Direct Measurement of theHyperfine Transition of Positronium
using High Power Sub-THz Radiation
Ph. D. thesis

Takayuki Yamazaki

Department of Physics
Facalty of Science, University of Tokyo
December 2011

Abstract
This thesis reports the ﬁrst measurement of the direct hyperﬁne transition of theground state of positronium using high power sub-THz radiation. We developed
a new optical system to accumulate sub-THz radiation of 11 kW (peak intensityof I = 8.3 × 107 W/m2 , peak energy density of  = 0.28 J/m3 ) in a Fabry-Pérot
resonant cavity and to cause hyperﬁne transition of the ground state of positronium.
We have observed clear transition signals at 5.4 σ level after 4.3 days of on-resonancedata taking. The transition probability is consistent with the QED calculation andno excess was observed in oﬀ-resonance data.

i

ii

Contents
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Properties of Positronium . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Hyperﬁne Structure of the Ground State of Positronium . . . . . . .
1.4 Previous Measurements of Ps-HFS . . . . . . . . . . . . . . . . . . .
1.5 Hyperﬁne Transition of the Ground State of Positronium . . . . . .

1
1
3
4
6
8

2 Experiment
2.1 Method of Our Experiment . . . . . . . . . . . . . . . . . . . . . . .
2.2 Optical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Gyrotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Mode Converter . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Fabry-Pérot Resonant Cavity . . . . . . . . . . . . . . . . . .
2.2.4 Power Estimation . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Positronium Assembly and γ-ray Detectors . . . . . . . . . . . . . .
2.3.1 Positronium Formation Assembly . . . . . . . . . . . . . . . .
2.3.2 Characteristics of Mixed Gas . . . . . . . . . . . . . . . . . .
2.3.3 γ-ray Detectors . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Electronics and Data Acquisition . . . . . . . . . . . . . . . . . . . .
2.5 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . .

15
15
15
16
22
31
39
44
48
49
51
53
54
58

3 Analysis
3.1 Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Calibration and Basic Cut . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Energy Spectrum of the Plastic Scintillator . . . . . . . . . .
3.2.2 Energy Spectrum of the LaBr3 (Ce) Scintillator . . . . . . . .
3.2.3 Time Walk Correction of the LaBr3 (Ce) Scintillator . . . . .
3.2.4 Time Walk Correction of the Plastic Scintillator . . . . . . .
3.2.5 Oﬄine Trigger Cuts . . . . . . . . . . . . . . . . . . . . . . .
3.3 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Signatures of the Transition Signal . . . . . . . . . . . . . . .
3.3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65
65
66
67
67
68
69
69
72
74
74

iii

3.4

3.5

3.3.3 Delayed Coincidence . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.4 Accidental Rejection . . . . . . . . . . . . . . . . . . . . . . . 76
3.3.5 γ-ray Energy Cut . . . . . . . . . . . . . . . . . . . . . . . . . 76
Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.4.1 Energy Scale and Energy Resolution . . . . . . . . . . . . . . 78
3.4.2 Ps Formation Probability and Pick-oﬀ Annihilation Probability 80
3.4.3 Accidental Rejection Eﬃciency . . . . . . . . . . . . . . . . . 80
3.4.4 Background Normalization . . . . . . . . . . . . . . . . . . . 82
3.4.5 Summary of the Systematic Errors . . . . . . . . . . . . . . . 82
Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4 Discussion
4.1 Various Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Accidental Rejection . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Energy Selection . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Time Spectrum of the Transition Signal . . . . . . . . . . . .
4.2 Comparison with QED Calculation . . . . . . . . . . . . . . . . . . .
4.3 Future Prospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85
85
85
85
86
86
87

5 Conclusion

95

Acknowledgements

97

iv

List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8

Historical plot of the Ps-HFS value. . . . . . . . . . . . . . . . . . . 2
The decay modes of Ps. . . . . . . . . . . . . . . . . . . . . . . . . . 4
Feynman diagram of positronium. . . . . . . . . . . . . . . . . . . . 5
Zeeman splitting of Ps. . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Plot of the X-electron coupling constant which can explain the PsHFS discrepancy vs. mass of X. . . . . . . . . . . . . . . . . . . . . 8
Level diagram of the hyperﬁne structure of positronium . . . . . . . 9
Lineshape functions without Doppler broadening (black) and with
Doppler broadening of 7 % (red). . . . . . . . . . . . . . . . . . . . . 11
Fraction of 2γ decays per o-Ps vs. radiation intensity. . . . . . . . . 12

2.1 A schematic of a gyrotron. . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 A schematic of an annular electron beam. . . . . . . . . . . . . . . . 19
2.3 A picture of the Gyrotron FU CW V. . . . . . . . . . . . . . . . . . 20
2.4 (left) TE03 mode, (right) TE02 mode. . . . . . . . . . . . . . . . . . 21
2.5 Space distribution of the gyrotron output (TE03 mode). . . . . . . . 22
2.6 A Gaussian beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 The schematic view and the picture of the mode converter. . . . . . 24
2.8 Cross-sectional view of the step-cut waveguide and the Vlasov mirror 25
2.9 Side view of the step-cut waveguide and the Vlasov mirror . . . . . . 26
2.10 Calculated intensity distributions at various distances from the Vlasov
mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.11 Measured intensity distributions at various distances from the Vlasov
mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.12 Polarization of the converted beam. . . . . . . . . . . . . . . . . . . . 30
2.13 Beam shape after the mode converter. . . . . . . . . . . . . . . . . . 30
2.14 A schematic of a Fabry-Pérot resonant cavity. . . . . . . . . . . . . . 32
2.15 Basic performance of Fabry-Pérot resonant cavity (calculation). . . . 34
2.16 The beam shape in the Fabry-Pérot cavity. . . . . . . . . . . . . . . 35
2.17 Pictures of the gold mesh mirror. . . . . . . . . . . . . . . . . . . . . 36
2.18 The reﬂectance (black) and the transmittance (red) of the Au plain
mesh mirror simulated using CST MICROWAVE STUDIO. . . . . . 37
2.19 Principal of operation of an SIDMTM (Smooth Impact Drive Mechanism)
actuator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38v

2.20 Structural drawing of the X-axis stage and its controller. . . . . . . . 38
2.21 Pyroelectric detectors (Spectrum Detector Inc. SPH-49). . . . . . . . 39
2.22 The accumulated power (left) and the reﬂected power (right) of the
Fabry-Pérot resonant cavity measured while changing cavity length. 40
2.23 Temperature of water measured while the water is exposed to theGaussian beam during 120 s. . . . . . . . . . . . . . . . . . . . . . . 41
2.24 The transmitted power of the Cu concave mirror Vtr normalized by
the input beam power Vin while changing the cavity length L by λ/2. 42
2.25 A picture of the PVC sheet taken by IR camera to estimate Phole /Pin . 43
2.26 Schematic diagram of the feedback stabilization system. . . . . . . . 45
2.27 Time variation of the gyrotron output power. . . . . . . . . . . . . . 46
2.28 Stabilization of the input beam power by controlling heater AC voltage. 47
2.29 Schematic of the rescan procedure. . . . . . . . . . . . . . . . . . . . 47
2.30 Control of the cavity length to keep resonance of the Fabry-Pérot
cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.31 A schematic and a picture of the positronium assembly chamber andγ-ray detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.32 A schematic of the positronium formation assembly . . . . . . . . . . 50
2.33 The decay scheme of the 22 Na isotope . . . . . . . . . . . . . . . . . 50
2.34 The lifetime of the slow positron as a function of the partial pressureof isobutane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.35 Absorption of sub-THz radiation by isobutane. . . . . . . . . . . . . 53
2.36 Four γ-ray detectors are arranged to make four back-to-back pairs. . 54
2.37 The schematic view of the whole DAQ system. . . . . . . . . . . . . 55
2.38 Schematic diagram of electronics for plastic scintillator system. . . . 57
2.39 Schematic diagram of electronics for γ-ray detector system. . . . . . 59
2.40 Schematic diagram of electronics for trigger system. . . . . . . . . . 60
2.41 Geometry of the Monte Carlo simulation. . . . . . . . . . . . . . . . 63
3.1
3.2
3.3
3.4

3.5

3.6
3.7
3.8
3.9

The timing diagram of the gyrotron output. . . . . . . . . . . . . . .
Plastic scintillator energy with long-gate vs that with short-gate . .
Energy spectrum of one of the LaBr3 (Ce) scintillators (la-0) . . . . .
The left and the right ﬁgure show the scatter plot of timing vs energy
deposited on one of the LaBr3 (Ce) scintillators (la-0) before and after
time walk correction, respectively. . . . . . . . . . . . . . . . . . . .
The left and the right ﬁgure show the scatter plot of timing vs energy deposited on the plastic scintillator before and after time walkcorrection, respectively. . . . . . . . . . . . . . . . . . . . . . . . . .
ps-1 energy (short-gate) vs. ps-0 energy (short-gate) . . . . . . . . .
Time diﬀerence between the ps-1 signal and the ps-0 signal . . . . .
Time diﬀerence between the back-to-back signals from the LaBr3 (Ce)scintillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time diﬀerence between the plastic scintillator and the coincidence
signal of the LaBr3 (Ce) scintillators. . . . . . . . . . . . . . . . . . .
vi

66
67
68

69

70
71
72
73
75

3.10 Time chart of the normal event (upper) and the accidental event(lower). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.11 Energy diﬀerence between long-gate and short-gate of the ps-0 signal. 77
3.12 Time spectrum before and after accidental rejection. . . . . . . . . . 78
3.13 Energy spectrum of on-resonance data (RUN I) when the delayedcoincidence and the accidental rejection are applied and a 511 keV
γ ray is observed at the opposite side of the LaBr3 (Ce) scintillator. . 79
3.14 The diﬀerence of the slow positron annihilation rate. . . . . . . . . . 81
3.15 The diﬀerence of the Ps formation probabilities. . . . . . . . . . . . . 81
3.16 The fraction of the transition signals vs. the accumulated power in
the Fabry-Pérot resonant cavity. . . . . . . . . . . . . . . . . . . . . 83
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9

Fitted time spectrum of the transition signals. . . . . . . . . . . . . .
The fraction of the transition signals is consistent with QED calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fitted decay rate (RUN I). . . . . . . . . . . . . . . . . . . . . . . .
A picture of a melted mesh mirror. . . . . . . . . . . . . . . . . . . .
Interference between input power and reﬂected power around resonance peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A schematic of a grating ring cavity. . . . . . . . . . . . . . . . . . .
Transition curve assuming I = 108 W/m2 . . . . . . . . . . . . . . . .
Simulation of Ps-HFS measurement assuming statistical accuracy of
10 % and power accuracy of 10 %. . . . . . . . . . . . . . . . . . . .
A schematic of an experimental setup with a slow positron beam. . .

vii

87
88
89
90
90
91
92
92
93

viii

List of Tables
1.1

The classiﬁcation of Ps according to its quantum numbers.

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8

Parameters and operational conditions of Gyrotron FU CW V . . . 19
Operating modes of Gyrotron FU CW V . . . . . . . . . . . . . . . . 19
Parameters of the step-cut waveguide and the Vlasov mirror . . . . . 27
Properties of NE-102 (plastic) scintillator. . . . . . . . . . . . . . . . 50
Properties of the PMT (HAMAMATSU R5924-70). . . . . . . . . . . 51
Properties of LaBr3 (Ce) scintillator . . . . . . . . . . . . . . . . . . . 54
Quantities recorded at the main trigger timing. . . . . . . . . . . . . 61
Quantities recorded in synchronization with the gyrotron output pulse. 61

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

Properties of the RUNs . . . . . . . . . . . . . . . . . . . . . . . . .
Energy resolutions of the LaBr3 (Ce) scintillators . . . . . . . . . . .
Summary of the event rates until the oﬄine trigger cut. . . . . . . .
Event rates after the delayed coincidence. . . . . . . . . . . . . . . .
Event rates after the accidental rejection. . . . . . . . . . . . . . . .
The positions and the widths of the 511 keV peaks of “beam ON”
events and “beam OFF” events. . . . . . . . . . . . . . . . . . . . . .
Summary of the systematic errors. . . . . . . . . . . . . . . . . . . .
Results of the four RUNs. . . . . . . . . . . . . . . . . . . . . . . . .

4.1
4.2

Check dependency on the upper limit of the accidental rejection cut. 85
Check dependency on the lower limit of the energy selection. . . . . 86ix

. . . . .

3

66
68
73
76
76
80
82
83

x

Chapter 1

Introduction
This thesis reports the ﬁrst measurement of the hyperﬁne transition of the groundstate of positronium using high power sub-THz radiation. In the ﬁrst section,
we describe the physics motivation of this experiment at the beginning. Next,
we introduce the properties of positronium. Then the theory and the previous
measurements of the hyperﬁne structure of the ground state of positronium aresummarized. Finally, the hyperﬁne transition of the ground state of positronium
are described.

1.1

Motivation

Positronium (Ps), a bound state of an electron and a positron, is a purely leptonic
system and is a good target to study Quantum Electrodynamics (QED). The triplet(13 S1 ) state of Ps is called ortho-positronium (o-Ps) and mainly decays into three
γ rays with long lifetime of 142 ns. On the other hand, the singlet (11 S0 ) state of
Ps is called para-positronium (p-Ps) and mainly decays into two γ rays promptly(lifetime is 125 ps). The energy level of the ground state o-Ps is higher than that of
the ground state p-Ps because of the spin-spin interaction. The diﬀerence is called
the hyperﬁne structure of the ground state of positronium (Ps-HFS). The Ps-HFS
is signiﬁcantly large (about 203 GHz) compared to the hyperﬁne structure of thehydrogen atom (about 1.4 GHz).
Precise measurement of Ps-HFS gives the direct information on QED, especiallyin the bound state. The precise measurements have been performed in 1970’s and1980’s, whose results are shown in Fig. 1.1 with the theoretical value. All previous
measurements of the Ps-HFS employed static magnetic ﬁeld (about 1 T) and thePs-HFS has been measured indirectly using Zeeman splitting (about 3 GHz). These
results are consistent with each other, and the combined value of the most accurate
two results [1, 2] is 203.388 65(67) GHz (3.3 ppm). This combined value is shownwith the green band.
New method to calculate the higher order corrections up to O(α3 log α−1 ) forthe bound state is established in 2000 [3]. The QED prediction is 203.391 69(41)12

1.1 Introduction
mg

Experimental average

Theory (2000)

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)

203.38203.385203.39ΔHFS (GHz)

Figure 1.1: Historical plot of the Ps-HFS value. Points with error bars show theexperimental results with references. The green and red bands show the averageof the measured values (the average of the latest two results) and the theoretical
calculation up to O(α3 log α−1 ), respectively.

GHz (2 ppm) shown with the red band. There is a large discrepancy (3.9 σ, 15 ppm)
between the measured and the theoretical value. It is very important to measure
the Ps-HFS again with a method totally diﬀerent from the previous experiments
since non-uniformity of the static magnetic ﬁeld is the most signiﬁcant systematic
error in the previous experiments.
Direct measurement of the Ps-HFS is free from systematic uncertainty of thestatic magnetic ﬁeld, which is one of the possible common uncertainties in theprevious experiments. The direct measurement has never been performed because
the rate of spontaneous emission (or Einstein’s A coeﬃcient A = 3.37 × 10−8[s−1 ]) is 14 orders of magnitude smaller than the decay rate of o-Ps (λo−Ps =7.0401(7) × 106 [s−1 ] [4]). High power sub-THz radiation is necessary to cause
enough stimulated emission but there was no high power radiation source so far,therefore even the hyperﬁne transition itself has not yet been observed. However,
the recent development of the gyrotron, which is a novel high power radiation source
for sub-THz to THz region, changes the situation. Its output is monochromatic,
the power is high (100 W - 1 kW), and it can operate in continuous wave (CW) or
long pulse wave. Although the direct output power of the gyrotron is not enoughhigh to cause suﬃcient amount of the hyperﬁne transition of the ground state ofPs, we can achieve it (∼ 10 kW) by accumulating the output radiation in an opticalresonator (Fabry-Pérot resonant cavity). The ﬁrst target of our experiment is toobserve the hyperﬁne transition for the ﬁrst time using the new optical system,3

1.2 Introduction
which is described in this thesis.

1.2

Properties of Positronium

Positronium (Ps), the bound state of an electron (e− ) and a positron (e+ ), is thelightest hydrogen-like atom bound by the electromagnetic interaction. Since Ps
is a purely leptonic system and eﬀectively free from hadronic and weak interaction eﬀects, precise measurements of decay rate and energy levels give us direct
information on bound state QED.
Positronium consists of two fermions as hydrogen atom, and its wave functions
are classiﬁed according to the principal quantum number n, a sum of the orbital
angular momentum l and the spin angular momentum s, and its projection on aquantization axis m. Basically, the energy levels of positronium are given by
En = −me c 2 α 2
,
4n2

(1.1)

where me is the electron mass, c is the speed of light, and α = e2 /(4π0 c) is theﬁne structure constant. The ﬁne structure due to the spin-orbit interaction andthe hyperﬁne structure due to the spin-spin interaction make shifts from the above
equation. E0 = −6.8 eV and there is no ﬁne structure in the case of the groundstate (n = 1) because the orbital angular momentum l = 0. However, the energy
level of the ground state is split by the total spin s of positronium, which is called
hyperﬁne structure. The detail of the hyperﬁne structure is described in the next
section. The singlet state with antiparallel spins (total spin s = 0, its projection
m = 0) is called para-positronium (p-Ps), and the triplet state with parallel spins(s = 1, m = 0, ±1) is called ortho-positronium (o-Ps).
nametotal spin s

projection mCPdecay mode

lifetime

p-Ps
o-Ps

0
1

0
0, ±1

+
−

−
−

2γ(, 4γ, 6γ, ...)
3γ(, 5γ, 7γ, ...)

125 ps142 nsTable 1.1: The classiﬁcation of Ps according to its quantum numbers.
A fermion-antifermion system with orbital angular momentum l and spin angular momentum s has a C-parity of (−1)l+s under a charge-conjugation transformation and a parity (P ) of (−1)l+1 . Thus o-Ps and p-Ps have odd and even C
parities respectively. Due to the C-invariance of the electromagnetic interaction,
this diﬀerence plays a crucial role on the decay modes. In fact, o-Ps decays only
into odd γ’s and p-Ps decays only into even γ’s since the system consisting of nphotons has a C parity of (−1)n . In addition, single photon decay in vacuum isprohibited by energy-momentum conservation. Then, the decay modes of o-Ps andp-Ps are as follows.

