学位論文
Development of a Transmission Electron Microscope
Adopting Two-Mode Superconducting RF Accelerating
Cavity for Achieving Low Energy Dispersion(低エネルギー分散を実現する２モード超伝導加速空洞を採用した透過型電子顕微鏡の開発研究)
平成 28 年 7 月博士 (理学) 申請
東京大学大学院理学系研究科
物理学専攻
東直

Abstract
This doctoral thesis reports the development and the study about the overall system design
and the fundamental technologies for the world’s first application of superconducting radio
frequency (SRF) technology to a transmission electron microscopes (TEM).
A TEM is a vital tool to image sub-nanometer structures of specimens. It has beenutilized in a wide range of fields including material science and biology. It is necessary toprocess specimens into layers as thin as a few tens of nm for the present TEM with theaccelerating voltage of a few 100 kV. This is because the imaging power with high spatial
resolution cannot be realized for thicker specimens as a consequence of the energy loss of theelectron beam due to inelastic scatterings with the specimen and the decrease of the electron
beam flux due to scatterings including elastic ones.
Recently a project that aims to adopt various technologies developed for high-energy
accelerators into TEM is being carried out at High Energy Accelerator Research Organization(KEK) in Tsukuba, Ibaraki, Japan. This employs Radio Frequency (RF) resonant cavity
which can generate accelerating fields as high as a few 10 MV/m in order to realize a new
type of TEM with higher accelerating voltage than ever. Moreover superconducting RF
cavity is adopted so that a high beam current comparable with conventional TEMs can beacquired and the accelerating field can be controlled more precisely than normal-conducting
cavities. This new type of TEM, superconducting RF transmission electron microscope (SRFTEM) is expected to enable the observation of thicker specimens than ever owing to its highaccelerating voltage.
The SRF-TEM is capable of realizing a significant increase in e− beam energy with areasonably compact facility. However, the RF acceleration scheme generally used in highenergy accelerators causes larger beam energy dispersion which degrades the spatial resolutionof the TEM. In this study, a new type of accelerating cavity named two-mode cavity
in which two oscillating accelerating fields coexist was developed. The two-mode cavity
is capable of flattening the sharp peak of the accelerating RF wave by superimposing one
resonant mode of TM010 whose resonant frequency is 1.3 GHz on its double resonant mode of
TM020 (2.6 GHz). A photocathode electron gun that is already developed is introduced into
the SRF-TEM instead of a thermal electron or a cold field emission gun normally used fora conventional TEM. This can generate ultra-short bunches of a few ps and supply electron
bunches precisely synchronized to the period of the accelerating field of the two-mode cavity.
This is also expected to provide the good time resolution to SRF-TEM.
In this thesis, I describe the overall system design of the SRF-TEM and the establishmentof the essential technologies for the SRF-TEM with 300 kV as a proof-of-principle prototype
and a future 3 MV SRF-TEM. As the system design, we confirmed that the two-mode cavity
can achieve low energy dispersion for the SRF-TEM with 300 kV and 3 MV by beam dynamics
simulations.
Essential technologies are 1) the two-mode cavity which can generate the high accelerating
gradient of about 10 MV/m with two resonant modes of TM010 and TM020 , and 2) the RF
feedback system which can control the accelerating fields generated in the two-mode cavity
precisely; (0.0012 ± 0.0003) % in terms of amplitude and (0.04 ± 0.01) mdeg. in terms ofphase for the TM010 mode and (0.0005 ± 0.0004) % and (0.03 ± 0.01) mdeg. for the TM020
mode.
With the above, we opened up the road to the fully new TEM, SRF-TEM.

Contents
1 Introduction
1.1 Significance of This Study . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Suggestion of Superconducting Radio Frequency Transmission Electron
Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Development of Superconducting Radio Frequency Transmission Electron Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Structure of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1

2 Present Situation of Transmission Electron Microscope and Limitation
2.1 History of Transmission Electron Microscope . . . . . . . . . . . . . . . . . .
2.2 System of Conventional Transmission Electron Microscope . . . . . . . . . .
2.3 Imaging Theory of Transmission Electron Microscope . . . . . . . . . . . . .
2.4 Limit of Thickness in Observation by Transmission Electron Microscope . . .

7
7
8
10
13

1
4
6

3 Concept of Superconducting Radio Frequency Transmission Electron Microscope15
3.1 System Configuration of Superconducting Radio Frequency Transmission Electron Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.1.1 Two-mode Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.1.2 Photocathode Electron Gun . . . . . . . . . . . . . . . . . . . . . . .
19
3.1.3 RF Feedback System for Two-mode Cavity . . . . . . . . . . . . . . .
19
3.2 Configuration Comparison between Conventional Transmission Electron Microscope and Superconducting Radio Frequency Transmission Electron Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.3 Advantage of Superconducting Radio Frequency Transmission Electron Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.3.1 Ability to Observe Thicker Specimen . . . . . . . . . . . . . . . . . .
21
3.3.2 Downsizing Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.3.3 Acquisition of Time Resolution . . . . . . . . . . . . . . . . . . . . .
26
4 System Design of Superconducting Radio Frequency Transmission Electron
Microscope with 300 kV and 3 MV by Numerical Simulation of Beami

Dynamics
4.1 Scheme of Simulation Study Using General Particle Tracer . . . . . . . . . .
4.1.1 Scheme of Numerical Simulation . . . . . . . . . . . . . . . . . . . . .
4.1.2 Scheme of Design Optimization . . . . . . . . . . . . . . . . . . . . .
4.2 Comparison of Two-mode Acceleration and One-mode Acceleration . . . . .
4.2.1 Method of Optimization . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Condition of Beam Dynamics Simulation . . . . . . . . . . . . . . . .
4.2.3 Optimization Result of 300 keV Acceleration . . . . . . . . . . . . . .
4.2.4 The optimization result of the 3 MV acceleration . . . . . . . . . . .
4.3 Requirement for Cavity Control . . . . . . . . . . . . . . . . . . . . . . . . .

29
29
30
31
33
33
34
38
40
42

5 Development of Two-mode Cavity
5.1 Fundamental Property and Design of Superconducting Cavity . . . . . . . .
5.1.1 Nature of Superconductivity . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Resonant Mode in Pillbox . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Performance Index of Superconducting Cavity . . . . . . . . . . . . .
5.1.4 Design of Two-mode Cavity . . . . . . . . . . . . . . . . . . . . . . .
5.2 Fabrication of Two-mode Cavity . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Fabrication of Niobium Material . . . . . . . . . . . . . . . . . . . . .
5.2.2 Electron Beam Welding . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2.1 Surface Polishing . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2.2 Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2.3 High-pressure Washing . . . . . . . . . . . . . . . . . . . . .
5.2.2.4 Assembling . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Performance Test of Two-mode Cavity . . . . . . . . . . . . . . . . . . . . .
5.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Measurement Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2.1 Cable correction . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2.2 QL measurement . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2.3 Power measurement . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Frequency Tuning Test of Two-mode Cavity . . . . . . . . . . . . . . . . . .

45
45
45
46
51
54
56
56
59
62
68
68
69
70
70
72
72
73
75
76
77

6 Development of Feedback Control System for Two-mode Cavity
6.1 Source of Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Setup of Feedback System For Two-mode Cavity . . . . . . . . . . . . . . . .
6.3 Perfomance Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Single feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Dual Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.3 Consideration of Ultimate Control Stability . . . . . . . . . . . . . .

81
81
82
87
87
91
96

ii

7 Ultimate Performance of Superconducting-Radio-Frequency Transmission
Electron Microscope
97
7.1 Performance Summary of Essential Technology for Superconducting Radio
Frequency Transmission Electron Microscope . . . . . . . . . . . . . . . . . .
97
7.1.1 Performance of the Two-mode Cavity . . . . . . . . . . . . . . . . . .
97
7.1.2 Performance of RF Feedback System . . . . . . . . . . . . . . . . . .
98
7.2 Estimation of Ultimate Performance of 300 kV Superconducting Radio Frequency Transmission Electron Microscope . . . . . . . . . . . . . . . . . . .
98
7.3 Future Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3.1 Fabrication of Cryostat for Two-mode Cavity . . . . . . . . . . . . . 100
7.3.2 Performance Evaluation of 300 kV SRF-TEM Prototype . . . . . . . 100
7.3.3 Future SRF-TEM with Higher Accelerating Voltage than 300 kV . . 101
7.4 Achievement in This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A Imaging Theory of High-Resolution Transmission Electron Microscope 105
A.1 Phase Change by Electrostatic Potential . . . . . . . . . . . . . . . . . . . . 105
A.2 Phase Shift at Objective Lens . . . . . . . . . . . . . . . . . . . . . . . . . . 108
B Response Calculation of Superconducting Cavity by Lumped-constant Circuit Modeling109
B.1 Relation Between Cavity and Coupler . . . . . . . . . . . . . . . . . . . . . . 109
B.2 Diﬀerential Equation for Superconducting Cavity and General Solution . . . 111
B.3 Stationary State of Superconducting Cavity . . . . . . . . . . . . . . . . . . 113
B.4 Response to Pulsed Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
B.5 Estimate Q0 and Vc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Bibliography

119

iii

Chapter 1
Introduction
In this chapter, the bases of this thesis are introduced. Firstly, the significance of this study
is summarized. Next, the structure of this thesis is explained.

1.1

Significance of This StudyIn this section, a fully new TEM called superconducting radio frequency transmission electron
microscope (SRF-TEM) is introduced. Our study to establish the fundamental technologies
for SRF-TEM is summarized.

1.1.1

Suggestion of Superconducting Radio Frequency Transmission Electron Microscope

Motivation of Observation of Thick Specimen
TEM is a powerful tool for imaging sub-nm structures of specimens. Generally, the observation object has to be sliced thin as O(10 nm) with the accelerating voltage of O(100 kV)
for the observation with good spatial resolution. Destruction of the original configuration isinevitable in the process of slicing specimens. There are phenomena which appear depending on the thickness of the specimen, for example, magnetic vortices in high-temperature
superconductors[1] and skyrmions[2].
Type II superconductors do not expel the applied magnetic flux completely and permit
the penetrations of the magnetic flux in the form of vortices. The movements of this vortices
generate heat dissipations. By observing these with TEMs, we understand the superconducting nature of the materials. However the appearance of this phenomenon depends on the
thickness of the material. Tonomura observed the movement of the magnetic vortices with amaterial of Bi-2212 and the thickness was 400 nm with the 1 MV TEM[1].
A new spin configuration called skyrmion was observed in 2010[2]. This skyrmion takes
the configuration like a vortex, and is expected to become a next-generation magnetic memory
element with its high-stability and the ultra-low driving current. Until this point, skyrmions
have been observed only in specimens as thin as O(10 nm). The condition for skyrmions to1

appear depends on the thickness of the specimen, and there is a prediction that skyrmionsappear in the 570 nm FeGe sample[3].
In order to image more genuine structures and study further new materials and phenomena with thicker specimens, we need to realize the accelerating voltage higher than 3 MV,
which is the highest values accomplished in the conventional TEM, to suppress the energy
spread of the electron beam at the specimen (Sec. 2.4). However, we cannot improve theaccelerating voltage with the DC acceleration utilized in conventional TEMs for the discharge
limitation1 .
Adoption of Radio Frequency Acceleration
In order to resolve this discharge limitation, we introduce the radio frequency acceleration
scheme (RF acceleration) with resonant cavities. Generally, resonant cavities can generatehigh accelerating fields up to 100 MV/m. It is easy to realize the accelerating energy higher
than 3 MV using RF acceleration with resonant cavities.
The normal conducting cavities made of copper can easily be fabricated. They do notneed to be cooled with liquid helium during the operation. However, the dissipating power
is too high to operate in continuos wave mode (CW mode) which means the cavity cannot
be operated at 100 % duty. Therefore, the beam current cannot be sustained as the samelevel as the conventional TEM. Additionally, it is diﬃcult to operate the normal conducting
cavities with accurate field strength and waveform due to the deformation of the cavity by
the heat.
Choice of Superconducting Cavity
In order to acquire high beam current, we choose superconducting resonant cavities. It cangenerate high accelerating field as O(10 MV/m), and be operated in CW mode. CW mode
realizes the stable accelerating field as well as the high beam current.
However, the oscillating electric field generated by the superconducting cavity inevitably
increases the energy spread. This is because the electron bunch with a finite length passes theelectric field whose distribution changes from moment to moment in the cavity. This makes
the spatial resolution of the TEM machine worse. According to the latest data[4], the stabilityof the electrostatic accelerating voltage ∆V /V = 3.0×10−7 . On the other hand, a general RF
accelerating mode accelerates electrons with the spread of ∆V /V > 1.0 × 10−4 . Therefore,
we have to develop a new SRF acceleration scheme in order to realize a high-voltage TEM
with a good spatial resolution.
Development of Two-mode Cavity
In order to suppress the energy spread of the beam, we developed a fully new superconducting
cavity named two-mode cavity . This cavity has two resonant modes; the first-order mode
1

The discharge limitation in the vacuum state is dependent on the material of electrodes and the surface
condition, however it is generally about 40-50 kV with 1 mm gap[5].

2TM010 (1.3 GHz) and the second-order mode TM020 (2.6 GHz) illustrated in Figure 1.1.
Superimposing the TM020 mode onto the TM010 mode, the flatter peak of the accelerating
electric field can be generated (see Figure 1.2). With the two-mode cavity, we can suppress
the energy spread compared with the conventional single mode acceleration with the TM010
mode. In addition, a photocathode electron gun is introduced instead of the thermal gun.
This enables electron bunches to synchronize the resonant frequency of two-mode cavity; 1.3
GHz.
We call the TEM employing this two-mode cavity SRF-TEM . The history of superconducting cavities is long, however the two-mode cavity is the world’s first superconducting
cavity that has two resonant mode. As described in Chapter 5, Its development is very challenging. In addition, in order to realize the energy dispersion low enough to get the good
spatial resolution, it is needed to control two resonant mode with high accuracy at the sametime. This is also main task for the SRF-TEM. The detail of the two-mode cavity and thetwo-mode acceleration are described in Chapter 3 .

