博士論文

New precise measurement of thehyperfine splitting of positronium(ポジトロニウムの超微細構造の新しい方法
による精密測定)石田明

Abstract
In this thesis I report a new precision measurement of the ground state
hyperfine splitting of positronium ∆HFS which is sensitive to high order corrections of quantum electrodynamics (QED) in a bound state. The theoretical prediction and the averaged experimental value for ∆HFS have a discrepancy of 15 ppm, equivalent to 3.9 standard deviations (s.d.). The new precise measurement, in which the systematic uncertainty from the positronium
thermalization eﬀect is reduced, has been performed. The non-thermalizationeﬀect was measured to be as large as 10 ± 2 ppm in the used timing window. When this eﬀect is taken into account, obtained new result becomes∆HFS = 203.394 2 ± 0.001 6(stat., 8.0 ppm) ± 0.001 3(sys., 6.4 ppm) GHz, whichfavors the QED prediction, on the other hand disfavors the previous experimental average by about 3 s.d.

iAcknowledgements
Firstly, I would like to thank Professor S. Asai (UTokyo) for suggesting this
topic and for the guidance in this work.
I also wish to thank Prof. T. Kobayashi and Dr. T. Namba (UTokyo) forvery useful suggestions and discussions. Without their supports, this work can
not be accomplished.
It is a pleasure to thank Dr. H. Saito (UTokyo) for useful discussions especially about theoretical work on the Zeeman transition.
Sincere gratitude is expressed to Dr. M. Yoshida (KEK) for the great support
for construction and operation of microwave system.
Sincere thanks are due to Dr. A. Yamamoto and Dr. K. Tanaka (KEK) for
providing excellent and comfortable experimental environment, and especiallyfor maintenance of the MRI superconducting magnet.
Sincere gratitude is expressed to Dr. T. Suehara (Kyushu U.), Dr. Y. Sasaki,
Mr. G. Akimoto (UTokyo), Prof. A. P. Mills, Jr. (UC Riverside), Dr. H. A. Torii,
Dr. T. Tanabe (UTokyo), and Prof. W. Hardy (TRIUMF) for useful discussions.
Warm thanks are due to facilities and the entire members of the Cryogenics
Science Center at KEK without whose excellent support this experiment could
not have been successfully performed.
I also would like to thank all staﬀs and colleagues of Department of Physics
and ICEPP (UTokyo) for all their support.
Finally, I thank my parents, grandparents, brother, and all my relatives fortheir warm and kind support.

ii

Contents
Acknowledgementsii

1 Introduction
1
1.1 Positronium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Hyperfine Splitting of Positronium . . . . . . . . . . . . . . . . .
2
1.3 Theoretical Calculation of Ps-HFS . . . . . . . . . . . . . . . . .
2
1.4 Measurements of Ps-HFS . . . . . . . . . . . . . . . . . . . . . .
3
1.4.1 Principle of indirect measurement using the Zeeman eﬀect 3
1.4.2 Theoretical resonance line . . . . . . . . . . . . . . . . . .
5
1.4.3 Previous Results of Measurements . . . . . . . . . . . . .
7
1.5 Discrepancy between Measurements and Theory . . . . . . . . .
91.5.1 Miscalculation of the theoretical value . . . . . . . . . . .
9
1.5.2 Common systematic uncertainties in the previous experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.3 New physics beyond the Standard Model . . . . . . . . . 10
2 Principle
11
2.1 Possible systematic uncertainties in the previous experiments . . 11
2.1.1 Possible non-uniformity of the magnetic field . . . . . . . 11
2.1.2 Non-thermalized o-Ps eﬀect . . . . . . . . . . . . . . . . . 11
2.2 New methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Large bore superconducting magnet . . . . . . . . . . . . 14
2.2.2 Timing information . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 High performance γ ray detectors . . . . . . . . . . . . . . 15
3 Experimental Setup
19
3.1 Overview of the apparatus . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Static magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Positron source and β-tagging system . . . . . . . . . . . . . . . 23
3.3.1 Source holder . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.2 Plastic scintillator and fused quartz light guide . . . . . . 27
3.3.3 Fine mesh PMTs . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 RF system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.1 RF supply . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.2 Monitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.3 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5.1 Gas-handling system . . . . . . . . . . . . . . . . . . . . . 37
iii

CONTENTS

CONTENTS

3.5.2 Slow positron background . . . . . . . . . . . . . . . . . .
3.5.3 Pickoﬀ background . . . . . . . . . . . . . . . . . . . . . .
3.6 Gamma ray detectors . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 LaBr3 (Ce) . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 UVT light guide . . . . . . . . . . . . . . . . . . . . . . .
3.7 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.1 Overview of DAQ . . . . . . . . . . . . . . . . . . . . . .
3.7.2 CAMAC . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . .
3.9 Ps Thermalization Measurement . . . . . . . . . . . . . . . . . .
3.10 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . .

37
38
38
38
41
44
44
50
50
53
53

4 Analysis
54
4.1 Basic cuts and calibrations . . . . . . . . . . . . . . . . . . . . . 54
4.1.1 Cut: Time diﬀerence of the two PS signals . . . . . . . . 54
4.1.2 Calibration: Energy of the plastic scintillator . . . . . . . 55
4.1.3 Calibration: Energy of the LaBr3 (Ce) scintillators . . . . 55
4.1.4 Time walk correction . . . . . . . . . . . . . . . . . . . . . 56
4.1.5 Cut: Energy diﬀerence of the plastic scintillator . . . . . . 58
4.1.6 Cut: Time diﬀerence of LaBr3 (Ce) . . . . . . . . . . . . 58
4.1.7 Summary of cuts . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Decay rate of Ps . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.1 Deduction of the true signal timing distribution . . . . . . 61
4.2.2 Fitting timing spectra of RF-OFF . . . . . . . . . . . . . 61
4.3 Timing windows . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Energy spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 The Zeeman resonance line . . . . . . . . . . . . . . . . . . . . . 64
5 Results and Discussion
66
5.1 Fitting of the Zeeman resonance lines . . . . . . . . . . . . . . . 66
5.2 Systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2.1 Material Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2.2 Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2.3 RF System . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2.4 Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 Ps-HFS value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Non-thermalized Ps eﬀect . . . . . . . . . . . . . . . . . . . . . . 86
5.5 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6 Conclusion

88

A Run Summary89
A.1 Temperature control . . . . . . . . . . . . . . . . . . . . . . . . . 89
A.2 Run table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B Ps Thermalization Measurement94
B.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 94
B.1.1 Overview of the apparatus . . . . . . . . . . . . . . . . . . 94
B.1.2 Electronics circuit . . . . . . . . . . . . . . . . . . . . . . 94
B.1.3 Monte Carlo simulation . . . . . . . . . . . . . . . . . . . 95
iv

CONTENTS

CONTENTS

B.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
C Final Fitting Parameters

108v

List of Figures
1.1
1.2
1.3

Energy levels of the ground state Ps . . . . . . . . . . . . . . . .
History plot of Ps-HFS measurements . . . . . . . . . . . . . . .
Positronium HFS . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
8
9

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8

Time evolution of Ps mean kinetic energy . . . . . . . . . . . . .
Time evolution of lifetime-weighted Ps mean velocity . . . . . . .
Non-thermalized Ps eﬀect . . . . . . . . . . . . . . . . . . . . . .
Non-thermalized Ps eﬀect on HFS . . . . . . . . . . . . . . . . .
Time evolution of 2γ annihilation rate (S2γ ) . . . . . . . . . . . .
Time evolution of 2γ transition rate . . . . . . . . . . . . . . . .
2γ decay rate without timing information . . . . . . . . . . . . .
2γ decay rate with a timing window of 35–155 ns . . . . . . . . .

13
13
14
15
16
16
17
17

Whole system of our experimental setup . . . . . . . . . . . . . .
Top view in the magnet . . . . . . . . . . . . . . . . . . . . . . .
Photographs of the experimental setup around the magnet . . . .
Static magnetic field distribution without compensation magnets
Static magnetic field distribution with compensation magnet (before Ps-HFS measurement) . . . . . . . . . . . . . . . . . . . . .
3.6 Static magnetic field distribution with compensation magnet (after Ps-HFS measurement) . . . . . . . . . . . . . . . . . . . . . .
3.7 Time fluctuation of the static magnetic field . . . . . . . . . . . .
3.8 Decay scheme of 22 Na . . . . . . . . . . . . . . . . . . . . . . . .
3.9 β-tagging system . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Polar angle to the magnetic field . . . . . . . . . . . . . . . . . .
3.11 Polar angle dependence of the gain of fine mesh PMTs in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12 Polar angle dependence of the energy resolution of fine mesh
PMTs in a magnetic field . . . . . . . . . . . . . . . . . . . . . .
3.13 Polar angle dependence of the timing resolution of fine mesh
PMTs in a magnetic field . . . . . . . . . . . . . . . . . . . . . .
3.14 RF circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.15 RF power at discharge as a function of gas pressure . . . . . . .
3.16 Photograph of the RF cavity . . . . . . . . . . . . . . . . . . . .
3.17 Magnetic field of the TM110 mode . . . . . . . . . . . . . . . . .
3.18 Relative power as a function of time . . . . . . . . . . . . . . . .
3.19 Distribution of AMP output power . . . . . . . . . . . . . . . . .
3.20 LaBr3 (Ce) scintillators . . . . . . . . . . . . . . . . . . . . . . . .

20
20
21
23

3.1
3.2
3.3
3.4
3.5

vi

24
25
26
26
27
28
29
29
30
31
32
32
33
35
36
39

LIST OF FIGURESLIST OF FIGURES

3.21 Background energy spectra of LaBr3 (Ce) . . . . . . . . . . . . .
3.22 Decay scheme of 138 La . . . . . . . . . . . . . . . . . . . . . . . .
3.23 UVT light guide . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.24 Light guide length dependence of light output . . . . . . . . . . .
3.25 Light guide bending angle dependence of light output . . . . . .
3.26 Change of light output and energy resolution with light guide . .
3.27 DAQ of β-tagging system . . . . . . . . . . . . . . . . . . . . . .
3.28 DAQ of LaBr3 (Ce) system . . . . . . . . . . . . . . . . . . . . . .
3.29 DAQ of trigger system . . . . . . . . . . . . . . . . . . . . . . . .
3.30 DAQ of gates generation system . . . . . . . . . . . . . . . . . .
3.31 DAQ of visual scaler system . . . . . . . . . . . . . . . . . . . . .
3.32 Timing chart of electronics . . . . . . . . . . . . . . . . . . . . .
3.33 DAQ system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.34 Side view of the geometry in the MC simulation . . . . . . . . .
3.35 Bird’s-eye view of the geometry in the MC simulation . . . . . .

40
41
42
43
43
44
45
46
47
48
49
49
51
51
52

4.1
4.2
4.3
4.4
4.5
4.6

Time diﬀerence of the PMTs of the plastic scintillator . . . . . .
Time walk correction of LaBr3 (Ce) . . . . . . . . . . . . . . . .
Time walk correction of the plastic scintillator . . . . . . . . . . .
Energy balance cut of the plastic scintillator . . . . . . . . . . . .
Timing spectra (n =0.881 amagat, B =0.865 733 6 T) . . . . . . .
Fitting of RF-OFF timing spectrum (n = 0.881 amagat, B =0.865 733 6 T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fitted Γpick (n, ∞) as a function of gas density . . . . . . . . . . .
Energy spectra of the timing window of 50–60 ns (n =0.881 amagat,
B =0.865 733 6 T) . . . . . . . . . . . . . . . . . . . . . . . . . . .

55
57
59
60
62

5.1 Resonance lines at a gas density of 0.129 amagat . . . . . . . . .
5.2 Resonance lines at a gas density of 0.133 amagat . . . . . . . . .
5.3 Resonance lines at a gas density of 0.167 amagat . . . . . . . . .
5.4 Resonance lines at a gas density of 0.232 amagat . . . . . . . . .
5.5 Resonance lines at a gas density of 0.660 amagat . . . . . . . . .
5.6 Resonance lines at a gas density of 0.881 amagat . . . . . . . . .
5.7 Resonance lines at a gas density of 0.969 amagat . . . . . . . . .
5.8 Resonance lines at a gas density of 1.193 amagat . . . . . . . . .
5.9 Resonance lines at a gas density of 1.353 amagat . . . . . . . . .
5.10 Resonance lines at a gas density of 1.358 amagat . . . . . . . . .
5.11 Resonance lines at a gas density of 1.366 amagat . . . . . . . . .
5.12 Time evolutions of ρ00 for various gas density. (Reproduced from
Ref. [1]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13 Time evolutions of mean velocity of Ps for various gas density.
(Reproduced from Ref. [1]) . . . . . . . . . . . . . . . . . . . . .
5.14 Time evolutions of pick-oﬀ annihilation rate for various gas density. (Reproduced from Ref. [1]) . . . . . . . . . . . . . . . . . . .
5.15 Time evolutions of ∆HFS for various gas density. (Reproduced
from Ref. [1]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.16 Spatial distribution of gas temperature in the RF cavity (n =0.881 amagat) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.17 Fitting-start-time dependence of ∆HFS . . . . . . . . . . . . . . .
5.18 Fitting-end-time dependence of ∆HFS . . . . . . . . . . . . . . .

68
69
70
71
72
73
74
75
76
77
78

4.7
4.8

vii

63
64
65

79
79
80
80
82
84
84

LIST OF FIGURESLIST OF FIGURES

5.19 ∆HFS at each gas density . . . . . . . . . . . . . . . . . . . . . .
5.20 Summary of ∆HFS measurements from pas experiments and thiswork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86
87

B.1 Side view of the setup of Ps thermalization measurement. . . . . 95
B.2 Front view of the setup of Ps thermalization measurement. . . . 96
B.3 DAQ of Ps thermalization (Plastic scintillator 1) . . . . . . . . . 97
B.4 DAQ of Ps thermalization (Plastic scintillator 2) . . . . . . . . . 98
B.5 DAQ of Ps thermalization (Trigger) . . . . . . . . . . . . . . . . 99
B.6 DAQ of Ps thermalization (Ge detector) . . . . . . . . . . . . . . 100
B.7 DAQ of Ps thermalization (Ge Main Gate) . . . . . . . . . . . . 101
B.8 DAQ of Ps thermalization (Ge TRP veto) . . . . . . . . . . . . . 102
B.9 DAQ of Ps thermalization (Ge ADCs’ gates) . . . . . . . . . . . 103
B.10 DAQ of Ps thermalization (Clock system) . . . . . . . . . . . . . 104
B.11 Side view of the geometry in the MC simulation (thermalization
measurement) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
B.12 Bird’s-eye view of the geometry in the MC simulation (thermalization measurement) . . . . . . . . . . . . . . . . . . . . . . . . 105
B.13 Pick-oﬀ ratio as a function of time (with aerogel) . . . . . . . . . 106
B.14 Pick-oﬀ ratio as a function of time (without aerogel) . . . . . . . 107
C.1
C.2
C.3
C.4

D1 (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
D2 (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
ϵ(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
BRF (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

viii

List of Tables
1.1
1.2

Rare decay modes of positronium . . . . . . . . . . . . . . . . . .
History of Ps-HFS measurement . . . . . . . . . . . . . . . . . .

2
8

3.1
3.2
3.3

Decay radiation of 22 Na . . . . . . . . . . . . . . . . . . . . . . .
Properties of LaBr3 (Ce) . . . . . . . . . . . . . . . . . . . . . . .
Decay radiation of 138 La . . . . . . . . . . . . . . . . . . . . . . .

26
39
40

4.1

Summary of cuts . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

5.1

Summary of systematic errors . . . . . . . . . . . . . . . . . . . .

81

A.1
A.1
A.1
A.1

Run table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Run table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Run table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Run table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90
91
92
93

ix

Chapter 1

Introduction
1.1

Positronium

Positronium (Ps) is a bound state of an electron (e− ) and a positron (e+ ). It
is the lightest bound state in the hydrogen-like atoms. It is an ideal systemfor precision tests of bound-state QED since it is a purely leptonic atom andeﬀectively free from the hadronic eﬀects and the weak interactions. It is also
suitable for searches of new physics, because it is a bound state of a particle
and an anti-particle.
States of Ps is classified according to the principal quantum number n, theorbital angular momentum L, the spin angular momentum S, and the angularmomentum projected on a quantization axis m. The ground state Ps (n = 1,
L = 0) is classified according to the total spin S into two states; the spin
singlet (S = 0) state called parapositronium (p-Ps, 11 S0 ) and the spin triplet(S = 1) state called orthopositronium (o-Ps, 13 S1 ). Positronium is sensitive tothe structure of vacuum and searches for new particles. Parapositronium has
the same quantum number as vacuum and o-Ps has the same quantum number
as a photon.
Positronium has a parity under a charge-conjugation C of
C = (−1)L+S .

(1.1)

Positronium is a fermion-antifermion system and decays into photons. Because aphoton has intrinsically odd-parity under the C-transformation, p-Ps can decay
only into even γ’s, on the other hand, o-Ps into odd γ’s.
p-Ps → nγ, n = 2, 4, 6, 8...,(1.2)

o-Ps → nγ,(1.3)

n = 3, 5, 7, 9....

p-Ps decays mostly into 2γ, whereas o-Ps into 3γ. Other decay modes are summarized in Table 1.1. There are many experiments on other decay modes,
but they are extremely suppressed. The decay rates of p-Ps (Γp-Ps ), and ofo-Ps (Γo-Ps ) are
Γp-Ps
Γo-Ps

= 7.990 9(17) ns−1 [2],= 7.040 1(7) µs

1

−1

[3].

(1.4)
(1.5)

2

1.3 INTRODUCTION

Table 1.1: Rare decay modes of positronium.
Decay mode
Branching ratio
Reference= 1.30(30) × 10−6[4]−6=
1.50(11)×
10[5]
Γp-Ps→4γp-Ps → 4γ= 1.48(17) × 10−6[6]
Γp-Ps→2γ= 1.19(26) × 10−6[7]−6=
1.14(39)
×
10
[8]
(
)
+2.6
−6
Γo-Ps→5γ= 2.2−1.8 × 10
[9]o-Ps → 5γ= 1.7(11) × 10−6[8]Γo-Ps→3γ˙Γo-Ps→νν < 5.8 × 10−4 (90%CL)[10]o-Ps → νν˙Γo-Ps→3γ < 2.8 × 10−6 (90%
CL)
[11]
−7 ˙
< 4.2 × 10 (90% CL)
[12]

1.2

Hyperfine Splitting of Positronium

Precise tests of positronium and other hydrogen-like atoms are summarized inRef. [13]. Many values such as decay rates and diﬀerences of energy levels
have been measured and most of them are consistent with QED predictions.
The 1S hyperfine splitting (HFS) of Ps (Ps-HFS, ∆HFS ) is the diﬀerence ofenergy levels between p-Ps and o-Ps . The energy of o-Ps is higher than p-Ps by∆HFS ≈ 203 GHz (0.84 meV). It is one of the most precisely tested values.
Ps-HFS is caused by the spin-spin interaction and the quantum oscillation ofo-Ps → γ ∗ → o-Ps. The spin-spin interaction becomes large because of the lightmass of Ps. The contribution of the quantum oscillation is about 30%.