4

1.3 Introductiono − Ps → nγ, n = 3, 5, 7, 9, · · ·(1.2)

p − Ps → nγ, n = 2, 4, 6, 8, · · ·(1.3)

As a number of photons increases, the width of the decay mode decreases by
a factor of α. Furthermore, phase space of the ﬁnal state is getting smaller as thenumber of the ﬁnal state photons increases. Thus, decay modes into many photons
are highly suppressed. As a result, p-Ps mainly decays into two γ rays and o-Ps
mainly decays into three γ rays. The Feynman diagrams of 2γ decay of p-Ps and3γ decay of o-Ps are shown in Fig. 1.2. The experimental values of the decay rateof p-Ps and o-Ps are
λp−Ps = 7.990 9(17) × 109 s−1 [5],λo−Ps = 7.040 1(7) × 10 s
6
e

(1.4)

[4].

(1.5)

   -−1

  



e

-




e+



e+



Figure 1.2: The decay modes of Ps.

1.3

Hyperﬁne Structure of the Ground State of Positronium

Since positronium is a particle-antiparticle system, not only scattering process (tchannel) but also annihilation process (s-channel) contribute to the energy levels asshown in Fig. 1.3. The total Hamiltonian for positronium to O(1/c2 ) is expressed

5

1.3 Introduction



























Figure 1.3: Feynman diagram of positronium. The left diagram is scattering process(t-channel) and the right diagram shows annihilation process (s-channel)
asp2e2+ V1 + V2 + V3 + V4−
me 4π0 r


μ2B
p4
e2
r · (r · p)p2p +δ(r) −
V1 = − 3 2 +4me c
 0 c28π0 m2e c2 rr2
3μ2B l · sV2 =2π0 c2 r3




3μ2B
μ2B 4 2s2
(s · r)(s · r)s − 2 δ(r)
− 3 +
V3 =
2π0 c2
r5
3r
 0 c2 3
μ2
V4 = B2 s2 δ(r),
0 c
H=

(1.6)
(1.7)
(1.8)
(1.9)
(1.10)

where μB = e/(2me ) is the Bohr magneton, 0 is the vacuum permittivity. V1 isorbital correction terms. V2 is the spin-orbit interaction, which causes ﬁne structure
splitting. V3 and V4 are the spin-spin interactions of t-channel and s-channel,
respectively. The spin-spin interaction causes hyperﬁne structure splitting.
Now we consider the ground state (n = 1, l = 0) of positronium. There is no
ﬁne structure since the orbital angular momentum l is zero. The energy diﬀerence
between the energies of the ground states of o-Ps and p-Ps (Ps-HFS) derives from
V3 and V4 . In the lowest order of α, Ps-HFS is
1
1
7
2 4
2 4
Δth
me c 2 α 4
0 = me c α + me c α =3412= 0.85 meV = 204 GHz.

(1.11)(1.12)

The ﬁrst term is the t-channel contribution and the second term is the s-channel
contribution.

6

1.4 Introduction

Higher order contribution is calculated in the framework of dimensionally regularized nonrelativistic QED.



5
1
α 32 6+ln2+ α2 ln(1.13)
1−
Δth = Δth0
π 21 714α α 2  1367 5197
159
6 221 22−π ++π ln 2 −
ζ(3)
+
π
378
2016
7
84
56

 3
62 68
α
1
3 α3 2 1
ln
+
−
ln 2
ln + · · ·
−
2 π
α
15
7
π
α
= 203.391 69(41) GHz [3].
(1.14)

1.4

Previous Measurements of Ps-HFS

Ps-HFS has been measured precisely in 1970’s and 1980’s as shown in Fig. 1.1, but
all of the previous measurements are indirect measurements using Zeeman eﬀect,because it was impossible to prepare high power (∼ 10 kW) sub-THz radiation tocause direct hyperﬁne transition at that time.
In a static magnetic ﬁeld, the o-Ps state with m = 0 and the p-Ps state mix, andthe resultant states have diﬀerent energies (Zeeman eﬀect) as shown in Fig. 1.4. On
the other hand, the o-Ps states with m = ±1 are not aﬀected at all. The mixture
depends on the strength of the static magnetic ﬁeld, and the energy diﬀerence
between the perturbed o-Ps state (|+) and the unperturbed o-Ps states (|1, +1,
|1, −1) is related to the Ps-HFS by the following equation,
ΔHFS 
1 + x2 − 1
22g  μB B
x=
,
hΔHFS

Δmix =(1.15)(1.16)

where g  is the bound state electron g-factor in Ps, B is the strength of the static
magnetic ﬁeld, and h is the Planck constant. Therefore ΔHFS can be obtained fromΔmix and B. Δmix becomes about 3 GHz when we apply a static magnetic ﬁeld of
about 1 T.
In the previous experiment, the strong static magnetic ﬁeld was produced bya resistive electromagnet and the microwave of about 500 W was produced by amagnetron [1] and a klystron [2]. Zeeman transition from the unperturbed o-Ps
states (|1, +1, |1, −1) to the perturbed o-Ps state (|+) occurs when the microwave
frequency matches Δmix . The number of the 2γ decays increases on resonancesince the |+ state mainly decays into 2γ rays. The transition curve is obtainedby measuring the increase of 2γ decays while changing the strength of the static
magnetic ﬁeld whereas the microwave frequency is ﬁxed. The combined value ofthe most precise two independent experiments [1, 2] isΔexp. = 203.388 65(67) GHz.

(1.17)

7

1.4 Introduction

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






Figure 1.4: All previous measurements of the Ps-HFS employed static magnetic
ﬁeld (about 1T) and the Ps-HFS has been measured indirectly using Zeeman splitting (about 3GHz).
There is a 3.9 σ (15ppm) discrepancy between the experimental value and thetheoretical calculation. Veriﬁcation of the discrepancy with a diﬀerent method isnecessary. Possible reasons of the discrepancy are as follows:
1. common systematic error in the previous experiments (e.g. non-uniformityof the static magnetic ﬁeld, slow thermalization of Ps)
2. new physics beyond the Standard Model
3. miscalculation of the theoretical value
The non-uniformity of the static magnetic ﬁeld is the most signiﬁcant systematic
error in the previous experiments. It is diﬃcult to prepare uniform magnetic ﬁeld
in the region where Ps is formed, whereas the uncertainty of the magnetic ﬁeld
strength contributes to the Ps-HFS directly by Eq. (1.15). The other possible
common systematic uncertainty is the unthermalized Ps contribution, which isindicated in [6, 7]. The previous experiments used all positroniums although thePs thermalization process is slow and the contribution from unthermalized o-Ps
gives rise to an underestimation of the material eﬀect, therefore the extrapolation
to Ps-HFS in vacuum can be wrong.
On the other hand, if the discrepancy is real, it indicates the existence of an
unknown particle [8]. Assume that there exists a neutral particle X whose mass is
mX and the coupling constant between X and electron is gXee . Figure 1.5 shows

8

1.5 Introduction

a plot of gXee vs. mX which can explain the discrepancy between the measured
and the theoretical value of Ps-HFS. Excluded regions from positronium decay [9],reactor axion [10], electron beam dump [11, 12, 13], and electron g-2 [14, 15, 16]
are also shown. If the discrepancy between the measured and the theoretical valueof Ps-HFS is real, it indicates the existence of a pseudoscalar, a vector, or an axial
vector particle with a mass of ∼ 1 MeV or an axial vector particle with a mass of< 1 keV.








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











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











Figure 1.5: Plot of the X-electron coupling constant which can explain the PsHFS discrepancy vs. mass of X. The black line shows the case that X is a
pseudoscalar or a vector particle. The red line corresponds to an axial vector
particle. Excluded regions from positronium decay [9], reactor axion [10], electron
beam dump [11, 12, 13], and electron g-2 [14, 15, 16] are also shown.

1.5

Hyperﬁne Transition of the Ground State of Positronium

Figure 1.6 is a level diagram for the hyperﬁne transition of the ground state of
positronium with a linearly polarized radiation of energy density ρ(ω) per unit
angular frequency. Only |1, 0 and |0, 0 states are aﬀected by the linearly polarized radiation, which has a spin |1, 0, due to the conservation law of the angularmomentum. If we denote the population of |i, j state of Ps as Ni,j , then the9

1.5 Introduction 











  

















  

Figure 1.6: Level diagram of the hyperﬁne structure of positronium
rate equations for the hyperﬁne transition of the ground state of positronium areexpressed as
dN0,0 (ω, t)= −3B(ω)ρ(ω)N0,0 (ω, t) + 3B(ω)ρ(ω)N1,0 (ω, t)dt+ A(N1,0 (ω, t) + N1,+1 (t) + N1,−1 (t)) − λp−Ps N0,0 (ω, t)
dN1,0 (ω, t)= 3B(ω)ρ(ω)N0,0 (ω, t) − 3B(ω)ρ(ω)N1,0 (ω, t)
dt− AN1,0 (ω, t) − λo−Ps N1,0 (ω, t)
dN1,+1 (t)= −AN1,+1 (t) − λo−Ps N1,+1 (t)dtdN1,−1 (t)= −AN1,−1 (t) − λo−Ps N1,−1 (t).
dt

(1.18)(1.19)(1.20)(1.21)

where A and B(ω) are the Einstein’s coeﬃcients of Ps-HFS. A is the spontaneousemission rate of Ps-HFS. The reason of the pre-factor 3 in the term of 3B(ω)ρ(ω)
is as follows. The Einstein’s coeﬃcients are deﬁned in the case of unpolarizedradiation where the absorption and the stimulated emission occur in all substatesof o-Ps (|1, −1, |1, 0, and |1, +1) with equal rate of B(ω)ρ(ω). In the case,the energy density in one space direction is 1/3 of the total energy density of theunpolarized beam. On the other hand, the energy density of the linearly polarizedradiation in the polarization direction is the same as the total energy density of theradiation.

10

1.5 Introduction

The hyperﬁne transition of the ground state of Ps is M 1 transition since E1
transition is forbidden since the parity of o-Ps and p-Ps is the same, therefore thetransition probability is about α2 times smaller than E1 transition. E2 transition
is about (ΔHFS /E0 )2 times smaller than M 1 transition, which is negligible in thecase of the hyperﬁne transition of the ground state of Ps. Therefore, the transition
probability can be expressed asA=

g 2 2 ω 3 α
ω3
2
|μ|
=
= 3.37 × 10−8 s−1 ,3π0 c5
3m2e c4(1.22)

where μ is the magnetic dipole moment of the electron of positronium, which is
twice as large as that of hydrogen atom since the reduced mass of the electron of
positronium is me /2. B(ω) is related to A by the following equation,
B(ω) = A

π 2 c3g(ω),
ω 3(1.23)

where g(ω) is the lineshape function which satisﬁesg(ω)dω = 1.

(1.24)The spectrum lineshape is never strictly monochromatic because the states of
Ps have ﬁnite lifetimes. The ﬁnite lifetimes lead to a Lorentzian lineshape expressedasΔωn /2
1,(1.25)gn (ω) =π (ω − ω0 )2 + (Δωn /2)2
where Δωn is the natural linewidth (FWHM) because of the ﬁnite lifetimes of thestates of Ps, and(1.26)
Δωn = A + λp−Ps + λo−Ps
In the above equation, the decay rate of p-Ps is dominant and the natural linewidthis 1.3 GHz in frequency. The eﬀect of so-called pressure broadening can be includedin the above equation by replacing the decay rates of Ps in vacuum with those ingas.
In addition, the Doppler broadening also aﬀects the lineshape. The normalized
Gaussian lineshape function due to the Doppler broadening is
mc2 (ω − ω0 )2m
c(1.27)exp −gD (ω) =ω0 2πkB T
2kB T ω02
and the FWHM of the Doppler broadening isΔωD = 2ω0

(2 ln 2)kB T
mc2(1.28)

where m is the positronium mass, kB is the Boltzmann constant and T is its
temperature. T is the same as room temperature (about 300 K) after positronium
is fully thermalized. Then the width of the Doppler broadening is about 0.08 GHz.

11

1.5 Introductiong(f)The lineshape function g(ω) in the Einstein’s B coeﬃcient B(ω) is the convolution of gn (ω) and gD (ω) (Voigt function). Figure 1.7 shows the lineshape functions
with and without Doppler broadening. In the case of this measurement, the eﬀectof the Doppler broadening is small as shown in red line.

0.5
 

0.4
0.3
0.2
0.1
201

202

203

204
205
206
frequency [GHz]

Figure 1.7: Lineshape functions without Doppler broadening (black) and with
Doppler broadening of 7 % (red). Pressure broadening is assumed to be 5 %.

N0,0 (ω, t), N1,0 (ω, t), N1,+1 (t), and N1,−1 (t) are obtained by providing appropriate initial conditions. In the case of unpolarized positroniums the conditions are
N0,0 (ω, 0) = N1,0 (ω, 0) = N1,+1 (0) = N1,−1 (0) = N0 /4, where N0 is the populationsof all positroniums at t = 0.
Extremely high power radiation is necessary to cause enough stimulated emission to observe the hyperﬁne transition of the ground state of positronium because
the decay rate of the upper state (o-Ps) is 1014 times larger than the rate of the
spontaneous emission. Figure 1.8 shows fraction of 2γ decays per o-Ps versus radiation intensity. The fraction is zero without radiation but increases as the radiation
intensity arises, and when I > 108 W/m2 , the fraction becomes larger than 10%. Finally, the fraction is saturated at 1/3 because only |1, 0 state of o-Ps caninteract with linearly polarized radiation, whose spin is |1, 0. I = 108 W/m2 canbe achieved if 10 kW power is concentrated on 10 mm × 10 mm cross-section. Thisintensity is the most important requirement of this experiment.

12

Fo-Ps→2γ

1.5 Introduction

0.35
0.3
0.25
0.2
0.15
0.1

  

0.05
0
106

107

108

109
1010
Intensity [W/m2]

Figure 1.8: Fraction of 2γ decays per o-Ps increases as the radiation intensity arises
and is saturated at 1/3 due to the conservation law of the angular momentum. Red
point shows required radiation intensity of this experiment.

1.5 Introduction

13

This thesis is organized as follows. Chapter 2 describes the experimental apparatus in detail. The calibration and the analysis of the measured data is describedin Chapter 3. The systematic errors and the result of the measurement are shown
there. In Chapter 4, the result is checked and compared to the QED expectation.
Finally, we conclude this thesis in Chapter 5.

14

1.5 Introduction

Chapter 2

Experiment
2.1

Method of Our Experiment

The method of this experiment is as follows:
1. Gyrotron radiates sub-THz radiation, whose stable power is about 300 W.
(→2.2.1)
2. The gyrotron output (TE03 mode) is converted to a Gaussian beam. (→2.2.2)
3. The Gaussian beam is accumulated in a Fabry-Pérot resonant cavity and theaccumulated power is about 10 kW. (→2.2.3)
4. Positroniums are formed inside the Fabry-Pérot resonant cavity and o-Ps
transits to p-Ps (hyperﬁne transition). (→2.3)
5. Para-positroniums increase due to the transition and decay into two γ rays.
The γ rays are detected by γ-ray detectors and the increase of the 2γ decays
are measured. (→2.3)

2.2

Optical System

The optical system is composed of three devices. In this section, details of thefollowing devices are explained.
1. Gyrotron as a sub-THz radiation source
2. Mode converter to convert gyrotron output into Gaussian beam
3. Fabry-Pérot resonant cavity to accumulate sub-THz radiation
15

16

2.2.1

2.2 Experiment

Gyrotron

Our target frequency is 203 GHz, in other words, the wavelength is 1.5 mm. Thereare BWO (Backward Wave Oscillator), FEL (Free Electron Laser), and gyrotron
as a radiation source in this frequency region. BWO is frequency-tunable in a wide
range, but its output power is very low (1 - 100mW). In the case of FEL, theaverage power is high (100 W - 1 kW), however, the apparatus is very large, its
output is restricted to very short pulse (< 1 μs), and its linewidth is very wide (0.1
- 1 %). On the other hand, gyrotron can produce high power radiation (100 W- 1kW) and can operate in CW or long pulse wave and its size is reasonable. In
addition, the linewidth of its output wave is very narrow (< 100 ppm). Therefore
we use a gyrotron as a radiation source.
Theory of gyrotron
Gyrotron is an electron cyclotron maser. Figure 2.1 is a cross-sectional view of agyrotron. An electron gun, a resonant cavity, and a collector are in vacuum.
At the electron gun, its cathode is heated and thermal electrons are emitted.
There is an anode in the wall around the cathode and it pulls the electrons in atransverse direction. The electrons are cylindrically-distributed.
The electrons go toward the resonant cavity while accelerated by the potential
diﬀerence and rotating around a magnetic line created by the superconducting
magnet and the gun coil. The frequency of the cyclotron motion in the resonantcavity is
Ω0
eB
=
,
(2.1)
ωc =
me γ
γ
where e is the electron charge, B is a magnetic ﬁeld strength in the resonant cavity,
me is the electron mass, and γ = 1/ 1 − (v/c)2 (v is a velocity of the electron) is
a Lorentz factor of an electron.
Let’s consider the phase bunching process of electrons in a reference frame inwhich the axial velocity of electrons becomes zero. The radius of the cycrotron
motion of electron is r = vt γ/Ω0  R, where vt is the transverse velocity ofelectron. The transverse electric ﬁeld (TEmn mode) of the cavity of the gyrotron
accelerates or decelerates the electrons. If the phase of the electrons is random, theenergy exchange between the electrons and the electric ﬁeld of the cavity is zero intotal.
Phase bunching as shown in Fig. 2.2 occurs only when the frequency of theelectric ﬁeld in the resonant cavity becomes slightly larger than the initial frequencyof the cyclotron motion of electrons, which is determined by the strength of themagnetic ﬁeld. Then the electron 2 shown in the left side of Fig. 2.2 is accelerated,and gains energy while its cyclotron frequency decreases because its Lorentz factor
γ increases. Thus the electron 2 goes out of resonance and therefore the energy
gain on each cycle becomes less and less. On the other hand, the electron 1 isdecelerated and loses energy while its cyclotron frequency increases until it reaches

17

2.2 Experiment







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



Figure 2.1: A schematic of a gyrotron.

