Direct Measurement of
the Hyperfine StructureInterval of Positronium
Using High PowerMillimeter wave Technology
Ph. D. thesis

Akira Miyazaki

Department of Physics
Faculty of Science, The University of Tokyo
December 2013

Abstract
In this thesis, positronium hyperfine structure (Ps-HFS) is directly measured. Thisis the first direct measurement of Ps-HFS. Although Ps-HFS has been indirectly
measured with a Zeeman splitting method since 1952, there is no precedents forthe direct measurement because Ps-HFS is in the millimeter-wave range (frequency∆HFS
Ps = 203 GHz, wavelength = 1.5 mm, energy diﬀerence = 0.83 meV). A gyrotron
oscillator is developed as a high-power millimeter-wave radiation source. A FabryPérot resonant cavity accumulates radiation to equivalent power of about 20 kW
to cause the induced transition between ortho-positronium and para-positronium.
The obtained Ps-HFS value is
+0.15
∆HFS
Ps = 203.39−0.14 (stat.) ± 0.11 (syst.) GHz(1)

At the same time, lifetime of para-positronium is directly measured as
τp–Ps = 89+18−15 (stat.) ± 10 (syst.) ps(2)

This is also the first direct measurement of τp–Ps . They are consistent with QED
calculations.

i

ii

Contents
1 Introduction
1.1 Positronium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Positronium Hyperfine Structure . . . . . . . . . . . . . . . . . . . .
1.3 Theory of Direct Measurement of Ps-HFS . . . . . . . . . . . . . . .
1.4 Millimeter-wave Technology and Our Requirements . . . . . . . . . .
1.5 Previous Measurement of Ps-HFS and Observed Discrepancy . . . .
1.5.1 Indirect Precision Measurement of Ps-HFS . . . . . . . . . .
1.5.2 Theoretical Prediction . . . . . . . . . . . . . . . . . . . . . .
1.5.3 Discrepancy between Experiments and Theory . . . . . . . .

1
1
3
5
8
10
10
11
13

2 Experiment
2.1 Experiment Overview . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Millimeter-wave Optics System . . . . . . . . . . . . . . . . . . . . .
2.2.1 Gyrotron Oscillator . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Fabry-Pérot Resonant Cavity . . . . . . . . . . . . . . . . . .
2.2.3 Estimation of Accumulated Power . . . . . . . . . . . . . . .
2.2.4 Power Stabilization . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Positronium Assembly and γ-ray Detectors . . . . . . . . . . . . . .
2.3.1 Positronium Formation Assembly . . . . . . . . . . . . . . . .
2.3.2 Selection of Gas . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 γ-ray Detectors . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Electronics and Data Acquisition . . . . . . . . . . . . . . . . . . . .
2.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Electronics for the Plastic Scintillator . . . . . . . . . . . . .
2.4.3 Electronics for the γ-ray Detectors . . . . . . . . . . . . . . .
2.4.4 Electronics for the Trigger System . . . . . . . . . . . . . . .

17
17
18
18
29
39
46
49
49
51
56
57
57
59
59
60

3 Analysis
3.1 Acquired Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Calibration and Oﬄine Trigger . . . . . . . . . . . . . . . . . . . . .
3.2.1 Energy Calibration and Correction of the Plastic Scintillator
3.2.2 Energy Calibration of the LaBr3 (Ce) Scintillator . . . . . . .
3.2.3 Time Calibration . . . . . . . . . . . . . . . . . . . . . . . . .

65
65
67
67
67
67

iii

3.2.4 Time-Walk Correction of the LaBr3 (Ce) Scintillator . . . . .
3.2.5 Time-Walk Correction of the Plastic Scintillator . . . . . . .
3.2.6 Oﬄine Trigger Cuts . . . . . . . . . . . . . . . . . . . . . . .
Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Time Window . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Accidental Rejection . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 γ-ray Energy Cut . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Direct Transition Signal and Background Estimation . . . . .
Analysis for Ps-HFS . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Preparation of MC Samples . . . . . . . . . . . . . . . . . . .
3.4.2 Correction of Space Fluctuation . . . . . . . . . . . . . . . .
3.4.3 Correction of Time Fluctuation . . . . . . . . . . . . . . . . .
3.4.4 Calculation of the Reaction Cross-section . . . . . . . . . . .
Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Power Estimation Uncertainty . . . . . . . . . . . . . . . . .
3.5.2 Gas Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . .
3.5.4 Another Systematic Uncertainty . . . . . . . . . . . . . . . .
3.5.5 Summary of the Systematic Errors . . . . . . . . . . . . . . .
Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69
69
71
73
73
74
75
76
77
77
78
78
79
83
83
83
85
86
87
89

4 Discussion
4.1 Consistency Check of Power Estimation . . . . . . . . . . . . . . . .
4.2 Cut Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Time Window . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Accidental Rejection . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Energy Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Power Estimation with 100 kW-Class Gyrotron . . . . . . . .
4.3.2 Ps Formation in Vacuum . . . . . . . . . . . . . . . . . . . .
4.3.3 Frequency Tunable Gyrotron . . . . . . . . . . . . . . . . . .

91
91
92
92
93
93
94
94
95
95

5 Conclusion

97

Acknowledgements

99

3.3

3.4

3.5

3.6

A Gyrotron Theory1
A.1 Calculation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 1
A.2 Theory of the RF Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 1
A.3 Theory of the Electron Beam . . . . . . . . . . . . . . . . . . . . . . 5
A.4 Excitation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
A.5 Soft and Hard Excitation . . . . . . . . . . . . . . . . . . . . . . . . 9
A.6 Alignment Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
A.7 Theory of Gaussian Mode Converter . . . . . . . . . . . . . . . . . . 10iv

B Theory of Positron Acceleration
15
B.1 Toy Model of the Interaction . . . . . . . . . . . . . . . . . . . . . . 15
B.2 Random Walk Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
C Data Summary

17

D Small Systematic Uncertainties

19

v

vi

List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8

Schematic diagram of the lowest energy levels of positronium. . . . . 2
Feynman diagram of the lowest order Ps-HFS. . . . . . . . . . . . . 4
Level diagram of the hyperfine structure of positronium . . . . . . . 5
Theoretical calculation of transition probability. . . . . . . . . . . . . 7
Schematic diagram of THz gap. . . . . . . . . . . . . . . . . . . . . . 8
Zeeman splitting of Ps. . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Schematic view of the previous experiment. . . . . . . . . . . . . . . 12
Historical plot of the Ps-HFS value. . . . . . . . . . . . . . . . . . . 13

2.1 Schematic view of our experimental setup. . . . . . . . . . . . . . . . 18
2.2 Schematic view of the gyrotron oscillator FU CW G1. . . . . . . . . 19
2.3 Schematic of an annular electron beam in the RF cavity. . . . . . . . 20
2.4 Simulated interaction between electrons and the RF field. . . . . . . 22
2.5 Photograph of FU CW G1. . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Measured frequency spectra of FU CW G1. . . . . . . . . . . . . . . 26
2.7 Schematic view of toroidal mirror. . . . . . . . . . . . . . . . . . . . 27
2.8 Calculated output pattern of FU CW G1. . . . . . . . . . . . . . . . 28
2.9 Measured output pattern of FU CW G1. . . . . . . . . . . . . . . . . 28
2.10 Measured power distribution of Gaussian beam. . . . . . . . . . . . . 28
2.11 Schematic view of a Fabry-Pérot resonant cavity. . . . . . . . . . . . 29
2.12 Calculated resonances of the Fabry-Pérot resonant cavity. . . . . . . 32
2.13 Segmental view of mesh-mirror structure on a substrate. . . . . . . . 33
2.14 Simulated property of the mesh mirror using CST MW Studio. . . . 34
2.15 Expected Gain of the Fabry-Pérot cavity in case of plane wave input. 35
2.16 Photograph of gold mesh on silicon substrate. . . . . . . . . . . . . . 35
2.17 Principle of operation of an SIDMTM (Smooth Impact Drive Mechanism)
actuator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.18 Pyroelectric detectors (Spectrum Detector Inc. SPH-49). . . . . . . . 37
2.19 Schematic view and photograph of Fabry-Pérot cavity arrangement. 38
2.20 Measured resonance of the Fabry-Pérot cavity arrangement. . . . . . 39
2.21 Mode pattern at the end mirror. . . . . . . . . . . . . . . . . . . . . 40
2.22 Schematic view and photograph of simultaneous measurement of Vtr
and Pin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.23 Data fitting for the power calibration. . . . . . . . . . . . . . . . . . 43
vii

2.24 Data fitting for the reflection correction. . . . . . . . . . . . . . . . .
2.25 Result of C measurement. . . . . . . . . . . . . . . . . . . . . . . . .
2.26 Electron beam current stabilized by heater-voltage feedback. . . . .
2.27 Accumulated power stabilized by control of the mirror position. . . .
2.28 Photograph of the chamber. . . . . . . . . . . . . . . . . . . . . . . .
2.29 Schematic picture of a gas chamber (top view). . . . . . . . . . . . .
2.30 Schematic picture of a gas chamber (side view). . . . . . . . . . . . .
2.31 Time spectra of slow positron and o-Ps in 1 atm nitrogen. . . . . . .
2.32 Calculated frequency dependence of power absorption by isobutane.
2.33 Increase of positronium formation with random walk model of positron
acceleration with millimeter-wave radiation. . . . . . . . . . . . . . .
2.34 Time spectra of increase of positronium formation in 1 atm nitrogengas by millimeter-wave radiation. . . . . . . . . . . . . . . . . . . . .
2.35 Photograph of LaBr3 (Ce) crystals . . . . . . . . . . . . . . . . . . . .
2.36 The schematic view of the whole DAQ system. . . . . . . . . . . . .
2.37 Circuit diagram for plastic scintillator system. . . . . . . . . . . . . .
2.38 Schematic diagram of electronics for γ-ray detector system. . . . . .
2.39 Schematic diagram of electronics for trigger system. . . . . . . . . .
2D energy spectra of plastic scintillator measured with the short-gate
and the long-gate. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Energy spectrum of a LaBr3 (Ce) scintillator. . . . . . . . . . . . . .
3.3 2D plots of time vs energy deposited in one LaBr3 (Ce) scintillator. .
3.4 2D plots of time vs energy deposited in the plastic scintillator. . . .
3.5 2D energy spectrum of plastic scintillator. . . . . . . . . . . . . . . .
3.6 Time coincidence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Time spectrum obtained by time diﬀerence of the LaBr3 (Ce) signal
and plastic scintillator one. . . . . . . . . . . . . . . . . . . . . . . .
3.8 Cut condition of accidental rejection of the plastic scintillator. . . . .
3.9 Expected energy spectrum of LaBr3 (Ce) scintillators with Monte
Carlo simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Histograms after event selections. . . . . . . . . . . . . . . . . . . . .
3.11 Positrion distribution of Ps formation with MC. . . . . . . . . . . . .
3.12 Pulse shape of detected power. . . . . . . . . . . . . . . . . . . . . .
3.13 Histogram of accumulated power fluctuation in pulse. . . . . . . . .
3.14 Fitting result of measured cross-section of Ps-HFS transition. . . . .
3.15 Correlation among three fitting variables. . . . . . . . . . . . . . . .
3.16 Measured Stark shift of ∆Ps
HFS vs nitrogen density. . . . . . . . . . .

44
46
47
48
49
50
50
52
53
54
55
56
58
61
62
63

3.1

4.1
4.2
4.3

67
68
69
70
71
72
73
74
75
76
78
79
79
81
82
85

Relative power estimation with the slow positron decrease. . . . . . . 92
Schematic view of a possible future setup. . . . . . . . . . . . . . . . 94
Schematic diagram of dispersion relation of electrons and RF cavity. 96

A.1 Cross-sectional geometry of the RF cavity used in FU CW G1. . . .
viii

2

A.2 Transverse structure of the electric field in the RF cavity (TE52 mode). 3
A.3 Axial structure of the electric field in the RF cavity. . . . . . . . . . 4
A.4 Simulated magnetic field Bz on the z-axis. . . . . . . . . . . . . . . . 5
A.5 Simulated electron trajectories from the MIG. . . . . . . . . . . . . . 6
A.6 Coupling factor of the TE52 mode. . . . . . . . . . . . . . . . . . . . 7
A.7 Calculated output power of gyrotron. . . . . . . . . . . . . . . . . . . 9
A.8 Oscillation eﬃciency vs beam current. . . . . . . . . . . . . . . . . . 10
A.9 Dependence of Ist on displacement. . . . . . . . . . . . . . . . . . . . 11
A.10 Rays of time-averaged propagating TEmn mode in a waveguide. . . . 11
A.11 Schematic view of the helical launcher. . . . . . . . . . . . . . . . . . 11
A.12 Schematic view of Gaussian converter. . . . . . . . . . . . . . . . . . 13
B.1 Simulated Ps formation dependence on power. . . . . . . . . . . . . . 16ix

x

List of Tables
1.1
1.2
1.3

Comparison of positronium and hydrogen atom. . . . . . . . . . . . . 1
The classification of Ps according to its quantum numbers. . . . . . 3
History of the Ps-HFS measurements using the Zeeman eﬀect. . . . . 10

2.1
2.2
2.3
2.4
2.5
2.6
2.7

Operation parameters of FU CW G1. . . . . . . . . . . . . . . . . .
Operating points used for this measurement. . . . . . . . . . . . . .
Example of the measured shape correction for some frequencies. . . .
Example of C measurement and Vin correction. . . . . . . . . . . . .
Measured absolute calibration constant C. . . . . . . . . . . . . . . .
Property of gas (1 atm). . . . . . . . . . . . . . . . . . . . . . . . . .
Properties of LaBr3 (Ce) scintillator . . . . . . . . . . . . . . . . . . .

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9

Data sets for analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Energy resolution of LaBr3 (Ce) crystal scintillators . . . . . . . . . 68
Summary of the event rates by the oﬄine trigger cuts (RUN ID A-1). 72
Summary of the peak width of σlong−short . . . . . . . . . . . . . . . . 75
The amount of the transition signal. . . . . . . . . . . . . . . . . . . 77
Summary of the reaction cross-section. . . . . . . . . . . . . . . . . . 80
Comparison of pick-oﬀ rate between MC and time spectra. . . . . . 86
Summary of systematic uncertainties. . . . . . . . . . . . . . . . . . 88
Uncertainties in this measurement. . . . . . . . . . . . . . . . . . . . 89

4.1
4.2
4.3
4.4

Relative power constant. . . . . . . . . . . . . . . . . . . . . . . . . .
Check of time window. . . . . . . . . . . . . . . . . . . . . . . . . . .
Check of the accidental rejection cut. . . . . . . . . . . . . . . . . . .
Check of the energy cut. . . . . . . . . . . . . . . . . . . . . . . . . .

92
93
93
93

A.1 Resonant frequency and QD of three modes shown in Fig. A.3. . . .

4

24
24
41
45
46
53
56

C.1 Quantities recorded at main trigger timing. . . . . . . . . . . . . . . 17
C.2 Quantities recorded in synchronization with gyrotron output pulse. . 18
C.3 Quantities monitored with the interlock. . . . . . . . . . . . . . . . . 18
D.1 Accidental rejection eﬃciency. . . . . . . . . . . . . . . . . . . . . . . 20
D.2 Normalization uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . 21
xi

xii

Chapter 1

Introduction
Quantum electrodynamics (QED) is the most successful quantum field theory. A lot
of experiments give results in good agreement with calculated predictions. However,there are two problems in tests of QED: g-2 anomaly of muon (> 3σ) [1] and Lamb
shift of muonic hydrogen (∼ 5σ, so called the proton charge radius puzzle) [2][3].
Uncertainties from the nuclear structure eﬀects might be systematic problems intheoretical calculations. To overcome this limitation, one possibility is to study
purely leptonic systems like positronium. Another problem exists between previous
indirect measurements of hyperfine structure of positronium and purely leptonic
QED predictions. In this thesis, positronium hyperfine structure is measured witha direct transition method. Direct measurement has not yet performed because oftechnological problems, and is a possible future candidate to solve this anomaly.

1.1

Positronium

mass
reduced mass
Bohr radius
I1st
HFS

Ps

Hydrogen1.022 MeV
me /2
0.105 nm
6.8 eV
203.39 GHz

939.0 MeVme
0.053 nm13.6 eV
1.42 GHz

Table 1.1: Comparison of positronium and hydrogen atom.
Positronium (Ps), the bound state of an electron and a positron, is a goodsystem with which to precisely study bound-state QED [4]. Positron and electron
are so much lighter than the lightest hadrons or muon that the eﬀects of strong
interactions are negligible compared with the accuracy of present and any conceivable future experiments. For this reason, Ps is the unique system which can be
described with very high precision by means of the QED only.
1

2

1.1 Introduction

Ps is the lightest hydrogen-like atom. Table 1.1 compares Ps with a hydrogenatom. The Bohr radius of Ps is twice as large as that of hydrogen, and the firstionization potential (I1st ) is half as small as that of hydrogen. The states of Ps
are classified according to the principal quantum number n, a sum of the orbital
angular momentum L, a spin angular momentum S, and a total angular momentumJ
n 2S+1 LJ .
(1.1)
A level structure of Ps are shown in Fig. 1.1. L = 1 states are unstable to emit a
Lyman-α photon (243 nm). Some transitions have been studied precisely [5][6][7][8].
!"#$""""""""%!"#"""""""""""&#"!""""""""""""""%#"#"""""""$&!"!"%"%!"!"%"'!"!&!
%8%7"µ*"'",(%! ,"γ!23456!
'"%#%!
,8%9"-*",8%9"-*"%8&"-*"'"γ!23456!

'",#'"'",#%!
'",#&!

'"%(&!

-"'(.%/!!
A%!
B%! →!
A%"DEFG"H!

%'0"1*"'"γ!23456!

'7,"-:"/6;α!

%7'"-*","γ!23456!

%"%(&"%",(%")!"##*"+! )"$##*"+!

%87<"::"'&,87"=>?"#*;>@(!

(C%!

(C&!

Figure 1.1: Schematic diagram of the lowest energy levels of positronium.
In this thesis, we discuss the magnetic dipole (M1) transition at the groundstate. There are two spin states in the ground state: the 1 3 S1 state (triplet) andthe 1 1 S0 state (singlet). The triplet state is called ortho-positronium (o-Ps), andthe singlet state is called para-positronium (p-Ps). One denotes the o-Ps states
and the p-Ps state with a ket notation |S, m⟩ |1, +1⟩ = | ↑⇑⟩|1, 0⟩ = √12 (| ↑⇓⟩ + | ↓⇑⟩)
o–Ps
(1.2)

|1, −1⟩ = | ↓⇓⟩
p–Ps

|0,

0⟩ =

√1 (| ↑⇓⟩ − | ↓⇑⟩)
2

(1.3)

where m represents projection of S to the quantization axis, ↑ and ⇑ represent
electron spin and positron spin, respectively. The energy level of the o-Ps state isslightly higher than p-Ps by a hyperfine structure interval (HFS) due to the spinspin interaction. As shown in Table 1.1, HFS of Ps (Ps-HFS) is about 203 GHz,and 140 times as high as that of hydrogen (H-HFS). The detail of the hyperfine
structure is described in the next section.

3

1.2 Introduction
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 1.2: The classification of Ps according to its quantum numbers.
L = 0 states decay into some γ rays as shown in Table 1.2. Charge parity ofo-Ps and p-Ps is odd and even, respectively (C = (−1)L+S ). Since a system of nphotons has C = (−1)n , o-Ps decays only into odd γ rays, and p-Ps decays only
into even γ rays due to C-conservation. Single photon decay of o-Ps is prohibitedby energy-momentum conservation. As the number of photons increases, the widthof the decay mode decreases by a factor of the fine structure constant (α ∼ 1/137),
and phase space of the final state is getting smaller. Thus, decay modes into many
photons are highly suppressed. As a result, p-Ps mainly decays into two γ rays ando-Ps mainly decays into three γ rays. The experimental values of the decay rate ofp-Ps and o-Ps areΓp–Ps = 7.990 9(17) × 109 s−1 [9],(1.4)

Γo–Ps = 7.040 1(7) × 106 s−1 [10].

(1.5)

These decay rates are remarkably smaller than hadronic systems like J/Ψ. It is one
advantage of Ps when one studies particle-antiparticle systems precisely.

1.2

Positronium Hyperfine Structure

Let us derive Ps-HFS (∆Ps
HFS ) in the lowest order calculation. An energy shift by
the spin-spin interaction EbPs is expressed as [11]−→ −→EbPs = −µ−e · B,(1.6)−→
where B is the magnetic field at the electron induced by the positron magnetic
moment
]
[
→
−
→1
−
→ −
→
1 −
+
µ ×∇ ,
(1.7)
B =∇×
4π e
r
−
→
and µ∓
e denotes the electron and positron magnetic moment with the Pauli matrix
→
−
σ
→
−
→ ∓e −
σ
µ∓
=
e
me 2

(1.8)
(1.9)

4

1.3 Introduction

Using wave function of Ps ϕ(r) at r = 0, one obtains
2πα −−→ −−→⟨EbPs ⟩ =⟨σe+ · σe− ⟩ |ϕ(0)|2 .
3m2e(1.10)For the ground state(me α)3.
(1.11)
8π
The left figure of Fig. 1.2 shows the diagram of this spin-spin interaction. In
addition, there is another contribution to Ps-HFS: a virtual annihilation channel
shown in the right of Fig. 1.2
→ ⟩
πα ⟨−
Ps
⟨EA
⟩ = 2 S 2 |ϕ(0)|2 ,(1.12)me−→where S denotes total spin. The annihilation channel only contributes to o-Ps
because of C conservation.
|ϕ(0)|2 =

e–

e–

e+

e+

Figure 1.2: Feynman diagram of the lowest order Ps-HFS. Left shows the spinspin interaction ⟨EbPs ⟩ (exchange of covariant photon). Right represents virtualPs ⟩.
annihilation channel ⟨EA
Ps ⟩ of p-Ps from that of o-Ps
Ps-HFS is obtained by subtracting ⟨EbPs ⟩ + ⟨EA∆Ps
HFS = spin exchange + annihilation
1
1= me α 4 + me α 43474= me α ∼ 204 GHz(1.13)
12
One can derive the value of H-HFS (∆H
HFS ) in a similar way except for the virtualannihilation channel
H∆H(1.14)
HFS = ⟨Eb ⟩ ∼ 1.42 GHz,
Ps-HFS is much larger than H-HFS because of its large magnetic moment orlight mass. Ps-HFS is sensitive to new physics via the virtual annihilation channel
which contributes to the Ps-HFS interval by about 43%. This is one advantage ofstudying Ps compared with a hydrogen atom. Although precise direct measurements of the H-HFS interval have been performed using hydrogen masers since
1960’s [12], that of Ps-HFS has not been performed. The first motivation of thisthesis is to measure Ps-HFS with a direct transition method. Theory of the directtransition and its diﬃculty are described in the next section.

5

1.3 Introduction

1.3

Theory of Direct Measurement of Ps-HFS

Figure 1.3 is a level diagram for the hyperfine transition of ground-state Ps withradiation of energy density ρ(ω) per unit angular frequency. The input radiation isassumed to be linearly polarized, and the oscillating magnetic field vector is parallelto |1, 1⟩ and |1, −1⟩. In this case, only |1, 0⟩ and |0, 0⟩ states are aﬀected by theradiation because of angular momentum conservation.

Eo-Ps|1, -1〉|1, 0〉

Γo-Ps|1, +1〉

Γo-Ps

Γo-Ps

3γ decayτ = 142ns

A

Ep-Ps

A 3Bρ

3Bρ

|0, 0〉AΓp-Ps
2γ decayτ = 125ps

Figure 1.3: Level diagram of the hyperfine structure of positronium
There are some methods to calculate transition probability, for example, anumerical calculation of Hamiltonian [13], or using a Green’s function [14]. The
simplest method is a rate equation method and is introduced here. When we denotethe population of |S, m⟩ states as NS,m , then the rate equations are expressed 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)) − Γp–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) − Γo–Ps N1,0 (ω, t)
dN1,+1 (t)= −AN1,+1 (t) − Γo–Ps N1,+1 (t)dtdN1,−1 (t)= −AN1,−1 (t) − Γo–Ps N1,−1 (t).
dt(1.15)(1.16)(1.17)(1.18)

6

1.3 Introduction

where A is the Einstein coeﬃcient of Ps-HFS (i.e. a spontaneous emission rate) [15]A=

4 ℏ2 α 3
ω = 3.37 × 10−8 s−1 ,3 m2e c4 0

(1.19)and ω0 = 2π∆Ps
HFS . Note that this formula is written in the SI unit while Ref. [15]uses the CGS unit. The spontaneous emission occurs independently of input radiation, and emitted photons (millimeter waves) are unpolarized. The Einstein B
coeﬃcient (i.e. the induced transition coeﬃcient) is related to A with a well known
formula [16][17],π 2 c3
B(ω) = A 3 g(ω),(1.20)ℏω0
where g(ω) is a line-shape function which satisfies∫
g(ω)dω = 1.

(1.21)

Induced photons are polarized corresponding to the linearly polarized input radiation. The pre-factor 3 in the term of 3B(ω)ρ(ω) is due to this polarizationeﬀect.
The spectrum line-shape is broadened, and peak height is suppressed as
gn (ω) =∆ωn /21,
π (ω − ω0 )2 + (∆ωn /2)2

(1.22)

where ∆ωn is the natural width (FWHM) because of the finite lifetimes of thestates of Ps, and is expressed as∆ωn = A + Γp–Ps + Γo–Ps .

(1.23)In the above equation, the decay rate of p-Ps (Γp–Ps ) is dominant and the natural
width is about 1.3 GHz in frequency. The pressure broadening can be includedin the above equation by replacing the decay rates of Ps in vacuum with those ingas. The Doppler broadening is very small (FWHMu = 80 MHz) because Ps is
produced in gas of 300 K and is thermalized enough at this level of precision (Wewill discuss the non-thermalization problem of Ps in Sec. 1.5.3).
Let us solve Eq. (1.15) (1.16) (1.17) (1.18) by the low power approximation inwhich the transition rate is small enough (low Rabi frequency) :Γtran ≡ 3B(ω)ρ(ω) ≪ Γp–Ps .

