Test of Bound State QED
Higher Order Correction :Precision Measurement ofOrthopositronium Decay Rate
Ph. D. thesis

Yousuke Kataoka
March 2007
Department of Physics
Faculty of Science, University of Tokyo

Abstract
This thesis reports a new measurement of the ortho-positronium decayrate λo-Ps . The improvements of the setup and the analytic techniques solve
the crucial systematic errors previously suﬀered. The obtained result is−1
λo-Ps = 7.0401 ± 0.0006(stat.)+0.0007−0.0009 (sys.) μs , where the ﬁrst error is thestatistical error, and the second represents the systematic error.
The measured value is consistent with the theoretical QED prediction
and the recent measurements from 1995. In addition, the world average ofthe decay rate favors the O(α2 )-corrected QED prediction rather than thatof O(α) correction.

iAcknowledgements
First, I would like to express my gratitude to Prof. Tomio Kobayashi (ICEPP
Univ. of Tokyo) for providing me with this work. Without his continuous
encouragement and supports, I could not have ﬁnished this study. I also wish
to express my great appreciation to Shoji Asai (ICEPP Univ. of Tokyo) for
his continuous assistance and guidance at this work. He is my collaborator
in this experiment and heads a series of lifetime experiment from 1995. I
would like to thank Osamu Jinnouchi (KEK) for many useful informationand suggestions. This work bases on his previous measurement in 2001.
I would also like to thank Toshio Namba (ICEPP Univ. of Tokyo) for his
usefull suggestions and supports. I would like to thank all the students of ourgroup, Mitiru Kaneda, Genta Koreki, Keiji Kishimoto, Kazuyuki Nishihara,
Yusuke Tomishima. Sincere gratitude is expressed to Tetsuro Mashimo andJunichi Tanaka. They construct and maintain the computing system used inthis analysis. I gratefully acknowledge Nippon Aerosil Ltd. for the generous
supply of the silica powders and their valuable information. I would liketo thank all the staﬀs and students of International Center for Elementary
Particle Physics (ICEPP) at University of Tokyo,iiContents
Acknowledgementsii

1 Introduction
1.1 Properties of Positronium . . . . . . . . . . . . . . . . . . . .
1.2 Theoretical Predictions of Orthopositronium Decay Rate . . .
1.2.1 The Lowest Order Decay Width . . . . . . . . . . . .
1.2.2 The O(α) Correction . . . . . . . . . . . . . . . . . . .
1.2.3 α2 ln(α−1 ), α3 ln2 (α−1 ), α3 ln(α−1 ) Contributions . .
1.2.4 O(α2 ) Correction . . . . . . . . . . . . . . . . . . . . .
1.2.5 Summary of the QED Prediction . . . . . . . . . . . .
1.3 Previous Measurements of Orthopositronium Decay Rate . .
1.3.1 Techniques of the Measurement . . . . . . . . . . . . .
1.3.2 History of the Measurement . . . . . . . . . . . . . . .

1
1
2
3
3
3
4
5
5
6
7

2 Experiment10
2.1 Method of the Experiment . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Pick-oﬀ Annihilation . . . . . . . . . . . . . . . . . . . 10
2.1.2 Previous Extrapolation Method . . . . . . . . . . . . . 10
2.1.3 Direct Measurement of the Pick-oﬀ Rate . . . . . . . . 11
2.1.4 Measurements in 1995 and 2001 . . . . . . . . . . . . 13
2.2 New Features of the Present Measurement . . . . . . . . . . . 14
2.2.1 Introduction of Fast Inorganic Scintillator . . . . . . . 14
2.2.2 New Design of Ps Formation Assembly . . . . . . . . . 16
2.2.3 Geant4 Simulation and its Validation . . . . . . . . . 16
2.3 Description of Apparatus . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Positronium Formation Assembly . . . . . . . . . . . . 17
2.3.2 Characteristics of SiO2 Powders . . . . . . . . . . . . 21
2.3.3 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.4 Timing system . . . . . . . . . . . . . . . . . . . . . . 23
2.3.5 Electronics . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.6 The CAMAC Data Acquisition System . . . . . . . . 37
2.4 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . 38
2.4.1 Positron and Photon Simulation . . . . . . . . . . . . 38
iii

CONTENTS
2.4.2
2.4.3

CONTENTS
Response Function of the Ge Detector . . . . . . . . .
Response Function of the YAP Scintillator . . . . . .

4040

3 Analysis
47
3.1 Calibration and Time Walk Correction . . . . . . . . . . . . . 47
3.1.1 Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.2 Time and Energy Spectrum of the γ-ray Detector . . 48
3.1.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.4 Event Selection . . . . . . . . . . . . . . . . . . . . . . 52
3.1.5 Time Walk Correction . . . . . . . . . . . . . . . . . . 56
3.2 Determination of λpick (t)/λ3γ . . . . . . . . . . . . . . . . . . 67
3.2.1 Time and Energy Spectra of the Ge Detectors . . . . . 67
3.2.2 Extraction of the Pick-oﬀ Contribution . . . . . . . . 69
3.2.3 Determination of λpick (t)/λ3γ . . . . . . . . . . . . . . 74
3.3 Decay Rate Fitting . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3.1 Time and Energy Spectrum of the YAP Scintillators . 80
3.3.2 Decay Rate Fitting . . . . . . . . . . . . . . . . . . . . 82
3.4 Various Checks . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.1 Energy Selection . . . . . . . . . . . . . . . . . . . . . 86
3.4.2 Base Cut . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.3 Narrow-Wide Cut . . . . . . . . . . . . . . . . . . . . 87
3.4.4 SRT Cut . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4.5 Data Sample . . . . . . . . . . . . . . . . . . . . . . . 88
3.5 Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.5.1 TDC Module Related Errors . . . . . . . . . . . . . . 88
3.5.2 Contamination of the Pile-up Events . . . . . . . . . . 91
3.5.3 Subtraction of 3γ Energy Spectrum . . . . . . . . . . 91
3.5.4 Detection Eﬃciencies . . . . . . . . . . . . . . . . . . 93
3.5.5 Other Sources of Systematic Errors . . . . . . . . . . . 96
3.5.6 Summary of the Systematic Errors . . . . . . . . . . . 97
3.6 Results of the Present Measurement . . . . . . . . . . . . . . 97
4 Discussion
100
4.1 Experimental Meanings . . . . . . . . . . . . . . . . . . . . . 100
4.1.1 Solution of the Systematic Increase near the Prompt
Peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1.2 Validation of the Fitting Start Time . . . . . . . . . . 101
4.1.3 Improvements of the Monte Carlo Simulation . . . . . 102
4.1.4 Dependence on the Two Types of Silica Material . . . 104
4.2 Theoretical Implications . . . . . . . . . . . . . . . . . . . . . 104
5 Conclusions

109
iv

CONTENTS

CONTENTS

A Stark Shift110
A.0.1 Stark Eﬀects on o-Ps Decay Rate . . . . . . . . . . . . 110
A.0.2 Contribution of Charge on Primary Grains . . . . . . 110
A.0.3 Contribution of Dipole Moment on the Surface of Grains111
B Expected Time Spectrum
115
B.1 Probability of Accidental Events . . . . . . . . . . . . . . . . 115
B.2 Expected Time Spectrum . . . . . . . . . . . . . . . . . . . . 117
B.2.1 Pile-up by the Accidental Events ∼ X(C + C)∞ . . 118
B.2.2 Pile-up Events with the Prompt Signal ∼ AB . . . 120
B.2.3 Contamination of Pile-up Events . . . . . . . . . . . . 120
B.3 Time Spectrums of γ-ray Detectors . . . . . . . . . . . . . . . 121
B.3.1 The Case of Ge Detector . . . . . . . . . . . . . . . . 121
B.3.2 The Case of YAP Scintillator . . . . . . . . . . . . . . 124v

List of Figures
1.1
1.2
1.3

The diagrams of one-loop corrections to o-Ps decay. . . . . .
The energy spectrum with the O(α) correction . . . . . . . .
The chronology of the o-Ps decay rate measurement. . . . . .

4
5
9

2.1
2.2
2.3
2.4
2.5
2.6

Signal shapes of the various scintillators . . . . . . . . . . . . 15
Schematic diagram of apparatus (side view). . . . . . . . . . . 18
Schematic diagram of apparatus (top view). . . . . . . . . . . 19
The level diagram of the 68 Ge isotope . . . . . . . . . . . . . 20
Internal structure of germanium detector. . . . . . . . . . . . 22
The energy resolution of the Ge detectors as a function of theγ-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 The relative peak eﬃciency of the germanium detectors. . . . 25
2.8 Schematic diagram of Phase Locked Loop (PLL) . . . . . . . 27
2.9 Schematic diagram of Direct Digital Synthesis (DDS) . . . . . 28
2.10 Schematic diagram of the process data-ﬂow. . . . . . . . . . . 30
2.11 Schematic diagram of electronics for trigger system. . . . . . 32
2.12 Schematic diagram of electronics for YAP scintillator system. 34
2.13 Schematic diagram of electronics for Ge detector system. . . . 36
2.14 Comparison of the measured Ge spectrum with the simulation
using 85 Sr source . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.15 The diﬀerence of the measured Ge spectrum and the simulation 42
2.16 Pile-up events of Ge detector . . . . . . . . . . . . . . . . . . 43
2.17 Comparison of the measured YAP spectrum with the simulation using 85 Sr source . . . . . . . . . . . . . . . . . . . . . . 45
2.18 Collection eﬃciency of optical photon in YAP crystal . . . . . 46
3.1
3.2
3.3
3.4
3.5
3.6
3.7

Time and Energy spectrum of the YAP scintillator . . . . . .
Time and Energy spectrum of the Ge detector . . . . . . . .
The relative bin width . . . . . . . . . . . . . . . . . . . . . .
The schematic of the Base cut . . . . . . . . . . . . . . . . . .
Base spectra for Ge detectors . . . . . . . . . . . . . . . . . .
Base spectra for YAP scintillator . . . . . . . . . . . . . . . .
Base spectrum for the plastic scintillator . . . . . . . . . . . .
vi

50
51
53
54
54
55
56

LIST OF FIGURESLIST OF FIGURES

3.8 The schematic of the Narrow-Wide (NW) cut . . . . . . . . .
3.9 Narrow-Wide spectra for Ge detectors . . . . . . . . . . . . .
3.10 Narrow-Wide spectra for YAP scintillators . . . . . . . . . . .
3.11 Time walk of YAP scintillator . . . . . . . . . . . . . . . . . .
3.12 Time walk of plastic scintillator . . . . . . . . . . . . . . . . .
3.13 The rise-time of the Ge detectors . . . . . . . . . . . . . . . .
3.14 Time walk of Ge detectors . . . . . . . . . . . . . . . . . . . .
3.15 Time walk of the slow rise-time component . . . . . . . . . .
3.16 Time spectra of Ge detectors without silica target . . . . . .
3.17 Time spectra of Ge detectors . . . . . . . . . . . . . . . . . .
3.18 3γ SRT cut eﬃciency of Ge detectors . . . . . . . . . . . . . .
3.19 Time spectrum of Ge detectors . . . . . . . . . . . . . . . . .
3.20 511 keV photoelectric peak of the prompt events . . . . . . .
3.21 Energy spectrum of the delayed region. . . . . . . . . . . . . .
3.22 Subtraction of the 3γ spectrum . . . . . . . . . . . . . . . . .
3.23 Energy spectrum of the pick-oﬀ events . . . . . . . . . . . . .
3.24 Pick-oﬀ ratio λpick (t)/λ3γ . . . . . . . . . . . . . . . . . . . .
3.25 Uncertainties of the parameters (RUN I) . . . . . . . . . . . .
3.26 Uncertainties of the parameters (RUN II) . . . . . . . . . . .
3.27 Energy spectrum of YAP scintillator in the delayed region . .
3.28 Time spectrum of the YAP scintillators . . . . . . . . . . . .
3.29 Fitted decay rate (RUNI) . . . . . . . . . . . . . . . . . . . .
3.30 Fitted decay rate (RUNII) . . . . . . . . . . . . . . . . . . . .
3.31 Decay rates of the four samples for RUN I. . . . . . . . . . .
3.32 Decay rates of the four samples for RUN II. . . . . . . . . . .
3.33 The pick-oﬀ spectrum with varied 3γ normalization . . . . . .
3.34 ξ-parameter with varied 3γ normalization . . . . . . . . . . .
3.35 The results of the two runs and the combined result . . . . .
4.1
4.2
4.3
4.4

57
57
58
59
60
61
62
64
65
66
66
68
69
71
72
73
76
78
79
80
81
84
85
89
90
92
94
99

The ﬁtted decay rate with the ﬁtting start time before 80 ns 103
The measured pick-oﬀ ratios with two types of silica material 105
The history of the o-Ps decay rate measurements including
the present measurement . . . . . . . . . . . . . . . . . . . . . 107
The relation between the resent measurements and the theoretical predictions . . . . . . . . . . . . . . . . . . . . . . . . . 108

A.1 Schematic diagram of SiO2 grain. . . . . . . . . . . . . . . . . 112
A.2 A deﬁnition of the variables used for calculation in the context113
B.1 Probability of accidental events around the trigger signal . . . 116
B.2 Time distribution of the the γ-rays . . . . . . . . . . . . . . . 117
B.3 The suppression region in which accidental events spoil theevent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
B.4 Suppression region of the pile-up cuts . . . . . . . . . . . . . 122vii

LIST OF FIGURESLIST OF FIGURES

B.5 Suppresion factor S(t) for Ge detector . . . . . . . . . . . . . 123
B.6 Suppresion factor S(t) for YAP scintillator . . . . . . . . . . . 124

viii

List of Tables
1.1
1.2
1.3
1.4

The classiﬁcation of Ps according to their quantum numbers.
The summary of the logarithmic contribution . . . . . . . . .
Summary of QED calculation for o-Ps decay rate . . . . . . .
The history of the o-Ps decay rate measurements . . . . . . .

1
4
6
7

2.1
2.2
2.3
2.4
2.5
2.6
2.7

Characteristics of two powders used in the measurement . . .
Crystal size of germanium detectors . . . . . . . . . . . . . .
Properties of YAP scintillator . . . . . . . . . . . . . . . . . .
Energy resolution of the YAP scintillator . . . . . . . . . . .
Quantities measured with ADCs . . . . . . . . . . . . . . . .
Quantities measured with TDCs . . . . . . . . . . . . . . . .
Speciﬁcations of the CAMAC modules . . . . . . . . . . . . .

21
23
26
26
38
38
39

3.1 Properties of the RUNs . . . . . . . . . . . . . . . . . . . . .
3.2 Energy resolution of the prompt events . . . . . . . . . . . . .
3.3 Typical stop rate of Ge detectors . . . . . . . . . . . . . . . .
3.4 Compton factor fcompton . . . . . . . . . . . . . . . . . . . . .
3.5 Peak value and resolution of the pick-oﬀ events . . . . . . . .
3.6 Conversion factor F . . . . . . . . . . . . . . . . . . . . . . .
3.7 Deﬁnition of the time windows . . . . . . . . . . . . . . . . .
3.8 Summary of λpick (t)/λ3γ ﬁtting result . . . . . . . . . . . . .
3.9 Uncertainties of the ﬁtted parameter (RUN I) . . . . . . . . .
3.10 Ratio pick /3γ of YAP scintillators . . . . . . . . . . . . . . .
3.11 Typical stop rate of YAP scintillators . . . . . . . . . . . . .
3.12 Summary of the ξ-parameters . . . . . . . . . . . . . . . . . .
3.13 The diﬀerence between the measured data and the Monte
Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . .
3.14 The diﬀerence between the measured data and the Monte
Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . .
3.15 Summary of the systematic errors . . . . . . . . . . . . . . . .

48
67
70
70
74
74
75
77
77
82
82
93

ix

95
96
98

Chapter 1

Introduction
The aim of this experiment is a test of bound state Quantum ElectroDynamics (QED). The predictions of the higher order corrections are veriﬁedby measuring the orthopositronium (o-Ps) decay rate precisely. As an introduction, properties of o-Ps are explained in the ﬁrst section. Then thetheoretical predictions of bound state QED and the previous measurementsof o-Ps decay rate are summarized in the following sections.

1.1

Properties of Positronium

Positronium (Ps), the bound state of the electron (e− ) and the positron (e+ ),is the lightest hydrogen-like atom bound by the electromagnetic interaction.
Since Ps is a purely leptonic system and eﬀectively free from hadronic andweak interaction eﬀects, precision mesurements of decay rate and energy
levels give us direct information about bound state QED. Therefore, Ps is
considered as the best testing ground for bound state formalism in quantum
ﬁeld theory.
A Ps consists of two fermions as hydrogen atom, and its wave functions
are classiﬁed according to the principal quantum number n, a sum of theorbital angular momentum  and the spin angular momentum s, and its
projection on a quantization axis m. In the case of the ground state, thatis n = 1 and  = 0, Ps can be simply classiﬁed to the total spin s as in
Table 1.1.
Total Spin s
01

Projection m
00, ± 1State
singlettripletC parity
even
odd

Name
parapositronium (p-Ps)
orthopositronium (o-Ps)

Table 1.1: The classiﬁcation of Ps according to their quantum numbers.
A fermion-antifermion system with orbital angular momentum  and spin
1

2

1.2 Introduction

angular momentum s has a parity of (−1)+s under a charge-conjugationtransformation (C). Thus o-Ps and p-Ps have odd and even C parities
respectively. Due to the C-invariance of the electromagnetic interaction,
this diﬀerence plays a crucial role on the decay modes. In fact, o-Ps decays
only into odd γ’s and p-Ps decays only into even γ’s since the the system
consisting of n photons have a C parity of (−1)n . In addition, single photon
decay in vacuum is prohibited by energy-momentum conservation. Then,the decay modes of o-Ps and p-Ps are as follows.
o-Ps → nγ, n = 3, 5, 7, 9 · · ·p-Ps → nγ, n = 2, 4, 6, 8 · · ·As a number of photon increses, a number of electromagnetic vertex
increases and the width of the decay mode decreases by a factor of α. Furthermore, the phase space of the ﬁnal state is getting smaller as the numberof ﬁnal state photon increases. Thus, decay modes into many photons arehighly suppressed. In fact, the dominant decay mode of o-Ps is 3γ, and thecontribution of the 5γ decay is only a ppm level. The theoretical branchingratio of the 5γ decay is as follows [1].
BRth (o-Ps → 5γ) = Γ(o-Ps → 5γ)/Γ(o-Ps → γγγ)
= 0.19(1) (α/π)2
 1.0 × 10−6

(1.1)

The prediction has been conﬁrmed by the measurement [2] within theexperimental accuracy.
−6
BRex (o-Ps → 5γ)  2.2+2.6−1.8 × 10 .

(1.2)

Thus, the inﬂuence of the 5γ decay contribution to the total width isnegligible.
This relation also suggests that the decay rate of p-Ps is approximately
3
10 times higher than the decay rate of o-Ps. In fact, the measured lifetimeof p-Ps is 125.14 ±0.03 ps [3] and 1130 times shorter than the o-Ps’s. Thisfacilitates the separation of o-Ps events from that of p-Ps in the experiment.

1.2

Theoretical Predictions of Orthopositronium
Decay Rate

Theoretical expression for o-Ps decay rate can be written in the form of
perturbation as follows,

λo-Ps =

λ(0)
o

 

1 + Ao

α
π

+ Bo

 2
α

π

3

1.2 Introduction


α3 2 1α3
1
1+Co α ln + Doln
+ Eo
ln + · · ·
α
π
α
π
α
2

(1.3)

(0)

where λo is the lowest order decay width of o-Ps. The higher order terms,
whose coeﬃcients are expressed as Ao , Bo , Co , Do , Eo , will be discussed
in the following sections. The ellipses denote unknown higher order terms
which are neglected in this thesis.

1.2.1

The Lowest Order Decay Width

The lowest order calculation of o-Ps have been done by Öre and Powell [4],and conﬁrmed later by other authors [5, 6]. The lowest order decay width
is given as,2(π 2 − 9)me α6= 7.211 17 μs−1 (138.67 ns)(1.4)
9π
where me is the electron mass. In this formula, the three γ-vertexes
contribute a factor of α3 . The remaining α3 factor is originated from theﬂux factor which is the probability that the distance between the electron
and the positron is zero.
λ(0)
o =

1.2.2

O(α) Correction

The O(α) corrections come from the single photon exchange processes asshown in Fig. 1.1. Many authors have evaluated the one-loop contributions
to the decay width of o-Ps, and they have obtained the consistent resultswith each other. The most accurate result of the O(α) correction for o-Ps
decay rate was obtained by Adkins [7] in an analytical form. The value is,
Ao = −10.286 606(10)(1.5)

which results in -2.389 % correction to the lowest order decay width.
The evaluation of the O(α) correction on the energy spectrum of theemitted photon is carried out as well. This correction is almost ﬂat overthe photon energy, and the relative variation does not exceed 1 % around
511 keV as shown in Fig. 1.2. The variation of the energy spectrum dueto the O(α2 ) correction is expected to be further small. Then, the O(α)
corrected matrix element is used in the Monte Carlo simulation.

1.2.3

α2 ln(α−1 ), α3 ln2 (α−1 ), α3 ln(α−1 ) Contributions

The logarithmic corrections can be considered as the relativistic eﬀects in the
wave function of Ps. The logarithmic two-loop corrections and the leading
logarithmic corrections at three-loop are already obtained by some authorsin analytical forms. Since ln(α−1 ) is actually about 5, the O(α2 ln(α−1 ))correction is as important as the O(α2 ) correction. The already known
coeﬃcients are summarized in Table 1.2

4

1.2 Introduction

(a)

(b)

(c)

(f)

(e)

(d)

Figure 1.1: Graphs contributing to the o-Ps decay amplitudes through orderα. Each ﬁgure represents (a) self-energy, (b) outer-vertex, (c) inner-vertex,(d) double-vertex, (e) ladder, and (f) annihilation contributions.

Co
Do
Eo

Coeﬃcient−1/3−3/2+5.517

correction to Γ(0)−87 ppm−4.4 ppm+3.3 ppm

reference[8, 9, 10][11, 12][13, 14]

Table 1.2: The summary of the logarithmic contribution

1.2.4

O(α2 ) Correction

The calculation of the O(α2 ) correction to the decay rate of Ps is so complicated if the traditional bound-state methods are employed such as BetheSalpeter analysis. This is because each term in a traditional expansion has
contribution from both nonrelativistic and relativistic energy scales. To
overcome this diﬃculty, a new procedure has been developed and applied tothe calculations of the coeﬃcient Bo .
The calculation is based on a rigorous nonrelativistic reformation of QED
called Nonrelativistic Quantum Electrodynamics (NRQED). In this framework of the eﬀective ﬁeld theory, the O(α2 ) correction terms are separated
into nonrelativistic (soft) parts and relativistic (hard) parts.
A complete calculation of Bo had not been carried out until quite recently. Only some partial results on the non-logarithmic O(α2 ) corrections
had been carried out [15, 16, 17]. But, in recent theoretical eﬀorts, a complete evalation of this correction have been performed by Adkins [18]. The
result is Bo = 45.06(26), and its contribution to the lowest order calculationis 243.1(1.4) ppm.

5dΓ(E)/dE

relative changes

1.3 Introduction

1
0.99
0.98
0.97
0.96
0.95
0.94

0

100

200

300

400

500

0.93
0

Energy (keV)

100

200

300

400

500

Energy (keV)

Figure 1.2: (a) The lowest order energy spectrum of 3γ decay (red line) andthe energy spectrum with the O(α) correction (blue line). (b) The relativechange of the energy spectrum by the O(α) correction.