1.3

Theoretical Calculation of Ps-HFS

The bound-state QED has diﬃculty in higher-order calculations because of their
non-perturbative nature, which is diﬀerent from the situation of free particles.
Recently a technique to calculate higher-order corrections using a theory of
non-relativistic eﬀective field has been developed. It is called non-relativistic
quantum electrodynamics (NRQED), which is advancing studies of bound-state
QED. The Ps-HFS value is calculated up to O(α3 ) correction as follows [14]:{()α 32 6
5
1
th
∆th
=
∆
1
−
+
ln
2
+ α2 ln
HFS
HFS 0
π 21 714α()]( α )2 [ 1367 5197
6 221 2
159
+
−
π2 +
+
π ln 2 −
ζ (3)
π
378
2016
7
84
56
(
) 3
}
(
)
3
3
1
62 68
α
1
3α
α
ln2 +
−
ln 2
ln + D−+ ... ,(1.6)
2 π
α
15
7
π
α
π
7
me α4 ,
(1.7)
∆th
=
HFS 0
12
where me is the electron mass, α is the fine-structure constant, and D is an
unknown constant. The leading order ∆th
HFS 0 was calculated firstly by J. Pirenne
in 1947 [15], and confirmed later [16, 17]. The O(α) correction is calculated
by Karplus and Klein in 1952 [18]. Many eﬀects are calculated in the O(α2 )1.4 INTRODUCTION

3

correction until 2000 [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32].
The O(α3 ln2 α−1 ) term is calculated by Karshenboim in 1993 [33]. It was not
until 2000s that the O(α3 ln α−1 ) correction is calculated firstly by Kniehl and
Penin[14] and confirmed later [34, 35]. Recently, some contributions of theO(α3 ) term have been calculated [36, 37] to be ≈ +1 ppm, although not allcontributions have been obtained yet. Including all the corrections calculated
up to O(α3 ln α−1 ), the theoretical value results in∆th
HFS = 203.391 69(41) GHz (2.0 ppm) [14, 34, 35],

(1.8)

where the error estimation of Ref. [14] is used.

1.4

Measurements of Ps-HFS

Ps-HFS was firstly measured by Deutsch and Brown in 1952 [38]. After thatmeasurement, precise measurements were performed by independent two groups.
All the previous precise measurements obtained Ps-HFS indirectly by stimulating the Zeeman transition under a static magnetic field in order to avoid adiﬃculty in precisely measuring Ps-HFS by stimulating ∆HFS directly.

1.4.1

Principle of indirect measurement using the Zeeman
eﬀect

The four spin eigenstates of Ps is described as|0, 0⟩ =|1, 0⟩ =|1, 1⟩ =|1, −1⟩ =1√ (| ↑⇓⟩ − | ↓⇑⟩) ,2
1√ (| ↑⇓⟩ + | ↓⇑⟩) ,2| ↑⇑⟩,| ↓⇓⟩,(1.9)(1.10)(1.11)(1.12)

where the Ps spin eigenstates with the total spin S and the magnetic quantum
number mz is described as |S, mz ⟩, ↑ and ↓ indicate the spin of the electron,and ⇑ and ⇓ indicate that of the positron. In this thesis, the quantization axis
is chosen to be the z axis.
Figure 1.1 shows the energy levels of the ground state Ps as a function of astatic magnetic field strength. In a static magnetic field, the |1, 0⟩ and the |0, 0⟩
states mix and energy eigenstates of Ps become |+⟩ and |−⟩ states. This is the
Zeeman eﬀect, and the energy splitting ∆Zeeman is the Zeeman splitting.
The relationship between ∆HFS and ∆Zeeman at a static magnetic field B isapproximately given by the Breit-Rabi expression 1∆Zeeman ≈)(√1
1 + 4q 2 − 1 ,∆HFS
2(1.13)

1 This equation is approximation without an eﬀect of Ps decay, so that the exact solution issmaller than this by about 10 ppm. This fact does not matter the final results of the previous
experiments since the two most precise experiments applied a correction of a diﬀerence between
the Breit-Wigner function and a theoretical resonance line calculated from the Hamiltonian.

1.4 INTRODUCTION

443+> Hmz =0L

2

Dmix

1

E @GHzD

0, Eo-Ps­Ý>, ¯ß> Hmz =± 1L

DHFS
-203

E p-Ps-204-205-> Hmz =0L

-206
-207
0.0

0.2

0.4

0.6

0.8

1.0

B @TD
Figure 1.1: Energy levels of the ground state Ps. The horizontal axis shows
strength of static magnetic field.

1.4 INTRODUCTION

5

(
)
5 2
where q is given as g ′ µB B/ (h∆HFS ), g ′ = g 1 − 24
α is the g factor of thepositron (electron) in Ps [39, 40, 41, 42], µB is the Bohr magneton, and h is the
Planck constant.
When microwave with a frequency of ∆Zeeman is applied, a transition between
the mz = ±1 states and the |+⟩ state occurs. This transition makes 2γ decayrate higher and 3γ decay rate lower since the |1, ±1⟩ states decay into 3γ whereas
the |+⟩ state, which is a mixed state of |0, 0⟩ and |1, 0⟩ states, decays mostly into
2γ. The experimental signature is this change in the annihilation rates causedby the Zeeman transition.

1.4.2

Theoretical resonance line

In our experiment, ∆HFS is directly determined using the theoretical resonanceshape of ∆Zeeman obtained using the density matrix of Ps spin states because theBreit-Rabi equation is not precise enough at ppm level. The line shape is slightly
diﬀerent from the Breit-Wigner function (Lorentzian) because of the non-linear
dependence on the static magnetic field strength. The following calculation isbased on Refs. [43, 44]. The basis for the four spin eigenstates of Ps is defined
as

 

ψ0
|0, 0⟩
 ψ1   |1, 0⟩  
(1.14)
 ψ2  =  |1, 1⟩  .
ψ3|1, −1⟩The Hamiltonian H0 including the Ps decay without any external fields or interactions becomesih̄Γp-Ps000− h∆2HFS −
2


h∆HFS
− ih̄Γ2o-Ps
0
0
0


2
H0 = 
,
ih̄Γ
h∆
HFS
o
Ps


0
0
− 2
0
2
ih̄Γo-Ps
h∆HFS
0
0
0
−
2
2
(1.15)
where the zero of energy is chosen midway between p-Ps and o-Ps levels. An
external magnetic field B(t) is applied as follows:
B(t) = Bẑ + BRF x̂ cos (ωt) ,(1.16)

where ẑ and x̂ are the unit vectors for the z and x directions respectively, BRF
is the magnetic field strength induced by the microwaves, ω is the frequencyof the microwaves, and t is the time after the Ps is formed. The phase of the
microwaves is randomly distributed for each Ps in this experiment, but the eﬀect
on the determination of ∆HFS is less than 0.1 ppm so that an arbitrary phasecan be taken in the calculation.
The interaction Hamiltonian V in the magnetic field isV= −µ · B= g ′ µB B · (S− − S+ )g ′ µBB · (σ− − σ+ )=2
g ′ µB=[B (σz− − σz+ ) + BRF cos (ωt) (σx− − σx+ )] ,2

(1.17)(1.18)(1.19)(1.20)

1.4 INTRODUCTION

6

where µ is the magnetic moment of Ps, S is the spin operator of the electron
and the positron respectively, and σ indicates the Pauli matrices which aredescribed as
(
)
(
)
(
)
0 1
0 −i
1 0
σx =
, σy =, σz =,(1.21)
1 0
i 0
0 −1
with the quantization axis of z, and subscripts + and − indicate the positron andthe electron, respectively. Substituting Eq. (1.21) in Eq. (1.20), V is calculatedas
0 q −r r
 q 0 0 0 V =(1.22)
 −r 0 0 0  h∆HFS ,r 0 0 0(√)
where r = g ′ µB BRF cos (ωt) / 2h∆HFS (t) . The total Hamiltonian H = H0 +V becomes
 1
− 2 − 2i γs
−q
r
−r
1
i


−q
0
0
2 − 2 γt,
H = h∆HFS (t) × (1.23)
i
1


0
r
0
2 − 2 γt1i−r
002 − 2 γtwhere γs = Γ′p-Ps (t)/ (2π∆HFS (t)), γt = Γ′o-Ps (t)/ (2π∆HFS (t)), Γ′p-Ps (t) =Γp-Ps +Γpick (t), Γ′o-Ps (t) = Γo-Ps +Γpick (t), and Γpick (t) is the pick-oﬀ (Ps+e− →2γ + e− ) annihilation rate. The time-dependence of ∆HFS and Γpick are causedby Ps thermalization, which is described later.
The 4 × 4 density matrix ρ(t) evolves with the time-dependent Schrödinger
equation,ih̄ρ̇ = Hρ − ρH † ,(1.24)
where the i, j elements of ρ(t) are defined as ρij (t) ≡ ⟨ψi |ψ(t)⟩⟨ψ(t)|ψj ⟩ and theinitial state ρ(0) with an assumption of electron unpolarized is given as [45]

iϕ
−iϕ 
1
θ
P sin
√θe
− P cos− P sin4√θe2
4
4
4 2
iϕ


1
P sin
θ
P sin√
θe−iϕ√θe − P cos

4
4
4
2
4
2

,
ρ(0) =  P sin θe−iϕ P sin θe−iϕ (1+P cos θ)(1.25)

√
√
0


4
4 2
4 2
iϕ
iϕ
(1−P cos θ)
P sin
√θe
√θe
0
− P sin4
4 2
4 2
where P is the polarization of the positron, θ is the polar angle of the positronpolarization vector, and ϕ is the azimuthal angle of that.
The 2γ annihilation probability (S2γ ), and the 3γ annihilation probability(S3γ ) are calculated between t = t0 and t = t1 as)∫ t1 (3∑′S2γ =Γp-Ps (t)ρ00 (t) + Γpick (t)ρii (t) dt,(1.26)
t0

i=1

∫ t1
S3γ

=

Γo-Ps
t0

3
∑

ρii (t) dt.

(1.27)

i=1

Furthermore, S3γ is divided into two components to obtain the experimental
resonance line shape because of the diﬀerent angular distributions of decay γ

1.4 INTRODUCTION

7

rays from Ps between |1, ±1⟩ and |1, 0⟩ states [46]. The annihilation probabilityof the |1, ±1⟩ state, S|1,±1⟩ ≡ S|1,1⟩ + S|1,−1⟩ , and the annihilation probabilityof the |1, 0⟩ state, S|1,0⟩ , are obtained by∫ t1
S|1,±1⟩=

S|1,0⟩=

Γo-Ps (ρ22 (t) + ρ33 (t)) dt,(1.28)

Γo-Ps ρ11 (t) dt.

(1.29)

t0∫ t1
t0

These calculations are used later in Eqs. (5.1) and (5.2) for fitting experimental
data.

1.4.3

Previous Results of Measurements

The history of the measurements is summarized in Table 1.2 and Fig. 1.2.

1.4 INTRODUCTION

8

Table 1.2: History of Ps-HFS measurement.
Year
∆HFS
Error
References
(GHz)
(ppm)
1952 203.2(3)
1500
[38]
1954 203.38(4)200[47]1955 203.35(5)250[47]1957 203.33(4)200[48]1970 203.403(12)58[49]1972 203.396(5)24[50]1975 203.387 0(16)8[51]1977 203.384(4)20[52]1977 203.384 9(12)6[53]1983 203.387 5(16)8[43]1984 203.389 10(74)
3.6
[44]

Phys. Rev. 94, 758 (1954)
Phys. Rev. 98, 223 (1955)
Phys. Rev. 106, 934 (1957)
Phys. Rev. A2, 707 (1970)
Phys. Rev. Lett. 29, 1059 (1972)
Phys. Rev. Lett. 34, 246 (1975)
Phys. Rev. A15, 241 (1977)
Phys. Rev. A15, 251 (1977)
Phys. Rev. A27, 262 (1983)
Phys. Rev. A30, 1331 (1984)

203.3

203.35203.4∆HFS (GHz)

Figure 1.2: History plot of Ps-HFS measurements. The red band indicates thetheoretical calculation described in Eq. (1.8). The first measurement [38] is notplotted since the error is large.

1.5 INTRODUCTION

9

Experimental
average

Theory(Kniehl et al., 2000)

Mills et al., 1983

Ritter et al., 1984

203.385203.387

203.389203.391203.393203.395
HFS [GHz]

Figure 1.3: Positronium HFS. The points with error bars are the experimental
values [43, 44], the green band is their average Eq. (1.32), and the red band is
the QED calculation of Eq. (1.8).

Two most precise results are∆HFS=

203.387 5(16) GHz (8 ppm) [43],(1.30)∆HFS=

203.389 10(74) GHz (3.6 ppm) [44].

(1.31)

These values include results of their own past measurements [51, 53], and averaging these two values results in∆exp
HFS = 203.388 65(67) GHz (3.3 ppm) [51, 43, 44].

1.5

(1.32)

Discrepancy between Measurements and Theory

Figure 1.3 shows the comparison of the measured values of Eq. (1.30), (1.31),
their average Eq. (1.32), and the QED calculation of Eq. (1.8). There is adiscrepancy of 3.04(79) MHz (15 ppm) between the previous measurements andthe theoretical calculation. The discrepancy is 3.9 standard deviations (s.d.)level, so it is not likely a statistical fluctuation. The following three reasons are
possible to explain this discrepancy.

1.5.1

Miscalculation of the theoretical value

NRQED might not be applied at higher orders. Although NRQED is validated
up to O(α2 ) corrections in the o-Ps lifetime measurement [3], no precise test ofO(α3 ) corrections has been performed. The other probability is a simple miscalculation, but this probability is quite small since the diﬀerent three calculations
show the same result.

10

1.5.2

1.5 INTRODUCTION

Common systematic uncertainties in the previous
experiments

The discrepancy occurs if there are any common systematic uncertainties which
are not properly treated in the previous experiments. Possible common systematic uncertainties are described in Sec. 2.1.

1.5.3

New physics beyond the Standard Model

If the discrepancy is real, it indicates the existence of an unknown particle. For
example, a new pseudo-scalar with a mass of O(MeV) and weak coupling of(α ≈ 10−8 ) can cause this discrepancy.

Chapter 2

Principle
In this chapter, possible systematic uncertainties of all previous experiments are
discussed. I propose the new method to reduce these uncertainties.

2.1

Possible systematic uncertainties in the previous experiments

There are the following two possible common systematic uncertainties in theprevious experiments.

2.1.1

Possible non-uniformity of the magnetic field

The precise measurements using Zeeman splitting acquires an uniformity andan accuracy of the static magnetic field to be ppm level, since an uncertainty
and a non-uniformity of the static magnetic field directly aﬀect ∆HFS . An inhomogeneity of the static magnetic field was mentioned as the most significant
systematic uncertainty in the previous experiments. In the previous experiments, the static magnetic field is provided using normal conducting magnets
and the field is carefully shimmed to obtain ppm-level uniformity over the relevant volume of ≈ 10 cm3 where Ps 2γ annihilations are observed. However,the Ps formation region is in fact a volume of ≈ 100 cm3 , and there is still a
possibility of more uncertainty from the inhomogeneity of the static magnetic
field.

2.1.2

Non-thermalized o-Ps eﬀect

Gas molecules are necessary to form Ps in all experiments, but they produce
electric fields around the Ps which in turn aﬀects ∆HFS . This material eﬀect(Stark eﬀect) must be properly accounted for to obtain ∆HFS in vacuum. In all
previous experiments, Ps-HFS is measured at various gas densities and linearly
extrapolated to vacuum in order to estimate this eﬀect. This method is valid
only when Ps is well-thermalized and the shift of energy levels is proportional to
the gas density. However, a calculation of the Stark eﬀect on atomic collisions
under the Lennard-Jones potential [54] shows that this eﬀect is proportional to
3/5n [v(t)] , where n is the number density of gas molecules and v(t) is the Ps
1112

2.1 PRINCIPLE

mean velocity. The time dependence of v(t) is caused by the Ps thermalization
process. o-Ps has a kinetic energy of O(eV) just after the formation, and slows
down by collision toward the ultimate thermal energy of about 0.03 eV at room
temperature. This process is called thermalization. It takes much time for
Ps to be well thermalized in materials with low density. Because of this fact,
non-thermalized Ps has to be taken into account carefully. The eﬀect due tonon-thermalized Ps has already been shown to be significant in the so-calledo-Ps lifetime puzzle [55, 56, 57], which was a history of a disagreement between
experimental o-Ps values and theoretical calculations which was finally solved
by taking into account the eﬀect. This eﬀect is a possible serious systematic
uncertainty in the Ps-HFS case. The non-thermalized Ps eﬀect is estimatedhereinafter based on Refs. [58, 59].
The diﬀerential equation of time evolution of Ps thermalization in gas is
(
) √
√
8
dEav (t)32 2σm n= − 2mPs Eav (t) Eav (t) − kB T
,
(2.1)
dt
2
3 3π M
where t is the time since Ps is formed, Eav (t) is the mean Ps kinetic energy
at time t, σm is the momentum-transfer cross section of Ps collision with gas
molecules, n is the number density of gas molecules, M is the mass of the gas
molecule, mPs is the Ps mass, kB is the Boltzmann constant, and T is thetemperature of the gas. The solution of this equation is()2
1 + Ae−bt
3
Eav (t) =
kB T,
(2.2)
1 − Ae−bt
2√8
2 2σm n √b=
3mPs kB T ,3 3π M
A is defined by the initial condition described as√√
E0 − 32 kB T√
A= √,E0 + 32 kB T

where

(2.3)

(2.4)

where E0 = Eav (0) is the initial kinetic energy of Ps.
Let’s discuss thermalization process in nitrogen gas with a typical E0 =2 eV [60], T = 273.15 K for example. Nitrogen gas is used in Ref. [44], and σmof nitrogen is σm (N2 ) = (25 ± 14) × 10−16 cm2 [61, 60, 59]. Figure 2.1 shows
the time evolution of the Ps mean kinetic energy calculated by Eq. (2.2) at 0.1,
3/5
1, and 2 atm. The term n [v(t)]has to be considered.
The Ps mean velocity√

2
is calculated from kinetic energy as v(t) ≈mPs Eav (t). The eﬀect of nonthermalized Ps without any timing information is approximately estimated by∫∞
3/5
n [v(t)] exp (−Γo-Ps t), where only the weight of decrease of the number of
0
Ps of exp (−Γo-Ps t) is taken into account. The time evolution of ∆HFS is ignored
here, although it has not so small eﬀect. Figure 2.2 shows the lifetime-weighted
3/5
Ps velocity3/5 [v(t)] exp (−Γo-Ps t). Each area of this figure shows the material
eﬀect at each gas density. Figure 2.3 shows the non-thermalized Ps eﬀect as afunction of gas density. It is estimated by integrating Fig. 2.2 in the timing t =0–1000 ns. A Non-linear structure is clearly seen at low pressure region. Figure 2.4

ENERGY (eV)

2.1 PRINCIPLE

13

2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

50

100 150 200 250 300 350 400 450 500
TIME (ns)

3/5((m/s) )Figure 2.1: Time evolution of Ps mean kinetic energy in nitrogen gas with an
initial kinetic energy of 2 eV and temperature of 273.15 K. The blue line is at
0.1 atm, the black line is at 1 atm, and the red line is at 2 atm.

Lifetime-weighted VELOCITY

3/5

2500
2000
1500
1000
500
0
0

50

100 150 200 250 300 350 400 450 500
TIME (ns)Figure 2.2: Time evolution of lifetime-weighted Ps mean velocity in nitrogengas; An initial kinetic energy is 2 eV, and temperature is 273.15 K. The blueline is at 0.1 atm, the black line is at 1 atm, and the red line is at 2 atm.

2.2 PRINCIPLE

MATERIAL EFFECT (1 @ 1 atm)

14

2.5
2
1.5
1
0.5
0
0

0.5

1

1.5

2
2.5
3
o
PRESSURE (atm @ 0 C)

Figure 2.3: Non-thermalized Ps eﬀect in nitrogen gas; An initial kinetic energy
is 2 eV, and temperature is 273.15 K. The vertical axis is normalized to unity at
1 atm.

shows a linear-fit of toy data based on Fig. 2.3. The toy data are plotted on theline of Fig. 2.3 at the gas densities of the data of Ref. [44], and the error bars aretaken from the same data normalized by the amount of the material eﬀect. The
linear-fit results in a slope of 0.83(10) /atm, and y-intercept of 0.140(72). The
gas density dependence of Ps-HFS in nitrogen is −33(4) ppm/atm according toRef. [44], which corresponds to the slope of 0.83(10) of Fig. 2.4. Using thisrelation, 0.83 corresponds to 33 ppm on Ps-HFS, the y-intercept of 0.140(72)corresponds to 5.6 ± 2.9 ppm on Ps-HFS. This causes underestimation of thematerial eﬀect and Ps-HFS is estimated to be 5.6 ± 2.9 ppm larger than theprevious experimental value. The uncertainty of σm in nitrogen gas is large,
which results in 2–10 ppm underestimation of Ps-HFS. Although this estimationof the material eﬀect is rough, the underestimation is large enough and suitable
to solve the 15 ± 4 ppm discrepancy between the previous experimental averageand the theoretical calculation.