(a) The TM010 mode.

(b) The TM020 mode.

Figure 1.1: The distribution of the electromagnetic field utilized in the SRF-TEM.

Figure 1.2: A schematic comparison of the single acceleration with only the TM010 modeand the dual acceleration with the TM010 mode and the TM020 mode. The time unit of thehorizontal axis and the amplitude unit of the vertical axis are arbitrary.
3Advantage of SRF-TEM
The main advantage of the SRF-TEM is the ability of observing thick specimens. A calculation based on the TEM imaging theory has shown that a 10 MV SRF-TEM has the ability
to observe about 100 times thicker specimens with 100 pm spatial resolution compared withthe state-of-the-art 1.2 MV TEM in the case of aluminum film specimen.
This ability of observing thick specimens is expected with a compact facility owing to theaccelerating field much higher than the conventional DC acceleration scheme. The state-ofthe-art 1.2 MV TEM is as tall as about 10 m[6]. In contrast, our 1.0 MV SRF-TEM can berealized with the size of about 4 m.
In addition, the SRF-TEM is expected to provide good time resolution thanks to thebunched electron beam generated by the photocathode electron gun. The SRF-TEM can
get the time resolution of 12 µs for an assumed bunch intensity of 0.1 fC. Moreover, if thepumping probe method is employed, our SRF-TEM is able to acquire 2 ps time resolution.

1.1.2

Development of Superconducting Radio Frequency Transmission Electron Microscope

In this section, the development plan of the SRF-TEM is explained. After that, the achievement in this thesis is summarized.
Development Plan of SRF-TEM
We are now developing the first prototype of SRF-TEM with 300 kV acceleration voltage.
This is based on the conventional 300 kV TEM, H-9000 NAR. Figure 1.3 illustrates ourdevelopment of the 300 kV prototype. In order to realize this, we have to design the overallsystem of the SRF-TEM. Technologies for the photocathode electron gun, the two-mode
cavity and the RF feedback system have to be established. IN addition, 3 MV and 10 MV
SRF-TEM are considered as the future machine that is superior to the present state-of-the-art
TEM.
Achievement in This Thesis
This thesis explains my work of the system design of the SRF-TEM of 300 kV and 3 MV
acceleration voltage and the establishment of the backbone technologies for the SRF-TEM.
My work is summarized briefly as follows;
1. Simulated 3D beam dynamics for SRF-TEMs and confirmed that two-mode acceleration
works eﬀectively to reduce the energy spread compared with one-mode acceleration in
the case of 300 kV acceleration and 3 MV acceleration,
2. Established the manufacturing process of the two-mode cavity and confirmed that the
manufactured two-mode cavity satisfies the requirements for 300 kV SRF-TEM withthe performance tests,4

3. Developed the feedback system for the accelerating field generated by the two-mode
cavity and confirmed that it satisfies the requirements for 300 kV SRF-TEM with theperformance tests.
With the above, we opened up the road to the fully new TEM, SRF-TEM. This has theability of observing specimens much thicker than the state-of-the-art TEM, and it is realizedwith more compact facility. The SRF-TEM is also expected to acquire the time resolutionof O(µs) - O(ps).

Figure 1.3: A concept of 300 kV SRF-TEM prototype development based on H-9000 NAR,
conventional 300 kV TEM.
5

1.2

Structure of This Thesis

Here, the structure of this thesis is explained.
In chapter 2, the fundamental of the conventional TEM is introduced. Firstly, the history
of TEM is described. After that, the general configuration of the present TEM is explained.
Next, the imaging theory of TEM is introduced. Finally, the limitation of the observation
for present TEMs in terms of energy losses of electron beams is described.
In chapter 3, the concept design and key technologies of the SRF-TEM are introduced.
Next, the advantages of the SRF-TEM are explained in detail. Finally, the issues to realize
the SRF-TEM is described and the quantitative requirements for the key technologies areset.
In chapter 4, the overall system designs of the SRF-TEM with the accelerating energy 300
kV and 3 MV are explained. These are based on the comparison of the two-mode accelerationand the one-mode acceleration. In addition, the requirements for the accelerating field of thetwo-mode cavity in the case of the 300 kV acceleration are defined. Next, the RF control
error eﬀects on the energy spread are investigated for the two-mode acceleration on the 300
kV SRF-TEM prototype.
Chapter 5 explains the fabrication of the two-mode cavity and its performance test.
Chapter 6 describes the configuration of the developed feedback system and its performance test.
Chapter 7 summarizes the results of the development of each fundamental technology andfeedback the acquired performance values which were measured in each performance test toderive the ultimate performance of SRF-TEM in the 300 kV acceleration.

6

Chapter 2
Present Situation of Transmission
Electron Microscope and Limitation
In this chapter,1. the history of TEMs,
2. the system of TEMs,
3. the imaging theory of TEMs,
4. the limitation of the observation for present TEMs in terms of energy losses of electron
beams.
are summarized.

2.1

History of Transmission Electron Microscope

TEM has been a strong tool to image various matters including industrial materials andbiological ones. It has become an indispensable imaging device for semiconductor studies,
batteries, magnets and biological matters (recently including iPS cells[7]). The development
history of TEMs started when a limitation of a spatial resolution due to diﬀraction aberrationsof optical microscopes would be overcome with the shorter wavelength of electrons. After theinvention of TEM, the long successful history has been supported by the Japanese leading
technology which gave a birth of an ultra-high voltage TEM (∼3 MeV) to continue to establish
a new record of the spatial resolution. However the improvement of the spatial resolution
which was realized along with the shortening of de Broglie wavelengths was saturated. Thiswas because spherical aberrations became the dominant contribution determining the spatial
resolution. In late 1990s, Haider and others established the technology to correct sphericalaberrations using the combination of quadrupole magnets or sextupole magnets[8]. With
this new technology, the spatial resolutions better than 200 pm has been achieved with low
voltage TEMs of up to 300 kV.
7

Now the best spatial resolution is recored by Hitachi Cooperation in Japan[4]. The
TEM has the gun of cold field emission and the accelerating energy is 1.2 MV. Its sphericalaberration is corrected to the third order. Using the 3 nm-thick specimen of GaN, the spatial
resolution of 43 pm was achieved.

2.2

System of Conventional Transmission Electron Microscope

Here a general system of TEMs is explained. Figure 2.1 shows the overview of a conventional TEM. It has an electron gun, condenser lenses, a specimen part, an objective lens,an intermediate lens, a projector lens, an imaging detector from the top to the bottom.
TEMs are electron beam accelerators as they have electron guns and accelerating parts aswell as magnets to bend and focus beams. Thermal electron gun has been conventionallyused, whereas the use of cold field emission gun is becoming more common. The radio frequency acceleration (RF acceleration) has rarely been used for TEMs, and generally TEMs
employ electrostatic acceleration systems using Cockcroft-Walton circuits. TEMs’ magnets
are called lenses in analogy of the optical microscopes. It is the objective lens that
determines TEM’s performance dominantly. This has a strong magnetic field in order tofocus electrons which pass thorough specimens and its aberrations (spherical and chromatic)
limit the spatial resolution of TEMs. Above the objective lens, there is an objective aperturewhich cuts oﬀ electrons to determine the TEM performance. In the past, films used to be
used as the imaging detector which can display images which are proportional to fluxes ofelectrons, though nowadays fluorescent screens are employed to convert electron signals into
lights that are recored digitally by the CCD cameras. In the following, key components of
TEMs necessary for later discussions will be explained in detail.

Electron Gun
The electron gun is a crucial part to determine the quality of the beam. Today there are twomain types of guns for TEMs; thermal electron guns which use LaB6 as cathodes and cold
field emission guns which derive electrons using high electric fields. Photocathode electronguns which have been developed for high spin-polarized electron beams have now been appliedin TEMs and our research is also employing this photocathode electron gun[9]. Table 2.1
compares these guns in several aspects. In this thesis, energy spread ∆Tini is the crucial
parameter to aﬀect the energy dispersion of the beam.

Specimen
It is necessary to process the specimens so that features to observe can be seen. Specimens
are needed to be thin enough to be observed as close as possible to the original structureswith energy losses or electron beam flux losses, which would impair spatial resolution, limited
as much as possible. Generally for TEMs with several 100 kV accelerating voltage, specimens
8

Table 2.1: A comparison of guns. Materials and each values are just typical ones. ∆Tini is anenergy spread at its cathode. Je is a current density and β is a brightness which is calculated
dividing Je by a solid angle.
Vacuum means a general value needed to cooperate eachgun.
Type of Source Thermal [10] Cold Field Emission [10] Photocathode [11] [12]Material
LaB6
W
GaAs-GaAsP∆Tini [eV]
1.00
0.30
0.15
2
6
9
Je [A/m ]
≈ 10
≈ 10
≈ 106
β [A/m2 sr]≈ 1010≈ 1012≈ 1011−4−8
Vacuum [Pa]
< 10
< 10
< 10−8Lifetime [hours]
103
104
10

have to be sliced to a few 10 nm of thickness1 . Furthermore, in TEMs’ column, vacuum level

Figure 2.1: The overview of a conventional TEM.
1

Needless to say, it depends on materials and density, more accurately

9

mass-thickness .

has to be under O(10−3 Pa) in order to avoid interferences of the electron beam with air anddischarge problems, then specimens cannot be in wet state. Therefore when biological bodies
are objects for observing, these have to be frozen.

Objective Lens
The objective lens plays an important role in TEMs to determine spatial resolution. It has
two aberration; spherical and chromatic. Spherical aberration arises from the diﬀerence of the
focal length for each path of the electron due to the imperfection of the lens (deviation fromthe perfect convex lens). On the other hand, chromatic aberration is due to the diﬀerenceof the focal length for diﬀerent electron energy, and is aﬀected by the finite energy spreadof the electron beam. ∆f , the quantity called defocus also influences the TEMs’ spatial
resolution as well as aberrations. There are some definition for the sign of ∆f . In this thesis,
the stronger excitation than usual has minus sign called under-focus (∆f < 0) and vice versa.

2.3

Imaging Theory of Transmission Electron Microscope

In this part, the imaging theory of TEMs is summarized. There are two ways to analyze the ability of TEM imaging; the scattering contrast scheme and the phase contrast
scheme[13][14][15]. The scattering contrast scheme explains that the TEM image formation
is made by the electron flux decrease due to scatterings. In contrast, the phase contrast
scheme utilizes the phase change of the incident electrons by nucleus electrostatic potentials
and lenses. Practical image simulation using the specimen structures is based on the phasechanges and the wave functions. Therefore, we carried out the studies and developments
based on the imaging performance analysis by the phase contrast scheme. Here a brief explanation of the theoretical basis of the phase contrast is made. More detail is described inAppendix A.
The phase contrast theory uses a wave function to express the phase change of electron
wave by a nucleus. This wave function ψi is written as
ψi = exp(iσVp ),(2.3.1)

where σ is the interaction coeﬃcient of electrons and specimens and Vp is the projection
potential which is the result of integration of all potentials along the beam axis (here, z),!
V (x, y, z)dz = Vp (x, y).
(2.3.2)
In practical usage, this wave function is modified with

10lens transfer function (LTF) , andTable 2.2: The notation for lens transfer functions.
SymbolDefinition
Unit
C3
coeﬃcient of the third spherical aberrationm
C5
coeﬃcient of the fifth spherical aberrationmCc
coeﬃcient of the chromatic aberrationm∆f
defocusm∆defocus spreadm
σ(x)image spreadm
βillumination tilt spreadradis expressed as
LTF = exp(−iχps ) exp(χfs ) exp(χis ) exp(χsc )ππ
χps = C3 λ3 u4 + C5 λ5 u6 + π∆f λu223
π2 2 2 4
χfs = − λ ∆ u2
χis = −2π 2 σ 2 (x)u2"#2
χsc = −π 2 u20 (C3 λ2 u2 + ∆f )λu ,(2.3.3)(2.3.4)(2.3.5)(2.3.6)(2.3.7)

where u is reciprocal space coordinate corresponding to x. Other notations are explainedin Table 2.2. Image spread σ(x) is the x axis component of the mechanical fluctuation at
the detector level due to stage deviation and column vibration. u0 is a physics quantitythat satisfies u20 = (λ/β)2 where β means the divergence angle of the electron beam at the
specimen. The term exp(−iχps ) is responsible for the spherical aberration of the objective
lens and the defocus, exp(χfs ) accounts for the chromatic aberration of the objective lens,the energy dispersion of the beam and the deviation of the focal length for the instability of
the current of the objective lens. Then defocus spread ∆ is given by
$%
&2 %
&2
∆T
∆J
∆ = Cc+ 2,(2.3.8)T
J
where ∆T is the energy spread, and therefore ∆T /T is the energy dispersion of the beam.
∆J/J is the stability of the current of the objective lens which is generally O(10−5 ). exp(χis )is determined by the fluctuation of the imaging level along the horizontal axis and exp(χsc )depends on the divergence angle of the beam against the specimen.
The phase wave function ψi is aﬀected by the LTF above and modified as
ψi = 1 + iσVp ⊗ F̂ [exp(−iχps + χR )] ,(2.3.9)where ⊗ represents convolution, F̂ stands for inverse Fourier transformation and χR satisfies
χR = χfs + χis + χsc . The image intensity I can be calculated as I = ψi ψi∗ , thus
I = 1 + 2σVp ⊗ F̂ [sin(χps ) · exp(χfs + χis + χsc )] .
11

(2.3.10)

Here the example of Hitachi 1.2 MV aberration-corrected FE-TEM is considered[4]. Figure
2.2 shows its theoretical lens transfer function (LTF). The horizontal axis stands for reciprocal space coordinate u whose unit is nm−1 . The notation of each E is defined as follows;
Eps = sin(χps ), Efs = exp(χfs ), Eis = exp(χis ) and Esc = exp(χsc ). The thick yellow line
represents sin(χps ) · exp(χfs + χis + χsc ) which corresponds to the term under the inverse
Fourier transformation. Therefore this quantity plays a role to carry the projected electrostatic potentials in the specimen to the image as well as the interaction coeﬃcient σ. The
inverse Fourier transformation of constant A isF̂ [A] = A · δ(x).