1.2.5

Summary of the QED Prediction

As described above, the higher order corrections are important for o-Ps
decay rate. Table 1.3 shows the summary of the calculation for o-Ps decayrate. Except for the last line of the table, that is the non-logarithmic O(α2 )term, the calculations are conﬁrmed by several authors. On the other hand,the value Bo has just been obtained by the recent work, and it need to be
veriﬁed by the other calculations. Thus it would be premature to accept theresult now.
In this thesis, the following value is referred to as the O(α2 ) corrected
QED prediction [19].
(1.6)Γo−P s = 7.039 979 μs−1 .
Comparison between the theory and the experiments will be performed withthis decay rate.

1.3

Previous Measurements of Orthopositronium
Decay Rate

The situation is more complicated in the experiment. The signiﬁcant discrepancy between the prediction and the measurement had existed sincelatter half of 1970’s. But, the recent experiments reported the consistent results with the prediction. Before we discuss their results, we will review the6

1.3 Introduction
terms
tree level
Ao ( απ )Co α2 ln(1/α)
Do (α3 /π) ln2 (1/α)
Eo (α3 /π) ln(1/α)
Bo ( απ )2correction (ppm)
0.0-23893.9
-87.4
-4.4
+3.3
243.1

decay rate (μs−1 )7.211 167
7.038 864
7.038 234
7.038 202
7.038 226
7.039 979

lifetime (ns)138.6738
142.0684
142.0811
142.0817
142.0813
142.0459

Table 1.3: Summary of QED calculation for o-Ps decay rate
experimental techniques of o-Ps decay rate mesurement in the next section.

1.3.1

Techniques of the Measurement

In all the previous measurements, a positron was injected into a target
material to form o-Ps, and the time interval between the positron injection
and the detection of γ-rays emitted from o-Ps was measured. This was acommon idea in all the measurements, and only the target materials werealtered in each measurement. The employed target materials are classiﬁed
into three types, the gas, the surface of cavity which is coated by oxidematerial (e.g. MgO), and the powders of SiO2 .
The Gas experiments are the most popular and the earliest techniques.
In this technique, injected positrons form Ps in gases at various pressures.
Then, the decay rate in vacuum is estimated by extrapolation to zero densityof the gas.
In the Cavity experiments, Ps is formed in the vacuum container when
the slow positron beam hits the MgO surface of the cavity. The intrinsic
decay rate in vacuum was estimated by extrapolation to inﬁnite cavity size.
The extrapolated values, in this case, are the surface area of the cavity
container divided by volume, (S/V ).
In the Powder experiments, positrons are injected from a radioactive
source and Ps is formed in SiO2 powders. Powder experiments are performed
at various powder density, and the extrapolation to zero density yields o-Ps
decay rate in vacuum.
Another type of powder experiment appeared in 1995 [20, 21]. This wasquite diﬀerent from all the previous experiments, because it no longer used
extrapolation. In this experiment, the eﬀect of target material, which isobserved as a increase of the decay rate, is directly estimated by detecting
the γ-rays from 2γ annihiration on the target material. We will closely
discuss this technique in the next chapter.

7

1.3 Introduction

1.3.2

History of the Measurement

The history of the o-Ps decay rate measurements are summarized in Table 1.4 and plotted in Fig. 1.3 to see the historical changes of the measured
value with its precision.
index

Year

1
2
3
4

1968
1976
1976
1978

Decay rate(μs−1 )
7.275
7.104
7.09
7.056

5
6
7
8
9
10
11

1978
1978
1978
1978
1982
1987
1987

7.067
7.050
7.121
7.045
7.051
7.031
7.0516

0.021
0.013
0.012
0.006
0.005
0.007
0.0013

2972
1844
1685
852
709
996
184

12
13
14
15
16

1989
1990
1995
2002
2003

7.05147.04827.03987.04047.03960.00140.00160.00290.00130.0016

199
227
412
182
230

Total error(μs−1 ) (ppm)
0.015
2062
0.006
844
0.02
2821
0.007
992

Methodgas(Freon)
SiO2 powder
cavitygas(Freon,
isobutane)
SiO2 powder
cavitycavitygas
gas(isobutane)
cavitygas(isobutane,
N2 ,etc.)gas(N2 , N e)
cavitySiO2 powder
cavitySiO2 powder

Lifetime(ns)
137.70140.77141.04141.72

Reference

141.50141.84140.41141.94141.84141.22141.81

[25]
[26]
[27]
[28]
[29]
[30]
[31, 32]

141.82141.88142.05142.04142.05[33][34, 35][20, 21][36][37, 38]

[22]
[23]
[24]
[25]

Table 1.4: The history of the o-Ps decay rate measurements

The ﬁrst stage
The ﬁrst measurement of o-Ps decay rate was performed in 1968 [22].
In those days, theory and experiment agreed at 0.2 % level of accuracy.
But in 1976, two experiments were performed and resulted in the large
discrepancy between the theory and the experiments. This caused thereevaluation of both the theory and the experiments.
The second stage
After the new calculation in 1977 [39], the new experiments drastically
improve the accuracy. In the group of Michigan ,which is sometimes
referred to as Ann Arbor, the cavity and the gas experiments werecarried out and remarkable progress were made. Especially, the gas
experiments were performed for various gases to check the gas independency. Here again, the discrepancy with the theory motivated a1.3 Introduction

8

further investigation on both the theory and experiments.
The third stage
The two types of measurements, the cavity [34] and the gas [31, 33]
experiments was performed at around the beginning of 1990’s. They
achieved the accuracy of 200 ppm level and the discrepancy becamedeﬁnite. The results were consistently higher than the theoreticalprediction by a 1000 ppm level. This discrepancies had been called
orthopositronium lifetime puzzle, and aroused various discussions forthe physical interpretations. Since most measurements resulted in thehigher decay rate than the theoretical prediction, it was consideredthat there may be some unknown rare decay channels. Thus the exotic decay modes had been searched for in that period. However, nosign of the exotic decay has been observed by now [40].
The fourth stage
In 1995, a new measurement was performed in our laboratory (Tokyo
group) using SiO2 powder [20] and resulted in a consistent result with
the QED prediction. Since the experiment employed the new techniquewhich directly measured the eﬀect of material, the systematic error
accompanying the extrapolation method was considered as the maincause of the discrepancy. In 2001, Tokyo group improved the accurary
to the level of 200 ppm [37] and their results are in agreement withthe prediction.
Ann Arbor group admitted the posibility that the incomplete thermalization in low-pressure gases distorted the measured value in their gas
experiment. For their cavity experiment, they found main problem
was caused by not fully thermalized Ps, which increased the observed
decay rate through collision quenching and escapes from the detection
area. In 2003, they reported the new measured value consistent withthe theoretical prediction [36].

This thesis is organized as follows.
Chapter 2 describes the experiment. The experimental techniques andapparatus are explained in detail. The Monte Carlo simulation used in thisexperiment is also explained.
Chapter 3 provides the analysis of the measured data. All the procedures of the analysis are explained step by step. The systematic errors
accompanying the measurement are also explained. Then, the ﬁnal result isshown.
In Chapter 4, the experimental meanings and the theoretical implicationsof the obtained results are discussed.
Finally, Chapter 5 concludes this experiment.

9

1.3 Introduction#4

Phys.Rev.Lett. 40,737 (1978)#5

Phys.Rev.Lett. 40,737 (1978)#6

Phys.Lett. A69,97 (1978)#8

J.Phys. B11,743 (1978)#9

Phys.Rev.Lett. 49,525 (1982)#10

Nuovo Cimento 97A,419 (1987)#11

Phys.Rev.Lett. 58,1328 (1987)#12

Phys.Rev. A40,5489 (1989)#13

Phys.Rev.Lett. 65,1344 (1990)

Phys.Lett. B357,475 (1995)

#14

#15

Phys.Rev.Lett. 90, 203402 (2002)#16

Phys.Lett. B572,117 (2003)

7.02

7.03

7.04

7.05

7.06

7.07

Decay rate (μ sec )-1

Figure 1.3: The chronology of the o-Ps decay rate measurement. The number on the right shoulder of each item indicates a corresponding index number in Table 1.4. Vertical band represents the O(α2 )-corrected NRQED
prediction of 7.039 979 μs−1 . The inside line on each error bar represents astatistics error.

Chapter 2

Experiment
In this chapter, the details of the experiment are described. First of all,the method adopted in this measurement is explained. Then, the detailed
descriptions on the experimental apparatus will follow.

2.1

Method of the Experiment

2.1.1

Pick-oﬀ Annihilation

In all the o-Ps decay rate measurements, positrons from the β + source are
injected into the target materials to form Ps. These materials such as gases,oxide powders (e.g. SiO2 ) or cavities whose inside walls are coated by oxidematerials (MgO or Al2 O3 ), serve as the source of electrons constituting the oPs. The o-Ps continues to collide with the surrounding target materials, andinevitably cause the pick-oﬀ annihilation with the atomic electrons inside thetarget materials. It also occurres that the electron of the o-Ps exchange its
spin with the electrons of the target materials. This conversion from o-Ps
to p-Ps results in the rapid annihilation into two photons, which is referred
to as spin-ﬂip.
In these circumstances, the observed decay rate is expressed as
λobs (t) = λ3γ + λpick (t).

(2.1)

where λ3γ is the intrinsic decay rate of o-Ps and λpick (t) is the contribution of
the pick-oﬀ annihilation, which is generally dependent on time. Thus, how todeal with the pick-oﬀ annihilation is a main issure in all the measurements.

2.1.2

Previous Extrapolation Method

Phenomenologically, the pick-oﬀ rate would be expressed as,
λpick (t) = nσa v(t),
10

(2.2)

11

2.1 Experiment

where n is the density of the target materials, σa is the annihilation cross section, and v is the velocity of the o-Ps. Since the pick-oﬀ rate is proportional
to the density of the target material, a extrapolation method is considered
to be eﬀective. In the extrapolation method, the decay rate was measured
at some points varying the density of the target material and extrapolation
to zero density yields the intrinsic decay rate. Indeed, the time dependencyof the o-Ps velocity is a trouble at this stage. But, the velocity of the o-Ps
must converge to a constant value corresponding to the room temperaturesof 0.03 eV after some time from the o-Ps formation.
In fact, a number of experiments had been performed on the above assumptions by the early 90’s. Unfortunately, they suﬀered from the large
systematic error, that was known as orthopositronium lifetime puzzle. The
reasons are as follows.
• It is diﬃcult to determine the time when the o-Ps are fully thermalized.
Since the number of o-Ps event decreases exponentially in the later
decay time region, one cannot ﬁgure out the time when the decay rate
will be constant. Therefore, the ﬁtting start time inevitablly has thelarge systematic uncertainty.
• The situation gets worse when the density of the target material isvaryed. The thermalization process is strongly dependent on the density of the material and the properties of the surface. Especially inthe lower density, the o-Ps may not have been fully thermalized by its
decay time which is several hundreds ns at most. This will disturb theassumption of the linear dependence on the density.
• The kinematic energy of the o-Ps in the apparatus obeys the Boltzmann equation in the large part. But especially in cavity experiment,
there are some components which are not fully slowed down in thetarget material. They have the high probability of the pick-oﬀ annihilation on the cavity wall and sometimes escape from the detection
area. Thus, the estimation of those component and understanding of
their behavior are required.

2.1.3

Direct Measurement of the Pick-oﬀ Rate

In order to overcome the diﬃculty of the extrapolation method, the othermethod is employed in this experiment. The idea is that if the pick-oﬀ
rate is directly measured with its time dependency, the intrinsic decay rate
will be determined from the observed decay rate without the extrapolation
procedures. In this case, no asumptions on the thermalization process andthe behavior of the o-Ps are required.
For this purpose, high energy resolution germanium detectors are usedin this experiment. They measure the energy of the γ-ray together with

12

2.1 Experimentits decay time. The energy distribution of the γ-rays from the pick-oﬀ
annihilation is the monochromatic 511 keV peak and totally diﬀerent fromthe continuous distribution of the 3γ. Therefore, the ratio of the pick-oﬀ
2γ to 3γ is easily obtained from the energy spectrum of the germanium
detector. Once the λpick (t)/λ3γ has been determined, the observed time
spectrum can be ﬁtted with the function including the ratio to extract the
intrinsic decay rate.
The steps to the function of the expected time spectrum is rather straightforward. The population of o-Ps at time t, N (t), is expected to obey thefollowing diﬀerential equation,dN (t) = −(λ3γ + λpick (t)) N (t),dt
the solution of this equation is,

N (t) = N0 exp −

 t
0



 

(2.3)



λ3γ + λpick (t ) dt



.

(2.4)

Then, the measured time spectrum, Nobs (t) is expected to be,
Nobs (t) = ελobs (t)N (t)= (ε3γ λ3γ + εpick λpick (t)) N (t)= 3γ λ3γ 1 +pick λpick (t)3γ λ3γ

× N0 exp −λ3γ

 t
0

1+

λpick (t )
dt ,
λ3γ

(2.5)

where εpick and ε3γ are the detection eﬃciencies for the pick-oﬀ annihilations and 3γ decays respectively.
Finally we obtain the following form,


 t



λpick (t )1+dt + C .
Nobs (t) = e
λ3γ
0
(2.6)
The constant term C comes from accidental hits whose stop signals do notcorrespond to the β trigger. Another factor exp(−Rstop t) in Eq. (2.6) comesfrom the fact that the TDC always accept the ﬁrst γ hit as a stop signal.
The size of the term is determined by the random counting rate Rstop , which
is proportional to a strength of the positron source.
In the end of this section, the pick-oﬀ rate in the thermalization process
will be treated in a rather analytic manner. If the fractional energy loss ofo-Ps per collision with the material and the collision rate are dependent on
its energy, time dependence of the average kinetic energy of o-Ps at time t,E(t), is derived from the Boltzmann equation as in [41, 42],
−Rstop tεpick λpick (t)1+N0 exp −λ3γ
ε3γ λ3γ13

2.1 Experiment∞dE(t) = − 2mP s E(t) E(t) − 3kB T
aj
dt
j=0

E(t)
kB T

j/2

,

(2.7)

where mP s is the mass of o-Ps, T is the room temperature, and the kB
is the Boltzmann constant. In the equation, the asymptotic value of E(t)is 3kB T and the momentum transfer cross section of the SiO2 is expanded
in terms of E(t). Then, the coeﬃcients aj include all the information of thethermalization process of o-Ps.
Since the pick-oﬀ rate is proportional to the average velocity of the o-Ps
as can be seen in Eq. (2.2), the ratio of the pick-oﬀ rate to the intrinsic o-Ps
decay rate, θ(t) ≡ λpick (t)/λ3γ , obeys the following diﬀerential equation,∞d2
θ(t) = −C θ(t)2 − θ∞aj θ(t)j ,
dt
j=0

(2.8)

where θ∞ ≡ θ (t → ∞), aj (j = 0, 1, 2, · · ·), and C are constant values.
It is well known from the previous measurements that it takes about
600 ns for o-Ps to be well thermalized in low-density SiO2 powders.

2.1.4

Measurements in 1995 and 2001

In 1995, the method which directly measures the pick-oﬀ rate was used forthe ﬁrst time [20, 21]. The result was consistent with the QED prediction.
In 2001, the more accurate experiment was performed by the same technique [37, 38]. In this experiment, the following progress was made.
• The two types of SiO2 powder with quite diﬀerent pick-oﬀ ratios wereused for the systematic test of the direct pick-oﬀ correction method.
The obtained values are consistent with each other.
• The timing system and the time walk correction were improved. In
the end, the dependence of the decay rate on the ﬁtting start time wasdisappeared by 100 ns ﬁtting start time.
• The total error of about 230 ppm was achieved. The obtained decayrate was consistent with the O(α2 )-corrected QED prediction.
However there were several remaining problems,• The experimental error was still large for the test of O(α2 ) correction
which account for 240 ppm of the decay rate.
• The ﬁtting start time of 100 ns was used for the decay rate ﬁtting
because the decay rate was sytematically increasing before 100 ns.
The origin of this increase was not understood at that time, and clear
explanation is needed to prove the validity of the method.

2.2 Experiment14• The energy window was set to the Compton free region. But, thedependence on the energy window was rather large.
• Some systematic uncertainties were not fully understood. For example, the detection eﬃciency was only estimated by the Monte Carlosimulation.

2.2

New Features of the Present Measurement

In order to clear up the remaining problems (see Sec. 2.1.4), the following
trials are performed in the present measurement and some improvements
are made in the end.

2.2.1

Introduction of Fast Inorganic Scintillator

The previous measurements adopted NaI or CsI scintillators. These scintillators have the good energy resolution. However, they have the rather
long time constants and the timing characteristics are not so good. Then,other fast inorganic scintillators such as YAP, GSO, YLSO are evaluated in
the laboratory. The signal shapes measured with Flash ADC are shown in
Fig. 2.1.
The main characteristics of the signal shapes are as follows.
NaI scintillator
The time constant is as long as 200 ns. In addition, the large tail isobserved by several μs.
YAP scintillator
The time constant is about 30 ns. The long tail is also observed. But,
the amount is a % level.
GSO scintillator
The time constant is about 60 ns. The second component with thetime constant of about 300 ns is clearly seen.
YLSO scintillator
The time constant is about 40 ns. The long component is hardly
observed. Unfortunately, the large β decay background is observed.
Thus, the YAP scintillator is found to be most suitable for the decay ratemeasurement. The detailed properties of the YAP scintillator are shown inSec. 2.3.3.
The setup of the measurement are newly constructed on the base of
the YAP scintillator. With the introduction of the YAP scintillator, thefollowing improvements are made.

15

2.2 Experiment

102

102

NaI

YAP

10

10

1

1

10-1

10-1

0

500 1000 1500 2000 2500 3000 3500
Time (ns)102

0500 1000 1500 2000 2500 3000 3500
Time (ns)

102

GSO

YLSO

10

10

1

1

10-1

10-1

0

500 1000 1500 2000 2500 3000 3500
Time (ns)0

500 1000 1500 2000 2500 3000 3500
Time (ns)Figure 2.1: Signal shapes of the various scintillators. Signal shapes aremeasured with Flash ADC.

2.2 Experiment16• The excellent time resolution is obtained. The resolution is less than
1 ns for the whole energy region.
• Due to the fast time constant of 30 ns, the eﬀect of the pile-up isdecreased by a factor of 10. It makes the system more tolerant to highevent rate.

2.2.2

New Design of Ps Formation Assembly

The previous measurement adopted 22 Na for positron source. But, it isfound that the large part of positrons emitted from 22 Na are absorbed in thetrigger plastic scintillator. To increase the eﬃciency of the o-Ps formation,
more energetic positron source 68 Ge is introduced. Note that, as shownin Fig.2.4. positrons are emitted from the daughter nucleus 68 Ga. In thecase of 68 Ge, about 80 % of the positrons path through the scintillator andthe aluminized mylar without being absorbed. Then, the o-Ps formationassembly are designed to meet the change. The detailed descriptions areshown in Sec. 2.3.1 and the main structure is composed of the following two
systems.
trigger system
The emission of positron is detected by the thin plastic scintillator,
whose thickness is optimized to 200 μm. The light of the trigger
scintillator is collected by a mirror made of alminized mylar.
anti-trigger system
Some positrons of 68 Ge are too energetic to stop in the silica target,
which is about 3 cm thick. The 1 mm thick plastic scintillator on thewall of the container detects these positrons and suppresses the trigger
signal. The system is refered to as the anti-trigger.

2.2.3

Geant4 Simulation and its Validation

The previous measurement used a γ-ray simulation they developed for theexperiment. The simulation is fast and correctly reproduces the absorption coeﬃcients of γ-ray. But, the present measurement adopts the Geant4
simulation [43] for the following reasons. The detailed procedures of thesimulation is shown in Sec. 2.4.
• The Geant4 packages are validated by many users and their functions
are established at present.
• The Geant4 handles positron. The simulation of positron is needed todetermine the decay point. In addition, the secondary particles of theemitted positron such as electrons, bremsstrahlung-γ’s sometimes hit
the detector or anti-trigger.

17

2.3 Experiment• The Geant4 simulates the secondary particles. The secondary electrons and bremsstrahlung-γ’s sometimes enter the detector or escapefrom the detector. The eﬀect is observed in the Compton free region.
• The Geant4 supports the various eﬀects in the low energy region. For
example, the Geant4 reproduces Pb X-rays and the broad Compton
edge, which is caused by the electron’s binding momentum.

The Geant4 correctly simulates all the physics interactions. But, somecharacteristics speciﬁc to the detector are not reproduced, such as chargecollection eﬃciency and optical photon collection eﬃciency. Therefore, theresponse function of the detector is modiﬁed by the measured data (seeSec. 2.4). Consequently, the realistic response function suppresses the systematic uncertainty of the detection eﬃciency which leads to the error of
the pick-oﬀ correction and the energy window dependence of the decay rate.

2.3

Description of Apparatus

In this experiment, two measurements are performed, that are referred toas ‘RUN I’ and ‘RUN II’. Some properties of the experimental setup suchas the target material, the number of detectors has been changed between
the RUN’s.

2.3.1

Positronium Formation Assembly

The positronium formation assembly used in this experiment is shown in
Fig. 2.2. The 68 Ge positron source (endpoint 1.9 MeV) is placed at the center of the assembly. The positrons are actually emitted from the daughter
nucleus 68 Ga as shown in Fig.2.4. The strength of the source is approximately 0.3 μCi for RUN I and 0.2 μCi for RUN II. Since the radioactive
half-life of 68 Ge is 271 days, the event rate is considered to be stable during
a RUN. The expected γ-ray from the source is a 1077 keV γ-ray, whose
emision ratio is about 1% of the β + decays. The 1077 keV γ-rays are madeuse of to calibrate the energy scale of detectors.
The 68 Ge source is sandwiched between two sheets of plastic scintillators (NE102). The scintillators are 200 μm thick for each. The light of thescintillator is collected by a cone made of aluminized mylar (25 μm thick)and 1/2-inch photomultiplier (H3165P-10VT) to trigger the β + events. Owing to the relatively high endpoint of the positron source, about 80 % of thepositrons path through the scintillator and the aluminized mylar without being absorbed. On the other hand, half of the positrons are too energetic tostop in the target material and they will immediately annihilate on the wallof the assembly. To suppress these events, 1 mm thick cylindrical plastic

18

2.3 Experiment

&,# - $)


  ' 
$)* #$ 







+
%  &#  ' 
$ 

 

 

 


 

 ./ /'  & &"  # $!  

 



 "($   






Figure 2.2: Schematic diagram of apparatus (side view).

19

2.3 Experiment      



 





 




      








  !

  



Figure 2.3: Schematic diagram of apparatus (top view).

20

2.3 Experimentscintillator is placed on the surface of the wall. The light of the scintillator is collected by the two 1-inch photomultiplier (R1924P) and gives theanti-trigger signal to kill the trigger signal.
The assembly container is made of a 1 mm thick glass beaker, whose
diameter and height are 67 mm and 110 mm respectively. This container isﬁlled with silica aerogel (RUN I) or silica powder (RUN II). The characteristics of these materials are discussed in the next section.
The assembly container is evacuated to 6 × 10−2 mbar by a rotary pump
(EDWARDS E2M5), which is connected to the one of the two holes on top of
the assembly. The other hole is connected to the pirani gauge to monitor thepressure of the container. Though O2 in the air has the unpaired electron
spins and easily convert the o-Ps into the p-Ps by spin-exchange collisions,the eﬀect is also corrected in the direct pick-oﬀ correction method. The pickoﬀ rate of o-Ps in 1 atm of oxygen was observed to be about 35 μs−1 [44],and one can expect 4.2 × 10−4 μs−1 increase of decay rate. This corresponds
to λpick /λ3γ = 0.006%, which is much smaller than silica pick-oﬀ rate.
Due to the Zeeman eﬀect, the mixing between the o-Ps and the p-Ps
can occur in the presence of a magnetic ﬁeld. The absolute strength ofmagnetic ﬁeld around the positronium formation assembly is measured with
3-Axis Hall Gaussmeter [45] to be 0.5 ± 0.1 gauss. This feeble ﬁeld hardly
contributes to the mixing of the two states, and estimated to be 3×10−11 [46].