2.2

New methods

In order to reduce these possible systematic uncertainties in all previous measurements, the following three totally new methods are used in this experiment.

2.2.1

Large bore superconducting magnet

1. A large bore superconducting magnet is used to apply a static magnetic
field of B ≈ 0.866 T. A bore diameter of the magnet is 800 mm, and its
length is 2 m. With compensation coils surrounding the RF cavity, theapparatus achieves 1.5 ppm RMS in uniformity of the magnetic field in
the large volume of cylinder 40 mm in diameter and 100 mm long, where
Ps are formed.
2. The superconducting magnet is operated in persistent current mode andMATERIAL EFFECT (1 @ 1 atm)

2.2 PRINCIPLE

15

3
2.5
2
1.5
1
0.5
0
0

0.5

1

1.5

2
2.5
3
o
PRESSURE (atm @ 0 C)

Figure 2.4: Estimation of the non-thermalized Ps eﬀect on Ps-HFS in nitrogengas with an initial kinetic energy of 2 eV and temperature of 273.15 K. The
vertical axis is normalized to unity at 1 atm. The red points with error bars are
the toy data, the blue line is the same calculation as Fig. 2.3, and the black line
is a linear-fit of the toy data.
a stability of the magnetic field is obtained to be better than ±1 ppm.

2.2.2

Timing information

1. The timing information between Ps formation and decay is newly obtainedin this experiment to investigate the non-thermalized Ps eﬀect.
2. The signal-to-noise ratio of the measurement is significantly improved bya factor of 20, since prompt annihilations and p-Ps can be removed using
time information. Figure 2.5 shows the time evolution of S2γ calculated by
Eq. (1.26). 2γ annihilation probabilities are compared with and without
RF. Figure 2.6 shows the diﬀerence of 2γ annihilation rate between RF-ON
and RF-OFF of Fig. 2.5, which corresponds to the 2γ transition rate. It isclearly shown that application of a timing window using timing information is a very eﬀective event selection. Figure 2.7 shows the S2γ integrated
between t = 0 and ∞ as a function of a static magnetic field strength. It
corresponds to measurements without timing information. It is shownthat a very large background exists under the transition curve at 0.83 T.
On the other hand, Fig. 2.8 shows the S2γ integrated between t = 35 and155 nsec as a function of a static magnetic field strength. The backgroundis shown to become much smaller with the timing window. The timing
information enables an eﬀective selection of the Zeeman transition signal,
which much enhances the signal-to-noise ratio of the measurement.

2.2.3

High performance γ ray detectors

1. LaBr3 (Ce) scintillators are used to detect γ rays. The energy resolution
is 8% FWHM at 511 keV and the decay constant is as short as 16 ns. The

2.2 PRINCIPLE

16

2γ ANNIHILATION (/ns)

Graph

1
10-1

10-2
10-3
10-4
10-5
10-6
10-7 0

100

200

300

400
500
TIME (ns)Figure 2.5: Time evolution of the 2γ annihilation rate (S2γ ). The red line isa calculation at BRF = 14.2 G (RF-ON) and the green line is a calculation at
BRF = 0 (RF-OFF).

2γ DECAY TRANSITION (/ns)

Graph0.00180.00160.00140.00120.001
0.00080.00060.00040.0002
0
0

100

200

300

400
500
TIME (ns)Figure 2.6: Time evolution of the 2γ transition rate.

2.2 PRINCIPLE

17

2γ ANNIHILATION PROBABILITYWITHOUT TIMING WINDOW
0.65

0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0

0.2

0.4

0.6
0.8
1
MAGNETIC FIELD (T)Figure 2.7: 2γ decay rate without timing information. The red line is a calculation at BRF = 14.2 G (RF-ON) and the green line is a calculation at BRF = 0(RF-OFF).
2γ ANNIHILATION PROBABILITY

WITH TIMING CUT 35--155 ns
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0

0.2

0.4

0.6
0.8
1
MAGNETIC FIELD (T)Figure 2.8: 2γ decay rate with a timing window of 35–155 ns. The red line isa calculation at BRF = 14.2 G (RF-ON) and the green line is a calculation at
BRF = 0 (RF-OFF).

18

2.2 PRINCIPLE
acceptance of the setup is greatly increased by the good energy resolution, since 2γ events are eﬃciently separated from 3γ events by energy
information alone, instead of a back-to-back geometry selection.
2. The fast response of LaBr3 (Ce) enables measurement with a strong radioactive source. A 22 Na source of 1 MBq is used for this experiment sinceindividual events from individual positrons have to be distinguished.

Chapter 3

Experimental Setup
In this chapter, details of our experimental apparatus are described. An overview
of the whole setup is explained at first, and then details of individual components are described.

3.1

Overview of the apparatus

Figure 3.1 shows the whole system of our experimental setup. An RF cavity
is located at the center of the magnet, and γ ray detectors are surrounding it.
High power RF is guided through the waveguide into the cavity. Figure 3.2
shows the enlarged top view in the large bore superconducting magnet. All ofthese measurement systems are located in the static magnetic field. Figure 3.3
is photographs of the apparatus around and inside the magnet. The static
magnetic field is little disturbed by the apparatus because all systems except for
PMTs located in the magnet are nonmagnetic. The basic setup is not changed
over the whole period of the measurement. Basic procedure is as follows:
1. A positron emits from the positron source.
2. Timing information of the positron emission is obtained by the β-tagging
system.
3. The positron enters the RF cavity spiraling around the static magnetic
field.
4. The cavity is filled with pure (> 99.9%) i-C4 H10 gas. The positron loses
its energy colliding with the gas molecules.
5. The slowed positron picks an electron from a gas molecule, and forms
positronium with a probability of about 40%.
6. The |−⟩ state of Ps promptly decays into 2γ with a lifetime of about
120 ps. The |+⟩ state of Ps mostly decays into 2γ with a short lifetimeof about 8 ns. These two states annihilate soon. On the other hand, the|1, ±1⟩ states of Ps thermalize well in the gas because of its long lifetime.
7. RF makes transitions occur from the |1, ±1⟩ to the |0, 0⟩ state of Ps whena microwave field is applied in the RF cavity to observe the Zeeman transition.
1920

3.1 EXPERIMENTAL SETUP

Figure 3.1: Whole system of our experimental setup.

Figure 3.2: Top view in the magnet. (Reproduced from Ref. [1])

3.1 EXPERIMENTAL SETUP(a) Large bore superconducting magnet.

(b) Setup inside the magnet. A β-tagging system is located oﬀside of the cavity, although it cannot be seen in the photograph.
Microwaves are guided from this side to the cavity through the
waveguide.

Figure 3.3: Photographs of the experimental setup around the magnet.

21

3.2 EXPERIMENTAL SETUP

22

8. A transited Ps decays into 2γ immediately and a not transited Ps decays
into 3γ.
9. γ-rays from a Ps decay are detected by γ ray detectors. The Zeemantransition is measured by the change of 2γ- and 3γ- decay rates at various
magnetic field strengths with fixed RF frequency and power. ∆HFS is
obtained by fitting data points with theoretical resonance lines.
Each part of the experimental apparatus is explained in the following sections.

3.2

Static magnetic fieldA static magnetic field to cause the Zeeman splitting is produced by a large
bore superconducting magnet (Fig. 3.3(a)). There are two benefits from usingit.
• The magnet is developed for MRI and located at KEK Cryogenics Science Center. A bore diameter of the magnet is 800 mm, and its length is
2,000 mm. With compensation coils surrounding the RF cavity, a static
magnetic field with a uniformity of 1.5 ppm RMS is successfully√applied toa large Ps formation volume of O(100 cm3 (cylindrical region x2 + y 2 <20 mm where Ps are formed). This suppresses a possible systematic uncertainty from the inhomogeneity of the static magnetic field. Figure 3.4
shows the magnetic field distribution in the RF cavity without compensation coils. We found the diﬀerence larger than 20 ppm without compensation coils. Figures 3.5 and 3.6 show the magnetic field distribution
with compensation coils before and after the Ps-HFS measurement, respectively. The distribution is stable within ppm. The distribution is measured
using a proton NMR magnetometer (ECHO Electronics EFM-150HMAX) with a clock synthesizer (NF Corporation√ CK1615 PA-001-0312) at
310 points at 10-mm intervals in the region ( x2 + y 2 ≤ 50 mm, −35 mm ≤z ≤ 50 mm). Measured magnetic fields are well fitted by the following empirical function B(x, y, z):B(x, y, z)=

wx Bx + wy By + wzx Bzx + wzy Bzy,
wx + wy + wzx + wzy

Bx
By
Bzx

= fx (z) + gx (z)r02 ,= fy (z) + gy (z)r02 ,= fzx (x′ ) + hzx (x′ )(z − zx0 ) + gzx (x′ )(z − zx0 )3 ,Bzy
wx

= fzy (y ′ ) + hzy (y ′ )(z − zy0 ) + gzy (y ′ )(z − zy0 )3 ,= 0.5 cos2 θ,

wy
wzx
wzy

= 0.5a sin2 θ,= 0.5b cos2 θ,= 0.5c sin2 θ,x′y′= x0 + |r0 / cos θ| cos θ,= y0 + |r0 / sin θ| sin θ,x0y0= fCx (z),= fCy (z),(3.1)

3.3 EXPERIMENTAL SETUP

23
m_h_B
Entriesm_h_B

Mean xMean yZ (AXIAL) POSITION (mm)

RMS x
RMS y

40

40401
0.7827-2.712
37.2
34.94

20
10

20

0

0

-10

-20

-20
-30

-40
-60

-40

-20

0

20
40
60
X POSITION (mm)-40

Figure 3.4: Static magnetic field distribution without compensation magnet at
y = 0 plane (a horizontal plane at the center of the cavity). The colors indicate
diﬀerences from the average value in ppm.

r0

= fCzx (x′ ),= fCzy (y ′ ),√=(x − x0 )2 + (y − y0 )2 ,θ= tan−1 ((y − y0 )/(x − x0 )),

zx0
zy0

where fCx (z), fCy (z), fCzx (x′ ), fCzy (y ′ ), fx (z), fy (z), fzx (x′ ), fzy (y ′ ),gx (z), gy (z), gzx (x′ ), gzy (y ′ ), hzx (x′ ), and hzy (y ′ ) are 5th-, 3rd-, 3rd-, 2nd, 5th-, 3rd-, 3rd-, 2nd-, 5th-, 3rd-, 3rd-, 2nd-, 3rd-, and 2nd-order polynomial functions respectively, a, b, and c are constants. Figures 3.5 and 3.6
show extra- or inter-polated map with this function.
• The magnet is operated in persistent current mode. Fig. 3.7 shows thatthe stability of the magnetic field is better than ±1 ppm. The resonancecurve of the NMR probe is monitored during the run. The magnetic fieldstrength is obtained by a probing output from which the diﬀerence ofthe current input clock frequency from the resonant frequency of NMR isprovided as a voltage.
Static magnetic fields in the range of 0.85–0.88 T are applied with this magnet
and the Zeeman transition lines are measured.

3.3

Positron source and β-tagging system

The positron source is 1 MBq of 22 Na (Eckert & Ziegler POSN-22). Figure 3.8
shows the decay scheme of 22 Na, and decay radiation are summarized in Table 3.1.
Figure 3.9 shows a photograph of the β-tagging system around the 22 Na
source. The system is located at the lid of the RF cavity. The 22 Na source islocated at the center, and the emitted positrons are tagged by a thin (0.2 mm

3.3 EXPERIMENTAL SETUP

24

Z POSITION (mm)Y = 0 mm
6

40

8 10

0

4

2
20
0 ppm
0
-20

-4

-2

-6

-8
-10

0
-40

4
-60

-40

2

-20

0

20
40
60
X POSITION (mm)(a) y = 0 plane.

Y POSITION (mm)Z = 0 mm
60
40

-6

-8

-2

-4

20
0
-20
0 ppm
-40

2
4

-60
6
-60 -40

-20

0
20
40
60
X POSITION (mm)(b) z = 0 plane.

Y POSITION (mm)X = 0 mm
60

-10
-6

6

40
2

-2
20
4
0

2 0

0 ppm
0

-20
-4

-40

4

-8

8 10

-60
-40

-20

0
20
40
Z POSITION (mm)(c) x = 0 plane.

Figure 3.5: Static magnetic field distribution with compensation magnet (before Ps-HFS measurement). The contours indicate diﬀerences from the averagevalue.

3.3 EXPERIMENTAL SETUP

25

Z POSITION (mm)Y = 0 mm
-2

40

6
20

8

10

2
0 ppm

0
-2
-4

-20

-2

0

-6

4

-40
-60

-40

-20

0

20
40
60
X POSITION (mm)(a) y = 0 plane.

Y POSITION (mm)Z = 0 mm
60
-8
-6

40

-4
-2

0

20

-2

0
0 ppm

-20
-40
-60
-60

-40

-20

2
4
6
0
20
40
60
X POSITION (mm)(b) z = 0 plane.

Y POSITION (mm)X = 0 mm
60

10

-4
6

40
2
20
0

4

0

0 ppm

-2
0

-20
-2

-40
-60

8

-6
-40

4

-20

0
20
40
Z POSITION (mm)(c) x = 0 plane.

Figure 3.6: Static magnetic field distribution with compensation magnet (after Ps-HFS measurement). The contours indicate diﬀerences from the averagevalue.

3.3 EXPERIMENTAL SETUPMAGNETIC FIELD at NMR PROBE (mT)

26866.75866.749866.748866.747866.746
21/12

28/12

04/01
TIME (2011-2012)

Figure 3.7: Time fluctuation of the static magnetic field. The magnetic fieldstrength on 2011/12/20–2012/01/04 (for more than 2 weeks) is shown.

+

β 90.326%
EC 9.618%22
11Na 3+2.6027 y+1274.62

β+ 0.056%00+stable22
10Ne

Figure 3.8: Decay scheme of 22 Na [62]Decay modeβ+

Table 3.1: Decay radiation of 22 Na [62]EnergyParticle(keV)
endpoint
545.7(4)215.54(21)e+endpoint 1820.3(4)835.00(23)e−Auger (K)0.82
X ray (Kα2 )0.848
γX ray (Kα1 )0.849
1274.537(7)Intensity(%)
90.326(15)0.056(14)8.74(3)0.053(5)0.107(11)99.941(14)3.3 EXPERIMENTAL SETUP

27

Figure 3.9: β-tagging system.

thick) plastic scintillator (PS). Scintillation light is guided through fused quartz
light guides to fine mesh PMTs. Details of each component are described in thefollowing subsections.

3.3.1

Source holder

A source holder is fabricated with Copper and it is nonmagnetic. It holds thesource and the light guides. The holder itself is screwed to the lid of the RF
cavity.

3.3.2

Plastic scintillator and fused quartz light guide

A fused quartz light guide (10 mm thick , 15 mm wide and 75 mm long) witha hole 10 mm in diameter at the center is locked by a groove of the source
holder. The source is sandwiched by two Ti foils 0.012 7 mm thick in order to besuitable for using in vacuum. The active diameter is 9.53 mm, and the outsideof the active region is sandwiched by two Ti plates 0.254 mm thick and 19.1 mm
in outer diameter. The hole of light guide is covered by a plastic scintillator(NE102A) 12 mm in diameter and 0.1 mm thick. Positrons emitted from thesource spiraling pass through the hole of the Teflon plate, the hole of lightguide, and the plastic scintillator. The plastic scintillator emits scintillation
light when a positron passes. The scintillation light is guided through lightguide and two cylinder quartz light guides 200 mm long and 15 mm in diameter
glued to the both side of light guide, and then reaches two fine mesh PMTs.

3.3.3

Fine mesh PMTs

Two fine mesh PMTs (HAMAMATSU H6614-70MOD) are used to detect scintillation light from the plastic scintillator. Fine mesh PMTs are used because
they are located in a static magnetic field. The measured resolutions and gainof fine mesh PMTs gets worse in a static magnetic field depending on the polar

3.4 EXPERIMENTAL SETUP

28

Figure 3.10: Polar angle ‘THETA’ to the magnetic field.

angle to the field shown in Figs. 3.11, 3.12, 3.13. The polar angle “THETA” isdefined in Fig. 3.10. The gain is the highest at 30◦ , but resolutions are the highest at 0◦ . These data are obtained using the same magnet used in the Ps-HFS
measurement with a static magnetic field of 0.866 T. 511 keV back-to-back γrays are detected with two of fine mesh PMTs (HAMAMATSU H8409-70 (1.5”,
×2), HAMAMATSU H6614-70 (2.0”, ×1)) located back-to-back and LaBr3 (Ce)scintillators (Saint-Gobain BrilLanCeTM 380 of cylinder 25.4 mm in diameterand 25.4 mm long) attached. Details of LaBr3 (Ce) scintillators are described inSec. 3.6.1. Gains of all PMTs became the highest at 30◦ , because the eﬀective
area of the dynodes is the highest at 30◦ for electrons gained and spiraling go
to the anode. On the other hand, the best resolutions of energy and timing
are obtained at 0◦ , because the probability of photoelectrons reaching the firstdynode is the highest when PMTs are parallel to the field.
Fine mesh PMTs are located parallel to the magnetic field in the Ps-HFS
measurement to obtain the highest energy and timing resolutions. PMTs aﬀect
the static magnetic field by 10 ppm with a distance of 10 cm, and 100 ppm witha distance of a few cm, because magnetic material of Kovar, which is an alloy
mainly of Fe, Ni, and Co, is used in fine mesh PMTs. This eﬀect is about+3 ppm and eﬀect on the uniformity is negligible in the Ps-HFS measurement
since the PMTs are located 15 cm away from the RF cavity.

3.4

RF system

Figure 3.14 shows the RF circuit of the apparatus. RF circuit consists of RF
supply, a monitor, feedback, and an interlock.

3.4.1

RF supply

RF supply system consists of a signal generator (SG), an amplifier (AMP),
waveguides, feedthroughs, and the RF cavity.

3.4 EXPERIMENTAL SETUP

RELATIVE GAIN

129

1.5 inch
2.0 inch

0.1

0.01
-10

0

10
20
30
THETA (deg)4050

ENERGY RESOLUTION FWHM (%)

Figure 3.11: Polar angle dependence of the gain of fine mesh PMTs in a magnetic
field. The definition of the angle is shown in Fig. 3.10. The gain is shown by
the ratio compared to the gain without any magnetic field. The red points arevalues of a 1.5” PMT, and the blue points are that of a 2” PMT. Error bars are
smaller than markers.

1412

1.5 inch (511keV)
2.0 inch (511keV)

10
8
6
4
2
0
-10

0

10

20

30

40

50

THETA (deg)Figure 3.12: Polar angle dependence of the energy resolution of fine mesh PMTs
in a magnetic field. The definition of the angle is shown in Fig. 3.10. The
resolution is shown by FWHM at 511 keV. The red points are values of a 1.5”
PMT, and the blue points are that of a 2” PMT. The resolution without any
magnetic field is shown at −5◦ .

3.4 EXPERIMENTAL SETUP

30

RESOLUTION (ns)1.5inch

0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
-10

1.5inch2.0inch

0

10

20

30

40

50

THETA (deg)Figure 3.13: Polar angle dependence of the timing resolution of fine mesh PMTs
in a magnetic field. The definition of the angle is shown in Fig. 3.10. The
resolution is shown by FWHM at 511 keV. The red points are values of a 1.5”
PMT, and the blue points are that of a 2” PMT. The resolution without any
magnetic field is shown at −5◦ .