(1.24)

This condition is satisfied if input power is less than 100 kW. In this approximation,
N0,0 promptly decays and decouples from these equations because the opposite
transition from p-Ps to o-Ps can be ignored. Thus, the solutions can be expressed

7

1.3 IntroductionasN0,0 (t) = N0,0 (0) exp [− (Γp–Ps ) t]

(1.25)

N1,0 (t) = N1,0 (0) exp [− (A + Γtran + Γo–Ps ) t]

(1.26)

N1,1 (t) = N1,1 (0) exp [− (A + Γo–Ps ) t]

(1.27)

N1,−1 (t) = N1,−1 (0) exp [− (A + Γo–Ps ) t]

(1.28)

Transition probability in a time window [t0 , t1 ] is
∫ t1
dtN1,0Po–Ps→2γ = Γtran(1.29)

t0

Po-Ps → 2γ

Po-Ps → 2γThis probability is measured by counting the number of 2γ rays emitted fromtransiting o-Ps. Figure 1.4 shows transition probability with time window [0, ∞].

0.14

0.25

0.12
0.1

0.2

0.08

0.15

0.06
0.1
0.04
0.05

0.02
0
201

202

203

0

204
205
206
Frequency [GHz]

1

10

102
Power [kW]Figure 1.4: Theoretical calculation of transition probability. No pressure broadening and Doppler broadening, power broadening are considered. No time window
is required. (left) Transition probability vs frequency (power = 20 kW). (right)
Transition probability vs power (frequency = 203.4 kW).
The induced transition rate Γtran can be reinterpreted with a reaction crosssection σ(ω) [kb] and energy flux I [W/m2 ] [17]
Γtran = σ ·I,ℏω(1.30)and the reaction cross-section and energy flux are given byσ(ω) = 3A

π 2 c2g(ω)ℏω02I = ρ(ω)c(1.31)(1.32)

8

1.4 Introduction

For example, when one focuses power of 10 kW on 1 cm2 , corresponding I is
about 108 [W/m2 ], or energy density of 0.3 [J/m3 ]. Since Po–Ps→2γ depends on
the power flux and is inconvenient, σ(ω) is calculated to obtain Ps-HFS using theindependently measured power flux.
Total transition probability of Ps and hydrogen with the same energy flux diﬀers
only by an eﬀect of diﬀerent reduced masses. The diﬃculty of Ps-HFS measurement compared with H-HFS is due to short lifetime of Ps and the frequency. The
transition is suppressed with the natural width of 2πΓp–Ps ∼ 1.3 GHz, and the
expected value of the transition is reduced with lifetime of o-Ps τo–Ps = 142 ns.
Thus, high-power 203 GHz radiation of over 10 kW power is required to force o-Ps
to transit into p-Ps. A technological challenge related to the frequency is discussed
in the next section.

1.4

Millimeter-wave Technology and Our Requirements

A millimeter wave or terahertz (THz) wave is the last electromagnetic wave thatremains under development. Figure 1.5 shows so called “THz gap” where we haveless understandings and devices than well-developed electronics, optics, or X- andγ-rays. In this frequency range, an electromagnetic wave has a characteristic of
both light and radio-wave. It travels in almost a straight line like a laser beam,
and is strongly diﬀracted like radio-wave at the same time. It is lossy in a waveguide, but is diﬃcult to be handled in free space. Short wavelength requires highmechanical accuracy of waveguides, mirrors, or other devices for its propagation. In
the millimeter-wave range, most of the materials have high reflectance and absorption with strong frequency dependence. It is very diﬃcult to produce high-powerand stable millimeter-wave radiation (>1 W). Even in computer simulations, calculation times of a finite element method can be readily divergent because of highfrequency.

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Figure 1.5: Schematic diagram of THz gap.
There are some applications of millimeter waves recently. Cosmic Microwave

1.4 Introduction

9

Background (CMB) is in the millimeter-wave range, and measurement of the Bmode polarization in CMB is one of the most interesting topics in cosmology [18].
One plans to measure neutrino decay and its absolute mass diﬀerence with sensitive millimeter-wave detectors in KEK and Tsukuba University. Millimeter-wave
radiation is also a probe to study molecules in astronomy [19] because millimeter(THz) wavelength appears in transitions between rotational (vibrational) levels of
many molecules (This phenomenon also appears in this thesis. See Sec. 2.3.2).
Above applications require very sensitive superconducting sensors, for examples,
Transition Edge Sensor (TES), Superconducting-Insulator-Superconducting (SIS)
hetero-dyne detector, Superconducting Tunnel Junction (STJ), Microwave Kinetic
Inductance Detectors (MKIDs).
Many application of superconductors to the millimeter-wave detection meansthat a gap energy of a superconductor is in millimeter-wave energy, and indicatesthat millimeter waves play an important role in material science. In fact, millimeterwaves can be a probe to study an electromagnetic response of a solid near the
Fermi energy. Phonon, exiton, polariton, and other exited modes in solids are in
the millimeter-wave range. Landau levels are also in the millimeter-wave or THz
range, and for example, an application for the quantum Hall eﬀect in graphene isstudied recently [20]. It is worth mentioning that Landau levels are also related toa gyrotron oscillator described in Sec. 2.2.1.
The Ps-HFS direct measurement is one of the greatest technological challenges
in the millimeter-wave field today. Especially, the requirement for very high power(>10 kW) is quite unique compared with other millimeter-wave science and conventional atomic or nuclear physics. Such high power is only studied to heat plasma
in the field of fusion, and not applied to other fundamental science fields [21]. The
direct measurement of Ps-HFS with an precision of about 0.1% requires
1. A high-power millimeter-wave oscillator producing over 100 W.
2. A resonant cavity accumulating equivalent power of over 10 kW.
3. A method to measure accumulating power with accuracy of better than 20 %.
4. Frequency stability of better than 200 MHz.
5. A method to scan frequency from 201 GHz to 206 GHz.
To achieve these requirements, we developed an optical system composed of agyrotron oscillator and a Fabry-Pérot resonant cavity.
A gyrotron is a millimeter-wave oscillator with the highest power developedin plasma physics. Frequency stability of a gyrotron is usually very monochromatic (FWHM is better than 100 kHz) [22]. Since output frequency of a normal
gyrotron is lower than 170 GHz, we developed a new gyrotron with 203 GHz output. It was a technical challenge to fabricate such a high frequency (sub-THz)
gyrotron. Output frequency is tuned by replacing an internal RF cavity. Detailsof our gyrotron are explained in Sec. 2.2.1.

10

1.5 Introduction

A Fabry-Pérot cavity is usually used as an etalon in many frequency regions,and also used as an amplifier in laser physics. Although a basic theory of thiscavity is well known [23], it is non-trivial to put this cavity to practical use in the
millimeter-wave range. Some ideas were needed to accumulate high-power radiationand to measure its power with the accuracy of 20%. Details of the Fabry-Pérot
cavity are described in Sec. 2.2.2. An estimation of accumulated power is one of
the key points in this thesis, and explained in Sec. 2.2.3.

1.5

Previous Measurement of Ps-HFS and Observed
Discrepancy

1.5.1

Indirect Precision Measurement of Ps-HFS
year
1952
1954
1955
1957
1970
1972
1975
1977
1977
1983
1984

∆Ps
HFS (GHz)
203.2(3)203.38(4)203.35(5)203.33(4)203.403(12)203.396(5)203.387 0(16)203.384(4)203.384 9(12)203.387 5(16)203.389 10(74)error (ppm)
1500
200
250
200
58
24
8
20
6
8
3.6

reference
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[13]
[33]

Table 1.3: History of the Ps-HFS measurements using the Zeeman eﬀect.
Ps-HFS has been measured indirectly since 1952 as listed in Table 1.3. All of
the previous measurements used the Zeeman eﬀect, and indirectly obtained thePs-HFS interval. The method of indirect measurement is as follows. In a static
magnetic field, the o-Ps state with m = 0 and the p-Ps state mix, and the resultant
energy level of m = 0 diﬀers from the m = ±1 states which is not aﬀected by themagnetic field as shown in Fig. 1.6. The mixture state depends on the strength of
the static magnetic field, and the energy diﬀerence between the mixed o-Ps state(|+⟩) and the unperturbed o-Ps states (|1, +1⟩, |1, −1⟩) is related to Ps-HFS by
the following equation,
(√
)
∆Ps
HFS
1 + x2 − 1
22g ′ µB B
x=
,
h∆Ps
HFS

∆mix =(1.33)(1.34)

11

1.5 Introductionenergy()
where g ′ = g 1 − 5α2 /24 is the bound state electron g-factor in Ps, B is thestrength of the static magnetic field. Therefore, ∆Ps
HFS can be obtained from ∆mix
and B. ∆mix becomes about 3 GHz when one applies a static magnetic field of
about 1 T.

|＋⟩

2γ decay

Δmix = 3GHz
|1, +1⟩, |1, -1⟩o-Ps

3γ decay

ΔHFS = 203GHz|0, 0⟩

p-Ps

|−⟩
0

~1 T

B [T]

Figure 1.6: Zeeman splitting of Ps.
Figure 1.7 shows a setup of one of the previous measurements. Here, the strong
static magnetic field was produced with an electromagnet, and the microwave ofabout 500 W was produced with a magnetron [13] and a klystron [33]. A Zeemantransition from the unperturbed o-Ps states (|1, +1⟩, |1, −1⟩) to the mixed o-Ps
state (|+⟩) occurs when the microwave frequency matches ∆mix . The number of 2γ
decays increases on resonance since the |+⟩ state mainly decays into 2γ rays. The
transition curve is obtained by measuring the increase of 2γ decays as changing thestrength of the static magnetic field. The combined value of the most accurate two
independent experiments [13, 33] is∆Ps
HFS (exp.) = 203.388 65(67) GHz.

(1.35)The relative accuracy is 3.3 ppm.

1.5.2

Theoretical Prediction

We derived the Ps-HFS value in the lowest order in Sec. 1.2. Higher order corrections except for the first correction [34] had not been calculated when precise
measurements were performed. Measured values and theoretical predictions are
consistent since 1950’s.

12

The magnetic field H, was produced by an elecwith tapered ring-shim pole pieces
tromagnet
and a 10.6-cm air gap. The field was regulated
by a nuclear magnetic resonance system" (Fig. 3).
The transmitter was a frequency synthesizer withfrequency stability better than 0. 1 ppm over thetime periods during which data were taken. With
the magnet cooling water controlled to +0.25'C
and with much of the surface area of the yokecovered with thermal insulation, the stability ofthe magnetic field when controlled by this system
was observed to be better than 0. 5 ppm.
The NMR probe was a 6-mm-diam sphericalsured with two NMR systems. The field at thecenter was 4. 6+ 1.0-ppm lower than at the probe
location. Thus the measured NMR frequency is
0. 9-ppm higher than that of a pure water sample
at the cavity center.
Measurements of the homogeneity of the magnetic field were made using two NMR systems,
one with a fixed probe to regulate the field andthe other with a probe mounted on a positioning
device. The NMR sample for regulation was D, O
doped with GdCl„and the movable NMR sample
was mineraldifferent nuclei were used to
1.5oil;Introduction
avoid interference. A magnetic field plot for a

FIG. 2. Schematic diagram of the experimental
apparatus.

0

CAl

25

Figure 1.7: Schematic view of the previous experiment [31].

There are two approaches used in the studies of bound states. One is the BetheSalpeter method based on an exact two-body relativistic wave equation [11]. The
other is nonrelativistic QED (NRQED), which is the eﬀective theory that emerges
by expanding the QED Lagrangian in electron velocity in the center-of-mass frame.
Thus, the NRQED takes advantage of nonrelativistic energy of the electron andpositron in Ps. However, the NRQED calculations lead to divergent results because
it is the application of the nonrelativistic expansion in the relativistic momentum
region. The divergence is cancelled if one includes additional hard-scale contributions, which is not obtained from the nonrelativistic expansion.
In addition to the diﬃculties in the bound-state QED, calculation of Ps-HFS has
diﬃculties from its symmetry compared with a hydrogen atom or muonium. In a
two-body system composed of mass M and m, recoil eﬀect corrections are generally
suppressed by m/M . Ps is strictly M = m (if CPT invariance is true), and thissuppression does not work. The fourth-order recoil terms of theory of muonium
become the third-order corrections in Ps. Furthermore, the virtual annihilation
channel (the right figure of Fig. 1.2) gives rise to many diagrams to be corrected.
A new method to calculate the higher order corrections up to O(α3 log α−1 )is established in 2000 [35][36][37]. To deal with the divergences in the NRQED,
these calculations employ dimensional regularization, in which contributions from
diﬀerent energy scales are matched automatically. When one denotes the lowest
order result calculated in Sec. 1.2 as ∆Ps
HFS(0) , corrected Ps-HFS can be expressed

13

1.5 Introduction
as [35]{()α 32 6
5
1
Ps
∆Ps
(th)
=
∆
1
−
+
ln
2
+ α2 ln(1.36)HFS
HFS(0)
π 21 714α()]( α )2 [ 1367 5197
6 221 2
159
2
+
−
π +
+
π ln 2 −
ζ(3)
π
378
2016
7
84
56
(
) 3
}
( α )33 α3 2 1
62 68
α
1
−
ln
+
−
ln 2
ln + D
+ ···
2 π
α
15
7
π
α
π
where the coeﬃcient D is estimated as an error of the current calculation using
hyperfine structure of muonium. The QED prediction of Ps-HFS is∆Ps
HFS (th) = 203.391 69(41) GHz.

(1.37)The relative accuracy is 2.0 ppm.

1.5.3

Discrepancy between Experiments and Theory
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)

203.38203.385203.39∆HFS (GHz)

Figure 1.8: 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.
Figure 1.8 shows a 3.9 σ (15ppm) discrepancy between the average of the mostaccurate two measurements (green band) and the theoretical calculation (red band).
This is the third anomaly related to QED tests today. Possible reasons of thediscrepancy are as follows:14

1.5 Introduction
1. new physics beyond the Standard Model
2. miscalculation of the theoretical value
3. common systematic uncertainties in the previous experiments

If the both QED calculation and experiment are correct, the discrepancy indicatescontribution of an unknown particle. Ps-HFS is sensitive to a pseudo-scalar, avector or an axial vector particle with a mass of 1 MeV via the virtual annihilation
channel (the right figure of Fig. 1.2). To test this possible new physics, we haveto check systematic uncertainties in the previous measurements. There are two
possible systematic uncertainties in the previous indirect methods.
• The non-thermalized o-Ps becomes a source of the uncertainties not considered in the previous experiment. The previous measurement used a gas
molecule as an electron donor to produce Ps. Ps-HFS in finite gas pressureis shifted by the Stark eﬀect from electric fields of the gas molecule. One has
to extrapolate Ps-HFS values of some pressure points to that of vacuum. In
the previous extrapolation procedure, they assumed that the collision rate between Ps and gas molecules is proportional to the gas pressure. It means thatmean velocity of Ps is the same for the various pressured gases, i.e. Ps is wellthermalized in its short lifetime. It was reported that this well-thermalized
assumption is not satisfied in the decay rate measurements [10] known as“o-Ps lifetime puzzle” in 1980’s and 90’s. At this late date, it was experimentally indicated that the non-thermalized Ps would also aﬀect on the HFS
measurement, and corrected Ps-HFS favors the theoretical calculation anddisfavors the previous experimental average by 2.7 standard deviations [38].
One has to re-measure Ps-HFS in vacuum to confirm this assumption.
• Any eﬀects related to the static magnetic field can be another source of uncertainties. One possible uncertainty is non-uniformity of the magnetic field.
The formed Ps is widely spread in the microwave cavity whose size is about
20 cm in diameter (Fig. 1.7). The size of the used magnet is limited to preparea uniform magnetic field in the region where Ps is formed. The uncertaintyof the magnetic field strength directly contributes to the measured Ps-HFS
value by Eq. (1.33). One possible approach is to produce a very uniform (ppmlevel) magnetic field by a large superconducting magnet as reported in [38].
The other way is to measure Ps-HFS without a static magnetic field.
It is necessary to verify the discrepancy with a new method. The second motivation of this thesis is to blaze a trail for re-measuring the observed discrepancy
with a method totally diﬀerent from previous experiments. Some independent experiments (using quantum interference [39][40][41][42], optical lasers [43], and a
precise magnetic field and correcting gas eﬀect [44][38]) have been performed, buthave not yet reached a suﬃcient level of precision to completely address the observed discrepancy. A direct measurement of the Ps-HFS transition is the only

1.5 Introduction

15

method free from systematic uncertainties from use of a static magnetic field. Thisnew method firstly reported in this thesis will be a future competitor to the precise
measurements for addressing the discrepancy (See Sec. 4.3).

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 are considered, and the result of the measurement is shown there. In Chapter 4, the power estimation and the condition of
event selections are checked. The future prospect is also described there. Finally,
we conclude this thesis in Chapter 5.

16

1.5 Introduction

Chapter 2

Experiment
2.1

Experiment Overview

Figure 2.1 shows a schematic view of our experimental setup. The experiment isperformed in Research Center for Development of Far-Infrared Region in Universityof Fukui.
The basic principle of this experiment is as follows:
1. A gyrotron oscillator radiates a millimeter-wave Gaussian beam of power from100 to 550 W.
2. A Fabry-Pérot resonant cavity accumulates the Gaussian beam to obtainequivalent power of over 10 kW.
3. A 22 Na source emits a positron, and Ps is formed inside the Fabry-Pérot
resonant cavity filled with gas as an electron source.
4. Millimeter-wave radiation causes the stimulated transition from o-Ps to p-Ps,
5. An increase of 2γ rays from transiting p-Ps decays is measured with γ-ray
detectors.
6. Frequency is tuned from 201 GHz to 206 GHz by changing a RF cavity in the
gyrotron oscillator to measure the cross-section of the Ps-HFS transition.
This chapter is organized as follows. First, an optical system composed of a gyrotron oscillator and a Fabry-Pérot resonant cavity are described. A power estimation method of the Fabry-Pérot cavity is described next. A procedure of powerstabilization is also described. Then, a Ps formation assembly and γ-ray detectors
are described. Special concern is required for gas selection. Finally, electronics anda data acquisition system are described.

17

18

2.2 Experiment

Top View

Gamma-Ray Detectors

Ps
Gyrotron Oscillator

1700 mm100-450 W

MirrorM1

MirrorM2

Positronium Formation AssemblySide View

Na

22

Ps
20 kW

neopentane
1atm
Gas
Chamber

Fabry-Pérot resonant cavity

Figure 2.1: Schematic view of our experimental setup.

2.2

Millimeter-wave Optics System

2.2.1

Gyrotron Oscillator

This experiment needs high power (> 100 W), monochromatic (< 0.1%), and highduty (> 30%) millimeter waves. The only solution today is a gyrotron oscillator.
Principle of gyrotron oscillator
A gyrotron oscillator is one of the cyclotron-resonance-maser fast wave devices[45][46][47]. It is not an amplifier but a self-exciting oscillator of millimeter waves.
Gyrotron oscillation is caused by transitions between inversely distributed Landaulevels of gyrating electrons instead of atomic energy levels used in another maser
device. Since quantum mechanical treatment is not practical, a gyrotron theory
is usually understood classically. A classical theory of gyrotron is described inAppendix. A. In this section, a highlight of the gyrotron mechanism is described.
Figure 2.2 shows a schematic view of the gyrotron oscillator used in this experiment (FU CW G1). The gyrotron is composed of three parts: an RF cavity (it is
not a radio wave but is conventionally called RF) in a superconducting solenoid, a19

2.2 Experiment@80A,!!

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#.0)$&/,%%(%&:!
7"0!,8#0%05(.,'&/,%%(%!

;<&'06,+=!

-0+$&60.6$!
?!+&0)(*$!

:)*&0)(*$!
$.$'+%()&5$0/&&$/,++$%&%,)-!

10-)$+%()&
2)3$'+,()&4")!

-")8'(,.&&/0-)$+!
'0+>(*$!

Figure 2.2: Schematic view of the gyrotron oscillator FU CW G1.
Magnetron Injection Gun (MIG), and a Gaussian beam converter. These all are inhigh vacuum (about 10−5 Pa). Electrons are emitted at an electron beam emitter
applied high voltage (cathode voltage Vk = −18 kV), and its trajectory is controlledwith a gun-coil magnet (∼0.1 T), and 1st anode voltage (∼10 kV). Electrons are
gyrating in the RF cavity where a magnetic field B0 is induced.
Figure 2.3 shows a cross-sectional view of electrons at the RF cavity. The
cyclotron angular frequency of an electron is expressed as
ωc =

eBz
Ω0
=
,
me γ
γ

(2.1)

√
where γ = 1/ 1 − (v/c)2 (v is velocity of the electron) is the Lorentz factor of anelectron. The electron beam of current Ib is cylindrically distributed with radius
Rb corresponding to the shape of the emitter ring at the MIG. The Larmor radius

20

2.2 Experiment(gyrating radius) is expressed as
rL =

v⊥
,
ωc

(2.2)

which is small enough compared with Rb and its thickness. Electrons are in cyclotron motion at random phase ψ because they are thermally emitted from theemitter.
Gyrating electrons and electric fields in RF cavity (radius 5 mm, length 24 mm)
interact with each other according to Maxwell’s equations and the Lorentz force.
The interaction excites strong millimeter-wave radiation (100-550 W) of a waveguide mode (TEmn mode). Radius of the RF cavity and an excited mode (TE52mode in this experiment) determine output frequency. A waveguide mode is converted to a bi-Gaussian beam by the Gaussian beam converter and is output froma window.

!!

!"!

(!
#$% ψ!

&'!

!"!

Figure 2.3: Schematic of an annular electron beam in the RF cavity.
Figures in Fig 2.4 demonstrate the interaction between electrons and the RF
field. Note that electrons are in momentum space here (they are in real space in
Fig. 2.3). We introduce relative phase θ between the electric field and the electron
momentum. At the entrance of the cavity z = 0 mm, momenta of electrons are
in random phase θ because of thermal emission from the MIG. They all have the2.2 Experiment

21

same energy (normalized as unity) determined by cathode voltage. When electrons
start to interact with the RF field at the entrance of the RF cavity (z = 6 mm),some electrons are accelerated (gain energy, or outside the circle of radius 1 in theleft of Fig. 2.4), and the others are decelerated (lose energy, or inside the circle)
corresponding to their initial phases. Because of the Lorentz factor in cyclotron
angular frequency ωc in Eq. (2.1), ωc of accelerated electron decreases while thatof decelerated electron increases.
If RF frequency ωr is slightly higher than the initial value of the cyclotron
frequency ωc , ωc of only decelerated electrons moves closer to exact resonance with
the RF field, thereby losing an increasing amount of energy on each successive
cycle. It is clearly seen that automatic bunching to the deceleration phase occurs
at z = 12 mm (center of the RF cavity) in Fig. 2.4. Finally, at z = 20 mm, theend of the interaction region in the RF cavity, most electrons lose initial energy
as shown in the right figure. The RF field ideally gains 30% of initial energy ofthe electron beam. This is the principle of gyrotron oscillation, and a classicalinterpretation of cyclotron-maser-resonance.
Let us summarize the excitation condition of the gyrotron oscillator:• Use (weakly) a relativistic electron beam
• Set magnetic field strength so that relativistic cyclotron frequency is slightly
smaller than resonance frequency of the RF cavity
The remarkable feature of the gyrotron oscillator is using the relativistic eﬀectof the electron cyclotron motion. The phase bunching itself occurs even if electrons are non-relativistic. However, non-relativistic electrons can never bunch to
the deceleration phase, and does not excite millimeter-wave radiation in gyrotron
operation.
The diﬀerence between cyclotron frequency and RF frequency is called frequency detuning. Frequency detuning determines non-linearity of gyrotron oscillation. Excitation eﬃciency is a single-valued function of electron-beam current Ib incase of small frequency detuning. Output power increases as Ib increases, which iscalled soft excitation. However, if frequency detuning is large, excitation eﬃciency
is a multivalued function of Ib , which is called hard excitation. High eﬃciency
is obtained in the hard excitation condition (See Appendix A.5). In our case,
operation condition is soft but nearly hard excitation, and results in non-trivial
behavior of output power. In case of hard excitation, power starts to be output
at large electron-beam current, and increases as Ib decreases for a while. Then,power increases as Ib increases (See Fig. A.8 for detail). A gyrotron determines
its plasma interaction, and one controls parameters to the required value. Thus, it
is very diﬃcult to set output power as a target of feedback controls and stabilize
output power for a long time (one week). Details of power-stabilization procedure
are described in Sec. 2.2.4

22

2.2 Experiment!()+,-./0+,-./0%*"#!

!

&'('%'$$!
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2()
2
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'()
'

)! '!! ')!

3
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!"!#$γ

!()

&'
&'

&!()

!

!()

!

'

%$"#!

'6
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7
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&!()

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&')! &'!! &)!

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1()

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1

&!()

'()

&'

+,-./0%*"#!

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+,-./0&'(')'$$!

&!()

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&')! &'!! &)!

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%$"#!

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7
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3%3"#4! %!
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+,-./0+,-./0%*"#!

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&'('*+'$$!
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4

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)
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7

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&'

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+,-./0+,-./0%*"#!

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6
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5
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2

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&!()

1

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&'

&!()

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)! '!! ')!

"!
! !(1 !(3 !(4 !(6 ' '(1 '(3 '(4 '(6 17%7"#8! %!
!"!#$γ!

Figure 2.4: Simulated interaction between electrons and the RF field. Electron’s
momenta are calculated at z = 0, 6, 12, 20 mm in the RF cavity (see Fig. A.1
and Fig. A.3) with cathode voltage Vk = −18 kV, electron-beam current Ib =500 mA, magnetic field strength B0 = 7.4 T, cavity radius R2 = 2.475 mm, beam
radius Rb = 0.7 · R2 , and pitch factor = 1.2. Output power Pout = 2.5 kW. The
left figures show normalized momenta of electrons initially in random phase. The
middle figures show histograms of electron phase. The right figures show normalized
electron energy decreasing as passing through the RF cavity (the electromagnetic
field gains energy).