68
32Ge

68
31Ga

270.8dEC = 100%

EC = 10.9%+Br(β )
1077.467.63m68
31Ga

67.63mstable

1.1%
88.0%

68
30Zn

Figure 2.4: The level diagram of the 68 Ge isotope. The 68 Ge nucleus decays
to the 68 Ga nucleus through the electron capture (left diagram). Then,the 68 Ga nucleus decays to the 68 Zn nucleus through the β + decay (right
diagram).

21

2.3 Experiment

2.3.2

Characteristics of SiO2 Powders

In this experiment, two types of SiO2 materials are used as a target. The
one type is silica powder provided by Nippon Aerosil Ltd.. The silica powder consist of the primary silica grain, whose size is about 16nm. The
primary grains generally form the group and the groups form the higher
class of group. Then, the aggregate structure reached the size of several
μm. The silica powder is originally hydrophilic and the grain is coverd
by the hydroxyl groups (−OH). But the company provides hydrophobic
products whose hydroxyl groups are substituted with the tri-methyl-silylgroups (−Si − (CH3 )3 ) in hydrophobic processing. it is better to use thehydrophobe rather than the hydrophile since the water on the surface of thegrain surely increase the pick-oﬀ annihilation probability. In this experiment, to remove a water absorbed at the surface of grains, the SiO2 powder
is heated for 4 hours long in 130 ◦ C just before evacuation of the container.
The other type of silica material is silica aerogel, which is a solid material and not powder state. In the silica aerogel, a chain-like aggregationof the silica makes 3-dimensional network structure. The 0.03g/cm3 silica
aerogel is provided by the Matsushita Electric Works Ltd.. The hydrophobic
processing is also applyed to the silica aerogel used in the measurement.
The characteristics of two silica materials are summarized in Table 2.1.

Grade (code name)primary grain size (nm)surface area (m2 /g)
density (g/cm3 )mean distance (nm)process

Powder
R972CF16
110 ± 200.035
660
hydrophobe

AerogelSP-7
10unknown
0.03
480hydrophobe

Table 2.1: Characteristics of two powders used in the measurement
The surface area of the grain is estimated by the BET (Brunauer, Emmett, and Teller) method which utilizes the physical adsorption of nitrogengas on the surface of SiO2 powders [47]. The mean distance between thegrains L has been calculated with the assumption of uniform size and spacial
distributions of the grains as,
L=

4
3





ρ0
−1 R
ρ

(2.9)

where ρ0 is the bulk density of amorphous silica (2.20 g/cm3 ), ρ is thedensity of the powder, and R is the mean radius of the primary grains.
Because the pick-oﬀ rate is proportional to the collisional frequency between
the o-Ps and the grains, it strongly depends on the mean distance L.

22

2.3 Experiment

2.3.3

Detectors

In this experiment, two types of γ-ray detectors are used. One type isthe germanium semiconductor detector (Ge), and the other is the YAP
scintillator. Ge detectors, which has excellent energy resolution, are used toprecisely determine the pick-oﬀ ratio λpick (t)/λ3γ as a function of time. YAP
scintillators, which have good time resolution, are used to obtain the time
spectrum of the o-Ps decay. Three germanium detectors are used throughout
two RUNs, while three (four) YAP scintillators are used for RUN I (II).
Germanium detectors
The germanium detectors used in the measurement are the P-type highpurity coaxial germanium detectors (Ortec GEM 38195-P-plus series).


         

Figure 2.5: Internal structure of germanium detector.

The design of Ge detectors is shown in Fig. 2.5, where the size of crystals
are bit diﬀerent between the individuals and the values are summarized inTable 2.2. The lithium-diﬀused layer which is denoted as inactive volume
in Fig. 2.5 has been formed over the outer surface of crystal. The layer isabout 700 μm thick. The arrangement of the three Ge detectors is shownin Fig. 2.2. From now on, they are referred to as Ge0, Ge1 and Ge2 as inthe ﬁgure.
The energy resolutions are measured at several energy points by using
line-γ peaks from the γ-ray sources: 152 Eu, 137 Cs and 85 Sr. These values

23

2.3 Experiment

Serial Number
Length(mm)Diameter(mm)Ge 028-TP 10070
73.8
58.3

Ge 1
28-TP 10096
67.4
60.3

Ge 2
45-TP 22020A
48.5
68.5

Table 2.2: Crystal size of germanium detectors

are ﬁtted with linear function as in Fig. 2.6, and used for smearing of theenergy spectrum of the Monte Carlo simulation.
Throughout the measurement, 2mm thick lead sheet is placed in frontof the detector as seen in Fig. 2.5. It suppress the small energy photons,
which pile up and disturb the energy spectrum around 511 keV.
The relative peak eﬃciencies with the lead shield are measured using
152 Eu. 152 Eu has various line-γ peaks and their relative strength are wellknown and can be found in the data sheets [48]. On the other hand, the eﬃciencies are estimated by the Monte Calro simulation implemented with thesame geometry. As shown in Fig. 2.7, the measurement and the simulationshows a good agreement throughout the whole energy range.
The solid angles from the point source are about 3.4%, 3.5% and 4.7%
(of 4π) for Ge0, Ge1 and Ge2 respectively.
YAP scintillators
YAP (YAlO3 ) scintillator is a fast inorganic scintillator, whose characteristics are summarized in Table 2.3 [49, 50]. The YAP scintillators used in
the measurement are 50mm × 50mm × 33mm large crystal made in Czech
Republic. The crystals are lapped by aluminized mylar and black sheet(200 μm thick). Then, they are connected 2-inch photomultipliers (R32902). Four YAP detectors are arranged as in Fig. 2.2, though the YAP3 had
not been deployed in RUN I. Their solid angles are 4.9%, 4.9%, 6.6% and4.8% (of 4π).
The energy resolutions are measured using line-γ peaks from several γray sources: 113 Sn,85 Sr,137 Cs. These values are summarized in Table 2.4.

2.3.4

Timing system

The timing system is an essential part of the experimental setup and thenon-linearity of the TDC directly aﬀects the ﬁnal result.
Among the various types of TDC, the direct clock count type is suitable
for this experiment. This is because,• This type of TDC counts a constant frequency oscillator during the
input time period. Then, the integral non-linearity (INL) is the orderof the oscillator’s precision and generally small (∼ 10−6 ).

24

2.3 Experiment

Energy resolution σ (keV)

0.9
0.8

Ge1
0.7
0.6

Ge0
0.5

Ge2

0.4
0.3
200 300 400

500 600 700 800 900 1000
Energy (keV)

Figure 2.6: The energy resoluton of the Ge detectors as a function of theγ-energy. Energy resolution is expressed as the one standard deviation in
keV unit. Obtained functions are, for Ge 0 : σ = 0.457(1) + 0.276(2) ×
10−3 × E(keV), for Ge 1 : σ = 0.466(1) + 0.276(3) × 10−3 × E(keV), for Ge
2 : σ = 0.371(1) + 0.327(2) × 10−3 × E(keV)25

relative efficiency

2.3 Experiment

0.012
0.01
0.008
0.006
0.004
0.002
0
100

200

300

400

500

600

700

800

900
1000
Energy (keV)

600

700

800

900
1000
Energy (keV)

600

700

800

900
1000
Energy (keV)

relative efficiency

Ge0
0.012
0.01
0.008
0.006
0.004
0.002
0
100

200

300

400

500

relative efficiency

Ge1
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
100

200

300

400

500

Ge2

Figure 2.7: The relative peak eﬃciency of the germanium detectors. Measured data using the 152 Eu source are plotted with closed circles as a functionof γ-ray energy. The continuous line represent the relative eﬃciencies estimated with the Monte Carlo simulation. The data is normalized so as to ﬁt
the simulation.

26

2.3 Experiment
name
dope
atomic number (Z)densityemision peaklight output
decay constantrefractive index (n)absorption length

YAP (YAlO3 )cerium (Ce)39
5.37 g/cm3
370 nm
40 % of NaI
30 ns
1.95
15∼20 cm

Table 2.3: Properties of YAP scintillator

SourceEnergy (keV)

113 Sn

391.7
514.0
662.0

85 Sr
137 CsResolution in σ (keV)
YAP0 YAP1 YAP2 YAP3
27.3
30.8
27.8
31.3
33.3
33.0
34.3
34.9
39.2
41.2
39.4
47.4

Table 2.4: Energy resolution of the YAP scintillator• The diﬀerential non-linearity (DNL) is caused by the transitions of
the ﬂip-ﬂops and typically the order of 1 %. But, the DNL has aperiodical structure as 2n (n=1,2,3...) interval and rapidly decreases
in the longer range.
Considering the above situation, the new direct clock count type TDC
was developed at the previous measurement [38] with the exhaustive cooperation of the staﬀs of an Electronics Facility of High Energy Accelerator
Research Organization (KEK). The main features of the new TDC are as
follows.
• A direct clock count type TDC with 8 ch LEMO inputs. The TDC
is a kind of 16 bit counter and the time range is determined by theexternal clock source, which is connected through a SMA connector
on the front panel.
• The external clock source is a DDS-PLL synthesizer, DPL-2.5G (Digital Signal Technology Inc.). The DPL-2.5G utilize Direct Digital Synthesis (DDS) and Phase Locked Loop (PLL) techniques, which are
explained below. For the present measurement, the frequency of theDPL-2.5G is adjusted to 2 GHz. Therefore, the time range of the
16 bit TDC is 32 μs. The stability of the internal clock frequency isguaranteed as ±2.5 ppm against the temperature −10 ◦ C ∼ 50 ◦ C and±3 ppm against one year.

27

2.3 Experiment• The diﬀerential non-linearity of the TDC is 0.4 ∼ 0.5 % for 2 bin widthand 0.1 ∼ 0.2 % for 4 bin width. But the time walk correction andthe rebinning operation in the analysis totally smear the DNL.
Phase Locked Loop (PLL) ˙
The Phase Locked Loop (PLL) is a circuit whose output is locked(syncronized) on the input oscillator. The PLL consists of phasecomparator, loop ﬁlter and voltage controlled oscillator (VCO)
as shown in Fig. 2.8. At ﬁrst, the phase comparator detects thephase diﬀerence between the input signal and the loop-backed
signal. Then, the frequency of the output signal is adjusted inthe VCO according to the detected phase diﬀerence. That is tosay, the frequency of the output signal follows that of the input
signal with some time constant. With this mechanism, the rapidchange of the input frequency and the noise component is totallyeliminated and the output frequency is highly stabilized. Anotherblock, the loop ﬁlter is a low pass ﬁlter which eliminate a ripple
on the signal and also determine the time constant of the PLL.
The frequency of the output signal can be changed by the divider
or the mixer inserted in the loop-backed circuit.

    



      
   


Figure 2.8: Schematic diagram of Phase Locked Loop (PLL).

Direct Digital Synthesis ˙Though the frequency of the PLL is variable, the large factor ofthe division distorts the signal shape and the ﬁne adjustment of
the frequency is also diﬃcult. To achive the high purity and theﬁne adjustable frequency, the Direct Digital Synthesis (DDS) is28

2.3 Experimentintroduced. The DDS consists of the several parts as described inFig. 2.9. The signal shape is prepared in advance as the Look up
table on the memory. Then the Digital Analog Converter (DAC)
and the some ﬁlters reconstruct the signal shape. The frequency
is determined by the Accumulator which adds the adjusted value
at every clocks. The output of the accumulator addresses the
Look up table, which represents the phase of the signal.

 "
   #"  # !" 
 
      

   

Figure 2.9: Schematic diagram of Direct Digital Synthesis (DDS)The frequency of the clock is adjusted to 2 GHz within 1 ppm accuracy
before the measurement. Since the TDC is a direct clock count type andprincipally miscount never occurs, the integral non-linearity is expected tobe less than 15 ppm corresponding to the bin width.
Eﬀect of the smearing operation
The detector resolution and the analysis operation such as the time walkcorrection are considered to be smearing operation with the gaussian function. The exponential function and the ﬂat distribution do not change its
slope in the convolution operation with the gaussian function as, ∞1(t − t )2
N exp(−λt ) · √ exp −
2σ 2
σ 2π
−∞

dt = N exp −σ 2 λ22exp(−λt)(2.10)
Therefore, the smearing operation does not aﬀect the measured decay rate.
Note that the fast periodical structure such as the intrinsic diﬀerential nonlinearity of the TDC is dropped in the smearing operation.

29

2.3 ExperimentEﬀect of the binning operation
When the exponential distribution is ﬁlled in the histogram, the content of
the bin (t − Δt/2 ∼ t + Δt/2) is expressed as, t+Δt/2
t−Δt/2N −λ(t−Δt/2)e− e−λ(t+Δt/2)
λ1∼ N exp(−λt)(Δt + λ2 Δt3 )24

N exp(−λt )dt =(2.11)(2.12)

while the value of the function is N exp(−λt) at the center of the bin. Thus
the binning operation is considered to be the change of the normalization,
which is independent of the point t. Therefore, the binning operation does
not aﬀect the measured decay rate.

2.3.5

Electronics

Data ﬂow
The data acquisition system of the experiment consists of the clusters of NIM
standard modules and the CAMAC system. Fig. 2.10 shows the data-ﬂow
of the total system. The system mainly consists of three part, the trigger
part, the YAP part, and the Ge part. Each part has a latch veto structure
where the system accepts the ﬁrst signal and immediately sets the latch byitself to block the second signal. Thus, only one trigger signal is produced
for each event cycle. The latch of the trigger part, denoted as the Main
latch in the ﬁgure, is released by a pulse from the output register module of
CAMAC system. The latchs of the YAP and Ge parts, denoted as the Sub
latch in the ﬁgure, are released by the latch reset signals from the trigger
part.
The trigger part also generates the start signal for the TDC and the each
γ-ray detector stops the TDC to record the detection time. The energy
information is also recorded with the ADCs for each detector. The gate
signals for these ADCs are made by the corresponding detector signals, notby the trigger signal. The following three ADC gates are prepared for eachsignal.
• Wide gete which is enough wide to cover the signal. The energy
information of the signal is measured with the Wide gate.
• Base gate which is set prior to the signal. The baseline condition ismeasured with the Base gate.
• Narrow gate which is narrower than the Wide gate and covers thepeak of the signal. The energy diﬀerence of the Wide gate and Narrowgate is used to reject the pile-up events.
Details of the data-ﬂows and the modules are described for each part.

30

2.3 Experiment    


 

"

 



"

 

   

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

# 

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 



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# 

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 
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 

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

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 





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 
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 

  !  Figure 2.10: Schematic diagram of the process data-ﬂow.

 



31

2.3 Experiment

Trigger system
A schematic diagram of electronics for the trigger part is shown in Fig. 2.11.
The following explanations enumerated by the indices are corresponding to
the label within Fig. 2.11.
(t.0) The trigger PMT is operated at -1160V and the two anti-trigger PMT
is operated at -1160V and -1110V respectively. These high voltages
are supplied by a negative high voltage module Repic RPH-031.
(t.1) The output of the trigger PMT are divided into three signals by alinear fanout module. One signal is sent to a discriminator to produce
the trigger signal. The other two signals are fed into a charge sensitive
ADC denoted as Trig ADC (LeCroy 2249A), through delay cables.
The Trig ADC measures the amplitude of the trigger PMT signal andthe baseline condition just before the signal, which is refered to as
Wide and Base from now on.
(t.2) The amplitude of the signal is about −50 mV for a typical energy
deposit of 50 keV. The threshold value is ﬁxed at −12 mV and morethan 90 % of β + events are triggered.
(t.3) The outputs of the two anti-trigger PMTs are summed in a linear
fanin fanout module and used as a veto signal for the trigger logic.
The amplitude of the signal is about −150 mV for a typical energy
deposit of 200 keV and the threshold value is ﬁxed at −40 mV . Abouta half of the positrons pass through the silica target and almost all ofthem produced the anti-trigger signals. In fact, the counting rate isdecreased by half at the logic unit.
(t.4) Only when the veto signal is not imposed, the trigger signal passes
through the logic unit. The margin of the timing between these two
signals is 20 ns.
(t.5) One output signal of the logic unit is used to set the main latch toinhibits the succeeding trigger signals. This main latch is released by
the CAMAC controller at the beginning of the event cycle. Another
output is used as the start signal for the main TDC. This TDC is oﬀen
called as KEK TDC since KEK and our group jointly developed it as
the 2 GHz clock count type TDC.
(t.6) One of the output signals from the logic unit provides the gate signal
for the Trig ADC. The gate width is set to be as narrow as 50 ns. Thisis required for the reason that the ADC value correlates the pile-up
activity within the gate width, and the time walk correction or cut
by this value results in disturbing the time spectrum of the accidental
events.

32

2.3 Experiment# )

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     !"# $%  & ( 3  2,"&. . ) / 2,"&. .%  #( ,  ( ,.
(  ' )*#6
'( +( , ) 2,"&. . )  2,7*"
'( +( ,Figure 2.11: Schematic diagram of electronics for trigger system.

33

2.3 Experiment(t.7) The trigger signals are distributed to all the systems as the latch reset
signals. The latch reset signal is as narrow as 20 ns since the longer
reset signal may block the latch set signal when the signal comes justafter the release of the veto signal.
(t.8) At the end of event cycle, the force veto signals are distributed to set
all the latches by force. In addition, some other signals are distributed
to stop all the TDCs.
Electronics for the YAP scintillators
A schematic diagram of electronics for a YAP scintillator part is shown in
Fig. 2.12. The logic described in the ﬁgure is for just one YAP system. Threesystems are prepared for RUN I and four systems for RUN II corresponding
to the number of YAP scintillators used. As in the description of the trigger
part, the following indices are corresponding to the labels in Fig. 2.12.
(y.0) The photomultipliers for the YAP scintillators are operated around-1.7kV. These high voltages are supplied by a negative high voltage
module Repic RPH-031.
(y.1) The output of the YAP PMT is divided into four lines by a linear
fanout module. One of the divided signals is fed into a discriminator.
The others are used to measure the amplitude of the signal (Wide)and the baseline condition (Base). The amplitude of the signal is alsomeasured by the narrower gate, which is referd to as Narrow, since
the diﬀerence to the longer gate is sensitive to the pile-up activity.
For the measurements of YAP Narrow and YAP Wide, the charge
sensitive ADC (phillips 7167) is used , which is denoted as YAP ADC
in the ﬁgure.
(y.2) The pulse-height of the 511 keV signal is about 0.5 V . The thresholdvalue is set to 70 mV , which is enough high to reject after pulse signals.
The noise level is far below the threshold.
(y.3) The baseline condition of the signal line is measured at the time justbefore the YAP signal (Base), which is performed by the charge sensitive ADC denoted as Base ADC (phillips 7167). In addition, thebaseline condition is also measured at the timing of the prompt (T0 )by the Trig ADC. The signals for the baseline measurements are ampliﬁed by a factor of 2.5. This is because the resolution of the Base
ADC value is originally as narrow as ADC bin width.
(y.4) The output of the discriminator is delayed by 100 ns in order to wait
the latch reset signal from the trigger part and fed into the logic unit.
The force veto signal from the trigger part is ORed in the logic unit
and the latch will be set by force at the end of event cycle.

34

2.3 Experiment5))/

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Figure 2.12: Schematic diagram of electronics for YAP scintillator system.

35

2.3 Experiment(y.5) The YAP part has its latch mechanism which prevent the double
trigger in a event cycle. This latch will be released by a logic signal
from the trigger part.
(y.6) One of the output from the logic unit provides the stop signal for the
KEK TDC. The others are used to make the gate signals for the YAP
ADC and the Base ADC. The gate widthes of the Base, Narrow and
Wide are 50 ns, 150 ns, and 500 ns respectively.
(y.7) The phillips 7167 ADC has individual gate for the each input signal
and the enable signal which covers all the gate signals is required tomake the gates active. Therefore, the timing signals of all the detectors
are fed into a fanin module and its output is used to produce the enable
signal.
(y.8) The enable signal is also used to generate the gate signal for thetemperature measurement.
Electronics for the germanium detectors
This section explains the electronics for a Ge detector section. The schematic
diagram is shown in Fig. 2.13.
(g.0) The bias voltages are supplyed by the bias supply modules (EG&G
ORTEC 459) through the HV buﬀers (SEIKO EG&G DS-1550). The
HV values are +2600 V for Ge0, +2000 V for Ge1, and +3500 V forGe2.
(g.1) The pre-ampliﬁer (or FET) of Ge detector provides the two identicalsignal outputs. The one of the pre-ampliﬁer outputs is sent to a spectroscopy ampliﬁer (EG&G ORTEC 673). The time constant of theshaping time is 6 μs. The output of the ampliﬁer is fed into a peakhold ADC (hoshin C011) for the measurement of the energy (Wide).
(g.2) The another output is fed into a timing-ﬁlter ampliﬁer (EG&G ORTEC 474). The time constants of the diﬀerentiation and integration
are set to be 500 ns and 20 ns respectively. The analog outputs of thetiming-ﬁlter ampliﬁer are used for the measurements of the baseline
condition (Base) and the amplitude (Narrow), and fed into the Base
ADC and the Ge ADC (phillips 7167) respectively.
(g.3) The outputs of the fanout module are fed into a discriminator to generate the timing signals. The three threshold values (30 mV , 50 mV ,150 mV ) are imposed to measure the risetime of the Ge signal. Thesetiming are recorded by the Ge TDC (Repic RPC-061) prepared foreach Ge detector. The timing signal discriminated by 50 mV threshold is also sent to the logic unit as the trigger signal.

36

2.3 Experiment

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/ =</*($%
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2 2/ =</*($%
 
) !"(%%

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24




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


 
!"8 $9#! : ) !"$%

!*# 3%4 
) !"5267

Figure 2.13: Schematic diagram of electronics for Ge detector system.

37

2.3 Experiment(g.4) The pre-ampliﬁer is a transistor reset type ampliﬁer, which releases
the collected charges periodically, and the timing of the transistor
reset is provided as the transistor reset pulse (TRP). The TRP isabout 10 Hz and it disturbs the signal lines for a while. Therefore,
the inhibit signal is used to veto the electronics of the Ge systems for
11ms.
(g.5) The trigger logics such as the latch mechanism and the force veto are
the same as in the YAP part. The one of the outputs provides thestop signal for the KEK TDC. The another output provides the startsignal for the Ge TDC.
(g.6) Just like the case in the YAP part, Base gate, Narrow gate, and Widegate are prepared. The widths are 100 ns, 2 μs and 40 μs respectively.
(g.7) The Ge ADC is also a individual gate type ADC and it requires thegate enable signal. Then, the timing signals of the Ge detectors are
ORed in the logic unit and the output is provided as the enable signal.
Electronics for the temperature measurement
The room temperature is measured with the temperature sensor IC (LM35DZ).
The output of the sensor is connected to the ADC (phillips 7167). Then, thelevel of the voltage is measured with 50 ns ADC gate as shown in Fig. 2.12(y.8).