(2.3.11)

Now replacing sin(χps ) · exp(χfs + χis + χsc ) with A, we get
I = 1 + 2σVp ⊗ F̂ [A]= 1 + 2σAVp (x).

(2.3.12)(2.3.13)

Here we define umax as the maximum value of u, where the LTF (sin(χps ) · exp(χfs + χis + χsc ))reduces to 1/e2 ≃ 0.135. This corresponds to the inverse of the spatial resolution ∆r of theTEM. For Hitachi 1.2 MV FE-TEM, umax = 23nm−1 , then ∆r is estimated as 43nm. In
the practical observation of 3 nm-thick GaN specimen, the spatial resolution of 43nm wasconfirmed.

1
0.9

E ps: Phase Contrast Transfer Function sinΧ0.8
E fs : Correction Term for Energy Dispersion

0.7
E is : Correction Term for Image Spread

0.6
0.5

E sc: Correction Term for Incident Angle

0.4

Modified Phase Contrast Transfer Function

0.3
0.2
0.1
0
0

5

10

15

20

25
30
35
40
-1
Spatial Frequency [nm ]Figure 2.2: The lens transfer function of Hitachi 1.2 MV FE-TEM [4].
12

2.4

Limit of Thickness in Observation by Transmission
Electron Microscope

In order to observe the object with the good spatial resolution, with the accelerating voltageof O(100 kV) we have to slice specimens thin as O(10 nm) . The allowed thickness isdetermined by the beam energy and the beam quality. The acceptable range of specimens’
thickness can be extended as the accelerating energy of TEMs increases.
Here ∆Ep is introduced as the most probable energy loss and can be calculated as' %&(NA e4 Zx
NA e4 Zx2∆Ep =ln
− β + 0.198 ,(2.4.1)8πϵ20 Ame c2 β 28πϵ20 I 2 A(1 − β 2 )and ∆EH is the FWHM of energy loss, written as∆EH = 4.02

NA e4 Zx
8πϵ20 Ame c2 β 2(2.4.2)by Landau’s theory[13]. The symbols appear in Eq.(2.4.1) and Eq.(2.4.2) are explained inTable 2.3. The spread of the energy loss by interacting with specimen can be approximatedby ∆EH . The FWHM for the energy loss as a function of the kinetic energy T is displayedin Figure 2.32 . It can be seen that FWHM of the beam energy loss decreases as increasingenergy and approaches to a constant value. Figure 2.4 shows the energy dispersion3 . As
described in Sec. 2.3, the defocus spread ∆ is determined by the energy dispersion ∆T/T ,not the energy spread ∆T . As the result, the higher accelerating energy can improve thespatial resolution. This can be understood by following the steps below;
1. the higher accelerating energy can reduce the energy dispersion (Figure 2.4),
2. the lower energy dispersion makes the lower defocus spread (Eq. (2.3.8)),
3. the lower defocus spread decelerate the decay speed of Ef s (Eq. (2.3.5) and Figure 2.2),
4. the umax gets larger, then the spatial resolution is improved.
It is necessary to calculate actual defocus spread and LTF in order to acquire the spatial
resolution in practice (Sec. 3.3.1).

2

Here the notation of T is used instead of E for the beam energy.
The definition of the energy spread for the phase contrast theory √is chosen as the standard deviation inplace of the FWHM, then now ∆TH is converted into ∆T as ∆TH = 2 2ln2∆T .
3

13

FWHM of Energy Loss ∆T H [eV]

Table 2.3: The notation for Landau’s theory [16].
SymbolDefinition
Value or Unit
NA
Avogadro’s number
6.02214129(27) × 1023 mol−1reclassical electron radius
2.8179403267(27) fm
me c2
electron mass
0.510998928(11) MeV
ρmaterial density- g/cm3Z
atomic number
no-unitA
atomic mass
- g/molαfine structure constant
1/137.035990074(44)Imean excitation energy
- eV

105

t = 3 nmt = 10 nm
t = 100 nm
t = 5 µm

4

10

103
102

10

1
10−1 −1
10102
10Kinetic Energy T [MeV]1

Energy Dispersion ∆T /T

Figure 2.3: The variation of FWHM of the energy loss to the kinetic energy of the electron
beam T . The thickness of the specimen t is chosen to 3 nm, 10 nm, 100 nm and 5 µm.

10−1

t = 3 nmt = 10 nm
t = 100 nm
t = 5 µm

10−2
10−3
10−4
10−5
10−6
10−7
10−8
10−9

10−10 −1
10102
10Kinetic Energy T [MeV]1

Figure 2.4: The variation of the energy dispersion to the kinetic energy of the electron beam
T . The thickness of the specimen t is chosen of 3 nm, 10 nm, 100 nm and 5 µm.
14Chapter 3
Concept of Superconducting Radio
Frequency Transmission Electron
Microscope
In this chapter, the configuration and the key technologies of the SRF-TEM are described indetail. Next, the advantages provided by the SRF-TEM are introduced.

3.1

System Configuration of Superconducting Radio
Frequency Transmission Electron Microscope

Figure 3.1 shows the overview of 300 kV SRF-TEM prototype. This is based on the conventional 300 kV TEM H-9000 NAR which was developed by Hitachi Cooperation. Because
its accelerating voltage is the same as the prototype, the optical system can be reused. A
newly developed 60 kV DC photocathode gun is introduced into the prototype instead of thethermal gun. The prototype has one two-mode cavity which can accelerate 60 keV-energy
electrons emerging from the gun to 1.0 MeV, while the existing optics can receive electrons
up to 300 keV. The Two-mode cavity is placed in a 4 K cryostat. In high energy accelerators
SRF cavities are generally operated at 2.0 K to reduce the dissipating power. However thisrequires additional pumping system in order to cool 4 K liquid helium to 2K. Therefore thefirst SRF-TEM prototype employs the higher temperature drive (4 K). There is a buﬀer for
liquid helium assuming that the circulation system for liquid He is not equipped in 300 keV
SRF-TEM prototype. There is a diﬀerential pumping chamber because there is a diﬀerenceof the required vacuum level between the conventional optical system and the photocathodeelectron gun. The required vacuum level for the optical system is O(10−5 ) Pa, while that forthe photocathode electron gun is better than O(10−8 ) Pa.
In the following sub-sections, each element will be explained in detail.

15

Figure 3.1: The overview of 300 kV SRF-TEM prototype.

16

3.1.1

Two-mode Cavity

The two-Mode cavity is one of the most important key technologies for SRF-TEM. A detailed
explanation is given in Chapter 5. A brief description is given below.
A two-mode cavity has two resonant modes; the TM010 mode and the TM020 mode whose
resonant frequencies are 1.3 GHz and 2.6 GHz respectively. Two mode cavity is designed sothat the ratio of resonant frequencies is exactly 2.0000 in order to acquire flatter accelerating
gradients in 100 % duty. Figure 3.2 shows the schematic overview of two-mode cavity. Twomode cavity is made of niobium, the superconducting material which is widely utilized foraccelerating cavities. Figure 3.3 displays the distributions of each resonant electric fieldcalculated by the 3D electromagnetic fields simulation software HFSS .
As will be explained in Chapter 5, the excited resonant fields gets lower when the gap isoccurred between the input RF frequency and the resonant frequency. This situation is called

Figure 3.2: The schematic overview of the two-mode cavity. The red arrow represents the
direction of the incoming electron.

(a) The TM010 mode.

(b) The TM020 mode.

Figure 3.3: The distribution of each resonant electric field of the two-mode cavity. The redparts indicates the stronger electric field than blue ones.

17

detuned. Therefore two-mode cavity has two frequency tuner. One is span tuner and theother is piezo tuner . Figure 3.4 shows the placements of tuners. Figure 3.5 indicates how
sensitive the resonant frequency when mechanically deformed at each position. Combining
these tuners, deviated resonant frequency is pulled back to the input RF frequency.

Figure 3.4: The span tuners and the piezo tuners equipped with the two-mode cavity. In thisfigure, only one set of tuners is depicted, however a two-mode cavity has two sets locatedsymmetrically.

Figure 3.5: The sensitiveness of the resonant frequency for the tuning.

18

3.1.2

Photocathode Electron Gun

The SRF-TEM employs a photocathode electron gun instead of the thermal gun alreadyinstalled in 300 kV H-9000 NAR. The photocathode gun adopts the mechanism that thecoherent light emitted from lasers irradiates cathodes, and electrons that gain energy byphotoelectric eﬀect are extracted. This employs a transmission-type so that the diameterof the beam at the cathode can be minimized to place lenses behind the cathode. Thistransmission-type photocathode gun has been already utilized for a TEM[9].
Our gun accelerates electrons with the energy of 60 kV using DC acceleration scheme.
Figure 3.6 is the overview of the entire of the gun. It is mainly composed of two chambers;
one is the main chamber in which the cathode is installed and electrons are accelerated andthe other is for cathode preparation.
The cathode is GaAs-GaAsP strained superlattice on a GaP substrate. This was originallydeveloped for a high spin-polarized electron source of high energy accelerators and achieved
the polarization of 90 %. Using this, the development of a spin-polarized TEM has beenpromoted[9]. Needless to say, SRF-TEM has this option.

(b) The inside of the main chamber.

(a) The overview of the gun.

Figure 3.6: The 3D CAD of our photocathode gun.

3.1.3

RF Feedback System for Two-mode Cavity

The SRF-TEM has to control 4 parameters in one two-mode cavity; each amplitude andphase of the TM010 mode and the TM020 mode. As will be considered in Chapter 4, if there
are some fluctuations of these parameters, the net accelerating field is deviated from the ideal,
designed one (Figure 3.7). As mentioned above, two-mode cavity has two tuners to control
the resonant frequency to deform its shape, though the speed of the operation of tuners arevery slow. In SRF-TEM, the span tuner is considered to be fixed or operated in O(1 Hz) by a
motor. Piezo tuner can be controlled electrically, then the operation speed is from DC to 200
Hz. The RF feedback system is a complementary tools to stabilize the resonant electric field
19

not to change the resonant frequency but adjust the amplitude and phase of the input power
to maintain the resonant fields stable. The detailed explanation will be done in Chapter 6.

Figure 3.7: A conceptual situation of a deviated field.

3.2

Configuration Comparison between Conventional
Transmission Electron Microscope and Superconducting Radio Frequency Transmission Electron Microscope

Here the new superconducting radio frequency transmission electron microscope (SRF-TEM)
is compared with conventional TEMs. Table 3.1 compares their specifications and performances. Performance values of SRF-TEM mean the targets of the development. With 300
kV SRF-TEM prototype, the establishment of the two-mode cavity and the RF feedback
system in order to satisfy the requirement of the energy spread ∆Tmac on Table 3.1 is thegoal.
Table 3.1: A comparison of the specifications and performances. ∆Tmac indicates the energy
spread which is determined by only of the machine, not including the interaction with thespecimen.
H-9000 NAR 300 kV SRF-TEM 3 MV SRF-TEM
Type of Gun
ThermalPhotocathode
Photocathode
Cathode
LaB6
GaAs-GaAsP
GaAs-GaAsP
Frequency of BeamDC
1.3 GHz
1.3 GHz
∆Tmac
1 eV
10 eV
30 eV

20

3.3

Advantage of Superconducting Radio Frequency Transmission Electron Microscope

Here the advantages of Superconducting Radio Frequency Transmission Electron Microscope(SRF-TEM) are explained. These advantages are three;
1. the ability to observe thicker specimen
2. the downsizing of TEM facilities
3. the acquisition of the time resolution
All advantages owes to the features mentioned in the previous section. The detailed explanations are given in the following.

3.3.1

Ability to Observe Thicker Specimen

As mentioned in Chapter 1, the most powerful advantage of the SRF-TEM is the ability ofobserving specimens thicker than ever. Here, one example is given to illustrate how SRFTEM can improve this ability compared with Hitachi 1.2 MV FE-TEM.
Figure 3.8 shows the Lens Transfer Function (LTF) arranged to maximize the defocus
spread ∆ keeping the spatial resolution 100 pm using the values achieved in Hitachi 1.2 MV
FE-TEM. Each parameter values used in order to calculate the LTF functions in Figure 3.8
are listed in Table 3.2[4]. Here ∆ = 9.0 nm. ∆ was already defined as
$%
&2 %
&2
∆T
∆J
∆ = Cc+ 2,(3.3.1)TJ
however this has to be modified by adding the energy spread occurred due to scattering at
the specimen as
$%
&2 %
&2 %
&2
∆Tini
∆Tspe
∆J
′
∆ = Cc++ 2(3.3.2)T
T
J
where ∆Tini is the energy spread generated at the cathode of the gun and the electrostatic
acceleration part and is as small as 0.3 ppm, and ∆Tspe is the energy spread occurred atTable 3.2: Parameters of Hitachi 1.2 MV FE-TEM[4].
C3 : Coeﬃcient of the third spherical aberration −23 µm
C5 : Coeﬃcient of the fifth spherical aberration
29 mm
σ(x) : Image spread
10 pm
β: Illumination tilt spread
0.05 mdeg.

21

1
Phase Contrast Transfer Function sinΧ

0.9

Correction Term for Energy Dispersion

0.8

Correction Term for Image Spread
Correction Term for Incident Angle

0.7

Modified Phase Contrast Transfer Function

0.6
0.5
0.4
0.3
0.2
0.1
0
0

5

10

15

20

25
30
35
40
-1
Spatial Frequency [nm ]Figure 3.8: The lens transfer function for Hitachi 1.2 MV FE-TEM. This is the case of the
maximum defocus spread ∆ with the spatial resolution of 100 pm.