18

2.2 Experiment

the frequency of the electric ﬁeld. Therefore, the energy loss of the electron 1
increases more and more. As a result, the energy exchange between the electrons
and the electric ﬁeld becomes maximum and the phase bunching occurs.
On the contrary, if the intial cycrotron frequency is larger than the frequencyof the electric ﬁeld of the cavity, the electron 2 approaches the resonance and gains
energy more and more, and the electron 1 goes out of resonance and loses energy
less and less. Consequently, the energy exchange between the electrons and theelectric ﬁeld becomes minimum.
The resonant cavity is a circular waveguide and the wave mode which interacts
with the electrons is TEmn mode, which has no electric ﬁeld parallel to the axial
direction (Ez = 0) and is expressed as
 

mωμ
jmn

J
r sin(mφ)e−iζmn z
Er (z) = iA 
(jmn /R)2 r m R
 

ωμ
jmn

r cos(mφ)e−iζmn z
JEφ (z) = iA 
(jmn /R)2 m R
Ez (z) = 0
 

ζmn
jmn

r cos(mφ)e−iζmn z
J
Hr (z) = −iA (jmn /R)2 m R
 

mζmn
jmn

J
r sin(mφ)e−iζmn z
Hφ (z) = iA 
(jmn /R)2 r m R
 

jmn
r cos(mφ)e−iζmn z
Hz (z) = AJm
R

(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)

 (x) is its 1st derivative, j 
where Jm (x) is the m-th Bessel function, Jm
mn is the n-throot of Jm (0) = 0 and ω 2  j  2
− mn
(2.8)
ζmn =
c
R

The resonance condition is ζmn = lπ/L (l = 0, 1, 2, . . . ), where L is the cavity
length. The resonant frequency of the circular cavity resonator of the gyrotron isωmn,l = c  2   2
jmn
lπ
+
.
R
L

(2.9)

Gyrotron FU CW V
We develop a gyrotron named “Gyrotron FU CW V” (Fig. 2.3). Its parameters andoperational conditions are summarized in Table 2.1. The peak power and frequency
of its operating modes are summarized in Table 2.2.
The frequencies are measured with a synthesizer as a reference. The sub-THz
radiation of frequency f and the reference RF radiation of frequency fRF are fed

19

2.2 Experiment
























Figure 2.2: A schematic of an annular electron beam. The left ﬁgure shows initial
random phasing of electrons. The right ﬁgure shows electrons bunched in phase.

cavity radiuscavity length
Q value of the cavity

2.39 mm
14 mm
1300

Cathode voltage
Anode voltage
Beam current
Repetition rate
Duty ratio−18 kV−12 kV300∼400 mA
5 Hz
30 %

Table 2.1: Parameters and operational conditions of Gyrotron FU CW V

mode (m, n)

B [T]fmeas [GHz]fcalc [GHz]P [W]eﬀ. [%]

0, 1
1, 2
2, 2
0, 2
3, 2
1, 3
4, 2
2, 3
0, 3

2.793
3.887
4.870
5.110
5.840
6.208
6.729
7.238
7.364

106.47134.00140.06159.95185.21
198.83202.8968.36
107.00134.45140.62160.54170.91
185.67199.43203.42

314
270
704
515
314
722
597
635
695

4.4
3.8
9.8
7.2
4.4
10.0
8.3
8.8
9.7

Table 2.2: Operating modes of Gyrotron FU CW V

20

2.2 Experiment

Figure 2.3: A picture of the Gyrotron FU CW V.

21

2.2 Experiment

2

2

1

1

y mm

y mm

to a Schottky barrier diode, and its output radiation of frequency |f − nfRF | ismeasured with a spectrum analyzer through low pass ﬁlter. In the case of TE03
mode, the linewidth of the gyrotron output is about 1 MHz (5 ppm) but its center
value varies by about 2 MHz (10 ppm) within a pulse (width = 15 ms). The
measured frequencies are uniformly smaller than the calculation despite the wave
mode. The diﬀerence can be explained if the radius is 12 μm larger than the design
value due to the manufacturing accuracy or transformation during assembly or
aging.
Power is estimated from the calorie deposited on water. All of the output
radiation is absorbed by water and its power is estimated from the temperature
increase. The absorption coeﬃcient of water for sub-THz radiation is very large(about 102 cm−1 [17]). The peak power is about 700 W at maximum as shown in
Fig. 2.2, but the stable peak power is down to only about 300 W.
We perform hyperﬁne transition measurements at oﬀ- and on-resonance frequencies for comparison. We use the TE03 mode (f = 202.89 GHz) for on-resonancemeasurement and TE02 mode for oﬀ-resonance measurement. We select TE02 modethough there are modes whose frequency is nearer resonance, because the mode converter described in the following subsection works well only in TE0n mode. TE0nmode has no electric ﬁeld in r-direction and is rotationally symmetric around z-axis.
Its space distribution looks n concentric circle as shown in Fig. 2.4

0

0

1

1

2

2

2

1

0
x mm

1

2

2

1

0

1

2

x mm

Figure 2.4: (left) TE03 mode, (right) TE02 mode.
Figure 2.5 is space distribution of the gyrotron output (TE03 mode) measured
at 360 mm from the window. The pattern is obtained by taking an infrared pictureof PVC (PolyVinyl Chloride) sheet (thickness = 1 mm) over the window. PVC isa good absorber of sub-THz radiation among solid and easily obtainable materials,
whose absorption coeﬃcient is about 1 cm−1 [18]. The reason of the asymmetricpattern is not clear, but a port near the window to evacuate the air is a possible
reason.

22

0.14

40
0.12

20

0.1
0.08

0

0.06

-20

Power [W/(0.68mm)2]

Y [mm]

2.2 Experiment

0.04
0.02

-40

0

-40

-20

0

20

40

X [mm]
Figure 2.5: Space distribution of the gyrotron output (TE03 mode).

2.2.2

Mode Converter

Gyrotron FU CW V operates in TE03 (or TE02 ) mode and its stable peak power isabout 300 W. The power is high but not enough to observe the hyperﬁne transitionof Ps. The radiation will be accumulated in the Fabry-Pérot resonant cavity but thewave mode of the Fabry-Pérot resonant cavity is Gaussian beam (Eq. (2.10)), whichis completely diﬀerent from TE0n mode. If we deﬁne the polarization direction of
the Gaussian beam as x-axis, the electric ﬁeld is expressed as
 

2

r
w0
kr2
−1 λz
exp −
− i kz − tan+Ex (r, z) = E0
(2.10)w(z)w(z)πw02 2R(z)
where

λz 2
w(z) = w0 1 +πw02 2 2 πw0R(z) = z 1 +.
λz(2.11)(2.12)

Figure 2.6 shows the shape of a Gaussian beam. The beam size w(z) is shown in a
red line.

23

2.2 Experiment







 

 







 
 
 

Figure 2.6: A Gaussian beam. Beam parameters are w0 = 8.39 mm and λ = 1.48
mm.
Consequently, the output of the gyrotron can not be coupled with the FabryPérot resonant cavity. The matching ratio η is expressed as





η=




∗
EFP dS 
Ebeam|Ebeam | dS
2

(2.13)|EFP | dS
2

and the scalar product in the numerator becomes zero since the electric ﬁeld of
TE0n mode is only in the φ direction but the Gaussian beam is linearly polarized.
Therefore, the gyrotron output must be converted to a Gaussian beam.

Design of the mode converter
Figure 2.7 is a schematic and a picture of the mode converter. Its main component
is the step-cut waveguide made of copper and the Vlasov mirror, which is a large
parabolic mirror made of aluminum. It converts the TE0n mode to plain wave (orbi-Gaussian beam) geometrically by matching the center of the step-cut waveguideand the focus point of the Vlasov mirror (Fig. 2.8). This is because the parabolais expressed as
y=

x2
−f
4f

(2.14)

24

2.2 Experiment&  '&( * '+%,(  .
%%%%&  '&(     '  ())%
%%%%/%  ' - (     

  ! "#" 
  !$%&  '&(!"$  /%Figure 2.7: The schematic and the picture of the mode converter.

25

2.2 Experimentand the path length L from the center of the waveguide to the wavefront at y = C
isL=x2 + y 2 + C − y  ==

4f (y  + f ) + y 2 + C − y (2.15)(y  + 2f )2 + C − y = 2f + C = const.

(2.16)

when the light is reﬂected at (x , y  ). The following two mirrors (M1 and M2) shapethe bi-Gaussian beam into a Gaussian beam and the aperture cuts the sidelobes.

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












  Figure 2.8: Cross-sectional view of the step-cut waveguide and the Vlasov mirror

Parameters of the Vlasov mirror are its length, its width and its focal length.
The length in a transverse direction is determined by the depth of the mirror andthe depth should be enough to reﬂect all the light from the step-cut waveguide. The
condition for the depth h in Fig. 2.8 would be h > f if there were no diﬀraction,
but because of the diﬀraction, the condition is modiﬁed as

2f
tan
h>f+
sin αmn(2f − a)λ sin αmn
16af


[19],

(2.17)

26

2.2 Experiment

where αmn is a dispersion angle of the gyrotron output wave, which satisﬁes
 /ajmn
2π/λmn /ajmn=  2  2

jmn
lπ
+
R
L
R
(if l = 0),=asin(αmn ) =(2.18)(2.19)(2.20)

where R is the radius of the cavity resonator, L is the length of the cavity resonator,
and a is the radius of the waveguide. The dispersion angle is independent of the
gyrotron mode if we use the same gyrotron in diﬀerent mode.
The length in a longitudinal direction is determined by the condition that it
reﬂects all the light from the step-cut waveguide. Figure 2.9 is a side view of thestep-cut waveguide and the Vlasov mirror.

 



 










Figure 2.9: Side view of the step-cut waveguide and the Vlasov mirror
The reﬂection conditions in the longitudinal direction areλ,L1 < (f − a) cot α +8a cos α


λ
[19].
L2 > (2f + 3a) cot α −8a cos α(2.21)(2.22)In the above equations, diﬀraction is considered. If the wavelength λ is enough short(λ  a), the above conditions become identical to those calculated by geometrical
optics. In addition, the step-cut length of the waveguide L is limited due to thecondition that it can reﬂect all the right toward the Vlasov mirror and it does not
obstruct the lights reﬂected by the Vlasov mirror. The condition is expressed as
2a cot α < L < (2f − a) cos α.

(2.23)

L can be solved when f > 1.5a, and parameters used in our experiment are summarized in Table 2.3.

27

2.2 ExperimentParameter

[mm]

f
L
L1
L2
h

2a = 28
220
73.6
554
51.5

Table 2.3: Parameters of the step-cut waveguide and the Vlasov mirror
Light reﬂected from the Vlasov mirror can be treated as if it comes from a plain
image source lying behind the Vlasov mirror with emission angle α. The light islinearly polarized. The direction of the polarization is deﬁned as the x-direction.
The ﬁeld over the image source calculated using geometrical optics [20] is

Ex =

x2
1+ 2
4f

− 12
exp(−iky cos α).

(2.24)

Field at a distant point, ψfar , can be calculated by the Huygens equation,
ψfar =

ik
4π

ψs
S

exp(−ikr)(r̂ + ŝ) · n̂dS,
r(2.25)

where r is a distance from a point on the source to the distant point, r̂ is a unit
vector from a point on the source to the distant point, ŝ is a unit vector of the
Poynting vector, and n̂ is a normal unit vector of the source plain. Figure 2.10
shows the result for various distances.
In order to convert the wave into the clear bi-Gaussian beam with maximum
conversion eﬃciency, light path of about 10 m is necessary. In that case the conversion eﬃciency is about 90 %. The reason of the loss of 10 % is that the converted
beam is not pure bi-Gaussian but an airy function and therefore there are manysidelobes around the main component. The spot size wx (0), wy (0) at the beam
waist of the bi-Gaussian beam (= the position of the plain image source) is estimated from the beam size wx (z), wy (z) of the far-ﬁeld by the following equation

wx,y (z) = wx,y (0)1+λzπwx,y (0)2,(2.26)and the estimated value is wx (0) = 44.5 mm and wy (0) = 22.5 mm.
There is a large discrepancy between the calculated (Fig. 2.10) and the measured space distribution (Fig. 2.11). Many sidelobes are observed in the measured
distribution at r = 710 mm. The space distribution at far-ﬁeld (r = 3380 mm) is
disturbed due to the interferences between these sidelobes, as shown in Fig. 2.11(c). Thus we use an aperture whose diameter is 50 mm and it is placed so that only

28

2.2 Experiment

1.6

50

1.4
1.2

0

1.6
1.4

50

1.2
1

0

1

0.8

0.8

0.6

0.6

-50

-50

0.4

0.4
0.2

0.2

-100
-100

-50

0

50
100
X [mm]

-100
-100

0

0.7
0.6

50

0.5

0
-50
-100
100 150
X [mm]

0



300

0.12

200

0.1

100

0.08

0.4

0.06

0.3

-100

0.04

-200

0.02

0.1

50

50
100
X [mm]

0

0.2

0

0

Y [mm]

0.8

100

-150
-150 -100 -50-50
power [A.U.]

Y [mm]

 
150

power [A.U.]

1.8



100

power [A.U.]

2

Y [mm]

 
power [A.U.]

Y [mm]

 
100

0

-300
-300 -200 -1000

100 200 300
X [mm]

0

Figure 2.10: Calculated intensity distributions at various distances from the Vlasov
mirror29

2.2 Experiment

the mainlobe can pass it. Figure 2.11 (d) is the space distribution at r = 3380 mm
when the aperture is used. The fraction of the beam which passes the aperture is
35 ± 2 %.

120
100
80

200

220

260

280
X [mm]

10

120

8
6

80
60

4

40

2

20
0

60 80 100 120 140 160 180 200
X [mm]

     

 

500

6

450

5

400

4

350

3

350

300

2

300

250

1

250

0
200
300 350 400 450 500 550 600
X [mm]

power [a.u.]

Y [mm]

140
100

Y [mm]

Y [mm]



240

power [A.U.]

140

power [a.u.]

Y [mm]

160

18
16
14
12
10
8
6
4
2
0



0




500

10

450

8

400

200
300 350 400 450 500 550 600
X [mm]

6

power [a.u.]

 
180

4
2
0

Figure 2.11: Measured intensity distributions at various distances from the Vlasov
mirror
The polarization of the converted beam is measured using a grid polarizer which
is made from tungsten wires (diameter = 15 μm, pitch = 60 μm). Only the electricﬁeld perpendicular to the wires can pass the polarizer. The polarizer behaves as
metal for the electric ﬁeld parallel to the wires because electrons of tungsten wires
can move freely in the wire direction, therefore the radiation of parallel polarizationis reﬂected by the polarizer. Figure 2.12 is the result of a polarization measurement.
θ is the angle of the electric ﬁeld from the horizontal direction.
The fraction of the beam polarized linearly in θ = 0◦ is 80 ± 6 %. The sidelobes
remaining even after the aperture are also linearly polarized, but the direction
is orthogonal to the mainlobe. Consequently, the overall eﬃciency is 28 ± 2 %combined with the eﬃciency of the aperture.
Since the shape of the mainlobe is wide in a horizontal direction, the shape isarranged by two parabolic mirrors (M1, M2) and the nearly circular beam expect

30

2.2 Experiment

140

5

7

150

6

140

5

8

160

7

150

6

140

5

130

4

130

4

130

4

120

3

120

3

120

3

110

2

110

2

110

2

100
90
60 70 80 90 100 110 120 130 140
X [mm]

1
0100
90
60 70 80 90 100 110 120 130 140
X [mm]

  

 

170

170

8

160

7

150

6

140

5

Y [mm]

Y [mm]

90
60 70 80 90 100 110 120 130 140
X [mm]

10

power [a.u.]

100

1
0

8

160

7

150

6

140

5

130

4

130

4

120

3

120

3

110

2

110

2

100
90
60 70 80 90 100 110 120 130 140
X [mm]

1
0100
90
60 70 80 90 100 110 120 130 140
X [mm]

power [a.u.]

6

160



170

power [a.u.]

150

8

Y [mm]

7 

170

power [a.u.]160

Y [mm]


8

power [a.u.]

Y [mm]


170

1
0

Figure 2.12: Polarization of the converted beam. θ is the angle of the electric ﬁeld
from the horizontal direction.

260

14
12

240

10
220

power [a.u.]

Y [mm]

for the tail at bottom right due to the sidelobe after the aperture is obtained asshown in Fig. 2.13.

8
6

200

4

180

2
160
80

100

120

140

160 180
X [mm]

0

Figure 2.13: Beam shape after the mode converter.

31

2.2 Experiment
Mode matching

When the waist sizes of the Gaussian beam and the Fabry-Pérot resonant cavity
are wbeam and wFP , respectively, and the distance of their waist positions is d, themode matching ratio η (Eq. (2.13)) becomes
η=

2
2
wFP
4wbeam22 )2 + (λd/π)2 .
(wbeam
+ wFP(2.27)

The waist of our Fabry-Perot cavity is at the front plain mirror of the cavity andthe waist size is determined by the curvature of the end mirror and the cavity
length. The waist size and its position of the Gaussian beam have to be matchedwith those of the Fabry-Pérot cavity in order to obtain good coupling. We use aspherical plano convex lens (curvature r = 120 mm) made of Teﬂon for the purpose
and the lens is placed between the mirror 3 (M3) and the Fabry-Pérot cavity. The
transmittance of lens can be calculated from the refractive index and the absorption
coeﬃcient of the material. The one-side power reﬂectance at the interface between
the air and the material (refractive index n, absorption coeﬃcient α) is


1−n 2
,
(2.28)
R=
1+nthus the transmittance of the lens (thickness L) is
Tlens = (1 − R)2 exp(−αL)(2.29)

Therefore, materials such as quartz are not suitable for sub-THz radiation because
its refractive index at frequency of sub-THz is large, for example, refractive index ofquartz at 202.9 GHz is about 2.0. On the other hand, the refractive index of Teﬂon
is 1.4 and the reﬂection is small. Transmittance of the Teﬂon lens is measured
89 ± 4 %, which is consistent with the calculation from the refractive index of
Teﬂon (n = 1.4 at 203 GHz) and the absorption coeﬃcient of Teﬂon (α = 0.05
[cm−1 ] at 203 GHz) [21]. The focal length of the lens is f = r/(n − 1) = 300 mm.

2.2.3

Fabry-Pérot Resonant Cavity

Theory of the Fabry-Pérot resonant cavity
A Fabry-Pérot resonant cavity is made with a plane front mirror and a concave end
mirror (Fig. 2.14). The incident light is resonant with the cavity when the cavity
length L is equal to the integral multiple of λ/2.
The input ﬁeld is deﬁned as Ein = E0 eikz . The front mirror is at z = 0 and theend mirror is at z = L. Then the transmittance of the cavity is
2ikL n]Etr = E0 eik(z−L) tf te eikL Σ∞
n=0 [rf re e

= E0 eik(z−L)

tf te eikL
1 − rf re e2ikL(2.30)(2.31)

32

2.2 Experiment







  

 

 





      

Figure 2.14: A schematic of a Fabry-Pérot resonant cavity.
where tf and te are the amplitude transmittance of the front mirror and the end
mirror, rf and re are the amplitude reﬂectance of the front mirror and the end
mirror, respectively, and eikL is a phase shift per way.
In the same way, reﬂectance of the cavity is
2ikL n]Ere = E0 e−ikz (−rf ) + E0 e−ikz re t2f e2ikL Σ∞
n=0 [rf re e
t2f re e2ikL.
= E0 e−ikz −rf +1 − rf re e2ikL(2.32)(2.33)

right
The internal wave going to the right direction Eacc
is

right
Eacc
=

tf
Etr
= E0 eikz,te
1 − rf re e2ikL(2.34)

left isand the internal wave going to the left direction Eacc

left
Eacc
=

Ere − E0 e−ikz (−rf )= E0 e−ikz
tftf re e2ikL
1 − rf re e2ikL

.