2.3.6

The CAMAC Data Acquisition System

Data acquisition is performed with the CAMAC system. The CAMAC
system on the CAMAC Crate (Toyo Corp. PS7500) is controlled by the
CAMAC Crate Controller CC/NET (Toyo Corp). The CC/NET is implemented as a single board computer where the Linux OS debian (ver. 3.1) isinstalled and the data taking program runs on it. The data read from the
CAMAC modules are compressed and stored as ROOT ﬁles (ver. 5.04 [51])on the other NFS host machine.
The procedures of data taking is as follows. At ﬁrst, all the CAMAC
modules are cleared by the logic signal from the output register (Technoland
C-RP202), which is triggered by the controller. Then, the main latch isreleased by the reset signal in the same manner. This makes all the systems
active and the controller waits the interrupt signal, so-called LAM (LookAt-Me) signal from the KEK TDC. The interrupt signal starts the datareading procedures for the KEK TDC. If no detector hits within the 10 μs,
the data is simply discarded and the controller proceeds to the next event
cycle. But if there is any hit, the controller read all the data from the ADCs
and the TDCs. After the reading procedures, the controller starts the next38

2.4 Experiment

event cycle by distributing the clear and latch reset signals to the system,and then saves the data through NFS.
The measured quantities and the used CAMAC modules are summarizedin Table 2.5, 2.6, 2.7.
conventional name
Pla Wide
Pla Base
Ge WideGe Narrow
Ge BaseYAP Wide
YAP Narrow
YAP Base
temperature

moduleLeCroy 2249A (AC)LeCroy 2249A (AC)hoshin C011 (AC)phillips 7167 (AC)phillips 7167 (DC)phillips 7167 (AC)phillips 7167 (AC)phillips 7167 (DC)phillips 7167 (DC)gate width
50 ns
50 ns
40 μs
2 μs
100 ns
500 ns
150 ns
50 ns
50 ns

Table 2.5: Quantities measured with ADCs

conventional nameGe Timing
YAP Timing
Ge T30 mV
Ge T50 mV
Ge T150 mV

moduleKEK TDCKEK TDC
Repic RPC-061
Repic RPC-061
Repic RPC-061
phillips 7186 for Ge2

Table 2.6: Quantities measured with TDCs

2.4

Monte Carlo Simulation

2.4.1

Positron and Photon Simulation

A Monte Carlo simulation plays a crucial role in this experiment. Its main
purposes are,
• The accurate 3γ energy spectrum of the Ge detector is required toseparate the pick-oﬀ contribution from the measured spectrum.
• The accurate detection eﬃciencies for both Ge detector and YAP scintillator are required. The ratio of the 3γ detection eﬃciency (3γ ) andthe pick-oﬀ 2γ detection eﬃciency (pick ) is used in the analysis.

39

2.4 ExperimentmoduleKEK TDC
Repic RPC-061
phillips 7166
LeCroy 2249A
phillips 7167
hoshin C011

bits
16
12
12
11
12
12

full range
32 μs
1 μs
800 nscomments
direct clock count type (2 GHz)
successive comparison typesuccessive comparison type
Wilkinson typesuccessive comparison type
Wilkinson peak-holding type

Table 2.7: Speciﬁcations of the CAMAC modules• Some of the systematic errors are estimated with the Monte Carlosimulation.
For the detector simulation, the Geant4 (ver. 8.0) package [43, 52] isused and the details of the present experimental setups are reproduced in
the simulation. In this package, the interactions of the positron and theγ-ray in various materials are implemented. Practically, the Low Energy
Electromagnetic Physics Library of the Geant4 package is used for the simulation. In this original library, binding eﬀect of Compton scattering is not
yet implemented. Therefore, the extra package called Low-Energy Compton
Scattering package (G4LECS ver. 1.06) [53] is introduced.
The Monte Carlo simulation is proceeded as follows. At ﬁrst, a positronis emitted from the source with the energy corresponding to the energy
spectrum of 68 Ge, which is reported in some articles [54, 55]. Then theinteractions of the positron in the materials are simulated step by step,
ionisation, bremsstrahlung, etc. The simulation of the secondary particles
is also included in this process. Some of the positrons are absorbed in
the trigger plastic scintillator or other materials in the positron formationassembly. But the positrons stopped in the silica target are selected as thecandidate for o-Ps.
The next step is the simulation of the γ-rays emitted from o-Ps. For thesimulation of the pick-oﬀ annihilation, two γ’s with the energy of 511 keV
are emitted from the point where the positron stopped in the positron simulation. Three γ’s from the o-Ps decay are also generated in the same manner, but their direction and energy are caluculated from the O(α) matrix
element [19]. Then, the interactions of all the γ-rays and their secondary
particles are simulated step by step, photoelectric eﬀect, Compton scattering, Rayleigh scattering, etc.
In the end, the energies depositted on various detectors are obtained
for the o-Ps decay and the pick-oﬀ annihilation respectively. The eﬀects of
the anti-trigger and the trigger threshold is negligible, though the events
where the positrons or their secondary electrons hit the anti-trigger havewider distribution of the decay point. The obtained energies are smeared by
the measured resolutions and the response functions of the corresponding2.4 Experiment40

detectors. 3.6 × 109 events of the o-Ps 3γ decay are generated for each RUN.

2.4.2

Response Function of the Ge Detector

The response function of Ge detector is measured by 514 keV single energy
γ-ray of 85 Sr source. The assembly beaker is replaced by the 85 Sr source
and the trigger signal is produced by the detector signal itself. The Fig. 2.14
is the comparison of the measured energy spectrum and the Monte Carlosimulation. The environmental background of the measured spectrum issubtracted by the spectrum taken without 85 Sr source. The two spectra
are in good agreement with each other except for the region around thephotoelectric peak. The observed tail of the photoelectric peak comes fromthe charge incollection in the germanium crystal. The amount of the tail isabout 1% of the photoelectric peak and depends on an individual crystal tobe precise. Therefore, the response functions of three Ge detectors are individually measured as in Fig. 2.15. The diﬀerence is ﬁtted with exponentialfunction and the function is used for the smearing of the Monte Carlo data.
Before the ﬁtting, the slow rise-time events are eliminated by the SRT cut(see Sec. 3.1.5) with the same condition as in the analysis of the run. Thisis because these events are more likely to form the charge incollection tail.
Another factor that disturbs the energy spectrum is pile-up events. Since
the pile-up eﬀect is dependent on the event rate and the energy of the γ-ray,
it is evaluated with the data taken for the decay rate measurement in thisexperiment. Fig. 2.16 shows the right side of the 511 keV photoelectric peakof the data. Since the Narrow-Wide cut, which is explained later, rejects theenergy deviation beyond ±15 keV, the pile-up tail of the photoelectric peakextends up to 525 keV. The amount of the tail is dependent on the detector
hit rate and typically 1 % of the photoelectric peak. The pile-up tail is
ﬁtted with exponential plus ﬂat function. The events used for the ﬁtting is
restricted within ±20 ns, where the simultanious hits of 3γ is negligible andthe background events are simply subtracted with the ﬂat function. The
obtained exponential function is used as a part of the response function.

2.4.3

Response Function of the YAP Scintillator

The response function of YAP scintillator is also measured by 85 Sr source.
The upper plot of the Fig. 2.17 is the comparison of the measured energy
spectrum and the simulation. The background is subtracted by the spectrum
taken without 85 Sr source as in the Ge case. The diﬀerence comes fromthe non-uniform collection eﬃciency of optical photon in the YAP crystal.
The non-uniformity of the eﬃciency is enhanced by the short absorption
length of optical photon in the YAP crystal, which is reported as about
20 cm [49]. The relative collection eﬃciency is evaluated by the opticalphoton simulation using the Geant4 package. The simulation only consider

41

2.4 Experiment

106
105
104
103
0

8000
7000
6000
5000
4000
3000
2000
1000
0
450

100

460

200

470

480

300

490

400

500

500
600
Energy (keV)

510 520 530
Energy (keV)

Figure 2.14: Comparison of the measured Ge spectrum with the simulation
using 85 Sr source. The blue line is the measured spectrum and the red lineis the spectrum generated by the Monte Carlo simulation. The two spectra
are in good agreement with each other except for the region around thephotoelectric peak. The lower ﬁgure is the enlarged spectrum around thephotoelectric peak. The exponential tail is observed below the photoelectric
peak.

42

2.4 Experiment

1200

1200

1200

1000

1000

1000

800

800

800

600

600

600

400

400

400

200

200

200

0

0

0

480 490

500 510 520
Energy (keV)

Ge0

480 490 500 510

520

Energy (keV)

Ge1

480

490 500 510 520
Energy (keV)

Ge2

Figure 2.15: The diﬀerence of the measured Ge spectrum and the simulation.
The tails of the photoelectric peaks are ﬁtted with exponential functions.
The amount of the tails are 1.1% for Ge0, 1.0% for Ge1, and 2.0% for Ge2.

43

counts/0.33keV

2.4 Experiment

107

6

10

5

10

104

505

510

515

520

525

530

535 540 545
energy (keV)

Figure 2.16: Pile-up events of Ge detector. The pile-up events disturb theenergy spectrum of Ge detector. Since the Narrow-Wide cut rejects theenergy deviation beyond ±15 keV, the pile-up tail of the photoelectric peakextends up to 525 keV. The amount of the tail is dependent on the event rate
and typically 1 % of the photoelectric peak. The pile-up tail is ﬁtted withexponential plus ﬂat function for each run. Then, the exponential function
is used as a part of the response function.

2.4 Experiment44

the absorption in the crystal and the reﬂection on the surface of the crystal.
The region in a crystal is divided into 2mm × 2mm × 1.5mm regions andthe collection eﬃciency is evaluated for each region. It is found that thecollection eﬃciency at the corner of YAP crystal is rather low by 40% at
most, as shown in Fig. 2.18. The obtained eﬃciency is incorporated into theγ-ray simulation and the non-gaussian response function is reproduced. In
the end, the simulation is in good agreement with the measured spectrum
as shown in the lower plot of the Fig. 2.17.

45

2.4 Experiment

12000
10000
8000
6000
4000
2000
0

100

200

300

400

500

600 700 800
Energy (keV)

100

200

300

400

500

600 700 800
Energy (keV)

12000
10000
8000
6000
4000
2000
0

Figure 2.17: Comparison of the measured YAP spectrum with the simulation
using 85 Sr source. The blue line is the measured spectrum and the red lineis the spectrum generated by the Monte Carlo simulation. The diﬀerence
is observed especially in the shape of photoelectric peak (the upper ﬁgure).
The simulation with non-uniform photon collection eﬃciency reproduces themeasured spectrum (the lower ﬁgure).

46

30

Z (mm)Z (mm)

2.4 Experiment

30

95%

25

25

20

20

15

85%

90%

15

95%

85%

90%

80%
10

10 60%

5

5

0-25 -20 -15 -10 -5 0 5 10 15 20 25
X=0, Y (mm)0-25 -20 -15 -10 -5 0 5 10 15 20 25
X, Y (mm)Figure 2.18: Collection eﬃciency of optical photon in YAP crystal. The left
ﬁgure shows the collection eﬃciency along the Y axis on the center of crystal(X=0). The right ﬁgure shows the collection eﬃciency along the diagonal
direction of X-Y plane. The photomultiplier is connected on the lower sideof X-Y plane (Z=0).

Chapter 3

Analysis
In this chapter, the details of the analysis are described. Calibration, several
cuts and time walk correction are explained in the ﬁrst section. Then, thedetailed description of the direct pick-oﬀ correction method will follow. The
last section contains a discussion of the systematic errors accompanying themeasurement. The various procedures in this chapter are explained for RUN
II since most of the parts are common to two RUNs.

3.1

Calibration and Time Walk Correction

3.1.1

Data Sets

The two RUNs have been performed in this experiment. Some properties of
the two RUNs are summarized in Table 3.1. Most of the parts are the same,but the types of the silica target are diﬀerent. The silica aerogel is used in
the RUN I, whereas the silica powder is used in the RUN II.
During the RUNs, data are taken with the CAMAC system. Totally
about 1.4×1010 events and 1.6×1010 events are taken for RUN I and RUN II
respectively. Only when the γ-detectors detect any γ-rays within 10μs fromthe β + trigger, data are stored in the disk storage. Totally about 5.7 × 109
events and 6.6 × 109 events are recorded for RUN I and II respectively. The
net data taking periods are about 2.3 month for RUN I and about 3.1 monthfor RUN II. Typical β + trigger rates before anti-trigger veto system are
10.8 kHz for RUN I and 6.8 kHz for RUN II.
Data are stored into the multiple calibration units for every 6 × 107
events. The calibration and the time walk correction are carried out within
one calibration unit. Since the data taking time for one unit is about 6hours, it is guaranteed that the ﬂuctuation and the shift of the measured
quantities by the environment do not propagate over that time scale.
During the measurements, the temperature of the experimental area
was controlled within ±0.5 ◦ C to suppress the ﬂuctuation of the measured
quantities.
47

48

3.1 Analysis
name
RUN I
RUN II

period
2.3 month
3.1 monthrate
10.8 kHz
6.8 kHzevents
1.4 × 1010
1.6 × 1010

targetsilica aerogelsilica powderdetectors
Ge×3, YAP×3
Ge×3, YAP×4

Table 3.1: Properties of the RUNs
Dependence on the event rate
There are two factors dependent on the event rate, the probability of theaccidental event and the probability of the pile-up event. However, thefollowing procedures in the analysis totally eliminate its propagation to theﬁnal result.
• The accidental ﬂat distribution is directly ﬁtted in the decay rate ﬁtting. Then the probability of the accidental event does not aﬀect theﬁnal result.
• The pile-up rejection cut is applied and the pile-up events are eliminated. The eﬀect of the contamination is taken into account as a small
modiﬁcation of the energy response function (see Sec.2.4.2).
• The time spectrum of the later time region is slightly suppressed by
the ﬁrst hit condition of the data taking system. The deviation isdependent on the detector hit rate and taken into account in the ﬁtting
function (see Eq.3.8).
Though the event rate does not aﬀect the ﬁnal result, the systematic
error of the pick-oﬀ correction is determined by the amount of the accidental
events and the event rate should be as low as possible. In addition, the time
range of decay curve region depends on the accidental level. In the present
measurement, the range of the decay curve is about ﬁve times lifetime of
o-Ps.

3.1.2

Time and Energy Spectrum of the γ-ray Detector

Before the details of the analysis is explained, the time and energy spectrumof the γ-ray detector is shown.
Spectrum of the YAP scintillator
Fig.3.1 shows the time and energy spectrum of the YAP scintillator. The
time spectrum is already corrected by the time walk correction, which is
explained in Sec.3.1.5. Due to the good time resolution of 1 ns, the time
spectrum of the YAP scintillator is very clear. The time spectrum is composed of the following three regions.

49

3.1 Analysis• prompt peak
A sharp peak called as the prompt peak is observed at T0 (t = 0).
The prompt peak consists of the p-Ps, e+ annihilation and 1077 keV
nuclei-γ.
• o-Ps decay curve
The o-Ps decay curve is observed after the prompt peak. The decay time is about the lifetime of o-Ps and decreased by the pick-oﬀ
contribution.
• accidental events
The ﬂat distribution is observed after the o-Ps decay curve. These
events consists of the accidental hits, that are not correlated with theβ + trigger.

On the other hand, the energy spectrum of the YAP scintillator is vague.
But the energy information of the YAP scintillator is basically not required.
The 2γ spectrum of the e+ annihilation is dominant and some part of thespectrum is the 3γ continuous spectrum.
Spectrum of the Ge detector
Fig.3.2 shows the time and energy spectrum of the Ge detector. The time
spectrum is already corrected by the time walk correction, which is explainedin Sec.3.1.5. The time resolution of the Ge detector is larger than that of
the YAP scintillator. But the resolution of 5 ns is good enough to extract
the time dependence of the pick-oﬀ ratio.
On the other hand, the energy spectrum of the Ge detector is very clear
and the resolution is about 0.5 keV around 511 keV photoelectric peak.
Making use of the good energy resolution, the Ge detector can separate thepick-oﬀ 2γ events from the 3γ events.

3.1.3

Calibration

ADC spectrum of YAP scintillators and Ge detectors
The calibration of the ADC spectra has made use of the 511 keV positronannihilation line and the accompanying 1077 keV nuclei-γ. Each peaks are
ﬁtted with normal gaussian function. Then, the ﬁtted peak values are used
to determine the energy scale of the ADCs. The Wide spectra and Narrowspectra are individually calibrated for both Ge and YAP detectors.
On the other hand, the absolute calibration is diﬃcult for Base spectra. Therefore, the peak is ﬁtted with normal gaussian and the standard
deviation is used to deﬁne the scale of Base spectrum.

50

counts/0.5ns

3.1 Analysis

910

prompt peak

8

10

107

o-Ps decay curve

6

10

accidental
5

10

104
0

200

400

600

800

1000 1200 1400 1600 1800

time (ns)counts/2keV

6

×10
25
20
15
10
5
0

100

200

300

400

500

600
700
energy (keV)

Figure 3.1: Time and Energy spectrum of the YAP scintillator. The upper
ﬁgure is the time spectrum of the YAP scintillator. The energy range isrestricted above 150 keV and the time walk correction is already applied.
The lower ﬁgure is the energy spectrum of the YAP scintillator. The time
window is restricted before 700 ns.

51

counts/ns

3.1 Analysis

107

prompt peak

6

10

o-Ps decay curve
5

10

accidental
104

3

10

0

200

400

600

800

1000 1200 1400 1600 1800

counts/0.33keV

time (ns)

107

6

10

5

10

100

200

300

400

500

600
700
energy (keV)

Figure 3.2: Time and Energy spectrum of the Ge detector. The upper ﬁgureis the time spectrum of the Ge detector. The energy range is restricted from300 keV to 520 keV and the time walk correction is already applied. The
lower ﬁgure is the energy spectrum of the Ge detector. The time window isrestricted before 700 ns. The lower energy region is suppressed by the slow
rise-time cut.

3.1 Analysis

52

TDC spectrum of YAP scintillators and Ge detectors
The absolute scale of the KEK TDC is determined by the external clock
source. The 2 GHz clock leads to 0.5 ns bin width and 32 μs full range.
The calibration of the absolute value makes use of the prompt events whichappears as a sharp peak and exactly stands at decay time t = 0 (T0 ) since itconsists of the p-Ps and 1077 keV nuclei-γ. After the time walk correction,
the prompt peak is ﬁtted with normal gaussian and the absolute decay timeis determined.
The diﬀerential non-linearity (DNL), which is the deviation of the relative bin width, is so small for the KEK TDC that the measured decay rate
is not aﬀected at all. But, the χ2 of the decay rate ﬁtting sums up the DNL
and the minimum value is increased. To avoid this eﬀect, the measured TDC
values are smeared with the gaussian function which has 1.5 ns standard
deviation.
Though the intrinsic DNL of the KEK TDC is negligible, the nonlinearity of the time spectrum is observed at the DNL test where the random
start signal and the stop signal of the clock are fed into the KEK TDC. As in
Fig. 3.3, the 0.05-0.1 % excess of the relative bin width is observed near theT0 . It is found that the transition of the veto input in the logic unit causes
the electric ﬂuctuation of the output line and leads to the non-linearity ofthe time spectrum near the T0 . Therefore, the non-linearity is corrected by
the relative bin width measured at the DNL test.
ADC spectrum of plastic scintillator
The calibration of Wide spectrum for the plastic scintillator is not needed
since no cut is applied for the spectrum. Wide spectrum is only used forthe time walk correction within a calibration unit. On the other hand, Base
spectrum of the plastic scintillator is calibrated by the standard deviation
as in YAP and Ge cases.

3.1.4

Event Selection

In order to extract the intrinsic timing of the γ-rays and the accurate energyof those, it is important to reject the pile-up event where the two or more
γ-rays hit the detector successively. The two types of cut adopted in themeasurement are explained in this section.
Base cut for Ge, YAP and the plastic scintillator
One type of the cuts is Base cut, which requires the clean baseline condition
just before the signal. The Base cut rejects the events where the preceding
signal disturbs the timing and the energy information of the succeeding53

3.1 Analysis

1.001
1.0008relative bin width

1.00061.00041.0002
1
0.99980.99960.9994T0

0.9992
0.999
0

500

1000

1500

2000

2500

3000

TDC count (0.5ns)

Figure 3.3: The relative bin width measured at the DNL test of the KEK
TDC. The 0.05-0.1 % excess of the relative bin width is observed near theT0signal as in Fig. 3.4. The events that are triggerd by the tail of the detector
signal are also gotten rid of.
The spectra of the Ge Base are shown in Fig.3.5. The slow rise-time
events, which are explained later, are not included in the ﬁgures. The higher
tail of the spectrum accounts for the pile-up events and the events triggeredby the tail of the signal. The lower tail comes from the hits on the undershootof the previous event. Thus, the events within ±4σ are selected as in theﬁgure.
For the YAP scintillators, the time dependence of the cut eﬃciency affects the time spectrum. In fact, the prompt hits such as 1077 keV nuclei-γ
and bremsstrahlung-γ of the positron deposit the energy in the Base gateof the delayed signal. If the Base cut ±4 σ is applied, the events near the
prompt are suppressed. In Fig. 3.6, these events are shown as the tail which
extends to about 140 σ. Then, the upper boundary of the Base cut is set
to 140 σ. The eﬀect of the contamination is calculated in Appendix B andassigned as a systematic error.
The Base cut is also applied for the plastic scintillator. The events within±4 σ are selected as in Fig. 3.7.

54

3.1 Analysis

Narrow Gate
Base Gate

Previous event

SignalFigure 3.4: The schematic of the Base cut. The Base cut rejects the events
where the preceding signal disturbs the timing and the energy informationof the succeeding signal.

107

107

107

106

106

106

105

105

105

104

104

104

103

103

103

-6 -4 -2 0 2 4 6 8 10

-6 -4 -2 0 2 4 6 8 10

-6 -4 -2 0 2 4 6 8 10Ge0 Base (σ)

Ge1 Base (σ)

Ge2 Base (σ)

Figure 3.5: Base spectra for Ge detectors. The slow rise-time events are
excluded in the ﬁgures. The events within ±4σ are selected. The twovertical lines in a ﬁgure show the boundaries of the cut.

55

3.1 Analysis

10

8

10

8

10

8

10

107

107

107

107

10

6

10

6

10

6

10

10

5

10

5

10

5

10

104

104

104

104

3

10

3

10

3

10

10

8

6

5

3

-6 -4 -2 0 2 4 6 8 10

-6 -4 -2 0 2 4 6 8 10

-6 -4 -2 0 2 4 6 8 10

-6 -4 -2 0 2 4 6 8 10

YAP0 Base (σ)

YAP1 Base (σ)

YAP2 Base (σ)

YAP3 Base (σ)

107

107

107

107

10

6

10

6

10

6

10

10

5

10

5

10

5

10

104

104

104

104

10

3

10

3

10

3

10

102

102

102

102

0

50

100 150 200

YAP0 Base (σ)

050

100 150 200

YAP1 Base (σ)

6

5

3

0

50

100 150 200

YAP2 Base (σ)

050

100 150 200

YAP3 Base (σ)

Figure 3.6: Base spectra for YAP scintillator. The events within the range
from −4 σ to +140 σ are selected. The upper ﬁgures show the lower boundaries of the cut as the vertical line. The lower ﬁgures show the upper boundaries of the cut as the vertical line.

56

3.1 Analysis

109
108
107
106
105
-5 0 5 10 15 20
Pla Base (σ)
Figure 3.7: Base spectrum for the plastic scintillator. The events within±4 σ are selected. The two vertical lines show the boundaries of the cut.
Narrow-Wide (NW) cut for Ge and YAP
The other type of the cuts is Narrow-Wide (NW) cut, which requires thesmall diﬀerence of the energies measured with Narrow gate and Wide gate.
The used quantity is Wide - Narrow in the keV unit. The NW cut rejects
the events where the succeeding signal disturbs the energy information of
the preceding signal as in Fig. 3.8.
The NW spectra for Ge detectors are shown in Fig. 3.9. The slow risetime events, which are explained later, are not included in the ﬁgures. The
higher tail account for the pile-up events. The events within ±15 keV are
selected as in the ﬁgure.
For the YAP scintillators, NW cut is also applied. The events within±30 keV are selected as in Fig. 3.10.