23

2.2 Experiment
Property of FU CW G1Figure 2.5 shows a photograph of the gyrotron (FU CW G1) developed specificallyfor this measurement [48]. Operation parameters of FU CW G1 are summarizedin Table 2.1. Table 2.2 summarizes strength of the superconducting magnetic field,
frequency, and power of each cavity radius. Cathode voltage is fixed during theexperiment. Strength of the superconducting magnetic field is adjusted to the value
determined by the cavity radius. 1st Anode voltage and the gun-coil magnetic field
is controlled depending on the situation to stabilize output power and line-width,
frequency drift.

Figure 2.5: Photograph of FU CW G1.

Output power was measured with a calorimeter, which is a 46 ml water pool ina thin Teflon case. A temperature increase of water was measured with a resistancethermometer Pt100. The calorimeter was placed at angle about 45◦ to prevent a
reflected beam from going back to the RF cavity because the reflected beam makes
a standing wave, and also interferes electron interaction in the RF cavity to reduce
accuracy of the power estimation (See Sec. 2.2.3). Expected output power is from
1 kW to 2 kW, while measured power was less than 600 W, and also varied by24

2.2 Experiment

Cathode voltage
1st Anode voltage
2nd Anode voltage
Gun coil magnetic field
Beam current
Repetition rate
Duty ratio−18 kV−8 ∼ −11 kV
GND
0.1∼0.14 T
300∼400 mA
5 Hz
30 %

Table 2.1: Operation parameters of FU CW G1.

mode

R2 [mm]B0 [T]

fcalc

Qcalc

fmeas

∆f [MHz]

P [W]

TE42
TE52
TE52
TE52
TE52
TE52
TE52
TE52

2.453
2.481
2.475
2.467
2.467
2.463
2.453
2.443

6.57
7.35
7.37
7.40
7.44
7.42
7.43
7.48

180.76202.51203.00203.66203.66203.99204.82205.65

2080
2400
2450
2500
2500
2530
2600
2650

180.56201.83202.64203.00203.25203.51204.56205.31

100
20
20
40
50
20
50
40

300
190
240
550
250
350
410
125

Table 2.2: Operating points used for this measurement.

25

2.2 Experimentfactor of four depending on the cavity. Condition of gyrotron oscillation is very
sensitive to the fabrication accuracy of the RF cavity, quality of the emitter surfaceof the MIG, and alignment between the cavity and the magnetic field. In general,
it is diﬃcult to reproduce the same condition for all gyrotron operations [49] (SeeAppendix A.6 for detail).
Output frequency was measured with hetero-dyne technique using a synthesizer (∼12 GHz) as a local oscillator. An even harmonic mixer (Virginia Diodes,
WR5.1EHM) made of a Schottky barrier diode was used. When we denote gyrotron frequency as fRF , synthesizer frequency as fLO , and harmonic number as n,intermediate frequency (IF) is expressed as
fIF = |fRF − nfLO | .

(2.3)

fIF was measured with an oscilloscope, and analyzed on-line with Fast Fourier
Transform (FFT). Lower Side Band (LSB) corresponds to IF when nfLO < fRF ,and Upper Side Band (USB) does when fRF < nfLO . Note that small LSB at thesame LO means small RF. A harmonic number n = 16 was selected to monitor
RF. Output power of the synthesizer for optimum measurement was from 9 dBm
to 11 dBm. According to Table 2.2, measured frequency fmeas is always smaller
than expected one fcalc by a few 100 MHz. Thermal expansion of cavity radius
may cause the frequency shift in steady state operation of the gyrotron.
Figure 2.6 shows an example of measured FFT spectra of fIF of LSB. Each line
shows a FFT spectrum at diﬀerent time in one gyrotron pulse duration. Measured
frequency line-width is more than 2 MHz, and is drifted randomly about 20 MHz.
Since these eﬀects are very small (<100 ppm), and gyrotron operation is in principle
complicated plasma interaction, many reasons are considered to explain these phenomena. The line-width may be due to large velocity spread ∆v// of the electron
beam or interaction with reflected electromagnetic waves [50][51]. The frequency
drift may be due to voltage depression by space charge or thermal expansion ofcavity radius in one pulse [52]. At the present stage of gyrotron technology, they
are inevitable and result in a systematic error. Fortunately, it is small enoughcompared with present level of accuracy needed in this thesis (0.1% or 200 MHz).
Better line-width and drift is needed to address the observed discrepancy (15 ppm
or 3 MHz) in the future. This technical improvement is feasible because one reported that line-width of a low frequency (< 170 GHz) gyrotron is better than
100 kHz [22].
In order to measure the cross-section curve of the Ps-HFS transition, input
radiation should be scanned from 201 GHz to 206 GHz. As shown in table 2.2,
this was achieved by changing a gyrotron cavity with diﬀerent radius R2 . A gate
valve was inserted between the cavity section and the MIG (Fig. 2.2) not for anemitter surface to be exposed to air during the cavity replacement. It took morethan 1 week after the replacement to flush the emitter, bake the cavity, carefullyalign all elements, and search for the best operation parameters. One exception
is fmeas = 180.56 GHz, where the operation mode is TE42 with the same cavity
radius as the fmeas = 204.56 GHz point to obtain a far oﬀ-resonant point.

26

2.2 Experiment

×10

power [a.u.]

-6

12
11

0.5-3.0 ms

1098

5.0-7.5 ms
10.0-12.5 ms

7
6

30.0-32.5 ms

5
4
3
-25

-20

-15

-10

-5
0
IF (LSB) [MHz]Figure 2.6: Measured frequency spectra of FU CW G1. IF of LSB is shown.
Diﬀerent lines correspond to diﬀerent delay time from the rising edge of pulse. A
time range for FFT is 2.5 ms. One spectrum is a result of the average of 100 cycles.

Output radiation from the window is designed to be a bi-Gaussian beam (SeeAppendix A.7). Power distribution of the bi-Gaussian was predicted with a Kirchhoﬀ integral method as shown in Fig. 2.8. The horizontally long distributionchanges to vertically long one during the propagation. Power distribution wasmeasured with a temperature increase of a thin (1 mm) polyvinyl chloride plate(PVC). Figures 2.9 show photographs of PVC taken with an Infrared (IR) camera.
Measured power distribution is in good agreement with the prediction. Parametersof the beam are beam waist size (wx , wy ) = (15.1 mm, 8.4 mm) and beam waist
position from the output window (z0x , z0y ) = (−102.7 mm, 56.8 mm)
The bi-Gaussian beam is converted to a Gaussian beam with toroidal mirrors
according to the Gaussian lens formula [23]. One should pay attention that focus
points are diﬀerent depending on two curvature radius R1 and R2 as shown in
Fig. 2.7. When the beam is reflected by a toroidal mirror at a reflection angle θ,
focus points F1 and F2 are
cos θ
2
1
F2 = R 2.
2 cos θF1 = R 1

(2.4)
(2.5)

Two mirrors M1 (R1 = 2902 mm, R2 = 1776 mm) and M2 (R1 = 679 mm,

27

2.2 Experiment

R2 = 1482 mm) made of aluminum are shown in Fig.2.1. The reflection angle isfixed at 45◦ . M1 is placed at a distance of 550 mm from the output window, andM2 is placed 600 mm from M1. A new beam waist (size w0 = 8.4 mm) is designed
to be at 600 mm from M2, where the input side mirror of the Fabry-Pérot resonantcavity is placed. Figures 2.10 show measured power distribution of the converted
beam. The beam is almost a perfect Gaussian beam, and eﬃciently couples to
Fabry-Pérot resonant cavity.

!#!"

Figure 2.7: Schematic view of toroidal mirror.

28

40

50

Y [mm]

50

Y [mm]

Y [mm]

2.2 Experiment

40

50
40

30

30

30

20

20

20

10

10

0

0

0

-10

-10

-10

-20

-20

-20

-30

-30

-30

-40

-40

-50
-50 -40 -30 -20 -10 0 10 20 30 40 50
X [mm]-50-50 -40 -30 -20 -10 0

10

-40
-50
-50 -40 -30 -20 -10 0 10 20 30 40 50
X [mm]

10 20 30 40 50
X [mm]

40

50

Y [mm]

50

Y [mm]

Y [mm]

Figure 2.8: Calculated output pattern of FU CW G1. Left figure is power distribution at 200 mm from the output window. Middle figure is at 400 mm. Right
figure is at 700 mm.

40

50
40

30

30

30

20

20

20

10

10

0

0

0

-10

-10

-10

-20

-20

-20

-30

-30

-30

-40

-40

-40

-50
-50 -40 -30 -20 -10 0 10 20 30 40 50
X [mm]-50-50 -40 -30 -20 -10 0 10 20 30 40 50
X [mm]-50-50 -40 -30 -20 -10 0 10 20 30 40 50
X [mm]

1020

30

X [mm]

30

X [mm]

X [mm]

Figure 2.9: Measured output pattern of FU CW G1. Left figure is power distribution at 200 mm from the output window. Middle figure is at 400 mm. Right figure
is at 700 mm.

20

30
20

10

10

0

0

0

-10

-10

-10

-20
-30
-30

10

-20

-20

-10

0

10

20
30
X [mm]

-30
-30

-20

-20

-10

0

10

20
30
X [mm]

-30
-30

-20

-10

0

10

20
30
X [mm]

Figure 2.10: Measured power distribution of Gaussian beam. Left figure is power
distribution at 600 mm from M2. Middle figure is at 750 mm. Right figure is at
900 mm.

29

2.2 Experiment

2.2.2

Fabry-Pérot Resonant Cavity

Output of the gyrotron is at most 550 W, and the Ps-HFS direct transition needs
over 10 kW. A Fabry-Pérot resonant cavity is developed to obtain such high equivalent power. In this section, a general theory of this resonant cavity is derived, andthen specific problems for millimeter-wave radiation are discussed.
Theory of the Fabry-Pérot resonant cavity
A Fabry-Pérot resonant cavity is composed of two parallel mirrors. Figure 2.11
shows a schematic view of the cavity. An electromagnetic wave whose electricfield is denoted as Ein is introduced from the left-hand side. A front mirror (lefthand side) is a half mirror of which the reflection coeﬃcient of amplitude is rf andtransmission one is tf . For simplicity, rf and tf are assumed to be independent of
the propagation direction. A right-bound propagating wave in the cavity is denoted
as Er and a left-bound one is denoted as El . A reflected wave from the cavity is Ere .
An end mirror (right hand side) whose reflection coeﬃcient is re has a very smalltransmission te to sample Er . A transmitted wave is denoted as Etr . Boundary
conditions for these electric fields are
Etr (d) = te Er (d),(2.6)

Ere (0) = −rf Ein (0) + tf El (0).

(2.7)

Note that the negative signs come from phase shift of π by reflection at each mirror.

!"#$%&'!

*.!

!($%&'!
(.!

(.!
!()$%&'!

()!

*)!

!*($%&'!

!+$%&'!

*.!

&$,$/!

&$,$-!

&!

Figure 2.11: Schematic view of a Fabry-Pérot resonant cavity.
Let us consider the simplest case, a monochromatic plane wave (angular frequency ω and wave number k) case. The right-bound propagating wave in the

30

2.2 Experimentcavity can be expressed as
Er (z) = tf Ein (0)e

ikz

∞ [
]j
∑
tf Ein (0)eikz.
re rf e2ikd =1 − rf re e2ikd

(2.8)

j=0

In the same way, the left-bound propagating wave in the cavity is
El (z) = tf re Ein (0)e−ik(z−2d)

∞ [
]j
∑
tf re Ein (0)e−ik(z−2d)
re rf e2ikd =.
1 − rf re e2ikd

(2.9)

j=0

According to the boundary conditions Eq. (2.6), (2.7), the transmitted wave andthe reflected wave can be calculated as
Etr (z) = te Er (d)eik(z−d) =tf teEin (0)eikz
1 − rf re e2ikd(2.10)and[−ikz

Ere (z) = [−rf Ein (0) + tf El (0)] e= −rf +

t2f re e2ikd
1 − rf re e2ikd]
Ein (0)e−ikz , (2.11)

respectively.
Using power reflection coeﬃcients Rf = rf2 , Re = re2 and transmission coefficients Tf = t2f , Te = t2e , transmitted power Ptr ∝ |Etr |2 and reflected powerPre ∝ |Ere |2 are expressed as
Tf Te
1
Ptr
√
,
=
2
Pin
(1 − Rf Re ) 1 + F sin2 kd[√√ ]2
Rf −(Tf +Rf ) Re√
+ (Tf + Rf )F sin2 kd
1− Rf Re
Pre
=
.
Pin
1 + F sin2 kd(2.12)(2.13)

where Pin ∝ |Ein |2 is input power from the gyrotron oscillator, and F is defined as√
4 Rf Re√F =.
(2.14)(1 − Rf Re )2
This cavity resonates in the case that cavity length d is half-integer multiple ofwavelength λ (k = 2π/λ) as shown in Fig. 2.12.
One does not know the reflection and transmission coeﬃcients of each mirror
which determine property of the cavity. Two observable parameters are introduced
in stead of these coeﬃcients. The first one is finesse F defined as
F=

π√
2π
F ∼
.
2
1 − Rf Re(2.15)

31

2.2 Experiment

Using Full Width at Half Maximum (FWHM) of Ptr at one resonance, F can beobtained asπ
FSR [µm]F==,(2.16)
FWHM [radian]
FWHM [µm]where FSR is Free Spectral Range which means a separating distance of resonances
as shown in Fig.2.12. Note that F is related to cavity quality factor Q and numberof axial nodes nQ
F= .
(2.17)n
The other parameter is an input coupling coeﬃcient β. To define β, we introduce an input reflection coeﬃcient |ρin | defined as√√
Rf − (Tf + Rf ) Re√√
1 − Rf Re
Pre (on resonance)√ .
= √(2.18)|ρin | = √Rf + (Tf + Rf ) Re
Pre (oﬀ resonance)√
1 + Rf Re
Then, β is expressed as
1 − |ρin |<1
1 + |ρin |1 + |ρin |β=>1
1 − |ρin |β=(under coupling)(2.19)(over coupling).

(2.20)

β corresponds to eﬀective power input into the resonator, and is obtained by measuring reflected power. One should determine under or over coupling before calculating β.
Cavity gain is expressed as a function of finesse F and coupling β. At first,
round-trip reflectance ρ is introduced as a function of F[(
) √ 2 (
)]2
π2
π
π2
2π
ρ(F) = Rf Re =
1+
−
1+
∼1−
.
2
2
2
2F
2F
2F
F

(2.21)The last approximation is valid if F >> 1 i.e. ρ ∼ 1. If we assume under coupling,ratio Tf /Rf is expressed as a function of β and ρTf1
β [1 − ρ(F)]√=√Rf
ρ(F) 1 + β ρ(F)(2.22)

Next, accumulated equivalent power Pacc inside the cavity can be expressed as(|Er + El |2
Pacc (z)== Tf
Pin|Ein |2√
1+√ Re
1− Rf Re)2

− √1 F sin2 k(z − d)Rf

1 + F sin2 kd(2.23)

32

2.2 Experiment

FSR

1

relative power [a.u.]

relative power [a.u.]

FSR

10-1

10-2

1
0.9
0.8

Ptr

0.7

Pre

0.6
FWHM

0.5
0.4
0.3
0.2

-3

10

0.1
146

146.5

147

0

147.5
148
cavity length [mm]146.96 146.98

147

147.02 147.04
cavity length [mm]Figure 2.12: Calculated resonances of the Fabry-Pérot resonant cavity. The left
shows neighboring resonances. The right represents comparison of Ptr /Pin (redline) and Pre /Pin (blue line). λ is 1.47 mm, F is 100, and β is 1 (critical coupling).

Cavity gain G at resonance is defined by averaging accumulated power ⟨Pacc ⟩(⟨Pacc ⟩1
G==
Pin reso
d

∫ d
Tf
0

√
1+
√ Re
1− Rf Re)2

− √1 F sin2 k(z − d)Rf

1 + F sin2 kd

dz

reso

(
)2
√
√
)
(
Tf
1 + Re2 Re
sin 2kd 

√
√
−
1+
=
2kd
1 + F sin2 kd
1 − Rf Re(1 − Rf Re )2

Tf (1 + Re )√∼.
(1 − Rf Re )2(2.24)

reso(2.25)The last approximation is valid if cavity length d is much longer than wavelength.
From Eq. (2.21) and Eq. (2.22), we obtain
√
)
(
ρ(F)
β [1 − ρ(F)]1√√
G(F, β) =.
(2.26)
1+Re(1 − ρ(F))2 1 + β ρ(F)If the resonance is over coupling, just replace β with 1/β. Since Re is usually very
close to unity, cavity gain can be uniquely determined by observable F and β.
So far, we considered response of a Fabry-Pérot cavity against a plane wave.
In order to stabilize resonance, the cavity should resonate a Gaussian beam [23].
The cavity has a plane mirror (front mirror) and a concave mirror (end mirror). A
Gaussian beam is uniquely determined if beam waist size w0 and its position are33

2.2 Experimentgiven. The waist position is at plane mirror, and w0 is expressed as
√
λ√
w0 =
d(R − d),π(2.27)

where R denotes curvature radius of the end mirror and d does cavity length.
Resonance has low diﬀraction loss if condition)(d<1(2.28)
0< 1−R
is fulfilled. A Fabry-Pérot cavity also couples to higher order Gaussian beams(Hermite-Gaussian beam). Input beam should be shaped (for instance with toroidal
mirrors) and carefully aligned (with accuracy of 1 mm) not to excite these higher
modes (mode matching).
Mirror design of the Fabry-Pérot resonant cavity
According to Eq. (2.26), a cavity should be optimized to have higher F and reasonable β. Coupling β should be near β = 1 (critical coupling), at which all poweris dumped in the cavity. To achieve these conditions, reflectivity and loss of thetwo mirrors should be considered. The stabilization condition Eq. (2.28) should beconsidered.
The end mirror should have as high reflectance Re as possible. A copper concave
mirror is selected (expected Re is 0.9985 and radius of curvature R is 300 mm).
Frequency dependence of reflectance is negligible from 201 GHz to 206 GHz. The
mirror has a small hole (diameter is 0.6 mm) to monitor transmitted power Ptr .
Designed cavity length d is approximately 156 mm, which satisfies Eq. (2.28) with
R = 300 mm.

"!
!!
#
#$%&!

!! "! ('!)*+&$$'%,!
!"#!$%&$'!

Figure 2.13: Segmental view of mesh-mirror structure on a substrate.
The front mirror is the half mirror which has high Rf and low loss Lf =1 − Rf − Tf . Although in the optical range high reflectance is achieved with coatingof multiple thin layers of dielectric materials, they are too lossy and useless in the
millimeter-wave region. Instead, a thin mesh mirror made of gold was developed.

1.8

0.994

0.009

0.993

0.008

0.992

0.007

0.991

0.006

0.99

0.005

600

0.989

0.004

550

0.988

0.003

500

0.987

199

200

201

202

203
204
205
Frequency [GHz]

800
1.6

750
700

Coupling β

0.995

Finesse

2.2 Experiment

Power Transmittance Tf

Power Reflectance Rf

34

1.4

650
1.2
1
0.8
199

200

201

202

203
204
205
Frequency [GHz]

Figure 2.14: Simulated property of the mesh mirror using CST MW Studio. (left)
Comparison of reflectance (red line) and transmittance (blue line). (right) Comparison of finesse (red line) and coupling (blue line).

Figure 2.13 shows a schematic view of a mesh pattern. It is a thin (thickness t)
gold film with many holes, of which line width is a and gap interval is g. The
mirror has high reflectivity with g << λ = 1.47 mm, and periodical holes prevent
diﬀraction loss of a Gaussian beam at the same time. Designed t is 1 µm, which is
thicker than skin depth of gold around 200 GHz (0.18 µm) and is enough thin to
reduce ohmic loss when a millimeter wave passes through this mirror.
The mesh is too thin to be free-standing. It is put on a substrate with thickness
tSub using photo-lithography and liftoﬀ technique. A high-resistivity silicon was
selected as a base material. Its thermal conductivity (150 WK−1 m−1 ) has adequatecooling power with water cooling. A problem is that high refractive index (3.54)
of silicon severely interferes reflectivity of the gold mesh. Reflectance inevitably
depends on frequency. Optimization was needed to obtain reasonable cavity gain
with minimum frequency dependence.
Mesh parameters (a, g, and tSub) were carefully optimized using CST MW Studio [53]. A plane wave was calculated with Frequency Domain Solver under periodic
boundary conditions. The left side in Fig. 2.14 shows the result of the simulation,of which optimized parameters are a=200 µm, g=140 µm, and tSub=1960 µm(t=1 µm). According to Eq. (2.15) and Eq. (2.19), expected finesse and coupling
are obtained (Re = 0.9985) and plotted in the right side of Fig 2.14. Finesse isalways over 450, and coupling is near critical (β ∼ 1). Expected gain is shown in
Fig. 2.15, and is about 400 around 203 GHz. Note that frequency dependence ofgain is only 10% around 203.4 GHz regardless of strong frequency dependence offinesse. If input power is 300 W, accumulated power is expected to be over 100 kW.
Figure 2.16 shows a photograph of the fabricated gold mesh on the silicon base.
A blank space around the mesh is for water cooling. Validation of the simulationneeds a resonance test of the Fabry-Pérot cavity with a cavity length control. Thisis described in the next section.

35

Gain

2.2 Experiment

460
450
440
430
420
410
400
390
380
370

198

199

200

201

202

203

204 205
Frequency [GHz]

Figure 2.15: Expected Gain of the Fabry-Pérot cavity in case of plane wave input.

%&"#'$!

!"#$$!

Figure 2.16: Photograph of gold mesh on silicon substrate.

36

2.2 Experiment

Test of the Fabry-Pérot resonant cavity
A cavity length control requires a long travelling range (∼ 1 mm) and high resolution (< 1 µm) at the same time. The Fabry-Pérot cavity resonates when cavity
length d is half-integer multiple of wavelength λ = 1.47 mm as shown in the left
side of Fig. 2.12. One has to search the exact resonant point by changing cavity
length more than FSR. Cavity length should also stay at the resonant point with
precision of better than FWHM as shown in the right side of Fig. 2.12. Simulatedfinesse F > 450 corresponds to FWHM of shorter than 1.5 µm.
The end mirror is mounted on a special X-axis stage which satisfies above requirements (NANO CONTROL TS102-G). It is an SIDMTM (Smooth Impact Drive
Mechanism) actuator using the piezoelectric element as a driving force. Figure 2.17
shows the principle of SIDM operation. In coarse movement, the X-stage moves
like an inchworm using friction of the piezoelectric element and inertia of the Xstage to achieve 15 mm travel distance. In slight movement, the position can becontrolled at a nanometer level by applying voltage to the piezoelectric element.
Coarse Movement
Driving
part

Piezoelectricelement

Movablebody

Slight Movement
position

position

Piezoelectricelement

Movablebody

Frictiontime
position
Piezoelectricelement

Piezoelectricelement

time

Movablebody

Movablebody

time

Figure 2.17: Principle of operation of an SIDMTM (Smooth Impact Drive
Mechanism) actuator.
Transmitted power Ptr and reflected power Pre should be monitored to control
cavity length. There are many power sensors in the millimeter-wave region, such asa Golay cell, a Schottky barrier diode, a bolometer, and a thermopile detector. A
detector should withstand 100 mW-class millimeter-wave radiation and also havea fast time response to catch up with the gyrotron pulse operation (60 ms, 5 Hz).
Most of the sensors saturate or respond slowly compared with our need.
A pyroelectric detector is selected that is a thermal detector made of lithium
tantalate (LiTaO3 ) and satisfies above requirements. Lithium tantalate is a pyro-

2.2 Experiment

37

electric crystal whose ends become oppositely charged when heated. The output
current of a pyroelectric detector, which is almost proportional to a change intemperature, is converted to voltage output via an operational amplifier and a
feedback resistance. A dynamic range can be controlled by changing the resistor(100 MΩ, 470 MΩ, 1 GΩ are usually used). Figure 2.18 shows a photograph of the
pyroelectric detector (Spectrum Detector Inc. SPH-49). Though expected to besmall, frequency dependence of the pyroelectric detector is unknown and discussed
in Sec. 2.2.3.

Figure 2.18: Pyroelectric detectors (Spectrum Detector Inc. SPH-49).
Figures 2.19 show a setup of the Fabry-Pérot resonant cavity including thepiezoelectric stage and the pyroelectric detectors. The front mirror, optimized
gold mesh, is fixed at the beam waist position of the Gaussian beam. The end
mirror, copper concave mirror, is mounted on the piezoelectric stage placed 156 mm
away from the front mirror. A Gaussian beam from the gyrotron is split into two
ways by a beam splitter (it is not shown in Fig. 2.1). 99% of the beam goesto the Fabry-Pérot cavity, and 1% is sampled by a pyroelectric detector as anincident beam (output voltage = Vin ). A reflected beam from the cavity is again
split and sampled by another pyroelectric detector (output voltage = Vre ). These
two detectors are arranged not to reflect the beam back to the gyrotron because
the reflected beam causes undesired interference. A part of accumulated power
passes through a small hole (diameter is 0.6 mm, and length is 0.5 mm) at thecenter of the end mirror. The other pyroelectric detector measures this transmitted
power (output voltage = Vtr ).
Figure 2.20 shows a result of the resonance test at 203.7 GHz. Measured FWHM
was about 1.7 µm, and according to Eq. (2.16), corresponding finesse is about

38

2.2 Experiment

430 (expected value is 600). The coupling coeﬃcient was 0.23 (expected valueis 1.2). One cannot explain this inconsistency with either underestimation of reflectance or additional loss. It turned out that the monochromatic assumption in
the theory is too ideal for the gyrotron used to test the Fabry-Pérot cavity. The
line-width and drift shown in Fig.2.6 reduce eﬀective finesse and coupling. As aresult, the simulation of the mesh mirror cannot to be validated by a measurementwith the current gyrotron. Moreover, the power estimation inside the Fabry-Pérot
resonant cavity needs a new method instead of using the gain formula given by
Eq. (2.26). This new method is described in the next section.

!,'!
6&"$%("&.('!

!#+!

6'*$%("&.('!

7*0%$/),&..*'$
889:;9!

6&"!

-03&.4$,*"1.5$-(".'(,!
;<=$%%!

)&*+(*,*-.'&-$/.01*!
6.'$$%("&.('!

