7-9: Upper panel: the corrected Gaussian width (keV) of the peaks in the 137Cs and the
60Co γ spectra as a function of peak energy. The red line is the result of a fit by Eq. 6.3. Lower
panel: the residuals of the fit.

shown in Eq. 7-9, the σ0 without rate corrections is overestimated by 0.01 keV1/2. According to
Table 6-1, this variation of σ0 introduces a -0.74 variation for bWM and a 3 × 10−5 variation for
bGT. The rate effect on σ0 has significantly smaller impacts on the physics observables than the
rate effect on the offset does.

The σ0 can also be extracted from a different set of calibration measurements, including a 40K
peak at 1461 keV, a 214Pb peak at 1764 keV, a 208Tl peak at 2615 keV, and the two 60Co peaks.
The peak width in keV as a function of the γ energy is shown in Fig. 7-9, where the widths are not
corrected based on the event rate since the rates in these measurement are low. A fit by Eq. 6.3

111

0.60.81.01.2310·Photon energy (chan)25303540 (keV)Es / ndf 2c 48.24 / 2 0s 0.003412– 1.077  / ndf 2c 48.24 / 2 0s 0.003412– 1.077 0.60.81.01.2310·Photon energy (keV)5-0Residualsgives a value of σ0:

σ0 = 1.063 ± 0.003(keV)1/2

.

(7.14)

Figure 7-10: Upper panel: the uncorrected Gaussian width (keV) of the peaks in 40K, 214Pb,
208Tl and 60Co γ spectra as a function of the peak energy. The red line is the result of a fit by Eq.
6.3. Lower panel: the residuals of the fit.

Compared with the new value of σ0 after the rate corrections shown in Eq. 7.13, this σ0 without
rate corrections is underestimated by 0.01 keV1/2. Combing the results in Eqs. 6.10, 7.13, and
7.14, for the γ peaks in the rate range of 0-14 kcps, the upper limit of the rate effect on σ0 is 0.01
keV1/2.

112

1.52.02.5310·Photon energy (chan)404550 (keV)Es / ndf 2c 9.791 / 4 0s 0.003328– 1.063  / ndf 2c 9.791 / 4 0s 0.003328– 1.063 1.52.02.5310·Photon energy (keV)2-1-012Residuals7.5.2 σ0 in the Analysis

In the analysis of the β-energy spectrum in 6He decay, the event rate varies in a wider range of 0-20
kcps. As mentioned in Sec. 7.2, each one-hour run is divided into six time windows to analyze data
in different rate ranges separately. The average rate of r for each time window is used to calculate
a new σ0 corresponding to rate r. Specifically, the values of σE(r) for the three γ peaks at rate r
are calculated with Eq. 7.10 and converted to keV according to the auto-calibration to the β-energy
spectrum. The new width in keV at rate r as a function of energy in keV is fitted by Eq. 6.3 to
extract the σ0 at rate r. For each time window, the absorbed spectrum from Geant4 is convolved
with a σ0 corresponding to the average rate of the time window.

7.5.3 Uncertainty of the σ0

The uncertainty of σ0 originates from three sources.

• δσ01 = 3 × 10−3 keV1/2 is the uncertainty in the fit shown in Fig. 7-9, which is related to
the error of the width in chan at the non-zero rate shown in Table 6-2.

• δσ02 is the uncertainty related to the error of the fitted aσ in Fig. 7-4. By replacing the aσ
in Eq. 7.12 by aσ − δaσ and aσ + δaσ, two sets of corrected widths are calculated and are
used to determine σ0. The resulting two values of σ0 are considered the upper and lower
limits of the σ0 related to the error of aσ. Their difference divided by two is an estimate of
δσ02 of 5 × 10−5 keV1/2.

• δσ03 is the uncertainty related to the rate variation within a time window since an average
rate of r is used to describe a range of rate in the time window. The average rate variation
within a time window can be calculated theoretically assuming an exponential decay. For
each bin of the time window in the exponential decay spectrum, the deviation of the rate
in this bin from r is calculated. The average rate variation within the time window is the
weighted average of these deviations, where the weight is the square root of the number of
events in each bin. The calculated average rate variation is 0.0023r.

113

A typical r in the first time window (beginning of the decay) is 18.5 kcps (Fig. 9-15), resulting
in an average rate variation of 0.04 kcps. Since a much wider rate variation of 0-14 kcps for
the γ peaks produced a 0.01 keV1/2 variation of σ0, the effect of a 0.04 kcps rate variation on
σ0 is much smaller than 0.01 keV1/2. The effect is even smaller in subsequent time windows
because of their narrower rate ranges. We use 0.01 keV1/2 as a very conservative estimate of
δσ03.

To summarize, the upper limit of uncertainty of σr is

(cid:113)(δσ01)2 + (δσ02)2 + (δσ03)2 = 0.01 keV1/2

δσ0 =

.

(7.15)

According to Table 6-1, the projected systematic uncertainties of bWM and bGT are 0.8 and 4×10−4,
respectively.

114

CHAPTER 8

FITTING THE β-ENERGY SPECTRUM

The previous chapters have determined an accurate description of the measured β-energy spectrum
in 6He decay, from which a function is constructed to fit the measured spectrum, where the observ-
ables, bWM and bGT, are extracted. This chapter determines an acceptable fit procedure including
three parts: constructing the fit function, finding a robust minimization tool, and determining a fit
condition that introduces no bias to the observables. The software tool used in this chapter is ROOT
[95], which is an object-oriented data analysis framework for data storage, statistical analysis, and
visualization developed at CERN.

8.1 Fit Function

8.1.1 The Expression of the Fit Function

The fit function includes the convolved spectrum in keV described in Chaps. 5, 6, and 7, an energy
calibration that transforms the spectrum in keV to a spectrum in chan described in Chaps. 6 and 7,
the contributions from the observables described in Eq. 1.26: a spectral function associated with
the bWM, and a factor associated with bGT. The expression of the fit function is

(cid:18)

(cid:19)

f(CH; N0, A, b) = N0 × f 0
CH − B

E =

,

A

c (E) × f1(E, bWM) ×

1 +

bGT
E

,

(8.1)

(8.2)

c (E) is a function representing the shape of the convolved spectrum. The expression of
where f 0
the spectral function is shown in Eq. 1.27. Equation 8.2 takes into account the energy calibration,
where E is the kinetic energy of a β particle in keV, B is the offset determined in Chap. 7, and A is
the gain determined during the fit with the auto-calibration described in Chap. 6.

Similar to the Monte Carlo studies and fits in Chapter 6, the fit function f(CH; N0, A, b) has
one variable, CH, which represents the channel number in the β-energy spectrum, and three free

115

parameters, N0 for the overall normalization, A for the gain, and b representing either bWM in the
SM fit or bGT in the BSM fit.

8.1.2 Constructing a Function from a Histogram

c (E) that represents the
The construction of the fit function requires constructing the function f 0
c (E) is
shape of the convolved spectrum. As the convolved spectrum is described by a histogram, f 0
not an analytic function. Three methods have been explored to construct a non-analytic fit function
from a histogram. The quality of each method is evaluated based on the consistency among the
statistical uncertainties of the fitted gain.

WM = 68.22, bGT = bSM

For each method, a Monte Carlo sample consisting of 100 β-energy spectra were generated
following Eq. 8.1, with N0 = 1, bWM = bSM
GT = 0, B = 20 chan, and A=AMC,
which a random number in the range of 1.5-1.8 chan/keV. Each spectrum (referred as the Monte
Carlo histogram) contains 106 events distributed among 256 bins. The spectra were then fitted
c (E) was constructed from a histogram using this method. N0, A, and an
by Eq. 8.1, where f 0
observable (bWM for SM fit or bGT for BSM fit) are the free parameters in the fit and B is fixed at
20 chan.

For the simplicity of the study, the f 0

c (E) in Eq. 8.1 was replaced by the phase space spectrum
of 6He decay as an approximation. When generating the spectra, the analytic expression of the
phase space spectrum was used; whereas when fitting the spectra, a histogram (refereed as the
function histogram) generated by Monte Carlo following the phase space factor with 109 events
distributed among 16384 bins was used. This makes the sample identical for the three methods,
c (E) constructed with each method, the only
leaving the fit function to the sample, specifically, f 0
difference in this study. The above approximation is valid because a robust fit method determined
with one fit function remains robust after introducing a small variation to the fit function. The
implementation of each method and the results from the fits are described below. The robustness of
the fits is evaluated based on the fraction of the converged fits ( fc), the fraction of fits with p-values
above 0.05 ( fp), and the consistency among the statistical uncertainties of the fitted gain.

116

8.1.2.1 GetBinContent

c (E) is the number of events in the bin corresponding to energy E,
The most direct estimate to f 0
which can be acquired with the GetBinContent method in the TH1 class in ROOT. From the 100
SM fits with this method, fc = 28% and fp = 10%. The fit results with p-values above 0.05 as a
function of the input gain in the Monte Carlo is shown in Fig. 8-1, where both bWM (left panel) and
the relative gain variation Ar (right panel) from the fits show inconsistent statistical errors among
the fits. From the BSM fits, fc = 13% and fp = 5%. The fit results with p-values above 0.05 are
shown in Fig. 8-2, where similar inconsistencies among the statistical errors are observed.