(2.35)The total internal ﬁeld is sum of these,
right
left
Eacc = Eacc
+ Eacc= E0 eikz + re e−ikz e2ikL(2.36)
tf.
1 − rf re e2ikL(2.37)

33

2.2 Experiment
From Eq. (2.31), (2.33) and (2.37), we obtain
Ptr
Tf Te
=
2 

√
Pin
1 − Rf Re
1 + F sin2 kL
√
√ 2
√
Rf − (Tf + Rf ) Re + 4(Tf + Rf ) Rf Re sin2 kL
Pre
=

2 

√
Pin
1 − Rf Re
1 + F sin2 kL√√Tf (1 + Re )2 − 4 Re sin2 k(z − L)
Pacc
=
,

2 

√
Pin
1 − Rf Re
1 + F sin2 kL(2.38)(2.39)(2.40)

where Pin , Pre , Ptr , and Pacc are the incident, the reﬂected, the transmitted, andthe accumulated power, respectively, Tf and Te are the power transmittance of thefront mirror and the end mirror, respectively, Rf and Re are the power reﬂectanceof the front mirror and the end mirror, respectively, and
√


2F 2
4 Rf Re
√=.
(2.41)F =π(1 − Rf Re )2
F is called ﬁnesse. If we denote ρ = Rf Re , which is called round-trip reﬂectivity,ﬁnesse is expressed as
√
π4ρ
2π
.
(2.42)
F=
√ ≈
1− ρ
1−ρ
Pacc depends on z. Average power over the cavity is√√Pacc 1 L Tf (1 + Re )2 − 4 Re sin2 k(z − L)
=
dz

2 

√
Pin
L 0
1 − Rf Re
1 + F sin2 kL
=

1−

√

Tf (1 + Re )
2 
,
1 + F sin2 kL
Rf Re(2.43)(2.44)

Therefore, given Ptr , Te , and Re , we can estimate Pacc  via following equation.
Pacc  =1 + RePtr
Te(2.45)

Pacc /Pin becomes maximum when resonance occurs (kL = nπ). The maximum
accumulated power is
max 
Pacc
Tf (1 + Re )=
(2.46)
2 ,√Pin
1 − Rf Re
and the transmitted power also becomes maximum,
Ptrmax
Tf Te
=
2 .
√
Pin
1 − Rf Re
On the other hand, the reﬂected power becomes minimum
√
√ 2
min
Rf − (Tf + Rf ) Re
Pre
=
.

2
√
Pin
1 − Rf Re(2.47)(2.48)

34

2.2 Experiment

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-4

450
400
350
300
250
200
150
100
50
0

-4

-3

-2

-1

0

Pre / Pin

Ptr / Pin

Pacc / Pin

Figure 2.15 is the basic performance (Pacc /Pin , Ptr /Pin , and Pre /Pin ) calculatedwith Rf = 99.38 %, Tf = 0.39 %, Re = 99.8 %, and Te = 0.2 %. Since Tf + Rf = 1

-3

-2

-1

0

1

2

3
4
L [μm]

1

2

3
4
L [μm]

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-4

-3

-2

-1

0

1

2

3
4
L [μm]

Figure 2.15: Basic performance of Fabry-Pérot resonant cavity calculated with
Rf = 99.38 %, Tf = 0.39 %, Re = 99.8 %, and Te = 0.2 %. The top ﬁgureshows accumulated power, the bottom left ﬁgure shows transmitted power, andthe bottom right ﬁgure shows reﬂected power.
due to the loss, sum of the transmitted power and the reﬂected power is less than
the incident power. The width of the peak is related to the ﬁnesse of the cavity by
the following equation
F=

λ/2
.
Γ

(2.49)

Next, we consider the beam shape in the Fabry-Pérot cavity. The lowest wave
mode of the Fabry-Pérot cavity is a Gaussian beam (or TEM00 mode). Figure 2.16
is a general Fabry-Pérot cavity with two mirrors of diﬀerent curvatures (R1 , R2 ).
The cavity resonator is stable when

0≤

L
1−
R1



L
1−
R2


≤ 1.

(2.50)

35

2.2 Experiment
 

 













  

Figure 2.16: The beam shape in the Fabry-Pérot cavity.
The waist position is at L1 from mirror 1 and L2 from mirror 2. L1 and L2 is
L(R2 − L)
R1 + R2 − 2L
L(R1 − L),
L2 =R1 + R2 − 2L

L1 =(2.51)(2.52)and the waist size w0 is

w0 =

λ
π

L(R1 − L)(R2 − L)(R1 + R2 − L).
R1 + R2 − 2L

The beam sizes at the mirror 1 and 2 areλL(R2 − L)
wz1 =  R1π
(R1 − L)(R1 + R2 − L)λL(R1 − L)
wz2 =  R2π
(R2 − L)(R1 + R2 − L)(2.53)(2.54)(2.55)

Performance of the Fabry-Pérot resonant cavity
Our Fabry-Pérot cavity is composed of a gold mesh plain mirror as a front mirror
and a Cu concave mirror as an end mirror. The curvature of the concave mirror isR = 300 mm and therefore the focal length f = R/2 = 150 mm. The cavity length
L = 136 mm. By substituting R1 = ∞ and R2 = R, the beam waist is at the front

36

2.2 Experimentmirror (L1 = 0, L2 = L) and the waist size is
w0 = wz1 =λπL(R − L) = 8.38 [mm](2.56)

wz = wz2 =

λ
R
π

L
= 11.34 [mm].
R−L(2.57)

The cavity resonator is stable since the parameters satisfy Eq. (2.50),



136
136
1−
= 0.55 ≤ 1.
0≤ 1−
∞
300

(2.58)The Au mesh mirror is a key component of the cavity. In order to obtain highgain resonator, both of the high ﬁnesse and the reasonable transmittance of thefront mirror are necessary. Figure 2.17 is a picture of the mesh mirror used in theexperiment.

Figure 2.17: Pictures of the gold mesh mirror. The right ﬁgure is its magniﬁed
picture.
The Au mesh is made on a SiO2 plate using conventional photolithography andliftoﬀ technique [22]. Thickness of the mesh pattern is 1 μm, which is 6 times
thicker than the skin depth of Au at 203 GHz. Skin depth δ is expressed as
δ=

1
[m],
πf μσ

(2.59)

where f is wave frequency, μ = 4π × 10−7 is the magnetic permeability, and σ iselectrical conductivity of metal. In our case, f = 203 GHz and σ = 41 × 106 S/m(Au), therefore δ = 0.17 μm.
The line width is 200 μm and the line separation is 160 μm, which is designed
to obtain high reﬂectivity and reasonable transmittance by 3D electromagnetic
ﬁeld simulation based on the ﬁnite integration technique (FIT) [23, 24] using CST

37

2.2 Experiment

0.998

0.012

0.996

0.01

0.994

0.008

0.992
0.006
0.99

power transmittance [%]power reflectance [%]MICROWAVE STUDIO (CST MWS). Figure 2.18 is the result of the simulation
whose parameters are the same as the mesh used in the experiment. The ﬂuctuation
with a cycle of about 25 GHz is due to the interference in the SiO2 substrate, whose
thickness is 3 mm. The cycle is equal to c/(2nL), where c is the speed of light invacuum, n is the refractive index of SiO2 (n = 2.0 at 202.9 GHz), and L is the
thickness of the SiO2 substrate. The reﬂectance basically increases with increaseof frequency because the larger wavelength results in less transmittance of themesh pattern. Its simulated reﬂectivity is Rf = 99.38 % and the transmittance is
Tf = 0.39 %.

0.004
0.988
0.002
0.986
150

160

170

180

190

0
200
210
frequency [GHz]

Figure 2.18: The reﬂectance (black) and the transmittance (red) of the Au plain
mesh mirror simulated using CST MICROWAVE STUDIO.
Cavity length is controlled by moving the Cu concave mirror mounted on an
X-axis stage (NANO CONTROL TS102-G). It is an SIDMTM (Smooth Impact
Drive Mechanism) actuator using the piezoelectric element as the driving force.
Its long stroke (15 mm) and its high resolution (10 nm) is achieved at the sametime by combining coarse motion and slight motion (Fig. 2.19). The closed loop
control with a linear encoder in the main body enables accurate control with 10
nm resolution (Fig. 2.20).
Figure 2.22 shows the accumulated power (left), which is estimated from thetransmitted power, and the reﬂected power (right) measured while changing cavity length of the Fabry-Pérot cavity. FWHM of the resonance peak is 1.19(6)μm, which corresponds to the ﬁnesse F = 623 ± 29 (or the round-trip reﬂectivity
ρ = Rf Re = 98.99(5) %). Relative power is measured with pyroelectric detectors(Spectrum Detector Inc. SPH-49, Fig. 2.21). Pyroelectric detectors are thermal

38

2.2 Experiment    
  
 

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



    
  
       
 


 

 

Figure 2.19: Principal of operation of an SIDMTM (Smooth Impact Drive
Mechanism) actuator.



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

Figure 2.20: Structural drawing of the X-axis stage and its controller.

2.2 Experiment

39

Figure 2.21: Pyroelectric detectors (Spectrum Detector Inc. SPH-49).
detectors made of lithium tantalate (LiTaO3 ). Lithium tantalate is a pyroelectric
crystal whose ends become oppositely charged when heated. The output currentof a pyroelectric detector, which is proportional to a change in temperature, isconverted to voltage output via an operational ampliﬁer and a feedback resistance.
Transmitted power is transmitted through a hole (diameter = 0.6 mm) at the centerof the Cu concave mirror. The hole diameter is smaller than half of the wavelength(λ/2), therefore its transmittance is much smaller than the fraction of the beam
within the hole area. This is important to keep high reﬂectivity of the Cu concave
mirror and obtain high ﬁnesse of the Fabry-Pérot cavity. The accumulated power
reaches ∼ 10 kW at the resonance peak of the Fabry-Pérot resonant cavity. The
accumulated power is estimated from the transmitted power of the Fabry-Pérot
resonant cavity. Detail of the power estimation is described in next subsection.

2.2.4

Power Estimation

Power accumulated in the cavity is estimated from the transmitted power, thatis the output voltage Vtr [V] of the pyroelectric detector behind the hole at thecenter of the Cu concave mirror, since they are related via Eq. (2.45). Calibration
constant Ccalib = Pacc /Vtr [W/V] is necessary to estimate accumulated power Pacc
[W].
We obtain the calibration constant in the following way because there is no radiation sources whose absolute power and the shape is well-known in the frequency
and power domain of our experiment. In this section, details of the following measurements are explained.
1. measure Pin /Vin [W/V] with water2.2 Experiment

10
9
8
7
6
5
4
3
2
1
0
100 102 104 106 108 110 112 114 116
ΔL [μm]Vre [V]

Pacc [kW]

40

0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
100 102 104 106 108 110 112 114 116
ΔL [μm]Figure 2.22: The accumulated power (left) and the reﬂected power (right) of the
Fabry-Pérot resonant cavity measured while changing cavity length.
2. measure Vtr /Vin of the hole of Cu concave mirror
3. measure Phole /Pin with PVC sheet
4. calculate Phole /Pacc from the cavity length, the curvature of the Cu concave
mirror, the hole diameter, and the reﬂectivity of the Cu concave mirror
The calibration constant is obtained from these by the following equation.
Ccalib =Pacc
Pin /Vin × Phole /Pin=
Vtr
Vtr /Vin × Phole /Pacc(2.60)

Measurement of Pin /Vin [W/V] with water
Water is a good absorber of sub-THz radiation (α is about 102 [cm−1 ] [17]). We ﬁll
V = 50 cc of water in a box made of Teﬂon. Cross section of water is 100 mm ×
100 mm and its thickness is 5 mm. The cross section is much larger than the beamsize, and 5 mm of water is enough thick to absorb all of the radiation which passthrough Teﬂon wall, whose thickness at the beam side is 3 mm. Transmittance ofTeﬂon wall at the beam side is TTeﬂon = 95.1 ± 1.4 %, which is calculated from thethickness, the refractive index (n = 1.44) and the absorption coeﬃcient (α = 0.05
[cm−1 ]) [21]. The error is mainly due to the uncertainty of the absorption coeﬃcientof the Teﬂon. Temperature increase ΔT [K] after beam exposure for Δt [sec] ismeasured. During beam exposure, the water is agitated. A fraction of the input
power Vin [V] during beam exposure is monitored with a pyroelectric detector.
The temperature of water increases in proportion to the integral of Vin over timeof exposure, but the water cools down as time passes due to diﬀusion as shown in
Fig. 2.23. In order to estimate the total beam power Pin normalized by Vin , thegraph is ﬁtted by the following equation




 t
1 t




,
(2.61)T
Vin (t )dt = p0 + p1 p2 1 − exp −Vin (t )dt
p2 0
0

41

2.2 Experimentand Pin /Vin can be estimated from its gradient at V · t = 0, which is equal to p1 ,because the gradient is equal toPin /Vin  · fduty · TPTFE
C ·V(2.62)°

T [ C]

where C (= 4.18(1) [J/K·g]) is the heat capacity of water at room temperatureand fduty (= 0.3) is duty ratio of the beam. Thus Pin /Vin = 694 ± 44 [W/V] isobtained.

32
30
28
26
24
22
0

2

4

6

8

10

12
14
Vin⋅t [V⋅s]

Figure 2.23: Temperature of water measured while the water is exposed to theGaussian beam during 120 s. The horizontal axis of the ﬁgure is the integral of the
input beam power measured with a pyroelectric detector.

Measurement of Vtr /Vin of the Cu concave mirror
A portion of radiation which comes within the hole area transmits the Cu concave
mirror is detected by the pyroelectric detector because the diameter of the hole issmaller than λ/2. Input beam power Vin [V] and transmitted power through the
Cu concave mirror Vtr [V] are measured with pyroelectric detectors at the sametime. Figure 2.24 shows Vtr normalized by Vin while changing cavity length. The
dependency on the cavity length in λ/2 cycle is due to the resonance with very
low ﬁnesse. In order to measure the transmittance, the Cu concave mirror must beplaced perpendicular to the beam, and then the beam reﬂected by the Cu concave
mirror can go back to the gyrotron. There is a Teﬂon lens in the light path. The
Cu concave mirror and the lens become an optical resonator with very low ﬁnesse
and the interference curve is observed in Fig. 2.24.

42

2.2 Experiment

In order to obtain the value if there is no interference, the graph is ﬁtted by
Eq. (2.38) with initial phase, that is,
Vtr
=
Vin

1+ρ+

√

Vtr /Vin x − Δxρ cos 2πλ/2(2.63)

Vtr / Vin

where Vtr /Vin  is the value without interference, Δx is the initial phase times λ/4π.
As a result, Vtr /Vin  = 0.09458(72) is obtained in the case of this beam alignment.

0.18

χ2 / ndf
0.02834 / 252〈Vtr / Vin〉 0.09458 ± 0.0007239
ρ
0.05195 ± 0.001786

0.16

Δx

335.8 ± 2.065

0.14
0.12
0.1
0.08
0.06
0.04
0

100 200 300 400 500 600 700 800
x [mm]

Figure 2.24: The transmitted power of the Cu concave mirror Vtr normalized by
the input beam power Vin while changing the cavity length L by λ/2.

Measurement of Phole /Pin with PVC sheet
Because space distribution of the input beam is similar to but not the same as thatof the accumulated electromagnetic ﬁeld in the cavity, correction of the diﬀerenceis necessary. How much fraction of the input beam comes within the hole area of
the Cu concave mirror is measured with PVC sheet. PVC sheet is placed at thesame position as the face of the Cu concave mirror and is exposed to the incident
beam. Thus the space distribution of the input beam at the Cu mirror is measured.
Relative total power of the input beam Pin is estimated from the integral of thetemperature increase over the PVC sheet. There is remaining background evenafter the space distribution before beam exposure is subtracted from that after
beam exposure. The background is estimated from sideband where the beam nevercomes, and subtracted from the integral. The error of Pin is estimated from theamount of the background in the integral region.

43

40
30
20
10
0
-10

25

ΔT [K]

Y [mm]

2.2 Experiment

20
15
10

-20
-30

5

-40
-40 -30 -20 -10 0 10 20 30 40
X [mm]

Figure 2.25: A picture of the PVC sheet taken by IR camera to estimate Phole /Pin .
The origin is the position of the hole at the center of the Cu concave mirror.

Relative power coming within the hole area Phole is estimated from productof the temperature increase at the center of the PVC sheet, which should be thesame as the center of the Cu concave mirror, and the hole size. This is because theresolution of the IR camera is not enough (pixel size is about 0.5 mm × 0.5 mm).
The error of Phole derives from the alignment uncertainty (±1 mm) of the centerof the PVC sheet. The other uncertainties are cancelled when divided by Pin . The
uncertainty of the hole diameter is almost cancelled when divided by Phole /Pacc .
−4
In the case of this beam alignment, Phole /Pin = 3.06+0.27−0.17 × 10 .

Calculation of Phole /Pacc
The internal waveform of the Fabry-Pérot cavity is Gaussian. Its space distribution is determined by its beam size and the beam size can be calculated with thefollowing three parameters, the cavity length L = 136 mm, the curvature of the Cu
concave mirror R = 300 mm, and the frequency of the radiation f = 202.9 GHz.
The beam size at the Au mesh mirror is w0 = 8.38 mm and that at the Cu concave
mirror is wz = 11.34 mm.
Phole /Pacc is estimated from the integral of the space distribution of the beam

44

2.2 Experimentover the hole area divided by (1 + Re ) (see Eq. (2.45)),
0.3

Phole
=
Pacc

0

0

∞



exp −2(r/wz )2 rdr
exp −2(r/wz )


2

rdr

·

1
= 7.02(24) × 10−4 .
1 + Re(2.64)

Re is evaluated from the round-trip reﬂectivity ρ = Rf Re = 98.99(5) % and thefact that the possible maximum of Re is 99.85 %, which is the reﬂectivity of Cu,
and as a result, Re = 99.42(43) %. The manufacturing accuracy of the hole is 0.01
mm in diameter.
Result of the power estimation
Ccalib is obtained from the results of these four measurements using Eq. (2.60),
Ccalib ==Pacc
Pin /Vin × Phole /Pin=
Vtr
Vtr /Vin × Phole /Pacc(2.65)−4694 ± 44 [W/V] × 3.06+0.27−0.17 × 10
0.09458(72) × 7.02(24) × 10−4

(2.66)= 3.20+0.37−0.29 [kW/V].