60--!
6.'!

!"#!

6'*!

$%#&"#&'()&*+#!

2'(".$%&''('! !"#$%&''('!

Figure 2.19: Schematic view and photograph of Fabry-Pérot cavity arrangement (top view).

39

monitored power [A.U.]

2.2 Experiment

1
0.8

Vtr
Vre

0.6
0.4
0.2
0
-10

-5

0

5

10

mirror position [μm]Figure 2.20: Measured resonance of the Fabry-Pérot cavity arrangement. The
red points are voltage of transmitted power Vtr , and the blue points are that ofreflected power Vre . The solid lines are the best fitting results of each data. The
peak position is adjusted to the origin of the horizontal axis. Input frequency is203.7 GHz.

2.2.3

Estimation of Accumulated Power

Transmitted power samples a part of accumulated power according to the boundary
condition Eq. (2.6)
1
Pacc .
(2.29)
Ptr = Te
1 + ReThis formula is valid even if input electromagnetic waves are not perfectly monochromatic. Transmittance of the end mirror through a small hole Te is too complicatedto be calculated. It is also non-trivial to estimate a response of the pyroelectricdetector and interference of the transmitted beam in the base material of the detector and the end mirror. An absolute power-calibration factor C including all
diﬃculties mentioned above is defined as
1 + RePtr = CVtr .
(2.30)
Pacc =Te
C depends on frequency and is experimentally determined as follows:• Prepare a known Gaussian beam with a shape correction• Simultaneously measure Vtr and power of the Gaussian beam Pin using achopper• Correct reflection eﬀects from the end mirror.
Details of the measurement are described from the next section.

40

2.2 Experiment

Shape correction with a PVC sheet and an IR camera
The only choice of the high-power Gaussian beam is one from the gyrotron. The
shape of the radiation is well converted and is almost a perfect Gaussian beam.
However, it is necessary to correct shape diﬀerence mainly due to limited accuracyof millimeter-wave beam alignment (more or less 3 mm). The left figure of Fig. 2.21
shows a typical measured mode pattern, and the right figure shows a theoretical
calculation. The data were fitted by bi-Gaussian, and were appeared to have larger
beam waist size than the prediction by factor of 1.4 and center displacement of
1 mm. They were corrected as follows. Temperature of the fitted function at theorigin T0 was obtained, and temperature T inside three standard deviations was
integrated. Then, a hole-to-beam power ratio Rbeam was calculated as
2 T
πrhole0,
Rbeam = ∫ 2π ∫ 3σ
dθ
T dr
0

(2.31)

0

50

15
10

40

5

20
15

50

10

40

5
30

0
-5

20

-10

30

0
-5

20

-10
10

-15
-20
-20 -15 -10Power [A.U.]

20

Y [mm]

Y [mm]

where rhole denotes radius of the small hole and is 0.3 mm. Systematic errors wereobtained by displacing the origin within the accuracy of alignment, 3 mm, andcalculating minimum and maximum relative change of Rbeam

10
-15

-5

0

5

10

15 20
X [mm]

0-20-20 -15 -10

-5

0

5

10

15 20
X [mm]

Figure 2.21: Mode pattern at the end mirror. (left) Measured mode pattern bytaking temperature distribution of a PVC plate with an IR camera. White line isa fitted bi-Gaussian function. (right) Theoretical mode pattern in the Fabry-Pérot
resonant cavity.
A hole-to-cavity-mode power ratio in the Fabry-Pérot cavity Rcavity was esti-

41

2.2 Experimentmated by integrating a theoretical function∫ rhole()
exp −2(r/wz )2 rdr
1Rcavity = ∫0 ∞·()
1 + Reexp −2(r/wz )2 rdr(2.32)

0{[]}= 1 − exp −2(rhole /wz )2

1,
1 + Rewhere wz denotes beam size at the end mirror expressed as
√
√
λ
d
wz =
R
.
π
R−d

(2.33)(2.34)

Table 2.3 summarizes examples of measured shape ratios at the same position as
the end mirror in the Fabry-Pérot cavity; therefore, Rbeam ∼ Rcavity is expected.
All Rbeam s look consistent with each other, and also in good agreement with Rcavity .
frequency

RcavityRbeam

201.83202.64203.51204.55205.31

6.10 × 10−4
6.12 × 10−4
6.15 × 10−4
6.18 × 10−4
6.21 × 10−4

−4
6.91+0.03−0.35 × 10
+0.08
6.16−0.39 × 10−4
−4
6.05+0.06−0.35 × 10
+0.02
6.49−0.28 × 10−4
−4
6.29+0.13−0.44 × 10

Table 2.3: Example of the measured shape correction for some frequencies. Measurements were performed at the end mirror position in the Fabry-Pérot cavity.

Simultaneous measurement of Vtr and Pin
The gyrotron is in pulse operation (duty ratio 30% and a repetition rate 5 Hz).
For simultaneous measurement, output power should be divided into two opticalpaths. A beam splitter like one used to sample Pin and Pre (See Fig. 2.19) isuseless because its frequency dependence is unknown. A chopper was developed tosynchronously divide gyrotron pulse into two as shown in Fig. 2.22. At one phase,
the Gaussian beam hits a Teflon box to be absorbed by water, and raises watertemperature (∆T ). At the other phase, the Gaussian beam hits the end mirror to
be detected by a pyroelectric detector (Vtr ). No power leak was observed at the
chopper. An eﬀective duty ratio is 15% and a repetition rate is 2.5 Hz. The pulse
width is still 60 ms so that response time eﬀects of pyroelectric detector are thesame as the original case.
Reflection at the Teflon box is 5% (measured and assigned as a systematic
uncertainty) and others are all dumped inside the box filled with 46 ml water. A

42

2.2 Experiment

1%2.#(*3#4!
560%&*789*//:!

!/!
'()*+,&&#&!

$%#&"#&'()**+,-*.!

$-&#%.%/0&,/*)%0%/0#&!

!"#!
#01023&'!
$,%;#%.%/0&,/*<06=%!
!"#$$%&!

6)>6(/%*5,0",(*6*2%5*λ!

Figure 2.22: Schematic view and photograph of simultaneous measurement of Vtr
and Pin (top view).

43

2.2 Experimenttemperature increase of water, carefully stirred with a stick, was measured with aresistance thermometer Pt100 during the experiment (60 sec). The left figure in
Fig. 2.23 shows a typical result of the temperature increase of water. The data
were fitted by a function including thermal diﬀusion of water, which is expressedasf (t < t0 ) = T0

(2.35)[]f (t > t0 ) = T0 + (Tinf − T0 ) 1 − e−α(t−t0 ) ,(2.36)

where t0 is a start time, T0 is initial temperature, Tinf is asymptotic temperature
after long time, and α is a constant determining velocity of thermal diﬀusion. Input
power is expressed asPin =(Tinf − T0 ) × α × 4.2 × 46,
0.95 × duty/2(2.37)

34

Vtr [V]temperature [ °C ]where 0.95 means the transmittance of the Teflon box, and duty ratio is divided
by 2 corresponding to chopping. In case of Fig. 2.23, Pin = 280 W.

32
30

Vtr

0.1753 ± 0.002779

ρ

0.08634 ± 0.004372

0.3

k0.004229 ±

0.25

θ

-701.5 ± 2.704

0.35

0

0.2
28
26

t0
t1

70 ± 1.868

T0

24.02 ± 0.005847

Tinf

43.87 ± 0.01998

α

24
0

15.29 ± 0.03364

0.01042 ± 1.296e-05

0.15
0.1
0.05
0

10 20 30 40 50 60 70 80 90
time

1000

2000

3000

4000 5000
position [µm]Figure 2.23: Data fitting for the power calibration. (left) Measurement of Pin withthe temperature increase of water. Black points are measured data, and a red lineis the best fit. (right) Measurement of Vtr by moving the piezoelectric stage near
the mesh mirror position. Black points are data when the chopper is closed, andred points are measured data. A red line is the best fit.
Systematic uncertainty of this measurement was estimated as follows:• Reflectivity of the Teflon box is 5%/95%.
• Accuracy of the amount of water is 2 ml/46 ml.
• Accuracy of time measurement is 2 s/60 s.
• Accuracy of temperature measurement is 0.2 ◦ C/10 ◦ C.

44

2.2 Experiment

They are combined with square sum.
Transmitted power Vtr was measured at the diﬀerent phase of the chopper.
The reflected beam at the end mirror is again reflected at the RF cavity in the
gyrotron or the output window to make up a standing-wave. The piezoelectric stage
advanced within a few wavelength to measure the standing-wave. The standingwave formula Eq. (2.12) was used here to fit Vtr and re-written as
f (x) =

Vtr
,
√ 2
√
[1 − ρ] + 4 ρ sin2 [k(x − θ)](2.38)

where ρ is round-trip reflectance, and k is wave number and fixed. The right figureof Fig. 2.23 shows the result of this fitting.
Correction of the reflection eﬀect

Vin

0.131 ± 0.0002284

0.2

Vin’
ρ

0.15

k
θ

0.1033 ± 0.001969
0.1034 ± 0.006003
0.004229 ± 0-703.3 ± 3.024

0.1
0.05
0

1000

2000

3000

4000 5000
position [µm]Vin [V]Vin [V]

There is one problem due to reflection at the end mirror other than the standingwave production. The reflected beam from the end mirror reduces oscillationeﬃciency only in measuring Vtr . One assumes no-reflection in gyrotron oscillation described in Sec. 2.2.1. Reflected radiation changes boundary conditions (SeeEq. (A.7) for detail), and disturbs the electric field in the RF cavity. This phenomenon is aﬀected by both electron-beam alignment and reflection conditions,and the same phenomenon is reported in a MW-class gyrotron [49].

0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0

1000

2000

3000

Vin

0.4366 ± 0.002022

Vin’ρ

0.1223 ± 0.0003246
0.0009998 ± 0.001633

kθ

0.004246 ±0

14.4 ± 7.449

4000 5000
position [µm]Figure 2.24: Data fitting for the reflection correction. Black points are data inphase of Pin measurement (no reflection), and red points are data in that of Vtr
measurement. Solid lines are the best fits for each data. The end point mirror
moves a few wavelength near the mesh mirror position of the Fabry-Pérot cavity.
(left) Small correction case. (right) Large correction case.
This eﬀect was corrected by sampling incident power (Vin ) with the beam splitter shown in Fig. 2.19 before the chopper. Although splitting eﬃciency itself is
unknown, one can use it to give Vtr a relative correction at the same frequency.
Vin at Vtr measurement (defined as Vin′ ) was fitted by Eq. (2.38), while Vin at thewater measurement phase was fitted by a constant. Pin value was corrected with45

2.2 Experiment′ in measuring V with reflection. Figures 2.24
Vin′ /Vin to simulate the real value Pintrshow two typical results of Vin measurement at diﬀerent conditions. Systematic
uncertainties were propagated from fitting errors of Vtr , Vin′ and Vin .

Result of the measurement for calibration constant C
Summarizing above discussion, the calibration constant C is expressed as
C=′Rbeam Pin
Rbeam Vin′ Pin
Pacc
=
=
.
Vtr
Rcavity Vtr
Rcavity Vin Vtr(2.39)We measured calibration constant C at three diﬀerent distances (600 mm, 750 mm,
900 mm from the toroidal mirror M2) to check the reflection correction with Vin .
Examples of C measurements are listed in Table 2.4. In case of 201.83 GHz,
reflection correction Vin′ /Vin is almost independent of the distance, and C is within
one standard deviation. Reflection corrections of 202.64 GHz diﬀer by factor offive for independent four measurements; however, C is still within two standard
deviations. It means the reflection correction is necessary to obtain proper C values,
and works well in all cases.
frequency

distance

Rbeam
Rcavity

Pin [W]′
Vin
Vin

Vtr [mV]C [kW/V]

201.83 GHz
201.83 GHz
202.64 GHz
202.64 GHz
202.64 GHz
202.64 GHz

600 mm
900 mm
600 mm
600 mm
900 mm
900 mm1.880+0.195−0.3670.516+0.002−0.015
1.842+0.077−0.242
1.638+0.007−0.1960.505+0.002−0.0150.583+0.002−0.039280(22)239(19)271(21)582(46)240(18)484(38)

0.793
0.864
0.634
0.280
1.010
0.388

17.53(28)5.41(09)
18.04(15)10.92(07)5.42(05)
4.17(03)23.7 +3.1−5.0
19.7 +1.6−1.7
17.5 +1.6−2.7
24.4 +2.7−3.7
22.5 +1.8−1.8
26.3 +2.1−2.7

Table 2.4: Example of C measurement and Vin correction.
The results are all averaged for all frequencies and distances, and summarizedin Table 2.5 and Fig. 2.25. C becomes roughly increasing in decreasing frequency
because transmittance of an evanescent mode is reduced (cutoﬀ frequency is about
290 GHz). A discrete structure near 203 GHz may be due to interference behind theend mirror, and is expected to give a proper result if one fixes the mirror structure.
The value of C is tested with an independent method in Sec. 4.1, and obtained
consistent results.
A systematic error was given by square root of the sum of squares of Rbeam ,Pin , Vtr , Vin , and Vin′ . In addition, standard deviation of each C value for diﬀerent
conditions was combined in a similar way to introduce remaining fluctuations ineach measurement. A systematic error in C measurement directly aﬀects the PsHFS value. The purpose of this experiment, the first direct measurement, can beachieved with this uncertainty. In the future, one should improve the precision by46

2.2 Experiment

100 times to address the observed discrepancy of Ps-HFS. A possible improvement
is discussed in Sec. 4.3.
frequency

C [kW/V]

relative accuracy

180.56201.83202.64203.00203.25203.51204.56205.3132.4+7.8−9.4
23.2+3.4−4.0
23.4+4.5−4.9
14.6+2.2−2.3
13.0+1.5−1.7
10.2+1.3−1.5
11.3+2.3−2.5
11.4+1.4
−1.6

27%
16%
20%
15%
12%
14%
21%
14%

C [kW/V]

Table 2.5: Measured absolute calibration constant C.

40
35
30
25
20
15
10
180

185

190

195

200
205
Frequency [GHz]

Figure 2.25: Result of C measurement.

2.2.4

Power Stabilization

Accumulated power in the Fabry-Pérot resonant cavity should be stabilized toproperly determine the cross-section of the Ps-HFS transition. Resonant power
is a result of output power of the gyrotron, stability of output frequency, andlength of the Fabry-Pérot cavity. The former two are automatically determinedby non-linear plasma-wave interaction in the gyrotron. Controllable parameters

47

2.2 Experiment

430

11.2

420

11
10.8

410

heater voltage [V]beam current [mA]

are magnetic field strength, electron-beam current, electron-beam trajectories, andlength of the Fabry-Pérot cavity.

10.6
400

10.4

390

10.2

380

10

0

10

20

30

40

50
time [sec]Figure 2.26: Electron beam current stabilized by heater-voltage feedback.

Stabilization of electron beam current
Electron beam current Ib in the gyrotron is controlled with heater temperature
at the beam-emitter ring (thermionic electron emission). Heater temperature isaround 1000◦ C if one imposes voltage VH ∼ 10 V to the heater. Ib was stabilizedwith Proportional Integral (PI) control method. The voltage of the heater wascontrolled with its AC high-power supply (MATSUSADA SRJ500) which can communicate with the PC via RS232C. The output voltage of the power supply VAC
was determined by the following equation.
VAC (n) = VAC (n − 1) + KP (Ib (n − 1) − Ib (n)) + KI (Ibtarget − Ib (n)),(2.40)

where VAC (n) is the supply voltage set at the n-th operation, Ib (n) is the beam
current at the n-th measurement, KP is a proportional gain, KI is an integral gain,
and Ibtarget is a target voltage. KP is set to 40 and KI is set to 0.08 (Typical voltage
is VAC ∼ 135 V corresponding to VH ∼ 10 V, Ib ∼ 400 mA). Figure 2.26 is amonitor plot of Ib and VH with this stabilization system. It was achieved that the
gyrotron beam current was stable within ±10 % for both a short time (1 minute)and a long time (one day). Gyrotron oscillation gradually becomes stable andoutputs steady power by constant beam current. Note that output power itself
is not a proper parameter to be stabilized because of strong non-linearity in thehigh-power gyrotron operation. As is mentioned in Sec. 2.2.1, output power can bea multivalued function of Ib in case of nearly hard excitation [46].

48

2.2 Experiment

Stabilization of resonance of the Fabry-Pérot resonant cavity

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
0

50

100

150

200

250
time [sec]

+5
+4
+3
+2
+1
0
–1
–2
–3
–4
–5

mirror position [μm]Vtr [V]

Cavity length is varied without moving the piezoelectric X-stage due to the fluctuation of the temperature of the gas chamber and long-term frequency drift ofgyrotron. Cavity length should be controlled to accomplish on-resonance for longtime. Accumulated power integrated with one pulse (60 ms) was stabilized as follows. Two thresholds Vhigh and Vlow were defined, and cavity length was controlled
so that Vtr was between Vhigh and Vlow . Cavity length was rescanned to search theresonant position only when the Vtr falled below Vlow /2. Figure 2.27 is a monitor
plot of Vtr and the mirror position.
Even if cavity length is well controlled, accumulated power in one pulse is notnecessarily stabilized because line-width (a few MHz) and frequency drift (about
20 MHz) are broader than bandwidth of the Fabry-Pérot cavity. Control of cavity
length cannot catch up with such a fast fluctuation. This resulted in power fluctuation in pulses depending on condition of gyrotron operation. This eﬀect wasinevitable with the current gyrotron, and was corrected in data analysis describedin Sec. 3.4.

Figure 2.27: Accumulated power stabilized by control of the mirror position.

49

2.3 Experiment

2.3

Positronium Assembly and γ-ray Detectors

Photographs of a Ps assembly and γ-ray detectors are shown in Fig. 2.28. Figure 2.29 and 2.30 show top and side view of a schematic picture of a gas chamber,respectively. Ps is formed by a positron from a 22 Na source and an electron in a gas
molecule. If it is formed in a beam region of the Fabry-Pérot cavity, the transition
from o-Ps to p-Ps occurs. γ rays from Ps decays are detected by γ-ray detectors.
front view	

rear view	

inside the chamber	
source & light-guide	

Figure 2.28: Photograph of the chamber.

2.3.1

Positronium Formation Assembly

A 22 Na positron source is mounted on a UV-transparent acrylic light-guide shownin Fig. 2.28. The source emits a positron (end point energy is 546 keV, probability
is 90%), then consecutively emits one 1275 keV γ ray. A thin plastic scintillator(NE102, thickness = 0.1 mm) is placed next to the source to tag emission time ofa positron. Optical photons go through the light-guide which is divided into rightand left to reach two photo-multipliers (PMT) shown in Fig. 2.30 as Pla-0 andPla-1. Coincidence of two PMTs reduces accidental events from dark current orelectric noise. A fine-mesh type PMT (HAMAMATSU R5924-70, 2 inch) is used toreduce eﬀects of a remaining magnetic field (1 mT) of the superconducting magnet
in the gyrotron. A lead plate (thickness = 20 mm) shields 1275 keV γ rays and
511 keV ones from positrons annihilating near the source.
The gas chamber is filled with 1 atm gas. The gold mesh mirror on a silicon
substrate is used as a window of the gas chamber. The silicon substrate is not

50

2.3 Experiment-../0*1)($+1!

'1(*(%$2*3'23.)!

8"9):&;<.=&#)>,*"2&,#34*322"*()!

+",&'()*!
8"0:! 8"0A!

#((234+&-2"4+.!

!"#$$%&'()*!

#(''.)&%3))()!
'(5.)&%(43*()&6(7!

+(2/&%.,1&%3))()!

-"4!

#((234+&'3'.!

8"0@! 8"0B!
?@@&%%!

Figure 2.29: Schematic picture of a gas chamber (top view).
!"#$#%&'$(!'()*!

'(0"$50&(4)!

?'/5>! ?'/5@!

112/-!#.($*#3-.#&*,)!

!'/.$(,-.,(3$(''/$#*!
')/4-."()'4!
!()+#)'),$*(,-.$/0)!

6/7*8-9:);,*<.$/'-.,(3$(''/$#*!
=>>-%%!

Figure 2.30: Schematic picture of a gas chamber (side view).

51

2.3 Experimenttransparent to optical photons so that it plays a role of a light shielding for theplastic scintillator and the light-guide. Gas temperature is cooled with a coolingpipe (20◦ C water flow) and a fan in the chamber as shown in Fig. 2.29. A positron
decelerates as ionizing gas molecules and finally stops in the chamber. The numberof stopping positrons at the beam region is about 2% of all emitted positrons and 7%of positrons tagged by the plastic scintillator. About half of the stopping positrons
becomes Ps. 25% of the formed Ps is p-Ps and immediately (lifetime = 125 ps)
decays into two γ rays (511 keV). 75% of them is o-Ps and is alive long time. Since
7.9% of o-Ps annihilates in colliding with an electron in a gas molecule (pick-oﬀ
annihilation), lifetime of o-Ps is reduces from 142 ns to 131 ns. Neopentane gas
was selected to eliminate backgrounds, and is discussed in more detail in the next
section.

2.3.2

Selection of Gas

Gas selection was one of the most important points in this experiment. Mixed gasof nitrogen and isobutane was used in previous measurements [54]. It appearedthat a non-negligible systematic error remains in this gas. We consider gas eﬀects
divided into following three steps:• Reduction of slow positron• Power absorption by gas• Positron acceleration by millimeter waves and collision with gas molecules
These are described in the following sections.
Slow positron annihilation
According to the Ps formation theory by Ore, a positron (ionization potential =6.8 eV) can produce Ps if its kinetic energy E satisfies
I1st − 6.8 eV < E < Eex ,(2.41)

where I1st and Eex are the first ionization potential and excitation energy of thegas molecule, respectively. This condition is called Ore gap. If kinetic energy of apositron is too small, the positron cannot capture an electron to become a Ps atom.
Less than half of positrons stopped in gas can make Ps. The others become slow
positron and remain in gas. The annihilation rate of slow positron is expressed asΓslow = πr0 cnZeﬀ ,(2.42)

where r0 is the classical radius of electron, n is number of density of gas, and Zeﬀ
is the eﬀective electron number which contributes to the annihilation. Zeﬀ depends
on gas molecules listed in Table 2.6 [55][56].

52

2.3 Experimentcounts / ns / sec

Figure 2.31 shows time spectra of slow positron and o-Ps in 1 atm nitrogen
separated with energy information. When a part of o-Ps transits into p-Ps, its
decay looks the same as slow positron (back-to-back 2γ rays of 511 keV). Signal
to noise ratio (S/N) is too low to observe this transition in case of pure nitrogen.
Lifetime of slow positron should be smaller than that of o-Ps.

10
1
10-1

o-Ps events

10-2
10-3
10-4

slow positron events

0 100 200 300 400 500 600 700 800
time [ns]Figure 2.31: Time spectra of slow positron and o-Ps in 1 atm nitrogen. The red line
shows slow positron (lifetime = 167 ns) by selecting back-to-back 511 keV γ ray.
The black line shows o-Ps (lifetime = 133 ns in nitrogen) by selecting Compton-free(from 360 keV to 450 keV) γ ray.

According to Table 2.6, alkane molecules are favored because of its short lifetime of slow positron. Isobutane gas was used as a quencher of slow positron insome experiments [38][54]. However, it is not proper to use pure isobutane in our
experiment because of power absorption described in the next section.
Absorption of 203 GHz radiation
Polyatomic gases have rotational levels and vibrational levels as an internal degree of freedom. If a molecule also has an electric dipole moment d, it absorbs
electromagnetic radiation via transitions between these levels. In case of 203 GHz(wavelength = 1.47 mm) radiation, the rotational transitions should be considered.
The vibrational transitions are higher than 1 THz and can be ignored.
Energy diﬀerence ∆E between rotational levels is expressed as [57]∆E = 2 ·ℏ2(J + 1) = 2B(J + 1),
2I(2.43)

53

2.3 Experiment
gas

Zeﬀ

lifetime [ns]B [GHz]d [Debye]

Ar
N2
CO2
methaneisobutane
neopentane

33.8
30.5
54.7
142
14,40021,400

147.9
163.9
91.4
35.2
0.35
0.23

no
59.91
11.70
157.13
7.789
4.4 (calc.)

0.00
0.00
0.00
0.00
0.132
0.00

Table 2.6: Property of gas (1 atm).

power absorption [a.u.]where I is a moment of inertia around one axis, and J = 1, 2, 3, ... is a rotational
quantum number. B = ℏ2 /2I is called rotational constant, and measured or calculated for many molecules. B and d for some gas molecules often used in Ps
experiments are listed in Table 2.6 [58][59]. Isobutane has absorption lines near
203 GHz, which are at 186.9 GHz, 202.5 GHz and 218.1 GHz.

0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
180

185

190

195

200

205

210 215 220
frequency [GHz]

Figure 2.32: Calculated frequency dependence of power absorption by isobutane.
Black line is 2 atm, red is 1 atm, green is 0.5 atm and blue is 0.1 atm.
If the absorption line is sharp enough, gyrotron frequency can be tuned tokeep away from the absorption peaks. Line-width ∆f of the power absorption isbroadened by collisions of gas molecules, and is expressed as [57]12πτ(2.44)

1
σvrel n(2.45)∆f = 2 ×τ=

where τ is lifetime of one rotational level determined by the collision, σ is a crosssection of the collision, vrel is relative velocity between the gas molecules, n is

54

2.3 Experiment

number of density of the gas molecule. If we assume elastic scattering and approximate isobutane radius as 3.5Å, σ is ∼ 1.5 × 10−18 m2 . In case of 1 atm and 300 K,
vrel = 465 m/s and n = 2.45 × 1025 m−3 . Estimated ∆f is about 5.4 GHz, which
covers from 197 to 209 GHz as shown in Fig. 2.32. It is fatal that power absorption
by pure isobutane causes depression of finesse of the Fabry-Pérot cavity. Althoughpure isobutane was studied in detail in other experiments [38], it is improper to be
used in this experiment.
Positron acceleration by millimeter waves

G [%]So far we rejected to use pure nitrogen and pure isobutane because of slow positronand power absorption, respectively. However, mixed gas of nitrogen and isobutane
seems to work well. Slow positron lifetime is 3.4 ns (1.9 atm nitrogen and 0.1 atm
isobutane), and power absorption is strongly suppressed by pressure broadening bymixed nitrogen. This mixed gas was previously used in the first observation of thePs-HFS transition at 202.9 GHz [54]. A new systematic problem becomes apparent
when we measure the Ps-HFS resonance curve. The high-power millimeter-wave
field (E ∼ 200 kV/m at 20 kW accumulation) used increases the amount of Ps
formed.