Signal generator and amplifier
Continuous microwaves of about 6.6 dBm are generated by a SG (ROHDE &SCHWARZ SMV 03). SG is controlled via GPIB interface. The frequency of
microwaves is monitored with a precision of better than 0.1 ppm. An AMP(R&K A2856BW200-5057-R) amplifies the microwaves from SG by 50 dB, andoutputs continuous microwaves of about 500 W. Interlock system is installed forthe safety. New GaN semiconductors are used in AMP. GaN semiconductors
have an energy gap of 3.3 eV, which is about 3 times larger than conventional
Si semiconductors then the low background level is expected. It has also higher
insulation resistance, lower ON resistance, lower energy loss, even it is usedat high voltage and high current, and extremely higher switching speed thanthose of conventional semiconductors. GaN semiconductors are expected to bea high-power devices of new generation because of these good properties.
Waveguides, feedthroughs, and the RF cavity
High power microwaves generated by the RF supply system are guided throughwaveguides fabricated with Al. An input feedthrough is an alumina ceramiccylinder 30 mm in diameter and 140 mm long, in which high power microwaves
pass through into the RF cavity. The input feedthrough is specially designed
to avoid gas discharges at low gas density. Fig. 3.15 shows the RF power atdischarge as a function of gas pressure (measured at ≈ 25◦ C). Even at 0.25 atm,
we could apply 500 W RF. RF power of the runs was chosen slightly below thelimit line (see Appendix A). The coupling of the input feedthrough is carefully
tuned to ∼ 1. On the other hand, an output feedthrough used for monitoring
power is a spherical antenna whose coupling is very low.
Figure 3.16 is a photograph of the RF cavity. The RF cavity is made withoxygen-free copper; the inside of the cavity is a cylinder 128 mm in diameter3.4 EXPERIMENTAL SETUP

Figure 3.14: RF circuit.

31

3.4 EXPERIMENTAL SETUPPOWER at DISCHARGE (W)

32

600
500
400
300
200
100
0
0

0.05

0.1

0.15
0.2
0.25
GAS PRESSURE (atm)Figure 3.15: RF power at discharge as a function of gas pressure. The line
indicates a fit with a simple assumption that the power is proportional to squareof the gas pressure.

Figure 3.16: Photograph of the RF cavity.

3.4 EXPERIMENTAL SETUP

33

TM110 H distribution
1.0

0.5

0.0

-0.5

-1.0
-1.0

0.0

-0.5

0.5

1.0

Figure 3.17: Magnetic field of the TM110 mode. The length of the arrows show
the strength of the field, and the direction of the field is shown by the directionof the arrows.

and 100 mm long. The γ rays pass through the side wall of the cavity eﬃciently,
because its thickness is only 1.5 mm. The cavity is operated in the TM110 modewith a resonant frequency of 2.856 6 GHz and a loaded quality factor QL of
14,700. Figure 3.17 shows the cross sectional view of the RF magnetic field of
the TM110 mode. The RF magnetic field is the two counter-rotating swirls andthe maximum at the center. The TM110 mode is selected since the Zeemantransition is caused by the oscillating magnetic field perpendicular to the static
magnetic field, which is longitudinal to the axis of the RF cavity. The diameteris determined by the wavelength of microwaves, and the length is determined
to avoid interferences with other modes. The intrinsic resonant frequency f0 of
TM110 mode cavity is described as

f0 =

cJ11
,
2πa

(3.2)

where c is the speed of light, J11 is the first zero of Bessel function of order
1, and a is the radius of the cavity. Ohmic loss at the cavity produces heat
when microwaves are supplied. The heat is removed by constant temperature(typically 25 ◦ C is used) water in order to avoid changes of the properties of thecavity. The water passes through cooling pipes silver brazed to the cavity andthe temperature is controlled by a chiller.

3.4 EXPERIMENTAL SETUP

34

3.4.2

Monitor

The RF system are RF frequency, SG output power, cavity input power, cavity
transmitted power, cavity reflection power, and the phase shift between input
and transmitted RF of the cavity are monitored as follows.
RF frequency
RF frequency is monitored by the setting value itself of the SG. The monitoring
precision is better than 0.1 ppm. The absolute accuracy is 0.6 ppm, which is
calculated using an aging of 0.1 ppm/year.
SG output power
The output RF from the SG is picked using a directional coupler (ARRA
N4164-10) for monitoring. This power is measured by a crystal detector (Nihon
Koshuha DM-NP-61L). A crystal detector is a device which converts an RF
input to a voltage output using a commutator. The characteristic of the output voltage depends on the input power. The output voltage is nearly linearly
proportional to the input power in the range of −40–0 dBm input power. The
output voltage non-linearity is calibrated by fitting the output-input curve by a
5th order polynomial. The output of the crystal detector is digitised by a digital
multimeter (DMM, KEITHLEY 2000), and recorded through a GPIB interface.
Cavity input power
The cavity input power is picked (by an amount of −30 dB) using a directional
coupler located between GaN AMP and the cavity. The picked power is measured by a power sensor (Agilent E9301A) read by a power meter (Agilent
E4419B). The output of the directional coupler is carried by a 5 m SUCOFLEX(SF) 104 cable and divided to 4 outputs by a divider. One of the output isused for measuring power, another output is used for measuring the phase shift,
and the other outputs are terminated by 50 Ω resistor. The measured value isdisplayed at the power meter, and recorded via a GPIB interface.
Cavity transmitted power
The transmitted RF from the antenna attached to the cavity is carried via 7.5 m
SF 104 cable, let through an isolator (TMT H2830), and divided by 4 with adivider. One of the outputs is −6 dB attenuated (Lucas Weinschel) and used
to measure the power using a crystal detector (Agilent 423B), another output
is also used to measure the power but using a power sensor (Agilent E4412A)
read by the same power meter as used for the input power, another output isused to measure the phase shift, and the other output is 50 Ω terminated. The
voltage of the crystal detector is read by the DMM and recorded via the GPIB
interface, and the measured power of power meter is also recorded via the GPIB
interface.
Cavity reflected power
The power of the RF reflected at the cavity is picked by an amount of −30 dB
using the same directional coupler as that used for picking the cavity input

RELATIVE POWER (dB)

3.4 EXPERIMENTAL SETUP

35

0.06
0.04
0.02
0
-0.02
-0.04
01-20h 01-21h 01-22h 01-23h 02-00h 02-01h
TIME (2010/07)

Figure 3.18: Relative power as a function of time. The red line indicates the
AMP output power, and the green line indicates the SG output power using
feedback.
power. It is −10 dB attenuated (Lucas Weinschel) and divided by 2 with adivider. One of the outputs is measured by a crystal detector (Hewlett-Packard
8471E) used for the interlock system, and the other output is measured byanother crystal detector (Agilent 423B) for monitoring. The measured powerby 423B is recorded as the same way as for the SG output power.
Phase shift between input and transmitted RF of the cavity
One of the outputs of the divided cavity input power and one of the outputsof the divided cavity transmitted power are used to measure the phase shift for
feedback. The output of the divider of the cavity input RF is −20 dB attenuated(Lucas Weinschel), delayed by 0.5 m SF 104 cable, isolated by an isolator (Nihon
Koshuha), and let into a mixer (J2080LB). On the other hand, the output of thedivider of the cavity transmitted RF is −20 dB attenuated (Lucas Weinschel),
isolated by an circulator (PASTERNACK PE8401), let through a phase shifter(Nihon Koshuha SRC-SMJ-205P), and let into the mixer. The output of themixer is measured by the DMM.

3.4.3

Feedback

The feedback system consists of power feedback and frequency feedback.
Power feedback
The RF power has to be stabilized within 0.3% to measure Ps-HFS with anaccuracy of O(ppm). The AMP output RF power is stabilized by changing
SG output power. Feedback is performed using the measured power of thecavity input RF. Figure 3.18 shows that the AMP output power is stabilized
by the power feedback system which tunes the SG output power. As shown inthis figure, power is stabilized. Figure 3.19 shows the measured distribution ofrelative RF power of typical run. It is clear that the power is well stabilized3.5 EXPERIMENTAL SETUP

COUNTS (A.U.)

36

10

3

Mean

0.00208

RMS

0.008565

102

10

1

-0.04 -0.03 -0.02 -0.01

00.01 0.02 0.03
RELATIVE POWER (dB)Figure 3.19: Distribution of AMP output power. The horizontal axis shows therelative value (dB) to a standard power.

within 0.2% fluctuation. Although it is well stabilized, the power can change bychanging the RF setup or change of an air temperature. This can be correctedin analysis process if the cavity input power is measured at enough relative
precision. Detail of analysis is described in Chap. 4.
Frequency feedback
The RF frequency ideally should be constant the same as the RF power, but theresonant frequency of the RF cavity always changes by the reason such as temperature. The change of the resonant frequency makes impedance mismatch,which decrease the input power to the cavity and increase the reflection power.
The increase of the reflection power is very dangerous for the RF circuit and
AMP. The impedance mismatching is avoided by matching the RF frequency of
SG output with the cavity resonant frequency. Ps-HFS still can be measured
at the required precision without making it constant when the RF frequency
is monitored at a suitable, correcting this eﬀect in analysis. The phase shift
described in Sec. 3.4.2 corresponds to the diﬀerence between SG output RF frequency and the cavity resonant frequency. The frequency matching is performedby changing RF output frequency to make the phase shift zero.

3.5

Gas

The RF cavity is filled with pure i-C4 H10 gas (Takachiho chemical industrial,> 99.9%). Isobutane gas has the following 4 important roles: 1. stopping
positrons in the RF supplied region of the cavity, 2. providing electrons to formPs, 3. quenching slow positrons which are the positrons with very low energy notenough for forming Ps, 4. thermalizing Ps to the room temperature. Especiallythe third role is quite important because the remaining slow positrons becomea huge experimental background. For example, in the nitrogen gas, which wasused in the previous Ps-HFS measurements, at 30 ◦ C and 1 atm, a slow positron
decay into 2γ with a lifetime of 186 ns, which is similar to that of o-Ps . This

3.5 EXPERIMENTAL SETUP

37

topology is the same as the Zeeman transition signal of “delayed 2γ decay”, sothat the backgrounds from slow positrons’ decay cannot be removed by analysis. Typical Ps formation fraction of stopped positrons is 20–40% in gases, andall the remaining positrons become slow positrons. The amount of this background is 15–40 times larger than the Zeeman transition signal since the Zeemantransition probability is typically 10%, which results in a few percent of all the
stopped positrons, whereas the slow positrons are 60–80% of all the stopped
positrons. Details of gas-handling system, the slow positron background, andthe pickoﬀ background are described in this section.

3.5.1

Gas-handling system

Gas-handling system is constructed by VCRs, KFs, and Swageloks. The diameter of most pipes are 1/2” from the RF cavity to the pumps. A valve islocated at the flange of the magnet in order to shut the heat-pumping withexperimental hall, but not inside the magnet because valves are magnetic. The
pumps are the set of a rotary pump and a turbo molecular pump, which canevacuate the cavity to a vacuum of 10−4 Pa. At the beginning of every run,the cavity is evacuated to that vacuum and then filled to 0.129–1.366 amagat1of i-C4 H10 . The whole gas-handling system is baked with 60 ◦ C water for two
weeks after every exposure to air. The absolute pressure of the gas is measured
by a pressure transmitter (setra MODEL 225) with a 0.1 kPa precision, whichis recorded with a portable, high-speed data logger (HIOKI MEMORY HiLOGGER Model 8420-50). The logger is connected to the Linux PC via RS-232 C
interface and the data are recorded to HDDs. The gauge pressure is also measured by a pressure transmitter (NAGANO KEIKI KM31-574) and a digital
display (NAGANO KEIKI GC95-911).

3.5.2

Slow positron background

The annihilation rate of slow positron Γslowpos is described using a parameter
Zeﬀ asΓslowpos ≡ πr02 cnZeﬀ ,(3.3)
where r0 is the classical electron radius, c is the speed of light in vacuum, and n
is a number density of gas molecules. Letting (N2 ) indicate the amount about
nitrogen, Zeﬀ (N2 ) = 29.75 ± 0.85 [63] andΓslowpos (N2 ) = 5.98(17) × 10−3 n(amagat) ns−1 ,(3.4)

where amagat is a unit of number density normalized by that of ideal gas at0◦ C, 1 atm. The lifetime of a slow positron in 1 amagat nitrogen is 167.3 ± 4.8 ns
which is similar to that of o-Ps (142.04(1) ns [3] in vacuum). On the other hand,letting (iso) indicate the amount about isobutane, Γslowpos /n(iso) = (107.9 ±2.1) × 10−12 cm3 /s [64] andΓslowpos (iso) = 2.899(56) n(amagat)ns−1 .

(3.5)

The lifetime of a slow positron in 1 amagat i-C4 H10 is 0.3449 ± 0.0067 ns, which
is much shorter than that of o-Ps. Even if in 0.1 amagat i-C4 H10 , it has stillshort enough lifetime of 3.4 ns.
1 the amagat unit is the number density normalized to that of an ideal gas at 0 ◦ C , 1 atm.

3.6 EXPERIMENTAL SETUP

38

3.5.3

Pickoﬀ background

The pickoﬀ annihilation is another source of backgrounds because the pickoﬀ
also has the same topology as the Zeeman transition. The annihilation rate of
pickoﬀ Γpickoﬀ is described using a parameter 1 Zeﬀ asΓpickoﬀ ≡ 4πr02 cn1 Zeﬀ .

(3.6)

For nitrogen, 1 Zeﬀ (N2 ) = 0.260 ± 0.005 [65] andΓpickoﬀ (N2 ) = 2.090(40) × 10−4 n(amagat) ns−1 .

(3.7)

The pickoﬀ annihilation rate in 1 amagat nitrogen is 2.968(57)% of that ofo-Ps in vacuum. On the other hand, about i-C4 H10 , 1 Zeﬀ (iso) = 0.729 ±0.002 [65] andΓpickoﬀ (iso) = 5.859(16) × 10−4 n(amagat) ns−1 .

(3.8)

The pickoﬀ annihilation rate in 1 amagat i-C4 H10 is 8.323(23)% of that of o-Ps in
vacuum.

3.6

Gamma ray detectors

In this section, details of γ ray detectors are described. Six γ ray detectors
consist of LaBr3 (Ce) scintillators, UVT (ultraviolet transmitting) light guides,
and fine mesh PMTs are used. Details of LaBr3 (Ce) scintillators and UVT lightguides are described in the following sections. The fine mesh PMT is describedin Sec. 3.3.3.

3.6.1

LaBr3 (Ce)Six cylinder LaBr3 (Ce) inorganic crystal scintillators (Saint-Gobain BrilLanCeTM
380) 38.1 mm in diameter and 50.8 mm long are used. A photograph of scintillators is shown in Fig. 3.20.
Properties of LaBr3 (Ce) scintillator
The properties of LaBr3 (Ce) are summarized in Table 3.2 [66]. LaBr3 (Ce) isused because of the following three reasons.
• It has excellent energy resolution (4% FWHM at 511 keV). The acceptanceof the setup is greatly increased by the good energy resolution, because
2γ events are eﬃciently separated from 3γ events with only energy information instead of a back-to-back geometry selection.
• It has good timing resolution (200 ps FWHM at 511 keV). It is importantto timing information at high precision for this experiment. LaBr3 (Ce)has high enough timing resolution.
• It has a decay constant as short as 16 ns, and no long component. It
reduces pileup events, enabling measurements with a high rate.

3.6 EXPERIMENTAL SETUP

39

Figure 3.20: LaBr3 (Ce) scintillators.

Table 3.2: Properties of LaBr3 (Ce) [66]. The properties of NaI(Tl) are also
listed for comparisons.
LaBr3 (Ce) NaI(Tl)Light yield (photons/keV)
63
38
1/e decay
time
τ
(ns)
16
250
(√
)
F. O. M.
(τ /lightyield)
0.5
2.6
Wavelength of maximum emission λmax (nm)380415
Refractive index at λmax
∼ 1.9
1.85
Density (g/cm3 )
5.08
3.67
Thickness for 50% attenuation at 662 keV (cm)1.8
2.5

3.6 EXPERIMENTAL SETUP

40
LaBr3
1

La-138_sim
Entries
Mean
RMS

10-1

2e+07
0.2246
0.3698

Ac-227_sim
BG
Entries 1599856 Entries 5582410
0.3102Mean
1.448 Mean
0.4752RMS
0.9059 RMS
BG_simCounts (/keV/s)

Entries 2.159986e+07
Mean
0.282610-2

RMS

0.4866

10-3
10-4
10-5
10

-6

0

0.5

1

1.5
Energy (MeV)

2

2.5

3

Figure 3.21: Background energy spectra of LaBr3 (Ce). The red line indicatesthe data, black line is simulated with an isotope 138 La, the green line is a MC
simulation of 227 Ac, and the blue line is sum of the black line and the green
line.

Table 3.3: Decay radiation of 138 La [69].
Decay mode

e−
EC

γ
65.6(5)%β−34.4(5)%

Energy(keV)Particlee−γAuger (L)Auger (K)Internal Conversion (K)X ray (L)X ray (Kα2 )X ray (Kα1 )X ray (Kβ3 )X ray (Kβ1 )X ray (Kβ2 )endpoint 255(11)

3.67
26.4
1398.354(10)4.47
31.81732.19436.30436.37837.2551435.795(10)95.7(41)788.742(8)Intensity(%)
49.0(5)4.16(18)
0.0485
6.1(3)
10.6(3)19.4(5)
1.86(5)3.59(9)1.14(3)65.6
34.4(7)34.4

Background of LaBr3 (Ce)
LaBr3 (Ce) has intrinsic backgrounds because it contains radioisotopes. It contains a natural radioisotope of 138 La (natural abundance of 0.09%) and a contaminant of 227 Ac. Figure 3.21 shows a self-counting energy spectrum of LaBr3 (Ce)shielded by 5 cm thick Pb compared with Monte Carlo (MC) simulations of 138 La
and 227 Ac. It can be explained by sum of 138 La and 227 Ac. The MC simulations
are performed with Geant4 [67, 68] release 4.9.0.p01.
138

La Figure 3.22 shows the decay scheme of 138 La. The radiations and intensity are summarized in Table 3.3. The MC simulation in Fig. 3.21 is3.6 EXPERIMENTAL SETUP

41

138
57 La

EC
65.6%
1435.82++00stable
138
56 Ba

+

5

1.02x1011 y

β34.4%
2+
788.7

+

0

0
138
58 Ce

Figure 3.22: Decay scheme of 138 La [69].

normalized by the natural abundance of 0.09%. The background of low
energy region can be explained by 138 La.
227

Ac A decay cascade of 227 Ac is the actinium series. The rate of MC simulation in Fig. 3.21 is normalized by the rate of measurement because
227
Ac does not exist naturally but it is just a contamination. The energy
deposit of α particle is multiplied by 0.3 because α particle deposits its
large energy in very short distance and the light yield is not proportional
to the energy deposit any more. The factor 0.3 is arbitrary defined. The
high energy region of LaBr3 (Ce) background can be explained by 227 Ac.

The total background rate is about 100 Hz and the rate is very low in energy
range of 350–550 keV which is used for analysis. Furthermore, this backgroundcan be subtracted as described in Chap. 4 since these events are accidental.

3.6.2

UVT light guide

Six UVT light guides of the same size and shape are used to guide scintillation
light of LaBr3 (Ce) to fine mesh PMTs. As shown in Fig. 3.23, each light guideconsists of 4 parts of cylinder 38.1 mm in diameter and 35 mm long and edges
obliquely cut at an angle of 15◦ , totally 140 mm and bent by 90◦ . There aretapped holes for light guide holders which ensure pressure joining of a LaBr3 (Ce)and a PMT. The UVT light guides are surrounded by a reflector of 0.5 mm thickGore-tex.
The long, 90◦ bent, and UVT light guides are used because of the following
reasons.
1. The fine mesh PMT disturbs the static magnetic field since it containmagnetic materials as described in Sec. 3.3.3. It must be located as far as
possible from the Ps formation volume of the RF cavity.

3.6 EXPERIMENTAL SETUP

42

Figure 3.23: UVT light guide.