Energy Dispersion ∆T spe /T

−6

2

×10

t = 25.5 nm
1.9

t = 27.0 nm

1.8

t = 28.5 nm
t = 30.0 nm

1.7
1.6
1.5
1.4
1.3
1.2
1

1.05

1.1

1.15

1.2

1.25
1.3
1.35
1.4
Kinetic Energy T [MeV]

Figure 3.9: The energy dispersion for an Al specimen magnifying the energy range of 1.2
MeV.
22

the specimen due to scattering. The first and the last terms are determined by the machine
performance, and the second depends on the specimen’s material, thickness and density.
∆Tspe /T can be written using Equations (3.3.1) and (3.3.2) as∆Tspe
1 √ ′ 2=∆ − ∆2 .
T
Cc(3.3.3)[4] reveals ∆ = 1.1 nm when the spatial resolution is recorded as 43 pm. If a degradationof the spatial resolution is allowed up to 100 pm, the maximum values of ∆ is 9.0 nm whichbecomes a new ∆′ . Then ∆Tspe /T is calculated as ∆Tspe /T = 1.65 × 10−6 using Eq. (3.3.3).
Figure 3.9 shows the energy dispersion in the case of that the specimen is Al. Reading this,for the beam energy of 1.2 MeV, the allowed thickness is about 28.5 nm.
Next the future 10 MV SRF-TEM will be considered. Figure 3.10 shows its LTF usingparameters in Table 3.2 and the ∆ =55 nm maximized under the situation that the spatial
resolution is 100 pm. For SRF-TEM, the defocus spread has to be modified again due to theenergy spread for RF acceleration. Eq. (3.3.2) is modified as
$%
&2 %
&2 %
&2 %
&2
∆Tini
∆TRF
∆Tspe
∆J
′′
∆ = Cc+++ 2(3.3.4)
T
T
T
J
where ∆TRF is the energy spread generated in the RF acceleration. Using Eq. (3.3.1),
$
%
&
∆Tspe
1
∆TRF
′′
2
2
=
∆ − ∆ − Cc.
(3.3.5)T
Cc
T
Here the value of ∆TRF is assumed as 30 eV1 . Then the maximum ∆Tspe /T to allow theobservation with the spatial resolution of 100 pm becomes 9.7 × 10−6 . Figure 3.11 showsthe energy dispersion for Al specimen, magnifying the range of T = 10MeV. With Figure
3.11, the maximum thickness to achieve spatial resolution of 100 pm is 2.6 µm for 10 MV
SRF-TEM.
Summarizing these considerations, 10 MV SRF-TEM has the ability to observe about
100 times thicker specimens with 100 pm spatial resolution compared with the state-of-theart Hitachi 1.2 MV FE-TEM. More precise calculations needs simulations to consider eachspecimen’s configuration.
Looking at this advantage of observing thicker specimens from another perspective, whenobserving specimens of the same thickness, SRF-TEM can achieve better spatial resolution.
As described above, the spatial resolution of 100 pm is sustained when the Al specimen’s
thickness is 28.5 nm with Hitachi 1.2 MV FE-TEM. Figure 3.12 shows the energy dispersion
from T =1 MeV to 10 MeV with 28.5 nm-thick Al specimen. It can be seen that the energy
dispersion is 1.0×10−7 when kinetic energy T =10 MeV. Therefore it leads to ∆′′ = 16.3 nm
using Eq. (3.3.4). Substituting this into the LTF equations, Figure 3.13 is acquired. This
spatial resolution is derived as 60 pm. This is because the phase contrast transfer functionsinχps decreases owing to the shortening of the de Broglie wave length.
1Later it will be discussed whether this value is proper.

23

1
Phase Contrast Transfer Function sinΧ

0.9

Correction Term for Energy Dispersion

0.8

Correction Term for Image Spread

0.7

Correction Term for Incident Angle
Modified Phase Contrast Transfer Function

0.6
0.5
0.4
0.3
0.2
0.1
0
0

5

10

15

20

25
30
35
40
-1
Spatial Frequency [nm ]Figure 3.10: The lens transfer function for the future 10 MV SRF-TEM. This is the case of
maximizing the defocus spread ∆ with the spatial resolution of 100 pm.

Energy Dispersion ∆T spe /T

−6

12

×10

t = 2.2 µm
t = 2.4 µm
t = 2.6 µm
t = 2.8 µm

11.5
11
10.5
10
9.5
9
8.5
8
8

8.5

9

9.5

10

10.5
11
11.5
12
Kinetic Energy T [MeV]

Figure 3.11: The energy dispersion for an Al specimen magnifying the energy range of 10
MeV.
24

Energy Dispersion ∆T spe /T

10−5
t = 28.5 nm

10−6

10−7

10−8

1

10
Kinetic Energy T [MeV]

Figure 3.12: The energy dispersion for a 28.5 nm-thick Al specimen.

1
Phase Contrast Transfer Function sinΧ

0.9

Correction Term for Energy Dispersion

0.8

Correction Term for Image Spread
Correction Term for Incident Angle

0.7

Modified Phase Contrast Transfer Function

0.6
0.5
0.4
0.3
0.2
0.1
0
0

5

10

15

20

25
30
35
40
-1
Spatial Frequency [nm ]Figure 3.13: The lens transfer function for 10 MV SRF-TEM with a 28.5 nm-thick Al specimen in the case of ∆′′ =16.3 nm and ∆f =13 nm. The spatial resolution is 60 pm.

25

3.3.2

Downsizing Facility

RF resonant cavities can generate much higher accelerating gradient so that the TEM facilities can be compact. For example, 1.2 MV FE-TEM of Hitachi is tall as about 10 m[6]. In our
SRF-TEM, only one two-mode cavity is needed to accelerate electrons to 1.0 MV even when
applying two-mode acceleration2 . In this case, the entire system is designed as Figure 3.1
and its height is about 4 m. This scale is comparable with a few hundred of keV acceleration
voltage TEMs which are the most common type. However, our SRF-TEM needs a helium
circulation system including a liquefaction facility and a helium gas collection line. Generallaboratories has its own helium facility. When the SRF-TEM is installed, it is necessary toconstruct new gas collection lines connected to the existing helium facility.

3.3.3

Acquisition of Time Resolution

Density Distribution [a.u]Conventional TEMs use DC electron beams with thermal guns or cold field emission guns.
Therefore it is diﬃcult to observe specimens with time resolution. In contrast, SRF-TEM
employs a photocathode gun which supplies the electron bunches in synchronization withthe period of the two-mode cavity’s acceleration; 1.3 GHz. This allows SRF-TEM to observe
specimens with time resolution. Figure 3.14 shows the density distribution of electrons at thecathode, calculated based on the thermal diﬀusion model[17]. The optimized bunch charge
is 0.1 fC which corresponds to 640 electrons for our SRF-TEM. In order to acquire one TEM
image, about 107 electrons are needed[18]3 . This indicates that about 15600 bunches areneeded for one image for our SRF-TEM. If bunches are supplied with 1.3 GHz (this situation
is called 100 % duty), SRF-TEM has the time resolution of 12 µs.
There is another observation scheme with time resolution; pumping probe method. Thisutilizes both exciting light pulse (pumping) and imaging electrons (probe). This type of

1.2

1

0.8

0.6

0.4

0.2

0
0

5

10

15

20

25

30
time [ps]Figure 3.14: The density distribution of electrons at the cathode.
2

Two-mode acceleration utilizes both of accelerations and decelerations with two resonant modes. Thismeans if the first priority to minimize the energy spread was abandoned, higher gradients can be acquired
and this enables the downsizing the facility.
3
Strictly speaking, it depends on the brightness of the beam and the number of the pixel of CCD sensors.

26

TEMs are called dynamic transmission electron microscopes (DTEMs). The transition of
the specimen is reconstructed virtually4 to change the phase diﬀerence between the pumping
light and probing electrons. Time resolution τ is estimated as)
2
2 + τ2 ,τ = τpum+ τpro(3.3.6)jit
where τpum represents the time width of the pumping laser, τpro represents the time widthof the imaging electron bunch and τjit represents the jitter of the phase between pumping
lights and imaging electron bunches. τpum and τjit can be suppressed to 100 fs. τpro is 2 psfor our SRF-TEM (see Figure 3.14). In this way, our SRF-TEM is able to acquire 2 ps time
resolution using the pumping probe method. It is noted that when pumping probe method
is employed, one electron bunch has to generate one image. This increase the bunch charge
which leads to additional energy dispersion.

4

It is noted that the pumping probe method cannot record one picked process with many temporal slices.
It records one image in one process. Therefore this method can be applied only for repeatable phenomena
and it just reconstructs the transition.

27

28

Chapter 4
System Design of Superconducting
Radio Frequency Transmission
Electron Microscope with 300 kV and3 MV by Numerical Simulation ofBeam Dynamics
This chapter shows how eﬀectively the two-mode acceleration realized by the two-mode cavity
can restrain the energy spread for 300 kV and 3 MV acceleration. In order to achieve thisgoal, the numerical simulation program, called General Particle Tracer (GPT) is utilized[19].
This employs the fifth order Runge-Kutta solver which is capable of calculating 3D beam
dynamics to trace the trajectory of each charged particle. It can account for space charge
eﬀect (not considered in this thesis).

4.1

Scheme of Simulation Study Using General Particle
TracerIn this section, a brief explanation concerning GPT is given, and this is the summary quoted
from GPT manual[19]. And, the explanation of modified Powell method is based on [20].

294.1.1

Scheme of Numerical Simulation

This part explains the method of the numerical simulation adopted by GPT. GPT calculates
the dynamics of charged particles based on the relativistic equation of motion,
dP i
= Fi
dt
dxi
pc
= vi = * 2 idtpi c2 + m2i c2(4.1.1)(4.1.2)where P i is the relativistic momentum of the i-th particle, F i is the power applied on theith particle, xi is the coordinate of the i-th particle, v i is the velocity of the ith particle, miis the mass of the ith particle and c is the speed of light. F i is given by
F i = q(E i + v i × B i )(4.1.3)where E i and B i are the electric and magnetic field applied on the ith particle respectively
and q is the particle charge.
GPT can consider the space charge eﬀect which is the mutual repulsion eﬀect of charged
particles inside the beam bunch. Including this eﬀect, Eq. (4.1.1), (4.1.2) and (4.1.3) aresummarized asdy(t)= f (t, y(t))
dt(4.1.4)

where y(t) is (N × 6)-dimensional vector of all particles coordinates and momentums.
GPT employs the fifth order embedded Runge-Kutta method in order to calculate Eq.(4.1.4).
This can automatically optimize the next step size h to consider whether the current step
size is adequate in each step. Each step is calculated as
k1 = hf (xn , yn )k2 = hf (xn + a2 h, yn + b21 k1 )...
k6 = hf (xn + a6 h, yn + b61 k1 + ... + b65 k5 )yn+1 = yn + c1 k1 + ... + c6 k6 + O(h6 ).

(4.1.5)The coeﬃcients of ai , bi and ci are given in Table 4.1. On the other hand, by replacing c∗iwith ci , Eq. (4.1.5) becomes 4th-order Runge-Kutta equation,∗
yn+1= yn + c∗1 k1 + ... + c∗6 k6 + O(h5 ).

(4.1.6)

Embedded Runge-Kutta method can always modify the step size to decide whether thecurrent step size is appropriate by calculating the diﬀerence between two answers yn+1 and
∗
yn+1
as
∗
∆ = yn+1 − yn+1
=

30

6
+
i=1

(ci − c∗i )ki(4.1.7)

Table 4.1: Coeﬃcients of embedded Runge-Kutta method utilized in General Particle Tracer.
i
ai
1
2 1/5
3 3/10
4 3/5
5
1
6 7/8
j=

bij
1/5
3/40
9/40
3/10
-9/10
-11/54
5/2
1631/55296 175/512
12

6/5-70/27575/13824
3

35/27
44275/110592
4

253/2096
5

ci
37/3780250/621125/5940
512/1771c∗i
2825/27648
018575/48384
13525/55296
277/14336
1/4

where ∆ is the sum of every sixth vector of all particles. Naming γβ and x of the particle
that has the maximum ∆ with (γβ)max and xmax respectively, then a new parameter of Λ isdefined as%&(γβ)max xmax
Λ = max,,(4.1.8)(γβ)err xerr
This is used in order to evaluate the adequacy of the current step size. Here (γβ)err and xerr
are the reference values and (γβ)err = 1.0 × 10−4 and xerr = 1.0 × 10−4 unless specified in the
GPT program. In each step, the calculation of Λ is done. In the case that Λ > 1, the newstep size h is suggested as,SΛ1/4 hold (SΛ1/4 > 0.10)
hnew =
0.10hold(otherwise)
and in the case that Λ ≤ 1, the new step size h is suggested as,SΛ1/5 hold (SΛ1/5 < 5.0)
hnew =
5hold(otherwise)
where S = 0.85 unless specified in the GPT program.
As explained above, GPT can minimize the consuming time required for calculation tomaintain the accuracy that users set. However this automation of the step size is done bycomparing the fifth-order accuracy with forth-order one, which means that users have to set
the step size anew when calculating the dynamics in the high gradient fields.

4.1.2

Scheme of Design Optimization

GPT has the optimization scheme to minimize or maximize the specified variable. The scheme
employed by GPT is the modified Powell method[20]. The second-order Taylor expansion of31

the function f (x) that has N variables is+ ∂f + ∂ 2f 1
f (x) = f (x0 ) +(xi − xi0 ) +(xi − xi0 )(xj − xj0 ) + ...
∂x
2
∂x
∂x
i
i
j
x=x0
x=x0
i
i,j
1
≈ c − b · (x − x0 ) + (x − x0 ) · A · (x − x0 )(4.1.9)2
wherec ≡ f (x0 )b ≡ − ∇f |x=x0∂ 2 f -[A]ij ≡.
∂xi ∂xj -x=x0(4.1.10)(4.1.11)∇f = A · (x − x0 ) − b(4.1.13)δ(∇f ) = A · δx.