100
80
60
40
20
0
0

10

20

30

40

50

60 70
Pacc [kW]Figure 2.33: Increase of positronium formation with random walk model of positron
acceleration with millimeter-wave radiation.
It was reported in the first indirect measurement of Ps-HFS that Ps formation
probability in nitrogen gas increases under high-power microwaves [24]. It wasinterpreted that slow positrons are accelerated to the energy over the lower edge of
Ore gap defined by Eq. (2.41). This eﬀect was studied in detail using a static electricfield [60][61], but not done for high frequency fields so far. The same phenomenon bya millimeter wave was firstly observed in this experiment. A positron is accelerated

55

2.3 Experimentcounts / 25ns / sec

or decelerated by a millimeter wave due to the Lorentz force. Since frequency ofthis interaction is about 200 GHz, mean energy gain of positron in vacuum is verysmall (about 1 meV). However, in about 1 atm gas, the energy gain can exceed theOre gap due to collisions (a few ps cycle) between a positron and gas molecules.
These collisions randomize the phase of a positron in acceleration and a part ofpositrons statistically obtains kinetic energy of a few eV within several ns. A
simple theory with a random-walk model is derived in Appendix B. Figure 2.33
shows simulated positronium formation probability G in case of 1 atm nitrogengas. It is clearly seen that an increase of Ps formation probability can occur inhigh-power millimeter-wave radiation.

0.08
0.06

o-Ps increase

0.04
0.02
0

-0.02
-0.04

slow positron decrease-0.06
100 200 300 400 500 600 700 800
time [ns]Figure 2.34: Time spectra of increase of positronium formation in 1 atm nitrogengas by millimeter-wave radiation. Millimeter-wave radiation of equivalent power
Pacc = 24 kW is confined in the Fabry-Pérot cavity.
The increase of Ps formation was observed as shown in Fig. 2.34, in whicho-Ps events increase while slow positron ones decrease by high-power millimeterwaves (24 kW). Slow positron events were selected by an energy cut 511 keV +3σ−2σ
whose energy is from 360 keV to 450 keV, and o-Ps events were selected by anenergy cut from 360 keV to 450 keV (Compton free from the 511 keV signal). The
Ps formation increase was probed by subtracting beam OFF events from ON ones(See Sec. 3.3.4). In case of simple gas like argon or nitrogen, the increase of Ps
formation was observed. The same phenomenon occurred in mixed gas of nitrogen and isobutane. We cannot use gas containing nitrogen in this experiment. In
contrast, Ps did not increase in pure alkane gas as suggested in [60][61]. An accelerated positron excites many internal modes (rotation and vibration) of complex gas
molecules at inelastic scattering to drastically reduce energy gain. Since isobutane
absorbs 203 GHz radiation, the only remaining possibility is pure neopentane.

56

2.3.3

2.3 Experimentγ-ray Detectors

Four LaBr3 (5% Ce) crystal scintillators (Saint-Gobain Crystals, BriLanCeTM 380)
are placed around the beam region denoted as La-0, La-1, La-2, and La-3 (Fig. 2.29).
Figure 2.35 shows a photograph of the scintillators. Size of LaBr3 (Ce) crystal are
ϕ 1.5 inch × 2.0 inch. The crystals are covered with an aluminum housing (0.5 mm
thick). They are connected to PMT (HAMAMATSU R5924-70). Assemblies of thecrystals and PMTs are out of the gas chamber protected against contamination ofneopentane.

Figure 2.35: Photograph of LaBr3 (Ce) crystals

The annihilation γ-rays were detected using a coincidence pair of scintillators.
Four back-to-back pairs are (La-0, La-2), (La-0, La-3), (La-1, La-2), and (La-1,
La-3). The number of events of each pair is almost the same because Ps formation
region in gas is wide (about 100 mm), and shape of 203 GHz beam accumulated
in the Fabry-Pérot cavity is long (156 mm). The energy resolution is higher thanother inorganic scintillators (about 4% in FWHM), and together with back-to-backcoincidence, two 511 keV γ-ray annihilation events can be enhanced by tight energy
selection. Its fast time response (τ = 16 ns) is also appropriate for experiments withhigh statistics compared with semiconductor detectors like a Germanium crystal.
High Z of La gives higher stopping power than a NaI scintillator. The characteristicsof the scintillator are summarized in Table 2.7.
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 2.7: Properties of LaBr3 (Ce) scintillator

2.4 Experiment

2.4

Electronics and Data Acquisition

2.4.1

Overview

57

A data acquisition (DAQ) and control system consist of two parts, a detector partand an optics part. The detector part is comprised of the NIM standard modules
and the CAMAC system to acquire electric signals from PMTs of the plastic scintillator and the LaBr3 (Ce) detectors. The optics part controls the optical system
and monitors experimental conditions (power, temperature, pressure, gyrotron parameters, a position of the piezoelectric stage, etc.). It consists of a data logger(HIOKI 8420-50), a piezoelectric X-stage (NANO CONTROL TS102-G), pyroelectric detectors, an ADC (National Instruments NI PCI-6225, 16 bits, 250 kHz), andan interlock (OMRON PLC Cj1). All data are sent to a Linux computer (PC) andstored in a hard disk drive. Figure 2.36 is a schematic view of the whole DAQ
system.
The resonant peak of the Fabry-Pérot cavity is searched by moving the piezoelectric stage, whose driver is connected to the PC via RS232C. Cavity length
is controlled so that the cavity can stay on resonance. Once the cavity stays on
resonance, then the electron-beam current Ib is stabilized by controlling the heater
voltage of the MIG (Sec. 2.2.4). The AC voltage of the heater is supplied by Matsusada Precision SRJ500 and its output voltage is remote-controlled from the PC
via RS232C. Accumulated power in the cavity gradually becomes stable as steadygyrotron oscillation achieved with the constant beam current.
After the stabilization, the PC sends a start command to the CAMAC controller (Toyo Corp. CC/NET), and the detector part begins to acquire data. All
the CAMAC modules are cleared and the latch is released by the reset signal fromthe output register. This makes all the systems active and the controller waits
the interrupt signal, so-called LAM (Look-At-Me) signal from a charge-sensitive
ADC (CAEN C1205). The interrupt signal comes only when the trigger condition is satisfied. The data acquisition is triggered when back-to-back γ-ray signals
from the LaBr3 (Ce) scintillators are coincident within 40 ns and then when thiscoincidence is within −100 ns to 1100 ns of the timing of the plastic scintillators. The charge-sensitive ADC (CAEN C1205) measures energy information of
the LaBr3 (Ce) detector. Two charge-sensitive ADCs (PHILLIPS 7167 and REPIC
RPC-022) are used to measure energy information of the plastic scintillator with
diﬀerent gate length. Time information of the plastic and LaBr3 (Ce) scintillators
is recorded using a direct clock (2 GHz) count type TDC (KEK GNC-060). Afterthe reading procedures, the controller saves the data through NFS and starts thenext event cycle.
Environmental conditions (temperature, pressure) are recorded with the datalogger (HIOKI 8420-50). The incident (V in), reflected (V re) and transmitted(V tr) power are monitored with the pyroelectric detectors as shown in Fig. 2.19.
Signal waveforms of the pyroelectric detectors (duty ratio 30%, a repetition rate
5 Hz, pulse width 60 ms) are read with the ADC (NI PCI-6225) whose sampling58

2.4 Experimentcathode power supply

temperature of collector
temperature of bodytemperature of window
temperature of flange
gyrotron vacuum
water flow for gyrotron coolingwater flow for chamber coolingInterlock
OMRON PLC CJ1

temperature of room
temperature of NIM bintemperature in chamberpressure in chambercurrent in superconducting coil

LoggerHIOKI 8420-50

1st anode amplifier

gun coil power supply

socketgyrotron trigger

socketgyrotron beam current
input power

pyroelectricdetectorreflected power

pyroelectricdetectortransmitted power

pyroelectricdetectorheater AC HVSRJ500PC (Linux)@ Fukui

RS232CX-stage
TS102-G

PMTs

ADC
NI PCI-6225

X-stage controller

NIM

CAMAC

backupPC (Linux)@ Tokyo

RS232Cssh, NFS

Figure 2.36: The schematic view of the whole DAQ system.

rate is set to 0.5 Hz. The trigger pulse for the gyrotron output and the level of theelectron-beam current (Ib ) are also recorded with the same ADC. These data per
gyrotron trigger are recorded to the PC every 100 cycles.
Some parameters important for safety operations are recorded with an Interlock(OMRON PLC Cj1). The interlock shuts down a cathode power supply, a voltage
amplifier of 1st anode, a gun-coil power supply, and a heater power supply incase of a bad operation or a sudden blackout. Temperatures of diﬀerent partsof the gyrotron are monitored to check abnormal electron-beam trajectories andprevent components from thermal destruction. Gyrotron vacuum is monitored not
to expose the emitter surface of the MIG to accidental out-gassing. Water-cooling
flow and temperature are also checked. These data per five seconds are sent to the
PC independently of the other DAQ systems.
Details of the detector part are described in the following sections.

2.4 Experiment

2.4.2

59

Electronics for the Plastic Scintillator

A circuit diagram for the plastic scintillator is shown in Fig. 2.37. Two PMTs forthe plastic scintillator are operated at +2215 V (Pla-0) and +2250 V (Pla-1). High
voltage are supplied by a positive high voltage module (iseg NHQ204M). Theirgains are 3.4 × 107 (Pla-0) and 3.8 × 107 (Pla-1).
Output of the PMT is divided into three lines by a linear fan-out module (RIS0255). One of the divided signals is fed into a discriminator (KN246). The thresholdvalue of the discriminator is set to 25 mV, which corresponds about 1 photoelectron.
The others are used to measure the signal amplitude with a charge-sensitive ADC(PHILLIPS 7167) of short gate (60 ns) and a charge-sensitive ADC (REPIC RPC022) of long gate (1000 ns). All analog transmission lines are covered with electricshield, and are prepared as short as possible to reduce electric noise from gyrotron’s
pulse operation.
One of the outputs of the discriminator is delayed by 200 ns and then provides
a stop signal for the TDC (KEK GNC-060). The other output is used to make acoincidence signal of the two PMTs (Pla-and). Dark current and another accidental
noises are 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 signalof the charge-sensitive ADC (PHILLIPS 7167), a long gate signal of the chargesensitive ADC (REPIC RPC-022), a fast clear, and a main trigger. These aredescribed in a trigger section.

2.4.3

Electronics for the γ-ray Detectors

A circuit diagram for the γ-ray detectors is shown in Fig. 2.38. High voltages ofthe four PMTs (HAMAMATSU R5924-70) are supplied by a positive high-voltage
module (MATSUSADA HEER-3R10) and distributed by a voltage divider (technoland 201A-SHV). The high-voltage values are +1450 V for La-0, +1330 V for
La-1, +1230 V for La-2, and +1290 V for La-3.
Output of the PMT is divided into two lines by a linear fan-out module (RIS255). One of the divided signals is fed into a discriminator (KN246). The thresholdvalue is set to 90 mV corresponding to about 50 keV. The other signal is used tomeasure the signal amplitude with a charge-sensitive ADC (CAEN C1250)of 150 ns
gate width.
One of the outputs of the discriminator is delayed by 200 ns and then provides
a stop signal for the TDC (KEK GNC-060). The other output is fed by a logic
fan-in/fan-out module, and then sent to a coincidence logic unit. The output of thelogic unit becomes 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 the charge-sensitive
ADC (CAEN C1205) and the main trigger.

60

2.4.4

2.4 ExperimentElectronics for the Trigger System

A circuit diagram for the trigger system is shown in Fig. 2.39. The coincidence
signal from the two PMTs for the plastic scintillator (Pla-and) is widened to 1200
ns (Pla-gate). The La-b2b signal is delayed by 100 ns, and as coincidence with
Pla-gate, it triggers data acquisition (main trigger). The main trigger is also a gate
signal of the charge-sensitive ADC (CAEN C1205) to record energy informationof the LaBr3 (Ce) detectors. The trigger signal provides a latch start at the sametime. The latch signal vetos the subsequent gate signals and fast clear signals.
After all the data are read and saved, the latch reset signals are produced by an
output register of CAMAC.
The short and long gate signals for the ADCs for the plastic scintillator are
produced by Pla-gate independently of the main trigger. The data are cleared by
the clear signal (fast clear) unless the main trigger signal is produced. The Pla-gatealso produces the common start signal for the TDC. The TDC data is cleared by
the fast clear in the same way.

RIS-0255
QUINT LINEAR FAN-IN/OUT
oﬀset = -20mV

RIS-0255
QUINT LINEAR FAN-IN/OUT
oﬀset = -20mV

Pla-0

Pla-1

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

170ns delay

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

170ns delay

170ns delay

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

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

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

KEK GNC-060
2GHz TDCKN410
OCTAL FANOUT

Pla-and

KN410
OCTAL FANOUTKEK GNC-060
2GHz TDCKC3122
12ch SCALER1860
VISUAL SCALER1860
VISUAL SCALERKC3122
12ch SCALER

2.4 Experiment
61

Figure 2.37: Circuit diagram for plastic scintillator system.

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

62
2.4 Experiment

Figure 2.38: Schematic diagram of electronics for γ-ray detector system.

La-b2b

Pla-gate

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

VETO

16CH CS ADC

VETO

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

Figure 2.39: Schematic diagram of electronics for trigger system.
3470
16-BIT OUTPUT REGISTER1500
DUAL GATE GENERATOR
through, width = 150ns

stop

start
1500
DUAL GATE GENERATOR
latch mode

16CH QADC

gate CAEN Mod. C1250

PHILLIPS 7167

gate enable 16 CHANNEL QDC

LeCroy MODEL620CL
8 CHANNEL DISCRIMINATOR
width = 20ns

short gate
PHILLIPS 7167
16 CHANNEL QDC

16 CHANNEL QDC

short gate PHILLIPS 7167

KC3122①

inhibit 12CH SCALER

VETO

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

KN246②OCTAL DISCRIMINATOR
thr = -100mV, width = 20ns

KN246②OCTAL DISCRIMINATOR
thr = -100mV, width = 20ns

2GHz TDC

2GHz TDC

LeCroy MODEL429A
LOGIC FAN-IN/FAN-OUT

600ns delay

KN1500
DUAL GATE GENERATOR
through, width = 1200ns

common stop KEK GNC-060

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

VETO

600ns delay

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

common start KEK GNC-060

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

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

OUT

OUT

gate REPIC RPC-022

1860
VISUAL SCALER

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

VETO

KN1500
DUAL GATE GENERATOR
through, width = 1200ns

KC3122
12CH SCALER①HOSHIN N010
CLOCK GENERATOR
500kHz, width = 20ns

LeCroy MODEL429A
LOGIC FAN-IN/FAN-OUT

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

KC347116-BIT OUTPUT REGISTER

Pla-and

KC3122①
12CH SCALER

fast clear 16CH CS ADC

REPIC RPC-022

16 CHANNEL QDC

fast clear PHILLIPS 7167

2GHz TDC

fast clear KEK GNC-060

Pla-gate

2.4 Experiment63

64

2.4 ExperimentChapter 3

Analysis
This chapter is organized as follows. First, A summary of acquired data is described. The data calibration and oﬄine trigger selections are described next. The
transition signals are obtained after some event selections. The reaction crosssection σ(ω) at one frequency point is obtained by comparing this signal amountwith a Monte Carlo simulation. Then, σ(ω) is fitted by a Breit-Wigner function.
Finally, systematic uncertainties are considered to obtain a result.

3.1

Acquired Data

Table 3.1 summarizes acquired data for analysis. This experiment was performed
from May to December in 2013. Data sets are divided into 12 RUNs corresponding
to diﬀerent frequencies and power conditions. The RUN with the same frequency
is named to have the same alphabet. We measured 201.83 GHz, 203.51 GHz,
205.31 GHz, 202.64 GHz, 204.56 GHz, and 203.00 GHz (203.25 GHz) in chronological order. Reference RUNs (B-0, C-0, D-0, E-0, F-0) were also acquired without
resonating the Fabry-Pérot cavity to check systematic troubles. A decrease of thetrigger rate in chronological order is due to contamination of neopentane gas to theacrylic light-guide and the plastic scintillator. Corrections of this eﬀect is describedin Sec. 3.3.2 and Sec. 3.4.4.
Data for each RUN are divided into some SETs by 30 minutes, and pedestal
data are taken between two SETs. Calibration including some corrections arecarried out for each SET. This short time (30 min) for each DAQ guarantees thatfluctuation and drift of the measured quantities by environments do not propagate
over one SET.
A gyrotron is in pulse operation of duty ratio 30% and a repetition rate 5 Hz, andbackground levels are slightly diﬀerent between beam ON and OFF. Data analysis
was carried out separately for beam ON and OFF events. Then, the diﬀerence isassigned as systematic uncertainties (It turns out to be negligibly small comparedwith other systematic uncertainties in Sec. 3.5.4).
65

66

3.2 Analysis

RUN ID

frequency

power

date (dd/mm)

live time

trigger rate

A-1
A-2
B-0
B-1
C-0
C-1
C-2
C-3
D-0
D-1
D-2
D-3
D-4
D-5
E-0
E-1
E-2
F-0
F-1
F-2
G-0

201.83 GHz
201.83 GHz
202.64 GHz
202.64 GHz
203.25 GHz
203.00 GHz
203.25 GHz
203.25 GHz
203.51 GHz
203.51 GHz
203.51 GHz
203.51 GHz
203.51 GHz
203.51 GHz
204.56 GHz
204.56 GHz
204.56 GHz
205.31 GHz
205.31 GHz
205.31 GHz
180.59 GHz

21.6 kW22.2 kW
0.0 kW
23.4 kW
0.0 kW
20.6 kW16.8 kW24.5 kW
0.0 kW
36.5 kW67.4 kW30.9 kW
19.0 kW79.3 kW
0.0 kW
14.5 kW24.6 kW
0.0 kW
23.8 kW23.4 kW40.5 kW

14/05-15/05
15/05-16/05
28/09
26/09-28/09
12/12
03/12-04/12
04/12-05/12
10/12-11/12
02/08
25/07-27/07
01/08
01/08
01/08-02/08
02/08
14/10-15/10
12/10-13/10
14/10
14/09
13/09-14/09
16/09-17/09
16/10-18/10

6.7 × 104 sec
6.7 × 104 sec
7.0 × 104 sec
1.1 × 105 sec
7.2 × 104 sec
6.6 × 104 sec
6.8 × 104 sec
9.9 × 104 sec
2.6 × 104 sec
7.2 × 104 sec
1.2 × 104 sec
1.3 × 104 sec
1.4 × 104 sec
1.3 × 103 sec
2.3 × 105 sec
6.1 × 104 sec
6.0 × 104 sec
5.8 × 104 sec
1.0 × 105 sec
8.0 × 104 sec
1.2 × 105 sec

667 Hz669 Hz531 Hz540 Hz
485 Hz494 Hz489 Hz491 Hz608 Hz627 Hz627 Hz628 Hz616 Hz610 Hz530 Hz526 Hz530 Hz530 Hz565 Hz598 Hz523 HzTable 3.1: Data sets for analysis

67

3.2 Analysis

3.2

Calibration and Oﬄine Trigger

3.2.1

Energy Calibration and Correction of the Plastic Scintillator

100
90
103

80
70
60

102

50
40
30

charge miscollection

10

energy (long gate) [p.e.]energy (long gate) [p.e.]

Energy information of the plastic scintillator was calibrated to the number of photoelectrons (p.e.) detected with a PMT. Two charge data, measured with a short-gate(60 ns) and a long-gate (1000 ns), were independently processed in this calibration.
Charge diﬀerence of these two were used to reduce accidental events in Sec. 3.3.2.
Charge miscollection of the short-gate energy depending on the signal amplitude
was corrected to obtain the original signal amplitude. Six histograms of the longgate energy were prepared for each short-gate energy divided into 5 p.e. widthfrom 15 p.e. to 45 p.e., and were fitted with Landau distributions. Then, peaksof the Landau distributions were fitted by a second polynomial function to obtainthe most probable charge miscollection with the short-gate. Figure 3.1 shows a 2D
energy plot of the plastic scintillator before and after the correction.

100
90
80

103

70
60

102

50
40
30

10

20

20

10

10

0
1
0 10 20 30 40 50 60 70 80 90 100
energy (short gate) [p.e.]010 10 20 30 40 50 60 70 80 90 100
energy (short gate) [p.e.]

Figure 3.1: 2D plot of the plastic scintillator energy measured with the short-gate
and the long-gate. (left) Before the charge correction. (right) After the correction.

3.2.2

Energy Calibration of the LaBr3 (Ce) Scintillator

Energy information of LaBr3 (Ce) crystal scintillators was calibrated to make apositron annihilation peak become 511 keV. Pedestal peaks and annihilation peaks
were fitted by Gaussian functions. Figure 3.2 shows a calibrated spectrum of asingle detector (La-0). Fitted resolutions are summarized in Table 3.2.

3.2.3

Time Calibration

An absolute scale of the TDC (KEK GNC-060) was determined by an external
clock source of 2 GHz (accuracy is 1 ppm). A full range of a time spectrum

68

counts / sec / 2 keV

3.2 Analysis

5
4
3
2
1
0
100 150 200 250 300 350 400 450 500 550 600
energy [keV]

Figure 3.2: Energy spectrum of a LaBr3 (Ce) scintillator (La-0).

Numberσpedestal

σ511

La-0
La-1
La-2
La-3

0.62 keV
0.59 keV
0.60 keV
0.53 keV

9.00 keV
7.84 keV
8.01 keV
8.46 keV

Table 3.2: Energy resolution of LaBr3 (Ce) crystal scintillators

69

3.2 Analysis

was determined by a common stop signal delayed 1200 ns from a common startsignal. To determine of the time origin, we made use of the prompt events fitted
by a Gaussian function. Since the prompt peak consists of p-Ps decay, positronannihilation, and others, it appears as a sharp peak and exactly stands at decay
time t = 0. Time walk corrections described in Sec. 3.2.4 and Sec. 3.2.5 were also
carried out to accurately calibrate time spectrum.

3.2.4

Time-Walk Correction of the LaBr3 (Ce) Scintillator

Signal risetime depends on its amplitude, and results in time walk of LaBr3 (Ce)signals as shown in the left side of Fig. 3.3. Time-walk was corrected to obtain aproper time spectrum with the following method. Time histograms were preparedfor each energy divided by 25 keV from 100 keV to 600 keV, and each prompt peak
was fitted by a Gaussian function. Fitted centers were then fitted by
f (x) = p0 +p1
+ p2 E,
E

(3.1)

20
103

15
10

20
103

15
10

5

102

0

5

102

0

-5

-5
10

-10
-15
-20

time [ns]time [ns]where E denotes the energy deposit in the LaBr3 (Ce) scintillator. The correction value for each energy was determined by this function, and the result of thiscorrection is shown in the right side of Fig. 3.3

10

-10
-15

100

200

300

400

500
600
energy [keV]

1

-20

100

200

300

400

500
600
energy [keV]

1

Figure 3.3: 2D plots of time vs energy deposited in one LaBr3 (Ce) scintillator.
(left) Before the time-walk correction. (right) After the correction.

3.2.5

Time-Walk Correction of the Plastic Scintillator

There is also a non-negligible time lag in plastic-scintillator signals as shown in
the left figure of Fig. 3.4. Time-walk corrections of the plastic scintillator were also
carried out. Pla-0 and Pla-1 signals were combined and divided into 16 regions from70

3.2 Analysis

4 p.e. to 100 p.e. Prompt time peaks for these regions were fitted by Gaussianfunctions. The peak centers were fitted byf (E) = p0 +p1
+ p2 E + p3 E 2 ,
E

(3.2)

10
8
6

103

4

8
6

2
102

-2
-4

10
103

4

2
0

time [ns]time [ns]where E denotes the number of summed photoelectrons of Pla-0 and Pla-1. The
correction value for each energy was determined by this function, and the result of
this correction is shown in the right side of Fig. 3.4

0

102

-2
10

-6

-4

10

-6

-8

-8

-10
1
0 10 20 30 40 50 60 70 80 90 100
energy [p.e.]-10
10 10 20 30 40 50 60 70 80 90 100
energy [p.e.]

Figure 3.4: 2D plots of time vs energy deposited in the plastic scintillator. (left)
Before the time-walk correction. (right) After the correction.

71

3.2 Analysis

3.2.6

Oﬄine Trigger Cuts

Oﬄine trigger cuts are required before event selections for the transition signal.
e+ tagging
A threshold for energy deposits in the plastic scintillator was set to 1 p.e. both for
Pla-0 and Pla-1. And both two, Pla-0 and Pla-1 of the short-gate, were required
to balance with each other.
√√
EPs–0 − EPs–1 < 3σ
(3.3)

energy (Ps-1) [p.e.]

These requirements eliminate cross-talk noise and events in which a positron hits
an edge of the scintillator. Figure 3.5 shows the 2D plot of Pla-0 and Pla-1 withthe e+ tagging condition. Times of signal detection with Pla-0 and Pla-1 werealso required to coincide within 3 standard deviations as shown in the left side ofFig. 3.6.

100

103

90
80
70
60

102

50
40
30

10

20
10
0
1
0 10 20 30 40 50 60 70 80 90 100
energy (Ps-0) [p.e.]

Figure 3.5: 2D energy spectrum of plastic scintillator. The region inside black lines
is selected.

Back-to-back γ-ray Hits
It was required that γ rays hit only two back-to-back LaBr3 (Ce) scintillators whoseenergy deposits were over 100 keV. Events in which energy deposits of over three72

3.2 Analysis

count / 0.5 ns

count / ns

scintillators exceed 100 keV were eliminated. These conditions reduce contamination of three γ-ray decay of o-Ps and accidental γ rays (1275 keV and 511 keV)from the positron source. Times of signal detection with back-to-back LaBr3 (Ce)crystals were also required to coincide within 3 standard deviations as shown in
the right figure of Fig. 3.6.