2. The light guides must be UVT since the wavelength of maximum emissionof LaBr3 (Ce) is 380 nm.
Detail reasons to choose the design as shown in Fig. 3.23 are the following
properties of light guides. The properties are investigated using a LaBr3 (Ce)of cylinder 25.4 mm in diameter and 25.4 mm long and UVT light guides of38.1 mm in diameter, and a 2” fine mesh PMT. As γ sources, 22 Na (511 keV,
1275 keV), 137 Cs (662 keV), and 60 Co (1173 keV, 1333 keV) are used.
Length and angle dependence
Figure 3.24 shows attenuation of light yield as a function of the length of lightguides. Firstly, up to 5 cm, the light output rapidly decreases with the length ofthe light guide. This is because the lights which do not satisfy a total reflection
condition are lost. Secondly, in the region of 5–10 cm, the loss related to thetotal reflection condition becomes smaller and the absorption loss appears to
be visible. Finally, at 20 cm, the light output is about a half, but it is not
so smaller than that at 10 cm. This indicates that the absorption loss is quite
small and the length can be determined arbitrary if the light guide is longer than
10 cm, which is the shortest possible length to avoid PMTs disturbing the static
magnetic field. This property does not depend on the absolute light outputs,
because it is the same at all γ ray energies.
Figure 3.25 shows the relative light outputs as a function of the bending angleof light guides. Values are normalized with the output of LaBr3 (Ce) directly
glued to the PMT. The light output is as large as 50% when the light guide isbended by 90◦ . The bending angle dependence is small so that bending by 90◦
is reasonable.
Light Output of the Final Design
Figure 3.26 shows energy spectra of 22 Na with the UVT light guide and without
light guide. A LaBr3 (Ce) of a cylinder 38.1 mm in diameter and 50.8 mm long

3.6 EXPERIMENTAL SETUP

43

RELATIVE LIGHT OUTPUT

Figure 3.24: Light guide length dependence of light output. 0 cm is the data of
LaBr3 (Ce) directly glued to the PMT.

1.2
511 keV
662 keV1173 keV
1275 keV
1333 keV

1
0.8
0.6
0.4
0.2
0

0

20

40
60
80
LIGHT GUIDE BENDING ANGLE (deg)Figure 3.25: Light guide bending angle dependence of light output. The dataaround (0 deg, 1) are measured with LaBr3 (Ce) directly glued to the PMT, andthe other data are measured with light guides of 20 cm long.

3.7 EXPERIMENTAL SETUP

44

boke
Entries292885boke

Mean 1487

COUNTS (A.U.)

RMS

1159

102
10
1
10-1
10-2
10-3

0

1000

2000

3000

4000

5000
6000
ADC CHANNEL

Figure 3.26: Change of light output and energy resolution with light guide. The
black line is the data without any light guides, the red line is the data with a
UVT light guide, and the blue line is the data with a non-UVT light guide. The
gains of the fine mesh PMT are the same and the arrows indicate 1,275 keV.

is used. For comparison, energy spectrum with non-UVT light guide is alsosuperimposed (blue line). The light output with UVT light guide is 0.3 ofthat without any light√ guides. The energy resolution only depends on photon
statistics, and is 1/ 0.3 with UVT light guide compared to that without any
light guides.

3.7

Electronics

In this section, the NIM and CAMAC system used for data acquisition (DAQ)
is described. A CLEAR-PULSE 607 NIM crate is used for High Voltage supply, and three REPIC RPN-011-053 NIM crate is used for others. CAMAC iscontrolled via Linux PC. Noise-cut transformers (DENKENSEIKI NCT-G) areused to reduce noises. Coaxial cables of FUJIKURA RG-174/U are used toconnect between modules. Firstly, the overview is shown. Secondly, a β-tagging
system and a LaBr3 (Ce) system are described and then a trigger and gates
generation system is explained. Finally, a CAMAC system to read all data is
described.

3.7.1

Overview of DAQ

Figures 3.27–3.31 show our DAQ system. Figure 3.32 shows the timing chartof main signals.
The DAQ system consists of 4 parts. The first one is aβ-tagging system, which receives signals from plastic scintillators and generates
the timing signal of Ps formation. The second one is a LaBr3 (Ce) system, which
receives signals from LaBr3 (Ce) , and generates the timing signal of Ps decay.
The third one is a trigger and gates generation system, which generates a trigger
through NIM logical circuit from the Ps formation signal of the β-tagging systemand the Ps decay signal of the LaBr3 (Ce) system. The fourth one is a CAMAC
system, which reads the charge and timing information from analog-to-digitalPlastic
Scintillator Light Guide
NE102A

Fine mesh PMT
Hamamatsu
H6614-70MOD

Gate Generator
Through, 3 µs
KAIZU KN1500

START

OUT

Discriminator-80 mV, 68 ns
Phillips 708

IN

OUT

x10 Amplifier
IN LeCroy 612A OUT

HV supply
REPIC RPH-021

Figure 3.27: DAQ of β-tagging system.
1.7 µs delay

CAMAC
ScalerTechnoland N-TS 112

Visual Scaler
Technoland N-OR 425

CAMAC
8 GHz TDCCAMAC
Q-ADC (PS-short)
RPC-022CAMAC
Q-ADC (PS-long)
Phillips 7166

Discriminator-80 mV, 68 ns
Phillips 708

To
Coincidence Unit
(1)

Gate & delay generator OUT
width 50 ns
delay 65 ns
Technoland N-TM 205
OUT

START

DiscriminatorIN -300 mV, 30 ns
Phillips 708

1.6 µs delay

LinearFAN-IN/FAN-OUT OUT-6 mV offset
IN LeCroy 428F
OUT

3.7 EXPERIMENTAL SETUP
45

LaBr3(Ce)Light GuideSaint-Gobain
TM
BrilLanCe 380

Fine mesh PMT
Hamamatsu
H6614-70MOD

x10 Amplifier
LeCroy 612A

IN

OUT

OUT

LinearFAN-IN/FAN-OUT
IN -6 mV offset
LeCroy 428F

HV supply
REPIC RPH-021

OUT

OUT

Figure 3.28: DAQ of LaBr3 (Ce) system.
START

CAMAC
Q-ADC (LaBr3-base)
Phillips 7166
(DC coupling)

CAMAC
ScalerTechnoland N-TS 112

Visual Scaler
Technoland N-OR 425

CAMAC
8 GHz TDCTo
FAN-IN/FAN-OUT(2)

200 ns delay

Gate & delay generator
OUT
width 50 ns
delay 65 ns
Technoland N-TM 205
OUT

IN

Discriminator-100 mV, 160 ns
OUT
LeCroy 623B

275 ns delay

250 ns delay

CAMAC
Q-ADC (LaBr3-energy)
CAEN C1205

46
3.7 EXPERIMENTAL SETUP

(VETO)

LaBr3-6

LaBr3-5

LaBr3-4

LaBr3-2
LaBr3-3

LaBr3-1

From
Discriminator
(2)

Logic
FAN-IN/FAN-OUT
LeCroy 429A

IN

Coincidence Unit
30 ns
LeCroy 465

Coincidence Unit
OUT (Self trigger mode)
50 ns
Technoland N-TM 103

From
PS-1
Discriminator
PS-2
(1)

From
OUT
FAN-IN/FAN-OUT
(6)

OUT

OUT

OUT

(VETO)

Coincidence Unit
30 ns
N-TM 103

To
Gate Generator(3)

Clock generator
1 kHz, 30 ns
Technoland N-TM203

Figure 3.29: DAQ of trigger system.
CAMAC
ScalerTechnoland N-TS 112

Visual Scaler
Technoland N-OR 425

From
OUT
FAN-IN/FAN-OUT (VETO)
Coincidence Unit
(7, 8)
30 ns
LeCroy 465

Logic
FAN-IN/FAN-OUT
LeCroy 429A

CAMAC
Output Register
Technoland C-RP 202

100 ns delay

Gate Generator
Through, 1.5 µs
START
Self VETOOUT
KAIZU KN1500 OUT

OUTTo
FAN-IN/FAN-OUT
(4)

Logic
FAN-IN/FAN-OUT
LeCroy 429A
OUT

To
Gate Generator(5)

3.7 EXPERIMENTAL SETUP
47

Figure 3.30: DAQ of gates generation system.

From
Coincidence Unit
(4)

From
Gate Generator
(3)

Logic
FAN-IN/FAN-OUT
LeCroy 429A

GATE
OUT

Gate Generator
Through, 100 ns
START KAIZU KN1500
Self VETO

OUT

CAMAC
Q-ADC (PS-short)
RPC-022CAMAC
Q-ADC (PS-long)
Phillips 7166

CAMAC
Q-ADC (LaBr3-base)
GATE Phillips 7166
(DC coupling)

GATE
OUT

Gate Generator
Through, 50 ns
N-TM 203

OUT

CLK

CAMAC
8 GHz TDCCAMAC
Q-ADC (LaBr3-energy)
GATE
CAEN C1205

COM STOP

Gate Generator
START Through, 1.75 µs
KAIZU KN1500
OUT
Self VETO

START

START

Clock generator
10 MHz, 20 ns
HOSHIN N010

To Coincidence Unit (6)To Coincidence Unit (7)To Coincidence Unit (8)COM START

8 GHz
Clock generator

OUT

Gate Generator
Through, 350 ns
KAIZU KN1500
Self VETO

Logic
FAN-IN/FAN-OUT
LeCroy 429ACAMACINHIBIT Scaler
Technoland N-TS 112

Gate Generator
OUT
1.8 µs delay, 100 ns
START
KAIZU KN1500
Self VETO

Coincidence Unit
100 ns
N-TM 103
Self VETO

OUT

CAMAC
Output Register
KAIZU KC3471

OUT

STOP

Gate Generator
Through, Latch
KAIZU KN1500

START

Logic
FAN-IN/FAN-OUT
LeCroy 429A

OUT

VETO

Gate Generator
START 1.5 µs delay, 300 ns
KAIZU KN1500
OUT
Self VETOOUTSTART Gate Generator
Through, 1.75 µs
KAIZU KN1500 OUT

Visual Scaler
Technoland N-OR 425

From
FAN-IN/FAN-OUT(5)48
3.7 EXPERIMENTAL SETUP

3.7 EXPERIMENTAL SETUP

49

Push button
START

Gate Generator
Through, 100 ns
KAIZU KN1500

OUT

Visual Scaler
START Technoland N-OR 425

OUT

START
Visual Scaler
Technoland N-OR 425

Carry out

START

STOP

Gate Generator
Through, 100 ns
KAIZU KN1500

OUT
Clock generator
1 MHz, 30 ns
Technoland N-TM203

Figure 3.31: DAQ of visual scaler system.

Figure 3.32: Timing chart of electronics.

OUT

STOP

3.8 EXPERIMENTAL SETUP

50

converters (ADCs) and time-to-digital converters (TDCs), and records themto HDD of a Linux PC. During the data reading of CAMAC, veto signals are
generated by NIM latch system to avoid further DAQ. The system are released
for the next event when the data reading finishes.
Figure 3.27 shows the DAQ of β-tagging system. The β-tagging system isused to make a signal of a positron emitted from the source and going to the Ps
formation volume in the RF cavity. This signal is a start signal, and it gives thetiming of Ps formation (t = 0). The energy deposit in the plastic scintillator by
the positron is also measured. The signals of the two fine mesh PMTs (let PS-1
and PS-2 denote them) are processed individually.
Figure 3.28 shows the DAQ system for LaBr3 (Ce) signal. The LaBr3 (Ce)system is used to make a signal of a detection of γ rays from a Ps decay. Thissignal generates a trigger by coincidence with a signal of the β-tagging system,
and it gives the timing of Ps decay. The energy of γ rays are also measured.
This energy information is quite important to enhance the signal-to-noise ratio
and obtain resonance lines of the Zeeman transition. The signals of the six
fine mesh PMTs (let LaBr3 (Ce) -1–LaBr3 (Ce) -6 denote them) are processedindividually.
Figure 3.29 shows the DAQ trigger system, and Fig. 3.30 shows the DAQ
gate generation system. The trigger signal for DAQ starting is generated bycoincidence of the β-tagging system and the LaBr3 (Ce) system. The gate andstart signals of the CAMAC modules are generated by this trigger signal.

3.7.2

CAMAC

The CAMAC system is used for DAQ. A CAMAC crate of TOYO PS-7500 isused. Information obtained via CAMAC is timing and energy of the β-tagging
system, timing and energy of the LaBr3 (Ce) system, and rates of signals of NIM
modules. Figure 3.33 shows the overview of the DAQ system.

3.8

Monte Carlo simulation

Monte Carlo simulations are used in fitting of the Zeeman resonance lines toestimate the ratio of detection eﬃciency of |1, 0⟩ decay normalized by that of
the |1, ±1⟩ decay and polarization of positrons. This section describes details of
the MC simulations.
Geant4 [67, 68] (release 4.9.6.p02) is used. Geant4 is a package which simulates the passage of particles and used in a wide areas including high energy
physics such as the LHC experiment, nuclear and accelerator physics, as wellas medical and space science. Validations of this package is widely performed.
The low energy physics package of “option 4”, which is cited as the most precise in low energy region by the Geant4 collaboration team, is used. All of theexperimental setup (materials, geometry, and the static magnetic field) are carefully input. Figures 3.34 and 3.35 show the geometry input in the simulation.
The leading order of the energy spectrum of 3γ decay was calculated firstly byÖre and Powell [70], which was confirmed later [71, 72]. The O(α) correction
was analytically calculated by Adkins [73]. In the MC simulation, the energy
spectrum including this O(α) correction is used, and the angular distribution of3.8 EXPERIMENTAL SETUP

Figure 3.33: DAQ system.

Figure 3.34: Side view of the geometry in the MC simulation.

51

52

3.8 EXPERIMENTAL SETUP

Figure 3.35: Bird’s-eye view of the geometry in the MC simulation

3.10 EXPERIMENTAL SETUP

53

decay γ rays are calculated using [46]. MC simulations are performed at everygas density and every static magnetic field strength of experimental data point.

3.9

Ps Thermalization Measurement

The
thermalization parameters in i-C) 4 H10 gas are measured to be(2
by DBS (Doppler Broadening Spectroscopy)
E0 = 3.1+1.0
eV,
σ
=
146
±
11
Å
m
−0.7
technique [60] in the range of 0.15–1.52 eV Ps kinetic energy. However, the
DBS result cannot be applied for o-Ps whose kinetic energy is less than 0.17 eV
since σm depends on the kinetic energy of Ps. As mentioned in Ref. [60], rovibrational excitations of the i-C4 H10 molecule increase σm of Ps with kinetic
energy above 0.17 eV because i-C4 H10 has a vibrational level at 0.17 eV. The‘pick-oﬀ technique’ [55, 56, 3], which can access o-Ps with lower energy than
0.17 eV, is a complementary method. This technique measures Γpick (t)/Γo−Ps =(2γ annihilation rate)/(3γ annihilation rate) as a function of time using γ-ray
energy spectra. The thermalization can be measured by this method because
Γpick (t) depends on the Ps velocity.
An experimental apparatus and results of our Ps thermalization measurement are described in Appendix B. The result of σm = 47.2 ± 6.7 Å2 for o-Ps below 0.17 eV has been obtained by our independent thermalization measurement
using the ‘pick-oﬀ technique’. In our analysis, the thermalization parametersfrom DBS measurement are used from t = 0 to the time at which the kinetic
energy of o-Ps reaches 0.17 eV, and then the σm is changed to our value.

3.10

Data acquisition

Duration of the measurement was from July 2010 to March 2013. The measurement was performed at the Cryogenic Science Center at KEK. In the overall
period, the signal rate for each LaBr3 (Ce) was around 800 Hz, the “OR” rate of
LaBr3 (Ce) was around 4.5 kHz, the signal rate for each plastic scintillator wasaround 100 kHz, the coincidence rate of plastic scintillators was around 40 kHz,the trigger rate was around 1.7 kHz, and the data acquisition rate was around
910 Hz. The pedestal-taking and the event-taking were performed alternately.
The mode was changed at about every 30 minutes. A typical length of pedestaltaking was 1 minute, and that of event-taking was 30 minutes.
The Zeeman transition was measured at various magnetic field strengths with
fixed RF frequency and power. The transition resonance lines were obtained at
11 gas densities (0.129, 0.133, 0.167, 0.232, 0.660, 0.881, 0.969, 1.193, 1.353,
1.358, and 1.366 amagat). Data were taken at two diﬀerent conditions, RFON and RF-OFF, at every gas density and magnetic field strength. RF-ON
data were taken with microwaves supplied. RF-OFF data were taken without
microwave by switching oﬀ the signal generator and the amplifier.

Chapter 4

Analysis
This chapter describes the analysis of the experimental data. All the analysis isperformed using ROOT [74], and MINUIT [75] implemented in ROOT is used
for fitting. Data are carefully analyzed to obtain time-evolutions of resonancelines, which will be used to see non-thermalized Ps eﬀect.

4.1

Basic cuts and calibrations

This section describes basic cuts and calibrations of the data. As a sample, thefigures and the numbers shown in this section is the data at the gas density of
0.881 amagat and the static magnetic field strength of 0.865 733 6 T. The same
analysis are applied to all of the data.

4.1.1

Cut: Time diﬀerence of the two PS signals

Events within a time diﬀerence between two PMT signals of the β-tagging system are selected in order to reject accidental noise. Details are as follows:
1. A spectrum of the time diﬀerence between the two PMTs (PS-2 − PS-1)
is made at every 10 s.
2. The spectra is fitted by a Gaussian with a fitting range of the center±HWTM(Half Width at Tenth Maximum). Figure 4.1 shows a typical fitting of thetime diﬀerence. Although the spectrum is not in a Gaussian shape because the timing resolution of plastic scintillator depends on the energy
deposit, it does not aﬀect the result since this Gaussian-fitting process isonly for the determination of the center, the width, and their fluctuation.

3. The following condition is imposed using the s.d. of the fitted Gaussian:|tPS-1 − tPS-2 | < 5s.d.,where tPS-1 is the timing of PS-1, and tPS-2 is the timing of PS-2.
Events remains after this cut is ≈98% of the total events.
54

(4.1)

COUNTS (/2ns/10s)

4.1 ANALYSIS

55

103

102

10
-20

-10
0
10
20
PS TIME DIFFERENCE (PS-2 - PS-1, ns)

Figure 4.1: Time diﬀerence of the PMTs of the plastic scintillator. The histogram shows the observed data, and the curved line is the best-fit by a Gaussian. The green area remains after the cut.

4.1.2

Calibration: Energy of the plastic scintillator

The energy of the plastic scintillator is calibrated using the mean value of the
Q-ADC as follows:
1. Simply 0 ch of Q-ADC is regarded as the pedestal because Phillips 7166
automatically corrects the pedestal to 0 ch.
2. The mean value of Q-ADC is regarded as 12.7 photoelectrons (p.e.). Thisvalue is determined by a measurement of photoelectrons without any static
magnetic fields, in which situation the PMTs are well calibrated using an
LED. Uncertainty of this calibration does not aﬀect the Ps-HFS value
because the absolute counts of photoelectrons is not used any more in theanalysis.

4.1.3

Calibration: Energy of the LaBr3 (Ce) scintillators

Pedestals of LaBr3 (Ce) energy are calibrated every 10 s. Pedestal information
is taken even in the event-taking mode because energy information of all the
LaBr3 (Ce) is recorded when at least one of the all LaBr3 (Ce) outputs signal,
which means that pedestals of the PMTs which do not output signal are recorded
simultaneously. A histogram of energy of every LaBr3 (Ce) is made every 10 s
livetime, and the pedestal peak is fitted by a Gaussian with a range of mean
± FWHM. The center of the Gaussian is treated as a pedestal. The gains of
LaBr3 (Ce) energy are calibrated using 511 keV peak every 100 s.