Figure 8-1: Left panel: values of bWM extracted from SM fits using GetBinContent. The dotted
line indicates the input value used in the Monte Carlo. Right panel: values of Ar calculated with
A extracted from the same fits.

The above discrepancy among the statistical errors originates from the statistical nature of
a histogram. As the number of events in a bin and in its neighbor bins is subject to statistical
fluctuations, the constructed fit function is not smooth. Figure 8-3 shows a typical SM fit to a
β-energy spectrum generated by Monte Carlo zoomed in near the top of the spectrum. The fit
function shows a zig-zag structure, causing inconsistent statistical errors among fits.

8.1.2.2 Linear Interpolation

c (E) using the
To improve the smoothness of the fit function, when estimating the value of f 0
bin content, one should take into account not only the exact bin corresponding to energy E, but

117

1.51.61.7 (chan/keV)MCA50100WMb1.51.61.7 (chan/keV)MCA0.5-0.00.53 10· rAFigure 8-2: Left panel: values of bGT extracted from BSM fits using GetBinContent. The dotted
line indicates the input value used in the Monte Carlo. Right panel: values of Ar calculated with
A extracted from the same fits.

Figure 8-3: A typical SM fit to a β-energy spectrum generated by Monte Carlo. The fit uses the
c (E). The spectrum is zoomed in
number of events in the bin corresponding to E to estimate f 0
near the top to show the details of the structure.

also the neighbor bins. One approach is to perform a linear interpolation with the Interpolate
method in the TH1 class in ROOT. Given an energy E, Interpolate looks for the bin number
c (E) by an interpolation based on the content of the two
(i) corresponding to E, and estimates f 0
nearest bins (i − 1 and i + 1).

With f 0

c (E) constructed using Interpolate, the new fit function results in fc = 99% and
fp = 96% from the 100 SM fits. The extracted bWM from fits with p-values above 0.05 is shown in
the left panel Fig. 8-4 with mostly consistent statistical errors among the fits. The distribution of
these statistical errors is plotted in the right panel, where the majority of the statistical errors are

118

1.51.61.71.8 (chan/keV)MCA024 (%)GTb1.51.61.71.8 (chan/keV)MCA0.2-0.00.23 10· rA2.53.0310·Electron kinetic energy (chan)17.518.018.5310·counts/bin-energy spectrumbFitting functionconsistent while there are a few fits that produced significantly lower statistical errors. From the
same fits, the calculated Ar from the fitted A has a similar behavior, as shown in Fig. 8-5, where
the left panel shows the values of Ar and the right panel shows the statistical errors of A. From the
BSM fits, fc = 100% and fp = 92%. The results from fits with p-values above 0.05 are shown in
Figs. 8-6 and 8-7, where both bGT and A have mostly consistent statistical errors among the fits
but with a few fits resulting in lower statistical errors.

Figure 8-4: Left panel: values of bWM extracted from SM fits using Interpolate. The dotted
line indicates the input value used in the Monte Carlo. Right panel: distribution of the statistical
errors of bWM from the same fits.

Figure 8-5: Left panel: values of Ar extracted from SM fits using Interpolate. Right panel:
distribution of the statistical errors of A from the same fits.

The above results indicate that the Interpolate method is robust with close to 100% converged
fits, but the statistical errors of the parameters extracted from the fits are not completely consistent.

119

1.51.61.71.8 (chan/keV)MCA50-050100150WMb203040WMbs01020counts/bin1.51.61.71.8 (chan/keV)MCA2-023-10·3 10· rA0.00.51.01.52.02.53-10· (chan/keV)As0510counts/binFigure 8-6: Left panel: values of bGT extracted from BSM fits using Interpolate. The dotted
line indicates the input value used in the Monte Carlo. Right panel: distribution of the statistical
errors of bGT from the same fits.

Figure 8-7: Left panel: values of Ar extracted from BSM fits using Interpolate. Right panel:
distribution of the statistical errors of A from the same fits.

The possible reason is that the linear interpolation is not sufficient to make the fit function smooth
enough, and a higher-order interpolation is required.

8.1.2.3 Spline Interpolation

To further improve the smoothness of the fit function, spline interpolations have been explored
c (E). The TSpline class in ROOT is capable of performing third (TSpline3) and
to estimate f 0
fifth (TSpline5) spline interpolations. For a given a histogram, a spline object can be created that
describes the shape of the histogram, and it acts as a smooth function. The value of the spline

120

1.51.61.71.8 (chan/keV)MCA5-05 (%)GTb81012143-10·GTbs0510counts/bin1.51.61.71.8 (chan/keV)MCA2-1-0123-10·3 10· rA0.00.51.01.52.02.53-10· (chan/keV)As051015counts/binobject at E, acquired with an Eval method, provides a smooth estimate of f 0

c (E).

With f 0

c (E) constructed using Tspline3, the new fit function results in fc = 100% and
fp = 98% from the 100 SM fits. The results extracted from fits with p-values above 0.05 are
shown in Fig. 8-8 for bWM and Fig. 8-9 for Ar, where the statistical errors of both bWM and A
are consistent. From the BSM fits, fc = 100% and fp = 89%. The result extracted from fits with
p-values above 0.05 are shown in Fig. 8-10 for bGT and Fig. 8-11 for Ar, where both bGT and A
show consistent statistical errors among the fits.

Figure 8-8: Left panel: values of bWM extracted from SM fits using Tspline3. The dotted line
indicates the input value used in the Monte Carlo. Right panel: distribution of the statistical errors
of bWM from the same fits.

Figure 8-9: Left panel: values of Ar extracted from SM fits using Tspline3. Right panel:
distribution of the statistical errors of A from the same fits.

121

1.51.61.71.8 (chan/keV)MCA0100200WMb203040WMbs01020counts/bin1.51.61.71.8 (chan/keV)MCA2-023-10·3 10· rA0.00.51.01.52.02.53-10· (chan/keV)As05101520counts/binFigure 8-10: Left panel: values of bGT extracted from BSM fits using Tspline3. The dotted line
indicates the input value used in the Monte Carlo. Right panel: distribution of the statistical errors
of bGT from the same fits.

Figure 8-11: Left panel: values of Ar extracted from BSM fits using Tspline3. Right panel:
distribution of the statistical errors of A from the same fits.

To compare the shape of the fit function in Fig. 8-3 where GetBinContent was used, Fig.
8-12 shows a SM fit to the same β-energy spectrum as that shown in Fig. 8-3 but with a spline
interpolation. The spectrum zoomed in near the top, a much smoother fit function is seen.

It is clear that among the three methods studied above, the spline interpolation, which gives
consistent statistical errors of the fitted parameters and a fc of 100%, is a correct approach to
c (E) from a histogram. For all three methods, fp is higher for the SM fits than for
construct f 0
BSM fits. The methods and conditions used in the fit with the spline interpolation are taken as a
reference in the rest of the chapter. Each of the rest of the studies was performed with one method

122

1.51.61.71.8 (chan/keV)MCA5-05 (%)GTb81012143-10·GTbs0510counts/bin1.51.61.71.8 (chan/keV)MCA2-1-0123-10·3 10· rA0.00.51.01.52.02.53-10· (chan/keV)As051015counts/binFigure 8-12: A typical SM fit to a β-energy spectrum generated by Monte Carlo. The fit uses a
c (E). The spectrum is zoomed in near the top to show the
spline interpolation at E to estimate f 0
details of the structure.

or condition different from this reference.

8.2 Fit Procedure

With a correct fit function constructed, one can use the TH1:Fit method to fit the data. It is

important to determine a robust fit procedure, i.e., the fit method and the minimization tool.

8.2.1

2 and Log-Likelihood

χ

The available fit methods in the ROOT::Fit::Fitter class include:
least-square fit, binned
likelihood fit, un-binned likelihood fit, and linear fit, which all require a minimization process.
2) and the binned likelihood (refereed as log-likelihood
This chapter has studied the least-square ( χ
2 fits.
below) fits. The results shown previously in Figs. 8-8, 8-9, 8-10, and 8-11 are from the χ
Similar fits are performed but with the log-likelihood method. The results for SM fits are similar to
Figs. 8-8 and 8-9, with fc = 100% and fp = 93%, and the results for BSM fits are similar to Figs.
8-10 and 8-11, with fc = 100% and fp = 92%. In both SM and BSM fits, consistent statistical
errors of both the observables and the fitted gain are observed. Therefore under this fit condition,
the χ

2 and the log-likelihood methods have similar performance in terms of the robustness.

123

2.53.0310·Electron kinetic energy (chan)17.518.018.5310·counts/bin-energy spectrumbFitting function8.2.2 Minimizer

2 and the log-likelihood fits, a few minimizer
For the minimization process required by the χ
libraries are available in ROOT. The ROOT website [96] recommends the use of Minuit (TMinuit
or Minuit2) for multi-dimensional minimizations. Since the fit function in Eq. 8.1 has three free
parameters, Minuit is used for minimizations in this study.