(2.67)We repeat these measurements 10 times in order to estimate the uncertainty
due to the discrepancy between the beam shape during the Phole /Pin measurementwith a PVC sheet and that during the Vtr /Vin measurement with the Cu concave
mirror. The arithmetic average of these 10 measurements is 2.91+0.52−0.36 kW/V, andthe standard deviation of these 10 measurements is 0.80 kW/V. We denote quadrature sum of the arithmetic mean error and the standard deviation as a total error.
Consequently, we obtain
Ccalib = 2.91+0.95−0.88 [kW/V].

(2.68)

It is diﬃcult to perform these two measurements at the same time because of
the diﬀerence between their measurement times. If we can perform these two
measurements at the same time with an appropriate beam splitter, the uncertaintyof the hole position becomes large.

2.2.5

Stabilization

We have to stabilize gyrotron output power and keep resonance of the Fabry-Pérot
cavity during the measurement of the hyperﬁne transition. Figure 2.26 shows theschematic diagram of the feedback stabilization system. The incident power andthe power reﬂected from the Fabry-Pérot cavity are measured with pyroelectricdetectors placed behind the Mirror 3, which is a beam splitter. The transmitted
power of the Fabry-Pérot cavity is also measured with a pyroelectric detector placed
behind the hole at the center of the Cu concave mirror.

45

2.2 Experiment" #    !   "  $%&'(  
# 





  
# 


 

      
# 




 )*   



 

)



Figure 2.26: Schematic diagram of the feedback stabilization system. The feedback
circuit for the gyrotron output power is shown in blue line. The red line shows the
feedback circuit for the Fabry-Pérot resonant cavity stabilization.
Stabilization of the gyrotron output power
The heater of the electron gun is controlled to stabilize gyrotron output power,
since the temperature of the heater is higher, more electrons are emitted from theelectron gun and the gyrotron output power becomes higher.
The gyrotron output power is measured as follows. The Gaussian beam is split
by a beam splitter (or Mirror 3 in Fig. 2.7). It is a gold mesh mirror whose powertransmittance is high (2.7 %). The line width is 50 μm and the line separation is
130 μm. The beam reﬂected by the beam splitter goes to the Fabry-Pérot resonantcavity. On the other hand, the beam through the beam splitter is monitored by apyroelectric detector. We stabilize the output of the pyroelectric detector (Vin [V]),since it is a sample of the gyrotron output power. The output of the pyroelectricdetector is measured with a 16-bit ADC (NATIONAL INSTRUMENTS NI USB6215). The gyrotron output without feedback stabilization is very unstable asshown in Fig. 2.27. It is necessary to stabilize it.
The temperature of the heater of the electron gun is determined by its AC
high voltage. The AC high voltage power supply (MATSUSADA SRJ500) can becontrolled from the PC via RS232C. The heater voltage Vh is determined by thefollowing equation (PI control).
Vh (n) = Vh (n − 1) + KP (Vin (n − 1) − Vin (n)) + KI (Vintarget − Vin (n)),(2.69)

where Vh (n) is the heater voltage set at the n-th operation, Vin (n) is the output ofthe pyroelectric detector at the n-th measurement, KP is a proportional gain, KI
is an integral gain, and Vintarget is a target voltage. In this experiment, KP is set to46

2.2 Experiment

Figure 2.27: Time variation of the gyrotron output power is very large unless the
feedback system is installed.
100 and KI is set to 0.05 (Typical voltage : Vh ∼ 200 V, Vin ∼ 0.2 V). Figure 2.28
is a monitor plot of the gyrotron output power and the heater voltage with thisstabilization system. The gyrotron output power is stable within ±10 %.
Stabilization of the resonance of the Fabry-Pérot resonant cavity
We control the cavity length to keep it resonant with the Gaussian beam. The
cavity length is varied without moving the piezoelectric X-stage due to the ﬂuctuation of the temperature of the gas chamber made of aluminum. The coeﬃcientof thermal expansion of aluminum is 2.3 × 10−5 /K, and the length of the surface
where the Fabry-Pérot cavity is mounted is 562 mm. The temperature ﬂuctuationof ±0.05 K results in ±0.6 μm (±1 ppm) ﬂuctuation of the length of aluminum,
which unstabilizes resonance of the Fabry-Pérot cavity. Note that 0.6 μm is equal
to the HWHM (Half Width at Half Maximum) of the resonance peak of the FabryPérot cavity. The resonance of the Fabry-Pérot cavity lasts for a few minutes even
if the cavity length is not controlled. We rescan the cavity length searching resonance peak position only when the transmitted power of the Fabry-Pérot cavity
falls below a threshold.
The rescan procedure is as follows. Figure 2.29 is its schematic.
high, rescan process starts and the cavity length is
1. If Vtr gets less than Vthr
low .
lengthened until Vtr gets less than Vthr
low of the other side of
2. We shorten the cavity length until Vtr gets less than Vthr

47

2.3 Experiment

Figure 2.28: Stabilization of the input beam power by controlling heater AC voltage.

the resonance curve, while searching the peak of it.
3. The cavity length is set at the peak position found in the 2nd step.
The 1st step is necessary since we can not ﬁgure out in which side of the resonancecurve the cavity is.
Figure 2.30 is a monitor plot of the transmitted power and the displacement of
the piezoelectric stage. The cavity rapidly returns to the resonance when resonancehighbreaks. The rescan threshold Vthr
is 80 % of the peak power.












 



 













   

Figure 2.29: Schematic of the rescan procedure.

 



48

2.3 Experiment

Figure 2.30: Control of the cavity length to keep resonance of the Fabry-Pérot
cavity.

2.3

Positronium Assembly and γ-ray Detectors

Figure 2.31 is a top view of the positronium assembly chamber and the γ-ray
detectors. A positronium atom is formed when a positron stops in the mixedgas (1.9 atm nitrogen (N2 ) and 0.1 atm isobutane (i-C4 H10 )). The gas is ﬁlled
and the positron source is placed in the sealed chamber. The Fabry-Pérot cavity
is also placed in the chamber. The Gaussian beam enters the chamber passing
the window made of high-resistive Si. We select high-resistive Si since it prevents
light whose wavelength is smaller than 1.2 μm entering the chamber and is enoughrigid (bending strength ∼ 80 MPa) to ﬁll 2 atm of the mixed gas in the chamber.
However, the refractive index is large (n = 3.4 [25]). The transmittance of an
optical parallel of thickness d cm, absorption coeﬃcient α cm−1 , and refractive
index n is
(1 − R)2,(2.70)
T =(1 − Re−2αd )2 + 4Re−4αd sin2 kd


1−n 2
where R =is a one-side reﬂectivity. We optimize the thickness of the1+nwindow to maximize the transmittance for 202.9 GHz radiation. The thickness of
the Si window is 1.96 mm (measured) and the transmittance is measured 98 ± 2%. Four γ-ray detectors are placed 25 mm away from the beam axis, which is theshortest distance so that the γ-ray detectors do not interrupt the resonance of the
Fabry-Pérot cavity.

49

2.3 Experiment        

    



!









 

  Figure 2.31: A schematic and a picture of the positronium assembly chamber andγ-ray detectors. The plastic scintillator and the 22 Na source is not placed in theright picture.

2.3.1

Positronium Formation Assembly

Positronium formation assembly used in this experiment is shown in Fig. 2.32.
Intensity of the 22 Na positron source (Eckert & Ziegler POSN-22) is approximately
780 kBq. Since the radioactive half-life of 22 Na is 950.8 days, the decrease of theevent rate is only 1.5 % during the experiment. The decay scheme of the 22 Na isshown in Fig. 2.33.
The 22 Na positron source is titanium packaged and 19.1 mm in diameter andhas an active diameter of 9.53 mm. The activity is placed between two layers of
0.0051 mm titanium foil, supported by two 0.25 mm titanium disks. This assemblyis sealed by electron beam welding.
Positrons are emitted from the 22 Na source, which is placed 47 mm away fromthe beam axis. Positrons emitted toward the beam direction pass a thin (100 μm)
plastic scintillator (OKEN NE-102). The characteristics of the plastic scintillator
are summarized in Table 2.4. About 88 % of the positrons annihilate into two γrays around the source, and these γ rays become source of accidental background.
In order to block such γ-rays, the source and the plastic scintillator are surrounded
by a 15 mm Pb shield. The light of the scintillator is collected by a light guide madeof PMMA and two 1 12 -inch ﬁne-mesh photomultipliers (HAMAMATSU R5924-70)
to tag e+ emissions. The characteristics of these PMTs is summarized in Table 2.5.

50



2.3 Experiment # $  
 !





""

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

Figure 2.32: A schematic of the positronium formation assembly

Figure 2.33: The decay scheme of the 22 Na isotope

Light output [% of NaI(Tl)]Primary decay time [ns]Density [g/cm3 ]Wavelength of emission max [nm]Refractive index
Melting point [◦ C]

29
2.4
1.03
423
1.58
75

Table 2.4: Properties of NE-102 (plastic) scintillator.

51

2.3 Experiment

Size
Active diameter
Dynode type
Wavelength
Typical HV
Typical gain
Typical Q.E.
Rise time
Transit time

φ51 mm
φ39 mmFine mesh (19 stages)
300 - 650 nm (peak 420 nm)
2000 V1 × 10720 %2.5 ns
9.5 nsTable 2.5: Properties of the PMT (HAMAMATSU R5924-70).
The ﬁne-mesh PMT is used because there is remaining magnetic ﬁeld (0.5 - 1.0mT) of the superconducting magnet of the gyrotron. About 5 % of the positrons
are tagged by the plastic scintillator and stop in the gas. The others stops atmaterials around the source or are too energetic to stop in the gas and annihilate
at the wall of the chamber.

2.3.2

Characteristics of Mixed Gas

We use the mixed gas of nitrogen 1.9 atm and isobutane 0.1 atm as a source ofelectron to form positronium. The mixing ratio is determined to satisfy following
conditions. Details are also given later.
1. Slow positron annihilation becomes negligible.
2. Absorption of 203 GHz radiation becomes small enough.
3. The formation rate of positronium becomes as high as possible.
4. The stopping power for positron becomes as high as possible.
The condition 3. and 4. is equal to as much isobutane as possible.
Slow positron annihilation
A positron, whose energy is under the ﬁrst ionization energy of a gas molecule,can not form a positronium any more, and is called slow positron. When sucha positron collides with an electron of the gas molecule, pair-annihilation occurs.
The rate of the slow positron annihilation is expressed as
λslow = πr02 cnZeﬀ = 0.201ρZeﬀ μs−1(2.71)

where r0 is the classical electron radius, c is the speed of light, n is number densityof the gas molecule, and Zeﬀ is the eﬀective number of electrons per gas molecule

52

2.3 Experiment[26]. ρ is the gas density in amagat, which is deﬁned as the number of ideal gas
molecules per unit volume at 1 atm and 0 ◦ C. Gas of P [atm] and T [◦ C] isρ=P·273.15[amagat]273.15 + T(2.72)

Nitrogen gas does not absorb 203 GHz radiation at all but Zeﬀ (N2 ) is small(Zeﬀ (N2 ) = 29.75(85) [27]), therefore λslow (N2 ) = 11.96(31) μs−1 and the lifetime
is long (τslow (N2 ) = 91.3(2.6) ns) in nitrogen gas at 2 atm and 25 ◦ C. In this case,the slow positron annihilation becomes serious background since the lifetime of theslow positron annihilation in nitrogen gas at 2 atm is comparable to that of o-Ps.
On the other hand, isobutane absorbs 203 GHz radiation but it is a good quencher
of the slow positron because its Zeﬀ is very large (Zeﬀ (iso) = 14400 [28]). In themixed gas of nitrogen at ρN2 amagat and isobutane at ρiso amagat, the rate of theslow positron annihilation λslow is expressed as
λslow = 0.201(ρN2 Zeﬀ (N2 ) + ρiso Zeﬀ (iso))(2.73)

τslow [ns]Figure 2.34 shows the lifetime of the slow positron in the mixed gas at total pressure
2 atm and 25 ◦ C while changing the partial pressure of isobutane. If the partialpressure of isobutane is larger than 0.03 atm, then the lifetime of slow positron
becomes smaller than 10 ns.

80
70

  


60
50
40
30
20
10
0
0

0.02

0.04

0.06

0.08
0.1
Piso / (P +Piso)
N2

Figure 2.34: The lifetime of the slow positron as a function of the partial pressureof isobutane.

53

2.3 Experiment
Absorption of 203 GHz radiation in isobutane

Since the sub-THz beam makes many round-trips when the cavity is on resonance,
absorption by the gas is not negligible. Figure 2.35 shows transmitted power on theresonance of the Fabry-Pérot cavity while changing the partial pressure of isobutane(total pressure is 1 atm). The graph is ﬁtted by the following function, which isthe natural extension of Eq. (2.47),
Ptr
C0
=
2
√
Pin
1 − ρ + 1 − exp(−αpiso L)(2.74)

where α is the absorption coeﬃcient of isobutane at 1 atm, L is the cavity length(136 mm) and piso is the pressure of isobutane. If the graph is ﬁtted by ﬁxing
the round-trip reﬂectivity ρ = 0.9899(5) (or F = 623 ± 29) in order to obtain
the absorption coeﬃcient, α = 2.41(50) × 10−4 [cm−1 ] at 1 atm (Fig. 2.35). If the

Ptr / Pin

χ2 / ndf35

3.146 / 9C00.0005967 ± 7.819e-05

α0.0002405 ± 5.007e-05

30
25
20
15
10
5
0

0

20

40

60

80
100
iso-C4H10 [%]Figure 2.35: Absorption of sub-THz radiation by isobutane.
partial pressure of isobutane is less than 0.1 atm, the loss of the accumulated power
is less than 20 %.
Considering both of the lifetime of slow positron annihilation and the absorption, we selected the mixed gas of nitrogen 1.9 atm and isobutane 0.1 atm.

2.3.3

γ-ray Detectors

Four LaBr3 (Ce) inorganic scintillators (Saint-Gobain Crystals, BrilLanCeTM 380)
are used to detect γ rays from decay of Ps. LaBr3 (Ce) detector has good energy resolution (FWHM = 4 % at 511 keV), which is good advantage to search

54

2.4 Experimentmonochromatic 511 keV γ rays. In addition, its fast time response (τ = 16 ns) isappropriate for high statistics experiment. The characteristics are summarized inTable 2.6.
Light yield [photons/keVγ]
Primary decay time [ns]Density [g/cm3 ]Wavelength of emission max [nm]Refractive index @ emission max
Thickness for 50 % attenuation (662 keV) [cm]

63
16
5.08
380
1.9
1.8

Table 2.6: Properties of LaBr33 (Ce) scintillator
The LaBr3 (Ce) detectors used in this experiment are φ 1.5 inch × 2.0 inchlarge crystal. The crystals are covered with aluminum housing (0.5 mm thick).
They are connected to 1.5 inch photomultipliers (HAMAMATSU R5924-70). Four
LaBr3 (Ce) detectors are arranged to make four back-to-back pairs as shown in
Fig. 2.36.









Figure 2.36: Four γ-ray detectors are arranged to make four back-to-back pairs.
The pairs are (la-0, la-2), (la-0, la-3), (la-1, la-2), and (la-1, la-3).

2.4

Electronics and Data Acquisition

Data ﬂow
The data acquisition system of the experiment consists of two parts, the detector
part and the optics part. The detector part acquires the data concerning the transition signal and consists of the clusters of NIM standard modules and the CAMAC
system. The optics part controls the optical system and monitors the condition of
the experiment (power, temperature, pressure, gyrotron parameters, position of thepiezoelectric stage, etc.). It consists of a data logger (HIOKI 8420-50), the piezoelectric X-stage (NANO CONTROL TS102-G), the pyroelectric detectors, and the55

2.4 Experiment-.3-.3&'0-.$/-0.$/-0
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0$ .


  1

8!:

/;& !

Figure 2.37: The schematic view of the whole DAQ system.
ADC (National Instruments NI USB-6215). Figure 2.37 is a schematic view of thewhole DAQ system.
First of all, the gyrotron output power is stabilized by controlling the heater
voltage of the electron gun. The AC voltage of the heater is supplied by Matsusada
Precision SRJ500 and its output voltage is remote-controlled from the PC via
RS232C. Once the gyrotron output power becomes stable, then the resonancepeak of the Fabry-Pérot cavity is searched by moving the piezoelectric stage, whose
driver is connected to the PC via RS232C. The cavity length is controlled so thatthe cavity can stay on resonance. Then the PC sends a start command to the
CAMAC controller (Toyo Corp. CC/NET). At ﬁrst, all the CAMAC modules arecleared and the latch is released by the reset signal from the output register. This
makes all the systems active and the controller waits the interrupt signal, so-called
LAM (Look-At-Me) signal from the CAEN ADC (CAEN C1205). The interrupt
signal comes only when the main trigger condition is satisﬁed. The data acquisitionis triggered when back-to-back γ-ray signals from the LaBr3 (Ce) scintillators arecoincident within 40 ns and then when this coincidence is within −100 ns to 1100
ns of the timing of the plastic scintillators. A charge ADC (CAEN C1205) isused to measure the energy information of the LaBr3 (Ce) detectors, a charge ADC(PHILLIPS 7167) and a charge ADC (REPIC RPC-022) are used to measure theenergy information of the plastic scintillator. The time information of the plastic
and LaBr3 (Ce) scintillators is recorded using a direct clock (2 GHz) count type