3.1.5

Time Walk Correction

The detection timings of the signals are depending on their amplitudes. Thislag of time called walk must be corrected to obtain the intrinsic decay time.
The following time walk corrections make use of the prompt events, which
consist of the p-Ps and the e+ annihilation etc, and occur at T0 (t = 0).
Time walk correction for YAP scintillator
Due to the fast rise-time feature of YAP scintillator, the time walk is as
small as 2 ns if the energy is above 150 keV. The procedures of the time

57

3.1 Analysis

Wide Gate
Narrow Gate

Pileup event
SignalFigure 3.8: The schematic of the Narrow-Wide (NW) cut. The NW cut
rejects the events where the succeeding signal diturbs the energy informationof the preceding signal.

108

108

108

107

107

107

106

106

106

105

105

105

104

104

104

-20-10 0 10 20 30 40 50-20-10 0 10 20 30 40 50-20-10 0 10 20 30 40 50

Ge0 Wide-Narrow (keV)

Ge1 Wide-Narrow (keV)

Ge2 Wide-Narrow (keV)

Figure 3.9: Narrow-Wide spectra for Ge detectors. The events within±15 keV are selected. The two vertical lines shows the boundaries of the
cut.

58

3.1 Analysis

10

8

10

8

10

8

10

107

107

107

107

10

6

10

6

10

6

10

10

5

10

5

10

5

10

104

104

104

104

3

10

3

10

3

10

10

8

6

5

3

-40-20 0 20 40 60 80100-40-20 0 20 40 60 80100-40-20 0 20 40 60 80100-40-20 0 20 40 60 80100

YAP0 Wide-Narrow (keV)

YAP1 Wide-Narrow (keV)

YAP2 Wide-Narrow (keV)

YAP3 Wide-Narrow (keV)

Figure 3.10: Narrow-Wide spectra for YAP scintillators The events within±30 keV are selected. The two vertical lines shows the boundaries of the
cut.
walk correction for YAP scintillator are as follows. At ﬁrst, the whole energy
range is divided into the 50 keV width regions and the prompt peaks forthese regions are ﬁtted with normal gaussian. Then, the peak values are
ﬁtted with seven order polynomial function as in Fig. 3.11. The relative
correction width is determined by this function at any energy. After thetime walk correction, the excellent time resolution of 1.3 ns is obtained forthe whole energy range above 150 keV.
Time walk correction for plastic scintillator
In order to extract the time walk of the plastic scintillator, the prompt
events of YAP scintillator is used. The energy region for YAP scintillator
is restricted within ±100 keV from 511 keV. Though the plastic scintillator
has a fast rise-time signal, the time walk is clearly seen in Fig. 3.12. This isbecause the typical amplitude of the signal is just above the threshold. As
in the YAP case, the whole Wide range of the plastic scintillator (Pla Wide)is divided into the 10 counts regions and the prompt peaks for these regions
are ﬁtted with normal gaussian. Then, the peak values are ﬁtted with sevenorder polynomial function as in the ﬁgure. The relative correction width isdetermined by this function at any Pla Wide value.
Time walk correction for Ge detector
For the Ge detectors, things are rather complicated due to the slow rise-timefeature of the Ge signal. The rise-time of the Ge detector is as large as 200 ns
and depending on the strength of the electric ﬁeld. Since the electric ﬁeld

TDC count (0.5ns)

59

3.1 Analysis

30
20
10
0
-10
-20
0

200

400

600

800

1000

1200
Energy (keV)

800

1000

1200
Energy (keV)

800

1000

1200
Energy (keV)

800

1000

1200
Energy (keV)

TDC count (0.5ns)

YAP0
30
20
10
0
-10
-20
0

200

400

600

TDC count (0.5ns)

YAP1
30
20
10
0
-10
-20
0

200

400

600

TDC count (0.5ns)

YAP2
30
20
10
0
-10
-20
0

200

400

600

YAP3

Figure 3.11: Time walk of YAP scintillator. The detection time of YAP
scintillator is plotted against the YAP Wide value. The prompt peaks forevery 50 keV regions of YAP Wide is ﬁtted with seven order polynomial
function. The blue line shows the ﬁtted polynomial function and the time
walk is corrected by this function.

60

TDC count (0.5ns)

3.1 Analysis

190
180
170
160
150
140
1300

20

40

60

80 100 120 140 160 180 200
ADC count

Figure 3.12: Time walk of plastic scintillator. The detection time of YAP
scintillator is plotted against the Wide value of the plastic scintillator (PlaWide). The prompt peaks for every 10 counts regions of Pla Wide is ﬁtted with seven order polynomial function. The blue line shows the ﬁtted
polynomial function and the time walk is corrected by this function.

varies in a crystal, the rise-time of the signal depends on the position wherethe γ-ray is absorbed. In fact, the rise-time of the signal with a single energy
widely ranges as shown in Fig. 3.13. The spectra in the ﬁgures are the timeinterval between two thresholds 50 mV and 150 mV , and the energy region
is restricted from 480 keV to 520 keV. The second peak in the spectrum,
which is refered to as SRT events, consist of the γ-rays absorbed at the front
cap of the crystal, where the electric ﬁeld is complicated and some places
are so weak. Since the time resolution of the SRT events is too bad, theSRT events have to be eliminated. The vertical lines in the ﬁgures show theboundaries of the SRT cut.
The time walk correction makes use of the measured rise-time, or thetime interval of two thresholds 50 mV and 150 mV . The energy region
is restricted from 480 keV to 520 keV. The whole range of the rise-time
variable is divided into 20 counts regions and the prompt peaks for theseregions are ﬁtted with normal gaussian. Then, the peak values are ﬁtted
with seven order polynomial function as in Fig. 3.14. Thus, the relative
correction width is determined by the function at any rise-time variable.
The obtaned time resolutions for the 511 keV γ-rays are 5.1 ns, 5.2 ns and3.2 ns for Ge0, Ge1 and Ge2 respectively.
Evaluation of the SRT cut eﬃciency
The 2γ and 3γ detection eﬃciencies must be known to determine the decayrate by the direct pick-oﬀ correction method. But, the SRT cut eﬃciency
is not resolved by the detector simulation. Then the SRT cut eﬃciency is

61

3.1 Analysis

6

35

6

×10

35

6

×10

35

30

30

30

25

25

25

20

20

20

15

T 150mV-T50mV15

T 150mV-T50mV

15

10

10

10

5

5

5

0
0

0
100 200 300 400 500 600 700

Ge0(count)0

×10

T 150mV-T50mV

0
100 200 300 400 500 600 700

Ge1(count)0 100 200 300 400 500 600 700 800 900

Ge2(count)Figure 3.13: The rise-time of the Ge detectors. The time interval between
two thresholds 50 mV and 150 mV are plotted. The unit of the horizontal
axis is corresponding to 0.25 ns for Ge0 (Ge1) and 0.2 ns for Ge2. Theenergy region is restricted from 480 keV to 520 keV. The slow rise-timecomponent, which is refered to as SRT events, is cut at the vertical line.

62

TDC count (0.5ns)

3.1 Analysis

200
150
100
50
0
0

100

200

300

400

500
600
T150mV -T50mV (count)

400

500
600
T150mV -T50mV (count)

500

600
700
T150mV -T50mV (count)

TDC count (0.5ns)

Ge0
200
150
100
50
0
0

100

200

300

TDC count (0.5ns)

Ge1
200
150
100
50
0
0

100

200

300

400

Ge2

Figure 3.14: Time walk of Ge detectors. The detection time of the prompt
peak is plotted against the rise-time variable (T150 mV -T50 mV ). The prompt
peaks for every 20 counts regions of the rise-time variable is ﬁtted withseven order polynomial function. The blue line shows the ﬁtted polynomial
function and the time walk is corrected by this function. The slow rise-timecomponent beyond the vertical line is eliminated.

63

3.1 Analysis

evaluated with the measured data.
The SRT cut eﬃciencies for Compton scatterd 2γ event and photoelectric
3γ event are slightly diﬀerent with each other even when the deposit energy
is the same. This is because the distributions of the absorbed positionin the crystal are diﬀerent for these two cases. In practice, the Compton
scattered 2γ events rather than the photoelectric 3γ events are more likely
to be the SRT events. Therefore, the 2γ SRT cut eﬃciency and the 3γ SRT
cut eﬃciency must be separately evaluated within the each dominant time
region.
The dominant time regions for the 3γ events and the 2γ events are
obviously the decay curve region and the prompt peak respectively. But
the some part of the prompt 2γ events enter the decay curve region becauseof the bad timing characteristics of the SRT events. To exclude these 2γ
events, the special time walk correction is applied for the SRT events andthe much later time region is used for the evaluation of the 3γ SRT cut
eﬃciency.
At ﬁrst, the time walk correction is applied for the SRT events. The
used quantity is the time interval between two thresholds 30 mV and 50 mV,
which is rather sensitive to the slow rise-time events. The Fig. 3.15 shows thecorrelation between the rise-time variable (T50 mV -T30 mV ) and the detection
time of the prompt events. The fast rise-time component is not included in
the ﬁgures. The linear functions shown in the ﬁgures are used to determine
the time walk correction.
The obtained time resolution for the SRT events is evaluated with thedata which is measured without the silica target. The Fig. 3.16 shows thetime spectra before the SRT cut and after the SRT cut. The prompt peaks
after the SRT cut are conﬁned within 50 ns from the T0 , though the o-Ps
in the air forms the rather long tail, which is simultaneously observed with
the YAP detectors. On the other hand, the prompt peaks before the SRT
cut spread over 200 ns for Ge0 (Ge1) and 600 ns for Ge2. Thus, it is foundthat the time region after 200 ns can be used to evaluate the 3γ SRT cut
eﬃciency for Ge0 and Ge1, whereas no time region is available for Ge2. The
diﬀerence between Ge0 (Ge1) and Ge2 comes from the shape of the crystal.
In fact, the Ge2 has the larger front cap part of the crystal than the others,
where electric ﬁeld is so complicated. In the circumstances, data of the Ge2
is not included in the analysis from now on.
For the 3γ SRT cut eﬃciency, the time spectra of Ge0 and Ge1 is sliced
out from 256 ns to 512 ns as in Fig. 3.17. The accidental contribution in thistime region is subtracted by the spectrum from 2 μs to 3.6 μs. Then, the 3γ
SRT eﬃciency is obtained as the ratio of the events before and after the SRT
cut. The Fig. 3.18 shows the 3γ SRT eﬃciency along the deposit energy.
The spectrum is ﬁtted with the linear function from 480 keV to 500 keV andthe obtained function is used for the 3γ SRT cut eﬃciency. This is because
the eﬃciency curve is slightly shifted from the linear relation in the region

64

700
600
500

TDC count (0.5ns)

TDC count (0.5ns)

TDC count (0.5ns)

3.1 Analysis

700
600
500

700
600
500

400

400

400

300

300

300

200

200

200

100

100

100

0
0

0
100 200 300 400 500 600

T50mV -T 30mV (count)00
100 200 300 400 500 600

0 100 200 300 400 500 600 700

T50mV -T 30mV (count)

T50mV-T 30mV (count)

Figure 3.15: Time walk of the slow rise-time component. The detection
time of the prompt events are plotted against the rise-time variable (T50 mV T30 mV ). Note that the small amount of 3γ events are not rejected. The
correlations are clearly seen in the ﬁgures. The linear function shown as theblue line is used to determin the time walk correction.
under 480 keV. In addition, the pick-oﬀ contribution is negligibly small in
the region under 500 keV.
On the other hand, the 2γ SRT cut eﬃciency is evaluated within thetime region from -256 ns to 1536 ns. The obtained values are 0.76626(9) forGe0 and 0.68378(8) for Ge1 in the region of 511 keV ±4 keV, where the 2γ
SRT cut eﬃciency is used in the analysis. The contamination of 3γ is small
around 511 keV and the 3γ SRT cut eﬃciency is the same as the 2γ’s at
511 keV from the beginning. Then, the values are considered to be reliable.
The 2γ SRT cut eﬃciency is also used to calculate the 2γ eﬃciency in theregion of 480 keV-505 keV. In this region, the contamination of 3γ increases
the obtained value by about 2 %. But the precision is not required in thiscase and the uncertainty of the eﬀect does not contribute to the ﬁnal result.

65

3.1 Analysis

107

107

107

106

106

106

10

5

10

5

10

104

104

104

3

10

3

10

10

-200

0

200

Ge0

400

600

5

3

-200

ns

0

200

Ge1

400

600

-200

ns

0

200

Ge2

400

600

ns

Figure 3.16: Time spectra of Ge detectors without silica target. The insidehistogram is the spectrum after the SRT cut. The long tail of the spectrum
after the SRT cut comes from the o-Ps in the air. The slow rise-time component of the prompt peak spreads over 200 ns for Ge0 (Ge1) and 600 ns
for Ge2. The vertical lines show the region used for the evaluation of the 3γ
SRT eﬃciency.

66

3.1 Analysis

108

108

107

107

106

106

105

105

104

104
0

200

400

600

800

Ge0

1000

0

ns

200

400

600

800

Ge1

1000

ns

1

3γ SRT efficiency

3γ SRT efficiency

Figure 3.17: Time spectra of Ge detectors. The inside histogram is the
spectrum after the SRT cut. The time region within the two vertical lines
is used for the evaluation of the 3γ SRT eﬃciency.

0.9
0.8
0.7
0.6

1
0.9
0.8
0.7
0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0
470

475

480

485

490 495

Ge0

500

505

510

Energy (keV)

0
470

475

480

485

490

Ge1

495

500

505

510

Energy (keV)

Figure 3.18: 3γ SRT cut eﬃciency of Ge detectors. The event ratio before
and after the SRT cut is plotted along the energy. The spectrum is ﬁtted
with linear function from 480 keV to 500 keV and the function is used as
the 3γ SRT cut eﬃciency. Ge0: 1.023(150)×10−3 E(keV)+0.247(73), Ge1:
1.063(153)×10−3 E(keV)+0.143(75)

67

3.2

3.2 Analysis

Determination of λpick (t)/λ3γ

In this section, the pick-oﬀ ratio λpick (t)/λ3γ is determined as a functionof time. Since the pick-oﬀ ratios for the two RUNs are diﬀerent from eachother, most of the plots are presented for each RUN.

3.2.1

Time and Energy Spectra of the Ge Detectors

To determine the pick-oﬀ ratio λpick (t)/λ3γ , the energy and the timing information of the Ge detectors are made use of. After the SRT cut andthe time walk correction, the time spectrum as shown in Fig. 3.19 is obtained. The three characteristic regions are observed in the ﬁgures, prompt
peak, o-Ps decay curve and accidental region. These regions have diﬀerent
contributions of γ-rays and diﬀerent energy spectra as follows.
prompt peak
A sharp peak, called as the prompt peak, is observed at T0 (t = 0). The
prompt peak consists of the p-Ps, the 1077 keV nuclei-γ and the e+annihilation, which is the events that positrons do not form Ps. Since
most of the contributions are 2γ, the energy spectrum is concentrated
on the 511 keV photoelectric peak. The 511 keV peak of the prompt
events is shown in Fig. 3.20. The energy resolution of the 511 keV peakis 1.1∼1.2 keV, which is summarized in Table 3.2, whereas the intrinsic
resolution of the Ge detector is 0.5∼0.6 keV. This is because electrons
in material have the bound momentum and the e+ annihilations areaﬀected by the Doppler eﬀect of this motion, and then observed as thebroad photoelectric peak.

Ge0
Ge1

RUN I
511 keV1077 keV
1.143 keV 0.734 keV
1.153 keV 0.752 keV

RUN II
511 keV1077 keV
1.160 keV 0.744 keV
1.183 keV 0.775 keV

Table 3.2: Energy resolution of the prompt events
o-Ps decay curve
The time region after the prompt peak, sometimes refered to as delayed region, is dominated by the o-Ps 3γ and forms the exponential
decay curve. Thus, the energy spectrum is a combination of the 3γ
continuous spectrum and the 2γ spectrum as shown in Fig. 3.21. The
latter comes from the pick-oﬀ annihilation and the accidental events.
accidental events
The time region far beyond the prompt peak is dominated by the

68

107

counts/ns

counts/ns

3.2 Analysis

prompt peak

6

10

107
6

10

o-Ps decay curve
510
5-40-20 0 20 40 60 80100

10time (ns)accidental
104

3

10

0

200

400

600

800 1000 1200 1400 1600 1800

time (ns)107

counts/ns

counts/ns

RUN I

prompt peak

6

10

o-Ps decay curve

107
6

10

5

10

5

-40-20 0 20 40 60 80100

10time (ns)accidental
104

3

10

0

200

400

600

800 1000 1200 1400 1600 1800

RUN II

time (ns)Figure 3.19: Time spectrum of Ge detectors. The upper ﬁgure is the spectrum for RUN I and the lower ﬁgure is the spectrum for RUN II. The
spectrum includes both Ge0 and Ge1 events in the energy region of 300 keV520 keV. The ﬁgure in the right coner is the enlarged view of the prompt
peak.

69

3.2 Analysis
accidental events and forms the ﬂat time spectrum. The γ-ray hitof the accidental event is not correlated with the β + trigger. Thus,
the accidental events include all the γ-ray contributions, p-Ps, o-Ps,
e+ annihilation and environmental background. The energy spectrumof the accidental events is nearly 2γ spectrum as shown in Fig. 3.21.
The spectrum is used to subtract the accidental contribution from thedelayed region.

6

counts/0.33keV

counts/0.33keV

6

×10
25

20

15

×10
25

20

15

10

10

5

5

0

507 508

509 510511 512RUN I

513 514 515
energy (keV)

0507 508

509 510511 512RUN II513 514 515
energy (keV)

Figure 3.20: 511 keV photoelectric peak of the prompt events. The left
ﬁgure is the spectrum for RUN I and the right ﬁgure is the spectrum for
RUN II. The spectrum includes the prompt events of both Ge0 and Ge1,
whose time region is restricted within ±10 ns.

3.2.2

Extraction of the Pick-oﬀ Contribution

The delayed time region is divided into several regions to determin the pickoﬀ ratio as a function of time. But in this section, the time region from
60 ns to 700 ns is used for the explanation of the procedures.
• First of all, the accidental contribution is subtracted from the delayed
region as in Fig.3.21. The energy spectrum of the accidental time region (2000 ns - 3600 ns) is prepared for the subtraction. The normalization factor of the spectrum is described as the following equation.
fnorm =

(700 − 60)
× exp(Rstop · Δt)(3600 − 2000)(3.1)

where the ﬁrst term is the width of the used region and the second termis the stop rate factor. The value Δt is the time diﬀerence between thecenter of the delayed region (60 ns + 700 ns)/2 and the center of the

70

3.2 Analysis

accidental region (2000 ns + 3600 ns)/2. The stop rate factor comesfrom the fact that the data taking system accepts only the ﬁrst hit ofthe detector, which is explained in Appendix B. The stop rates of the
Ge detectors are summarized in Table 3.3.

RUN I
RUN IIGe0
880 Hz550 HzGe11010 Hz640 HzTable 3.3: Typical stop rate of Ge detectors• The next procedure is the subtraction of the 3γ spectrum as shown in
Fig.3.22. The 3γ spectrum is produced by the Monte Carlo simulationand multiplyed by the 3γ SRT cut eﬃciency prepared in Sec. 3.1.5.
Then, the 3γ spectrum is normalized by the number of events (N3γ )within 480 keV-505 keV. However, the region containes the Compton
scattered events of the pick-oﬀ 2γ. Therefore, the normalization factor
is decreased by this amount as following equation.
N3γ = N (480 keV ∼ 505 keV) − fcompon × Npick(3.2)

where Npick is deﬁned as the number of events within ±4 keV from511 keV after the subtraction of the 3γ spectrum. The coeﬃcient
fcompton is evaluated by the simulation and the values are summarized
in Fig. 3.4.
detector
Ge0
Ge1
Ge0
Ge1

RUN
RUN I
RUN I
RUN II
RUN II

fcompton
0.03420.03370.03490.0356

Table 3.4: Compton factor fcompton
In this step, it is found that the pick-oﬀ ratios for the two RUNs diﬀer
from each other by about a factor of three, which is seen in Fig. 3.22.
• After the subtraction of the 3γ spectrum, the remaining spectrum isthe pick-oﬀ contribution. As shown in Fig. 3.23, the spectrum is a
Doppler broadning 2γ spectrum like the prompt events. The peakvalue and the resolution of the 511 keV peak are summarized in Table 3.5.

counts/0.33keV

71

3.2 Analysis

×10

3

400
350
300
250
delayed200
accidental

150

o-Ps spectrum

100
50

counts/0.33keV

0
470

×10

480

490

500
RUN I

510
520
energy (keV)

3

200

150
delayed100

accidental
o-Ps spectrum

50

0
470

480

490

500
RUN II

510
520
energy (keV)

Figure 3.21: Energy spectrum of the delayed region. The upper ﬁgure is
the spectrum for RUN I and the lower ﬁgure is the spectrum for RUN
II. The spectrum includes both Ge0 and Ge1 events in the time region of
60 ns-700 ns. The contribution of the accidental events is subtracted by thespectrum of the accidental region from 2000 ns to 3600 ns (dotted line).

72

3.2 Analysis

counts/0.33keV

3

×10
160
140
120

o-Ps spectrum

100
80
60

3γ spectrum

40

pick-off 2γ

200counts/0.33keV

470

480

490
500
RUN I

510
520
energy (keV)o-Ps spectrum

60000
50000
40000

3γ spectrum

30000
20000

pick-off 2γ

10000
0

470

480

490
500
RUN II

510
520
energy (keV)

Figure 3.22: Subtraction of the 3γ spectrum. Energy spectrum of the delayed region. The upper ﬁgure is the spectrum for RUN I and the lower
ﬁgure is the spectrum for RUN II. The ﬁgure shows the spectrum after thesubtraction of the accidental events. The 3γ spectrum prepared by the simulation is normalized in the region from 480 keV to 505 keV. The diﬀerenceis the contribution of the pick-oﬀ 2γ events.

73

3.2 Analysis

counts/0.33keV

3

120

×10

100
80
60
40
20
0

counts/0.33keV

507 508

509 510511 512RUN I

513 514 515
energy (keV)

509 510511 512RUN II513 514 515
energy (keV)

25000
20000
15000
10000
5000
0

507 508

Figure 3.23: Energy spectrum of the pick-oﬀ events. The upper ﬁgure is the
spectrum for RUN I and the lower ﬁgure is the spectrum for RUN II. The
ﬁgure shows the spectrum after the subtraction of the 3γ spectrum. The
spectrum is a Doppler broadning 2γ spectrum like the prompt events.