Here we have studied the TMinuit and the Minuit2 fits. The results shown in Figs. 8-8,
8-9, 8-10, and 8-11 are from Minuit2 fits. Similar fits can be performed but with the TMinuit
method. The results for SM fits are similar to Figs. 8-8 and 8-9 and the results for BSM fits are
similar to Figs. 8-10 and 8-11. Both SM and BSM fits resulted in fc = 100% and fp = 91% and
consistent statistical errors of both the observables and the fitted gain were observed. Therefore in
this study, TMinuit and the Minuit2 minimization methods have similar performances in terms
of the robustness.

8.3 Fit Condition

The fit results depend on the properties of the histograms, such as the statistics and binning. In
the previous fits, 106 events were generated for each Monte Carlo spectrum. Both the Monte Carlo
histogram and the function histogram had 16384 initial bins and were rebinned by a factor of 64
before the fit. This section studies the impacts of the statistics and the binning on the fit results,
from which we have determined acceptable fit conditions, i.e., the conditions that result in a high
fc, a high fp, and a bias smaller than one standard deviation. The bias of a variable X is defined as

Bias =

X − Xi
σX

,

(8.3)

where X is the mean value of X extracted from the fits, Xi is the input value of X in the Monte
Carlo, and σX

is the standard deviation of the fitted values.

124

8.3.1 Number of Events

To study the impact of the number of events (Ne) in each Monte Carlo spectrum on the results,
similar simulations and fits to those in Sec. 8.1.2.3 were performed but with the different number of
2 method and Minuit2 minimizer
events generated in each spectrum. The results fitted with the χ
are shown as a function of Ne in Figs. 8-13 and 8-14 for the SM and the BSM fits, respectively.
The left panels of both figures show the results for fc, where most fits to the spectra with more
than 104 events have acceptable convergence performances. However, fp highly depends on Ne,
as shown in the middle panels, where 105 and 106 number of events produce the highest fp. The
absolute value of the bias decreases as a function of Ne in both SM and BSM fits, as shown in the
right panels. The values of Ne that result in both high fp and bias under one standard deviation in
the SM and the BSM fits are 105 and 106.

Figure 8-13: Results from SM fits with the χ
fits as a function of the number of events in each Monte Carlo histogram. Middle panel: the
fraction of fits with p-values above 0.05 as a function of the number of events. Right panel: the
bias as a function of the number of events.

2 method. Left panel: the fraction of the converged

Similar results fitted with the log-likelihood method are shown in Figs. 8-15 and 8-16 for the
SM fits and the BSM fits, respectively. The left panels of both figures show that the fits to spectra
with more than 105 events have acceptable convergence performances. Similar to the results with
2 method, fp highly depends on Ne, as shown in the middle panels, with 105 and 106 number
the χ
of events producing the highest fp. The absolute value of the bias decreases as a function of Ne
in the BSM fits but shows no trend in the SM fits. The value of Ne resulting in both high fp and
values of bias under one standard deviation in the SM and the BSM fits is 106.

125

310410510610710810eN0.20.40.60.81.0cf310410510610710810eN0.00.51.0pf310410510610710eN1234Bias2 method. Left panel: the fraction of the converged fits
Figure 8-14: Results from BSM fits with χ
as a function of the number of events in each Monte Carlo histogram. Middle panel: the fraction
of fits with p-values above 0.05 as a function of the number of events. Right panel: the bias as a
function of the number of events.

Figure 8-15: Results from SM fits with the log-likelihood method. Left panel: the fraction of the
converged fits as a function of the number of events in each Monte Carlo histogram. Middle
panel: the fraction of fits with p-values above 0.05 as a function of the number of events. Right
panel: the bias as a function of the number of events.

Figure 8-16: Results from BSM fits with log-likelihood method. Left panel: the fraction of the
converged fits as a function of the number of events in each Monte Carlo histogram. Middle
panel: the fraction of fits with p-values above 0.05 as a function of the number of events. Right
panel: the bias as a function of the number of events.

126

310410510610710810eN0.20.40.60.81.0cf310410510610710810eN0.00.51.0pf310410510610710eN4-2-0Bias310410510610710810eN0.900.951.00cf310410510610710810eN0.00.51.0pf310410510610710eN2-02Bias310410510610710810eN0.60.81.0cf310410510610710810eN0.20.40.60.81.0pf310410510610710eN012BiasFrom the above results of the χ

2 and the log-likelihood fits, the optimal number of events for

the fitted histogram is 106 when the function histogram has a statistics of 109 events.

8.3.2 Number of Bins

The numbers of bins for both the Monte Carlo histogram and the function histogram have impacts
on the quality of the fit. These impacts are studied with similar simulations and fits to those in Sec.
8.1.2.3 but with different combinations of the rebin factor of the function spectrum (RF) and the
rebin factor of the Monte Carlo histogram (RMC) . Here both histograms have an initial number of
bins equal to 16384.

For the SM fits with χ

2, the fc and fp results are show as a function of the log2 RMC and
log2 RF in Fig. 8-17. he result of fp suggests that the (RF, RMC) pairs that satisfy RF ≥ RMC are
preferred. Therefore for the bias results shown in Fig. 8-18, only binning combinations satisfying
RF ≥ RMC are of interest. The left panel of Fig. 8-18 shows the bias from the SM fits with χ
2, and
the right panel is a zoomed-in view of the left graph. Most of the (RF, RMC) pairs on the diagonal
have biases lower than one standard deviation. For the BSM, similar results to the SM fits were
observed. The combinations of RF and RMC that satisfy RF ≥ RMC are preferred based on the fc
and fp results, and most of the (RF, RMC) pairs on the diagonal have biases lower than one standard
deviation.

For the log-likelihood fits, similar results to the χ

2 fits were observed. The results for the SM
fits with log-likelihood are shown in Figs. 8-19 and 8-20, where combinations of RF and RMC that
satisfy RF ≥ RMC are preferred based on the fc and fp results, and most of the (RF, RMC) pairs on
the diagonal have biases lower than one standard deviation. Similar results were observed in the
BSM fits with log-likelihood.

8.4 Determined Fit Condition

To summarize, in this study, both χ

2 and log-likelihood fits are acceptable fit methods, and
both TMinuit and Minuit2 are acceptable minimizers. For high fc and fp, 105 and 106 events

127

2 as a function of the function
Figure 8-17: Left panel: the fraction of convergent SM fits with χ
spectrum rebin factor (RF) and the Monte Carlo histogram rebin factor (RMC). Right panel: the
fraction of the fits with p-values over 0.05 as a function of the function spectrum rebin factor (RF)
and the Monte Carlo histogram rebin factor (RMC).

2 as a function of the
Figure 8-18: Left panel: the bias of bWM extracted from SM fits with χ
function spectrum rebin factor (RF) and the Monte Carlo histogram rebin factor (RMC). Right
panel: zoomed view of the left panel.

128

0.70.80.91.0Cf02468)MCR(2log02468)FR(2log0.00.20.40.60.81.0Pf02468)MCR(2log02468)FR(2log0510Bias02468)MCR(2log02468)FR(2log3-2-1-01Bias45678)MCR(2log45678)FR(2logFigure 8-19: Left panel: the fraction of convergent SM fits with log-likelihood as a function of the
function spectrum rebin factor (RF) and the Monte Carlo histogram rebin factor (RMC). Right
panel: the fraction of the fits with p-values over 0.05 as a function of the function spectrum rebin
factor (RF) and the Monte Carlo histogram rebin factor (RMC).

Figure 8-20: Left panel: the bias of bWM extracted from SM fits with log-likelihood as a function
of the function spectrum rebin factor (RF) and the Monte Carlo histogram rebin factor (RMC).
Right panel: zoomed view of the left panel.

129

0.20.40.60.81.0cf012345678)MCR(2log012345678)FR(2log0.00.20.40.60.8pf012345678)MCR(2log012345678)FR(2log5-05Bias02468)MCR(2log02468)FR(2log4-3-2-1-0123Bias345678)MCR(2log345678)FR(2login the Monte Carlo histogram are preferred. Under the same condition with that in Sec. 8.1.2.3,
histograms with 106 events can consistently be fitted with less than one standard deviation. Finally,
the same rebin factor for the function histogram and the Monte Carlo histogram is recommended,
and the most reliable choices of (RF, RMC) pairs are (32, 32) and (64,64), assuming both the
function histogram and the Monte Carlo histogram have 16384 initial bins before rebinning.

8.5 Pile-Up Contribution to the Fit Function

In the experimental spectrum, in addition to the expected β-decay events, a pile-up spectrum is
introduced by the digital DAQ system. As seen in the right panel of Fig. 3-11, the events above the
endpoint energy are pile-up events. Therefore the pile-up spectrum is also included when fitting the
experimental data. The function used to fit the experimental data including the pile-up spectrum
has the expression

(cid:18)

(cid:19)

f(CH; N0, A, b)PL = N0 × f 0
CH − B

c (E) × f1(E, bWM) ×

1 +

bGT
E

+ N1 × fpl(E),

(8.4)

,

A

E =

(8.5)
where fpl(E) is a function corresponding to the shape of the pile-up spectrum. The shape of a pile-
c (E) using the parameters and algorithms
up spectrum was calculated according to the shape of f 0
in the digital DAQ system. Figure 4-2 shows the shape of the pile-up spectrum (black) calculated
based on the shape of the convolved spectrum (green). The event rate has no impact on the shape
pl(E) but affects the normalization factor N1, or the weight of the pile-up spectrum. With the
of f 0
above four-parameter function the experimental spectrum shown in the right panel of Fig. 3-11 is
2 to the number of degree of freedom ratio from the fit
fitted and shown in Fig. 8-21, where the χ
equal to 0.98. The measured β-energy spectrum in 6He decay can now be understood over than
range of four orders of magnitude.