56

2.4 Experiment

TDC (KEK GNC-060). After the reading procedures, the controller saves the data
through NFS and starts the next event cycle.
Environmental conditions (temperature, pressure, and gyrotron vacuum) arerecorded with the data logger (HIOKI 8420-50). The incident, reﬂected and transmitted power are monitored with pyroelectric detectors as shown in Fig. 2.26. The
signal waveform of the pyroelectric detectors are read with an ADC (NI USB-6215).
The trigger pulse for the gyrotron output and the level of the cathode current of
the gyrotron are also recorded with the ADC. These data per gyrotron output
pulse are recorded to the PC.
In the following subsections, details of the detector part are described.
Electronics for the plastic scintillator
A schematic diagram of electronics for the plastic scintillator is shown in Fig. 2.38.
The photomultipliers for the plastic scintillator are operated at +2215 V (ps-0)
and +2250 V (ps-1). These high voltage are supplied by a positive high voltage
module (REPIC RPH-022). Their gains are 3.4 × 107 and 3.8 × 107 .
The output of the plastic scintillator PMT is divided into three lines by a linear
fanout module. One of the divided signals is fed into a discriminator. The others
are used to measure the amplitude of the signal with short (60 ns) and long (1000ns) gate.
The threshold value is set to 25 mV, which corresponds about 1 p.e. One of theoutput of the discriminator is delayed by 200 ns and then provides the stop signal
for the TDC (KEK GNC-060).
The other output is used to make a coincidence signal of the two PMTs (psand). The noise is suppressed by requiring the coincidence. The coincidence signal
is used to make a common start signal of the TDC (KEK GNC-060), a short gate
signal of a charge sensitive ADC (phillips 7167), a long gate signal of a charge
sensitive ADC (REPIC RPC-022), the fast clear, and the main trigger. These aredescribed in the explanation of the trigger part.
Electronics for the γ-ray detectors
A schematic diagram of electronics for the γ-ray detectors is shown in Fig. 2.39.
The high voltages of the photomultipliers are supplied by a positive high voltage
module (REPIC RPH-022). The HV values are +1310 V for la-0, +1220 V for la-1,+1440 V for la-2, and +1270 V for la-3.
The output of the γ-ray detector PMT is divided into two lines by a linear
fanout module. One of the divided signals is fed into a discriminator. The other isused to measure the amplitude of the signal with a charge sensitive ADC (CAEN
C1250). The gate width is 150 ns. The pulse height of the 511 keV signal is about
0.9 V. The threshold value is set to 90 mV, which corresponds about 50 keV.
One of the outputs of the discriminator is delayed by 200 ns and then provides
the stop signal for the TDC (KEK GNC-060). The other outputs of the fourKC3471
16-BIT OUTPUT REGISTERPHILLIPS MODEL740
QUAD LENEAR FAN-IN/OUT
offset = -20mV

ps-1

HOSHIN N010
CLOCK GENERATOR
500kHz, width = 20ns

170ns delay

N-TM 103
3CH 4-FOLD 1-VETO COINCIDENCE
width = 20ns

OUT

OUT

N-TM 103
3CH 4-FOLD 1-VETO COINCIDENCE
width = 20ns

KEK GNC-060
2GHz TDCKN410
OCTAL FANOUT

ps-andKN410
OCTAL FANOUTKEK GNC-060
2GHz TDC

N-RY 007
2CH GATE&DELAY GENERATOR
through, width = 2100ns

VETO

KN246②OCTAL DISCRIMINATOR
thr = -200mV, width = 20ns

LeCroy MODEL365AL
4-FOLD LOGIC UNIT
2/2, width = 20ns

KN246②OCTAL DISCRIMINATOR
thr = -200mV, width = 20ns

N-RY 011
3CH 4-FOLD MAJORITY LOGIC
1/2, width = 20ns

REPIC RPC-022
16ch CS ADC

170ns delay

KC3122①
12CH SCALER

PHILLIPS 7167
16ch QDC

200ns delay

200ns delay

PHILLIPS 7167
16ch QDCREPIC RPC-022
16ch CS ADC

170ns delay

KN246①OCTAL DISCRIMINATOR
thr = -45mV, width = 20ns

KN246①OCTAL DISCRIMINATOR
thr = -45mV, width = 20ns

LeCroy MODEL429A
LOGIC FAN-IN/FAN-OUT

ps-andPHILLIPS MODEL740
QUAD LENEAR FAN-IN/OUT
offset = -20mV

ps-0

170ns delay

KN1500
DUAL GATE GENERATOR
through, width = 1200ns

KC3122
12ch SCALER1860
VISUAL SCALER1860
VISUAL SCALERKC3122
12ch SCALER

ps-gate

2.4 Experiment57

Figure 2.38: Schematic diagram of electronics for plastic scintillator system.

58

2.5 Experimentdiscriminators are fed by a logic fan-in/fan-out module and a logic unit, and theoutput of the logic unit is the back-to-back coincidence signal of the four γ-ray
detectors (la-b2b). The la-b2b signal is used to make a gate signal of a charge
sensitive ADC (CAEN C1205) to record the energy information of the LaBr3 (Ce)detectors and the main trigger. These are described in the explanation of thetrigger part.
Electronics for the trigger system
A schematic diagram of electronics for the trigger system is shown in Fig. 2.40.
The coincidence signal from the two plastic scintillator PMTs (ps-and) is widened
to 1200 ns (ps-gate). The la-b2b signal is delayed by 100 ns, and when the delayed
signal coincides with ps-gate, then the data acquisition is triggered (main trigger).
The main trigger is also the gate signal of the charge sensitive ADC (CAEN C1205)
to record the energy information of the LaBr3 (Ce) detectors. The trigger signal
works also as the latch start. The latch signals veto the second gate signals andthe fast clear signals. After all the data are read and saved, the latch reset signals
are produced by an output register.
On the other hand, the short and long gate signals for the ADCs which record
the energy information of the plastic scintillator is produced as long as the signals
from the two PMTs of the plastic scintillator are coincident within 40 ns. Though
the coincidence rate is very high (about 200 kHz), the data are cleared by the clear
signal (fast clear) unless the main trigger signal is produced.
In the same way, the coincidence signal from the two PMTs of the plastic
scintillator is the common start signal for the TDC. The TDC data is also clearedby the fast clear signal unless the main trigger signal is produced.
Data summary
The quantities recorded at the main trigger timing are summarized in Table 2.7,
and the quantities recorded in synchronization with the gyrotron output pulse aresummarized in Table 2.8.

2.5

Monte Carlo Simulation

In this experiment, a Monte Carlo simulation is used to estimate the transition
probability and the systematic error because the ratios of the 3γ detection eﬃciency
and the 2γ detection eﬃciency for the LaBr3 (Ce) scintillators are necessary.
For the detector simulation, the Geant4 (ver. 4.9.3) package [29] is used and thedetails of the experimental setup (Fig. 2.41) are reproduced in the simulation. In
this package, the interaction of e± and γ-rays in various materials are implemented.
Practically, the Penelope low-energy electromagnetic models for e± and γ-rays areused.

RIS-0255
Quint Linear Fan-in/Fan-out
offset = -10mV

RIS-0255
Quint Linear Fan-in/Fan-out
offset = -10mV

RIS-0255
Quint Linear Fan-in/Fan-out
offset = -10mV

RIS-0255
Quint Linear Fan-in/Fan-out
offset = -10mV

la-0

la-1

la-2

la-3

200ns delay

KN246①OCTAL DISCRIMINATOR
thr = -100mV, width = 20ns

KN246①OCTAL DISCRIMINATOR
thr = -100mV, width = 20ns

200ns delay

200ns delay

KN246①OCTAL DISCRIMINATOR
thr = -100mV, width = 20ns

KN246①OCTAL DISCRIMINATOR
thr = -100mV, width = 20ns

200ns delay

CAEN Mod. C1205
16CH QADC

200ns delay

200ns delay

CAEN Mod. C1205
16CH QADCCAEN Mod. C1205
16CH QADC

200ns delay

200ns delay

CAEN Mod. C1205
16CH QADC

KN246②OCTAL DISCRIMINATOR
thr = -200mV, width = 20ns

LeCroy MODEL429A
LOGIC FAN-IN/FAN-OUT

KN246②OCTAL DISCRIMINATOR
thr = -200mV, width = 20ns

KN246②OCTAL DISCRIMINATOR
thr = -200mV, width = 20ns

LeCroy MODEL429A
LOGIC FAN-IN/FAN-OUT

KN246②OCTAL DISCRIMINATOR
thr = -200mV, width = 20ns

KEK GNC-060
2GHz TDCKN410
OCTAL FANOUTKN410
OCTAL FANOUTKEK GNC-060
2GHz TDCKEK GNC-060
2GHz TDCKN410
OCTAL FANOUTKN410
OCTAL FANOUTKEK GNC-060
2GHz TDCKC3122
12ch SCALER1860
VISUAL SCALER1860
VISUAL SCALERKC3122
12ch SCALERKC3122
12ch SCALER1860
VISUAL SCALER1860
VISUAL SCALERKC3122
12ch SCALER

LeCroy MODEL365AL
4-FOLD LOGIC UNIT
2/2, width = 20ns

la-b2b

2.5 Experiment
59

Figure 2.39: Schematic diagram of electronics for γ-ray detector system.

la-b2b

ps-gate

N-RY 011
3CH 4-FOLD MAJORITY LOGIC
2/2, width = 20ns

VETO

VETO

VETO

3470
16-BIT OUTPUT REGISTER

stop

1500
DUAL GATE GENERATOR
latch mode

16CH QADCPHILLIPS 7167

gate CAEN Mod. C1250

start

short gate

LeCroy MODEL429A
LOGIC FAN-IN/FAN-OUT
KC3122①

OCTAL DISCRIMINATOR
thr = -100mV, width = 20ns

600ns delay KN246②

inhibit 12CH SCALEROCTAL DISCRIMINATOR
thr = -100mV, width = 20ns

600ns delay KN246②

KEK GNC-060
2GHz TDCPHILLIPS 7167
16 CHANNEL QDC

16 CHANNEL QDC

short gate PHILLIPS 7167

common start
LeCroy MODEL620CL
8 CHANNEL DISCRIMINATOR
width = 20ns
gate enable 16 CHANNEL QDC

1500
DUAL GATE GENERATOR
through, width = 150ns

1860
VISUAL SCALER

16CH CS ADC

gate REPIC RPC-022

KN1500
DUAL GATE GENERATOR
through, width = 1200ns

TKY-0240
Dual Gate Generator
through, width = 1000ns

KN246①OCTAL DISCRIMINATOR
thr = -500mV, width = 20ns

KC3122
12CH SCALER①

N-RY 007
2CH GATE & DELAY GENERATOR
through, width = 60ns

VETO

TKY-0240
Dual Gate Generator
through, width = 20ns

2GHz TDC

common stop KEK GNC-060

REPIC RPC-022

fast clear 16CH CS ADC

16 CHANNEL QDC

2GHz TDC

fast clear PHILLIPS 7167

fast clear KEK GNC-060

60
2.5 Experiment

Figure 2.40: Schematic diagram of electronics for trigger system.

61

2.5 Experimentname

moduleevent ID
real time-

gyrotron ON/OFF

input register (LeCroy C005)

ps-energy (short gate), 2ch
ps-energy (long gate), 2chla-energy, 4ch

CS ADC (phillips 7167)
CS ADC (REPIC RPC-022)
CS ADC (CAEN C1205)

ps-time, 2ch
la-time, 4ch

TDC (KEK GNC-060)
TDC (KEK GNC-060)

live time
ps-rate, 2ch
ps-and-rate
la-rate, 4ch

SCALAR (Kaizu KC3122)
SCALAR (Kaizu KC3122)
SCALAR (Kaizu KC3122)
SCALAR (Kaizu KC3122)

Table 2.7: Quantities measured at the main trigger timing.

nameinput power
reﬂected powertransmitted power
trigger pulse of gyrotron
cathode current of gyrotron
heater voltage of gyrotron
vacuum of gyrotron
cavity length
room temperaturetemperature of NIM bintemperature in the Ps assembly chamberpressure in the Ps assembly chamber

devicepyroelectric detector
pyroelectric detector
pyroelectric detector
ADC (NI USB-6215)
ADC (NI USB-6215)
logger (HIOKI 8420-50)
NANO CONTROL TS102-Glogger (HIOKI 8420-50)
logger (HIOKI 8420-50)
logger (HIOKI 8420-50)
logger (HIOKI 8420-50)

Table 2.8: Quantities measured in synchronization with the gyrotron ouput pule62

2.5 Experiment

The Monte Carlo simulation is proceeded as follows. At ﬁrst, a positron isemitted from the source with the energy corresponding to the energy spectrum of
22 Na. Then the interactions of the positron in the materials are simulated stepby step. The simulation of the secondary particles is also included in this process.
The positrons which stopped in the gas are selected as the candidate for Ps.
The next step is the simulation of the γ rays emitted from Ps. For the simulationof the 2γ annihilation, back-to-back γ rays with the energy of 511 keV are emitted
toward in a random direction from the point where the positron stopped in thepositron simulation. The 3γ annihilation from the o-Ps decay are also generated in
the same manner, but their direction and the energy are calculated from the O(α)
matrix element [30]. Then, the interactions of all the γ rays and their secondary
particles are simulated step by step.
In the end, the energies deposited on the LaBr3 (Ce) scintillators and the plastic scintillator are obtained for o-Ps→ 3γ (normal 3γ decay), o-Ps→ 2γ (pick-oﬀ
annihilation), and o-Ps→p-Ps→ 2γ (stimulated emission). The obtained energies
are smeared by the measured resolutions of the corresponding detectors.

2.5 Experiment

Figure 2.41: Geometry of the Monte Carlo simulation.

63

64

2.5 ExperimentChapter 3

Analysis
The details of the analysis are described in this chapter. First, data sets are described. Calibration, several cuts, time walk correction and charge leak correction
are explained in Sec. 3.2. The detailed description of the event selections to selectthe transition events will follow. In the next section, the systematic errors are
discussed in Sec. 3.4. Finally, the result of this experiment is shown.

3.1

Data Sets

Four RUNs have been performed and detail information of the four RUNs are
summarized in Table 3.1. Most of the parts are the same, but the accumulated
power and the radiation frequency are diﬀerent. RUN I, III and IV use 202.9 GHz
radiation (TE03 mode) and they are diﬀerent in accumulated power in the FabryPérot resonant cavity. RUN I is the most high power RUN and the accumulated
power is 11.0 kW in the average during the DAQ. The accumulated power of 11.0
kW corresponds to the peak intensity of 8.3 × 107 W/m2 and the peak energy
density of 0.28 J/m3 at the center of the Fabry-Pérot resonant cavity. RUN IV is
operated at about half power of RUN I, and RUN III is the power OFF data by
not resonating the Fabry-Pérot cavity. RUN II is the oﬀ-resonance data and uses
140 GHz radiation (TE02 mode) by setting the magnetic ﬁeld strength to about
5.07 T to check systematic uncertainties due to the absorption of the radiation in
the mixed gas.
The pulse frequency of the gyrotron output used in the four RUNs is 20 Hz
and the duty cycle is 30 % as shown in Fig. 3.1. We measure the transition signal
by comparing the data during “beam ON” and “beam OFF”. Duration of total
data taking and live time during “beam ON” and “beam OFF” are summarizedin Table 3.1. The duration of data taking and the live time are not proportional
because the data acquisition is not performed when the Fabry-Pérot cavity is outof resonance (except for RUN III, which is non-resonance RUN).
Data are stored into the multiple calibration units for every 30 minutes. The
calibration, some corrections, and oﬄine trigger cuts described in the next section
65

66

3.2 Analysis

are carried out within one calibration unit. Since the data taking time for one
unit is only 30 minutes, it is guaranteed that the ﬂuctuation and the shift of themeasured quantities by the environment do not propagate over that time scale.
During the measurements, the temperature of the experimental area is controlled
within ±1.3 ◦ C to suppress the ﬂuctuation of the measured quantities due to theelectronics. The diﬀerence of the trigger rate is due to the diﬀerence of alignmentof the source and Pb shield, the pressure and the temperature of the mixed gas.

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











Figure 3.1: The timing diagram of the gyrotron output. The pulse frequency is 20
Hz and the duty ratio is 30 %.

ID

frequency

power

durationlive time (ON)live time (OFF)trigger rate

I
II
III
IV

203 GHz140 GHz203 GHz203 GHz11.0 kW
3.3 kW
0.0 kW
5.6 kW

4.3 days
3.3 days
2.4 days
2.8 days

7.0 × 104 sec
4.3 × 104 sec
4.1 × 104 sec
3.8 × 104 sec

1.6 × 105 sec
1.0 × 105 sec
9.6 × 104 sec
8.9 × 104 sec

949 Hz949 Hz936 Hz932 HzTable 3.1: Properties of the RUNs

3.2

Calibration and Basic Cut

In this section, the energy and time calibration, some corrections, and oﬄine trigger
cut are described. They are carried out within one calibration unit per 30 minutes,
and the “beam ON” data and the “beam OFF” data are processed separately because there is a diﬀerence (less than 2 %) in the width of the pedestal peak between“beam ON” data and “beam OFF” data due to the electrical noise coincident withthe gyrotron output pulse.

67

3.2 Analysis

3.2.1

Energy Spectrum of the Plastic Scintillator

100
80

103

60
102
40

long gate [p.e.]long gate [p.e.]

Energy spectra of the plastic scintillator are calibrated in the number of photoelectrons by the gain of the PMTs. The left ﬁgure of Fig. 3.2 shows the spectrum
measured with long-gate (ps-0) versus that measured with short-gate (ps-0). The
charge incollection of the short-gate energy depends on the amplitude and must
be corrected to obtain the intrinsic amplitude because the diﬀerence of energies ofshort-gate and long-gate is used for accidental rejection. The correction for shortgate energy is as follows. At ﬁrst, the short-gate energy range from 15 p.e. to 45
p.e. is divided into 5 p.e. width regions and the long-gate energy spectra for theseregions are ﬁtted with Landau distribution. The long-gate energy spectra for theregions where the short-gate energy is less than 15 p.e. are not used to determine
the correction curve, since there are many accidental events and it is diﬃcult toﬁnd the peak position without accidental events. Then, the most probable energy
loss values are ﬁtted with second order polynomial functions as in the left ﬁgure of
Fig. 3.2. The short-gate energy is corrected with this function. The right ﬁgure of
Fig. 3.2 shows the 2D energy distribution after this correction.

100
80

103

60
102
40

10

20
0
0

20

40

60
80 100
short gate [p.e.]

1

10

20
0
0

20

40

60
80 100
short gate [p.e.]

1

Figure 3.2: Plastic scintillator energy with long-gate vs that with short-gate. (left)
before the correction of short-gate charge incollection, (right) after the correction

3.2.2

Energy Spectrum of the LaBr3 (Ce) Scintillator

The calibration of the ADC spectra has made use of the pedestal line and the 511
keV positron annihilation line. Each peak is ﬁtted with normal Gaussian function.
Then the ﬁtted center values are used to determine the energy scale of the ADCs.
Figure 3.3 shows energy spectrum measured with one of the LaBr3 (Ce) scintillators(la-0). The time window is restricted within prompt peak (from −3 ns to 1.5 ns),
where positron annihilations are dominant, in order to enhance 511 keV peak. The

68

3.2 Analysis

absolute time calibration is described in the following two subsections. The energy
resolutions of the LaBr3 (Ce) scintillators are summarized in Table 3.2.

Counts / keV

3

×10
400
350
300
250
200
150
100
50

0
100 150 200 250 300 350 400 450 500 550 600
Energy [keV]

Figure 3.3: Energy spectrum of one of the LaBr3 (Ce) scintillators (la-0). The time
window is restricted within prompt peak (from −3 ns to 1.5 ns).
LaBr3 (Ce)

σped

σ511

la-0
la-1
la-2
la-3

0.56 keV
0.57 keV
0.58 keV
0.56 keV

7.99 keV (3.68% FWHM)
8.08 keV (3.72% FWHM)
9.05 keV (4.17% FWHM)
8.29 keV (3.82% FWHM)

Table 3.2: Energy resolutions of the LaBr3 (Ce) scintillators

3.2.3

Time Walk Correction of the LaBr3 (Ce) Scintillator

The absolute scale of the KEK TDC is determined by the external clock source. 2
GHz clock leads to 0.5 ns bin width and 32 μs full range. The calibration of theabsolute value makes use of the prompt events which appears as a sharp peak andexactly stands at decay time t = 0 since it consists of the p-Ps, e+ annihilation,etc., and at the same time, time walk of LaBr3 (Ce) scintillator is corrected. Timewalk is the lag of detection timings of the signals depending on their amplitudes.
The procedures of the time walk correction for LaBr3 (Ce) scintillator are as
follows. At ﬁrst, the energy range from 100 keV to 600 keV is divided with a stepof 25 keV, and the prompt peak of each region is ﬁtted with normal Gaussian.