4.1 ANALYSIS

56

4.1.4

Time walk correction

The time walk is corrected for LaBr3 (Ce) energy and β energy subsequently.
Each timing spectrum of 12 diﬀerent combination of LaBr3 (Ce) and β is corrected independently.
Time walk correction of LaBr3 (Ce)The higher the energy of LaBr3 (Ce) , the earlier the timing becomes, since thetiming of LaBr3 (Ce) is a minuend when the timing spectrum is made. The
detail procedure is as follows:
1. A timing histogram is made every 200 s of livetime for every 50 keV energy
window with the energy range of 250–450 keV.
2. Prompt peak of the histogram is fitted by a Gaussian where the fitting
range is from mean−5 s.d. to mean+HWHM (half width at half maximum).
3. Fitted prompt peak is fitted by the following function f (E):√f (E) = t − σ

(
2 ln

)
E
,
E0

(4.2)

where the shape of the PMT signal is approximated to be a Gaussian withthe center timing of t and s.d. of σ, and E0 corresponds to the thresholdof the discriminator under this approximation.
Figure 4.2(a) shows the time-energy 2D plots, and black line show the fitted
curve. Figure. 4.2(b) shows walk is corrected.
Time walk correction of the plastic scintillator
In the similar way, the walk of plastic scintillator is corrected as follows:
1. An energy histogram of each PMT (PS-1 or PS-2) is made for the total
events with a bin width of 0.125 p.e. to determine the energy regions formaking timing histograms.
2. Counting rate of the three bins, the maximum bin and the bins of bothsides of that bin, is calculated. It is denoted as Nmax hereafter.
3. The energy histogram is divided into several windows with the same counting rate as Nmax .
4. A timing histogram is made every 200 s of livetime for every energy window.
5. The prompt peak of the histogram is fitted by a Gaussian by the same
way as the time walk correction of LaBr3 (Ce) .
6. The prompt timing is plotted as a function of energy.

4.1 ANALYSIS

57

χ 2 / ndf 15.13 / 6
Probt

0.01926
35.53 ± 6.048

Sigma
8.846 ± 0.7576

TIME (ns)E

0.08269 ± 0.1143

25
103

20
15
10

102

5
0
10

-5
-10
-15

200 400 600 800 1000 1200 1400
ENERGY (keV)

1

TIME (ns)

(a)

25
103

20
15
10

102

5
0
10

-5
-10
200 400 600 800 1000 1200 1400
ENERGY (keV)

1(b)

Figure 4.2: Time walk correction of LaBr3 (Ce) . (a) Before the time walk correction. The black solid line shows the result of the fitting. (b) After the time
walk correction.

4.2 ANALYSIS

58

Table 4.1: Summary of cuts.
Cut onCondition

Remaining events(%)

Time diﬀerence ofplastic scintillator
Time walk correction
Energy balance ofplastic scintillator±5 s.d.

98

Energy above “Threshold”

73

sqrt-diﬀerence in ±5 s.d.

73

Time diﬀerence of LaBr3 (Ce)All combinationof LaBr3 (Ce) ±5 s.d.

72

7. The plot is fitted by the following function g(E):√g(E) = t0 + σ tanh(τ (E − E0 )) + C E,

(4.3)

where t0 , σ, τ , E0 , C are constants. It fits to the data very well rather
than using a function as Eq. (4.2), although Eq. (4.3) is not derived byany theoretical assumptions, i.e. it is only an empirical expression.
8. The timing is corrected by simply subtracting the value of the function
g(E) from the original timing.
Figure 4.3(a) shows the time-energy 2D plots and black line show the fitted
curve. Fig. 4.3(b) shows walk is corrected.

4.1.5

Cut: Energy diﬀerence of the plastic scintillator

To reduce noise event, energy-balance deposited on both plastic scintillators is
required. Figure 4.4(a) shows the 2D plot of two plastic scintillators.
Both ener√
gies
have
good
correlation.
Figure4.4(b)
showsthediﬀerence(
energyof PS-1−√
energy of PS-2). It is required to be within 5 s.d. of this Gaussian.
Almost all of the events pass this cut, but a little fraction of ≈ 10−6 of thedata are rejected.

4.1.6

Cut: Time diﬀerence of LaBr3 (Ce)If more than one LaBr3 (Ce) TDC stop signals are recorded, it is required thattiming of every combination of two LaBr3 (Ce) signals falls within 5 s.d. of thetime resolution. 99.7% of the remained events passes this cut.

4.1.7

Summary of cuts

Table 4.1 shows a summary of cuts and their acceptance fractions. About 70%of the total events remain after all the cut. These events are used for further
analysis.

4.2 ANALYSIS

59

2

χ / ndf 132.2 / 16Prob2.369e-20
t -1.008 ± 0.05404
0
σ 1.222 ± 0.07974
τ0.1524 ± 0.006324

TIME (ns)E 0 4.003 ± 0.3308
C 0.114 ± 0.007775

25
20
15

102

10
5
10

0
-5
-10
-15
0

1
10

20

30
40
50
60
PS-1 ENERGY (p.e.)

TIME (ns)

(a)

25
20
15
102

10
5
0

10

-5
-10
0

10

20

30
40
50
60
PS-1 ENERGY (p.e.)

1(b)Figure 4.3: Time walk correction of the plastic scintillator. (a) Before the time
walk correction. The black solid line shows the result of the fitting. (b) Afterthe time walk correction.

4.2 ANALYSIS

PS-2 ENERGY (p.e.)

60

105

160
140

104

120
100

103

80
102

60
40

10

20
0
0

20 40 60 80 100 120 140 160
PS-1 ENERGY (p.e.)

1

COUNTS (A.U.)(a) Distribution of charge of plastic scintillator PMTs. The blacksolid line shows the selection.

105
104
103
102
10
1
-10
-5
0
5
10
PS ENERGY DIFFERENCE ( p.e.)(b) Energy-sqrt-diﬀerence between two plastic scintillator PMTs.
The heavy black line shows a fitting with a Gaussian.

Figure 4.4: Energy balance cut of the plastic scintillator.

4.2 ANALYSIS

4.2

61

Decay rate of Ps

In order to check quality of the data, decay rate and formation fraction of Ps areevaluated. All analysis in this section is performed with RF-OFF data. Analysis
is performed at every gas density and every static magnetic field strength. The
decay rate and formation fraction of Ps are determined by fitting the timing
spectrum. The timing spectrum consists of four components; slow positrons,|+⟩ state, mz = ±1 states, and accidentals.

4.2.1

Deduction of the true signal timing distribution

The true signal timing distribution is deduced using Ref. [76]. Figure 4.5 showsa typical timing spectra between the β-tag and the γ-signal. The timing spectra without accidental contributions are obtained by subtracting the accidental
spectra from the raw timing spectra. The diﬀerence from a simple exponential
shape arises because of a change in eﬃciency of accidental events, which depends
on the true signal shape. There are two origins of the accidental spectra. One istotally random stop pulses which are not correlated to the start pulses. It makes
a simple exponential, almost flat, structure. Another is random stop pulses butcorrelated to the start pulses, which occurs as follows. Firstly, a positron makes
a start pulse. Then, another positron makes a β-tagging signal and a stop signal
earlier than a possible stop signal from the first start pulse. This is diﬀerent
from the totally random pulses, and the shape depends on the true signal timing
spectrum. The true timing spectrum with true rate is obtained by correcting
the suppression caused by the dead time of the electronics. The diﬀerence ofthe slope between RF-ON and RF-OFF is caused by the Zeeman transition.
A timing window of 50–440 ns is applied to select o-Ps events. The window
is divided into 11 sub-windows in the analysis, and the time evolution of the
Zeeman transition is confirmed.

4.2.2

Fitting timing spectra of RF-OFF

Each signal timing spectrum is fitted with the following equation for N (t) which
includes Ps thermalization eﬀect [55, 56]:
[
) ]
∫ t(
Γpick (t′ )N0 exp −Γo-Ps
1+
dt′
Γ
o
Ps
0
) ]
[
∫ t(
Γpick (t′ )
1
+
dt′ ,
+ N1 exp −Γ|+⟩Γ|+⟩0

N (t) =

(4.4)

where N0 and N1 are normalization constants, Γ|+⟩ is the decay rate of the |+⟩state of Ps. The component of |−⟩ state is ignored because of its short lifetime.
The time dependence of Γpick (t) is estimated using the following thermalizationeﬀect as
[
]0.6
v(t)
Γpick (n, t) = Γpick (n, ∞) ×
,
(4.5)
v(∞)
0.6

in i-C4 H10 gas. The [v(t)] dependence is obtained by reading thermalization
data from Fig. 2(c) of Ref. [77] and fitting them by a power-law function ofvelocity, which results in an exponent of ≈ 0.6. The power-law dependence is4.2 ANALYSISCOUNTS (/keV/ns/s)

62

10-1
10-2
10-3

RF-OFF
RF-OFF Raw

10-4
RF-OFF Accidental
RF-ON
10-5
0

200

400

600

800 1000 1200 1400
TIME (ns)Figure 4.5: Timing spectra (n =0.881 amagat, B =0.865 733 6 T). The solid
arrow shows the timing window used for transition lines, and the dashed arrowshows the accidental timing window used for subtraction of energy spectra. The
accidental contribution has been already subtracted in the black ‘RF-OFF’ and‘RF-ON’ lines. The data in the γ-ray energy range of 285–575 keV are shown.
(Reproduced from Ref. [1])

COUNTS (/keV/ns/s)

4.3 ANALYSIS

63

1
10-1

10-2
-3

10

-4

10

-5

10

-6

10

10-7
-8

10

0

200

400

600

800

1000

1200
1400
TIME (ns)Figure 4.6: Fitting of RF-OFF timing spectrum (n = 0.881 amagat, B =0.865 733 6 T). The black histogram is the data, and the red line is the fitting
result. The data in the γ-ray energy range of 285–575 keV are shown.

indicated in Fig. 5 of Ref. [78]. Uncertainty in the power 0.6 is not large enoughto aﬀect ∆HFS . Γ|+⟩ is calculated as a function of the static magnetic fieldstrength as√√Γp-Ps + Γo-Ps
I2 + J2 − IΓ|+⟩ =− 2π∆HFS
,
(4.6)
2
2
J2
,
(4.7)
I = 4x2 + 1 −4
Γp-Ps − Γo-Ps
J=
,
(4.8)
2π∆HFS
where x is given as g ′ µB B/ (h∆HFS ), the same as in Chap. 2. Equations (4.6)–(4.8) are used in the fitting.
Figure 4.6 shows an example of the fitting, and Fig. 4.7 shows fitted Γpick (n, ∞)as a function of gas density. Γpick (n, ∞) at 0.167, 0.232, 0.660, and 0.881 amagat
are apparently larger than others because vacuum level obtained before theseruns were poorer. It is assumed to be the eﬀect of residual gas, mainly water, inthe cavity. This assumption is consistent with the fact that the residuals fromthe straight line are similar among these gas densities.

4.3

Timing windows

The diﬀerence of the slope of Fig 4.5 between RF-ON and RF-OFF is causedby the Zeeman transition. A timing window of 50–440 ns is used to select oPs events. This window is divided into 11 sub-windows: 50–60 ns, 60–70 ns,
70–80 ns, 80–90 ns, 90–105 ns, 105–120 ns, 120–140 ns, 140–165 ns, 165–200 ns,
200–260 ns, and 260–440 ns for the study of time-evolution. These sub-windows

4.5 ANALYSIS

Γpick (µs-1)

64

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.2

0.4

0.6

0.8
1
1.2
1.4
GAS DENSITY (amagat)

Figure 4.7: Fitted Γpick (n, ∞) as a function of gas density. The straight line isthe linear best-fit result.

are determined so that each sub-window has roughly the same number of events.
They are introduced to get better precision and confirm the time evolution of
the Zeeman transition.

4.4

Energy spectra

The energy spectra are obtained by subtracting the accidental contribution fromthe raw spectra as shown in Fig. 4.8. The accidental spectra are estimated using
the energy spectra in a timing window of t =1,000–1,430 ns. RF-OFF spectra
are also superimposed. It is clear that 2γ decay events (observed as a peak at
511 keV) are enhanced by the Zeeman transition.

4.5

The Zeeman resonance line

The resonance lines are obtained by (NRF-ON − NRF-OFF ) /NRF-OFF as a function of the static magnetic field strength, where NRF-ON is the counting rate of
the events in the energy window of 511 keV ± 1 s.d.(≈ 17 keV) of RF-ON, and
NRF-OFF is that of RF-OFF. Obtained resonance lines with a best-fit function
are discussed in the next chapter.

COUNTS (/keV/ns/s)

4.5 ANALYSIS

65

0.022
0.02

RF-ON

0.018
0.016
0.014
RF-OFF Raw

0.012
0.01
0.008
0.006
0.004
0.002
0
300

RF-OFF
RF-OFF Accidental
350

400

450

500

550
600
ENERGY (keV)

Figure 4.8: Energy spectra of the timing window of 50–60 ns (n =0.881 amagat,
B =0.865 733 6 T). The accidental contribution has already been subtracted in
the black ‘RF-OFF’ and ‘RF-ON’ lines. The transition lines are obtained bycomparing the areas of RF-ON and RF-OFF spectra, inside the energy window
indicated by the arrow. (Reproduced from Ref. [1])Chapter 5

Results and Discussion
The Zeeman resonance lines are fitted for each gas density in this chapter. Ps
thermalization eﬀect is carefully treated in our fitting. It is shown that thenon-thermalized Ps eﬀect is crucial for measurement of ∆HFS and our result is
consistent with QED prediction, disfavoring the previous experimental results
by about 3 s.d.

5.1

Fitting of the Zeeman resonance lines

Figures 5.1–5.11 show the transition rate (NRF-ON − NRF-OFF ) /NRF-OFF as afunction of the static magnetic field strength. Following the calculations in
Sec. 1.4.2, resonance lines are fitted by the following function F (t, n, B):
F (t, n, B)

R(t, n, B)

RRF-ON (t, n, B) − RRF-OFF (t, n, B)
RRF-OFF (t, n, B)+ D2 (n),≡ ϵ(n)S2γ (t, n, B) + S|1,±1⟩ (t, n, B)=

D1 (n)+ ϵ′ (n)S|1,0⟩ (t, n, B),

(5.1)
(5.2)
(5.3)

where n is the number density of gas molecules, D1 (n) is a normalization factor, D2 (n) is an oﬀset, ϵ(n) and ϵ′ (n) are the relative detection eﬃciencies of
2γ and |1, 0⟩ decay, respectively, normalized by that of the |1, ±1⟩ decay. Annihilation probabilities, S2γ , S|1,±1⟩ , and S|1,0⟩ are calculated numerically from
Eqs. (1.26), (1.28), and (1.29), respectively. In the fitting process, D1 (n) andD2 (n) are treated as free parameters for each gas density because of the following reason. The normalization of the counting rate using RF-ON and that of
RF-OFF is sensitive to gas density, since i-C4 H10 slightly absorbs microwaves
which makes the gas temperature high (the density low). ϵ(n) and BRF (n)(Eq. (1.16)) are also treated as free parameters since the distribution of Ps formation position in the cavity depends on the gas density and this dependency
makes the detection eﬃciency and the eﬀective BRF depend on the gas density. A typical value of ϵ is 6.5. The eﬀective BRF is typically decreased byabout 10% from maximum value (typically 15 G) because of this distribution.
ϵ′ is estimated by simulations in which all the materials are reproduced and66

5.1 RESULTS AND DISCUSSION

67

Ps formation position are also carried out. A typical value of ϵ′ is 1.139. The
polarization of the positrons is also estimated by simulation. Estimated values
are within the range between 0.23 (low gas density) and 0.42 (high gas density).
The Doppler broadening eﬀect is taken into
√ account by a convolution with aGaussian distribution of ω with a s.d. of ω kT / (mPs c2 ), where c is the speedof light in vacuum.
The time dependence of ∆HFS (t) and Γpick (t) are estimated using the following thermalization eﬀect and they are taken into account in the evolution of
S2γ , S|1,±1⟩ , and S|1,0⟩ as∆HFS (n, t) =Γpick (n, t) =3/5

∆0HFS − Cn [v(t)] ,
(
)0.6
v(t)
Γpick (n, ∞) ×
,
v(∞)

(5.4)
(5.5)

where ∆0HFS is the Ps-HFS in vacuum and C is a constant. ∆0HFS and C arecommon free parameters for the fitting of all data points. The obtained valuesof Γpick (n, ∞) in Sec. 4.2.2 are used in the fitting process. As shown in Figs. 5.1–
5.11, the data points are well fitted by Eq. (5.1).
By fitting all of the data points (11 gas densities × 11 timing windows ×
4–7 magnetic field strengths) simultaneously, the best-fit value of∆0HFS = 203.394 2(16) GHz(5.6)

is obtained with χ2 /ndf = 633.3/592 and a p-value of 0.12. The fitting freeparameters are D1 (n), D2 (n), ϵ(n), BRF (n) (11 parameters for each), ∆0HFS ,and C, totally 46 parameters. Fitted values of D1 (n), D2 (n), ϵ(n), and BRF (n)are shown in Appendix C. Time evolution of some parameters: ρ00 (t), v(t)/c,
Γpick (t), and ∆HFS (t), at several gas density using our final fitting results areshown in Figs. 5.12–5.15. The graphs are drawn at the static magnetic fieldstrengths of the nearest data points to the centers of the resonances at 0.129,
0.881, and 1.358 amagat gas density.
Fig. 5.12 shows the time evolutions ofa component ρ00 (t) of the density matrix ρ. The 2γ annihilation rate is mainly
proportional to this function (see Eq 1.26). The graphs are drawn for the RFON condition, and the shape depends on the microwave power. The prompt |−⟩decay, the short components of |+⟩ decay, and then the long components of the
Zeeman transition are reproduced well. At low gas densities, the measurements
were performed with low microwave power to avoid gas discharges, which makes
the slope of the long component gradual.
Figure 5.13 shows the time evolution of the normalized Ps velocity v(t)/c.
It shows that thermalization takes a long time at low gas density. Small kinks
where σm changes because the Ps energy crosses the 0.17 eV threshold can beseen but not aﬀect the final results. Fig. 5.14 shows the time evolution of the
0.6
pick-oﬀ annihilation rate Γpick (t). The n [v(t)] dependence is caused by thePs thermalization. The ‘pick-oﬀ technique’ measures this function to obtain thethermalization parameters. Fig. 5.15 shows the time evolution of ∆HFS (t). A
dramatic change of O(100) ppm is shown in the timing range earlier than 50 ns
which we do not use for the analysis. A slow change of O(10) ppm is also evident
at low gas density: this clearly shows the eﬀect of non-thermalized o-Ps on thevalue of ∆HFS .

5.1 RESULTS AND DISCUSSION

1.4

50- 60 ns
70- 80 ns
90-105 ns
120-140 ns
165-200 ns
260-440 ns

1.2
1
0.8
0.6

(N

RF-ON

- NRF-OFF) / N

RF-OFF

68

0.4
0.2
0.866

0.867

1.4

0.868
0.869
MAGNETIC FIELD (T)60- 70 ns
80- 90 ns
105-120 ns
140-165 ns
200-260 ns

1.2
1
0.8
0.6

(N

RF-ON

- NRF-OFF) / N

RF-OFF

0

0.4
0.2
0

0.866

0.867

0.868
0.869
MAGNETIC FIELD (T)Figure 5.1: Resonance lines at a gas density of 0.129 amagat. The markers
with error bars indicate obtained data, and the lines indicate the best-fit result. Eleven dataset for various time-windows are divided into two figures for
improved visibility.

(N

RF-ON

- NRF-OFF) / N

RF-OFF

5.1 RESULTS AND DISCUSSION

69

50- 60 ns
70- 80 ns
90-105 ns
120-140 ns
165-200 ns
260-440 ns

1.4
1.2
1
0.8
0.6
0.4
0.2

(N

RF-ON

- NRF-OFF) / N

RF-OFF

0

0.866

0.867

0.868
0.869
MAGNETIC FIELD (T)60- 70 ns
80- 90 ns
105-120 ns
140-165 ns
200-260 ns

1.4
1.2
1
0.8
0.6
0.4
0.2
0

0.866

0.867

0.868
0.869
MAGNETIC FIELD (T)Figure 5.2: Resonance lines at a gas density of 0.133 amagat. The markers
with error bars indicate obtained data, and the lines indicate the best-fit result. Eleven dataset for various time-windows are divided into two figures for
improved visibility.