8.6 Complete Fit Function

Equation 8.4 is the fit function used in the current analysis and is, in fact, an approximation
form of a complete fit function since it ignored the effects of the detector simulation response and

130

Figure 8-21: A β-energy spectrum in 6He decay measured in the CsI(Na) detector (the right panel
of Fig. 3-11) fitted by Eq. 8.4.

the energy-resolution convolution on the two factors associated with the observables shown in Eq.

1.26: the spectral function and the(cid:16)1 + bGT

(cid:17) factor. A complete fit function will be used in future

W

analysis, which has the expression

f(CH; N0, A, b) =N0 × pW(W0 − W)2

η(W)dW × f1(E, bWM) ×

+ N1 × fpl(E),
CH − B

,

A

E =

(cid:19)

(cid:18)

1 +

bGT
W

(cid:126) Fc(E)

(8.6)

(8.7)

where Fc(E) is the detector response function including the correction based on the detector
simulations and the convolution with the Gaussian width. The above expression can be rewritten

131

0123456110210310410510counts/bin0123456Electron kinetic energy (MeV)3−2−1−0123Residualsc

(E) + bGTC−1 f −2
c (E) + (C−1 + bGT) f −1

c

c

(E)(cid:3) + N1 fpl(E),
(E)(cid:3) + N1 fpl(E),

by substituting f1(E, bWM) with Eq. 1.27.

f(CH; N0, A, b) =N0(cid:2)(1 + C0 + bGTC1) f 0

c (E) + C1 f 1

c (E)

≈N0(cid:2)(1 + C0) f 0

+ (C−1 + bGT + bGTC0) f −1
c (E) + C1 f 1
η(W)dW (cid:126) Fc(E),
η(W)WdW (cid:126) Fc(E),
η(W)dW
(cid:126) Fc(E),
η(W)dW
(cid:126) Fc(E),

c (E) =pW(W0 − W)2
f 0
c (E) =pW(W0 − W)2
f 1
pW(W0 − W)2
f −1
(E) =
c
W
pW(W0 − W)2
f −2
(E) =
W2
c

(8.8)

(8.9)

(8.10)

(8.11)

(8.12)

(8.13)

f −1
c

c (E),

c (E),
f 1

(E), and f −2
(E) are the convolved spectra, i.e., the absorbed spectra
where f 0
c
η(W)dW
η(W)dW, pW(W0−W)2
simulated with Geant4 with pW(W0−W)2
,
and pW(W0−W)2
as the source spectra that are convolved with the Gaussian width. The
W2
complete function also has four free parameters but is based on four convolved spectra and a pile-up
spectrum. Therefore more Geant4 simulations will be performed to complete the analysis.

η(W)WdW, pW(W0−W)2

η(W)dW

W

132

CHAPTER 9

BACKGROUND STUDY

In addition to the β-decay events, the experimental data also include background events. This
chapter focuses on eliminating the impact of the background on the shape of the measured β-
energy spectrum. The first half of this chapter studies the origin of the background, estimates
the size of the background contribution, and provides a solution to include the background in the
analysis. The second half of the chapter studies the corrections to the background through both
Monte Carlo simulations and the analysis of the experimental data, based on which the impacts of
the background corrections on the physics observables are estimated. The systematic uncertainties
of the observables associated with the background are provided.

9.1 Background in the Analysis

9.1.1 Background Component

The background originates from both the beam-induced background and the ambient background.
Figure 9-1 shows the decay spectrum from the run shown in Fig. 3-11, where the events in the left
panel of Fig. 3-11 are projected to the time axis. The dominant feature of the decay spectrum is an
exponential decay of the number of events as a function of time. Towards the end of the cycle (12.5
s), the number of events converges to a constant with the events dominated by the background.
The red and blue dashed lines indicate the levels of the beam-induced background and the ambient
background measured in dedicated background runs.

The ambient background was measured with the CsI(Na) detector for twelve hours. The
resulting energy spectrum is shown in Fig. 9-2, where the γ peaks of 40K, 214Bi, 214Bi, and 208Tl
are observed. The beam-induced background originates from the nuclear reactions between the
beam and the detector. Immediately after stopping the beam, the beam-induced background was
measured for eighteen minutes. The energy spectrum measured with the CsI(Na) detector is shown

133

Figure 9-1: The 6He decay spectrum recorded during the beam-off interval. The data is collected
during a single one-hour run. The red and blue dashed lines indicate the levels of the
beam-induced background and the ambient background.

in Fig. 9-3, where the peaks are from the decays of the reaction products. Theoretical calculations
have predicted that the reaction products are dominated by 124Cs, 126Cs, 122I, and 128Cs.

Figure 9-2: The energy spectrum of the ambient background measured with the CsI(Na) detector
with peak identifications.

134

4681012Time (s)310410510Counts/bin0246310×Energy (chan)110210310410counts/bin40K (1461 keV)214Bi (1764 keV)214Bi (2204 keV)208Tl (2615 keV)Figure 9-3: The energy spectrum of the beam-induced background measured with the CsI(Na)
detector

9.1.2 Background Subtraction

The contribution of the background is eliminated by subtracting the background spectrum from
the measured signal+background (S+B) spectrum that includes both the β-decay events and the
background events. As the true background contribution to the S+B spectrum is unknown, the
energy spectrum near the end of the decay cycle, dominated by background events, is taken as an
estimate to the true background spectrum. As shown in the left panel of Fig. 9-4, the time interval
of 3.43-3.61 s is selected as the S+B window, during which the measured spectrum is shown in
the upper right panel of Fig. 9-4 in black; and the time interval of 9.43-9.61 s is selected as the
background window, during which the measured spectrum is shown in the upper right panel of
Fig. 9-4 in blue. The background spectrum is subtracted from the S+B spectrum, resulting in the
spectrum shown in the lower right panel of Fig. 9-4.

For each one-hour run like Fig. 9-1, six pairs of S+B and background windows are defined.
When defining the time windows, we require the event rate to be below a maximum acceptable
rate rmax, and the number of events after the background subtraction to be about 106 to satisfy
the fit condition determined in Chap. 8. For each pair of time windows, the background window
starts six seconds after the S+B window starts, and the two windows have the same width. The
first S+B time window in a run starts at the first bin with an event rate below rmax, and stops at the

135

0123456310·Energy (chan)1-10110210310Counts/binFigure 9-4: Left panel: an example of selected S+B and background time windows. The time
interval of 3.43-3.61 s (left vertical band) is selected as the S+B window, and time interval of
9.43-9.61 s (right vertical band) is selected as the background window. Right panel: the energy
spectra measured with the CsI(Na) detector during 3.43-3.61 s (black) and 9.43-9.61 s (blue)
corresponding to the two time windows marked in the left panel.

first bin where the background-subtracted spectrum has over 106 events. Each subsequent signal
window starts at the next bin after the end of its previous window and stops at the bin where the
background-subtracted spectrum has over 106 events.

The deviation of the estimated background spectrum from the true background spectrum in-
troduces a distortion to the background-subtracted spectrum and has a potential impact on the
extracted observables. This deviation originates from the differences in the gain and the sigma (the
Gaussian width) between the true and the estimated background spectra. Due to the after-glow pile
up discussed in Chap. 7, the estimated background at low rate has a lower gain and a lower sigma
than the true background at a high rate does.

136

4681012Time (s)310410510Counts/bin01234567310·Energy (chan)0.00.20.40.60.81.0310·counts/binS+B window: 343.00 - 361.00 (sec)S+B rate = 18529 (cps)B window: 943.00 - 961.00 (sec)B rate = 497 (cps)0246310·Energy (chan)0.00.20.40.60.81.0310·counts/binFigure 9-5 shows the background-subtracted spectra from the run shown in Fig. 7-2, with the
S+B spectra from a high-rate time window and the background spectra from a low-rate time window.
The structure near 1000 chan shows that the events in the background γ peak are populated at lower
channels than the events in the S+B γ peak, indicating lower gains in the background spectra
than in the S+B spectra. The presence of this structure indicates that the background has not
been successfully subtracted.
In order to successfully subtract the background, gain-and-sigma
corrections were applied to the estimated background spectra, producing corrected background
spectra with the same gain and sigma with the S+B spectra. The methods of these background
corrections are first demonstrated with Monte Carlo simulations.

Figure 9-5: The background-subtracted spectrum for the β spectra with an extra 137Cs γ peak,
where the S+B spectra were projected from four time windows.