(4.1.14)(4.1.12)

A is called Hessian matrix and is a symmetric square matrix if f (x) is continuous for every
variable. Take gradient of Eq. (4.1.10),

is given, thenIf the minimum point is found in a direction of u, using u · δ(f ) = 0,
u · A · δx = 0(4.1.15)

is given. Based on this, the next direction to find the minimum is δx that satisfies Eq.
(4.1.15). These directions which are more than one are called conjugated direction.
Here, the process that finds the minimum point in the case of two variables is illustrated
in practice (Figure 4.1).
1. Find the minimum point x1 in the direction of u1 from the initial point x0 .
2. Find the minimum point x2 in the direction of u2 from the previous minimum pointx.
3. Set x2 − x0 = u2 and u2 = u1 , then find the new minimum point x0 .
Repeating the process, the calculation is finished when the current minimum point isconverged. The minimization of the one direction is conducted by successive parabolic interpolation method of golden section search method. It is necessary to set the appropriate
initial values in order to minimize the variable correctly. Without this, the minimization isconverged at a metastable point or the calculation diverges. To prevent from these problems,
it is useful to sweep the variable and estimate the appropriate initial values to some extent.
GPT has the function of MR where the variable can be swept, and using this, incorrect
minimization can be avoided.
32

Figure 4.1: The concept of modified Powell method.

4.2

Comparison of Two-mode Acceleration and Onemode Acceleration

In this section, the capability of the two-mode acceleration in reducing the energy spreadis confirmed by carrying out GPT simulations. The objective accelerating energy is 300
keV for the first SRF-TEM prototype and 3 MeV which is the same value as the world’s
highest energy of TEMs. For each accelerating energy, the energy spread ∆T and the energy
dispersion ∆T /T is calculated in the both case of the one-mode acceleration which employs
only the TM010 mode and the two-mode acceleration which employs both the TM010 modeand the TM020 mode.

4.2.1

Method of Optimization

For the one-mode acceleration, there are two variables to control in order to minimize theenergy spread, ; the amplitude and the phase of the TM010 mode. For the two-mode acceleration, there are four variables to control; each amplitude and phase of the TM010 mode andthe TM020 mode. It is diﬃcult to find the minimum point in this optimization. In order toresolve this, I used the MR function which is equipped with GPT. This program sweeps thedesignated variables to observe the variation of the target value which is the energy spread∆T here. After obtaining the information around the minimum point, the optimization iscarried out around the estimated point.
Figure 4.2 shows an example of the result of MR function. This is the one-mode acceleration employing only the TM010 mode and the goal accelerating energy is 300 keV. In thisexample, the designated variable to be swept is the phase of the TM010 mode, whereas theamplitude of the TM010 mode is fixed. The most probable point where the energy dispersion
33

converges to the minimum value is estimated to be about 28.6 degree (0.50 radian1 ). This isfollowed by optimization.
The example above is the easiest one. In the case of four free parameters, the basis of
the optimization process is the same.

Figure 4.2: An example of MR function. Here the one-mode acceleration for 300 keV is thecase. The accelerating field is fixed as 7.46 MV/m and the phase is swept, then the energy
spread ∆T and the accelerating energy T are recorded. The minimum point of the energy
dispersion which corresponds to the maximum point of the accelerating energy is about 28.6
degree of the phase of the TM010 mode.

4.2.2

Condition of Beam Dynamics Simulation

In this sub-section, the conditions utilized to simulate the beam dynamics are explained.
The initial condition is defined by the cathode and is common for all simulations. The initial
parameters are summarized in Table 4.2. The initial distribution of electrons are depicted in
Figure 4.3.
1

GPT describes the phase in the unit of radian.

34

h1

150
100

14

β x [×10-4]10y [µm]

200

8

h1

8

Entries

624

Mean x− 1.6096

Mean y0.00404

Std Dev x 49.16

Std Dev x

49.16

Std Dev y

4

Std Dev y

4.442

Entries

624

Mean x−1.609Mean y−2.305
50.6

50

12
10
8
6

−50

0

4

−2

3

−4

4

−100

6
5

2

0

7

2
−6

2

−150
−200
−200 −150 −100 −50

0

50

100

150 200x [µm]

1

−8

0

−10
−200 −150 −100 −50(a) x vs. y.

8h1
Entries624

150 200 0x [µm]

100

Mean x189.8

6Mean y0.00404

Std Dev x

34.49

4

Std Dev y

4.442

7

7
h1

8
6

Entries

624

Mean x0.00404

4

Std Dev x

4.442

Std Dev y

4.417

2

0

4

0

−2

3

−2

3
2

2
−6

−6
1

−8
−100

5
4

−4

−4

6

Mean y −0.02012

6
5

2

10

β y [×104]

β x [×10-4]50(b) x vs. βx .

10
8

0

50

100

150

200

250

300
350 0
-4
β z [×10 ]

1

−8
−10
−10 −8

−6

−4

−2

0

2

(d) βx vs βy .

(c) βz vs βx .

Figure 4.3: Electron Distribution at t = 2.0 × 10−12 .

35

4

6

8 10
β x [×104]0Table 4.2: The initial condition for GPT simulation. C represents the charge of one bunch,rini represents the initial radius on the cathode, f010 and f020 represent the resonant frequencyof the TM010 mode and the TM020 mode respectively and ∆Eini represents the initial energy
spread at the cathode.
C-0.1 fCrini
100 µmf010 1.3 GHz
f020 2.6 GHz∆Eini 0.15 eV

The distribution of electric fields of 60 kV DC gun and the two-mode cavity are installed
into GPT simulations. Those are calculated by the software Superfish. Figure 4.4 shows the
distributions of the electric field for the two-mode cavity.
The geometry of the simulation is depicted in Figure 4.5. This is the case of 300 kV
acceleration and the required number of the two-mode cavity is one. Figure 4.6 shows thegeometry in the cace of 3 MV acceleration and the number of the two-mode cavity is three.
The spaces between each component are determined to reflect the actual size of components
and the vacuum tools.

(b) The TM020 mode.

(a) The TM010 mode.

Figure 4.4: The distribution of the electric fields of the two-mode cavity.

36

Figure 4.5: The geometry of the simulation for 300 kV.

Figure 4.6: The geometry of the simulation for 3 MV.

37

4.2.3

Optimization Result of 300 keV Acceleration

Table 4.3 summarizes the simulation result for 300 keV acceleration. Owing to the two-mode
acceleration, the energy dispersion ∆T /T is reduced by a factor of twenty compared to thecase of one-mode acceleration. Table 4.4 shows the optimal values in order to minimize theenergy spread ∆T . The electric field on the beam axis Ez in the two-mode cavity is built in
the form of
Ez = Eap cos(ωt + θ)

(4.2.1)in GPT simulations. Table 4.4 indicates the result for the amplitude Eap and the phaseθ. ω is 2π × 1.3 GHz and 2π × 2.6 GHz for the TM010 mode and the TM020 mode respectively. t indicates the elapsed time starting from the firing of the gun. The values of Eap
directly become the performance requirements for the practical two-mode cavities in orderto manufacture the 300 kV SRF-TEM prototype.
Figure 4.7 - 4.8 show the variations of the accelerating energy T for the optimal one-mode
acceleration and two-mode acceleration respectively.
Table 4.3: The simulation result of the 300 kV acceleration.
T
∆T
∆T /T
One-mode Acceleration 330 keV 57.6 eV 1.75 × 10−4
Two-mode Acceleration 272 keV 2.19 eV 7.91 × 10−6

Table 4.4: The optimal vales of the accelerating modes for the 300 kV acceleration.

One-modeTwo-mode

Amplitude of TM010
6.98 MV/m
7.46 MV/m

Phase of TM010
0.456 rad.
0.454 rad.

38

Amplitude of TM020
7.20 MV/m

Phase of TM020
0.500 rad.

Figure 4.7: The variation of the accelerating energy T for the optimal 300 kV one-mode
acceleration.

Figure 4.8: The variation of the accelerating energy T for the optimal 300 kV two-mode
acceleration.
39

4.2.4The optimization result of the 3 MV acceleration

Table 4.5 summarizes the simulation result for 3 MeV acceleration. In this case, the system
has three two-mode cavities. In order to minimize the energy dispersion ∆T /T just below
the last cavity, I optimized the system in each step to add the cavity one by one. Owing to
the two-mode acceleration, the energy dispersion ∆T /T can be reduced by ten times thanthe one-mode acceleration to compare the values at the outlet of Cavity3.
Table 4.6 shows the optimal values in order to minimize the energy spread ∆T . These
Amplitudes directly become the performance requirements for the practical two-mode cavities
in order to manufacture the future 3 MV SRF-TEM
Figure 4.9 - 4.10 show the variations of the accelerating energy T for the optimal one-mode
acceleration and two-mode acceleration respectively.
Table 4.5: The simulation result of the 3 MV acceleration. Each value is at the outlet ofeach cavity.
Cavity1
Cavity2Cavity3T
1.17 MeV
2.38 MeV
3.59 MeV
One-mode Acceleration∆T
163 eV269 eV368 eV−4−4∆T /T 1.39 × 10
1.13 × 10
1.03 × 10−4T
1.13 MeV
2.17 MeV
3.07 MeV
Two-mode Acceleration∆T
23.3 eV29.4 eV29.5 eV−5−5∆T /T 2.06 × 10
1.35 × 10
9.61 × 10−6

Table 4.6: The optimal vales of the accelerating modes for the acceleration for 3 MV
Cavity1
Cavity2Cavity3
Amplitude of TM010 24.8 MV/m 24.8 MV/m 24.8 MV/m
One-mode
Phase of TM010
0.695 rad.
1.700 rad.
2.800 rad.
Amplitude of TM020
Phase of TM020
Amplitude of TM010 18.3 MV/m 24.7 MV/m 24.7 MV/m
Two-mode
Phase of TM010
0.900 rad.
1.100 rad.
2.800 rad.
Amplitude of TM020 28.1 MV/m 6.2 MV/m 10.8 MV/m
Phase of TM020
3.200 rad.
2.100 rad.
5.520 rad.

40

Figure 4.9: The variation of the accelerating energy T for the optimal 3 MV one-mode
acceleration.

Figure 4.10: The variation of the accelerating energy T for the optimal 3 MV two-mode
acceleration.

41

4.3

Requirement for Cavity Control

In the previous sections, it is confirmed that the two-mode acceleration is eﬀective to reduce
the energy spread ∆T . And the set values in order to function the two-mode acceleration
correctly are derived. In practical operations, we only have to set the previous values foreach amplitude and phase. However, it is impossible to set theses values without any errors.
In this section, the control error eﬀects on the energy spread ∆T are investigated for thetwo-mode acceleration on 300 kV SRF-TEM prototype.
Now two types of the energy spread are introduced; ∆T1 and ∆T2 . ∆T1 is the same as theprevious section, which indicates the energy spread in one electron bunch. ∆T2 is the energy
spread resulting from the deviation of the averaged accelerating energy T . The ultimate
energy dispersion ∆T is calculated by)∆T = ∆T12 + ∆T22 .
(4.3.1)Figure 4.11 - 4.14 show each control error eﬀect on the energy spread and the energy dispersion. The horizontal axis indicates the quantity of the error. The top-left figure shows thedeviation of averaged accelerating energy T . The top-right figure shows ∆T1 . The bottomleft figure shows ∆T2 . The top-right figure shows ∆T1 . The bottom-left figure shows ∆T2 .
The bottom-right figure shows the energy dispersion ∆T /T where ultimate energy spread ∆T
is calculate by Eq. (4.3.1). Here some goals of the energy dispersion ∆T /T are set. ∆T /T of
1.0 × 10−4 is the first goal for 300 kV SRF-TEM prototype. And ∆T /T of 4.0 × 10−5 is theultimate goal for 300 kV SRF-TEM prototype after commissioning. Table 4.7 summarizes
the requirements on each parameter control for both energy dispersion goals. These values
are used to define the targets for the RF control system for SRF-TEM.
Table 4.7: The control requirement for the two-mode acceleration of 300 kV SRF-TEM
prototype.
Target1.0 × 10−4
4.0 × 10−5

Amplitude of TM010
0.010 %
0.004 %

Amplitude of TM020
0.037 %
0.014 %

42

Phase of TM010
0.320 deg.
0.150 deg.

Phase of TM020
0.120 deg.
0.055 deg.

2.19

Energy Spread ∆ T 1 [eV]

Accelerating Energy T [eV]272.49272.48272.47272.46

2.18
2.17
2.16
2.15

272.452.14
272.442.13− 0.008− 0.006− 0.004− 0.002

0

0.002

2.12
−0.01

0.004
0.006
0.008
0.01
Deviation of TM010 Amplitude [%]30-5∆ T / T [× 10 ]Energy Deviation ∆ T 2 [eV]272.43
−0.01

20

8

6

− 10

4

− 20

2

− 0.006− 0.004− 0.002

0

0.002

0
−0.01

0.004
0.006
0.008
0.01
Deviation of TM010 Amplitude [%]− 0.004− 0.002

0

0.002

0.004
0.006
0.008
0.01
Deviation of TM010 Amplitude [%]− 0.008− 0.006− 0.004− 0.002

0

0.002

0.004
0.006
0.008
0.01
Deviation of TM010 Amplitude [%]100− 0.008− 0.006

12

10

− 30
−0.01

− 0.008

272.4918

Energy Spread ∆ T 1 [eV]

Accelerating Energy T [eV]

Figure 4.11: The control error eﬀect on the energy spread for the amplitude of the TM010
mode.

272.48272.47272.46

16
14
12
10
8

272.456
4

272.44
2
− 0.1

−0.05

0

0.05

0
−0.15

0.1
0.15
Deviation of TM010 Phase [deg.]