5.1 RESULTS AND DISCUSSION- NRF-OFF) / N

RF-OFF70

1.8
50- 60 ns
70- 80 ns
90-105 ns
120-140 ns
165-200 ns
260-440 ns

1.6
1.4
1.2

(N

RF-ON

1
0.8
0.6
0.4
0.2
0

- NRF-OFF) / N

RF-OFF

-0.2

0.864

0.866

0.868
0.87
MAGNETIC FIELD (T)1.8
60- 70 ns
80- 90 ns
105-120 ns
140-165 ns
200-260 ns

1.6
1.4
1.2
1

RF-ON

(N

0.862

0.8
0.6
0.4
0.2
0

-0.2

0.862

0.864

0.866

0.868
0.87
MAGNETIC FIELD (T)Figure 5.3: Resonance lines at a gas density of 0.167 amagat. The markers
with error bars indicate obtained data, and the lines indicate the best-fit result. Eleven dataset for various time-windows are divided into two figures for
improved visibility.

- NRF-OFF) / N

RF-OFF

5.1 RESULTS AND DISCUSSION

71

50- 60 ns
70- 80 ns
90-105 ns
120-140 ns
165-200 ns
260-440 ns

2.5
2

(N

RF-ON

1.5
1
0.5
0

- NRF-OFF) / N

RF-OFF

-0.5

0.864

0.866

0.868
0.87
MAGNETIC FIELD (T)

0.866

0.868
0.87
MAGNETIC FIELD (T)60- 70 ns
80- 90 ns
105-120 ns
140-165 ns
200-260 ns

2.5
2
1.5

RF-ON

(N

0.862

1
0.5
0

-0.5

0.862

0.864

Figure 5.4: Resonance lines at a gas density of 0.232 amagat. The markers
with error bars indicate obtained data, and the lines indicate the best-fit result. Eleven dataset for various time-windows are divided into two figures for
improved visibility.

5.1 RESULTS AND DISCUSSION- NRF-OFF) / N

RF-OFF72

2

50- 60 ns
70- 80 ns
90-105 ns
120-140 ns
165-200 ns
260-440 ns

1.5

(N

RF-ON

1
0.5
0

- NRF-OFF) / N

RF-OFF

-0.5

0.864

0.866

2

0.868
0.87
MAGNETIC FIELD (T)60- 70 ns
80- 90 ns
105-120 ns
140-165 ns
200-260 ns

1.5
1

RF-ON

(N

0.862

0.5
0

-0.5

0.862

0.864

0.866

0.868
0.87
MAGNETIC FIELD (T)Figure 5.5: Resonance lines at a gas density of 0.660 amagat. The markers
with error bars indicate obtained data, and the lines indicate the best-fit result. Eleven dataset for various time-windows are divided into two figures for
improved visibility.

- NRF-OFF) / N

RF-OFF

5.1 RESULTS AND DISCUSSION

73

50- 60 ns
70- 80 ns
90-105 ns
120-140 ns
165-200 ns
260-440 ns

1.5

0.5

(N

RF-ON

1

0

- NRF-OFF) / N

RF-OFF

-0.5

0.862

0.864

0.866

60- 70 ns
80- 90 ns
105-120 ns
140-165 ns
200-260 ns

1.5

1

0.5

(N

RF-ON

0.868
0.87
MAGNETIC FIELD (T)

0

-0.5

0.862

0.864

0.866

0.868
0.87
MAGNETIC FIELD (T)Figure 5.6: Resonance lines at a gas density of 0.881 amagat. The markers
with error bars indicate obtained data, and the lines indicate the best-fit result. Eleven dataset for various time-windows are divided into two figures for
improved visibility. (Reproduced from Ref. [1])

5.1 RESULTS AND DISCUSSION- NRF-OFF) / N

RF-OFF

74

50- 60 ns
70- 80 ns
90-105 ns
120-140 ns
165-200 ns
260-440 ns

1.5

0.5

(N

RF-ON

1

0

- NRF-OFF) / N

RF-OFF

-0.5

0.865

0.866

0.867
0.868
MAGNETIC FIELD (T)60- 70 ns
80- 90 ns
105-120 ns
140-165 ns
200-260 ns

1.5

0.5

(N

RF-ON

1

0

-0.5

0.865

0.866

0.867
0.868
MAGNETIC FIELD (T)Figure 5.7: Resonance lines at a gas density of 0.969 amagat. The markers
with error bars indicate obtained data, and the lines indicate the best-fit result. Eleven dataset for various time-windows are divided into two figures for
improved visibility.

- NRF-OFF) / N

RF-OFF

5.1 RESULTS AND DISCUSSION

1.4
1.2
1

75

50- 60 ns
70- 80 ns
90-105 ns
120-140 ns
165-200 ns
260-440 ns

0.8

(N

RF-ON

0.6
0.4
0.2
0
-0.2
-0.4

- NRF-OFF) / N

RF-OFF0.86450.86550.866

0.86650.867
MAGNETIC FIELD (T)0.86550.866

0.86650.867
MAGNETIC FIELD (T)60- 70 ns
80- 90 ns
105-120 ns
140-165 ns
200-260 ns

1.4
1.2
1
0.8
0.6

RF-ON

(N

0.865

0.4
0.2
0

-0.2
-0.4
0.86450.865

Figure 5.8: Resonance lines at a gas density of 1.193 amagat. The markers
with error bars indicate obtained data, and the lines indicate the best-fit result. Eleven dataset for various time-windows are divided into two figures for
improved visibility.

5.1 RESULTS AND DISCUSSION- NRF-OFF) / N

RF-OFF

76

1.4

50- 60 ns
70- 80 ns
90-105 ns
120-140 ns
165-200 ns
260-440 ns

1.2
1
0.8

(N

RF-ON

0.6
0.4
0.2
0
-0.2
-0.4

- NRF-OFF) / N

RF-OFF

0.865

1.4

0.867
0.868
MAGNETIC FIELD (T)60- 70 ns
80- 90 ns
105-120 ns
140-165 ns
200-260 ns

1.2
1
0.8
0.6

RF-ON

(N

0.866

0.4
0.2
0

-0.2
-0.4
0.865

0.866

0.867
0.868
MAGNETIC FIELD (T)Figure 5.9: Resonance lines at a gas density of 1.353 amagat. The markers
with error bars indicate obtained data, and the lines indicate the best-fit result. Eleven dataset for various time-windows are divided into two figures for
improved visibility.

- NRF-OFF) / N

RF-OFF

5.1 RESULTS AND DISCUSSION

1.4
1.2
1

77

50- 60 ns
70- 80 ns
90-105 ns
120-140 ns
165-200 ns
260-440 ns

0.8

(N

RF-ON

0.6
0.4
0.2
0
-0.2
-0.4

- NRF-OFF) / N

RF-OFF

0.864

1.2
1

0.866

0.867
0.868
MAGNETIC FIELD (T)

0.866

0.867
0.868
MAGNETIC FIELD (T)60- 70 ns
80- 90 ns
105-120 ns
140-165 ns
200-260 ns

0.8
0.6

RF-ON

(N

1.4

0.865

0.4
0.2
0

-0.2
-0.4
0.864

0.865

Figure 5.10: Resonance lines at a gas density of 1.358 amagat. The markers
with error bars indicate obtained data, and the lines indicate the best-fit result. Eleven dataset for various time-windows are divided into two figures for
improved visibility.

5.1 RESULTS AND DISCUSSION- NRF-OFF) / N

RF-OFF78

1

50- 60 ns
70- 80 ns
90-105 ns
120-140 ns
165-200 ns
260-440 ns

0.8

(N

RF-ON

0.6
0.4
0.2
0
-0.2

- NRF-OFF) / N

RF-OFF

0.865

1

0.867
0.868
MAGNETIC FIELD (T)60- 70 ns
80- 90 ns
105-120 ns
140-165 ns
200-260 ns

0.8
0.6

RF-ON

(N

0.866

0.4
0.2
0

-0.2
0.865

0.866

0.867
0.868
MAGNETIC FIELD (T)Figure 5.11: Resonance lines at a gas density of 1.366 amagat. The markers
with error bars indicate obtained data, and the lines indicate the best-fit result. Eleven dataset for various time-windows are divided into two figures for
improved visibility.

ρ00

5.1 RESULTS AND DISCUSSION

79

10-1
10-2
10-3
10-4
0.129 amagat

10-5

1.358 amagat

10-6
10-7

0.881 amagat

10-8
0

100

200

300

400

500
600
TIME (ns)v/cFigure 5.12: Time evolutions of ρ00 for various gas density. (Reproduced from
Ref. [1])0.00250.002
0.00150.001

0.881 amagat
0.129 amagat

0.00051.358 amagat
0
0

100

200

300

400

500
600
TIME (ns)Figure 5.13: Time evolutions of mean velocity of Ps for various gas density.
(Reproduced from Ref. [1])

5.1 RESULTS AND DISCUSSION

Γpick (ns-1)

800.00140.00120.001

1.358 amagat

0.00080.881 amagat
0.00060.00040.129 amagat

0.0002
0
0

100

200

300

400

500
600
TIME (ns)∆HFS (GHz)

Figure 5.14: Time evolutions of pick-oﬀ annihilation rate for various gas density.
(Reproduced from Ref. [1])203.40.129 amagat

203.38203.36

0.881 amagat
203.34
1.358 amagat
203.32203.3
203.28
0

100

200

300

400

500
600
TIME (ns)Figure 5.15: Time evolutions of ∆HFS for various gas density. (Reproduced
from Ref. [1])

5.2 RESULTS AND DISCUSSION

5.2

81

Systematic errors

Systematic errors are summarized in Table 5.1.
Table 5.1: Summary of systematic errors. (Reproduced from Ref. [1])SourceErrors in ∆HFS (ppm)
Material Eﬀect:
o-Ps pick-oﬀ
3.5
Gas density measurement
1.0
Temperature measurement
0.1
Spatial distribution of density
and temperature inside the RF cavity
Thermalization of Ps:
Initial kinetic energy E0
DBS result σmpick-oﬀ result σm

2.5
0.2
0.5
1.8

Magnetic Field:
Non-uniformity
Oﬀset and reproducibility
NMR measurement

3.0
1.0
1.0

RF System:
RF power
QL value of RF cavity
RF frequency
Power distribution in the cavity

1.2
1.2
1.0
< 0.1

Others:
Choice of timing window
Choice of energy window
Polarization of e+
Phase of microwaves
o-Ps lifetime
p-Ps lifetime

Quadrature sum

5.2.1

1.8
0.6
< 0.2
< 0.1
< 0.1
< 0.1
6.4

Material Eﬀect

The largest contribution is an uncertainty of the material eﬀect.
• The uncertainty of the o-Ps pick-oﬀ rate (Γpick (n, ∞)) is estimated using
the fitting error of the o-Ps decay curve.
• The uncertainty in the gas density is computed from the uncertainties
of the gas pressure and temperature to be 0.2%, resulting in 1.0 ppm
uncertainty in ∆HFS .
• The uncertainty of the gas temperature is 0.1 K, which corresponds to
0.1 ppm in ∆HFS .

5.2 RESULTS AND DISCUSSIONY POSITION (m)

82

200

0.06

180

0.04

160
0.02

140
120

0

100

-0.02

80
-0.04
-0.06
-0.06 -0.04 -0.02

60
40
0

0.02 0.04 0.06
X POSITION (m)Figure 5.16: Spatial distribution of gas temperature in the RF cavity (axial
view, n = 0.881 amagat). The color indicates the gas temperature in ◦ C, andthe black solid circle indicates the cavity wall.
• In estimating the systematic uncertainty due to spatial distributions ofgas density and temperature in the RF cavity, distributions with the extreme condition of no gas convection were used. It is assumed that thestrength of RF power absorbed by the gas is proportional to the energy
density of electric field of the TM110 mode. As a result, a gas temperature
distribution of ≈ 170 K range is produced in the RF cavity, and the fitting result of ∆HFS shifts by +2.5 ppm. A typical calculated temperature
distribution is shown in Fig. 5.16. This shift is conservatively considered
as a systematic error.
• The uncertainty in the Ps thermalization eﬀect is estimated by the errorsof the thermalization parameters: initial kinetic energy E0 , DBS result of
σm , and pickoﬀ-result of σm . These parameters are considered as independent, and eﬀects of errors of them are 0.2 ppm, 0.5 ppm, and 1.8 ppm,
respectively.

5.2.2

Magnetic Field

The second largest contribution is the uncertainty of the static magnetic field.
• The distribution of the static magnetic field is measured by an NMR magnetometer with the same setup as Ps-HFS measurement both before andafter the Ps-HFS measurement. These two measurements are consistent
with each other and the non-uniformity is weighted by the RF magnetic
field strength and distribution of Ps formation positions, which results in5.2 RESULTS AND DISCUSSION83

1.5 ppm RMS inhomogeneity.
• The strength of the static magnetic field is measured outside of the RF
cavity during the run. An oﬀset value at this point is measured during
the measurement of the magnetic field distribution, and its uncertainty
including reproducibility is 0.5 ppm.
• The precision of magnetic field measurement is 0.5 ppm, which comes fromthe polarity-dependence of the NMR probe.
These uncertainties are doubled because ∆HFS is approximately proportional to
the square of the static magnetic field strength.

5.2.3

RF System

Uncertainties related to the RF system are estimated with all the RF parameters
included in the fitting: power, cavity QL , and frequency.
• A long-term stability of 0.06% and a relative measurement uncertainty of
0.08% are associated with the power, which results in 0.10% total uncertainty, corresponding to 1.2 ppm error in ∆HFS .
• A long-term stability of 0.08% and a relative uncertainty of measurementof 0.06% are associated with the QL value, which also results in 0.10%
total uncertainty, corresponding to 1.2 ppm error in ∆HFS .
• A long-term stability of 0.8 ppm and an absolute uncertainty of 0.6 ppm areassociated with the frequency, which results in 1.0 ppm total uncertainty,
corresponding to 1.0 ppm error in ∆HFS .
• In our final global-fitting, Eq. (1.24) is solved for a given average RF
power. However, depending on their position the Ps see diﬀerent power
and the final results should be given by an average of several Eq. (1.24) for
diﬀerent power. A fitting with this method has been performed to estimate
this eﬀect. The distribution of Ps formation position is estimated using
GEANT4 MC simulation. Time dependence of RF power for each Ps isignored because the diﬀusion length of Ps within its lifetime is less than
1 mm and there is no large diﬀerence of RF power at this distance. A
free parameter of proportionality coeﬃcient to this distribution has beenused instead of BRF . The shift of ∆HFS has been less than 0.1 ppm.
An estimation including spatial distribution of gas density has also been
performed and the shift has also been less than 0.1 ppm. This shift isconsidered as a systematic error.

5.2.4

OthersOther systematic uncertainties are related to the analysis.
• Fittings with a starting time of 40 ns and 60 ns with a fixed fitting end timeof 440 ns are performed in order to study the systematic error associatedwith the choice of the timing window (Fig. 5.17). Fittings with an endingtime of 260 ns and 620 ns with a fixed start time of 50 ns are also performed(Fig. 5.18). There is no significant systematic dependence. The maximum
shift in ∆HFS is 1.8 ppm and is considered to be a systematic error.

5.2 RESULTS AND DISCUSSION∆ HFS (GHz)84203.397203.396203.395
203.394203.393203.392
203.391

40

45

50

55
60
FITTING START TIME (ns)∆ HFS (GHz)

Figure 5.17: Fitting-start-time dependence of ∆HFS . The fitting ending time isfixed to 440 ns.

203.397203.396203.395
203.394203.393203.392
250

300

350

400

450 500 550 600
FITTING END TIME (ns)Figure 5.18: Fitting-end-time dependence of ∆HFS . The fitting starting time isfixed to 50 ns.

5.3 RESULTS AND DISCUSSION

85• The gain and oﬀset of the detectors are calibrated every 10 minutes
and these uncertainties are negligible. Analysis with energy window of
511 keV ± 1.5 s.d.(≈ 26 keV) has been performed, and the result shifted∆HFS by 0.6 ppm. This shift is taken into account as a systematic error forthe choice of the energy window. Other systematic errors from detectors
are considered to be cancelled out by the subtraction of RF-OFF datafrom RF-ON data and the normalization by RF-OFF data.
• A comparison with an assumption that a positron is unpolarized has been
performed, but the change in the final fitted Ps-HFS value is less than
0.2 ppm. This shows that the uncertainty of the dependence of polarizationis less than 0.2 ppm.
• The uncertainty in the lifetime measurements of Ps aﬀect ∆HFS by lessthan 0.1 ppm.
• The uncertainty in the MC estimation of ϵ′ is negligible because the contribution of the |1, 0⟩ state is small.
• An eﬀect of excited states can be estimated using the Hamiltonian asshown in Ref. [79] and it is negligible.
• Other systematic errors such as the motional Zeeman and Stark eﬀects,
the spin-conversion quenching of Ps, the quadratic Zeeman eﬀect, andsmaller correction to g factor are negligible.
• Completely separate analysis which determine ∆HFS value at each gas
density has been performed to provide additional insight into the complete experimental data set and confirm their quality. Figure 5.19 shows
the result. It is obtained by fitting the resonance lines at each gas density
without considering the time evolution of ∆HFS , i.e. ∆HFS is treated asa constant at each gas density instead of using Eq. (5.4). This method
is similar to the method used in the previous experiments except thatour data use timing information, which was not taken in the previous
measurement, and 11 resonance lines within 50–440 ns timing window are
simultaneously fitted at each gas density. It is impossible to include thetime evolution of ∆HFS in this method. It is evident that the data fluctuations from the linear-fit function are reasonable compared to the error
bars. It is important to say that determination of ∆HFS using our dataneeds our new global-fitting method to treat the time evolution correctly.
• Comparing Fig. 5.19 and Fig. 4.7, there is no significant correlation between vacuum quality and fluctuation of ∆HFS value. It shows that aneﬀect of residual gas is negligible.

5.3

Ps-HFS value

Since the systematic errors discussed in the previous section are regarded as
independent, and the total systematic error is calculated as a quadrature sum.
The final result including systematic errors is:∆HFS = 203.394 2 ± 0.001 6(stat.) ± 0.001 3(sys.) GHz.

(5.7)

5.5 RESULTS AND DISCUSSION

86

χ 2 / ndf 10.85 / 9
Prob

0.286

Constant
203.4 ± 0.001738∆HFS (GHz)

Slope-196.6 ± 7.986

203.42203.4
203.38203.36203.34
0

0.2

0.4

0.6

0.8
1
1.2
1.4
GAS DENSITY (amagat)

Figure 5.19: ∆HFS at each gas density. The circles with error bars are thedata, and the solid line is the best-fit with a linear function. (Reproduced from
Ref. [1])Figure 5.20 is a summary plot of ∆HFS measurements. Our new result favors the
QED calculation within 1.2 s.d., although it disfavors the previous experimental
average by 2.6 s.d.

5.4

Non-thermalized Ps eﬀect

In order to evaluate the non-thermalized Ps eﬀect, which was not consideredin previous experiments, fitting of the Zeeman transition lines without taking into account the time evolution of ∆HFS and Γpick is performed. The
fitted Ps-HFS value with an assumption that Ps is well thermalized resultsin 203.392 2(16) GHz (χ2 /ndf = 721.1/592, p = 2 × 10−4 ). Comparing thiswith Eq. (5.6), the non-thermalized o-Ps eﬀect is evaluated to be as large as
10 ± 2 ppm in the used timing window. This eﬀect might be larger if no timing
window is applied, since it depends on the timing window used for the analysis.
In the timing window of 0–50 ns, which is not used for the analysis, Ps-HFS is
dramatically changing because the Ps is not well thermalized and the Ps velocity
is still rapidly changing. It strongly suggests that the reason of the discrepancyin ∆HFS is the eﬀect of non-thermalized Ps.

5.5

Future prospects

In our experiment, it is shown that the non-thermalized Ps eﬀect is crucial for
precise measurement of ∆HFS . To obtain further precision, it is necessary to
reduce the systematic error associated with the material eﬀect. Unfortunately,
it is diﬃcult to achieve with gas experiment because the lower the gas density,
the lower the eﬃciencies of Ps formation and transition. Instead of using gas,

5.5 RESULTS AND DISCUSSION
Previous experimental
average87

O(α3ln α-1) QEDa
Old methodbThis measurement

203.386 203.388

203.39203.392 203.394 203.396∆HFS (GHz)

Figure 5.20: Summary of ∆HFS measurements from past experiments and thiswork. The circles with error bars are the experimental data (a−[43], b−[44]),
the hatched band is the average of the previous experiments (a and b), and thedotted band is the QED calculation [14, 34, 35]. (Reproduced from Ref. [1])performing whole measurement in vacuum will be a quite eﬃcient way. There
are at least three benefits from vacuum experiment:• High statistics can be easily obtained because measurement at only one
density (vacuum) is enough.
• It is completely free from material eﬀect, because ideally Ps does notcollide with anything.
• Short measurement period gives us small systematic errors associated withinstability of the system.
The key point in such an experiment is eﬃcient Ps formation in vacuum withenergy which enables easy treatment. Short (ns) pulsed slow positron beam is
one of the most possible choice for positron source. Many eﬀorts must be done
for further precise measurement.