137

0246310·Energy (chan)0200400600counts/binS+B window: 343.00 - 370.00 (sec)S+B rate = 18078 (cps)B window: 943.00 - 970.00 (sec)B rate = 1505 (cps)0246310·Energy (chan)0200400600counts/binS+B window: 371.00 - 407.00 (sec)S+B rate = 14128 (cps)B window: 971.00 - 1007.00 (sec)B rate = 1478 (cps)0246310·Energy (chan)0200400600counts/binS+B window: 408.00 - 461.00 (sec)S+B rate = 10106 (cps)B window: 1008.00 - 1061.00 (sec)B rate = 1444 (cps)0246310·Energy (chan)0200400600counts/binS+B window: 462.00 - 563.00 (sec)S+B rate = 5981 (cps)B window: 1062.00 - 1163.00 (sec)B rate = 1419 (cps)9.2 Corrections to the Background: Monte Carlo

In the Monte Carlo study, instead of generating β-energy spectra only, energy spectra similar to
Fig. 7-2 were generated in order to use the Gaussian peak to evaluate the quality of the background
subtraction. The S+B spectrum, shown in the upper panel of Fig. 9-6 in black, was generated with
a higher gain and a higher sigma than the background spectrum (blue) shown underneath the S+B
spectrum.

Figure 9-6: Upper panel: the S+B (black) and background (blue) spectra generated with Monte
Carlo. The S+B spectrum has a higher gain and a higher sigma than the background spectrum.
Lower panel: the spectrum after background subtraction.

The S+B spectrum is a sum of a β-energy spectrum with 1.05× 106 events and a Gaussian peak
with 104 events. The β-energy spectrum was generated following the distribution of a convolved

138

012345678310·Energy (chan)00.511.522.5310·counts/bin012345678310·Energy (chan)0.00.51.01.52.02.5310·counts/bin  spectrum. The convolved spectrum is from the convolution of the absorbed spectrum from Geant4
with a Gaussian whose centroid is E and width is ησ0
E, where E is the energy in keV, and η is
a factor larger than 1. The Gaussian peak was generated following the distribution of a Gaussian
whose centroid is the energy of the 137Cs peak in keV and width is ησ0. The S+B spectrum was
generated in chan with a gain of AH.

√

The background spectrum is a sum of a β-energy spectrum with 4 × 104 events and a Gaussian
peak with 104 events. The β-energy spectrum was generated following the distribution of a
convolved spectrum. The convolved spectrum is from the convolution of the absorbed spectrum
E. The Gaussian peak was
from Geant4 with a Gaussian whose centroid is E and width is σ0
generated following the distribution of a Gaussian whose centroid is the energy of the 137Cs peak
in keV and width is σ0. The background spectrum was generated in chan with a gain of AL that is
smaller than AH and was subtracted from the S+B spectrum, producing the spectrum shown in the
lower panel of Fig. 9-6. The structure near 1000 chan indicates that the background is not correctly
subtracted.

√

9.2.1 Background Gain Correction

Since the background gain is underestimated, we have performed corrections to the background to
match its gain to that of the S+B spectrum. A gain-corrected background spectrum is generated
following the distribution of the uncorrected background shown in the upper panel of Fig. 9-6 with
an additional gain factor of AH/AL, resulting in the same gain with the S+B spectrum.

The left panel of Fig. 9-7 shows the S+B spectrum (black), the uncorrected background
spectrum (blue), and the gain-corrected background spectrum (red) zoomed in near the Gaussian
peak. The Gaussian peak is shifted to higher channels after the gain correction. The right panel
of Fig. 9-7 shows the full view of the S+B spectrum and the gain-corrected background spectrum
in the upper panel, and the new background-subtracted spectrum in the lower panel. The structure
near 1000 chan is still present because the S+B and background spectra have different sigmas, but
is reduced compared with Fig. 9-6 because the S+B and background spectra have the same gain.

139

Figure 9-7: Left panel: zoomed-in view of the S+B (black), uncorrected background (blue), and
gain-corrected background (red) spectra generated with Monte Carlo. Right panel: the full view
of S+B (black) and gain-corrected background (red) spectra in the upper panel, and the new
background-subtracted spectrum in the lower panel.

9.2.2 Background Gain-and-Sigma Correction

Since the background sigma is underestimated, we have performed corrections to the background
to match its sigma to that of the S+B spectrum. A gain-and-sigma corrected background spectrum
is generated following the distribution of the gain-corrected background shown in the upper panel
of Fig. 9-7, with an additional convolution with a Gaussian whose centroid is energy in chan and

√

E, resulting in the same sigma with the S+B spectrum.

width is(cid:112)

2 − 1AH σ0

η

The left panel of Fig. 9-8 shows the S+B spectrum (black), the uncorrected background
spectrum (blue), and the gain-and-sigma corrected background spectrum (red) zoomed in near the

140

00.511.522.53310·Energy (chan)210310counts/bin  012345678310·Energy (chan)00.511.522.5310·counts/bin  012345678310·Energy (chan)00.511.522.5310·counts/bin  Gaussian peak. The Gaussian peak is stretched wider after the sigma correction. The right panel
of Fig. 9-8 shows the full view of the S+B and the gain-and-sigma corrected background spectra
in the upper panel, and the new background-subtracted spectrum in the lower panel. The structure
near 1000 chan has disappeared because the S+B and background spectra have the same gain and
sigma, and the background is fully subtracted.

Figure 9-8: Left panel: zoomed-in view of the S+B (black), uncorrected background (blue), and
gain-and-sigma corrected background (red) spectra generated with Monte Carlo. Right panel: the
full view of S+B (black) and gain-and-sigma corrected background (red) spectra in the upper
panel, and the new background-subtracted spectrum in the lower panel.

141

00.511.522.53310·Energy (chan)210310counts/bin  012345678310·Energy (chan)00.511.522.5310·counts/bin  012345678310·Energy (chan)00.511.522.5310·counts/bin  9.3 Corrections to the Background: Experiment

The gain+sigma background correction procedure successfully demonstrated with Monte Carlo
simulations are applied to the experimental data. We have first performed background corrections
to the same experimental data shown in Figs. 9-5 and 7-2 in order to use the Gaussian peak
to evaluate the quality of background subtraction. These corrections were then applied to the
measured β-energy spectra from the run shown in Fig. 3-11.

9.3.1 Corrections to the β + γ Spectrum

The background-subtracted spectra shown in Fig. 9-5 were calculated with the S+B (black) and
background (blue) spectra shown in Fig. 9-9 below. gain-and-sigma corrections to these background
spectra have been performed. For the convenience of future comparisons, a zoomed-in view of
Figs. 9-5 is provided in Figs. 9-10.

9.3.1.1 Gain Correction

Similar gain correction method to that in Sec. 9.2.1 has been applied to the background spectra
shown in Fig. 9-9. The four S+B spectra in Fig. 9-9 were first fitted to extract the S+B gain
(AH) of each time window. AH as a function of the rate is then used to extrapolate the background
gain (AL) in each time window. A gain-corrected background spectrum is generated following the
distribution of the uncorrected background shown in Fig. 9-9 with an additional gain factor of
AH/AL, resulting in the same gain with the S+B spectrum.

Figure 9-11 shows the S+B spectrum (black), the uncorrected background spectrum (blue), and
the gain-corrected background spectrum (red) zoomed in near the Gaussian peaks. The Gaussian
peaks were shifted to higher channels after the gain corrections. Figure 9-12 shows the new
background-subtracted spectra. The structure near 1000 chan is still present because the S+B and
background spectra have different sigmas, but are significantly reduced compared with Fig. 9-10
because the S+B and background spectra have the same gain.

142

Figure 9-9: The S+B (black) and background (blue) spectra for the β spectra with an extra 137Cs
γ peak.

9.3.1.2 Gain-and-Sigma Correction

Similar sigma correction method to that in Sec. 9.2.2 has been applied to the gain-corrected
background spectra shown in Fig. 9-11. The four S+B spectra shown in Fig. 9-9 were first fitted
to extract the S+B sigma for the 611 keV peak (σH) in each time window. σH as a function
of the rate is used to extrapolate the background sigma for the 611 keV peak (σL) in each time
window. The gain-and-sigma corrected background spectra were then generated following the
distributions of the gain-corrected background spectra shown in Fig. 9-11, with an additional
AH,
convolution with a Gaussian whose centroid is energy in chan and width is
where σE
H

(cid:112)E/661, and σE

(cid:112)E/661.

(cid:17)2 −(cid:16)

σE
H

σE
H

= σH

= σL

L

(cid:114)(cid:16)

(cid:17)2

143

012345310·Energy (chan)0200400counts/binS+B window: 343.00 - 370.00 (sec)S+B rate = 18078 (cps)B window: 943.00 - 970.00 (sec)B rate = 1505 (cps)012345310·Energy (chan)0100200300400counts/binS+B window: 371.00 - 407.00 (sec)S+B rate = 14128 (cps)B window: 971.00 - 1007.00 (sec)B rate = 1478 (cps)012345310·Energy (chan)0200400600counts/binS+B window: 408.00 - 461.00 (sec)S+B rate = 10106 (cps)B window: 1008.00 - 1061.00 (sec)B rate = 1444 (cps)012345310·Energy (chan)0.00.51.0310·counts/binS+B window: 462.00 - 563.00 (sec)S+B rate = 5981 (cps)B window: 1062.00 - 1163.00 (sec)B rate = 1419 (cps)Figure 9-10: Zoomed-in view of the background-subtracted spectra for the β spectra with an extra
137Cs γ peak (Fig. 9-5).