5

10

5

10

104

4

10

103
103
102

102

10
-20 -15 -10-50
5 10 15 20
time (Ps-0) - time (Ps-1) [ns]10
-20 -15 -10-50
5 10 15 20
time (La-0) - time (La-2) [ns]Figure 3.6: Time coincidence. (left) Between Pla-0 and Pla-1. (right) Between
La-0 and La-2. Cut region is shown as dashed black lines.
An example of event rates after each selection is summarized in Table 3.3.
About 59 % of the online-triggered events pass the oﬄine trigger cuts.
selection

rate [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

667.34(18)660.91(18)618.89(17)410.12(14)391.33(14)Table 3.3: Summary of the event rates by oﬄine trigger cuts (RUN ID A-1).

73

3.3 Analysis

3.3

Event Selection

Event selections are needed to measure the amount of the direct transition signals.
The signal is delayed two back-to-back γ rays of 511 keV. Three selections wererequired:• Time window (delayed coincidence)• Accidental rejection• γ-ray energy cut
There are two remaining backgrounds after these three event selections: three γray contamination and pick-oﬀ annihilation. They all are discussed in the following
sections.

3.3.1

Time Window

counts / 0.5ns / sec

The oﬄine-triggered data contain backgrounds of a prompt peak at t = 0 and
accidental events as shown in Fig. 3.7. They were eliminated with a time window
from 50 ns to 250 ns. This range of the time window was determined with Monte
Carlo simulations to obtain the best signal-to-noise ratio (S/N) in case of 30 kW
accumulation. S/N was improved by factor of six by this cut.

prompt peak

10
1

original accidental
reduced accidental

10-1
10-2
10-3

0 100 200 300 400 500 600 700 800
time windowtime [ns]Figure 3.7: Time spectrum obtained by time diﬀerence of the LaBr3 (Ce) signal
and plastic scintillator one. Time window for delayed coincidence is shown as aregion between two black dashed lines. The black and red solid lines show before
and after accidental rejection, respectively.

74

3.3 Analysis

3.3.2

Accidental Rejection

A part of accidental background due to pileup events of the plastic scintillator
still remains after requiring the time window (Fig. 3.7). We made use of charge
diﬀerence of the plastic scintillator signal measured with the short-gate and thelong-gate. The left side of Fig. 3.8 shows a scatter plot of energy deposits measured
with the short-gate and the long-gate. The long-gate stores more charges than the
short-gate in case of accidental pileup events (shown as a triangle region) whilethese two charges balance with each other in signal events. A cut condition usedwas
−5σlong−gate < Elong−gate − Eshort−gate < 3σlong−short .

(3.4)

as shown in the right figure of Fig. 3.8. The peak width of the Elong−gate −Eshort−gatespectrum changes in chronological order because contamination of neopentane tothe light-guide gradually decreases amount of light from the plastic scintillator.
Table 3.4 summarizes σlong−short of each frequency.
Figure 3.7 compares time spectra before and after the accidental rejection cut.
S/N was enhanced by a factor of two. Events in an accidental time window from
750 ns to 850 ns were subtracted from that in the signal time window (from 50 ns
to 250 ns) to reduce the remaining accidental background which was very small.

signal region

90

103

80
70
60

102

50
40
30

signal region

counts / 0.1 p.e. / secenergy (long gate) [p.e.]

accidental due to pileup

100

10
8
6
4

10

20

2

10
0
1
0 10 20 30 40 50 60 70 80 90 100
energy (short gate) [p.e.]

0
-10

-5
0
5
10
energy (long gate - short gate) [p.e.]

Figure 3.8: Cut condition of accidental rejection of the plastic scintillator. (left)
2D plots of energy deposits measured with the short-gate and the long-gate. (right)
Diﬀerence of energy deposits measured with the short-gate and the long-gate. The
signal region is expressed as a region between two black lines.

75

3.3 Analysis
frequencyID

σlong−short

201.83 GHz
202.64 GHz
203.00 GHz
203.51 GHz
204.56 GHz
205.31 GHz
180.59 GHz

A-1, A-2
B-0, B-1
C-0, C-1, C-2, C-3
D-0, D-1, D-2, D-3, D-4, D-5
E-0, E-1, E-2
F-0, F-1, F-2
G-0

0.90 p.e.
0.47 p.e.
0.53 p.e.
0.73 p.e.
0.50 p.e.
0.63 p.e.
0.51 p.e.

Table 3.4: Summary of the peak width of σlong−short .

3.3.3

γ-ray Energy Cut

Large background from three γ-ray contamination exists without energy cuts. An
energy cut around 511 keV was required
511 keV − 2σ511 < Eγ < 511 kev + 3σ511 .

(3.5)

counts / 5keV / sec

Figure 3.9 shows predicted energy spectra required this cut only at the opposite
side of the back-to-back pairs. Four back-to-back pairs (La-0, La-2) (La-0, La-3)(La-1, La-2) (La-1, La-3) were combined (Fig. 2.29). When the cut condition was
also required at this side of the pairs (shown as dashed lines in Fig. 3.9), S/N was
improved by a factor of 70 compared with no energy cuts.

0.12

MC totalsignal3 gammapick-off

0.1
0.08
0.06
0.04
0.02
0
100

200

300

400

500

600

LaBr3 energy [keV]

Figure 3.9: Expected energy spectrum of LaBr3 (Ce) scintillators with Monte Carlosimulation. f =203.4 GHz and Pacc =20 kW. 511 keV is required at the opposite
side of the back-to-back pairs.

76

3.3 Analysis

3.3.4

Direct Transition Signal and Background Estimation

counts / 10 ns / sec

counts / 5keV / sec

Although we required some event selections, remaining backgrounds shown in
Fig. 3.9 are still dominant contribution to the 511 keV peak, and the transition
signal is about 10% of total events. Three γ-ray contamination background is due
to decays of non-transiting o-Ps. According to kinematics, a part of this decay isvery similar to two γ decays of 511 keV, and is inseparable with non-segmented
crystals. Pick-oﬀ background is caused by annihilation between a positron in oPs and an electron in a gas molecule. This is mainly two γ-ray annihilation, andinevitable in use of gas as an electron source.

0.14
0.12

beam OFFbeam ON

0.1
0.08
0.06

10-1

beam OFFbeam ON

10-2

0.04
10-3

0.02
0
460

480

500

520

540
560
energy [keV]

100

200

300

400

500
time [ns]Figure 3.10: Histograms after event selections. Histograms of RUN D-2 are shownas an example. (left) Energy spectrum. (right) Time spectrum. Solid lines show
the best fit.
Remaining backgrounds were estimated with real data. The gyrotron was inpulse operation of duty ratio 30% and a repetition rate 5 Hz. Background eventswere obtained with beam OFF events, and the transition signal was obtained bysubtracting OFF rate from ON rate. Since beam ON and OFF are within 200 ms
sequentially, one can safely ignore that environmental conditions such as temperature do not change between beam OFF and ON.
Figure 3.10 compares energy and time spectra of beam OFF and beam ON
events. The beam OFF and ON spectra were normalized with the prompt peakof the time spectra, which was consistent with livetime normalization. A clear
enhancement around 511 keV for beam ON events was obtained (the left side).
At the same time, lifetime of beam ON events decreases from beam OFF events
because of the transition (the right side). The amounts of the direct transition
signals for each RUN are summarized in Table 3.5. The transition was observed inhigh-power RUNs, and not in 0 kW RUNs. No signals at far oﬀ-resonance pointwith high power (G-0) guarantees that there is no fake signals from unexpected
systematic eﬀects. Corrections of power fluctuations are needed to obtain theproper resonance curve (the reaction cross-section σ(ω)). This method is discussed
in the next section.

77

3.4 Analysis
RUN ID

frequency

power

ON rateOFF rateON − OFF

A-1
A-2
B-0
B-1
C-0
C-1
C-2
C-3
D-0
D-1
D-2
D-3
D-4
D-5
E-0
E-1
E-2
F-0
F-1
F-2
G-0

201.83 GHz
201.83 GHz
202.64 GHz
202.64 GHz
203.25 GHz
203.00 GHz
203.25 GHz
203.25 GHz
203.51 GHz
203.51 GHz
203.51 GHz
203.51 GHz
203.51 GHz
203.51 GHz
204.56 GHz
204.56 GHz
204.56 GHz
205.31 GHz
205.31 GHz
205.31 GHz
180.59 GHz

21.6 kW22.2 kW
0.0 kW
23.4 kW
0.0 kW
20.6 kW16.8 kW24.5 kW
0.0 kW
36.5 kW67.4 kW30.9 kW
19.0 kW79.3 kW
0.0 kW
14.5 kW24.6 kW
0.0 kW
23.8 kW23.4 kW40.5 kW520.2(5.2)
525.0(5.2)
340.5(4.2)
383.3(3.5)
414.6(7.5)
379.5(4.5)
362.0(4.4)
397.0(3.8)
414.6(7.5)
499.0(4.9)
547.8(12.4)
481.4(11.1)
460.3(10.7)
495.2(11.5)
369.5(4.8)
372.1(4.7)
408.9(5.0)
399.5(7.5)
431.7(3.8)
420.1(4.3)
374.4(3.4)

498.9(3.3)
510.0(3.4)
331.8(2.7)
361.6(2.2)
406.9(4.9)
350.5(2.9)
337.4(2.8)
358.1(2.4)
406.9(4.9)
452.5(3.1)
454.5(7.3)
432.3(7.0)
423.7(6.6)
416.4(7.0)
374.8(3.1)
358.9(3.0)
390.3(3.2)
395.4(4.9)
412.6(2.5)
416.4(2.8)
375.9(2.2)

21.2(6.1)
14.9(6.2)
8.8(5.0)
21.7(4.2)
7.8(8.9)29.0(5.4)
24.6(5.2)
38.9(4.5)
7.8(8.9)
46.5(5.8)
93.3(14.4)
49.1(7.0)
36.6(12.6)
78.8(13.4)−5.3(5.7)
13.1(5.6)
18.6(5.9)
4.1(8.9)
19.1(4.6)
3.7(5.1)−1.7(4.3)

Table 3.5: The amount of the transition signal. Rates are shown in mHz.

3.4

Analysis for Ps-HFS

In this section, we consider space-time fluctuations of accumulated power and detection eﬃciency to obtain the reaction cross-section curve of Ps-HFS. A Monte Carlosimulation (MC) is introduced to estimate detection eﬃciency. The transition signal (the ON − OFF rate) is aﬀected by the decreasing trigger rate; therefore, S/N(normalized by the OFF rate) is compared with MC. Ps-HFS, lifetime of p-Ps, andthe Einstein A coeﬃcient of the Ps-HFS transition are simultaneously determined.
This is the first direct measurement of Ps-HFS and p-Ps lifetime.

3.4.1

Preparation of MC Samples

Geant4 simulations were performed after the experiments with proper conditionof temperature and gas pressure. To save time, MC is divided into two parts.
Firstly, stopping positions of positrons were simulated (106 events) as shown in
Fig. 3.11. Secondly, γ rays were simulated from the obtained distribution of the

78

3.4 Analysis

Z [mm]

50

La-3

40

25
20

250

15

180

La-2

10

30

160

5

20

140

-5

200

La-2, 30

10 3σ
0 1σ

-25
-50

100
80

-10

50

accumulated Gaussian beam

-20
-40

-30

-20

-10

0

10

20

30

40

25
20

←front mirror

50
X [mm]

end mirror →

1560-20 accumulated-30

5Gaussian beam

-50
-50

40

La-0

-40
-40

-30

20

La-1
-20

10

-10

0

10

20

30

0

40 50
Y [mm]

0

200
180
160

3σ
2σ
1σ

140
120
100

-5

80

-10

60

-15

0

150

100

-15

120

2σ

La-0, 1

1σ
2σ
3σ

-10

Z [mm]

X [mm]

positron-stopping position. We simulated two γ annihilation for signal and pick-oﬀ
events (25 × 106 events) and three γ annihilation for o-Ps events (500 × 106 events).
Diﬀerent sets of MC were prepared for all RUNs.

40

accumulated Gaussian beam

-20
-25
-50

-40

-30

-20

-10

0

10

20

20

30

40

50
Y [mm]

Figure 3.11: Position distribution of Ps formation with MC. (left) Top view. (topright) Cross-sectional view of the Fabry-Pérot cavity. (bottom right) Side view of
the Fabry-Pérot cavity. See Fig. 2.29 and Fig. 2.30 as a reference.

3.4.2

Correction of Space Fluctuation

Shape eﬀects of the Gaussian beam and the standing-wave reduce transition probability compared with an expectation with averaged power. The shape of theGaussian beam is shown in Fig. 3.11 with white lines. A detection eﬃciency of γrays was estimated by making diﬀerent MC histograms corresponding to the position of Ps formation divided into 25 parts from 0σ to 5σ, and summing themwith power weight of the Gaussian distribution. The shape of the standing-wave
was estimated by simply calculating power weight of 2 sin2 ky and was integrated
because this structure is so small (750 µm) that no detectors are sensitive to its
geometrical eﬀect.

3.4.3

Correction of Time Fluctuation

Figure 3.12 shows a typical pulse shape recorded with 500 Hz. As is described inSec. 2.2.4, the electron beam current Ib was stabilized with a feed-back control of
the heater voltage. Input power to the Fabry-Pérot cavity Vin was also more or lessstable under steady-state gyrotron oscillation with constant Ib . Transmitted powerVtr integrated in one pulse was stabilized by the control of cavity length (Fig. 2.27).
However, as shown in Fig. 3.12, Vtr fluctuates in one pulse because of unstable
gyrotron frequency (Fig. 2.6). This fluctuation reduces transition probability, and79

3.4 Analysis

0.4
0.3
0.2
0.1
0

–0.1
–0.2

0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
–0.02
–0.04
–0.06
0

1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6

Vin

Ib
Vtr

Vtr [V]

0.5

Vin [V]Ib [A]

depends on diﬀerent gyrotron conditions for each RUN. This eﬀect was estimatedby taking a histogram of Vtr sampled per 2 ms. Figure 3.13 shows two examples ofthe obtained histogram filled per 2 ms of this power fluctuation. Frequency linewidth of the gyrotron was stable in RUN A-1 (left), but quite unstable in RUN
D-1 (right). The transition probability was corrected by weighting this histogram.

10 20 30 40 50 60 70 80 90 100
time [ms]Figure 3.12: Pulse shape of detected power. Beam current Ib (black), monitored
incident power Vin (blue), and transmitted power Vtr (red) are shown together.

×10
350

350
300

300

250

250

200

200

150

150

100

100

50

50
0

×10

3

countscounts

3

400

0

0.5

1

1.5

2
Vtr

0

0

2

4

6

8

10
Vtr

Figure 3.13: Histogram of accumulated power fluctuation in pulse. (left) RUN A-1.
(right) RUN D-1.

3.4.4

Calculation of the Reaction Cross-section

Reaction cross-sections of each RUN were calculated by comparing the ratio oftransition signals with MC predictions. The obtained detection eﬃciencies of 2γ
rays and 3γ rays with MC are expressed as η2γ and η3γ , respectively. We also

80

3.4 Analysis

introduce fraction of the signal (Fsig ) and background, pick-oﬀ (Fpick ) and o-Ps
(Fo–Ps ), using Eq. (1.26), (1.27), and (1.28).
∫ t1
1
dtN1,0Fsig = Γtran (P )4t0]∑ [∫ t1
1
Fpick = Γpick
dtN1,m4t0
m=−1,0,1]∑ [∫ t11
dtN1,mFo–Ps = Γo–Ps
4
t0

(3.6)
(3.7)
(3.8)

m=−1,0,1

Coeﬃcient 1/4 means that the number of each |S, m⟩ state formed is the sameprovided Ps is unpolarized. These three fractions are functions of accumulated
power because Γtran and N1,0 depend on power. Predicted S/N is expressed as∑nON =wP {η2γ [Fsig (P ) + Fpick (P )] + η3γ · Fo–Ps (P )}(3.9)
PnOFF = η2γ [Fsig (0) + Fpick (0)] + η3γ · Fo–Ps (0)nON − nOFF,
S/N(MC) =nOFF(3.10)(3.11)∑
where
P means corrections of power fluctuation with weight wP described inSec. 3.4.2 and Sec. 3.4.3. We solved equation of S/N (data listed in Table 3.5) =S/N (MC) taking a reaction cross-section as an unkown variable. The results
are listed in Table 3.6. They were fitted by a Breit-Wigner function defined by
Eq. (1.32). Figure 3.14 shows the fitting result with statistical errors only. Correlations among three fitting variables (∆HFS
Ps , τp–Ps , A) are shown in Fig. 3.15 We
estimated the fitting errors as the maximum and the minimun of the correlation
region of one standard deviation (red).
Frequency [GHz]σ [kb]

180.59(10)201.83(02)202.64(04)203.00(04)203.25(05)203.51(02)204.56(05)205.31(04)−0.45(1.21)
8.88(2.50)
18.22(4.99)
34.18(10.9)
38.66(7.33)
29.09(6.02)
13.07(4.17)
5.81(2.22)

Table 3.6: Summary of the reaction cross-section.

81

σ [kb]

3.4 Analysis

40
30

χ2 / ndf

1.581 / 5-83.654 ±0.4692

A [×10 /s]

Ps-HFS [GHz] 203.3 ±0.1377

20

p-Ps life [ps]89.33 ±15.12

100

σ [kb]

180

185

190

195

200
205
Frequency [GHz]

202

203

204
205
206
Frequency [GHz]

50
45
40
35
30
25
20
15
10
5
201

Figure 3.14: Fitting result of measured cross-section of Ps-HFS transition. (top)Full range. (bottom) Enlarged view near the resonance.

82

A [ × 10 -8 s-1]

3.4 Analysis

5
4.5
4
3.5
3
2.5
2
203.2

203.4

203.6 203.8
[GHz]
∆Ps
HFS

40

60

80

100 120 140 160 180
τp-Ps [ps]

40

60

80

100 120 140 160 180
τp-Ps [ps]

Ps
[GHz]
∆HFS

203

203.8

203.6

203.4

203.2

A [ × 10 -8 s-1]

203

5
4.5
4
3.5
3
2.5
2

Figure 3.15: Correlation among three fitting variables. The red, blue, green regions
are one, two, three standard deviations, respectively.

3.5 Analysis

3.5

83

Systematic Errors

Systematic errors for the transition signal of each RUN are considered. They are
categorized into four types:• power estimation uncertainty• gas eﬀect• Monte Carlo simulation• another
They are described in the following sections.

3.5.1

Power Estimation Uncertainty

A systematic uncertainty from the power estimation has been already discussed inSec. 2.2.3 and summarized in Table 2.5. Relative accuracy is better than 21% except
for the far oﬀ-resonance point F-0. An eﬀect for the Ps-HFS value was estimatedby scanning C of each frequency with an asymmetric Gaussian distribution.

3.5.2

Gas Eﬀect

Ps Formation Probability
In this experiment, it turned out that the Ps formation probability is enhanced inuse of nitrogen under high-power millimeter-wave radiation (See Sec. 2.3.2). Thisenhancement can be a systematic error if it still exists in case of neopentane. An
energy cut from 360 keV to 450 keV is free from the Compton scattering of 511 keV
γ rays, and is required to study an increase of o-Ps.
In oﬀ-resonance RUNs (ID = A and F) and the far frequency point G-0, S/N
is zero consistent and no Ps increases are observed. In on-resonance RUNs (ID D),
a decrease from the Ps-HFS transition o-Ps → p-Ps hides the eﬀect of Ps increase.
The systematic uncertainty was estimated to be −0.1 06(76)% with G-0 data whichis free from the signal contamination. An increase of Ps formation probability may
depend on frequency and power. Frequency dependence can be estimated as (SeeEq. (B.3))
δE(203 GHz)
(2π · 180 GHz)2 + ωc2∼.
(3.12)(2π · 203 GHz)2 + ωc2
δE(180 GHz)If we approximate the neopentane molecule as an elastic ball of radius 3.5Å, collision
frequency ωc is about 1 THz at 300 K and 1 atm. The resulting correction factor
by frequency change from 180 GHz to 203 GHz is only 0.9. Thus, an increase of
Ps formation is zero consistent and negligible.

84

3.5 Analysis

Pick-oﬀ Annihilation Probability
If pick-oﬀ annihilation probability depends on beam OFF or ON, it can be a systematic error. The pick-oﬀ rate is determined by velocity of Ps, gas density, andthe eﬀective electron numbers in a gas molecule which contributes to the annihilation. A neopentane gas molecule does not interact with millimeter-wave radiation
at all because it does not have an electric dipole moment (See Table 2.6). Fluctuation of gas density due to heat does not follow a short pulse of gyrotron (60 ms).
However, initial velocity of Ps might be aﬀected by millimeter waves because wefound positron acceleration by millimeter-wave radiation (See Sec. 2.3.2). Pick-oﬀ
annihilation probability can be changed by accelerated Ps velocity.
Far oﬀ-resonance data of RUN G-0 was used to eliminate signal contamination.
We required accidental rejection and Compton free energy selection (from 360 keV
to 450 keV). Time spectra were fitted from 50 ns to 850 ns, and obtained pick-oﬀ
probability Ppick = Γpick /Γo–Ps is
Ppick (OFF) = 8.43(17)%(3.13)

Ppick (ON) = 8.06(26)%(3.14)

It is consistent with each other. Near oﬀ-resonance data (RUN A, F) were also
fitted, and no significant diﬀerences between beam OFF and ON were observed.
Thus, this eﬀect turned out to be negligible in this experiment.
Stark eﬀect
The Ps-HFS value is shifted by the Stark eﬀect due to static electric fields of gas
molecules. Since non-thermalization eﬀect can be ignored at the current level of′ Ps
precision, the shifted ∆HFS
linearly depends on gas density D as shown in Fig. 3.16
′

Ps
∆HFS
= ∆Ps
HFS [1 + a(σm )D] ,(3.15)

where coeﬃcient a(σm ) is introduced.
Although a(σm ) of nitrogen (Fig. 3.16) or isobutane was measured in previous
experiments, there are no experimental data of a(σm ) in neopentane gas. We canassume that a(σm ) is proportional to σm if we ignore non-thermalization eﬀect of Ps(about 10 ppm). An elastic scattering cross-sections σm of nitrogen and neopentane
are measured with Doppler-broadening measurements [62]N2σm
= 13.0(0.5) Å2(3.16)

neoσm= 228(13) Å2(3.17)

In case of nitrogen gas, the averaged value of a(σm ) in two independent measurements [30][33] isaN2 (σm ) = −28.7(8.7) ppm · amagat−1(3.18)

2.5102.820line center. The value of Av derived using Sc must be increased by 18+1 ppm, in rough agreement with Ref. 11.
averaging the values of bv obtained from the individua
resonance lines. Figure 7 and Table II show the datafrom the present experiment and from Ps IV. In calculating the average for hv(D) we have discarded any line fitsthat had less than one percent chance of occurring according to g statistics. If we fit all of these data to Eq. (10),we obtain

bv=203. 38910(56) 6Hz (2. 8 ppm),

3.5a=Analysis—3.3(4)X10

atm

'

(14)

(0 C),

203 370.1
203 371.6

4.9
2.9

where our X for the fit is 18.7 with 18 degrees of freedom; the error quoted is the statistical error only. The
data were also fit with both a term linear in D and one
quadratic in D; the X fit was no better than that with thelinear term alone and hence only the linear fit is used.
The sources of error in hv are listed in Tab l e III.. Theeerrors for Ps IV are also shown. The principal error is'd ue t o coun ting statistics. The next largest error is due
85 tothe static magnetic field Ho and is associated wit t e

0. 40—

0.39—
hJ

Q
C9

O

CV
I

O.

38—

0.37

0

I

I

1

NITROGENDENSITY (amagat)

FIG. 7. Measured values of Av vs N~ density (11 ama
am g at is the density of 1 atm N 2 a t 0 C). The open circles are from Ps IV andthe closed circles are from the present work. The straig
rai ht line is Ps
the best fit described in Eq. ( 4).

Figure 3.16: Measured Stark shift of ∆HFS vs nitrogen density. The open circles
are from [32] and the closed circles are from [33]. The straight line is the best
fit [33]

As a result, the value of this eﬀect to neopentane of 1 atm and 300 K is estimatedas
σ neo 273
∼ −460 ppm.
(3.19)
aneo (σm ) = aN2 (σm ) × m
×
N2
300
σm
The measured Ps-HFS value is corrected by +460 ppm, and ±460 ppm is also
assigned as a systematic uncertainty. This is a common systematic error to all
RUNs, and was combined at last.
Note that linear extrapolation is invalid if Ps is not well thermalized. This is oneof the possible systematic uncertainties in previous measurements (See Sec. 1.5.3)
and results in about 10 ppm correction in case of isobutane gas [38]. A similarcorrection is expected for neopentane because its molecular structure is almost thesame as isobutane. It is negligible in this measurement at the current level ofprecision and should be considered in the future (See Sec. 4.3).