69

3.2 Analysis

15
10

102

5

time [ns]time [ns]The ﬁtted center values of these regions are ﬁtted with a function expressed as
p0 + p1 /E + p2 E as shown in the left ﬁgure of Fig. 3.4. The relative correction
width is determined by this function at any energy (the left ﬁgure of Fig. 3.4).

15
10

102

5

0

0
10

-5

10
-5

-10

-10
1

-15
100

200

300

400 500 600
energy [keV]

1
-15
100

200

300

400 500 600
energy [keV]

Figure 3.4: The left and the right ﬁgure show the scatter plot of timing vs energy
deposited on one of the LaBr3 (Ce) scintillators (la-0) before and after time walkcorrection, respectively.

3.2.4

Time Walk Correction of the Plastic Scintillator

The time walk of the plastic scintillator is clearly seen in the left ﬁgure of Fig. 3.5,
though the plastic scintillator has a fast rise-time signal. As in the LaBr3 (Ce) case,the energy range of the plastic scintillator (ps-0 short + ps-1 short) from 4 p.e. to
100 p.e. is divided into 16 regions and the prompt peaks for these regions are ﬁtted
with normal Gaussian. Then, the peak values are ﬁtted with a function expressed
as p0 + p1 /E + p2 E + p3 /E 2 as shown in the left ﬁgure of Fig. 3.5. The relative
correction width is determined by this function at any energy (the right ﬁgure of
Fig. 3.5).

3.2.5

Oﬄine Trigger Cuts

Oﬄine trigger cuts are applied before main analysis. The oﬄine trigger is similar
to the online trigger but requires more strict conditions. This cut is applied to eachcalibration unit per 30 minutes.
e+ tagging
To select events such that e+ hits the plastic scintillator, the following energy
selection is applied on the signals of two PMTs placed at the both side of the3.2 Analysis

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

102

10

1
0

50

100
energy [p.e.]time [ns]time [ns]

70

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

102

10

1
0

50

100
energy [p.e.]

Figure 3.5: The left and the right ﬁgure show the scatter plot of timing vs energy deposited on the plastic scintillator before and after time walk correction,
respectively.
plastic scintillator (Fig. 3.6). The second cut is the condition on the energy balance,
which rejects the events such that e+ does not hits near the center of the plastic
scintillator and cross-talk noise.
psps> 2 p.e. & ESG,1> 2 p.e.
• ESG,0




ps
ps 
•  ESG,0− ESG,1
 < 3σ,ps
pswhere ESG,0and ESG,1
are the energy deposited on the plastic scintillator measured
with short-gate with one PMT (ps-0) and the other PMT (ps-1), respectively. Notethat the online threshold for the signals from the two PMTs of the plastic scintillatoris 1 p.e.
In addition, we require that the two signals from the plastic scintillator is coincident within ±3σ (σ = 1.2 ns) (Fig. 3.7).
ps• |tps0 − t1 | < 3σ,pswhere tps
0 and t1 are the timing of the plastic scintillator hit measured with one
PMT (ps-0) and the other PMT (ps-1).

Back-to-back γ-ray hits
To select events such that only two back-to-back γ rays are detected by the LaBr3 (Ce)scintillators, the following energy selection is applied on the energy of the LaBr3 (Ce)scintillators.
• (E0la > 100 keV | E1la > 100 keV) & (E2la > 100 keV | E3la > 100 keV)

71energy (ps-1) [p.e.]

3.2 Analysis

100
103

80
60

102

40
10

20
0

0

20

40
60
80 100energy (ps-0) [p.e.]

1

Figure 3.6: ps-1 energy (short-gate) vs. ps-0 energy (short-gate). The energy
selection is shown in black line.

72

counts / ns

3.3 Analysis

105
104
103
102
10
-25 -20 -15 -10 -5 0 5 10 15 20 25
time (ps-0) - time (ps-1) [ns]Figure 3.7: Time diﬀerence between the ps-1 signal and the ps-0 signal. The
coincidence cut region is shown in black dash line.
• Nhits = 2
Eila is the energy deposited on the i-th LaBr3 (Ce) scintillator. Nhits is the numberof hits on the LaBr3 (Ce) scintillators. The oﬄine threshold is 100 keV.
In addition, we require that the timing of two LaBr3 (Ce) scintillator hits arecoincident within ±3σ (σ = 0.55 ns) (Fig. 3.8). The two LaBr3 (Ce) scintillators
make a back-to-back pair as shown in Fig. 2.36.

la  < 3σ,• tla−tij where tlai is the timing of the i-th LaBr3 (Ce) scintillator hit. Note that the coincidence width of the electronics is 40 ns.
Event rates after each selection are summarized in Table 3.3. About 57 % of
the online-triggered events pass the oﬄine trigger cuts and there is no diﬀerence
between “beam ON” and “beam OFF” so far.

3.3

Event Selection

In this section, the signature of the transition signal and the background are described in the ﬁrst two subsections. Then, event selections to reduce the backgroundare described.

73

counts / 0.5ns

3.3 Analysis

104
103
102
10
1
-25 -20 -15 -10 -5 0 5 10 15 20 25
time (la-0) - time (la-2) [ns]Figure 3.8: Time diﬀerence between the back-to-back signals from the LaBr3 (Ce)scintillators. The coincidence cut region is shown in black dash line.

selection

ON [Hz]OFF [Hz]

online trigger
e+ tagging : energy cut
e+ tagging : time coincidence
back-to-back γ-ray hits : energy cut
back-to-back γ-ray hits : time coincidence

948.55(11)771.75(10)724.90(10)491.51(8)457.44(8)948.59(7)771.94(7)724.97(7)491.56(5)457.47(5)Table 3.3: Summary of the event rates until oﬄine trigger cut.

74

3.3 Analysis

3.3.1

Signatures of the Transition Signal

The transition signals are the events that p-Ps (τ = 125 ps) transited from oPs (τ = 142 ns) decays into two back-to-back monochromatic (511 keV) γ rays.
Therefore the transition signals have distinctive features as follows:• long lifetime of o-Ps (142 ns)• two back-to-back monochromatic 511 keV γ rays
In order to select such events, delayed coincidence and 511 keV energy selection
are applied to the oﬄine-triggered events. Detail will be discussed in Sec. 3.3.3
and 3.3.5.

3.3.2

BackgroundThe background processes are as follows:• pick-oﬀ annihilation of o-Ps
• 3γ decay of o-Ps
• accidental background
The pick-oﬀ annihilation of o-Ps is caused by a collision of o-Ps with the atomic
electrons inside the target gas. It also occurs when the electron of the o-Ps exchanges its spin with electrons of the target materials. This conversion from o-Ps
to p-Ps results in the rapid annihilation into two γ rays (spin-ﬂip). This 2γ decay
has the same kinematics as the transition signals and becomes background.
3γ decays of o-Ps make continuous energy spectrum and some of them become
background because of the ﬁnite resolution of the γ-ray detectors. In addition,some of the 3γ decays are such events that two of the three γ rays are emitted in
the same direction and the other γ ray is emitted in the opposite direction. This
event mimics back-to-back 511 keV γ decay events.
The accidental background is dominant (about 2/3 of the total background)after the delayed coincidence and the 511 keV energy selection are applied. Thisbackground events removed as mentioned in Sec. 3.3.4. After accidental rejection,
the remaining small accidental background estimated from the sideband in the Ps
decay time distribution is subtracted.
Background estimation
Since the gyrotron output is pulse wave of duty 30 %, positroniums are not exposedto the radiation for 70 % of the operation period. The events during the “beamOFF” period are used to estimate background.
In order to avoid the uncertainty of normalization due to the pulse shape (width)
and the other systematics by the oﬄine trigger cuts, “beam OFF” events are normalized to “beam ON” events using the number of prompt decay events (time75

3.3 Analysis

window is set from −3.0 ns to 1.5 ns). About 80 % of the prompt decay eventsare due to the e+ annihilation, therefore the uncertainty from the diﬀerences in Ps
formation probability, transition probability, and pick-oﬀ probability is small.

3.3.3

Delayed Coincidence

Figure 3.9 shows the time diﬀerence between the plastic scintillator and the coincidence signal of the LaBr3 (Ce) scintillators. A sharp peak, called as the prompt
peak, is observed at t = 0, where e+ annihilations and p-Ps decays are dominant.
The time region after the prompt peak is dominated by the o-Ps events and forms
the exponential decay curve. The ﬂat time spectrum far beyond the prompt peakis dominated by the accidental events. The γ-ray hit of the accidental event is notcorrelated with the e+ hit.
Back-to-back 511 keV γ rays are mainly due to the prompt peak. Delayedcoincidence is applied in order to improve S/N. The time window is set from 50 ns
to 350 ns (Fig. 3.9). The event rates after the delayed coincidence is summarizedin Table 3.4.
• 50 ns < t < 350 ns,
where t is the time diﬀerence between the plastic scintillator and the coincidence
signal of the LaBr3 (Ce) scintillators.

counts / 0.5ns / sec



 

102




10 1 

10-1
10-2
0

200

400

600

800

1000
time [ns]Figure 3.9: Time diﬀerence between the plastic scintillator and the coincidence
signal of the LaBr3 (Ce) scintillators. Time window is set from 50 ns to 350 ns(black dash line). Note that this spectrum is a spectrum of the mean time of thetwo back-to-back LaBr3 (Ce) hits.

76

3.3 Analysis

selection
Delayed coincidence

rate (ON) [Hz]rate (OFF) [Hz]67.933(29)67.797(19)Table 3.4: Event rates after the delayed coincidence.

3.3.4

Accidental Rejection

Even if delayed coincidence is applied, accidental events are dominant source ofthe back-to-back 511 keV γ rays. Accidental events are subtracted afterwards but
subtraction of a lot of events spoils statistics. Thus, accidental rejection cut isapplied before subtraction.
Figure 3.10 is time chart of the normal event (upper) and the accidental events(lower). In the case of the accidental events, there is another plastic scintillator
hit at the timing of the γ-ray hit. As a result, the energy deposit on the plastic
scintillator measured with long-gate becomes larger than that measured with shortgate.
The energy diﬀerence of the long-gate energy and the short-gate energy canbe used to reject the accidental events. Figure 3.11 shows the energy diﬀerence
between long-gate and short-gate of the ps-0 signal. Cut window is set from −2.5
p.e. to 1.7 p.e. and the cut is applied to both ps-0 and ps-1. Resolution of thezero peak is about 0.85 p.e.. The event rates after the accidental rejection aresummarized in Table 3.5.
In order to show the eﬃciency of the accidental rejection, time spectrum before(black) and after (blue) accidental rejection are shown in Fig. 3.12. Accidental
coincidence are suppressed and o-Ps decay curve is clearly enhanced.
selection
Accidental rejection

rate (ON) [Hz]rate (OFF) [Hz]29.116(19)29.079(13)Table 3.5: Events rate after the accidental rejection

3.3.5

γ-ray Energy Cut

Finally, the number of events such that back-to-back 511 keV γ-rays are observed
are counted. Figure 3.13 is the energy spectrum when the delayed coincidence andthe accidental rejection are applied and, in addition, a 511 keV γ-ray hit on the
LaBr3 (Ce) scintillator at the opposite side is required (the energy window is set
from 494 keV to 536 keV). The accidental events are only 4 % in time window
after all selections are applied. Remaining accidental background estimated fromthe events in another time window (accidental window) set from 850 ns to 900 ns
is subtracted.

77

3.3 Analysis

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





  



   counts / 0.1p.e. / sec

Figure 3.10: Time chart of the normal event (upper) and the accidental event(lower).




10
8
6
4
2
0

-4

-2

0

2
4
6
8
10
long gate - short gate [p.e.]

Figure 3.11: Energy diﬀerence between long-gate and short-gate of the ps-0 signal.
The cut is shown in black dash line.

78 

3.4 Analysis!
!


 
 
 




Figure 3.12: Time spectrum before (black) and after (blue) accidental rejection.
The signal region is from 494 keV to 536 keV and transition signals are clearlyobserved. The diﬀerence in rate between “beam ON” events and “beam OFF”
events is
RON −ROFF = 303.7(2.3) [mHz] −288.6(1.5) [mHz] = 15.1±2.7(stat.) [mHz] (3.1)

3.4

Systematic Errors

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

3.4.1

Energy Scale and Energy Resolution

If the energy scales and the energy resolutions are diﬀerent between “beam ON”
events and “beam OFF” events, fake signals appear. In order to estimate theuncertainties, the 511 keV peak of “beam ON” events and “beam OFF” events
after the energy calibration are ﬁtted again with Gaussian function. The positions
and the resolutions of the peaks are summarized in Table 3.6. The width includes
pedestal uncertainty. These values are put into the Monte Carlo simulation, andthe diﬀerence of the number of events in the signal region without power is foundto be +0.08 %, thus the systematic error for the transition signal is −0.08 % of the
background.

79

counts / 5keV / sec

3.4 Analysis

0.05
0.04




0.03
0.02
0.01

counts / 5keV / sec

0
460 470 480 490 500 510 520 530 540
energy [keV]

0.004
0.003
0.002
0.001
0

-0.001
460 470 480 490 500 510 520 530 540
energy [keV]

Figure 3.13: Energy spectrum of on-resonance data (RUN I) when the delayedcoincidence and the accidental rejection are applied and a 511 keV γ ray is observed (energy window is set from 494 keV to 536 keV) at the opposite side of the
LaBr3 (Ce) scintillator. Accidental background estimated from the events in thetime window set from 850 ns to 900 ns is subtracted. (top) ON (red line), OFF(black line), (bottom) ON − OFF80

3.4 Analysis

LaBr3 (Ce)

Eoﬀ
511 [keV]oﬀ [keV]
σ511

Eon
511 [keV]on [keV]
σ511

0
1
2
3

511.047(6)511.049(6)511.136(9)511.085(7)7.994(8)8.084(8)9.052(12)8.291(9)511.072(10)511.050(10)511.153(14)511.089(11)7.951(12)8.074(12)9.016(18)8.275(13)Table 3.6: The positions and the widths of the 511 keV peaks of “beam ON” eventsand “beam OFF” events.

3.4.2

Ps Formation Probability and Pick-oﬀ Annihilation Probability

Ps formation probabilities of the “beam ON” and the “beam OFF” data are different when sub-THz radiation resonant with the Fabry-Pérot cavity. Figure 3.14
is the magniﬁed view of the time spectrum of “beam ON”−“beam OFF” just after
the oﬄine trigger cut. Except for the no power data (RUN III), the slow positronannihilation rate of “beam ON” data is smaller than that of “beam OFF” data.
Since we adopt the delayed coincidence, the diﬀerence of the slow positron annihilation rates does not contribute to the systematic error, but it means that the Ps
formation probability of “beam ON” data is larger than that of “beam OFF” data,
because the slow positron is the positron which is not able to form positronium.
Diﬀerence of the Ps formation probability is a source of systematic error.
Actually, the diﬀerence of the Ps formation probability is estimated by counting
the number of events in the time window (accidental events are subtracted) before
energy cut is applied and is found to be +0.27 % in RUN I. Since the Ps formation
probability is not dependent on the γ-ray energy cut, the systematic error is foundto be −0.27 %. The reason of the diﬀerence of the Ps formation probability is not
sure, but the amount of the diﬀerence correlates with the temperature of the gas
ﬁlled in the chamber, as shown in Fig. 3.15. The diﬀerence of the Ps formation
probability is the largest in RUN II, which is oﬀ-resonance RUN using 140 GHz
radiation, because the gas temperature of RUN II is also the highest among thefour RUNs. In other words, we can conclude that if no excess is observed in RUN
II, the uncertainty of the Ps formation probability is not crucial to this experiment.

3.4.3

Accidental Rejection Eﬃciency

The accidental rejection cut also rejects the signal events to some extent. The
eﬃciency of the accidental rejection depends on the rates of the plastic scintillator
signals which go over the discriminator threshold (∼ 1 p.e.). If the “beam ON”
rate is smaller than the “beam OFF” rate, less events are rejected. In order toestimate this systematic eﬀect, the accidental rejection cut eﬃciency is checked
and the diﬀerence between “beam ON” events and “beam OFF” events is −0.17%, which is not dependent on the γ-ray energy cut, thus the systematic error is

81

counts / 2.5ns / sec

3.4 Analysis

0.1
0
-0.1
        

-0.2
-10

-5

0

5

10

15

20




 

25 30
time [ns]Figure 3.14: The diﬀerence of the slow positron annihilation rate.

(Fon-Foff ) / F

off

[%]

2 / ndf
χ

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






3.47 / 3
0.04014 ± 0.007712

p0
  

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

28

30

32

34

36
°
T [ C]

Figure 3.15: The diﬀerence of the Ps formation probabilities. It correlates with thetemperature in the gas chamber. The ﬁtting function (red line) is p0 (T − 27).

82

3.5 Analysis

found to be +0.17 %.

3.4.4

Background Normalization

The background is estimated from the “beam OFF” events and its normalizationis performed using the number of events in the prompt time window (from −3 ns
to 1.5 ns), where the usual e+ annihilation is 77 %, the decay of p-Ps is 14 %,and the slow positron annihilation is 8 %. Since the fraction of the p-Ps eventsis larger than that of the slow positron events, this normalization is conservative
about the Ps formation probability. These events are statistically independent of
the events in the signal region. The uncertainty of the background normalization
can be estimated from the statistical error of the normalization factor, which isonly ±0.03 %.

3.4.5

Summary of the Systematic Errors

Above systematic errors are summarized in Table 3.7. The total systematic error is
calculated as a quadrature sum of them since they are considered to be independent
with each other. The values are percentage of the rate of “beam OFF” events.
sourceRUN I

RUN II

RUN IIIRUN IV

Energy scale and resolution
Ps formation probability
Accidental rejection eﬃciency
Background normalization−0.08 %
−0.27 %
+0.17 %
±0.03 %
+0.17
−0.29 %

+0.06 %
−0.39 %
+0.05 %
±0.04 %
+0.08
−0.39 %

−0.11 %
+0.20 %
+0.13 %
±0.04 %
+0.24
−0.12 %

−0.02 %
−0.13 %
+0.23 %
±0.04 %
+0.24
−0.14 %

Total

Table 3.7: Summary of the systematic errors. The values are percentage of the rateof “beam OFF” events.