Figure 9-13 shows the S+B spectrum (black), the uncorrected background spectrum (blue),
and the gain-and-sigma corrected background spectrum (red) zoomed in near the Gaussian peaks.
The Gaussian peaks are stretched wider after the sigma corrections. Figure 9-14 shows the new
background-subtracted spectra. The structure near 1000 chan is still present and has not significantly
reduced compared with Fig. 9-12, which indicates that the sigma correction has a much lower
impact on the background shape than the gain correction. The background spectra are not fully
corrected because the gain and the sigma extracted by fitting the S+B spectra are not accurate due
to the presence of background in the S+B spectra.

144

0.00.51.01.52.0310·Energy (chan)0200400600counts/binS+B window: 343.00 - 370.00 (sec)S+B rate = 18078 (cps)B window: 943.00 - 970.00 (sec)B rate = 1505 (cps)0.00.51.01.52.0310·Energy (chan)0200400600counts/binS+B window: 371.00 - 407.00 (sec)S+B rate = 14128 (cps)B window: 971.00 - 1007.00 (sec)B rate = 1478 (cps)0.00.51.01.52.0310·Energy (chan)0200400600counts/binS+B window: 408.00 - 461.00 (sec)S+B rate = 10106 (cps)B window: 1008.00 - 1061.00 (sec)B rate = 1444 (cps)0.00.51.01.52.0310·Energy (chan)0200400600counts/binS+B window: 462.00 - 563.00 (sec)S+B rate = 5981 (cps)B window: 1062.00 - 1163.00 (sec)B rate = 1419 (cps)Figure 9-11: The S+B (black), background (blue), and gain-corrected background (red) spectra
for the β spectrum with an extra 137Cs γ peak.

9.3.2 Corrections to the β Spectrum

For the β-energy spectra from Fig. 3-11, the S+B (black) and background (blue) spectra are shown
in Fig. 9-15. The resulting background-subtracted spectra are shown in Fig. 9-16, where a structure
near 400 chan is observed. We have performed corrections to these background spectra. Since the
gain in the β-energy spectra can be determined with high accuracy, the background spectra here
are expected to be better corrected than the β+γ background spectra.

145

0.00.20.40.60.81.01.21.41.61.82.0310·Energy (chan)0200400600counts/binS+B window: 343.00 - 370.00 (sec)S+B rate = 18078 (cps)B window: 943.00 - 970.00 (sec)B rate = 1505 (cps)0.00.20.40.60.81.01.21.41.61.82.0310·Energy (chan)0200400600counts/binS+B window: 371.00 - 407.00 (sec)S+B rate = 14128 (cps)B window: 971.00 - 1007.00 (sec)B rate = 1478 (cps)0.00.20.40.60.81.01.21.41.61.82.0310·Energy (chan)0.00.20.40.60.81.0310·counts/binS+B window: 408.00 - 461.00 (sec)S+B rate = 10106 (cps)B window: 1008.00 - 1061.00 (sec)B rate = 1444 (cps)0.00.20.40.60.81.01.21.41.61.82.0310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 462.00 - 563.00 (sec)S+B rate = 5981 (cps)B window: 1062.00 - 1163.00 (sec)B rate = 1419 (cps)Figure 9-12: Zoomed-in view of the new background-subtracted spectra for the β spectra with an
extra 137Cs γ peak after gain corrections to the background spectra.

9.3.2.1 Gain Correction

A similar gain correction method to that in Sec. 9.3.1.1 has been applied to the background spectra
shown in Fig. 9-15. The only difference is that the values of the S+B gain (AH) in each time
window were extracted from the background-subtracted spectra shown in Fig. 9-16 instead of the
S+B spectra.

Figure 9-17 shows the S+B spectra (black), the uncorrected background spectra (blue), and
the gain-corrected background spectra (red) zoomed in near the structure near 400 chan. The
background peaks in the structure were shifted to higher channels after the gain corrections. Figure
9-18 shows the new background-subtracted spectra. The structure near 400 chan is significantly

146

0.00.51.01.52.0310·Energy (chan)0200400600counts/binS+B window: 343.00 - 370.00 (sec)S+B rate = 18078 (cps)B window: 943.00 - 970.00 (sec)B rate = 1505 (cps)0.00.51.01.52.0310·Energy (chan)0200400600counts/binS+B window: 371.00 - 407.00 (sec)S+B rate = 14128 (cps)B window: 971.00 - 1007.00 (sec)B rate = 1478 (cps)0.00.51.01.52.0310·Energy (chan)0200400600counts/binS+B window: 408.00 - 461.00 (sec)S+B rate = 10106 (cps)B window: 1008.00 - 1061.00 (sec)B rate = 1444 (cps)0.00.51.01.52.0310·Energy (chan)0200400600counts/binS+B window: 462.00 - 563.00 (sec)S+B rate = 5981 (cps)B window: 1062.00 - 1163.00 (sec)B rate = 1419 (cps)Figure 9-13: The S+B (black), background (blue), and gain-and-sigma corrected background
(red) spectra for the β spectra with an extra 137Cs γ peak.

reduced after the gain correction to the background. The background appears to be fully subtracted.

9.3.2.2 Gain-and-Sigma Correction

The same sigma correction method with that in Sec. 9.2.2 has been applied to the gain-corrected
background spectra shown in Fig. 9-17 using the same σH versus rate relation with that in Sec.
9.2.2. Figure 9-19 shows the S+B spectra (black), the uncorrected background spectra (blue), and
the gain-and-sigma corrected background spectra (red) zoomed in near the background peaks. The
background peaks are stretched wider after the sigma corrections. Figure 9-20 shows the new
background-subtracted spectra. The background again seems fully subtracted. The structure near

147

0.00.20.40.60.81.01.21.41.61.82.0310·Energy (chan)0200400600counts/binS+B window: 343.00 - 370.00 (sec)S+B rate = 18078 (cps)B window: 943.00 - 970.00 (sec)B rate = 1505 (cps)0.00.20.40.60.81.01.21.41.61.82.0310·Energy (chan)0200400600counts/binS+B window: 371.00 - 407.00 (sec)S+B rate = 14128 (cps)B window: 971.00 - 1007.00 (sec)B rate = 1478 (cps)0.00.20.40.60.81.01.21.41.61.82.0310·Energy (chan)0.00.20.40.60.81.0310·counts/binS+B window: 408.00 - 461.00 (sec)S+B rate = 10106 (cps)B window: 1008.00 - 1061.00 (sec)B rate = 1444 (cps)0.00.20.40.60.81.01.21.41.61.82.0310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 462.00 - 563.00 (sec)S+B rate = 5981 (cps)B window: 1062.00 - 1163.00 (sec)B rate = 1419 (cps)Figure 9-14: Zoomed-in view of the new background-subtracted spectra for the β spectra with an
extra 137Cs γ peak after gain-and-sigma corrections to the background spectra.

400 chan has no significant change compared with Fig. 9-18, indicating a much lower impact
of the sigma correction on the background shape than the gain correction, which agrees with the
observation in Sec. 9.2.2.

9.4 The Impact of the Background

The impact of the background corrections on the observables have been estimated. The CsI(Na)
dataset in Table 3-1 were fitted to extract bWM and bGT under the three background conditions
shown in Table 9-1.

148

0.00.51.01.52.0310·Energy (chan)0200400600counts/binS+B window: 343.00 - 370.00 (sec)S+B rate = 18078 (cps)B window: 943.00 - 970.00 (sec)B rate = 1505 (cps)0.00.51.01.52.0310·Energy (chan)0200400600counts/binS+B window: 371.00 - 407.00 (sec)S+B rate = 14128 (cps)B window: 971.00 - 1007.00 (sec)B rate = 1478 (cps)0.00.51.01.52.0310·Energy (chan)0200400600counts/binS+B window: 408.00 - 461.00 (sec)S+B rate = 10106 (cps)B window: 1008.00 - 1061.00 (sec)B rate = 1444 (cps)0.00.51.01.52.0310·Energy (chan)0200400600counts/binS+B window: 462.00 - 563.00 (sec)S+B rate = 5981 (cps)B window: 1062.00 - 1163.00 (sec)B rate = 1419 (cps)Figure 9-15: The S+B (black) and background (blue) spectra for 6He β decay.

Table 9-1: List of background conditions applied when fitting the observables.

Condition

bWM
0bWM
no background correction
gain background correction 1bWM
2bWM
gain-and-sigma corrections

bGT
0bGT
1bGT
2bGT

149

01234567310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 343.00 - 361.00 (sec)S+B rate = 18529 (cps)B window: 943.00 - 961.00 (sec)B rate = 497 (cps)01234567310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 362.00 - 384.00 (sec)S+B rate = 15622 (cps)B window: 962.00 - 984.00 (sec)B rate = 474 (cps)01234567310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 385.00 - 412.00 (sec)S+B rate = 12712 (cps)B window: 985.00 - 1012.00 (sec)B rate = 457 (cps)01234567310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 413.00 - 448.00 (sec)S+B rate = 9802 (cps)B window: 1013.00 - 1048.00 (sec)B rate = 440 (cps)Figure 9-16: Zoomed-in view of the background-subtracted β-energy spectra in 6He decay.