3.5.3

Monte Carlo Simulation

The reaction cross-section was calculated with detection eﬃciency obtained with
the MC simulation. As defined by Eq. (3.11), use of S/N reduces eﬀects from parameters with large uncertainty, such as an absolute source rate, a stopping positionin gas, positronium formation probability, or fine structure of energy spectrum. A
possible remaining systematic uncertainty is a ratio of detection eﬃciency η2γ /η3γ
due to improperly idealized MC modeling. This ratio is nothing but the pick-oﬀ
annihilation probability Ppick of beam OFF events. We estimated systematic errors
from MC by comparing pick-oﬀ probability obtained from OFF histograms withones obtained by fitting time spectra. The Compton free energy cut (from 360 keV
to 450 keV) and accidental rejection were required to draw time spectra. Table 3.7

86

3.5 Analysis

summarizes the relative diﬀerence of the two pick-oﬀ annihilation probabilities.
Worst values of the relative diﬀerence were assigned as a systematic uncertainty.
RUN ID

frequency

Ppick (MC)Ppick (fit)relative diﬀerence

A-1
A-2
B-1
C-1
C-2
C-3
D-1
D-2
D-3
D-4
D-5
E-1
E-2
F-1
F-2
G-0

201.83 GHz
201.83 GHz
202.64 GHz
203.00 GHz
203.25 GHz
203.25 GHz
203.51 GHz
203.51 GHz
203.51 GHz
203.51 GHz
203.51 GHz
204.56 GHz
204.56 GHz
205.31 GHz
205.31 GHz
180.59 GHz

7.46(38)
7.71(39)
7.41(35)
7.84(48)
7.59(47)
8.49(44)
7.96(40)
8.31(92)
8.95(98)
7.37(80)
8.95(99)
8.99(53)
7.68(45)
8.75(45)
7.72(39)7.90(36)

7.99(18)
7.89(18)
8.50(17)8.55(21)9.01(22)7.91(22)
8.24(18)
8.74(43)
7.63(43)
8.41(43)
9.03(46)
9.01(23)
8.32(21)8.35(16)
8.11(18)
8.43(16)

6.67(29)%
2.34(30)%
12.79(23)%8.35(40)%
15.78(37)%7.43(34)%
3.40(29)%
4.89(1.39)%
17.25(1.78)%
12.40(1.26)%
0.88(1.47)%
0.25(41)%7.72(38)%
4.82(26)%
4.79(29)%
6.29(23)%

Table 3.7: Comparison of pick-oﬀ rate between MC and time spectra.

3.5.4

Another Systematic Uncertainty

Energy Scale and Energy Resolution of γ-ray detectors
If diﬀerences of the energy scales and the energy resolutions exist, they make fakesignals. In order to estimate this uncertainties, the 511 keV peaks after the energy
calibration were fitted again with Gaussian functions. In almost all the RUNs,
the diﬀerence is zero consistent within statistical uncertainty (less than 2 standard
deviation). Three exceptions were found. Since there are 48 samples (12 RUNs
and 4 detectors), these three exceptions can be statistical fluctuations. Thus, thiseﬀect is negligible in this experiment.
Accidental Rejection Eﬃciency
If the peaks used for accidental rejection (Fig. 3.8) change their position or width,
cut eﬃciency becomes diﬀerent between beam OFF and ON. The cut condition
was taken loose (from −5σ to 3σ) to reduce this eﬀect. This eﬀect was estimatedusing data before requiring the back-to-back 511 keV energy cut because the real
signal is enhanced with this energy cut and causes the under-estimation of the cut
eﬃciency of beam ON events. It turned out that its contribution is at most +0.68%−0.22% ,3.5 Analysis

87

and negligible in calculating Ps-HFS at the current level of precision. Table D.1summarizes the results of this estimation.
Background Normalization
The remaining background was estimated with beam OFF events. The beam OFF
and ON spectra were normalized with the number of events at the prompt peak(from −4.0 ns to 3.0 ns) of time spectra. The prompt peak contains normal positronannihilations, p-Ps events, and slow positron events. These events are statisticallyindependent of ones in time window (from 50 ns to 250 ns). A statistical error of
the number of events in the prompt peak becomes a systematic uncertainty. Evenin the RUN with the least statistics, the uncertainty due to normalization was only
0.17%. This is negligible in calculating Ps-HFS at the current level of precision.
Table D.2 summarizes this uncertainty.

3.5.5

Summary of the Systematic Errors

Table 3.8 summarizes systematic uncertainties for each RUN. Eﬀects of the power
estimation uncertainty was estimated by scanning C of each frequency with anasymmetric Gaussian function when we fit the data with MC. The Stark eﬀect(−460 ppm shift for Ps-HFS) is a common systematic error and was consideredat last. Other errors were combined with statistical uncertainty of S/N of data
and MC. Another uncertainty for each RUN has negligible eﬀect when obtaining
Ps-HFS and p-Ps lifetime.

88

3.5 Analysis

RUN IDPower Estimation

MC

Accidental Rejection

Normalization

A-1
A-2
B-1
C-1
C-2
C-3
D-1
D-2
D-3
D-4
D-5
E-1
E-2
F-1
F-2
G-0

+14.7 / −17.2%+14.7 / −17.2%+19.2 / −20.9%+12.7 / −14.7%+12.7 / −14.7%+12.7 / −14.7%+12.7 / −14.7%+12.7 / −14.7%+12.7 / −14.7%+12.7 / −14.7%+12.7 / −14.7%+20.4 / −22.1%+20.4 / −22.1%+12.3 / −14.0%+12.3 / −14.0%+24.0 / −29.0%

6.67%
2.34%
12.79%
8.35%
15.78%
7.42%
3.40%
4.89%
17.25%
12.40%
0.88%
0.25%
7.72%
4.82%
4.79%
6.29%

0.10%
−0.10%
negligible−0.23%−0.04 %
−0.07 %
−0.02%
0.68%−0.02%−0.20%−0.11%
0.06%
0.22%
0.22%
0.09%
0.13%

0.07%
0.07%
0.06%
0.08%
0.08%
0.07%
0.12%
0.17%
0.16%
0.15%
0.16%
0.08%
0.09%
0.06%
0.07%
0.06%

Table 3.8: Summary of systematic uncertainties.

89

3.6 Analysis

3.6

ResultWe obtained the following results from the analysis described in this chapter. PsHFS and the Einstein’s A coeﬃcient are
+0.15
∆HFS
Ps = 203.39−0.14 (stat.) ± 0.11 (syst.) GHz(3.20)

A = 3.69 ± 0.48 (stat.) ± 0.29 (syst.) × 10−8 s−1 .

(3.21)

This result contains the Stark shift correction. This is the first direct measurementof Ps-HFS. An asymmetric errors mean the correlations among three fitting variables. These two results are consistent with the QED predictions of O(α3 log α−1 )for Ps-HFS and the tree level calculation for A, within 1 standard deviation∆Ps
HFS (th) = 203.391 69(41) GHz [35][36][37]A(th) = 3.37 × 10

−8 −1

s

(3.22)[15](3.23)

Lifetime of p-Ps (τp–Ps ) is
τp–Ps = 89+18−15 (stat.) ± 10 (syst.) ps(3.24)

This is the first direct measurement of p-Ps lifetime. This result is consistent with
the QED prediction of O(α3 log α−1 ) within 1.7 standard deviations
τp–Ps (th) = 125.164 7(2) ps [63][64](3.25)A summary of uncertainties is shown in Table 3.9

Counting statistics
Power estimation uncertainty
Stark eﬀect
Monte Carlo simulation
Quadrature sum

Ps-HFS

p-Ps life

A coeﬃcient+740 ppm−690 ppm
430 ppm
460 ppm
280 ppm

+20 %
−17 %
10%
negligible
5.5%

+13 %
−13 %
7.2%
negligible
3.0%+1010 ppm
−980 ppm

+23%
−20%

+15%
−15%

Table 3.9: Uncertainties in this measurement.

90

3.6 Analysis

Chapter 4

Discussion
4.1

Consistency Check of Power Estimation

The most problematic systematic error is the uncertainty of power estimation unestablished in the millimeter-wave range. This uncertainty should be checked witha totally independent method from one described in Sec. 2.2.3. In this research,
we discovered an unexpected increase of Ps formation probability in gas containing
nitrogen molecules (This trouble was solved by using neopentane gas as describedin Sec.2.3.2). Therefore, we can estimate accumulated power in the Fabry-Pérot
resonant cavity with this increase.
The increase of Ps is statistically limited because of the small detection eﬃciencyof 3γ from the cavity region. Instead, a ratio of the slow positron decrease f (Vtr )was studied in pure 1 atm nitrogen. Figure 4.1 shows results, in which required cut
conditions are the same as that described in Chap. 3. It is diﬃcult to properly fit
the data with a toy model shown in Fig. 2.33 because this model ignores inelastic
scattering. A few data points cannot constrain complex models with many unknown
parameters. Therefore, a simple empirical formula is introduced√(4.1)
f (Vtr ) = Crel Vtr
The coeﬃcient Crel can be used to estimate power calibration constant C as summarized in Table 4.1. We estimated new C (denoted as C ′ ) by fixing C ′ = C
at 201.8 GHz and relatively calculated other C ′ by Crel ratio. The results are allconsistent with the absolute power calibration constant C.
Signal contamination of the Ps-HFS transition is a systematic error for thisrelative power estimation. One can estimate upper limit of signal contamination
using the neopentane data in which no slow positrons exist in time window. In
RUN A-1 (oﬀ-resonance), the rate of beam OFF events (background) is 499 mHz,and the transition rate is 21 mHz at 21.6 kW. Slow positron data at a similar
condition (203.81 GHz, 24.3 kW) has the background rate of 1.82 Hz, and the slow
positron decrease rate of 298 mHz. Thus, the signal contamination was estimatedat most 10% at oﬀ-resonance. In RUN C-2 (on-resonance), the background rate is91

92

4.2 Discussion+

slow e decrease [%]453 mHz, and the transition rate is 93 mHz at 67.4 kW. Slow positron data at asimilar condition (203.56 GHz, 56.2 kW) has the background rate of 1.66 Hz, andthe slow positron decrease rate of 422 mHz. Therefore, the signal contamination
was estimated at most 20% at on-resonance. These are less than the errors whichhave been already assigned to the relative power calibration.

35
30
25
20
201.8 GHz15

202.4 GHz10203.0 GHz5

203.6 GHz
204.3 GHz0205.5 GHz

0

1

2

3

4

5

6
Vtr [V]

Figure 4.1: Relative power estimation with the slow positron decrease.

Frequency

Crel

C ′ [kW/V]

C [kW/V]

201.83 GHz
202.64 GHz
203.00 GHz
203.51 GHz
204.56 GHz
205.31 GHz

232(28)209(29)97(15)
88(17)
95(11)
123(16)23.2 (fixed)
20.9(3.8)
9.7(1.9)
8.8(2.0)
9.5(1.6)
12.3(2.2)

23.2+3.4−4.0
23.4+4.5−4.9
14.6+2.2−2.3
10.2+1.3−1.5
11.3+2.3−2.5
11.4+1.4−1.6

Table 4.1: Relative power calibration constant.

4.2

Cut Validation

The result should be independent of the event selection within statistical uncertainties. We slightly changed cut conditions to validate it. Only statistical uncertainties(no Stark eﬀect correction) were considered in the following sections.

4.2.1

Time Window

We required the time window from 50 ns to 250 ns. Table 4.2 summarizes when
the upper bound which determines remaining accidental events of this condition is93

4.3 Discussion

varied. Integral of fitted time spectra should return a consistent result comparedwith cut-based values, and is listed in the same table. All results are consistent
with each other.
upper bound [ns]A coeﬃcient [×108 s−1 ]Ps-HFS [GHz]p-Ps life [ps]

200
250
300
350
400

3.64±0.47
3.65±0.47
3.58±0.47
3.62±0.48
3.79±0.49

203.29±0.12203.30±0.14203.36±0.15203.43±0.16203.44±0.15

97.2±17.3
89.3±15.1
85.6±15.1
77.9±15.2
84.0±15.0

fitting3.68±0.45

203.33±0.13

86.0±13.9

Table 4.2: Check of time window.

4.2.2

Accidental Rejection

We required accidental rejection condition from −5σ to 3σ to obtain the result.
Table 4.3 summarizes when the upper bound which determines remaining accidental
events is varied. All results are consistent with each other.
upper boundA coeﬃcient [×108 s−1 ]Ps-HFS [GHz]p-Ps life [ps]

2σ
3σ
4σ

3.43±0.49
3.65±0.47
3.89±0.45

203.35±0.15203.30±0.14203.37±0.13102.5±21.3
89.3±15.185.9±13.0

Table 4.3: Check of the accidental rejection cut.

4.2.3

Energy Cut

We required energy cut condition from −2σ to 3σ to obtain the result. Table 4.4
summarizes when the lower bound which determines 3γ-ray contamination is varied.
All results are consistent with each other.
lower boundA coeﬃcient [×108 s−1 ]Ps-HFS [GHz]p-Ps life [ps]

3σ
2.5σ
2σ

3.46±0.49
3.59±0.48
3.65±0.47

203.33±0.19203.36±0.15203.30±0.14

91.3±17.1
88.9±15.4
89.3±15.1

Table 4.4: Check of the energy cut.

94

4.3 Discussion

4.3

Future Prospects

The Ps-HFS value was directly measured for the first time in this research. The next
target is to reach a suﬃcient level of precision to address the observed discrepancy(15 ppm). Figure 4.2 shows a possible future setup. Three major improvements
are required: i) A estimation with a 100 kW-class gyrotron, ii) Ps formation in
vacuum, iii) A frequency tunable gyrotron. These three points are all technicalchallenges for each field, and are described from the next section.
455(67(83+""(
9&*:.*'8;(%.'+)3*(/;&#%&#'!
*H!

*C()*+,!
D5E(F@G(&+0$+%$#'!

<=>55?@=44(>+%8A$'/(#B%$8"(.'$%!

!#"$%&#'()*+,!
-+3#&$,*%*&!

!"! !"!
-#&&./+%*0(
1+2*/.$0*!

!"(+""*,)3;!

*99$8$*'%(!"(8#'2*&%*&!

Figure 4.2: Schematic view of a possible future setup.

4.3.1

Power Estimation with 100 kW-Class Gyrotron

According to Table 3.9, one of the dominant errors is from the power estimation
uncertainty. We need relative accuracy of better than 0.3% for the power estimation(currently, 10-20%). There are two problems at diﬀerent levels. One problem isabsence of established power standards in the high-power millimeter-wave range.
The other problem is the power uncertainty caused by use of the Fabry-Pérot
resonant cavity. The only reliable method is to measure REAL power with watertemperature under well-controlled interference conditions.
The most realistic solution with current technology is to develop a 100 kW-levelgyrotron and abandon the Fabry-Pérot cavity. Real power is always dumped intoa calorimeter, in which steady state between water flow and power absorption isachieved as shown in the right hand side of Fig. 4.2. We need essential changes ofthe power supply, the MIG, the electron beam collector, water cooling systems,theGaussian converter, and the output window. We should also develop an over-sized4.3 Discussion

95

corrugated waveguide system to safely handle really high-power millimeter-wave
radiation. It is necessary to correct a phase from TEM00 mode (Gaussian beam)
to HE11 mode (an eigenmode of the corrugated waveguide) with a matching optics
unit [65]. Such kind of system is very costly but already be in practice for fusion
applications. They are shown in the left hand side of Fig. 4.2.

4.3.2

Ps Formation in Vacuum

Statistics should be improved by over four orders of magnitude. Positrons, without static magnetic fields, are scattered randomly by gas molecules as shown in
Fig. 3.11. Small amounts of Ps are formed in a high-power radiation region andcontribute to the transition signal. Others are in an exterior region and resultin background (pick-oﬀ annihilation in gas and o-Ps decay). One requires either
focusing Ps in the beam region or eliminating background annihilations. This is
impossible with a commercial positron source and gas used in this thesis.
One has some diﬃculties in use of gas as an electron source. We cannot use
gas with really high-power gyrotrons because of discharge problems. Systematic
uncertainties (Ps formation increase, Stark eﬀect and non-thermalized Ps discussed
in Sec. 1.5.3) from gas should be also eliminated. Any problems due to gas handlingare undesirable in precise spectroscopy measurements.
Ps should be formed in vacuum with an intense positron beam. The pickoﬀ background does not exist in vacuum. Three γ-ray contamination from o-Ps
decay can be reduced by broadening high-power beam region and collimate the Ps
formation region. Ps collimation is also needed to eliminate annihilation at a wall.
There are some intensive studies to produce high Ps yield in vacuum [66][67].
An intense positron beam is produced with tungsten foils irradiated by an electron
beam [68]. A positron beam is introduced to a Ps assembly between the corrugatedwaveguides as shown in middle of Fig. 4.2. An eﬃcient Ps converter which emits
Ps into vacuum is required. Some materials with porous structure are candidates
for the Ps converter. Another option is use of a Ps beam instead of the positron
beam [69]. An well-controlled Ps beam can be easily collimated at the beam region.

4.3.3

Frequency Tunable Gyrotron

In this thesis, oscillation frequency of the gyrotron is changed by replacing the RF
cavity. Even though we treated the MIG with the greatest care to protect it fromair contamination, replacement of the RF cavity inevitably determines ultimate
lifetime of the MIG (less than one year). Irreproducibility of alignment (better
than 0.5 mm is required for a few meter size object as described in Appendix A.6)
is also problematic for a precision measurement. Gyrotron output frequency should
be continuously changed without open a vacuum case.
Theoretically, frequency-tunability can be obtained in use of backward-wave
oscillation. In a normal gyrotron operation, the electron beam couples to an axial
fundamental mode TEmn1 (See Fig A.3 and Table A.1). Other axial modes TEmn2 ,96

4.3 Discussion

TEmn3 , · · · have slightly diﬀerent resonant frequency, but are not excited becauseof high phase velocity and low Q-factor. Backward-wave oscillation makes use of
the Doppler shift of gyrating electrons to continuously connect higher axial modes.
Figure 4.3 shows operation conditions of gyro-devices. A dispersion relation of theelectron beam including the Doppler shift shown as a solid line in this figure isexpressed asω = ωc + kz v// ,(4.2)
where kz denotes wave number of electrons parallel to the magnetic field, and v//
does parallel velocity. A hyperbola represents approximate dispersion relation of
the RF cavity (See Appendix A.2 for detail) expressed as√( ′ )2j2
ωr = c kz + mn ,(4.3)
Rwhere R denotes radius of the RF cavity. Oscillation of gyro-devices occurs near
the cross points between above two curves. Gyrotron oscillation means kz ∼ 0, andis monochromatic. Interaction with backward waves at kz < 0 has a potential tocontinuously tune output frequency. Such devices are called gyro-BWO.

ω+,*-.*)/0&(+&12&/34"%0&!

 %# 
ω ! ! " #$" !  &'  ($$% "+,*-.*)/0&(+&*5&6*37!

ω ! ω ! " "# $!!

8.)36"9"%0&#0,(:;<=&(>*,3?()&!"#$%&'()*!

!"#0,(%,()&(>*,3?()&@7()(/$,(73?/A&

Figure 4.3: Schematic diagram of dispersion relation of electrons and RF cavity.

Many gyro-BWO devices have been tested in independent institutes [70]. Unfortunately, obtained output power is very low (much less than 100 W). High Q-factor
for higher axial modes causes ineﬃcient power extraction, and needs complicatedinternal mode converters. A new idea to get high-power extraction is a reflective
gyro-BWO with a tapering RF cavity. The next step in gyrotron development is
to test this new device and make it in practical use.

Chapter 5

Conclusion
Ps-HFS was firstly measured with a direct transition method. Measurement of thePs-HFS is diﬃcult because the Ps-HFS transition is suppressed with finite lifetimeof Ps, and Ps-HFS is in the millimeter-wave region. It requires development of ahigh-power millimeter-wave system, which is a big challenge for current technologyof plasma physics and millimeter-wave optics.
A gyrotron oscillator was developed as a high-power (> 100 W) millimeter-wave
radiation source. It was a technical challenge to fabricate and control the gyrotron
with such high frequency (203 GHz). We changed output frequency of the gyrotron
by replacing an internal RF cavity. A long time (more than a week) operations
were the first trial with the high-power gyrotron.
A Fabry-Pérot resonant cavity accumulates radiation to equivalent power ofabout 20 kW to cause the induced transition from o-Ps to p-Ps. In the millimeterwave range, a great eﬀort was required to design a half mirror with high reflectivity,
low loss, and withstanding against high-power dumping. A gold mesh mirror on asilicon substrate was developed.
There are no established methods to measure high-power millimeter-wave radiation today. Accumulated power is, especially, diﬃcult to estimate because cavity
finesse (Q-factor) and coupling cannot be measured with rather wide line-width ofgyrotron output. This thesis reported a new way to directly measure the accumulated power. We calibrated output of the pyroelectric detector with temperature
increase of a water load. We corrected reflected-beam eﬀects which reduced theoscillation eﬃciency of the gyrotron.
Output power from the gyrotron was stabilized with a feed-back control of theelectron-beam current. Accumulated power wan the Fabry-Pérot cavity is alsostabilized by controlling length of the cavity. There are inevitable fluctuations inthis accumulated power in one pulse because of line-width and drift of the gyrotron
frequency. We corrected this eﬀect with a Monte Carlo simulation in analysis.
A Ps formation assembly and γ-ray detectors were carefully arranged to enhance S/N of the direct transition. Use of neopentane gas was a key to reduce
an increase of Ps formation probability due to positron acceleration by millimeter97

98

5.0 Conclusion

wave radiation. LaBr3 (Ce) scintillators are used to detect γ rays with high energy
resolution.
The reaction cross-section of the Ps-HFS transition is studied using MC, andobtained Ps-HFS value ∆HFS
Ps , lifetime of p-Ps τp–Ps , and the Einstein A coeﬃcient
are
+0.15
∆HFS
Ps = 203.39−0.14 (stat.) ± 0.11 (syst.) GHz

τp–Ps = 89+18−15 (stat.) ± 10 (syst.) ps
A = 3.69 ± 0.48 (stat.) ± 0.29 (syst.) × 10

(5.1)
(5.2)
−8 −1

s

.

(5.3)

This is the first direct measurement of Ps-HFS and lifetime of p-Ps. These areconsistent with the QED calculations.
Precise but indirect measurements of Ps-HFS were performed in 1970’s and1980’s. Recent QED calculations of Ps-HFS diﬀer from the measured value by
3.04(79) MHz (15 ppm), which is unlikely to be due to a statistical fluctuation(3.9 standard deviations). A candidate in the future to test this discrepancy isthe direct measurement of Ps-HFS firstly performed in this thesis. Some technicalimprovements are needed to reach a suﬃcient level of precision to address theobserved discrepancy. This thesis pointed out development of a frequency tunablegyrotron of 100 kW and a method of eﬃcient Ps production in vacuum with apositron beam.

Acknowledgements
This experiment is a collaboration of the University of Tokyo and Fukui University.
I would like to thank my collaborators and Ph.D. advisers.
I would like to express my gratitude to my supervisor Prof. Shoji Asai who
supported me during past six years. My researcher life suddenly started just twodays after I firstly met him. This earlier start resulted in publication of the first
paper when I was in the master course. The themes he introduced to me werealways very unique. Especially, the experiment reported in this thesis was quite
challenging and worthwhile.
I wish to express my gratitude to collaborators in Research Center for Development of Far-Infrared Region, University of Fukui (FIR-FU). Prof. Takatoshi Idehara
always encouraged me, and sometimes literally helped me with a screw wrench toadjust the gyrotron. He also gave me many chances to present my study in theinternational arena. I would like to thank Prof. Yoshinori Tatematsu for developing
the new gyrotron oscillator with the internal Gaussian mode converter. It was agreat turning point to bring this experiment to a successful conclusion. Prof. Isamu
Ogawa was so kind to give me practical advices to operate the gyrotron and measure output frequency. I would like to give my special thanks to Mr. Tomohiro
Kanemaki and members in the factory for fabrication of the gyrotron.
I also wish to express my great appreciation to collaborators in the Universityof Tokyo. Prof. Tomio Kobayashi never complained about this experiment andalways encouraged me. Useful discussions with Prof. Haruo Saito pushed on thisexperiment at the earlier stage when I did not know anything about positronium
and its transition. Dr. Toshio Namba looked at me with hard eyes to carry out
this experiment in realistic time and cost required. Dr. Taikan Suehara is a pioneerof this experiment. He is always very positive even when we reached a deadlock(it happened every one month in this challenging experiment). He taught neversay-die mind to me. Dr. Takayuki Yamazaki was a great and smart collaborator.
He always finished his work before I asked him something. I usually come up withmany miscellaneous ideas and sink into a bog. He is mature experimentalist andgives me many useful suggestions. I wish to express my appreciation to Mr. Shigemi
Otsuka and members in the factory. Finally, I would like to thank Mr. Akira Ishida
and all staﬀs in department of Physics, and International Center for Elementary
Particle Physics for their frequent help.

99

100

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104

Appendix A

Gyrotron Theory
A.1

Calculation Procedure

Gyrotron physics is one of the most interesting topics in plasma physics. It is notpractical to quantum-mechanically consider the maser oscillation between Landaulevels. Instead, classical consideration using Maxwell’s equations and the Lorentz
force is usually taken [71][72]. Firstly, we introduce electromagnetic fields in the
RF cavity absence of an electron beam. Electron-beam handling is discussed next.
Then, interaction between millimeter waves and the electron beam is calculated.
This formalism is called a Cold Cavity method in which perturbation from electrons to a field profile of radiation is ignored. Another formalism is called a Self
Consistent method in which electron motion and RF behavior are simultaneously
solved. Cold cavity formalism is a good approximation when quality factor of the
RF cavity is high enough.

A.2

Theory of the RF Cavity

The RF cavity is an open-ended cylindrical resonator whose resonance angularfrequency is ωr . It can be considered as an irregular waveguide, of which radius
R varies slowly along the z-axis as shown in Fig A.1. Electrons from the MIG
are introduced from the left hand side of this figure, where the resonator is cutoﬀ
preventing RF fields from leaking and damaging the MIG. The intermediate region
is a flat waveguide of radius R2 and length L2 . The right side is an output section
and open.
The resonant mode of the RF cavity is a solution of Maxwell’s equations in acavity wall. Let us take cylindrical coordinates (r, ϕ, z). Only Transverse Electric(TE) modes are considered because gyrating electrons are coupled with transverse
electric fields, not with magnetic fields. Introducing an azimuthal index m =0, ±1, ±2, ±3, · · · , a radial index n = 1, 2, 3, · · · , and an axial index l = 1, 2, 3, · · · ,the eigenmodes of this cavity (TEmnl resonant modes) are products of a transverse−→
vector function E (r, ϕ) and an axial scalar function Ψ(z). One can approximate the12,"#$$%

A.2 Gyrotron Theory
'-0/
'-0
'-)/
'-)
'-//
'-/
'-(/
'-(
'-./
41!
'-.
&

43!
$"#("#&#'()*&!"#$%%&&#'()*!

+',-&!'.,#/&

42!

θ"

θ!
02!

01!

'

(

)

*

03!