74

3.2 Analysis

RUN I
RUN II

peak value
510.995(3) keV
511.013(9) keV

resolution
1.108(3) keV
1.110(9) keV

Table 3.5: Peak value and resolution of the pick-oﬀ events

3.2.3

Determination of λpick (t)/λ3γ

The pick-oﬀ ratio λpick /λ3γ is calculated with the observed counts as thefollowing equation.
λpick
Npick
=
×F
(3.3)
λ3γ
N3γ
where the counts N3γ and Npick are introduced in the previous section as• N3γ = number of events within 480 keV-505 keV
for the normalized 3γ spectrum• Npick = number of events within ±4 keV from 511 keV
for the pick-oﬀ spectrum.
and the conversion factor F is deﬁned as, 505 keV
keV
F =  480
515 keV507 keVP3γ (E) dE(3.4)
Ppick (E) dE

P3γ (E) and Ppick (E) are the detection eﬃciencies at energy E for 3γ decay and 2γ decay events respectively. That is, the observed counts Npick , N3γ
are converted into the intrinsic decay rate λpick , λ3γ with their detection efﬁciencies.
The factor F is evaluated with the Monte Carlo simulation. The spectra
for 3γ decay and 2γ decay events are produced and multiplied by their SRT
cut eﬃciency. Then, the probability P3γ and Ppick are integrated over theabove regions. The obtained F factor is summarized in Table 3.6.
detector
Ge0
Ge1
Ge0
Ge1

RUN
RUN I
RUN I
RUN II
RUN IIF factor
0.14700(15)0.14632(15)0.14846(14)0.14760(14)Table 3.6: Conversion factor F

75

3.2 Analysis

To extract the dependency of the decay time, the delayed region is divided into 16 regions as in Table 3.7. Then, the pick-oﬀ ratio λpick /λ3γ iscalculated for each region.
time range
60-72 ns
72-88 ns
88-112 ns
112-148 ns
148-204 ns
204-288 ns
288-416 ns
416-608 ns

window width
6 ns
8 ns
12 ns
18 ns
28 ns
42 ns
64 ns
96 ns

# of windows
2
2
2
2
2
2
2
2

Table 3.7: Deﬁnition of the time windows
The obtained ratio is shown in Fig. 3.24. The curve in the ﬁgure represents the thermalization process of o-Ps. It takes about 600 ns that o-Ps
is fully thermalized in silica aerogel and silica powder. The pick-oﬀ ratio of
the RUN I (silica aerogel) is about 3 times higher than the one of RUN II(silica powder). The diﬀerence is considered to originate in the characteristics of the silica material such as the grain size, the aggregate structure, thesurface.
The pick-oﬀ ratio is ﬁtted with the analytic function for later uses. The
analytic form of the thermalization process is discussed in Sec. 2.1.3. But,
the next rather simple form is adopted for technical reasons.
d2
θ(t)2βθ(t) = −C θ(t)2 − θ∞
dt

(3.5)

where θ(t) ≡ λpick (t)/λ3γ and numerous parameters are substituted with
one new ﬁtting parameter β. This notation is identical to the one which
can be found in reference [41]. The equation is numerically solved by the
Runge-Kutta method, which is implemented as the subroutine DDEQMR
in CERN LIB [56]. Since the ﬁrst-order diﬀerential equation requires aboundary condition, one more variable, the pick-oﬀ ratio at 60 ns (θ60 ), is
introduced. The result of the ﬁtting is summarized in Table 3.8.
The uncertainties of the ﬁtted parameters are estimated by the MINOS
error analysis routine in the MINUIT package [57]. However, the parameter
C and 2β are tightly correlated with each other and the correlation coeﬃcient is almost 1. This is caused by the small magnitude of the pick-oﬀ rate
θ(t), which is the order of 10−2 . Therefore, the rather diagonal form is used
for the error estimation.

76

3.2 Analysis

λpick/ λ3γ

0.045

0.04

0.035

0.03

0.025

0.02

0.015

100

200

300

400

600

time (ns)RUN I

λpick/ λ3γ

500

0.02
0.018

0.016
0.014
0.012
0.01
0.008
0.006
100

200

300

RUN II

400

500

600

time (ns)Figure 3.24: Pick-oﬀ ratio λpick (t)/λ3γ . The upper ﬁgure is the spectrum
for RUN I and the lower ﬁgure is the spectrum for RUN II. The slow curve
represents the thermalization process of o-Ps. The pick-oﬀ ratio of the RUN
I is about 3 times higher than the one of RUN II.

77

3.3 Analysis
parameters
C
θ∞
2β
θ60
reduced χ2

RUN I
2.71 × 103
1.24 × 10−2
3.02
4.04 × 10−2
1.00

RUN II
5.87 × 104
02.75
1.61 × 10−2
1.11

Table 3.8: Summary of λpick (t)/λ3γ ﬁtting resultd 2θ (t) = −C  θ (t)2 − θ ∞ θ (t)2β
dt

(3.6)

where θ (t) = f θ(t), C  = f −1−2β C, θ∞ = f θ∞ , f = 40 for RUN I
and f = 100 for RUN II. The estimated uncertainties of the parameters
are summarized in Table 3.9. The uncertainty of the parameter θ∞ in thenegative direction is not deﬁned since the deviation of χ2 does not reach +1
even if θ∞ is set to be 0.
RUN I
parameters
C
θ∞
2β
θ60

negative−2.68 × 10−4
not deﬁned−1.07−2.96 × 10−4

RUN II

positive+7.86 × 10−4
+7.22 × 10−3
+0.60
+2.96 × 10−4

negative−1.26 × 10−4
not deﬁned−1.13−3.03 × 10−4

positive+1.43 × 10−3
+6.50 × 10−3
+0.37
+3.17 × 10−4

Table 3.9: Uncertainties of the ﬁtted parameter (RUN I).
The propagation of the uncertainties is evaluated as the following procedures. At ﬁrst, one of four parameters is ﬁxed at the value which is deviatedfrom the best ﬁt value by 1 σ. The rest parameters are ﬁtted as the free parameters again. The obtained parameters and pick-oﬀ function are preparedto evaluate the uncertainty of the parameter. Thus, the deviated functions
for the four parameters are obtained as shown in Fig. 3.25 and Fig. 3.26.
Then, the total propagation of the uncertainties to the decay rate is deﬁned
as follows.
σtotal 4
  σ 2 (pos) + σ 2 (neg)
i
i
=
i=1

2

(3.7)

where σi (pos), σi (neg) are the deviation of the decay rate when theparameter i is deviated in the positive direction and the negative direction
and then the corresponding function is used for the decay rate ﬁtting.

78

3.3 Analysis

λpick(t)/ λ3 γ

0.045
p(1) +1 σp(2) +1 σp(3) +1 σp(4) +1 σ

0.04
0.035
0.03
0.025
0.02

100

200

300

400

500

600
time (ns)

400

500

600
time (ns)positive
λ pick(t)/ λ 3 γ

0.045
p(1) -1 σp(2) -1 σp(3) -1 σp(4) -1 σ

0.04
0.035
0.03
0.025
0.02

100

200

300

negative
residual

-3

0.5
0.4
0.3
0.2
0.1
-0
-0.1
-0.2
-0.3
-0.4
-0.5

×10

p(1) +1 σp(2) +1 σp(3) +1 σp(4) +1 σ

100

200

300

400

500

600
time (ns)

200

300

400

500

600
time (ns)positive

residual-3

0.5 ×10
0.4
0.3
0.2
0.1
-0
-0.1
-0.2
-0.3
-0.4
-0.5
100

p(1) -1 σp(2) -1 σp(3) -1 σp(4) -1 σnegative

Figure 3.25: Uncertainties of the parameters (RUN I). The upper two ﬁgures
are the deviated function when one parameter is ﬁxed at +1 σ and at 1 σ. The lower two ﬁgures show the deviation of the pick-oﬀ ratio for eachfunction.

79

3.3 Analysis

λpick(t)/ λ3 γ

0.02
0.018
0.016
0.014
0.012
0.01
0.008
0.006

p(1) +1 σp(2) +1 σp(3) +1 σp(4) +1 σ

100

200

300

400

500

600
time (ns)

400

500

600
time (ns)positive
λ pick(t)/ λ 3 γ

0.02
0.018
0.016
0.014
0.012
0.01
0.008
0.006

p(1) -1 σp(2) -1 σp(3) -1 σp(4) -1 σ

100

200

300

negative
residual

-3

0.5
0.4
0.3
0.2
0.1
-0
-0.1
-0.2
-0.3
-0.4
-0.5

×10

p(1) +1 σp(2) +1 σp(3) +1 σp(4) +1 σ

100

200

300

400

500

600
time (ns)

200

300

400

500

600
time (ns)positive

residual-3

0.5 ×10
0.4
0.3
0.2
0.1
-0
-0.1
-0.2
-0.3
-0.4
-0.5
100

p(1) -1 σp(2) -1 σp(3) -1 σp(4) -1 σnegative

Figure 3.26: Uncertainties of the parameters (RUN II). The upper two ﬁgures are the deviated function when one parameter is ﬁxed at +1 σ and at-1 σ. The lower two ﬁgures show the deviation of the pick-oﬀ ratio for eachfunction.

80

3.3 Analysis

3.3

Decay Rate Fitting

In this section, the time spectrum of the YAP scintillators is ﬁtted to obtain
the intrinsic decay rate. The ﬁtting function includes the pick-oﬀ ratio
measured in the previous section.

3.3.1

Time and Energy Spectrum of the YAP Scintillators

After the time walk correction, the time spectrum of the YAP scintillators
is obtained as in Fig. 3.28. As in Fig. 3.27, the energy region is restrictedabove 150 keV since the time walk and the resolution are large in the energy
region below 150 keV. The three characteristic regions, prompt peak, o-Ps
decay curve and accidental region, are observed as in the case of Ge detector.
But, the prompt peak of the YAP scintillators is more sharp than the oneof the Ge detectors. In addition, the number of events is much larger thanthe one of Ge. This is attributed to the good timing characteristics of YAP
scintillator.

counts/2keV

3

×10
3500
3000
2500
2000
1500
1000
500
0

100

200

300

400

500

600
700
energy (keV)

Figure 3.27: Energy spectrum of YAP scintillator in the delayed region. The
time region is restricted from 60 ns to 700 ns. The vertical line shows theenergy threshold of 150 keV.

81

3.3 Analysis

9

counts/ns

counts/0.5ns

10

8

10

prompt peak

9

10

8

10

107

7

10

6

10

o-Ps decay curve-20-10 0 10 20 30 40 50

6

10time (ns)accidental

5

10

104
0

200

400

600

800 1000 1200 1400 1600 1800

time (ns)910counts/ns

counts/0.5ns

RUN I

8

10

prompt peak

9

10

8

10

107

7

10

6

10

o-Ps decay curve6-20-10 0 10 20 30 40 50

10time (ns)accidental

5

10

104
0

200

400

600

800

1000 1200 1400 1600 1800

RUN II

time (ns)Figure 3.28: Time spectrum of the YAP scintillators. The upper ﬁgure is
the spectrum for RUN I and the lower ﬁgure is the spectrum for RUN II.
The spectrum include the events of all the YAP scintillators in the energy
region above 150 keV. The ﬁgure in the right coner is the enlarged view ofthe prompt peak.

82

3.3 Analysis

RUN I
RUN II

YAP0
0.718
0.730

YAP1
0.717
0.714

YAP2
0.701
0.700

YAP3
—
0.723

Table 3.10: Ratio pick /3γ of YAP scintillators
YAP01530 Hz750 HzRUN I
RUN IIYAP11440 Hz590 Hz

YAP21540 Hz850 HzYAP3—
680 HzTable 3.11: Typical stop rate of YAP scintillators

3.3.2

Decay Rate Fitting

The analytic form of the observed time spectrum is discussed in Sec. 2.1.3
and given by,




 t

λpick (t )Nobs (t) = e
1+dt + C .
λ3γ
0
(3.8)
The ratio pick /3γ is estimated by the Monte Carlo simulation. The
values are summarized in Table 3.10. The stop rate Rstop of YAP scintillators
are summarized in Table 3.11. Thus, the free parameters in Eq. (3.8) are
just three, N0 , C and λ3γ .
The part of Eq. (3.8) containing time integration can be rewritten as,−Rstop t

N0 exp

−λ3γ

 t
0

εpick λpick (t)1+N0 exp −λ3γ
ε3γ λ3γ1+

λpick (t )dt
λ3γ= N0 exp −λ3γ
= N0 exp −λ3γ

 tstart

1+

0

 t
tstart1+

λpick (t )dt − λ3γ
λ3γ

 ttstart1+

λpick (t )dt
λ3γ

λpick (t )dtλ3γTherefore, the integration before tstart is absorbed into the normalization
factor N0 and does not have any inﬂuence on the decay rate ﬁtting.
The diﬀerent size of pick /3γ factor is another problem when the time
spectra of the YAP scintillators are combined. To overcome this diﬃculty,
the following procedures are performed.
1. At ﬁrst, the constant term C is ﬁxed by ﬁtting the accidental time
region from 2000 ns to 3600 ns. Then, the accidental contribution issubtracted from the time spectrum.

(3.9)

83

3.4 Analysis
2. The detector dependent term,1+

εpick λpick (t)ε3γ λ3γ(3.10)

is divided from the time spectrum. Then the following spectrum isobtained for each detector.
N (t) = N0 exp −λ3γ

 ttstart1+

λpick (t )dt
λ3γ(3.11)

3. The subtracted term C is added to each time spectrum. At this stage,
the time spectrum is independent of the detector eﬃciency as,

N (t) = N0 exp −λ3γ

 ttstart1+

λpick (t )dt + C
λ3γ

(3.12)

4. The obtained time spectra are simply summed up into one spectrum.
Time spectrum ﬁtting for RUN I
The time spectrum is ﬁtted by the minimization package MINUIT [57]. The
ﬁtted decay rate and reduced χ2 is shown in Fig. 3.29. The ﬁtted decay rateis plotted against the ﬁtting start time to see the dependence on the ﬁtting
start time. The ﬁtting end time is ﬁxed at 3600 ns. It is found that theﬁtted decay rate is ﬂat along the decay time and no signiﬁcant dependence
is observed. The reduced χ2 is also ﬂat along the ﬁtting start time and thevalue is reasonable.
Therefore, the ﬁtting start time is ﬁxed at 60 ns and the obtained decayrate for RUN I is determined to be,
λ3γ = 7.03876 ± 0.0009(stat.) μs−1 ,(3.13)

where only the statistic error is shown. The propagated error from λpick (t)/λ3γ
ﬁtting is already included.
Time spectrum ﬁtting for RUN II
The time spectrum for RUN II is ﬁtted by the same procedures as for RUN
I. The result is shown in Fig. 3.30. No signiﬁcant dependence on the decay
time is found in both the ﬁtted decay rate and reduced χ2 .
Then, the ﬁtting start time is ﬁxed at 60 ns and the obtained decay ratefor RUN II is determined to be,
λ3γ = 7.04136 ± 0.0009(stat.) μs−1 ,where only the statistic error is shown as for RUN I.

(3.14)

84

reduced χ2

3.4 Analysis

1.03
1.02
1.01
1
0.99
0.98

decay rate λ3γ(μ s-1)

0.97

100

150

200

250

300
350
fitting start time (ns)

100

150

200

250
300
350
fitting start time (ns)

7.055
7.05
7.045

7.04
7.035
7.03
7.025

Figure 3.29: Fitted decay rate (RUNI). The ﬁtted decay rate is plottedalong the ﬁtting start time in the lower ﬁgure. The error bars show only thestatistic error and the inner lines represent the statistic error of the YAPtime spectrum. The reduced χ2 along the ﬁtting start time is also shown in
the upper ﬁgure. Since the bin width of the histogram is 0.5 ns, the degreeof freedom is 7080–3 at 60 ns ﬁtting start time.

reduced χ285

3.4 Analysis

1.03
1.02
1.01
1
0.99
0.98

decay rate λ3γ(μ s-1)

0.97

100

150

200

250

300
350
fitting start time (ns)

100

150

200

250
300
350
fitting start time (ns)

7.055
7.05
7.045

7.04
7.035
7.03
7.025

Figure 3.30: Fitted decay rate (RUNII). The ﬁtted decay rate is plottedalong the ﬁtting start time in the lower ﬁgure. The error bars show only thestatistic error and the inner lines represent the statistic error of the YAPtime spectrum. The reduced χ2 along the ﬁtting start time is also shown in
the upper ﬁgure. Since the bin width of the histogram is 0.5 ns, the degreeof freedom is 7080–3 at 60 ns ﬁtting start time.

86

3.4 Analysis

3.4

Various Checks

The several cuts are applied to the obtained data in order to reject the pileup events and the slow rise-time events. The measured decay rate must be
independent of the cut condition since the choise of the bounrary is rather
arbitrary. To make sure of this independency, the cut conditions are varied
to some extent.
In addition, to check the quality of the data sample, data of each run
are divided into four samples.

3.4.1

Energy Selection

The energy windows for YAP scintillator and Ge detector are varied and thedeviations of the decay rate are shown in the following tables. The eﬃciency
is also reevaluated according to the energy window. The observed deviationof the decay rate is less than one third of the statistic error for both YAP
scintillator and Ge detector. Considering the fact that about 20 % of theevents are varied for both YAP scintillator and Ge detector, the deviation of
the decay rate is considered to be statistical ﬂuctuation. Therefore, one canconclude that the obtained decay rate is not biased by the energy window.
The energy window for the pick-oﬀ 511 keV peak is not varied since
the wider window sums up the uncertainty of the 3γ subtraction. The
uncertainty of the 3γ is guranteed for the window of 511 keV ±4 keV in
Sec.3.5.3.
for YAP scintillator normal threshold: 150 keV
RUN I
RUN II

strict cut
200 keV ∼+14 ppm
200 keV ∼–20 ppmloose cut
100 keV ∼–30 ppm
100 keV ∼–9 ppmfor Ge detector normal range: 480 keV ∼ 505 keV
RUN I
RUN II

3.4.2

lower boundary
485 keV ∼ 505 keV–13 ppm
485 keV ∼ 505 keV+37 ppm

higher boundary
480 keV ∼ 500 keV–33 ppm
480 keV ∼ 500 keV–0 ppmBase Cut

The Base cut conditions for YAP scintillator and Ge detector are varied
and the deviations of the decay rate are shown in the following tables. The
deviation of the decay rate is considered to be the statistical ﬂuctuation.

87

3.4 Analysis

The eﬀect of the contamination below 140 σ boundary is discussed in thenext section.
for YAP scintillator normal range: –4 σ ∼ +140 σ
RUN I
RUN II

strict cut–3 σ ∼ +130 σ+0 ppm–3 σ ∼ +130 σ–1 ppmloose cut
–6 σ ∼ +160 σ+0 ppm–6 σ ∼ +160 σ+7 ppmfor Ge detector normal range: ±4 σ
RUN I
RUN II

3.4.3

strict cut–3 σ ∼ +3 σ
+9 ppm–3 σ ∼ +3 σ
+4 ppmloose cut
–6 σ ∼ +6 σ
+7 ppm–6 σ ∼ +6 σ
+4 ppmNarrow-Wide Cut

The Narrow-Wide cut conditions for YAP scintillator and Ge detector are
varied and the deviations of the decay rate are shown in the following tables.
The deviation of the decay rate is considered to be the statistical ﬂuctuation.
The eﬀect of the contamination is small as discussed in Appendix B.3.1.
But the deviation of Ge energy spectrum is incorporated into the response
function (see Sec. 2.4).
for YAP scintillator normal range: ±30 keV
RUN I
RUN II

strict cut–25 keV ∼ +25 keV+13 ppm–25 keV ∼ +25 keV+0 ppmloose cut
–40 keV ∼ +40 keV+7 ppm–40 keV ∼ +40 keV+3 ppmfor Ge detector normal range: ±15 keV
RUN I
RUN II

strict cut–12 keV ∼ +12 keV–4 ppm
–12 keV ∼ +12 keV–0 ppmloose cut
–20 keV ∼ +20 keV+4 ppm–20 keV ∼ +20 keV+4 ppm88

3.5 Analysis

3.4.4

SRT Cut

The SRT cut conditions for Ge detector are varied and the deviations of the
decay rate are shown in the following table. The eﬃciency is also reevaluated according to the cut condition. The deviation of the decay rate isconsidered to be the statistical ﬂuctuation and the deviation of the response
function. The response function of the Monte Carlo, which is evaluatedwith the normal cut condition, is not changed at this time. In practice, theSRT cut correlates with the response function and the decay rate slightly
increases at loose cut for the underestimation of F factor.
for Ge detector normal boundary: 62.5 ns (250 counts)
RUN I
RUN II

3.4.5

strict cut∼ 57.5 ns (230 counts)–7 ppm∼ 57.5 ns (230 counts)–6 ppmloose cut
∼ 67.5 ns (270 counts)+27 ppm∼ 67.5 ns (270 counts)+9 ppmData Sample

To check the quality of the data sample, data of each run are divided into
four samples. The decay rates of the four samples are shown in Fig. 3.31
for RUN I and Fig. 3.32 for RUN II. The decay rates of the four samples
are statistically ﬂuctuated in the ﬁgure. Since the same Monte Carlo data
is used for the four samples, the systematic error of the detector eﬃciency
and the shape of the 3γ spectrum is completely correlated among the four
samples.

3.5

Systematic Errors

In this section, the systematic errors accompanying the measurement is discussed. They are put into several categories according to their origins.

3.5.1

TDC Module Related Errors

The KEK TDC used in the measurement is jointly developed by KEK and
our group. The KEK TDC is a direct clock counting type TDC and all
the systematic errors can be attributed to the external clock source. The
characteristics of the KEK TDC and the external clock source is explainedin Sec.2.3.4.
Integral non-linearity (deviation of the full range)
The external clock source is adjusted to 2 GHz within 1 ppm accuracy.
Therefore, the integral non-linearity is considered to be less than ±15 ppm

89

3.5 Analysis

7.045
7.044

Decay rate (μ sec-1)

7.043
7.042
7.041
7.04
7.039
7.038
7.037
7.036
7.035

1

2

3

RUN I

4

sampleFigure 3.31: Decay rates of the four samples for RUN I. The bar of the data
point represents the statistic error. The horizontal line shows the decay rateof the run.

90

3.5 Analysis

7.045
7.044

Decay rate (μ sec-1)

7.043
7.042
7.041
7.04
7.039
7.038
7.037
7.036
7.035

1

2

3

RUN II4

sampleFigure 3.32: Decay rates of the four samples for RUN II. The bar of the data
point represents the statistic error. The horizontal line shows the decay rateof the run.

91

3.5 Analysis

corresponding to the bin width. The frequency of the clock is guaranteed to
be stable within 3 ppm in one year, and within 2.5 ppm in the temperature−10 ◦ C ∼ 50 ◦ C.
Diﬀerential non-linearity (deviation of the bin width)
The intrinsic non-linearity is totally smeared in the time walk correction andthe rebinning operation. Therefore, the eﬀect of the DNL is considered tobe negligible at the decay rate ﬁtting.

3.5.2

Contamination of the Pile-up Events

For the YAP scintillator, the Base cut cannot be applied with ±4 σ boundary. This is because the time dependence of the Base cut distorts the time
spectrum as mentioned in Sec. 3.1.4. Then, the upper boundary of the Basecut is set to +140 σ.
The eﬀect of the contamination (about 0.1 %) can be estimated as follows. The energies of these events are slightly increased. Then the eﬀective
eﬃciency above 150 keV energy threshold is raised by 10 %. Consequently,
the contribution to the decay rate is found to be less than 10 ppm (seeAppendix B.3.2 for detailed discussion).

3.5.3

Subtraction of 3γ Energy Spectrum

The pick-oﬀ ratio λpick /λ3γ is calculated with the counting method. The
separation of the pick-oﬀ events and the 3γ events is crucial for the counting
method. The uncertainty of the separation comes from the statistics of the3γ events and the shape of the 3γ spectrum. The shape of the 3γ spectrum
is related to the following factors.
• Monte Carlo simulation (all factors such as geometry, material, etc)
• 3γ SRT cut eﬃciency, which is evaluated with the data.
• resolution of Ge detector, which is measured with single γ-rays.
• response function, which is measured with 85 Sr source.
These factors varies the 3γ spectrum in their own manner and the propagations to the decay rate are diﬀerent from each other.
But, one can estimate the uncertainty of the subtraction by making useof the following relation. Since the 3γ spectrum sharply falls oﬀ at 511 keV,
the deviation of the 3γ spectrum varies only the left side of the residual pickoﬀ peak. This relation is easily seen in Fig.3.33, where the normalizationof the 3γ spectrum is varied by ±1 %. Therefore, the asymmetry of thepick-oﬀ peak can be used to validate the subtraction of the 3γ events.