The corrections to bWM and bGT resulting from gain background correction are

1bWM −0 bWM = −7.2 ± 1.0,
1bGT −0 bGT = 0.0031 ± 0.0005,

(9.1)

(9.2)

where the uncertainties are from the uncertainties of AH and AL. The corrections to bWM and bGT
resulting from background gain-and-sigma corrections are
2bWM −0 bWM = −8.6 ± 2.3,
2bGT −0 bGT = 0.0046 ± 0.0016,

(9.3)

(9.4)

where the uncertainties are from the uncertainties of AH, AL, σH, and σL. As shown in Eqs. 10.1
and 10.2, the statistical uncertainties of bWM and bGT from the CsI(Na) dataset are 4.9 and 0.0031,

150

0.00.51.01.5310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 343.00 - 361.00 (sec)S+B rate = 18529 (cps)B window: 943.00 - 961.00 (sec)B rate = 497 (cps)0.00.51.01.5310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 362.00 - 384.00 (sec)S+B rate = 15622 (cps)B window: 962.00 - 984.00 (sec)B rate = 474 (cps)0.00.51.01.5310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 385.00 - 412.00 (sec)S+B rate = 12712 (cps)B window: 985.00 - 1012.00 (sec)B rate = 457 (cps)0.00.51.01.5310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 413.00 - 448.00 (sec)S+B rate = 9802 (cps)B window: 1013.00 - 1048.00 (sec)B rate = 440 (cps)Figure 9-17: The S+B (black), background (blue), and gain-corrected background (red) spectra
for 6He β decay.

respectively. Therefore these corrections are non-negligible. In summary, the results in Eq. 9.3
and 9.4 are considered corrections to the observables resulting from the background corrections,
and the uncertainties are the systematic uncertainties related to the background.

151

0.00.20.40.60.81.01.21.4310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 343.00 - 361.00 (sec)S+B rate = 18529 (cps)B window: 943.00 - 961.00 (sec)B rate = 497 (cps)0.00.20.40.60.81.01.21.4310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 362.00 - 384.00 (sec)S+B rate = 15622 (cps)B window: 962.00 - 984.00 (sec)B rate = 474 (cps)0.00.20.40.60.81.01.21.4310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 385.00 - 412.00 (sec)S+B rate = 12712 (cps)B window: 985.00 - 1012.00 (sec)B rate = 457 (cps)0.00.20.40.60.81.01.21.4310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 413.00 - 448.00 (sec)S+B rate = 9802 (cps)B window: 1013.00 - 1048.00 (sec)B rate = 440 (cps)Figure 9-18: Zoomed-in view of the new background-subtracted β-energy spectra in 6He decay
after gain corrections to the background.

152

0.00.51.01.5310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 343.00 - 361.00 (sec)S+B rate = 18529 (cps)B window: 943.00 - 961.00 (sec)B rate = 497 (cps)0.00.51.01.5310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 362.00 - 384.00 (sec)S+B rate = 15622 (cps)B window: 962.00 - 984.00 (sec)B rate = 474 (cps)0.00.51.01.5310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 385.00 - 412.00 (sec)S+B rate = 12712 (cps)B window: 985.00 - 1012.00 (sec)B rate = 457 (cps)0.00.51.01.5310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 413.00 - 448.00 (sec)S+B rate = 9802 (cps)B window: 1013.00 - 1048.00 (sec)B rate = 440 (cps)Figure 9-19: The S+B (black), background (blue), and gain-and-sigma corrected background
(red) spectra for 6He β decay.

153

0.00.20.40.60.81.01.21.4310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 343.00 - 361.00 (sec)S+B rate = 18529 (cps)B window: 943.00 - 961.00 (sec)B rate = 497 (cps)0.00.20.40.60.81.01.21.4310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 362.00 - 384.00 (sec)S+B rate = 15622 (cps)B window: 962.00 - 984.00 (sec)B rate = 474 (cps)0.00.20.40.60.81.01.21.4310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 385.00 - 412.00 (sec)S+B rate = 12712 (cps)B window: 985.00 - 1012.00 (sec)B rate = 457 (cps)0.00.20.40.60.81.01.21.4310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 413.00 - 448.00 (sec)S+B rate = 9802 (cps)B window: 1013.00 - 1048.00 (sec)B rate = 440 (cps)Figure 9-20: Zoomed-in view of the new background-subtracted β-energy spectra in 6He decay
after gain-and-sigma corrections to the background.

154

0.00.51.01.5310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 343.00 - 361.00 (sec)S+B rate = 18529 (cps)B window: 943.00 - 961.00 (sec)B rate = 497 (cps)0.00.51.01.5310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 362.00 - 384.00 (sec)S+B rate = 15622 (cps)B window: 962.00 - 984.00 (sec)B rate = 474 (cps)0.00.51.01.5310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 385.00 - 412.00 (sec)S+B rate = 12712 (cps)B window: 985.00 - 1012.00 (sec)B rate = 457 (cps)0.00.51.01.5310·Energy (chan)0.00.51.01.5310·counts/binS+B window: 413.00 - 448.00 (sec)S+B rate = 9802 (cps)B window: 1013.00 - 1048.00 (sec)B rate = 440 (cps)CHAPTER 10

SENSITIVITIES TO PHYSICS AND BEYOND

10.1 Sensitivities

The systematic effects of bWM and bGT estimated in previous chapters are summarized in Table

10-1.

Table 10-1: Systematic effects considered and the corresponding uncertainties in the error budget.

Uncertainty on bWM Uncertainty on bGT (10−3)

Source
Theory
δbWM (CVC)
Detector simulation
Gain
Offset
Energy resolution
Non-Linearity
Fit method
Background
Total

0.08
N/A
4.4

9.2
0.80

2.3
11

included

included

negligible
negligible

negligible
negligible

0.03
0.29
1.8

4.0
0.40

1.6
4.7

The statistical uncertainties extracted from fits to the CsI(Na) dataset in Table 3-1 are 4.93 for

bWM and 3.1×10−3 for bGT. Therefore the estimates of bWM and bGT are

bWM = ? ± 4.9stat ± 11sys,

bGT =(cid:2)? ± 3.1stat ± 4.7sys(cid:3) × 10−3

,

(10.1)

(10.2)

where the central values of bWM and bGT will be determined after completing the Monte Carlo
study mentioned at the end of Chap. 2 as well as applying the complete fit function given in Eq.
8.13. The final result will provide the first measurement of the weak magnetism form factor in
6He decay and a new constraint on the Fierz term in the Gamow-Teller transition. The predicted
uncertainty of bGT satisfies the precision goal of O(10−3), a level which allows new constraints on
tensor couplings beyond the SM.

155

10.2 Summary and Outlook

This work has explored a new calorimetric technique for the measurement β-energy spectrum
and provides the current most precise and accurate measurement of the β-energy spectrum in 6He
decay. The next steps of this work include the Monte Carlo study of the detector’s response to
inner bremsstrahlung photons, the application of a more complete fit function, and the analysis of
the NaI(Tl) dataset mentioned in Table 3-1. The final results of bWM and bGT will include data
collected with both detectors.

The studies of the systematic effects associated with the new technique provide a solid foundation
for next-generation β-decay experiments with this technique. Future experiments will focus on
reducing the systematic uncertainties, including developing new approaches to determine the offset,
improving the control of the rate effects, and exploring new experimental techniques. These high-
sensitivity fundamental tests at the low energy scale open a unique window to physics beyond
the SM. It is possible that new physics would first appear in the low-energy and high-precision
nuclear-physics experiments before being discovered in high energy physics experiments.

156

APPENDIX

157

APPENDIX

DERIVATIONS IN CHAPTER 2

A.1 Derivation of Eq. 2.5

The screening component βs has the following form for the Hartree potential [54]:

3


j=1

βs = 2

aj βj,

where βj is given in Ref. [54]:

βj = 1.13bj αZ1/3

.

In addition, βs can be written in the usual form [51]:

βs = 2C(| ˜Z|)α| ˜Z|1/3

.

3


j=1

aj bj,

(.1)

(.2)

(.3)

(.4)

Therefore,

C(| ˜Z|) = 1.13

where the aj and bj are parameters determined in Ref. [53] as a result of fitting 19 Hartree curves.
To benchmark this result, let us calculate the C(| ˜Z|) value for Hg (Z = 80). Following Ref.

[51], the values of aj and bj for Hg can be found in Table 1 of Ref. [53]. This gives

C(80) (cid:39) 1.441,

(.5)

which agrees with the value of C(80) reported in Ref. [51].

Finally, with the values of aj and bj for He from the same table, the calculated value of C(| ˜Z|)

for He (Z = 2) is

C(2) (cid:39) 1.386.