Chapter 6

Conclusion
A new precision measurement of Ps-HFS free from possible common uncertainties from Ps thermalization eﬀect was performed to check the Ps-HFS discrepancy. The eﬀect of non-thermalized o-Ps was evaluated to be as large as
10 ± 2 ppm for the used timing window. This eﬀect might be larger than 10 ppm
if no timing window is applied, since it depends on the timing window. Including this eﬀect, obtained new experimental value is ∆HFS = 203.394 2 ±0.001 6(stat., 8.0 ppm) ± 0.001 3(sys., 6.4 ppm) GHz. It favors the O(α3 ln α−1 )QED calculation within 1.2 s.d., although it disfavors the previous measurements by 2.6 s.d. Our new result shows that the material eﬀect is crucial forthe measurement.

88Appendix A

Run Summary
This appendix describes summary of the runs.

A.1

Temperature control

Temperature information is obtained using platinum resistance thermometer
sensors (Pt100). Temperatures are measured by four-terminal method using
the data logger (HIOKI 8420-50) at the precision of 0.01 K and recorded to theLinux PC.
The following temperatures are measured:• Surface of the RF cavity at the center• Positron source-end surface of the RF cavity• Directional Coupler at the microwave waveguide• Microwave feedthrough at the cavity• NMR probe• Air around the RF circuit• Air around the DAQ electronics• Intake air of the RF amplifier• Exhaust air of the RF amplifier• Air of the experimental hall of the Cryogenics Science Center at KEK• Air in the bore of the superconducting magnet• Inside wall of the superconducting magnet• Flange of the superconducting magnet
89

A.2 RUN SUMMARY

90

The temperature information is used for correction of parameters of the RF
system and the interlock system. Air temperature is carefully controlled by air
conditioners, heat insulators, and the constant temperature water controlled
by the chiller. The water temperature is controlled within ±0.1 K (RMS is
0.02 K), and the air temperature is controlled within ±2 K (RMS is 0.7 K).
The air temperature around the RF circuit (TRFcirc ) is used for correcting RF
power measurements as follows: (1) RF amplifier output power measured by apower meter is corrected by +1.0 × 10−3 × (TRFcirc − 20◦ C) dBm. (2) RF cavitytransmitted power measured by another power meter is corrected by +2.5 ×
10−3 × (TRFcirc − 20◦ C) dBm.

A.2

Run table

All runs only used in analysis are summarized in Table A.1. If there wereaccidental power cuts or earthquakes during runs, pairs of event-taking andpedestal-taking data (about 30 minutes per each pair) which contained themwere removed from analysis. RF-instable data, i.e. in which sudden jumps ofparameters occurred, were also removed. When there was a sudden jump in
RF parameters, data only before or after the jump were used. There existed
magnetic-field-scan data at 0.1323 amagat and 0.1361 amagat, but they werealso removed because of their overall bad data quality.
Table A.1: Run table.
Magnetic field
RF state RUN start date
RUN finish date(T) (ON/OFF) (yyyy/mm/dd)(yyyy/mm/dd)
SCAN I: 0.880 9 amagat, 423.68 W, 2.852 445 1 GHz, QL = 10302.7
0.865 733 64
OFF
2010/07/02
2010/07/05
ON
2010/07/02
2010/07/02
0.867 265 02
ON
2010/07/05
2010/07/07
OFF
2010/07/07
2010/07/15
0.863 900 5
ON
2010/07/15
2010/07/16
OFF
2010/07/16
2010/07/20
0.869 994 2
ON
2010/07/20
2010/07/21
OFF
2010/07/21
2010/07/22
0.861 243 5
ON
2010/07/22
2010/07/23
OFF
2010/07/23
2010/07/25
0.864 752 7
OFF
2010/07/26
2010/07/27
ON
2010/07/27
2010/07/29
0.866 485 84
ON
2010/07/29
2010/07/30
OFF
2010/07/30
2010/08/02
SCAN II: 0.659 7 amagat, 445.52 W, 2.853 538 5 GHz, QL = 10993.3
0.865 662 42
ON
2010/08/02
2010/08/04
OFF
2010/08/04
2010/08/05
0.867 394 57
ON
2010/08/05
2010/08/06
OFF
2010/08/06
2010/08/09
0.863 959 22
OFF
2010/08/09
2010/08/11
ON
2010/08/09
2010/08/11
0.870 053 48
OFF
2010/08/23
2010/08/24

A.2 RUN SUMMARY

91
Table A.1: Run table.

Magnetic field(T)

RF state RUN start date
RUN finish date
(ON/OFF) (yyyy/mm/dd)(yyyy/mm/dd)
ON
2010/08/24
2010/08/26
0.861 219 99
ON
2010/08/26
2010/08/27
OFF
2010/08/27
2010/08/30
0.864 751 07
OFF
2010/08/30
2010/08/30
ON
2010/08/30
2010/08/31
SCAN III: 0.231 7 amagat, 438.60 W, 2.861 905 0 GHz, QL = 13667.2
0.866 465 30
ON
2010/10/04
2010/10/08
OFF
2010/10/08
2010/10/12
0.863 882 99
ON
2010/10/12
2010/10/15
OFF
2010/10/15
2010/10/18
0.867 319 00
ON
2010/10/19
2010/10/22
OFF
2010/10/22
2010/10/23
0.861 153 64
ON
2010/10/25
2010/10/29
OFF
2010/10/29
2010/11/01
0.869 899 85
ON2010/11/01
2010/11/05
OFF
2010/11/02
2010/11/08
0.864 693 10
ON2010/11/08
2010/11/12
OFF
2010/11/12
2010/11/15
0.864 693 97
ON
2010/11/15
2010/11/19
OFF
2010/11/19
2010/11/22
0.865 705 1
ON
2010/11/24
2010/11/26
OFF
2010/11/26
2010/11/29
SCAN IV: 0.167 amagat, 220.36 W, 2.862 400 4 GHz, QL = 14096.5
0.870 010 3
ON
2011/01/18
2011/01/21
OFF
2011/01/21
2011/01/24
0.861 255 81
ON
2011/01/24
2011/01/28
OFF
2011/01/28
2011/01/31
0.868 159 75
ON
2011/01/31
2011/02/04
OFF
2011/02/04
2011/02/07
0.870 912 22ON
2011/02/07
2011/02/10
OFF
2011/02/10
2011/02/14
SCAN V: 1.358 1 amagat, 469.48 W, 2.857 588 5 GHz, QL = 11060.0
0.867 884 00
ON
2011/07/05
2011/07/07
OFF
2011/07/07
2011/07/11
0.864 046 11
OFF
2011/07/11
2011/07/12
ON
2011/07/12
2011/07/14
0.867 296 47
ON
2011/07/14
2011/07/15
OFF
2011/07/15
2011/07/17
0.864 916 02
ON
2011/07/19
2011/07/20
OFF
2011/07/22
2011/07/25
0.866 313 94
ON
2011/07/25
2011/07/27
OFF
2011/07/27
2011/07/28
0.866 314 37
ON
2011/07/28
2011/07/29
OFF
2011/07/29
2011/08/03
SCAN VI: 0.969 2 amagat, 485.25 W, 2.859 258 4 GHz, QL = 12268.4

A.2 RUN SUMMARY

92
Table A.1: Run table.
Magnetic field(T)
0.865 485 28

RF state RUN start date
RUN finish date
(ON/OFF) (yyyy/mm/dd)(yyyy/mm/dd)
ON
2011/11/07
2011/11/09
OFF
2011/11/09
2011/11/10
0.868 125 57
ON
2011/11/10
2011/11/11
OFF
2011/11/11
2011/11/14
0.864 828 73
ON
2011/11/14
2011/11/16
OFF
2011/11/16
2011/11/17
0.867 784 54
ON
2011/11/17
2011/11/18
OFF
2011/11/18
2011/11/21
0.866 458 68
ON
2011/11/21
2011/11/22
OFF
2011/11/22
2011/12/05
SCAN VII: 1.352 5 amagat, 493.65 W, 2.857 666 6 GHz, QL = 11368.4
0.866 460 79ON
2011/12/20
2011/12/21
OFF
2011/12/22
2012/01/10
0.865 359 53
ON
2012/01/10
2012/01/11
OFF
2012/01/13
2012/01/16
0.867 101 34
ON
2012/01/16
2012/01/18
OFF
2012/01/20
2012/01/23
0.864 485 19
ON
2012/01/23
2012/01/25
OFF
2012/01/27
2012/01/30
0.867 890 8
ON
2012/01/30
2012/02/01
OFF
2012/02/03
2012/02/06
SCAN VIII: 1.365 8 amagat, 319.51 W, 2.857 677 6 GHz, QL = 11368.5
0.866 460 24ON
2011/12/21
2011/12/22
OFF
2011/12/22
2012/01/10
0.865 360 52
ON
2012/01/11
2012/01/13
OFF
2012/01/13
2012/01/16
0.867 101 71
ON
2012/01/18
2012/01/20
OFF
2012/01/20
2012/01/23
0.864 485 48
ON
2012/01/25
2012/01/27
OFF
2012/01/27
2012/01/30
0.867 890 24ON2012/02/01
2012/02/03
OFF
2012/02/03
2012/02/06
SCAN IX: 1.192 5 amagat, 466.39 W, 2.858 301 9 GHz, QL = 11582.9
0.864 573 50
ON
2012/02/15
2012/02/17
OFF
2012/02/17
2012/02/23
0.867 085 29
ON
2012/02/23
2012/02/24
OFF
2012/02/24
2012/02/27
0.865 440 03
ON
2012/02/27
2012/02/28
OFF
2012/02/28
2012/02/29
0.866 281 08
ON
2012/02/29
2012/03/02
OFF
2012/03/02
2012/03/05
SCAN X: 0.128 7 amagat, 117.91 W, 2.862 585 0 GHz, QL = 15355.2
0.867 481 3
ON
2012/05/15
2012/05/18
OFF
2012/05/18
2012/05/21
0.868 909 4
ON
2012/05/21
2012/05/25

A.2 RUN SUMMARY

93
Table A.1: Run table.

Magnetic field(T)

RF state RUN start date
RUN finish date
(ON/OFF) (yyyy/mm/dd)(yyyy/mm/dd)
OFF
2012/05/25
2012/05/28
0.865 497 14
ON
2012/05/28
2012/06/01
OFF
2012/06/01
2012/06/04
0.868 112 72
ON
2012/06/04
2012/06/08
OFF
2012/06/08
2012/06/11
0.865 944 34
ON
2012/06/11
2012/06/15
OFF
2012/06/15
2012/06/18
SCAN XI: 0.133 3 amagat, 126.48 W, 2.862 513 8 GHz, QL = 15310.3
0.865 942 42
ON
2012/06/18
2012/06/22
OFF
2012/06/22
2012/06/25
0.868 043 11
ON
2012/06/25
2012/06/29
OFF
2012/06/29
2012/07/02
0.865 385 04
ON
2012/07/02
2012/07/05
OFF
2012/07/06
2012/07/09
0.868 876 92
ON
2012/07/09
2012/07/13
OFF
2012/07/13
2012/07/17
0.867 215 37
ON
2012/07/17
2012/07/20
OFF
2012/07/20
2012/07/23
0.866 294 41
ON
2012/07/23
2012/07/27
OFF
2012/07/27
2012/07/30
0.868 436 81
ON
2012/07/30
2012/08/03
OFF
2012/08/03
2012/08/06

Appendix B

Ps Thermalization
Measurement
In this appendix, an experimental apparatus and results of our Ps thermalization
measurement using ‘pick-oﬀ technique’ are described. They are only roughly
explained here because this measurement is quite similar to the older ‘pick-oﬀ
technique’ thermalization measurements used in o-Ps lifetime measurement [80,81, 82].

B.1

Experimental setup

B.1.1

Overview of the apparatus

Figures B.1 and B.2 show the overview of the experimental setup. The gaschamber which is similar to the cavity used in Ps-HFS measurement was used.
The chamber was filled with the same i-C4 H10 gas as the Ps-HFS measurement,
and Ps were formed inside the chamber. Two separate measurement with andwithout aerogel were performed. With aerogel setup, most of positrons emitted
to aerogel were stopped so that Ps were formed very eﬃciently. Without aerogel,
only some fraction of positrons could be stopped with gas so that the Ps formation eﬃciency was worse than with-aerogel setup. Without-aerogel setup wasused to understand whether or not aerogel did aﬀect the result of thermalization parameters of i-C4 H10 . A β-tagging system similar to that used in Ps-HFS
measurement was used to measure timing information. For γ-ray detection,a Ge detector (ORTEC PopTop GEM40P-PLUS-S, Serial No.=45-TP22020A)
was used.

B.1.2

Electronics circuit

Electronics circuit used for data acquisition (DAQ) is shown in Figs. B.3–B.10.
Detail explanation is not given here because it is essentially the same as the
setup of the previous o-Ps lifetime measurement [82].
94B.1 Ps THERMALIZATION MEASUREMENT

95

Gas inlet, Vacuum pumpPressure sensor

Gas chamber
22

Na source

Plastic scintillator
Light guide
AerogelGe crystal

Filled with i-C4H10 gas

Ge detector
0

5

10 cm

Figure B.1: Side view of the setup of Ps thermalization measurement.

B.1.3

Monte Carlo simulation

Monte Carlo simulations were used to estimate the energy spectra of 3γ decayof Ps. The same Geant4 package as used in Ps-HFS measurement was used.
Most of the experimental setup (materials and geometry) are carefully input.
Figures B.11 and B.12 show the geometry input in the simulation.

B.1 Ps THERMALIZATION MEASUREMENT

96

2-inch PMT

2-inch PMT

Gas chamber
22

Na source

Plastic
scintillator

Light guide
AerogelFilled with i-C4H10 gas

0

5

10 cm

Figure B.2: Front view of the setup of Ps thermalization measurement.

B.1 Ps THERMALIZATION MEASUREMENT

Figure B.3: DAQ of Ps thermalization (Plastic scintillator 1).

97

98

B.1 Ps THERMALIZATION MEASUREMENT

Figure B.4: DAQ of Ps thermalization (Plastic scintillator 2).

B.1 Ps THERMALIZATION MEASUREMENT

Figure B.5: DAQ of Ps thermalization (Trigger).

99

100

B.1 Ps THERMALIZATION MEASUREMENT

Figure B.6: DAQ of Ps thermalization (Ge detector).

B.1 Ps THERMALIZATION MEASUREMENT

Figure B.7: DAQ of Ps thermalization (Ge Main Gate).

101

102

B.1 Ps THERMALIZATION MEASUREMENT

Figure B.8: DAQ of Ps thermalization (Ge Transistor-Reset-Pulse (TRP) veto).

B.1 Ps THERMALIZATION MEASUREMENT

Figure B.9: DAQ of Ps thermalization (Ge ADCs’ gates).

103

104

B.1 Ps THERMALIZATION MEASUREMENT

Figure B.10: DAQ of Ps thermalization (Clock system).

B.1 Ps THERMALIZATION MEASUREMENT

105

Figure B.11: Side view of the geometry in the MC simulation (thermalization
measurement)

Figure B.12: Bird’s-eye view of the geometry in the MC simulation (thermalization measurement)B.2 Ps THERMALIZATION MEASUREMENT

2γ / 3γ RATIO

106

0.09
0.08
0.92atm
0.37atm
0.16atm (Run1)
0.16atm (Run2)
0.00atm (Run1)
0.00atm (Run2)
0.00atm (Run3)

0.07
0.06
0.05
0.04
0.03
0

50

100

150

200

250

300

350 400
TIME (ns)Figure B.13: Pick-oﬀ ratio as a function of time (with aerogel). The dots witherror bars are the data, and the solid lines indicate the best-fit function.

B.2

Results

Proceeding an analysis similar to that of Ref. [82],Γpick (t)/Γo−Ps = (2γ annihilation rate)/(3γ annihilation rate) as a function oftime is obtained using γ-ray energy spectra.
Figures B.13 and B.14 show
the pick-oﬀ ratio of with- and without-aerogel measurement, respectively, as afunction of time. The data are fitted by the following function:
Γpick (t)0.6
= (Caero + Ciso n) [v(t)] ,
Γo−Ps

(B.1)

where Caero is a constant associated with the pick-oﬀ annihilation with aerogel,
Ciso is a constant associated with the pick-oﬀ annihilation with i-C4 H10 gas, andn is the gas density. v(t) is determined using the following diﬀerential equation
similar to Eq. (2.1):
(
)[ √
(
)β ]
√
3
8
Eav (t)dEav (t)2 2σm n= − 2mPs Eav (t) Eav (t) − kB T
+α
,
dt
2
3 3π M
kB T(B.2)
where α and β are constants associated with Ps thermalization in aerogel. Caeroand α are zero if there is no aerogel.
In our analysis, the thermalizationpa()
2
rameters from DBS measurement E0 = 3.1+1.0
eV,
σ
=
146
±
11
Å
are
used
m
−0.7
from t = 0 to the time at which the kinetic energy of o-Ps reaches 0.17 eV, andthen the σm is changed to free parameter.
The data are well-fitted by Eq. (B.1). σm = 46.0 ± 4.5 Å2 for with-aerogel2
measurement and σm = 50.7+8.8−7.9 Å for without-aerogel measurement are ob-

2γ / 3γ RATIO

B.2 Ps THERMALIZATION MEASUREMENT

107

0.1
0.09
0.08
0.07
0.06

0.657 amagat

0.05
0.04
0.03

0.277 amagat

0.02

0.138 amagat

0.01
0
0

100

200

300

400

500

600
700
TIME (ns)Figure B.14: Pick-oﬀ ratio as a function of time (without aerogel). The dotswith error bars are the data, and the solid lines indicate the best-fit function.

tained. These results are consistent with each other, and the combined resultof σm = 47.2 ± 3.9 Å2 is obtained for o-Ps below 0.17 eV.
In Eq. (B.1), it is assumed that the pick-oﬀ annihilation rate with aerogel is0.6
also proportional to [v(t)] . However, there is no evidence for this assumption.
To estimate a systematic error of this assumption, a fitting in which the pick-oﬀ
annihilation rate with aerogel is assumed to be proportional to v(t) was performed. The result was changed by +5.4 Å2 , which is considered as a systematic
error. The final result of the Ps thermalization measurement is obtained asσm = 47.2 ± 6.7 Å2 ,which is used in the analysis of the ∆HFS measurement.

(B.3)

Appendix C

Final Fitting Parameters

D1

In this appendix, the final fitting parameters of Eq. 5.1 and associated equations: D1 (n), D2 (n), ϵ(n), and BRF are shown as a function of gas density in
Figs. C.1, C.2, C.3, and C.4, respectively.

1.1
1
0.9
0.8
0.7
0.6
0.2

0.4

0.6

0.8
1
1.2
1.4
GAS DENSITY (amagat)

Figure C.1: D1 (n)

108

D2

C.0 FINAL FITTING PARAMETERS

0.4
0.3
0.2
0.1
0
-0.1
0.2

0.4

0.6

0.8
1
1.2
1.4
GAS DENSITY (amagat)∈

Figure C.2: D2 (n)

10
9
8
7
6
5

0.2

0.4

0.6

0.8
1
1.2
1.4
GAS DENSITY (amagat)

Figure C.3: ϵ(n)109C.0 FINAL FITTING PARAMETERSBRF (G)

110

19
18
17
16
15
14
13
12
11
0.2

0.4

0.6

0.8
1
1.2
1.4
GAS DENSITY (amagat)

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