92

3.5 Analysis

counts/0.33keV

3

120

×10

MC 0.0%MC +1.0%100

MC -1.0%
80
60
40
20
0

counts/0.33keV

507 508

509 510511 512RUN I

513 514 515
energy (keV)

MC 0.0%25000

MC +1.0%
MC -1.0%

20000
15000
10000
5000
0

507 508

509 510511 512RUN II513 514 515
energy (keV)

Figure 3.33: The pick-oﬀ spectrum with varied 3γ normalization. The upper
ﬁgure is the spectrum for RUN I and the lower ﬁgure is the spectrum for
RUN II. The ﬁgure shows the pick-oﬀ spectra when the normalization of the
3γ spectrum is varied by ±1 %. The left side of the peak is shifted according
to the variation of the normalization.

93

3.5 Analysis
The observed asymmetry of the pick-oﬀ peak is parameterized as,ξasym =⎛

E<511.0
 keV

N (E) / ⎝E=507.0 keV

E=515.0
 keV⎞

N (E)⎠(3.15)

E>511.0 keV

where N (E) represents the spectrum of the pick-oﬀ peak. Then, the ξasym
parameter of the ±1 % normalization is calculated as in Table 3.12, wherethe ξasym parameter for the prompt events is also shown. These relations
Normalization
RUN I
RUN II−1.0% MC
1.016
±0.0051.025±0.014±0.0% MC
1.001
±0.0050.984±0.014+1.0% MC
0.987
±0.0050.937±0.014

Prompt0.998
0.997

Table 3.12: Summary of the ξ-parameters
are also shown in Fig. 3.34. Since the prompt events mainly consist of2γ, the ξasym parameter of the prompt events is a good indicator for theproper subtraction of the 3γ events. In the ﬁgure, it is found that theξasym parameter at the ±0.0 % normalization is consistent with the ξasym
parameter of the prompt events within ±1 σ error. The ±1 σ error ofthe ξasym parameter corresponds to ±0.33 % normalization for RUN I and±0.34 % normalization for RUN II respectively. These values aﬀect the
decay rate by ±89 ppm for RUN I and ±91 ppm for RUN II, which is
assigned as a systematic error.

3.5.4

Detection Eﬃciencies

Detection eﬃciency of Ge detector
The pick-oﬀ ratio λpick /λ3γ is calculated from the event counts with thedetection eﬃciency of Ge detector, which is introduced as F factor. The
uncertainty of F factor comes from the following factors.
• response function, which is measured with 85 Sr source.
The response function of Ge detector is evaluated with 514 keV single
photon of 85 Sr source. To estimate the uncertainty of the Monte Carlosimulation, the following parameter is introduced,
Rpeak ==

photoelectric peak
photoelectric peak + compton free region
Number of events(514 keV ± 4 keV)
Number of events(483 keV ∼ 518 keV)

(3.16)(3.17)

94

ξ-parameter

3.5 Analysis

1.06± 0.33%

1.04

1.02

1

0.98

0.96

ξ-parameter

0.94

-1

-0.5

0
RUN I

0.5
1
normalization (%)

1.06± 0.34%

1.04
1.02

1
0.98
0.96
0.94

-1

-0.5

0
RUN II

0.5
1
normalization (%)

Figure 3.34: ξ-parameter with varied 3γ normalization. The upper ﬁgureis the spectrum for RUN I and the lower ﬁgure is the spectrum for RUN
II. The boundaries of the ±1 σ error from the ξ-parameter of the ±0.0 %normalization is shown as dotted lines. This error corresponds to ±0.33 %normalization for RUN I and ±0.34 % normalization for RUN II. The ξparameter of the prompt events is also shown as a solid line.

95

3.5 Analysis
This is because the counting eﬃciency of the pick-oﬀ events is related to the event number within ±4 keV from photoelectric peak andsensitive to the tail of the photoelectric peak, whereas the counting
eﬃciency of 3γ events in the Compton free region is not sensitive tothe response function around the photoelectric peak.
The diﬀerence between the measured data and the Monte Carlo simulation is summarized in Table 3.13. The maximum diﬀerence of 0.04%
is assigned as a systematic uncertainty of F factor.

RUN I
RUN II

Ge0
0.03±0.02 %0.04±0.02 %Ge1
0.01±0.02 %0.01±0.02 %Table 3.13: The diﬀerence between the measured data and the Monte Carlosimulation• Monte Carlo simulation (geometry, density, etc)
The response function of Ge detector is evaluated with 514 keV single photon. Then, the deviation of geometry and material, if exist, isincorporated into the response function. Therefore, remaining uncertainty is related to the silica material, which is introduced at the startof RUN. The density of the silica material is known at the start of
RUN. The variation of the density during the RUN is conservatively
estimated as 10 %. The 10 % variation of density leads to 0.12±0.12 %deviation of F factor, which is assigned as a systematic error. For the
RUN I, the systematic error is not assigned since the silica aerogel isin the solid state and the deviation of the density is negligibly small.
• SRT cut eﬃciency, which is evaluated with the data.
The precision of the 3γ SRT cut eﬃciency is statistically limited since
only the delayed time region can be used for the evaluation. The
uncertainty of the absolute value in the region 480 keV∼505 keV is0.08 % for RUN I and 0.09 % for RUN II.
Above systematic errors and statistic error of Monte Carlo simulation
0.07 % are quadratically added. Then, the total error of F factor is estimatedas 0.11 % for RUN I and 0.27 % for RUN II. These errors of F factor aﬀect
the decay rate by 33 ppm for RUN I and 28 ppm for RUN II, which is
assigned as a systematic error.
Detection eﬃciency of YAP scintillator
The detection eﬃciency of YAP scintillator is used for the decay rate ﬁttingin the form of the ratio εpick /ε3γ . The main factor of the uncertainty is the

96

3.5 Analysis

response function of the YAP scintillator. The response function of the YAP
scintillator is evaluated with 514 keV single photon of 85 Sr source.
To estimate the uncertainty of the Monte Carlo simulation, the spectrumof 514 keV γ-ray is parameterized as,
Rpeak ==

photoelectric peak
photoelectric peak + compton region
Number of events(400 keV ∼ 700 keV)
Number of events(150 keV ∼ 700 keV)

(3.18)(3.19)The diﬀerence between the measured data and the Monte Carlo simulation
is summarized in Table 3.14. The mean value of the diﬀerence is 2.0 % for
RUN I and 1.3 % for RUN II, which is considered as a systematic uncertaintyof the eﬃciency ratio εpick /ε3γ . When the eﬃciency ratio εpick /ε3γ is variedby that, the deviation of the decay rate is 64 ppm for RUN I and 19 ppm
for RUN II, which is assigned as a systematic error.

RUN I
RUN II

YAP0
1.1 %
1.1 %

YAP1
3.4 %
1.2 %

YAP2
1.5 %
2.2 %

YAP3
—
0.7 %

Table 3.14: The diﬀerence between the measured data and the Monte Carlosimulation

3.5.5

Other Sources of Systematic Errors

The Zeeman eﬀect
In the presence of a magnetic ﬁeld B, one of the o-Ps states is perturbed.
The decay rate of the state is estimated as [32],λo-Ps = λo-Ps 1 +η 2 λp-Ps
4 λo-Ps(3.20)

where η = 4μB/h̄ω = B(gauss)/36287  1, μ is magnetic moment of theelectron, h̄ω is the hyperﬁne structure splitting between o-Ps and p-Ps. As
in Sec. 2.3.1, the absolute magnetic ﬁeld is measured around the assembly.
The measured value B = 0.5 gauss contributes to the decay rate by +5 ppm,
which is assigned as a systematic error.
Three-photon annihilation of an electron-positron pair
The three-photon pick-oﬀ process can occur with a small relative ratio tothe two-photon process. The relative ratio σ3γ /σ2γ can be estimated from97

3.6 Analysis

the decay rate ratio,
σ3γ /σ2γ =λo−Ps × 3
∼ 1/378.
λp−Ps × 1

(3.21)

where λo−Ps and λp−Ps are the decay rate of o-Ps and p-Ps respectively.
The factor 3 in numerator comes from the triplet states of o-Ps.
On the other hand, the pick-oﬀ events by the spin-ﬂip eﬀect (see Sec.2.1.1)
do not have the 3γ decay process. If the contribution of the spin-ﬂip is neglected, the 3γ pick-oﬀ events increase F factor by 0.3 % and fcompton factor
by 1 %. These changes decrease the decay rate by 91(33) ppm for RUN I(II).
The Stark eﬀect
As mentioned in Appendix A, the electric ﬁeld on the grain surface of the
SiO2 powder can decrease the decay rate. These contributions are estimatedwith simple models in Appendix A. For RUN I, typically –3 ppm contribution is expected, while –4 ppm is expected for RUN II.
Excited state of Ps
The excited state (n=2) of Ps is observed in the experiment utilizing the slow
positron beam with a fraction of 10−3 -10−4 [58, 59]. 2P state immediatelytransits to 1S state (τp = 3.2 ns). However, 2S 3 state, whose fraction is
3/16 of excited states, decays with a low decay rate of λ3γ /8.
But in this experiment, positrons are much energetic and deeply enter
into the bulk of the target material. Then, the positrons diﬀusing to thesurface are considered to be well thermalized. In this case, the positrons
cannot be emitted as the excited state Ps since its work function is positive
owing to the small binding energy of 1.7 keV.
In addition, the excited state Ps is likely to be deexcited at the ﬁrst
collision on the SiO2 grain surface. Consequently, the excited state Ps cannot
aﬀect the time spectrum in the delayed region.

3.5.6

Summary of the Systematic Errors

Above systematic eﬀects are summarized in Table 3.15. The total systematic
error is calculated as a quadratic sum of them since they are considered to
be independent with each other.

3.6

Results of the Present Measurement

From the analysis in this chapter, the following o-Ps decay rates are obtained
for two runs,

98

3.6 Analysis
Source of the Contributions
TDC module related error– Integral Non Linearity
Contamination of pile-up events– for Base cut
Pick-oﬀ Correction– 3γ subtraction– Ge detector eﬃciency– YAP scintillator eﬃciency
Other Sources– Zeeman eﬀect– Three-photon annihilation– Stark eﬀect
Total

For RUN I (ppm)

For RUN II (ppm)< ± 15< ± 15< − 10< − 10

±89
±33
±64

±91
±28
±19

−5
−91
+3
−147 and +115

−5
−33
+4
−104 and +98

Table 3.15: Summary of the systematic errors
for RUN I,−1
λo-Ps (RUN I) = 7.03876 ± 0.0009(stat.)+0.0008−0.0010 (sys.) μs(3.22)

and for RUN II,−1
λo-Ps (RUN II) = 7.04136 ± 0.0009(stat.)+0.0007−0.0007 (sys.) μs(3.23)

where the ﬁrst error represents a statistic error, and the second one is for
systematic. The values are shown in Fig. 3.35. The O(α2 )-corrected QED
prediction is also shown in the ﬁgure.
The discrepancy of the two runs is 1.6 σ. Then it can be said that thetwo measurements with diﬀerent types of silica material are consistent. In
the calculation, the systematic error of the pick-oﬀ correction is considered
to be not correlated between two runs.
The ﬁnal result is obtained by combining these results of the two runs.
The weighted average value with respect to the error is,−1
λo-Ps = 7.0401 ± 0.0006(stat.)+0.0007−0.0009 (sys.) μs(3.24)The total error is about 150 ppm and 1.6 times more precise than that ofthe previous measurement [37].

99

3.6 Analysis

QED prediction
with O( α2) correction

RUN I

RUN II

Combined

7.037

7.038

7.039

7.04

7.041

7.042

7.043

Decay rate (μ sec-1)
Figure 3.35: The results of the two runs and the combined result. The
whole error bar shows the total error and the inside line represents the sta
tistic error. The O(α2 )-corrected NRQED prediction of 7.039 979μs−1 isalso shown as the red vertical line.

Chapter 4

Discussion
In this chapter, the experimental meanings and the theoretical implicationsof the obtained results are discussed.

4.1

Experimental Meanings

4.1.1

Solution of the Systematic Increase near the Prompt
Peak

The direct pick-oﬀ correction method adopted in this experiment guarantees that all the pick-oﬀ contributions are completely corrected even when
the pick-oﬀ ratio cannot be estimated in advance, or the pick-oﬀ ratio istime dependent. At this point, the direct correction method has the great
advantage over the previous extrapolation method.
But unfortunately, the previous two measurements by this method observed the rapid rise of the decay rate near the prompt peak. The deviation
reached 500 ppm∼1000 ppm at 60 ns ﬁtting start time. This systematic
error was not understood at that time and left room to doubt the obtained
decay rate.
However, the results of the present measurements show no indication ofthis systematic rise. The change is attributed to the following improvementsof the measurement.
• In the present measurement, the timing characteristics of γ-ray detectors are improved. For the measurement of the o-Ps time spectrum,
fast inorganic scintillator YAP is introduced. Consequently, one candeny the posibility that the tail of the prompt events or the pile-up
events distort the time spectrum near the prompt peak. For the measurement of the pick-oﬀ ratio, time walk correction and treatment of
the SRT component is thoroughly studied. Then, more conservative
approach is taken. Only the direct information such as measured risetime is used to correct the time walk and the validity of the time walk
100

101

4.1 Discussion
correction is evaluated with the measured data without silica target.

• In the present measurement, the validity of the electronics systemis thoroughly investigated. In the end, the systematic deviation of
the relative bin width is found in the KEK TDC. The cause of thedeviation is a logic unit module ﬂuctuated by the veto release signal
just before T0 . The veto release signal is produced by the trigger
system and the output of the logic unit is slightly correlated withthe trigger timing. Consequently, the detector hit signal suﬀers fromthe ﬂuctuation correlated with the trigger timing. Once the reason isunderstood, the deviation can be corrected with the calibration databy the random stop RUN.
In these improvements, the present measurement successfully conﬁrms thevalidity of the direct pick-oﬀ correction method. Then, the obtained decayrate is considered to be reliable.

4.1.2

Validation of the Fitting Start Time

The pick-oﬀ ratio can be obtained in the time region after 60 ns. This isbecause the prompt peak of the Ge detector extends up to 50 ns, whichis conﬁrmed with the measured data without silica target. However, theexcellent time resolution of the YAP scintillator makes it possible that thetime spectrum before 60 ns is ﬁtted with the analytic form of the pick-oﬀ
ratio, which is refered to as the thermalization function.
Fig. 4.1 shows the ﬁtted decay rate with the ﬁtting start time before
60 ns. It is found that the ﬁtted decay rate is stable around 60 ns ﬁtting
start time. However the ﬁtted decay rate is rapidly deviated before 40 ns
for RUN I and 45 ns for RUN II. It can be understood as the following
systematic eﬀects.
Statistic error of the thermalization function
Before 60 ns, the pick-oﬀ ratio is calculated with the extrapolation of
the thermalization function. As seen in Fig. 3.25 and Fig. 3.26, theuncertainty of the thermalization function is rapidly increased.
Systematic error of the thermalization function
For the thermalization function, the simplest form is introduced in
Eq. 3.5. It is too simple to reproduce the thermalization process before
60 ns. Then, the systematic uncertainty is considered to be large inthe extrapolated region. In fact, the thermalization function rapidly
increases in the hatched region of the Fig. 4.1. Consequently, the decayrate ﬁtting fails.
Systematic errors proportional to the pick-oﬀ ratio
Some systematic errors such as the error of the detector eﬃciency

4.1 Discussion

102

are proportional to the pick-oﬀ ratio. Since the pick-oﬀ ratio rapidly
increases before 60 ns, the propagation of these errors is much larger
than the assigned value, which is evaluated at 60 ns ﬁtting start time.
Suppression by the anti-trigger coincidence
A trigger may be suppressed by the anti-trigger hit of the accidental
event. The margin of the anti-trigger coincidence is set to 30 ns in themeasurement. Consequently, the time spectrum of the YAP scintillator before 30 ns is suppressed with the probability of the accidental
event.
Time walk correction of the plastic scintillator
The time walk correction of the plastic scintillator is carried out with
the Wide ADC value of the plastic scintillator. However the Wide
ADC value correlates with the accidental activity within the gate
width of 50 ns. Consequently, the small dip of the time spectrum
is expected just before 50 ns. The amount does not exceed the accidental probability.
The prompt peak of the YAP scintillator
Practically, the prompt peak of the YAP scintillator is not normal
gaussian function. This is because the time constant of the e+ annihilation in the plastic scintillator or the glass beaker is about 2 ns.
For the simplicity, all the prompt events are assumed to annihilate
with the time constant of 2 ns. It takes 20 ns for the prompt peak to
decrease to the level of accidental distribution. Then, it takes another20 ns to totally disappear.
Thus, one can conclude that indeed the ﬁtted decay rate is rapidly deviated before 40 ns or 45 ns. the deviation is totally understood as the above
systematic eﬀects. Then, one can safely set the ﬁtting start time at 60 ns
where the above systematic eﬀects are highly suppressed or do not exist.

4.1.3

Improvements of the Monte Carlo Simulation

The Monte Carlo simulation plays a crucial role in this experiment. In
the present measurement, the Geant4 based Monte Carlo simulation is introduced. It can handle the simulation of the positron and the simulationof the secondary particles. They are implemented in the previously used
Monte Carlo simulation. But after some studies, it is found that the simulation does not reproduce the measured data and the discrepancy originates inthe detector speciﬁc nature such as the optical photon collection eﬃciency,
the charge collection eﬃciency and the slow rise-time component. Then, thefollowing measures are taken.

decay rate λ3 γ (μ s -1)

103

4.1 Discussion

7.045
7.044

thermalization function
rapidly increases

7.043
7.042

used point

7.041
7.04
7.039
7.038
7.037
7.036
7.035

30

40

50

decay rate λ3 γ (μ s -1)

RUN I

60
70
80
fitting start time (ns)

7.045
7.044

thermalization function
rapidly increases

7.043
7.042
7.041
7.04
7.039

used point

7.038
7.037
7.036
7.035

30

40

50

RUN II60
70
80
fitting start time (ns)Figure 4.1: The ﬁtted decay rate with the ﬁtting start time before 80 ns.
The upper ﬁgure shows the decay rate for RUN I. The lower ﬁgure shows
the decay rate for RUN II. The error bars show only the statistic error andthe inner lines represent the statistic error of the YAP time spectrum. In
the left hatched region, the thermalization function rapidly increases.

4.2 Discussion

104• The response function is evaluated with the measured data and incorporated into the Monte Carlo simulation.
• The slow rise-time component of the Ge detector is cut with the measured rise-time information in the oﬄine analysis. Then, the cut eﬃciency is evaluated with the run data and applied to the Monte Carlosimulation data. In fact, this strategy is employed in the previous
measurement.
The obtained Monte Carlo simulation more realistically estimates thedetection eﬃciency and the shape of the 3γ spectrum. It can suppress theuncertainty of the pick-oﬀ correction and the energy window dependence.
In fact, the previously observed energy window dependence is disappearedin the present measurement.

4.1.4

Dependence on the Two Types of Silica Material

In the previous measurements, the two types of silica powder was used for asystematic test of the method. One is the hydrophilic silica powder and low
density, another is the hydrophobic silica powder and higher density. The
results of the two measurements are consistent with each other.
In the present measurements, the two types of silica material are used.
One is the silica aerogel and solid state, another is the silica powder. They
have the almost same density. However, the pick-oﬀ ratios are diﬀerent witheach other by a factor of 3 as clearly seen in Fig. 4.2. The reason of thisdiﬀerence is not fully understood. But it can be said that the pick-oﬀ ratio isdependent on not only density but also internal structure, such as the grainsize, the aggregate structure, the surface, etc. The measured decay rate isconsistent with each other. Then, one can conclude that the direct pickoﬀ correction method is a robust technique for the various materials whose
internal structure is not fully known, whereas the extrapolation method
which assumes the linearity to the density is not safe in this situation.

4.2

Theoretical Implications

As seen in Fig.4.3, the present measurement agrees with the recent threemeasurements from 1995. They include the two measurements of our groupby the direct pick-oﬀ correction method and the one measurement of Michigan group by the improved extrapolation method. Then it can be said thatthe experimental values by the diﬀerent method does not conﬂict with eachother in about 200 ppm level.
The present measurement disagrees with the old measurements before
1990. The reason of the discrepancy is the one mentioned in Sec. 2.1.2. The
orthopositronium lifetime puzzle is already solved.

λpick/λ3γ

105

4.2 Discussion

0.05

0.045
0.04

0.035
0.03

RUN I (aerogel)

0.025
0.02
0.015

RUN II (powder)

0.01
0.005
0

100

200

300

400

500

600

time (ns)Figure 4.2: The measured pick-oﬀ ratios with two types of silica material.
The pick-oﬀ ratio of silica aerogel is 3 times larger than that of silica powder.

106

4.2 Discussion

The result of the present measurement is in good agreement with theO(α2 )-corrected QED prediction as in the ﬁgure. In addition, the precision of the present measurement, about 150 ppm, enables us to discuss the
non-logarithmic α2 correction on the o-Ps decay rate. As in Fig. 4.4, theresult of the present measurement diﬀers from the O(α2 )-corrected QED
prediction by 0.1 σ while diﬀers from the QED prediction without O(α2 )correction by 1.7 σ. When the recent four measurements after the solutionof the orthopositronium lifetime puzzle are combined, the world average ofthe decay rate is obtained as,
λo-Ps = 7.0401 ± 0.0007(total)(4.1)

where the errors of the four measurements are considered to be not correlatedwith each other. The world average of the decay rate is in good agreementwith the O(α2 )-corrected QED prediction and the discrepancy to the QED
prediction without O(α2 ) correction is 2.6 σ. Then, it can be said that theworld average favors the O(α2 )-corrected QED prediction rather than thatof O(α) correction.
In fact, the absolute value of the O(α2 ) correction is not proved with theaccuracy of the present measurement. Though the theoretical group whichhave performed the complete calculation of the O(α2 ) term states a ppm
level uncertainty, the calculation has not been veriﬁed by other authors.
Then, the more accurate measurements of the o-Ps decay rate are expected.

107

4.2 Discussion

Phys.Rev.Lett. 40,737 (1978)

Phys.Rev.Lett. 40,737 (1978)

Phys.Lett. A69,97 (1978)

J.Phys. B11,743 (1978)

Phys.Rev.Lett. 49,525 (1982)

Nuovo Cimento 97A,419 (1987)

Phys.Rev.Lett. 58,1328 (1987)

Phys.Rev. A40,5489 (1989)

Phys.Rev.Lett. 65,1344 (1990)

Phys.Lett. B357,475 (1995)

Phys.Rev.Lett. 90,203402 (2002)

Phys.Lett. B572,117 (2003)

Present Measurement

7.02

7.03

7.04

7.05

7.06

7.07

Decay rate (μ sec )-1

Figure 4.3: The history of the o-Ps decay rate measurements including thepresent measurement. The each error bar shows the total error of the measurement and the inside line represents the statistic error. The red verticalline shows the O(α2 )-corrected NRQED prediction of 7.039 979μs−1 .

108

4.2 Discussion

without O(α2)
correction

QED prediction
with O( α2) correction
Phys.Lett. B357,475 (1995)

Phys.Rev.Lett. 90, 203402 (2002)

Phys.Lett. B572,117 (2003)

Present Measurement

7.037

7.038

7.039

7.04

7.041

7.042

7.043

Decay rate (μ sec-1)

Figure 4.4: The relation between the resent measurements and the theoretical predictions. The red vertical line shows the O(α2 )-corrected NRQED
prediction of 7.039 979μs−1 . The blue vertical line shows the QED prediction without O(α2 ) correction, that is 7.038 226μs−1 .