+& +' +( +) +* '& '' '(!"#$$%

Figure A.1: Cross-sectional geometry of the RF cavity used in FU CW G1.
−→
transverse electric field E (r, ϕ) by propagating TEmn modes in a circular waveguide
expressed as
( ′ )
R
j
Er (r, ϕ) = E0 m Jm mn r sin(mϕ),
(A.1)
r
R
( ′ )
jmn
′
r cos(mϕ),(A.2)
Eϕ (r, ϕ) = E0 Jm
R
Ez (r, ϕ) = 0,
(A.3)
where E0 is normalized strength of the electric field, Jm (x) is the m-th Bessel′ (x) is its 1st derivative, and j ′function of the 1st kind, Jm
mn is the n-th root of′Jm (x) = 0. Since taper angles of the cavity are small, changes of R(z) can beignored, and the mode mixing with another TEm′ n′ is also negligible. Figure A.2−→
shows E (r, ϕ) of the TE52 used in this experiment. Note that there are two degenerated states according to sign of the azimuthal index m except for m = 0 case,which corresponds to phase rotation around the z-axis. When gyrating electrons
exist, this degeneracy is removed to co-rotating or counter-rotating with electron
cyclotron motion in a magnetic field.
An axial function Ψ(z) obeys a nonuniform string equationd2 Ψ(z)2+ ζmn(z)Ψ(z) = 0,
dz 2√( ω )2 ( j ′ )2mn
ζmn (z) =
−
,
c
R(z)

(A.4)
(A.5)

(
)
where ω = ωr 1 + 2Qi D is complex RF frequency, and QD is a diﬀraction quality
factor. An open-end correction is calculated at cutoﬀ and output sections to estimate ωr and QD simultaneously. Equation (A.4) is solved numerically under the

3

A.3 Gyrotron Theory32

y [mm]

1

0

-1

-2

-3
-3

-2

-1

0
x [mm]

1

2

3

Figure A.2: Transverse structure of the electric field in the RF cavity (TE52 mode).
following boundary conditions:dΨ(0) = iζmn ΨdzdΨ(zout ) = −iζmn Ψ,
dz

(A.6)
(A.7)

where zout is an end-point of the output taper (zout = L1 + L2 + L3 ). One solves
Eq. (A.4) with Eq. (A.6) as an initial condition, for example using the RungeKutta formula. Then, two variables (ωr , QD ) are varied until the second boundary
condition Eq. (A.7) is satisfied. Figure A.3 shows the result of this calculation for
the RF cavity in this experiment. Three diﬀerent lines correspond to diﬀerent axial
indices (l = 1, 2, 3). The solution is listed in Table A.1. A quality factor due to
Ohmic loss QΩ and combined Q are also shown. The Ohmic quality factor is given
by
[
(
)]
m
R2
1−
QΩ =
,
(A.8)
′
δ
jmn
where δ = 0.18 µm is the skin depth of copper at 200 GHz. In a normal operationof the gyrotron (gyromonotron), TE521 (shown as a solid line in Fig. A.3) mode isexcited.
Interaction with backward-wave is needed for electrons to couple with higher
axial modes. This device is called gyro-BWO, and has potential to continuously
tune output frequency. Output power of this device is currently less than 100 W,
which is insuﬃcient for the Ps-HFS measurement. A proposal for higher power issuggested in the future prospects (Sec. 4.3).

4

Ψ(z) [a.u.]

A.3 Gyrotron Theory

120
100
80
60
40
20
0
0

5

10

15

20

25
z [mm]

Figure A.3: Axial structure of the electric field in the RF cavity. Solid, dots, anddashed lines show TE521 , TE522 , TE523 , respectively.

modefrequency [GHz]

QD

QΩ

Q

TE521
TE522
TE523

203.01203.59204.56

3200
810
380

10600
10600
10600

2500
750
370

Table A.1: Resonant frequency and QD of three modes shown in Fig. A.3.
Cavity parameters are (L1 , L2 , L3 ) = (5 mm, 14 mm, 5 mm), and (R1 , R2 , R3 ) =(2.345 mm, 2.475 mm, 2.735 mm).

5

A.3 Gyrotron Theory

A.3

Theory of the Electron BeamBz [ T ]An electron beam should be properly controlled and introduced to a right positionin the RF cavity to obtain maximum coupling between electrons and the RF fields.
A sophisticated method of electron control is developed in gyrotron physics. The
simplest approximation is described in this section.
Electrons are emitted from an emitter ring on the cathode, accelerated between
1st and 2nd anodes (Fig. 2.2), and guided by a magnetic field to reach the RF
cavity as gyrating where magnetic field strength is maximum (about 7.5 T) as
shown in Fig. A.4. At last, the electron beam is dumped by a beam collector(Fig 2.2). Figure A.5 shows electron trajectories simulated by CST PARTICLESTUDIO [73]. Radius of the electron beam is compressed by the strong magnetic
field at the cavity.

8
7
6
5
4
3
2
1
0

0

200

400

600

800

1000
z [mm]

Figure A.4: Simulated magnetic field Bz on the z-axis (by CST PARTICLE STUDIO). The emitter ring is at z=0 mm, and the RF cavity is at 650 mm.
Beam radius Rb should provide maximum electron coupling to the one mode
(TEmn1 ) of the RF cavity discussed in Sec. A.2. This coupling strength is determined by a function
J 2 (j ′ Rb /R)
Gmn = ′2m∓1 mn,(A.9)
2 (j ′2 )(jmn − m2 )Jmmn
where the sign in m ∓ 1 corresponds to co-rotating or counter-rotating waves compared with the direction of electron gyration. Figure A.6 shows calculated G52 . It
is clearly seen that the degeneracy of co- and counter-rotating modes is removed.
When one operates a gyrotron oscillator in the counter-rotating TE52 mode, Rbshould be controlled to be 0.7 of the cavity radius R2 .

6

A.4 Gyrotron Theory

Figure A.5: Simulated electron trajectories from the MIG (by CST PARTICLESTUDIO).
Beam radius Rb is compressed depending on magnetic field strength as demonstrated in Fig. A.5. This adiabatic compression is expressed as√
Rb (z) =Bc
Rc ,Bz (z)(A.10)

where Bc ∼ 0.1 T is z−component of the magnetic field at the emitter ring on thecathode, and Rc ∼ 14 mm is the ring radius (Fig. 2.2). A factor b = Bz (z)/Bc iscalled compression ratio. Bc is optimized by a normal conducting gun-coil solenoid
around the MIG so that Rb becomes the proper value (i.e. 0.7 of the cavity radius),
The magnetic field induced by the gun-coil also aﬀects the E × B drift near
the cathode [74]. The E × B drift determines initial perpendicular velocity v⊥c ofelectrons emitted thermally from the cathode surfacev⊥c ∼Ec ϕEB,
Bc(A.11)

where Ec is electric field strength at the cathode, and ϕEB is the angle between Ec
and Bc . Ec is controlled with diﬀerence of voltage between cathode (Vk ∼ −18 kV)and 1st anode (Va ∼ −10 kV). Electrons start gyrating because of this E × B drift.
2 /B
According to an adiabatic approximation, the magnetic momentum me v⊥z
is constant [74]; therefore, v⊥ becomes faster and faster along the trajectory corresponding√ to the increasing magnetic field (Fig. A.4). Velocity parallel to Bz ,v// =2 is correspondingly reduced in the RF cavity since total energyv 2 − v⊥

2 . Theis conserved. Energy of cyclotron radiation increases proportionally to v⊥
optimum value of the velocity ratio α = v⊥ /v// (called pitch factor) at the cavity
is from 1.2 to 1.5. The pitch factor is controlled with the magnetic field of the
gun-coil and 1st anode voltage.

7

A.4 Gyrotron Theory

0.025

G

0.02
0.015
0.01
0.005
0
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Rb/R

1

Figure A.6: Coupling factor of the TE52 mode. Solid line shows the counterrotating mode, dashed one shows the co-rotating mode.

A.4

Excitation Theory

So far, RF fields in the RF cavity is obtained, and the optimum electron beam
is introduced to the RF cavity. Now we calculate an interaction between themand think of excitation condition of the gyrotron [71]. We focus on a steady-state
operation.
Gyrotron oscillation is estimated by calculating electron motion aﬀected by the
Lorentz force from the RF fields described in Sec. A.2. Electron’s equation of
motion in general is
→
−
dE
→
= −e−
v ·E
dt
→
−
→ e→ −→d−p= −e E − −v × B,
dtc(A.12)(A.13)

where E = me γc2 is energy of electron.
One can consider Eq. (A.13) in a coordinate system with origin at the electron→
gyrocenter. In complex notation, one denotes −p → px + ipy = peiθ , where θ
is a relative phase in momentum space between an electron momemtum and theelectromagnetic wave. Electron energy and momentum is normalized by initialenergy before the interaction (denoted as me γ0 c2 ). Assume constant propagating
velocity in the RF cavity, and independent variable is transformed from time to a

8

A.5 Gyrotron Theory

normalized axial position expressed as
ζ=

2ω
β⊥
c
z.
2β// c(A.14)−→
With the small argument expansion of the Bessel functions in E (r, ϕ), resulting
equation of motion is
(
)
dp
Ψ(ζ)
,
+ ip ∆ + |p|2 − 1 = iG √∫dζ|Ψ|2 dζwhere ∆ is frequency detuning defined as[]
2 ωr − ωc
∆= 2
,
ωr
β⊥
and G is coupling parameter defined as√
1
QD · PoutG=
0.47 × 10−3
2 Gmn .
2
γ0 Vk ηel β// β⊥(A.15)(A.16)(A.17)

Acceleration voltage Vk (cathode voltage), output power Pout , and electron eﬃciency
2
β⊥
ηel =
(A.18)
2(1 − 1/γ0 )are also introduced.
One can understand steady-state gyrotron oscillation by solving Eq. (A.15) for
given Pout . Note that Pout determines an absolute value of the RF field whichis calculated without a source term in Cold Cavity formalism. The results ofEq. (A.15) for some electrons in diﬀerent phases are shown in Fig. 2.4. Orbital
eﬃciency is obtained by averaging the results of electrons initially in diﬀerent phases
pj (0) = exp (iθj ) (0 < θj < 2π, j = 1, 2, · · ·, N ,)
η⊥ = 1 −1∑|pj (ζout )|2 .
N(A.19)

jThen, electron-beam current Ib required to achieve given Pout is calculated by
Pout = Vk · Ib · η⊥ · ηel ·QΩ.
QΩ + QD(A.20)

This is an enegy conservation condition between the electron beam and the RF field.
The RF field gains energy from electrons calculated by Eq. (A.15) and Eq. (A.19).
Figure A.7 shows an example of the solution for diﬀerent magnetic field strength.
Output power suddenly increases and gradually decreases when one raises magnetic
field strength. This is a typical behaviour of the gyrotron oscillator.

9

Pout [kW]A.6 Gyrotron Theory

2.5
2
1.5
1
0.5
0

7.38

7.4

7.42

7.44

7.46
B [T]

Figure A.7: Calculated output power of the gyrotron. Operation parameters arecathode voltage Vk = −18 kV, beam current Ib = 500 mA, magnetic field B =7.4 T, cavity radius R2 = 2.475 mm, beam radius Rb = 0.7 · R2 , and pitch factor
α = 1.2.

A.5

Soft and Hard Excitation

There are two oscillation conditions: soft excitation (ωc ≤ ωr ) and hard excitation(ωc ≪ ωr ). Figures A.8 show calculation of oscillation eﬃciency η = η⊥ · ηel forthese two conditions. η is a single-valued function of Ib in case of soft excitation(left figure). η is a multivalued function of Ib in case of hard excitation (rightfigure). High power is obtained in hard excitation. A diﬃculty of hard excitationis that the gyrotron start to oscillate at large Ib , and η increases as Ib decreases fora while. After η exceeds ∼ 0.1, η increases as Ib increases. Stabilization of output
power with a simple feed-back control is diﬃcult because of this non-linear behaviorof the gyrotron (See Sec. 2.2.4). Moreover, two diﬀerent η for the same Ib and B0(the same operation condition) can cause a sudden jump of output power withoutchanging any external parameters. Careful operations are generally required incase of hard excitation.

A.6

Alignment Eﬀect

Gyrotron oscillation is sensitive to displacement between the RF cavity and theelectron beam. We introduce starting current Ist which is the smallest electronbeam current required for the gyrotron oscillation [72]. Ist can be calculated by
the method described in Sec. A.4 with a condition Pout > 0. Change of Ist by10!
&0#1""*2&'(!3#1!'#4((η&0#1""*2&'(!3#1!'#4((η!

A.7 Gyrotron Theory
&'.-

$%↓!

&'.
&'(&'(&',-$%↓!

&'(&'(&',&',

&',

&'&-

&'&&
&'(

&'.

&')

&'*

&'+

,

,'(

,')

,'*

,'+

(

!"!#$%&'()!*+(#,%%!'$(!"#!"#$%
-./(

&
&'(

&')

&'*

&'+

,

,'(

,')

,'*

,'+

(

!"#$%
!"!#$%&'()!*+(#,%%!'$(!"#-./(

Figure A.8: Oscillation eﬃciency vs beam current. Left shows soft excitation andright shows hard excitation. A solid line shows oscillation with the same magnetic
field or frequency detuning.

displacement d mm is expressed as [75]
)
( 2π
2
Jm±1
λ · RbIst (d) = Ist (0) ∞)(∑ ( 2π )2π22· d · Jm−q±1
· Rb
Jq
λ
λ
q

(A.21)

Results of the calculation are shown in Fig. A.9. In case of no displacement (left),only counter-rotating TE52 can oscillate when B0 is set at 7.45 T and Ib is 0.4 A(the same condition as this experiment). In case of displacement of 0.5 mm (right),
structure of Ist changes and both counter-rotating and co-rotating TE52 satisfy
Ist > Ib = 0.4 A. These two modes compete with each other, and eﬃciency ofelectron’s phase bunching is reduced (mode competition). Consequently, carefulalignment better than 0.5 mm is required to obtain high-power gyrotron output.
This is technically diﬃcult because the size of gyrotron is rather large (Fig. 2.2).
This is one of the reason of bad reproducibility of gyrotron operation (Table 2.2).
Note that the gyrotron previously used in this experiment [54] is operated in TE03
mode of which counter- and co-rotating modes are degenerated. No mode competition occurs. The previous gyrotron is insensitive to the displacement. We selectTE52 mode because conversion eﬃciency of the Gaussian mode converter is better
than TE03 mode. The principle of the converter is described in the next section.

A.7

Theory of Gaussian Mode Converter

Output from the gyrotron RF cavity is TEmn mode in a circular waveguide. One has
to convert it to a linearly polarized Gaussian beam (TEM00 mode) to accumulate
it in the Fabry-Pérot cavity [47]. This is achieved by a Gaussian mode converter
developed in application for plasma heating.
When we consider the time-averaged Poynting vector using Eq. (A.1), the energy flux of TEmn modes can be treated as rays bouncing along the inside wall of a

11

)
1(,
1(+
1(*
1()
1
%(,
%(+
%(*
%() !"#$%&'()
%

'

'()

!"#$%&*')
0./!"-$

0./!"-$

A.7 Gyrotron Theory!+$%&(()!+$%&,-)!"#$%&*()
'(*

'(+

'(,

,

,()
&%!"#$!"#$%&*') !"#$%&!+$%&'()
,-)
)
1(,
1(+
1(*
1()
1
!+$%&(()%(,!"#$%&
*()
%(+
%&.()
%(*
!+$%&,-)%() !"#$%&'()!+$%&*')%''() '(* '(+ '(,,,()&%!"#$

Figure A.9: Dependence of Ist on displacement. Left figure shows Ist with perfectalignment. Right figure shows Ist with displacement of 0.5 mm.

waveguide in a polygonal helix as shown in Fig. A.10. This picture reflects the factthat TEmn modes can be decomposed into a series of linearly polarized (helicallyrotating) plane waves. The envelope of helically reflecting rays in a waveguide withradius RW becomes cylindrical caustic with radius RC
RC = RW ×

m
.
′
jmn

(A.22)

In case of TE52 mode we use, RC is 0.475 of RW .
&"'()#*+(!
,"-!
!"#$%!!

Figure A.11: Schematic view of the helical launcher.
Figure A.10: Rays of time-averaged propagating TEmn mode in a waveguide.
The waveguide opening should be cut open in a helical shape with the helicalcut following the ray trajectories in the waveguide. Figure A.11 shows a picture

12

A.7 Gyrotron Theoryof this helical launcher used in this experiment. The radiation launched from thehelical antenna expands in a radial and twisted fashion corresponding to helicalrotation in the waveguide. A quasi-parabolic reflector is introduced to focus theradiation and shape its profile. The left figure of Fig. A.12 shows a top view of thelauncher and the reflector. The reflector shape calculated with a requirement ofequal optical path length for all rays between the phase front and the focal point.
For parallel light (focus point is at infinity), parametric representation (xr , yr ) of
the reflector shape can be expressed as [47]( )1
ϕ(A.23)xr (ϕ) = RC − (2f0 − RC ϕ) tan
2
( ) (
)[
( )
]
1
1
2 1
yr (ϕ) = RC tan
ϕ − f0 − RC ϕ tan
ϕ −1 ,(A.24)
22
2
where f0 is the maximum value of yr and ϕ is a parameter. In special case of m = 0,
caustic radius RC = 0 and the reflector shape becomes
yr = −

1 2
x + f0 .
4f0 r(A.25)

This type of reflector is used in the previous experiment in which the oscillation
mode is TE03 [54]. The shape for m ̸= 0 is similar to parabola, but slightly
diﬀerent (quasi-parabola). The reflected rays compose linearly polarized beam ofwhich electric field vectors are parallel to the x-axis shown in the left figure of
Fig. A.12.
The right figure of Fig. A.12 shows a side view of the launcher and the reflector.
Propagating angle θB is called the Brillouin angle expressed as
)
( ′
jmn λ
−1
θB = sin(A.26)
2πRW
The axial structure should be fabricated so that reflected rays do not hit thelauncher head for all frequency used.
One can make a linearly polarized bi-Gaussian beam with the launcher and thequasi-parabolic reflector. Two additional plane mirrors are located over the reflector to extract beam from a small bore of the superconducting solenoid (Fig. 2.2).
An ellipsoidal mirror re-shapes the beam to focus at near the output window.

13

A.7 Gyrotron Theory

y [mm]

linearly polarized
beam

15

quasi-parabolic reflectorE

causitic

10
quasiparabolicreflector

5
0
-5
-10

θB

-15
linearly polarized beam

-20

-20

-15

-10

-5

0

5

10

launcherx [mm]

causitic

Figure A.12: Schematic view of Gaussian converter. (left) Top view. (right) Side
view.

14

A.7 Gyrotron Theory

Appendix B

Theory of Positron Acceleration
B.1

Toy Model of the Interaction

Interactions between a positron, an oscillating electric field E0 sin (ωt), and gas
molecules can be expressed as the following diﬀerential equation:medv= eE0 sin (ωt) − mωc v,
dt

(B.1)

where v is positron velocity, ωc is averaged angular frequency of collisions, andmωc v is a relaxation term by collisions. An analytical solution of Eq. (B.1) is√
eE0v(t) =
me

[
( )]
1
ω
−1
.
sin ωt − tan22ω + ωcωc(B.2)

We estimate mean energy gain δE during collision cycle τc∫ t=τcδE =eE(t)v(t)dt =
t=0

e2
ωc2
× E02 .
2me ωc2 ω 2 + ωc2(B.3)

In case of 1 atm nitrogen gas and 1 eV positron, τc is about 3.3 ps. For 203 GHz
millimeter waves, expected energy gain of a positron is()2δE ∼ 0.51 meV × E0 /100kVm−1(B.4)

per one collision. This is very small compared with Ore gap.

B.2

Random Walk ModelWe can model a sequence of collisions by random-walk. Since millimeter-wave
radiation in the Fabry-Pérot resonant cavity is linearly polarized, it is adequate
to only consider one dimensional random walk. If collision is random enough and15

16

B.2 Theory of Positron Acceleration

not correlated to the electric field oscillation, probability distribution of v after ncollisions can be expressed as[]
1(v − v0 )2g(v) = √
exp −
,
(B.5)
2σ 2
2πσ
√
σ = nδv,(B.6)
where v0 is an initial velocity of slow positron (a √few eV), and δv is acceleration or
deceleration per one collision determined by δv = 2me δE. What we are interested
in is probability to exceed the lower edge of Ore gap (Ethr = I − 6.8 eV). Whenvthr denotes corresponding positron velocity, this probability can be calculated by∫ −vthr
G(v > vthr ) = 1 −g(v)dv.
(B.7)
vthr

G [%]

Figure B.1 shows G(v > vthr ) in 1 atm nitrogen case (Ethr ∼ 7.7 eV). Powerdependence is related to δE, which is determined by gas property at scattering.
Previous experiments studied an eﬀect of static electric fields with a Boltzmann equation [60][61]. One can also construct a Monte Carlo simulation to model
this phenomenon. Even with these seemingly more quantitative approaches, uncertainties in inelastic scattering are required to be determined through fitting thedata. Comparison between models and data needs many data points at diﬀerent
accumulated power. This is diﬃcult with current stability of the gyrotron.

100
80
60

δE0 = 0.05 meV
δE0 = 0.1 meV
δE0 = 0.5 meV
δE0 = 1.0 meV
δE0 = 5.0 meV

40
20
0
0

20

40

60

80

100
120
Pacc [kW]Figure B.1: Simulated Ps formation dependence on power. Calculations of diﬀerent
δE0 are plotted.

Appendix C

Data Summary
The quantities recorded at the main trigger timing are summarized in Table C.1.
The quantities recorded in synchronization with the gyrotron output pulse are summarized in Table C.2. The quantities monitored with the interlock are summarizedin Table C.3.
namemoduleevent ID
real time-

gyrotron ON/OFF

input register (LeCroy C005)

Pla-energy (short gate), 2ch
Pla-energy (long gate), 2ch
La-energy, 4ch

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

Pla-time, 2chLa-time, 4ch

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

live time
Pla-rate, 2ch
Pla-and-rate
La-rate, 4ch

SCALER (Kaizu KC3122)
SCALER (Kaizu KC3122)
SCALER (Kaizu KC3122)
SCALER (Kaizu KC3122)

Table C.1: Quantities measured at main trigger timing.

17

18

C.0 Data Summary

namedeviceinput powerreflected powertransmitted power
trigger pulse of gyrotron
beam current of gyrotron
heater voltage of the MIG
cavity length
room temperaturetemperature of NIM bintemperature in the gas chamberpressure in the gas chamber

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

Table C.2: Quantities recorded in synchronization with gyrotron output pulse.

namereasontemperature of the collector
temperature of the gyrotron bodytemperature of the flange on the MIG
temperature of the output window
vacuum of the gyrotron
water flow for gyrotron coolingwater flow for chamber coolingcheck abnormal electron beam operation
check abnormal electron beam operation
check abnormal electron beam operation
protect window glass
protect the emitter surface of the MIG
protect gyrotron vacuum vessel
prevent the mesh mirror from melting

Table C.3: Quantities monitored with the interlock.

Appendix D

Small Systematic Uncertainties
Eﬃciency of accidental rejection and background normalization are negligible incalculating Ps-HFS compared with other errors. The diﬀerences are small but
significant, and listed in Table D.1 and Table D.2 for the future precise measurement. They may be due to electric noise of the pulse operation of the gyrotron(voltage=−18 kV, current=500 mA, width=60 ms, frequency=5 Hz). RUNs inwhich Fabry-Pérot cavity is oﬀ-resonance (-0) are also shown as a reference.

19

20

D.0 Small Systematic Uncertainties

RUN IDeﬃciency (beam ON)

eﬃciency (beam OFF)diﬀerence

A-1
A-2
B-0
B-1
C-0
C-1
C-2
C-3
D-0
D-1
D-2
D-3
D-4
D-5
E-0
E-1
E-2
F-0
F-1
F-2
G-0

56.544(76) %56.916(77) %52.083(86) %52.574(65) %56.464(96) %56.156(69) %54.35(10) %56.576(79) %54.46(13) %53.047(73) %54.69(18) %54.41(18) %54.34(17) %54.63(19) %54.580(93) %54.434(92) %55.143(90) %56.10(14) %56.381(69) %52.779(72) %55.217(65) %56.486(50) %56.973(51) %52.179(56) %52.570(43) %56.362(65) %56.283(63) %54.370(66) %56.614(52) %54.418(87) %53.060(48) %55.07(12) %54.40(11) %54.45(11) %54.69(12) %54.590(61) %54.402(60) %55.025(60) %56.104(92) %56.258(45) %52.732(47) %55.146(43) %0.10333(17)%−0.09990(16)%−0.18416(36)%0.00799(1)%0.18097(37)%−0.22565(38)%−0.03678(8)%−0.06712(11)%0.07673(22)%−0.02458(4)%−0.6838(27)%0.01758(7)%−0.20478(78)%−0.11336(47)%−0.01841(4)%0.05929(12)%0.21576(42)%−0.002552(8)%0.21881(32)%0.08910(14)%0.12797(18)%

Table D.1: Accidental rejection eﬃciency.

21

D.0 Small Systematic Uncertainties

RUN ID

livetime

uncertainty

A-1
A-2
B-0
B-1
C-0
C-1
C-2
C-3
D-0
D-1
D-2
D-3
D-4
D-5
E-0
E-1
E-2
F-0
F-1
F-2
G-0

6.7 × 104 sec
6.7 × 104 sec
7.0 × 104 sec
1.1 × 105 sec
7.2 × 104 sec
6.6 × 104 sec
6.8 × 104 sec
9.9 × 104 sec
2.6 × 104 sec
7.2 × 104 sec
1.2 × 104 sec
1.3 × 104 sec
1.4 × 104 sec
1.3 × 104 sec
2.3 × 105 sec
6.1 × 104 sec
6.0 × 104 sec
5.8 × 104 sec
1.0 × 105 sec
8.0 × 104 sec
1.2 × 105 sec

0.07 %
0.07 %
0.08 %
0.06 %
0.08 %
0.08 %
0.08 %
0.07 %
0.07 %
0.12 %
0.17 %
0.16 %
0.15 %
0.16 %
0.08 %
0.08 %
0.09 %
0.12 %
0.06 %
0.07 %
0.06 %

Table D.2: Normalization uncertainty.