3.5

ResultFrom the analysis in this chapter, the following result is obtained for RUN I (202.89
GHz, 11.0 kW).
ON − OFF

(3.2)= 303.7 ± 2.3 (stat.) +0.5
−0.8 (sys.) − 288.6 ± 1.5 (stat.) ± 0.1 (sys.)+0.5= 15.1 ± 2.7 (stat.) −0.8 (sys.) [mHz]

(3.3)
(3.4)

where the ﬁrst error represents a statistical error and the second one is for systematic. The signiﬁcance of the transition signal is 5.4 σ.

83

3.5 Analysis

The results of four RUNs are summarized in Table 3.8 and the power dependence of on-resonance data is shown in Fig. 3.16. The amount of the transition is
proportional to the accumulated power and the oﬀ-resonance data (RUN II) gave
a null result though the systematic uncertainty of the Ps formation probability is
the largest among the four RUNs.
χ2 / ndf
0.1329 / 2
p0
0.4835 ± 0.08465

S/N [%]

7
6
5
4
3
2
1
0
-1
-2

0

2

4

6

8

10
12
Pacc [kW]Figure 3.16: The fraction of the transition signals is proportional to the accumulated power in the Fabry-Pérot resonant cavity.

ID
I
II
III
IV

frequency
203 GHz140 GHz203 GHz203 GHz

power
11.0 kW
3.3 kW
0.0 kW
5.6 kWpressure
2.011 atm
2.015 atm
2.003 atm
2.008 atm

temperature◦

33.94 C
36.55 ◦ C
27.00 ◦ C
30.51 ◦ C

Table 3.8: Results of the four RUNs.

S/N
5.2 ± 1.0 %1.0 ± 1.2 %0.2 ± 1.2 %3.1 ± 1.3 %84

3.5 Analysis

Chapter 4

Discussion
4.1

Various Checks

The several cuts are applied to reject the accidental events and the 3γ decay events.
The measured signal to noise ratio must be independent of the cut condition of theaccidental rejection. On the other hand, it must depend on the energy selection.
To make sure of these relationship, the cut conditions are varied to some extent.
In addition, lifetime of the transition signal is checked for conﬁrmation of theresult.

4.1.1

Accidental Rejection

The default cut window of the accidental rejection is from −2.5 p.e. to 1.7 p.e.,which corresponds to from −3σ to +2σ. The upper limit determines the rejection
power (c.f. Fig. 3.10) and therefore we set the more strict upper limit than the lower
limit. The upper limit is varied to check if the S/N (after remaining accidental
events are subtracted) is independent of it. The result is summarized in Table
4.1. The deviation of the S/N can be explained by the statistical ﬂuctuation (0.7σ,−0.6σ).
upper limit
S/Ndefault
1.7 p.e. (= +2σ)

tight
1.4 p.e. (= +1.64σ)

loose
2.5 p.e. (= +3σ)5.24 ± 0.96(stat.) %5.38 ± 0.98(stat.)5.43 ± 0.91(stat.)

Table 4.1: Check dependency on the upper limit of the accidental rejection cut.

4.1.2

Energy Selection

The default energy window for the 511 keV peak is from 494 keV to 536 keV,which corresponds to from −2σ to +3σ. The lower limit determines the 2γ/3γ
85

86

4.2 Discussion

ratio, therefore the S/N depends on the energy window. The lower limit is varied
to check if the S/N depends on it and the dependency is consistent to the Monte
Carlo (MC) expectation. The result is summarized in Table 4.2. The dependence on
the lower limit of the energy selection is observed and the dependency is consistent
to the MC expectation. The input parameters of the MC simulation is the sameas those determined in the next section.
upper limit

default
494 keV (= −2σ)tight
497 keV (= −1.64σ)loose
486 keV (= −3σ)

S/N
S/N (MC)

5.24 ± 0.96(stat.) %5.37 %

5.43 ± 0.99(stat.) %5.76 %

4.82 ± 0.86(stat.) %4.81 %

Table 4.2: Check dependency on the lower limit of the energy selection.

4.1.3

Time Spectrum of the Transition Signal

Figure 4.1 is the time spectrum of the transition signal, that is the diﬀerence
between the time spectrum of the “beam ON” events and that of the “beam OFF”
events after the accidental rejection and the back-to-back 511 keV cut. The time
spectrum is ﬁtted with exponential function and the lifetime is obtained to be
133 ± 34 ns, which is consistent with the lifetime of o-Ps. The reduced χ2 of the
ﬁt is 0.89 and the probability is 64 %. That is, the transition signals are properly
distributed as expected.

4.2

Comparison with QED Calculation

The energy spectra of the most high power on-resonance data (RUN I) is ﬁtted
with MC spectra to estimate Einstein’s A coeﬃcient of the hyperﬁne transition of
positronium and compare it with QED calculation (A = 3.37 × 10−8 [s−1 ]).
If we set the beam intensity distribution which can be calculated from theaccumulated power Pacc during DAQ and the beam shape inside the Fabry-Pérot
resonant cavity, parameters of the MC spectra are normalization factor C, theEinstein’s A coeﬃcient A, and the probability of the pick-oﬀ annihilation Ppick .
The MC spectra are expressed as follows :SOFF = C[f3γ (0, Ppick , A)S3γ + fpick (0, Ppick , A)S2γ ](4.1)

SON = C{f3γ (I, Ppick , A)S3γ + [fpick (I, Ppick , A) + ftrans (I, Ppick , A)]S2γ }(4.2)

where f3γ (I, Ppick , A), fpick (I, Ppick , A), and ftrans (I, Ppick , A) can be calculated
from the solution of the rate equation for the hyperﬁne transition of the groundstate of Ps (Eq. (1.18)).
The “beam ON” spectrum and the “beam OFF” spectrum are ﬁtted at thesame time as shown in Fig. 4.2. The ﬁt range is from 460 keV to 545 keV and87counts / 25ns / sec

4.3 Discussion
χ2 / ndf
p0
p1
p2

0.004

27.7 / 31
0.004306 ± 0.001392
133.4 ± 33.9
-0.0001139 ± 0.0000540

0.003
0.002
0.001
0
-0.001

100 200 300 400 500 600 700 800 900
time [ns]Figure 4.1: Fitted time spectrum of the transition signals.
the best ﬁt parameters are Ppick = 5.45(22) % and A = 3.07(93) × 10−8 s−1 when
Pacc = 11.0 [kW] (I = 8.3 × 107 W/m2 ,  = 0.28 J/m3 ) is put as the accumulated
power. The reduced χ2 is 0.91 and the probability is 77 %. In order to estimate thesystematic error of A due to the uncertainty of the absolute power estimation, wechange Pacc within its error and check how much A changes. Finally, the Einstein’s
A coeﬃcient is estimated to be
−8 −1
[s ],
A = (3.1+1.6−1.2 ) × 10

(4.3)

which is consistent with the QED calculation 3.37 × 10−8 s−1 .
Note that Ppick is also consistent with the pick-oﬀ probability estimated byﬁtting the time spectrum after the accidental rejection and the back-to-back 511
keV cut are applied. From Fig. 4.3, the o-Ps decay rate of data is 7.417(52) μs(ﬁtting start time = 50 ns), thus Ppick = 5.35(73) %.

4.3

Future Prospect

We are planning to measure the Ps-HFS directly for the ﬁrst time by the endof 2012. In order to measure the Ps-HFS, we have to measure the shape of theresonance curve, whose center value is the Ps-HFS. Therefore we have to performthe measurement of the hyperﬁne transition while changing the radiation frequency.
We plan to measure ﬁve points over a frequency range from 201 GHz to 205 GHz.

88

coutns / 5keV / sec

4.3 Discussion

0.08
0.07
0.06

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

0.05
0.04
0.03
0.02
0.01

coutns / 5keV / sec

0
460 470 480 490 500 510 520 530 540
Energy [keV]

0.08
0.07
0.06
0.05

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

0.04
0.03
0.02
0.01
0
460 470 480 490 500 510 520 530 540
Energy [keV]

Figure 4.2: The fraction of the transition signals is consistent with QED calculation.
The top ﬁgure shows the “beam OFF” spectrum and the bottom ﬁgure shows the“beam ON” spectrum (RUN I).

89

0.008

χ2 / NDF

Γo-Ps [ns-1 ]4.3 Discussion

0.00780.007632.5
20.00741.5
0.0072
1
0.007

20

40

60

80

100 120 140 160
fitting start time [ns]

20

40

60

80

100 120 140 160
fitting start time [ns]Figure 4.3: Fitted decay rate (RUN I). The ﬁtted decay rate is plotted along theﬁtting start time in the left ﬁgure. The reduced χ2 along the ﬁtting start time is also
shown in the right ﬁgure. The accidental rejection and the 511 keV back-to-backselection are performed on the time spectrum used in this ﬁtting.
The output frequency of gyrotron is determined by its cavity size, therefore now
we are developing a new gyrotron whose cavity can be quickly replacable while
keeping vacuum of the gyrotron.
In addition, the transition measurement is expected to improve. There are two
problems in the current optical system. One problem is that the mesh patternmelts when the accumulated power exceeds about 15 kW. Figure 4.4 is a pictureof a melted mesh mirror. This is due to local elevation of temperature, but it
is diﬃcult to cool mesh with water since the mesh pattern is formed on a SiO2
substrate, whose heat conductivity is small (∼ 1.4 W/m·K). The other problem isthe reﬂection from the Fabry-Pérot resonant cavity. The reﬂection goes back to the
gyrotron and interferes with the oscillation in the gyrotron. The interference makes
power estimation diﬃcult and the rapid decrease of reﬂection when the Fabry-Pérot
cavity resonates causes rapid change in the input beam power as shown in Fig. 4.5,
which leads to destabilization.
In order to solve these two problems, we plan to replace the Fabry-Pérot cavity
with a grating ring cavity. Figure 4.6 is a schematic of a ring cavity. The grating
made of Cu corresponds to the Au mesh mirror of the Fabry-Pèrot cavity. The
1st reﬂection of the grating enters the ring cavity and makes many round-trips in
the ring cavity when the ring cavity resonates. The direction of the reﬂection fromthe cavity is the same as that of the 0th reﬂection of the input beam, therefore no
beam goes back to the gyrotron. In addition, the grating is easy to chill since it ismade of Cu only.
Next, let us estimate the accuracy of the Ps-HFS measurement in 2012. From
the result of the transition measurement, we can measure a point at 202.89 GHz
with an accuracy of 20 % with 4 days of DAQ. Figure 4.7 is a transition curve
assuming I = 108 [W/m2 ]. The transition probability at 202.89 GHz is 75 % ofthat at the peak of the transition curve, therefore we can measure a point at the90

4.3 Discussion

0.14
0.13

Vre [V]Vin [V]

Figure 4.4: A picture of a melted mesh mirror.

0.07
0.06
0.05

0.12

0.04

0.11

0.03
0.02

0.1
0.09
100 102 104 106 108 110 112 114 116
ΔL [μm]0.01
0100 102 104 106 108 110 112 114 116
ΔL [μm]Figure 4.5: Interference between input power and reﬂected power around resonancepeak. The left ﬁgure shows input power while changing the cavity length and theright ﬁgure is reﬂected power.

91

4.3 Discussion    

Figure 4.6: A schematic of a grating ring cavity.
peak with 10 % accuracy with 5 days of DAQ. Two points at the half maximum
can be measured with an accuracy of 10 % with 20 days of DAQ. Therefore, if we
can measure the accumulated power to an accuracy of about 10 % relatively, thePs-HFS can be determined with an accuracy of about 400 ppm (Fig. 4.8). As forthe power estimation, the absolute power estimation is quite diﬃcult as describedin this thesis, but what we have to measure in the Ps-HFS measurement is therelative power accumulated in the cavity resonator. One idea of the relative power
measurement is that the position of the hole is displaced from the center of themirror and the hole size is made larger than the half of the wavelength. Then thetransmittance of the hole becomes independent of the wavelength, and determined
only on the beam size at the Cu mirror, which depends on the wavelength. For
example, if we knock four holes per 90 degrees at 10 mm from the center, theﬂuctuation of the transmittance is smaller than ±5 % since the beam size varies
only from 11.28 mm to 11.37 mm within the frequency range of 201.5 GHz - 205.0
GHz. Stability of accumulated power is also necessary. Introduction of a grating
ring cavity is expected to improve the stability of the gyrotron output power, but
we also have to optimize parameters of the feedback control. In addition, we haveto chill the gas chamber in order to stabilize the resonance of the Fabry-Pérot
cavity.
Finally, we address the precise measurement of the Ps-HFS of O(ppm) level.
There are many challenges to the precise measurement of the Ps-HFS with thedirect transition method. Of course, we have to improve the relative accuracy ofthe power estimation to O(0.1 %), which needs further study. Besides, we have to92

2γ annihilation probability

4.3 Discussion

0.05
0.04
  

0.03
0.02
0.01
0
201

202

203

204
205
frequency [GHz]

Figure 4.7: Transition curve assuming I = 108 W/m2 . The measurement of the
red point is described in this thesis.

2γ annihilation probability

χ2 / ndf
p0

0.04

2.979 / 2
0.0366 ± 0.005094

HFS [GHz]

203.3 ± 0.08408

Γ [GHz]1.716 ± 0.2517

0.03
0.02
0.01
0

202

203

204
205
frequency [GHz]

Figure 4.8: Simulation of Ps-HFS measurement assuming statistical accuracy of 10% and power accuracy of 10 %.

4.3 Discussion

93

improve statistics and reduce systematic errors of the transition measurement. The
error of the transition measurement should be also O(0.1 %). We plan to use theslow positron beam and make positroniums in vacuum using a thin metal foil toachieve this accuracy. Figure 4.9 is a schematic of the setup. The slow positron (∼ 1keV) are injected onto a metal foil [31]. Some of the positrons lose their energies
in the foil, diﬀuse back to the surface, and are emitted as Ps. The kinetic energyof Ps is about 1 eV, then only o-Ps can reach the sub-THz radiation region since
the lifetime of o-Ps (142 ns) is 1000 times longer than that of p-Ps (125 ps). The
γ rays from p-Ps and e+ annihilation are shielded by Pb shielding. In addition,
the pick-oﬀ background also vanishes since the Ps are formed in vacuum. As aresult, the energy spectrum of the γ-ray detectors becomes transition signals + 3γ
backgrounds only except for few pileup events of the γ-ray detectors. Therefore
S/N is expected to be improved as well as the statistics thanks to the high intensityof the positron beam. Besides, there is no absorption of the sub-THz radiation,
therefore the main systematic error of the current transition measurement vanishes
and, of course, no loss of the beam power in the gas is an improvement itself.

Figure 4.9: A schematic of an experimental setup with a slow positron beam.

94

4.3 Discussion

Chapter 5

Conclusion
The hyperﬁne transition of the ground state of positronium has been observed forthe ﬁrst time using high power sub-THz radiation. It is M1 transition between
the hyperﬁne structure of the ground state of positronium (Ps-HFS), which is theenergy splitting between the ground state ortho-positronium (o-Ps) and the groundstate para-positronium (p-Ps) due to the spin-spin interaction. The Ps-HFS issigniﬁcantly large (about 203 GHz). Extremely high power sub-THz radiation isnecessary to cause observable amount of stimulated emission between the Ps-HFS,
since the rate of the spontaneous emission between the Ps-HFS is 14 orders ofmagnitude smaller than the decay rate of o-Ps. We develop a new optical system
which consists of a gyrotron, which is a novel high power radiation source for subTHz to THz region, a mode converter, and a Fabry-Pérot resonant cavity, in orderto accumulate sub-THz radiation of 11 kW (peak intensity I = 8.3 × 107 W/m2 ,peak energy density  = 0.28 J/m3 ) in the Fabry-Pérot cavity. Positroniums areformed in gas and the high power radiation in the Fabry-Pérot cavity causes thehyperﬁne transition of the ground state of positronium. We have observed clear
transition signal, which is observed as the increase of 2γ decay of positronium
detected by surrounding γ-ray detectors. The signiﬁcance of the transition signal is
5.4 σ and the amount of the transition signal is consistent with the QED calculation.
We have also performed oﬀ- and half-power measurements. The amount of thetransition signal is proportional to the accumulated power. In addition, there is
no excess in oﬀ-resonance measurement. As a result, we conclude that we haveobserved the hyperﬁne transition of the ground state of positronium.
This measurement is also a great step toward the direct measurement of thePs-HFS. Precise measurements of the Ps-HFS give the direct information on thebound state Quantum Electrodynamics (QED) and have been performed in 1970’s
and 1980’s, but all of them are indirect measurement using Zeeman splitting causedby static magnetic ﬁeld. The most signiﬁcant common systematic uncertainty ofthe previous measurements is non-uniformity of the static magnetic ﬁeld, and there
is a large discrepancy (3.9 σ, 15 ppm) between the measured and the theoreticalvalue of the Ps-HFS. It is necessary to measure the Ps-HFS again with a diﬀerent
95

96

5.0 Conclusion

method free from the uncertainty of static magnetic ﬁeld. We plan to performthe ﬁrst direct measurement of the Ps-HFS with an accuracy of O(100 ppm) by
the end of 2012. A new gyrotron whose cavity can be quickly replacable is underdevelopment since the radiation frequency is determined by the size of the cavity ofgyrotron. In the future, we plan to measure the Ps-HFS precisely with an accuracyof O(ppm) by using a slow positron beam and creating positroniums in vacuum.

Acknowledgements
First, I would like to express my gratitude to my supervisor Prof. Tomio Kobayashi(ICEPP, Univ. of Tokyo) for providing me with this work and his continuous encouragement and supports.
I also wish to express my great appreciation to Prof. Shoji Asai (Univ. of Tokyo)
for his continuous assistance and guidance at this work.
My special thanks goes to Prof. Toshio Namba (ICEPP, Univ. of Tokyo) for
guiding the work in the right direction with his experience.
Of course I did not perform all the work in this thesis alone and, in fact, a lotof it would not have been possible without the support of many other people. FirstI would like to thank Dr. Taikan Suehara (ICEPP, Univ. of Tokyo) who have beenplaying a leading role in this experiment from the scratch. Special thanks also goesto Akira Miyazaki (Univ. of Tokyo) for his excellent cooperation.
The work presented in this thesis is a collaboration study with Research Center for Development of Far-Infrared Region, University of Fukui (FIR-FU). Here I
would like to express my gratitude to Prof. Toshitaka Idehara for developing thegreat radiation source Gyrotron FU CW V and for his support during the experiment at FIR-FU. I also wish to express my great appreciation to Prof. Isamu Ogawa
for many useful discussions and advises on the development of the mode converter.
Special thanks also goes to Prof. Svilen Sabchevski for his great suggestion about
our experimental setup.
At this point, I would like to thank all the staﬀs and students of International
Center for Elementary Particle Physics (ICEPP) at University of Tokyo,
Last but not least, I would like to thank my family for supporting my decision
to study physics and their constant encouragement.

97

98

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