(.6)

A.2 Table of Parameters in the Electrostatic Finite Size Correction

The finite size correction described by Eq. 2.20 requires bx-values provided in Table A-1:

158

Table A-1: Coefficients for the parameterization of L0(Z, W) for electrons copied from Ref. [3].

b1
0.115
-0.00062
0.02482
-0.14038
0.008152
1.2145
-1.5632

b2

-1.8123
0.007165
-0.5975
3.64953
-1.15664
-23.9931
33.4192

b3

8.2498
0.01841
4.84199
-38.8143
49.9663
149.9718
-255.1333

b4

-11.223
-0.53736
-15.3374
172.1368
-273.711
-471.2985
938.5297

b5

-14.854
1.2691
23.9774
-346.708
657.6292
662.1909
-1641.2845

b6

32.086
-1.5467
-12.6534
288.7873
-603.7033
-305.6804
1095.358

a−1
a0
a1
a2
a3
a4
a5

A.3 Order Zα

2 Radiative Correction

A.3.1 Uniform Charge Distribution

For uniformly charged sphere of radius R [61],

(cid:18) Λ

(cid:19)

∆2 =

∆F1 = ln
mp
3
√
2
10π
3√
10π

∆3 =

(cid:18)
− κ2 − 3√
10π
1 − π
√
Λ
2
10
mp
gA(1 + µV) Λ
mp

(cid:20)

√

(cid:17)(cid:21)

(cid:20)1
(cid:19)

2 + γE + ln

10 + ln(cid:16) mp
Λ
mp
(cid:17)
10 + ln(cid:16) mp
Λ
mp
√
γE − 1 + ln
√
π
10
√
10 = 0.395, and Λ ≡ √

Λ

Λ

4

+

,

,

(cid:21)

,

Λ
mp
10/R.

where gA = 1.27, 1 + µV = 4.70, κ2 = γE − 4/3 + ln

(.7)

(.8)

(.9)

A.3.2 Modified Gaussian Charge Distribution
For modified Gaussian charge distribution, i.e., ρ(r) = const × (1 + Ak2r2/a2)e−r2k2/a2 [61],

− κ1(Z) − 8√
6π

2 + 3A

3

1 + A −

1 +

k

Λ
mp
3
π
96 k

(cid:115)
(cid:18)
(cid:26)(cid:20) γE − 1

Λ
mp

1 +

k

2 + 2A

Λ
mp

∆2 =

∆F1 = ln Λ
mp
4√
6π
8√
6π

3

k

∆3 =

3 gA(1 + µV)
where A = (Z − 2)/3, k2 = 3
2

Λ
2 + 3α
mp
, and κ1(Z) = 1
2+5A2+3A
2

2

+ ln

(cid:26)(cid:20)

(cid:18)√

(cid:19)(cid:21)

(cid:27)

,

6
k

A
2

A
2

γE
2 + ln

mp
Λ

(1 + A) +

(cid:19) ,
(cid:115)
(cid:18)√
(cid:2)γE + ln(3/2k2) + 2A/(2 + 3A)(cid:3).

(1 + A) +

A
2 +

(cid:19)(cid:21)

mp
Λ

3
π
6

6
k

k
8

(.10)

(.11)

(.12)

(cid:27)

.

Λ
mp

159

A.4 Derivation from Eq. 2.58 to Eq. 2.62

(cid:112)(E − kmax)2 − 1
Under the condition kmax = E − 1 it can be shown that
(cid:18)E − kmax −(cid:112)(E − kmax)2 − 1
(cid:19)
√E − 1
E −(cid:112)E2 − 1
(cid:34)E2 − 1 − Ekmax +(cid:112)E2 − 1 ·(cid:112)(E − kmax)2 − 1
(cid:35)

−2

2
β

ln

2 ln

2(E2 − 1)

Replacing 1

2 ln 1+β
1−β

by tanh−1

β gives

= 0,

=

2
β

tanh−1

β,

= −2 ln[2(E + 1)].

(cid:19)

,

β − 1

=β tanh−1

β,

(cid:18) 1

tanh−1

=2 ln(E − 1)
= − 2

(tanh−1

β

β
β)2

.

(.13)

(.14)

(.15)

(.16)

(.17)

(.18)

(.19)

(.20)

(.21)

[70].

(.22)

(cid:20) 1

2β

(cid:19)

(cid:18)1 + β

1 − β

ln

(cid:21)

− 1

(cid:18)

− 2

2 ln λ

, I0
k

Þ kmax
Þ kmax
Þ kmax

0

0

I0
k

I1
k
I−1
k

=

=

=

= 2I−1, I0
k

Here I−1
k
Therefore,

β

λ

2

ln

(cid:19)

1 − β

1 + β
1 − β

1
2 β ln
ln

(cid:18)2kmax
(cid:19)
(cid:18)1 + β
(cid:19)(cid:21)2
Þ E
Þ E
(cid:19)
= 2
· 1 − β
1 + β

= 2

1

1 − tanh−1
(cid:20)1
− 1

β

β

(cid:19)
(cid:18)1 + βe
(cid:19)
(cid:18)1 + p
(cid:18)1 + βe

ln

The three integrals I−1
k

, and I1
k

can be rewritten in the following expressions

dk ln

dkk

dk

1
k

dE ln(E + p),

1

dE(E − E) ln(E + p),

1 − βe
E
1 − p
E
1 − βe
= 2I1 in which I−1, I0, and I1 are defined in Ref.

(E − E) ln E + p
E(1 + β) .

Þ E

= 2

dE

1

0
= 2I0, and I1
k

(cid:18)

(cid:19)

1
β

k − 1
I−1
E β

I0
k

+

1
2E2

β

I1
k

=

1
β

2I−1 +

I1

E2 − 2I0
E

.

Summing up the above terms and the constants in gv(E) and gb(E) gives the following expression

160

for g(E):

(cid:18) 1
(cid:19)

g(E) =gv(E) + gb(E)
=2 ln 2[(E − 1)]

(cid:18) 2β

1 + β

+

2
β

L

β − 1

β

tanh−1
+ β tanh−1

β +

(cid:19)

− 2 ln[2(E + 1)] +
3
tanh−1
β

β +

1
β

3
2

13

(cid:19)
8 − 2
ln mp +
E2 − 2I0
E

I1

β

2I−1 +

(cid:18)

(tanh−1

β)2

(.23)

A.5 Derivation from Eq. 2.62 to Eq. 2.69

g(E) can be rewritten by replacing the integrals in g(E) by the closed forms from Eq. 2.65-2.67

:

(cid:20)
(cid:20)

(cid:26) 1
(cid:26) 1

ln

β by 1

g(E) = ln

and replacing tanh−1
2 ln 1+β
1−β
1 + β
2(E − 1) · 2(1 +
1 − β
− 2 ln[2(E − 1) · 2(E + 1)] − 4
1 + β
1 − β
− 2 ln

2(E − 1) · 2(E + 1)

= ln

(cid:20)

(cid:21)

ln

β

β

− 4

β

1 + β
2(E − 1) · 2(E + 1) · 1 + β
1 − β

As

2(E − 1) · 2(E + 1) = 4

(cid:26) 1
(cid:26) 2

β

β

ln

ln

1 + β
1 − β
1 + β
1 − β

g(E) = ln

= ln

(cid:18) 4β
(cid:18) 2β

2
1 − β

1 + β

(cid:19)

(cid:19)
2 · 1 + β
1 − β
3
8 β +

+

+

7
8β

+

β

+

+

+

+

8

(cid:19)

Li2

Li2

1 + β

(cid:18)1
(cid:21)
1(cid:112)1 − β
(cid:19)
(cid:18)13
(cid:19)
(cid:18) 2β
2 − 1
2) · 1 + β
1 − β
(cid:27)
(cid:21)
8 − 3
4 + 2
(cid:18) 2β
(cid:19)
7
3
8 β +
8β
3
23
8 +
2
(cid:33)(cid:32)
1(cid:112)1 − β
(cid:27)

1(cid:112)1 − β
(cid:18) 4β
(cid:33)

2 − 1
(cid:27)
(cid:32) 4β

7
3
8 β +
8β
− 2 ln

2
1 − β
− 4
Li2

− 2 ln
2
1 − β

+ 1

(cid:33)

ln mp.

(cid:19)
(cid:18) 2β

− 4

=

β

β

2

2

2

1 + β

(cid:32)

(cid:18)3

(cid:19) 1

(cid:27)

3
8 − 1

β +

2 +

3
2

ln mp

β

(.24)

(.25)

2
4β
1 − β

2,

(cid:18) 2β

(cid:19)

1 + β
23
8 +

+

Li2

(cid:19)

23
8 +

3
2

+

ln mp

3
2

ln mp.

(.26)

23
8 +

3
2

+

ln mp.

(.27)

Replacing −Li2[(2β)/(1 + β)] by L[(2β)/(1 + β)] gives
g(E) = ln

− 2 ln

ln

+

3
8 β +

7
8β

1 + β
1 − β

(cid:19)

(cid:18) 2β

1 + β

(cid:18) 4β

2
1 − β

(cid:19)

2

(cid:19)

(cid:18) 2β

1 + β

+

L

4
β

(cid:20) 2

β

(cid:21)

161

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