Chapter 5

Conclusions
To test the bound state QED, especially the O(α2 ) correction term, theorthopositronium decay rate is newly measured. The features of the present
measurement are as follows.
• The setup is newly constructed on the base of the fast inorganic scintillator YAP.
• The positronium formation assembly and the trigger system are totally
reﬁned.
• The Monte Carlo simulation is made on the base of the Geant4 package. The detector speciﬁc nature is also reproduced by incorporating
the response function.
In the present measurement, the two runs are performed with the diﬀerent types of silica target. The following results are obtained.
• The systematic increase near the prompt peak is understood andsolved.
• Other systematic errors such as the energy window dependence andthe uncertainty of the detection eﬃciency are highly suppressed.
• The results of the two runs are consistent with each other, even though
they have the totally diﬀerent pick-oﬀ ratios.
• The combined result achieves the total error of about 150 ppm andagrees with the O(α2 )-corrected QED prediction.
In addition, the world average of the decay rate is in good agreementwith the O(α2 )-corrected QED prediction and the discrepancy to the QED
prediction without O(α2 ) correction is 2.6 σ. Then, one can conclude thatthe world average of the decay rate favors the O(α2 )-corrected QED prediction rather than that of O(α) correction.
109

Appendix A

Stark Shift
In this appendix, Stark shift of the decay rate is discussed. Most of thecalculations are quoted from the reference [38].

A.0.1

Stark Eﬀects on o-Ps Decay Rate

The decay rate of o-Ps is proportional to the Ps wave-function at the origin,which is a ﬂux factor of 3γ decay.
λ3γ ∝ |φ(r = 0)|2 .

(A.1)

If a uniform electric ﬁeld E is applied to the Ps, the wave function of thePs, ψ, is perturbed to be,ψ = ψ0 + Eψ1 + E 2 ψ2 + · · · ,(A.2)

where ψ0 is the unperturbed ground state wave function, and ψ1 , ψ2 , · · ·stand for the higher order perturbed terms. Simple calculation by the perturbation method shows the relative shift of the decay rate as a function ofE [60],λ3γ
λ3γ|ψ1 |2|ψ0 |2(A.3)= 248 · (E/E0 )2 ,

(A.4)

= E2

where E0 = m2e e5 /h̄4 ≈ 5.14 × 109 V /cm, and E is the electric ﬁeld sensed
by the Ps during its lifetime.

A.0.2

Contribution of Charge on Primary Grains

The charges on the silica materials are measured with a static electricity
sensor [61]. The measured values are 3(1) × 10−9 [C/g] for silica aerogel andless than 1 × 10−9 [C/g] for silica powder.
110

111

A.0 Stark Shift

Provided that the primary grains and the electric charges are uniformly
distributed, the volume L3 which one electron occupy on average is estimatedas,
1.6 × 10−19 [C]
L3 =
(A.5)
∼ 1.8 × 10−9 [cc]3 × 10−9 [C/g] · 0.03[g/cc]
If the distance between the Ps and the electron is x, the square of the electricﬁeld sensed by the Ps would be,|E|2 ==e
4πε0 x2


ẽ2
x4

2


ẽ = 1.4 × 10−9 [V · m ] .

(A.6)

Since the Ps is freely moving across the whole area between the grains, theaverage ﬁeld sensed by the Ps should be estimated by averaging over thewhole volume as,|E|2 =

= 4π
dV |E|2 /L3
 L

ε
2
4πẽ

εL3

dxx2

ẽ2 3
/L
x4

,

(A.7)

where ε is a minimum distance between the Ps and electron. Bohr radius of
the Ps rP s = 1.1 × 10−10 [ m ] (twice the Bohr radius of atomic hydrogen) isa good approximation for ε, since the Ps in the free space keeps away fromthe grain surface with a potential as [62],V (x) = V0 e−x/rP s(A.8)

where V0 is the work function of Ps (of order 1eV).
Thus, an average of the squared electric ﬁeld would be, |E|2 = 1.3 ×
8
10 [ V 2 /m2 ]. From Eq. (A.4), this could contribute to the decay rate only
by 1.2 × 10−7 ppm level. Clearly we can neglect this term.

A.0.3

Contribution of Dipole Moment on the Surface of Grains

SiO2 powders have mainly two types of functional groups, silanol groups(−Si (OH)) and siloxan groups (−Si (O) Si−). The hydrophilia of the powder is attributed to these silanol groups. In order to obtain the hydrophobecharacteristics, these groups are substituted with other functional groups
such as, the tri-methyl-silyl groups, the dimethyl-silyl groups, etc.
The typical density of the silanol groups on the grain surface are investigated, 2.5 /nm2 for hydrophilic silica and 0.44 /nm2 for hydrophobic silica.
Figure A.1 shows the schematic diagram of one SiO2 grain. The silanol

112

A.0 Stark Shift
OH

R

R
O

Si
O O

OH
Si
O

O

Si O
O

Si O
Si O
Si
O
O
O

Si
O Si

R=Si (CH3)3

Si
O O

OH
Si

Si

O

O
O Si O Si O
Si O Si OROO Si O Si R
O Si O Si O Si O
O
O
Si
Si
O
O
O
O
O
Si
Si
Si O Si O Si
R
R
O
O
O
O
O
Si
Si
O Si OOH
R
R

Figure A.1: Schematic diagram of SiO2 grain. Part of the surface functionalgroups are substituted, while there are still silanol groups left.

groups are existing on the surface of the grain. A typical size of dipole
moment for the groups −(OH) is 1.7 × 10−18 [esu · cm] [63].
For the sake of the simplicity, it is assumed that the dipole moment p isdirected outward on the surface of the grain as illustrated in Fig. A.2. Then,the electric ﬁeld at the point O outside the grain is expressed as, =E

13 (n · p) · n − p
·
4πε0
R03

(A.9)

The squared electric ﬁeld would be,p2
3 cos2 α + 1
R06
= ···
p2
=
R02 + 3 (r − R cos θ)2
R08|E|2 =(A.10)

Deﬁning the area density of the silanol-group as σ and integrating over thesurface of one grain, the squared electric ﬁeld at the position O would be,|E|2o = 2πσr2

 π
0

2 2

= 2πσr psin θdθ|E|2

 1
−1

dx

(r + Rx)2
1
+
3
R06
R08

(substituted x as x = − cos θ).

(A.11)

113

A.0 Stark Shift

α

p
r

R0

n

θ

O
R

Figure A.2: A deﬁnition of the variables used for calculation in the context.
n is a unit vector directing from dipole moment p to the observing point O.

In the end, the following equation is obtained.
|E|2o

= πσp

2 r

R

 

3
2







1
1
1
1
1
−
−
−
4
4
3
(R − r)(R + r)
2r (R − r)
(R + r)3


1
1
1
+
−
(A.12)
2
4Rr (R − r)
(R + r)2

As in the previous section, the Ps is assumed to be freely moving acrossthe grains. However in this case, the Ps continues to collide with the grains.
Then, average distance is calculated as a mean free path between the grains,
L, which has been already obtained as in Table 2.1. Thus the integration
is carried out in one dimension from the nearest distance to the powder ε(again the Bohr radius of Ps is used as ε), to the mean distance between thegrains L.
Integrating Eq. (A.12) by variable R (r + ε ≤ R ≤ L), Almost all contribution comes from the ﬁrst term of Eq. (A.12) as R approaches to r + ε.
Thus, the average squared electric ﬁeld would be,|E|2o =
=



 L
r+ε

dR|E|2o / L

3πσp2 r2L
3πσp2 r
2L
πσp2
.
2Lε3

 L
r+ε



·

dx

1
x(x − r)4

L
1
1
r −3(x − r)3 r+ε(A.13)For the silica aerogel (SP-7), the average squared electric ﬁeld |E|2o =A.0 Stark Shift114

2.2 × 1016 [V 2 /m2 ] is obtained. From the Eq. (A.4), one can estimate thecontribution of Stark shift to be 2.6 ppm of the decay rate.
For the silica powder (R972CF), the average squared electric ﬁeld |E|2o =2.2 × 1016 [V 2 /m2 ] is obtained. Then, one can estimate the contribution ofStark shift to be 3.7 ppm of the decay rate.

Appendix B

Expected Time Spectrum
The expected time spectrum is discussed in the following sections. The
situation is rather complicated when the conditions of the trigger and theeﬀects of the several cuts are taken into account.

B.1

Probability of Accidental EventsFirst of all, the probability of accidental events around a trigger is discussed
in this section. The trigger scheme of this experiment is• The ﬁrst hit of the trigger plastic scintillator from the main latch resetis a trigger candidate.
• Only when the event does not hit the anti-trigger, the trigger is accepted.
• The second trigger signal is blocked by the main latch system, even if
the above trigger condition is met.
The accidental events which hit the anti-trigger are uniformly distributed
around the trigger. This is because the accidental event with anti-trigger
hit does not make a trigger and does not correlate with the trigger signal
just like environmental background events. But, considering the eﬀect of theanti-trigger, the accidental event with anti-trigger hit cannot exist within thewidth of anti-trigger veto from a trigger. The width of anti-trigger veto is
150 ns and its margin from the trigger signal is 30 ns. Then, the probabilityof the accidental event with anti-trigger hit isPwith (t) =

⎧
⎪
⎨ 0

when − 120 ns< t < 30 ns⎪⎩ C

when t < −120 ns, 30 ns< t

with

(B.1)

where Cwith is a constant value determined by the event rate accompanying
anti-trigger hit.
115

116

B.1 Expected Time Spectrum

On the other hand, the accidental event which does not hit the antitrigger is a candidate of trigger and correlates with the trigger signal. Indeed
the probability after the trigger is uniform like the events with anti-trigger
hit, the probability before the trigger must be small due to the ﬁrst hit
condition. The accidental events without anti-trigger hit are allowed only
when the events occur before the main latch reset. The main latch reset
occurs before the trigger signal with the following probability.
Preset (t) = Creset exp(Rtrig · t)

(B.2)

where the exponential function comes from the ﬁrst hit condition and Rtrig
is the trigger rate. An accidental event without anti-trigger hit at t = tis allowed only when the main latch reset comes in the time region from t
to 0. Thus, the probability of accidental events without anti-trigger hit isexpressed as,



⎧
 0
⎪

⎪
Preset (t)dt ∼ −Cwithout
Rtrig · t
⎨ Cwithout

Pwithout (t ) =

⎪
⎪
⎩

t

when t < 0
Cwithout

when t > 0(B.3)
The probability Pwith (t) and Pwithout (t) is also shown in Fig. B.1. In theﬁgure, the probability Pwithout (t) remains at 0 in the time region from -50 ns
to 0 ns. This is because the trigger logic signal has the width of 50 ns andthe succeeding signal within 50 ns does not produce a trigger edge in thediscriminator.

  

          
      " !

 

Figure B.1: Probability of accidental events around the trigger signal. The
accidental events with and without anti-trigger hit have the diﬀerent probability.

117

B.2 Expected Time Spectrum

In the above discussion, the trigger eﬃciency of β + decay events is considered as 100 %. Practically, less than 10 % of them are not triggered andforms the ﬂat distribution over the trigger.
Another factor comes in the oﬄine analysis. The Base cut of the trigger
plastic scintillator suppress the accidental events just before the incident
trigger. But, the accidental events before the trigger are already suppressed
as seen in the ﬁgure.

B.2

Expected Time Spectrum

In this section, the expected time spectrum of γ-ray detectors are discussed.
Firstly, the γ-rays hit the detectors in this experiment are classiﬁed into
following three groups according to their time distribution, which is shownin Fig. B.2.
A.

The γ-rays of Ps which correlate with the trigger. These events consistof the 2γ events at T0 and the 3γ events which decay exponentiallyfrom T0 .

B.

The 1077 keV nuclei-γ, the positron and their secondary particles suchas electrons and bremsstrahlung-γ’s which correlate with the trigger.
These events are concentrated at T0 .

C.

The accidental events which do not correlate with the trigger. These
events consist of the positron source related events and the environmental backgrounds. The former is discussed in the previous section
and the latter is rather small and completely ﬂat distribution.













Figure B.2: Time distribution of the the γ-rays. The γ-rays which hit thedetectors in this experiment are classiﬁed into three groups. A. Ps events,
B. source related events, C. accidental events.

The group A and B are restricted to at most one event for a trigger,
whereas the group C can occur many times for a trigger. The probabilities

118

B.2 Expected Time Spectrumof these group are denoted by PA , PB and PC . PC is much smaller than PA
and PB .
Then, all the cases of event are expressed as(A + A)(B + B)(C + C)∞(B.4)

where A, B and C denote that the detector does not detect A, B and C
respectively. The term which includes A, B or C decrease its probability
by a factor of PA =1–PA , PB =1–PB , PC =1–PC ∼ 1. Thus, the above form
leads to
PB A(C + C)∞ + PA B(C + C)∞ + PA PB C(C + C)∞ + AB

(B.5)

The multiplied terms such as AB express the pile-up events and have theirown time distribution.

B.2.1

Pile-up by the Accidental Events∼ X(C + C)∞

The ﬁrst three terms of the form X(C+C)∞ denote the time distribution of
X under the accidental backgrounds. If two signals overlap with each other,
the pile-up cuts reject the event in the oﬄine analysis. Then, the proban
bility of X without the accidental event, denoted by XC , is a dominant
contribution on the observed spectrum. The probability of X(C + C)∞ is
already known such as a exponential function for X = A. The probabilitynof XC is considered to be a modiﬁcation of the intrinsic probability. Thisis expressed as∞(B.6)
XC = X(C + C)∞ × S(t)where the function S(t) is a suppresion factor depending on the detection
time of X.
The accidental events spoil the event in the following two situations.
• The accidental events hit the detector around the timing of X hit. The
overlapped events are suppressed by the Base cut and the NarrowWide cut in the oﬄine analysis. These range are denoted by tb andtnw respectively• The accidental events hit the detector before the X hit. The detector
system accepts only the ﬁrst hit from the latch reset.
Therefore, the time region in which the accidental hit spoils the X eventat t is0 ∼ t + tnw

when t > tb

(B.7)

t − tb ∼ t + tnw

when t < tb

(B.8)

which is also shown in Fig. B.3. Practically, the boundary between thesetwo cases is shifted by the margin of latch reset from T0 , which is 50 ns in

119

B.2 Expected Time Spectrum











  

  Figure B.3: The suppression region in which accidental events spoil theevent. The events with accidental hits are suppressed by the pile-up cuts.
The events are also suppressed through the ﬁrst hit condition.
this experiment. When the above time region is denoted by tstart ∼ tend ,the suppression factor is calculated with
S(t) = 1 − tendtstart(Cn terms) ∼ 1 − tendtstartC(B.9)

The probability of the accidental C is complicated around the trigger signal.
But, if the probability is a ﬂat distribution, the suppresion factor leads toS(t) = e−(tend −tstart )(B.10)

Then, the time spectrum of X is obtained as∞

XC= PX (t) × S(t)(B.11)To be precise, the energy distribution and the solid angle must vary among
the accidental events, such as the events with anti-trigger hit, the events
without anti-trigger hit and the environmental backgrounds. But, the eﬀect
is considered to be small.
The another term XC is usually rejected as a overlapped event. But, thisterm may remain in the following two cases. The eﬀect of the contaminations
which pass through the pile-up cuts are discussed later.
• If the detector has fast timing nature like YAP scintillator, the datataking of the accidental event may be completed before the hit of X.

120

B.2 Expected Time Spectrum
The probability is expressed as
XC = PC (t) ×

 ∞
t+tnw

PX (t )dt

(B.12)

where t is the timing of the accidental hit and the measured energy is
the one of the accidental event.
• when the amplitude of the signal X is less than the threshold value,
the accidental hit after the X is accepted as a accidental events. The
time spectrum is not just the accidental’s ﬂat distribution and aﬀectedby the time distribution of X through the Base cut. The probability
is expressed as,
XC = PC (t) × t−tb−∞dt

 Ethr
0

dE PX (t , E)

(B.13)

Therefore, a part of the accidental’s ﬂat distribution is considered tobe suppressed near the prompt peak.

B.2.2

Pile-up Events with the Prompt Signal∼ AB

The pile-up events with the prompt signal is not a problem since the events
are triggered at T0 and do not aﬀect the time spectrum in the delayed region.
But, the special care must be taken if the amplitude of the prompt signal
is less than the threshold level. In this case, the time spectrum of A is
suppressed near the prompt by the Base cut, just as the case of XC. Then,the probability is expressed as,
⎧
0
⎪
⎪
⎨

AB =

⎪
⎪
⎩ PA (t)when 0 ns< t < tb
 Ethr0(B.14)PB (E)dE when t > tb

The probability is usually larger than the one in the XC case, where theprobability of the accidental is so small from the beginning. If the boundaryof the Base cut is set above the energy threshold Ethr , the distortion of thetime spectrum is avoided. But in this case, the measured energy is deviatedby the prompt signal.

B.2.3

Contamination of Pile-up Events

The overlapped event is assumed to be eliminated by the pile-up cut inthe above discussion. But practically, the pile-up cuts have dependency on
the energy of the accidental γ-ray. In addition, the completely overlapped
events are not rejected by the pile-up cuts. Thus, the eﬀect of the pile-up
cuts is restricted as shown in Fig. B.5. The allowed region can be divided
into three typical regions.

121

B.3 Expected Time Spectrum

1. The overlapped signal is so small that the Narrow-Wide cut cannot
reject the event. The energy of the incident γ-ray is deviated by the
cut margin at most, which is 15 keV for Ge and 30 keV for YAP.
2. The two signals are completely overlapped and the Narrow-Wide cut
cannot distinguish these signals. The energy is largely deviated by theoverlapped γ-ray.
3. The preceding signal is so small that the Base cut cannot reject theevent. Especially near the prompt peak, the energy deviation and thecut eﬃciency must be closely looked at.
The probability of the contamination is obtained as a integral over the allowed region. When X and C are overlapped. there are two cases according
to the leading signal.
XC = PX (t) ×
CX = PC (t) ×

 ∞
0
 ∞

d(ΔE)
d(ΔE)

 tnw
−tb
 tnw

0

−tb

d(Δt) PC (t + Δt)Cont(Δt, ΔE) (B.15)
d(Δt) PX (t + Δt)Cont(Δt, ΔE) (B.16)

where the detection time is the timing of the leading signal X and C respectively. The measured energy is a sum of the leading signal’s energy E anda part of the pile-up signal’s energy ΔE.

B.3

Time Spectrums of γ-ray Detectors

B.3.1

The Case of Ge Detector

suppression factor S(t)Since tb is a scale of the detector signal, the value reaches several μs in thecase of Ge detector. On the other hand, tnw is determined by the widthof Wide gate, which is 40 μs for Ge detector. Thus, the probability of theaccidental events around the trigger does not aﬀect the time dependencyof the suppression factor. Only the diﬀerence of the probability before andafter the trigger, which is the probability of the accidental events withoutanti-trigger hit, determines the time dependence. Thus, the suppression
factor S(t) for Ge detector is expressed as
S(t) ∝⎧⎪⎨ exp(Rwithout · t)⎪
⎩ exp((Rwhen t < several μs(B.17)

with + Rwithout ) · t)when t > several μs

where Rwith and Rwithout are the detector hit rate of the event with andwithout anti-trigger hit. They are approximately half of the total hit rate.

122

B.3 Expected Time Spectrum 

 











    !

     








Figure B.4: Suppression region of the pile-up cuts. The upper ﬁgure shows
the suppresion region when the pile-up cuts are assumed to be perfect. ΔE
and Δt are the energy of the accidental signal and the time interval from theincident signal. The lower ﬁgure shows the suppresion region in considering
the energy dependence of the pile-up cuts. In this case, some regions areallowed.

123

B.3 Expected Time Spectrum         

Figure B.5: Suppresion factor S(t) for Ge detector. The events in the later
region are suppressed through the ﬁrst hit condition.

other factors
The time dependent deviation of the intrinsic time spectrum is not a problem
for Ge detector, since only the ratio of 2γ and 3γ is important in the analysis.
But, the time dependence of the accidental distribution is sensitive throughthe subtraction of the accidental spectrum. The energy of the accidental
event near the prompt is varied by the prompt hit with the probabilityof about 0.2 %. These tails, which extend to 50 keV threshold energy, iseliminated by the pile-up cut in the later accidental region. Most of thediﬀerence is in the region above the counting window 507 keV - 515 keV anddoes not contribute to the decay rate.
The slow rise-time component of Ge detector plays some role when theaccidental events pile up on the event. The event may be not rejected by the
SRT cut since the accidental signal provides the fast rise-time of the event.
Thus, the number of accidental events is increased by about 1 % near the
prompt peak. However, the energies of these events are not localized aroundthe 511 keV peak and widely spreaded over 1 MeV range. Therefore, theeﬀect is considered to be small in the analysis.
The energy deviation by the pile-up of small signals (region 1.) is nottime dependent. But, the deviation of the energy spectrum is a problem incalculating the 2γ and 3γ eﬃciencies. Therefore, the deviation is measured
with the events around the 511 keV peak and incorporated into the response
function of the Monte Carlo simulation.

124

B.3 Expected Time Spectrum

B.3.2

The Case of YAP Scintillator

suppression factor S(t)The suppresion factor for YAP scintillator is rather complicated by the fast
timing characteristics. For YAP scintillator, tb is about 200 ns and tnw is
500 ns. Then, the kink of the suppression factor comes in the delayed region
around 100 ns as
S(t) ∝⎧⎪⎨ exp(Rwithout · t)when t < about 100 ns(B.18)⎪⎩ exp((Rwith + Rwithout ) · t)when t > about 100 ns

Practically, the kink is smeared by the energy dependence of tb and thesensitivity of the point is less than 10 ppm.

         

 

Figure B.6: Suppresion factor S(t) for YAP scintillator. The events in thelater region are suppressed by the ﬁrst hit condition.

other factors
The YAP scintillators have fast timing characteristics. Then, the accidental
events near the prompt remain if A hits the detector after 500 ns or later(special case of XC). Otherwise, the events are rejected by the pile-up cuts.
The probability that A hit the detector after 500 ns is less than 0.1 %. The
0.1 % excess of the accidental events near the prompt does not contributeto the decay rate.

125

B.3 Expected Time Spectrum

The activities of the prompt events aﬀect the time spectrum near the
prompt peak through the Base cut. About 1 % of the accidental events
have the prompt hit under the energy threshold. Even for the delayed o-Ps
events, the prompt hits under the threshold exist with the probability of
about 0.1 %. Most of the contribution is bremsstrahlung of the positron
and the contribution of 1077 keV nuclei-γ is small due to its tiny emmision
ratio of 1 %.
To avoid the distortion of the time spectrum, the Base cut condition is
set to 140 σ, which correspond to the threshold value. Instead of that, theenergies of these events are getting larger near the prompt by the energyof the prompt hit. The eﬀect of this energy deviation is roughly estimatedas follows. The energy deviation is less than the energy threshold of about
70 keV and the mean value is considered to be around 30 keV. If the energy
is increased by 30 keV, the eﬃciency above the energy boundary 150 keV isincreased by about 10 %. Then, the eﬃciency is considered to shift exponentially with the time constant of about 30 ns, which is the signal shape of
YAP scintillator. The Wide gate will cover the whole signal of the prompt
hit at t = 30 ns, which is the margin of Wide gate. Finally, It is found thatthe increase of the events near the prompt contributes to the decay rate by+6 ppm.
The contamination of the completely overlapped event (region 2.) is not
so much due to the narrow ADC gates. The probability is less than 10−4of the o-Ps delayed events and about 1 % of the accidental events near the
prompt. In addition, the time dependence is considered to be corresponding
to the decay rate. Then, these eﬀects do not contribute to the decay rate.

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