25-5∆ T / T ( × 10 )Energy Deviation ∆ T 2 [eV]272.43
−0.15

20
15
10

−0.05

0

0.05

0.1
0.15
Deviation of TM010 Phase [deg.]

− 0.1

−0.05

0

0.05

0.1
0.15
Deviation of TM010 Phase [deg.]

10
9
8
7

5

6

0

5

−5

4

− 10

3

− 15

2

− 20
− 25
−0.15

− 0.1

1
− 0.1

−0.05

0

0.05

0
−0.15

0.1
0.15
Deviation of TM010 Phase [deg.]

Figure 4.12: The control error eﬀect on the energy spread for the phase of the TM010 mode.

43

Energy Spread ∆ T 1 [eV]

Accelerating Energy T [eV]272.5
272.49272.48272.47

2.157
2.156
2.155
2.154
2.153

272.462.152
272.452.151

272.44

2.15

−0.03

−0.02

−0.01

0

0.01

2.149
−0.04

0.02
0.03
0.04
Deviation of TM020 Amplitude [%]40-5∆ T / T [× 10 ]Energy Deviation ∆ T 2 [eV]272.43
−0.04

30
20

−0.03

−0.02

−0.01

0

0.01

0.02
0.03
0.04
Deviation of TM020 Amplitude [%]

−0.03

−0.02

−0.01

0

0.01

0.02
0.03
0.04
Deviation of TM020 Amplitude [%]

12

10

8

10
0

6

− 10
4
− 20
2

− 30
− 40
−0.04

−0.03

−0.02

−0.01

0

0.01

0
−0.04

0.02
0.03
0.04
Deviation of TM020 Amplitude [%]

272.5

22

Energy Spread ∆ T 1 [eV]

Accelerating Energy T [eV]

Figure 4.13: The control error eﬀect on the energy spread for the amplitude of the TM020
mode.

272.49272.48272.47

20
18
16
14
12
10

272.468272.456
4

272.44

2
−0.3

−0.2

− 0.1

0

0.1

0
−0.4

0.2
0.3
0.4
Deviation of TM020 Phase [deg.]

30-5∆ T / T [× 10 ]Energy Deviation ∆ T 2 [eV]272.43
−0.4

20

−0.3

−0.2

− 0.1

0

0.1

0.2
0.3
0.4
Deviation of TM020 Phase [deg.]

−0.3

−0.2

− 0.1

0

0.1

0.2
0.3
0.4
Deviation of TM020 Phase [deg.]

14
12
10

10
8
0
6
− 10
4
− 20

− 30
−0.4

2

−0.3

−0.2

− 0.1

0

0.1

0
−0.4

0.2
0.3
0.4
Deviation of TM020 Phase [deg.]

Figure 4.14: The control error eﬀect on the energy spread for the phase of the TM020 mode.

44

Chapter 5
Development of Two-mode Cavity
This chapter describes the development of the two-mode cavity. First comes a fundamental
description of superconducting cavities, followed by an explanation of design of the twomode cavity. The fabrication process of the two-mode cavity is diﬀerent from that of theconventional superconducting cavities. Finally the performance tests of the two-mode cavity
is described.

5.1

Fundamental Property and Design of Superconducting Cavity

5.1.1

Nature of Superconductivity

The phenomenon of superconductivity was found by Kammerlingh Onnes. The DC resistance of superconducting matter drops to zero when it is cooled. Superconductivity wastheoretically explained by Bardeen, Cooper and Schrieﬀer[21]. Pairs of two electrons makeCooper pairs owing to the interaction with phonons and indicates the phenomena of BoseEinstein condensation in lower energy state. This superconducting component carries thecurrent with zero resistance. However in case of AC, normal conducting components also
carry the current in the superconducting material, which leads to finite power dissipation.
This dissipating power Pdis is defined as
1
Pdis = Rs H 2
2(5.1.1)

Rs = RBCS + R0 .

(5.1.2)

where

RBCS is the component which is explained by BCS theory, and satisfies the following equation;
%
&
A 2
∆(0)
RBCS = f exp −(5.1.3)T
kB T
45

Table 5.1: The characteristics of niobium[22][23][24].
Atomic Number
41
Atomic Weight
92.91
Density
8.570 kgm−3
Melting Point
2750 K
Biting Point
5017 K
Transition Temperature
9.3 K
Specific Heat Capacity@298 K
265 JK−1 kg−1
Thermal Conductivity@273 K
53 Wm−1 K−1
Coeﬃcient of Linear Expansion@293 K 7.3 × 10−6 K−1
Resistivity@273 K
273 nΩm
Re-crystallization Temperature
1200 K - 1600 K
Stress Release Temperature1100 K

where A is a constant and ∆(0) is a half of the energy required to destroy one Cooper pair at
absolute zero temperature. kB is Boltzmann constant. R0 represents the contribution fromlattice defects and impurities and never disappear even at absolute zero temperature.
There are several materials which indicate the characteristics of superconductivity. The
most suitable material for accelerating cavities is niobium (Nb). The transition temperature
Tc is as high as 9.3 K, and it is possible to acquire niobium as highly pure state. Furthermore
it is easy to process the material. The RBCS of niobium satisfies the following empiricalequation well when T < Tc /2;
RBCS = 2 × 10

−4 1

T

%

f
1.5

&2

%

17.67
exp −
T

&

.

(5.1.4)

Therefore we employ pure niobium as the material of the two-mode cavity. Table 5.1 summarizes several characteristics of niobium.

5.1.2

Resonant Mode in Pillbox

In this sub-section, a brief explanation of the RF acceleration principle by superconducting
cavities is given (based on [25] and [26]) There are two types of RF accelerating cavities;
normal conducting and superconducting. The diﬀerence lies in the boundary condition on
the surface of the cavity. This sub-section features the boundary conditions for the superconducting cavities. In order to calculate the resonant electromagnetic fields, and Maxwell
equations and the boundary conditions led by the nature of the superconductivity are used.

46

First,divD = ρdivB = 0∂D∂t∂B
rotE = −∂trotH = j +(5.1.5)(5.1.6)(5.1.7)(5.1.8)andD = εEB = µH
j=0
ρ=0(5.1.9)(5.1.10)(5.1.11)(5.1.12)are given. Here, considering a traveling wave along the z-direction in a cylindrical waveguideof radius a called pillbox like Figure 5.1,
E(x, t) = E(x, y) exp(ikx − iwt)
H(x, t) = H(x, y) exp(ikx − iwt).

(5.1.13)(5.1.14)Substituting these with E and H in Eq. (5.1.8), (5.1.7), and calculating each rotation,(∇2 − ω 2 εµ)E = 0(∇2 − ω 2 εµ)H = 0(5.1.15)(5.1.16)

are given. Here using a relational expression of ∇2t = ∇2 − ∂ 2 /∂z 2 , Eq. (5.1.15) and (5.1.16)
can be modified as.
/{∇2t + ω 2 εµ − k 2 }E = 0(5.1.17).
/{∇2t + ω 2 εµ − k 2 }H = 0.
(5.1.18)

Figure 5.1: A pillbox-type cavity.
47

Table 5.2: The values of ρmn .
m\n
1
2
3
0
2.405 5.520 8.654
1
3.832 7.016 10.173
25.136 8.417 11.620

Here, defining ω 2 εµ − k 2 = γ 2 , and focusing on Ez in cylindrical coordinate system, Eq.
(5.1.17) gives
% 2
&
∂
1 ∂
1 ∂2
2
2
2
∇t E z + γ E z =+++ γ Ez = 0,
(5.1.19)∂r2 r ∂r r2 ∂θ2where r and θ are new coordinates of cylindrical coordinate system, andx = r cos θ
y = r sin θ(5.1.20)(5.1.21)are satisfied with the original Cartesian coordinate system. Here, assuming that Ez can be
written in the separable form as
Ez = R(r)Θ(θ)(5.1.22)

and, that Θ(θ) satisfies the equation of a simple harmonic motion asd2 Θ(θ)+ m2 Θ(θ) = 0, (m = 1, 2, 3, ...)dθ2

(5.1.23)%&d2 R(ρ) 1 dR(ρ)
m2++ 1 − 2 = 0.
dρ2
ρ dρ
ρ

(5.1.24)then , we get

Here ρ = γr. Eq. (5.1.24) is the Bessel equation, whose solutions are the attenuated sineand cosine functions (Figure 5.2). In the case of superconducting, the cylindrical waveguide
becomes a state close to a perfect conductor. Therefore at r = a, E∥ = 0 and H⊥ = 0
are approximately established, and Jm (γa) = 0 is also satisfied. Figure 5.2 shows roots ofJm (γa) = 0. Defining the n-th root of the m-th order Bessel function as ρmn , γa = ρmn isestablished. Each value of ρmn is listed in Table 5.2.
Using the above discussion, Eq. (5.1.22) is written as0ρ1mn
Ez = E0 eimθ Jmr ei(kz−ωt) .
(5.1.25)a48

In addition, modifying Eq. (5.1.17),
1(∇2t E t )
% %
&
&
1
∂Ez
= 2
∇t
− iωµez × ∇t Hz
ω εµ − k 2
∂z

Et =

ω 2 εµ − k 2(5.1.26)

is given, where Et is expressed only by Ez and Hz . In a similar way, Eq. (5.1.18) is written
as
% %
&
&
1
∂Hz
Ht = 2∇t− iωεez × ∇t Ez .
(5.1.27)
ω εµ − k 2∂zAccelerating modes which give the energy to charged particles satisfy both Ez ̸= 0 and
Hz = 0. These modes are called TM mode (transverse magnetic field modes) . The
complementary modes are TE modes (transverse electric field modes) which kick charged
particles. Here focusing attention on TM modes, Eq. (5.1.26) and (5.1.27) are simplified as
ik(∇t Ez )γ2ik
H t = 2 (iωεez × ∇t Ez )γ
Et =Figure 5.2: Bessel Functions.

49(5.1.28)(5.1.29)

respectively. Substituting Eq. (5.1.25) into these equations,
0ρ
1
ik ∂
ik
mn
imθ ′
E r = 2 E z = E 0 e Jmr ei(kz−ωt)
γ ∂r
γ
a
0ρ
1
ik 1 ∂k m
mn
Eθ = 2Ez = − 2 E0 eimθ Jmr ei(kz−ωt)γ r ∂θ
γ ra0ρ
1iωε 1 ∂
ωε mmn
Hr = 2Ez = − 2 E0 eimθ Jmr ei(kz−ωt)γ r ∂θ
γ r
a
0
iωε ∂
iωε
ρmn 1 i(kz−ωt)′Hθ = − 2
Ez = −
E0 eimθ Jm
r e
γ ∂r
γ
a

(5.1.30)(5.1.31)(5.1.32)(5.1.33)

are given. These express TM modes propagating in the cylindrical waveguide. Furthermore,a pillbox-like cavity like Figure 5.1 is considered. The resonant field in this pillbox is limitedby the boundary conditions of the cut-oﬀ walls at z = 0 and d. When considering thesuperconducting state, the cavity becomes the state close to a perfect conductor. Therefore
at z = 0 and d, E ⊥= 0, H ∥= 0 is approximately established, then Er |s = 0 and Eθ |s = 0
become the boundary conditions. Then the fields of TM modes in the superconducting cavity
like Figure 5.1 are expressed as0ρ
1
0 pπz 1mn
E z = E 0 Jm
r cos(mθ) cos(5.1.34)ad0ρ
10 pπz 1kmn′
E r = E 0 Jm
r cos(mθ) sin(5.1.35)γad0ρ
10 pπz 1k m
mn
E θ = − 2 E 0 Jm
r sin(mθ) sin(5.1.36)γ rad
Hz = 0(5.1.37)0ρ
10 pπz 1
ωε mmn
H r = − 2 E 0 Jm
r sin(mθ) cos(5.1.38)
γ r
a
d
0ρ
1
0 pπz 1ωεmn′
H θ = − E 0 Jm
r cos(mθ) cos(5.1.39)γadwhere γ = ρmn /a and k = pπ/d are established. And m, p = 1, 2, 3, ..., and n = 0, 1, 2,
3,....Therefore the resonant frequency of TMmnp in vacuum is2
c 0 ρmn 12 0 pπ 12f=+.
(5.1.40)2πadThis is the function of the radius and full-length of the cavity. The lowest-order mode of TM
mode is TM010 , and its fields are
%
&
2.405
E z = E 0 J0r
(5.1.41)
a
%
&
ωε
2.405
′
H θ = − E 0 J0r ,(5.1.42)γa50Table 5.3: The values of ρ′mn .
m\n
1
2
3
0
3.832 7.106 10.173
11.841 5.331 8.536
2
3.054 6.706 9.969

and other four components are all zero. High energy accelerators generally employ this TM010
mode, whose fields are distributed as (a) of Figure 1.1.
Calculating TE modes in the same way as TM modes, the electric-magnetic fields are
Ez = 0% ′ &0 pπz 1ωε m
ρmn
E r = 2 E 0 Jm
r sin(mθ) sin
γ r
a
d
% ′ &
0
ωε
ρmn
pπz 1
Eθ =
E 0 Jmr cos(mθ) sin
γ
a
d
% ′ &
0
1
ρmn
pπz
H z = E 0 Jm
r cos(mθ) sin
a
d
% ′ &
0 pπz 1kρmn
H r = 2 E 0 Jm
r cos(mθ) cos
γ
a
d
% ′ &
0 pπz 1k m
ρmn′
H θ = − 2 E 0 Jm
r sin(mθ) sin
γ rad(5.1.43)(5.1.44)(5.1.45)(5.1.46)(5.1.47)(5.1.48)

where γ = ρ′mn /a and ρ′mn /a is the n-th root of the first derivative of m-th order Bessel
function (Table 5.3).The resonant frequency of TEmnp in vacuum is$%&2 0 1
c
ρ′mn
pπ 2
f=
+
.
(5.1.49)2πadGenerally, the TM010 mode is called the fundamental mode, and plays a huge role in accelerating charged particles. Other higher order modes are called HOM, and disturb a stableacceleration and destabilize the beam.

5.1.3

Performance Index of Superconducting Cavity

Using the performance indices for superconducting cavities, we can evaluate the cavity performance quantitatively. In this sub-section, general performance indices and the importantones for this thesis are explained.

51

Accelerating Voltage
When a charged particle path through the cavity where some resonant mode is existing, thetotal accelerating voltage Vacc that the particle receives is expressed as-! diωz/cβVacc = -Ez |r=0 e
dz -- .
(5.1.50)0

Vacc is dependent on the relativistic β and the phase of the electric field. In general highenergy accelerator developments, β = 1 is chosen and the phase is set so that Vacc gets the
maximum value. Therefore the averaged accelerating electric field Eacc becomes

Vacc.
(5.1.51)d
Generally, Eacc is an essential value to indicate the accelerating performance of the cavity.
However it is not the most favorable one for our study, because the incident energy into thetwo-mode cavity is low as 60 kV (β = 0.45). Therefore we use the maximum electric field on
z-axis Eap . This has no dependency on β and the incident phase.
Eacc =

Quality Factor Q0
Even if we cool a superconducting cavity, there must be a normal conducting component.
This carries a finite current, then the stored energy in the cavity is lost. This dissipating
power is called Pc , then the dissipating power per unit area is defined asdPc
1= Rs |H|2 .
(5.1.52)
ds
2
Rs is about 10 nΩ in the case of superconducting cavies. The stored energy in the cavity isdefined as!
1
U = µ0|H|2 dv(5.1.53)
2
!V
1
= ϵ0
|E|2 dv.
(5.1.54)
2
V
Here, a new performance index ofquality factor (Q0 )
Q0 =

ω0 U
.
Pc

is defined as(5.1.55)

This is the ratio of the stored energy in the cavity and the dissipating power on the cavity
wall, and indicates how eﬀective the energy can be stored in the cavity. Using Eq. (5.1.52)
and (5.1.55),3
ω0 µ0 V |H|2 dv
Q0 =3(5.1.56)
Rs S |H|2 ds
G=(5.1.57)
Rs
52

Table 5.4: The performance indices of the designed two-mode cavity.
TM010
TM020
Frequency [GHz] 1.30015224 2.59994554
Q01.86 × 108 1.00 × 108
RBCS [nΩ]7.54×102
3.01×103
Pc [W]
3.01
4.84
Ra /Q0 [Ω]
1.23 ×102
4.53 ×101
Eap [MV/m]
5.33
5.14
1/2 −1A[Ω m ]
225.2
233.1

is established where G is called form factor which is not dependent on the size of thecavity but its shape. This value of G is calculated by electromagnetic simulations.
Other Index
Shunt impedance

Ra
Ra =

2
Vacc
Pc

(5.1.58)

indicates the accelerating eﬃciency. The ratio with Q0
Ra
V2
= acc
Q0
ωU

(5.1.59)

is independent of the size of the cavity and the resonant frequency, only determined by thecavity shape. Substituting Eq. (5.1.51) into Eq. (5.1.58) gives*
Ra /Q0 *Eacc =
Q0 · Pc .
(5.1.60)d*This Ra /Q0 /d can be calculated by the electromagnetic simulation software Superfish .
Furthermore, the relation of
Eap ∝ Eacc

(5.1.61)is established, which leads to
Eap = A*Q 0 · Pc ,where this coeﬃcient A can be calculated by Superfish as well.

53(5.1.62)5.1.4

Design of Two-mode Cavity

In this subsection, the design of the two-mode cavity is described. The requirements for thedesign are the following;
1. The Q0 of both the TM010 mode and the TM020 mode are almost the same value.
2. The ratio of the resonant frequency of the TM010 mode and the TM020 mode can reach
2.0000 using frequency tuners.
Conventionally, the higher-order resonant modes called HOM are unfavorable since they
disturb accelerating beams. However, for our two-mode cavity, the TM020 mode, the secondharmonic resonant mode, is necessary in order to suppress the energy spread of the beam.
Therefore, the quality factor Q0 which indicates the power eﬃciency to resonate the cavity
has to be high for the TM020 mode as the fundamental TM010 mode. Additionally, in order toaccelerate the electron bunch with 1.3 GHz, we have to set f020 , the resonant frequency of theTM020 mode just twice as much as f010 , the resonant frequency of the TM010 mode. However,in the process of the manufacturing the two-mode cavity, there must be some errors for the
resonant frequencies. In order to correct these, we have two frequency turners as describedin Sec. 3.1.1.
Superfish employs the finite diﬀerence method with non-uniform triangular meshes[27].
Using this, the two-mode cavity which satisfies the above requirements is designed. Figure
5.3 shows the section views and the electric field distributions of the designed two-mode
cavity. Performance indices are listed in Table 5.4. These parameters are calculated by thecondition that β = 1, accelerating length d = 1m, Eacc = 1 MV/m and the temperature
is 4.4 K. The temperature of 4.4 K is a little higher than the temperature of liquid helium(4.2 K), and this situation is realized when cooling the cavity with higher pressure than theatmosphere.

54

(a) The TM010 mode(b) The TM020 mode

Figure 5.3: The section views and the electric field distributions of the two-mode cavity.

55

5.2

Fabrication of Two-mode Cavity

We manufactured the two-mode cavity of Figure 5.4 with the process as in Table 5.5. The
main body of the cavity is made of seven parts described in Figure 5.4 - 5.5. The original
niobium parts were processed in the form of ingots and sheets by the suppliers. After thedelivery of materials, all fabrication processes were conducted in High Energy Accelerator
Research Organization (KEK). In the following, a detailed explanation for each process isgiven.

5.2.1

Fabrication of Niobium Material

The cavity is made of pure niobium. The niobium comes from Brazil and Canada where
niobium oxide is included in minerals. These minerals are crushed to powder, then melted
in a vacuum furnace by electron beams in order to increase the purity of niobium. In thisprocess, the cooled product called ingot includes high-purity niobium. Niobium sheets
are produced by pressing and annealing the ingots. Niobium cavities are generally made ofsheets. Our two-mode cavity is composed of ingots and sheets. Ingot shaving is conductedby a lathe, and the sheet is machined by the spinning process. Niobium sheets are fitted ona rotating iron mold, and deformed by a brass rod. Figure 5.5 shows all niobium parts that
configures the body of the two-mode cavity.

Figure 5.4: The overall design of the two-mode cavity.
56

Table 5.5: The fabrication process of the two-mode cavity.
Fabrication of Element from Nb Ingot and Sheet⇓
Fabrication of Half-Cell by Electron Beam Welding (EBW)⇓
Grind Polishing inside Half-Cell⇓
Electric Polishing (EP)⇓EBW on Equator of Cell⇓
Annealing⇓
Chemical Polishing (CP)⇓High-pressure Washing⇓
Assembling⇓
Vacuuming⇓
Performance Test

57

(a) Half-cell 1.

(b) Half-cell 2.

(c) Beam-pipe 1.

(d) Beam-pipe 2.

(e) Beam-pipe 3.

(f) Port 1.

(g) Port 2.

Figure 5.5: The elements for the two-mode cavity fabricated from Nb ingots and sheets.

58

5.2.2

Electron Beam Welding

Generally, superconducting cavities are welded using the technique of electron beam welding(EBW) . EBW is performed in a vacuum furnace, where it is less likely for the welding
point to be contaminated. Further, the penetration depth of EBW is large, thus it can weld
thick material easily.
When using the technique of EBW, the following conditions have to be considered;
1. Electron beam current [mA]
2. Accelerating voltage of the electron beam [kV]
3. Feed rate of the welding point against the electron beam [mm/s]
4. Distance between the welding point and the electron gun (work distance) [mm]5. Focal length of the EBW machine [mm].
It is important to find out the welding condition that leaves the clearest trace. However, it ismore essential to find out the most stable condition, because there can be some deviations,for example, due to the non-uniformity of the cavity, the instability of the EBW machines.
It is almost impossible to evaluate these deviations. On the other hand, it is possible to findout the suﬃciently stable conditions.
To find out the stable conditions, we conducted the following;
1. Beam profile measurement with a comb-shaped sample
2. Plate EBW
3. Sample EBW.
We carried out the above steps for each welding point.
Beam Profile Measurement With Comb-shaped Sample
It is important to know the beam size at the welding point in order to find out the stableconditions. However this is almost impossible to measure this directly. Instead, we did abeam profile measurement with a comb-shaped sample to know the beam size on the welding
point indirectly. This comb-shaped sample is a long plate with a regular-pattern asperities
resembling a washboard. Installed at 45 deg. angle with respect to the beam, this sample
was sent in parallel to the gradient direction. The beam size was estimated by evaluating thewelded traces. This enables us to determine the relationship between the bead width andthe work distance at some focal length (Figure 5.6, 5.7).

59

Figure 5.6: The setup for the comb measurement.
Figure 5.7: The weld bead made by the comb
measurement.

Plate EBW
The appropriate conditions of EBW are highly dependent on the thickness of the material.
Therefore, using the plate sample with the same thickness as the cavity, it is possible tofind a suitable condition to observe the welded beads. It is only a bead width that can beevaluate quantitatively. Other check points are; the presence of the back bead, the deepnessof the depression of the bead, the uniformity of the bead surface. However it is diﬃcult toevaluate these points quantitatively, then the observation depends on the subjective view ofthe observer. Figure 5.8 shows the result of a plate EBW test with several conditions for abeam current and a feed rate. Other conditions are; acceleration voltage of 140 kV, focal
length 635 mm, work distance 270 mm.
Sample EBW
A sample EBW welds two materials on the boundary. It is preferable for the sample tohave the work distance and the volume same as the real EBW. Figure 5.9 shows the tested
material for a beam-pipe and a half-cell. After confirming that the condition is appropriate,
the real EBW is conducted with the same current or higher current in the stable range. Thisis because the sample’s volume is generally smaller than the real niobium materials.
Real Production of EBW
The real production of EBW was conducted in the process of Table 5.6. Each production
EBW was conducted under the condition that was acquired by the above three steps.

60(a) The front bead.

(b) The back bead.

Figure 5.8: An example of the plane EBW.

Figure 5.9: An example of the sample EBW.

Figure 5.10: EBW for an ingot Material.

Table 5.6: The real production of EBW.
EBW Beam Pipe with Port⇓EBW Half-cell with Beam Pipe⇓
Grind Polishing⇓
EBW on Cell Equator

Ingot Material
General superconducting cavities uses 2.8 mm-thickness sheets for cavity walls. On the other
hand, our two-mode cavity adopts the ingot material for the cavity wall in order to ensure
61the strength enough to withstand the deforming of the tuning. Compared with sheets, thegrain boundaries is much larger in ingot materials, and this may change the appropriate
conditions for EBW. In order to investigate this eﬀect, we conducted a EBW test with an
ingot material. As a result, the bead is almost the same pattern as plates, and no conspicuous
problem occurred at grain boundaries (Figure 5.10). Therefore, we treated ingots as well as
plates with the same appropriate conditions.
EBW on Equator of Cavity
The last EBW is for the equator of the two-mode cavity to bring together two half-cells.
However, The back bead is inside the cavity, then it cannot be evaluated whether the welding
is successful or not outside the cavity. KEK has an observation system to watch inside
cavities called Kyoto Camera , however this was developed only for TESLA-like cavities,
then cannot be utilized for our two-mode cavity due to the cavity’s narrow beam aperture.
Therefore, a simple observation system was set up (Figure 5.11). In this system, a mirror isinstalled inside the cavity, and a telecamera was used to observe the back bead reflected on
the mirror. Figure 5.12 shows the picture of the back bead, acquired by the system. With
this picture, we decided that the back bead was observed, and the conducted EBW was
successful. Based on these decision using this system, we conducted five discontinuous EBW
at the equator.

Figure 5.11: The observation system for the Figure 5.12: The observed back bead at theequator of the two-mode cavity.
equator.

5.2.2.1

Surface Polishing

It is desirable that there are less impurities and irregularities as much as possible on thesurface of the cavity inside. If there are much impurities, these make the cavity’s resistancevalue large, then the destruction of superconducting state would occur locally. Additionally,62

an electric field at irregularities gets larger, and emits field emission electrons, then this leads
to the destruction of the superconducting state. Furthermore, much irregularities means anincrease of the area, then this leads to the large resistance value as a whole cavity. Therefore,
we tried to manufacture the better surface state to perform the following three processes.
Grind Polishing
Figure 5.13 shows the back bead at the boundary between a half-cell and a beam-pipe. In
order to polish mm-order irregularities like this picture, we did a grind polishing. This was
conducted by a tool called leutor whose tip rotates at a high speed. On its tip, a carbide
diamond cutter or grindstone are mounted (Figure 5.14). Half-cells are axial-symmetric.
Then we set these on the electric pottery wheel, then polished stably by rotating it (Figure5.15).

Figure 5.14: The Leutor with Figure 5.15: The half-cell on
Figure 5.13: The back bead on
a carbide blade and a grind an electric pottery wheel after
the half-cell after the EBW.
the grind polishing.
stone.

Electro Polishing
Electro polishing (EP) is a technique where the object niobium is set as an anode, an aluminum rod is set as a cathode, and a container is filled with EP solution, then some voltage
is applied (Figure 5.16). Our EP solution is composed of sulfuric acid (concentration: 98 %)and hydrofluoric acid (46 - 48 %), and the volume ratio is 9:1. The chemical reactions are
the following three steps[28];+2−−2Nb + 5SO2−
4 + 5H2 O −→ Nb2 O5 + 10H + 5SO4 + 10e%&
13
Nb2 O5 + 6HF −→ H2 NbOF5 + NbO2 F ·H2 O + H2 O
2
2
%
&
1
3
NbO2 F ·H2 O + 4HF −→ H2 NbOF5 + H2 O.
2
2

63(5.2.1)(5.2.2)(5.2.3)(5.2.4)

The first step is an oxidation of niobium. The second and the third steps are dissolution of
niobium oxide. The reaction rate of EP can be adjust by changing the applied voltage, andthe polishing amount can be calculated by the integrated current. The polishing proceeds
in proportion to the local strength of electric fields[29]. It is impossible to polish the surface
uniformly, and the calculated polishing amount is only an approximate value. The reaction
rate is strongly dependent on the temperature of the EP solution. The temperature increaseof the EP solution is suppressed by a chiller.
Before the real EP, we conducted EP tests in order to determine the appropriate current
density. This time, 20 mA/cm2 and 50 mA/cm2 were tested . The total polished amount wasestimated to be 52 µm and 40 µm respectively by comparing the mass of the sample before
and after the test. Figure 5.17 shows the tested sample. As can be seen, while the amount ofpolishing is the same level, the surface state are quite diﬀerent. In order to observe in moredetail, we utilized a laser microscope (Figure 5.18). Using its function, the average roughness
was measured 0.178 µm at 20 mA/cm2 and 0.161 µm at 50 mA/cm2 . We chose the currentof 50 mA/cm2 .
In the real EP, we aimed the 70 µm as the polishing amount. The polishing amount wasmeasured by the integrated current in real time. The EP solution was kept at 40 ◦ C by thechiller. Figure 5.19 shows the half-cell after 70 µm EP. As can be seen, the surface was likea mirror, and the side of the beam aperture was recessed slightly. This is the trace of thegrind polishing, and the concentration of the electric field never occur owing to its smoothsurface.

Figure 5.17: The test sample of EP.

Figure 5.16: The setup of EP.

Chemical Polishing
Chemical polishing (CP) utilizes the reaction of niobium with CP solution[30]. CP solution
is composed of nitric acid (65 - 66 %), hydrofluoric acid (46 - 48%) and phosphoric acid (8564(a) The surface condition of 20 mA/cm2 .

(b) The surface condition of 50 mA/cm2 .

(c) The 3D surface condition of 20 mA/cm2 . (d) The 3D surface condition of 50 mA/cm2 .
Figure 5.18: The observation of the EP test sample by a laser microscope.

%), and the volume ratio is 1: 1: 2 this time. The reaction proceeds as
4Nb + 10HNO3 −→ 2Nb2 O5 + 10NO2 + 5H2%&
1
3
Nb2 O5 + 6HF −→ H2 NbOF5 + NbO2 F ·H2 O + H2 O
2
2
%
&
1
3
NbO2 F ·H2 O + 4HF −→ H2 NbOF5 + H2 O.
2
2(5.2.5)(5.2.6)(5.2.7)(5.2.8)The first reaction is the oxidation of niobium by nitric acid, and the second and the third
reactions are the dissolution of niobium by hydrofluoric. The rate of CP reactions is much
faster than EP, and increased exponentially depending on the solution temperature as is thecase of EP. This saves time of polishing, however the control is diﬃcult. In order to control
the CP rate easily, we adjusted the amount of phosphoric acid which does not contribute to
the reactions.
Another feature of CP is the non-uniform polishing due to the diﬀerent rate at each grain
65boundaries, and this makes the etchings of the grain boundaries. Our CP process was after
the EP process in order to polish the equator of the cell, however the surface state worsened
a little at other areas.
This time, we aimed 40 µm CP polishing. As the volume rate was set as 1:1:2, thereaction rate was slow. Figure 5.20 shows the set up for CP. Ports of the cavity were closed
and a valve was installed at the bottom beam-pipe. CP process was carried out for threeminutes after closing this valve, filled with CP solution in the cavity, and functioning themixer. After this, the valve was opened to dump the CP solution, then the inside of thecavity was cleaned with pure water. The polishing of 40 µm was conducted repeating thisprocess four times. After CP, we conducted an ultrasonic cleaning for thirty minutes with acleaning solution called FM-20 filled in the cavity and installing the cavity in the 50 ◦ Cbathtub in order to remove niobium oxide (Figure 5.21). After this, we washed the cavity
with pure water for about fifteen minutes, closed the beam-pipes with aluminum foils, thencarried the cavity to a cleanroom. The real polishing amount was estimated as 33 µm by theamount of the niobium dissolved in the CP solution.

Figure 5.19: The half-cell after the EP process.
66

Figure 5.20: The setup of CP.

Figure 5.21: The ultrasonic cleaning after the CP process.

675.2.2.2

Annealing

Annealing is to heat the cavity installed in a vacuum furnace in order to release the stressin the cavity which was made in the process of EBW, and encourage the degasifying on thecavity surface. This time, we annealed the cavity at 800 ◦ C for three hours. Figure 5.22
shows the cavity installed in the vacuum furnace. The cavity was set in a stainless jig inorder to avoid nun-uniform heating by direct radiation by heaters. Figure 5.23 shows the
manufactured two-mode cavity after the annealing process.

Figure 5.22: The two-mode cavity installed in Figure 5.23: The manufactured two-mode cavity after the annealing.
a vacuum furnace.

5.2.2.3

High-pressure Washing

We conducted a high-pressure washing in order to remove impurities on the cavity made in
the process of EBW and polishings. This was conducted in a class 10 clean room1 After thispoint, the inside of the cavity never touch the ambient air.
In order to wash the cavity uniformly, it was placed at a seat, and a bar was installed
through the bottom beam-pipe (Figure 5.24). This bar has two holes with a diameter of 1
mm, located at 180 degrees opposite sides. Passing through these holes, high-pressure ultrapure water was ejected with 3 - 5 MPa pressure2 . By rotating the seat, and lowering the bar,
we washed all inside area of the cavity. We wash the cavity for thirty minutes, and then, we
assembled the cavity with jigs and flanges.

1

Class 10 is a standard stipulated by the United States Federal Standard FS209, and this indicates theenvironment where there are less than 10 particles whose size is 0.5 µm or more in 1 ft3 . Now, FS209 is deadand replaced with JISB9920 compatible with international standards, and class 4 in JISB9920 responds toclass 10 in FS209 [31].
2
The purity of this water was maintained by the method using electric conductivity[32].

685.2.2.4

Assembling

Impurities in the cavity cause residual resistance and become sources of field emission electrons that works as obstacles to put high power in the cavity. In order to shut out these,
installing of jigs and flanges were carried out in the cleanroom, and the following processesof vacuuming and performance tests were conducted sealing the cavity. The cavity has two
ports where each power antenna used for performance tests was installed. The cavity has two
beam-pipe apertures, and the bottom one was closed out, and the vacuuming was carriedout through the top one. Additionally, the frequency tuners were set in this step.
As the cavity is cooled to the temperature of liquid helium (4.2 K), the niobium bodyand other stainless jigs shrinks. The coeﬃcient of linear thermal expansion of niobium andstainless are diﬀerent 3 . In order to avoid the vacuum leak due to this diﬀerence of theshrink rate, we utilized an indium ribbon as a seal at the flange instead of a gasket (Figure5.25). This indium ribbon is much flexible than usual gaskets, then reduces the probabilityof the leak. After assembling the cavity, it was vacuumed. The final vacuum level before theperformance test was 2 × 10−6 Pa.

Figure 5.24: The seat for the high-pressure
Figure 5.25: The indium ribbon for the sealing.
washing.

3

Linear expansion coeﬃcient of niobium at 293 K is 7.3−6 K −1 , and stainless steel is 14.7−6 K −1 [22][33].

69

5.3

Performance Test of Two-mode Cavity

In this section, the performance test of the manufactured two-mode cavity is described.
Performance tests for superconducting cavities are called vertical tests where the cavities
are settled in the 4.2 K (liquid He temperature) or 2 K (decompression) cryostat vertically,
and cooled to be superconducting state. In this situation, the input power Pin , the reflection
power Pr and the transmitted power Pt are measured to estimate the quality factor Q0 andthe axial accelerating field Eap 4 .
The cavity and the measurement system can be modeled as a lumped-constant circuit. A
detailed description is given in Appendix B. Using that result, we estimated the performanceindices.

5.3.1

Setup

The measurement system consists of the cooling system and the RF analog circuit (Figure5.26). The two-mode cavity is cooled in the 4.2 K cryostat filled with liquid helium. Thisis surrounded by two heat-insulating layers. The inner one has a liquid nitrogen circulation
system. The cavity hangs from the flange of the He layer, and has two ports where the RF
powers come and go. These RF power Pin , Pr and Pt are detected at the RF analog circuit bypower sensors and monitors. The signals attenuate during propagating in cables. Therefore
power values have to be corrected. This is called cable correction . The correction is madeusing correction factors
Pin = Cin Pin′
Pr = Cr Pr′
Pt = Ct Pt′ ,(5.3.1)(5.3.2)(5.3.3)where the power value with the prime symbol indicates the attenuated one measured at the
RF analog circuit, and the bare one indicates the estimated power value at the port of thetwo-mode cavity. C indicates each correction factor.
The RF analog circuit is composed of several elements. In the following, a brief explanations is made for each component.
Signal Generator
This is the source of the RF signal, and determine the RF frequency. In vertical tests, thisfrequency can be changed in order to match the resonant frequency of the two-mode cavity.
This signal generator receives the signal made at the mixer explained later. The amplitudeof this signal is proportional to the frequency diﬀerence between the two-mode cavity andthe signal generator.
4

Generally, Eacc is calculated to discuss the cavity performance. However this is eﬀective for β = 1, thenwe deal with Eap which is the simple physical quantity.

70

Frequency Counter
This measures the frequency of the output of the signal generator. Our frequency generator
HP 5350B has a single digit accuracy[34].
Linear RF Modulator
This can control the input RF power with an arbitrary factor from 0.000 to 10.000.
RF Switch
This receives the signal which comes from a function generator, and output the product of
the RF input from the linear RF moderator and the signal from the function generator. The
received signal levels is TTL. When measuring the loaded quality factor QL of the two-mode
cavity, square waves become the input and pulsed RF goes to the two-mode cavity. Whenmeasuring Pin , Pr and Pt , 0 V becomes the input as logic 0. In the case of high voltage aging
for discharge problems, the pulsed mode is utilized.

Figure 5.26: The setup for the vertical test.
71

Directional Coupler
This can pick up the signal power of both the input RF Pin (going to the cavity) and thereflection RF Pr (coming from the cavity). The coupling constant is about 10 dB (1/10).
Mixer
As described in the explanation of the signal generator , this output the signal whose
amplitude is proportional to the frequency diﬀerence between the two-mode cavity and thesignal generator. This signal goes to the signal generator, then the new frequency around 1.3
GHz goes around the system.

5.3.2

Measurement Scheme

The quality factor Q0 and the axial-peak accelerating field Eacc are estimated in the following
steps;
1. Cable correction
2. QL measurement
3. Power measurement.
5.3.2.1

Cable correction

As explained in the previous subsection, in order to estimate Q0 and Eacc , the RF valuesof Pin , Pr and Pt at the port of the two-mode cavity are needed. However it is impossible
to measure these directly, then using Pin′ , Pr′ and Pt′ measured at the RF analog circuit, weestimate the real Pin , Pr and Pt . Figure 5.27 shows an example of the cable correction. In
1 to ⃝.
2
this case, we would like to know the attenuation of the path from ⃝First i), withan output signal from the signal generator, the power loss of the measurement system of
the left side from the circulator is measured. Second ii), removing the reflection board, the
1 and ⃝
2 is included. It is noted that during the cable correction, it
target path between ⃝is necessary to set the RF frequency detuned which means displaced from the resonant
frequency of the two-mode cavity so that the RF signal cannot come into the cavity. In thiscase, the cable loss is calculated as
10.44 dBm − 12.95 dBm = −2.51 dB= 10−2.51/10
1
≃
.
1.78
Here the unit of dBm is the absolute unit calculated asx dBm = 10x/10 mW(5.3.4)(5.3.5)(5.3.6)(5.3.7)

For all paths to monitor Pin , Pr and Pt , cable corrections have to be conducted in the same
scheme.
72

5.3.2.2

QL measurement

As explained in Sec. 5.1.3, the accelerating eﬃciency is expressed by the quality factor Q0 as
Q0 =

ω0 U
.
Pc

(5.3.8)In practice, there are other power losses; dissipating power at the input port Pe and transmitted power though the cavity Pt . Then the total dissipating power Ptot is calculated as
Ptot = Pc + Pe + Pt .

(5.3.9)And a new parameter called the loaded quality factor QL is defined as
QL =

ω0 U
.
Ptot

(5.3.10)

In a similar way, Qin and Qt is defined as
ω0 U
Pe
ω0 U
Qt =
.
Pt

Qin =

(5.3.11)(5.3.12)Figure 5.27: An example of cable correction.
C indicates a circulator which has threeports, and only permits the counterclockwise path of the signal. Reflection indicates the
RF reflection board where all power are reflected.
73

Using these parameters, QL is described as
1
Ptot
=
QL
ω0 U
Pc + Pe + Pt
=
ω0 U
1
1
1
=
+
+
Q0 Qin Qt
%
&
1
Q0
Q0
=
1+
+
Q0
Qin Qt1=(1 + βin + βt )Q01=
(1 + β ∗ ) ,Q0(5.3.13)(5.3.14)(5.3.15)(5.3.16)(5.3.17)(5.3.18)

where the new parameters βin and βt indicates the coupling strength of the power input
coupler and output coupler respectively. β ∗ can be estimated using the values of Pin and Pr
as described in Appendix B.
When the pulsed input power Pin is given into the cavity for a time enough to saturate
the stored energy U , the transmitted power Pt decays with the rate asPt =4β ∗ βtPin e−ω0 (t−toﬀ )/QL ,(1 + β ∗ )2

(5.3.19)

where t indicates the elapsed time from Pin is turned on, and toﬀ indicates the time Pin isturned oﬀ (Figure 5.28). The decay time t − toﬀ = τL , elapsing to reach a half value of thepeak satisfies1e−ω0 τL /QL = .
2(5.3.20)

QL ≃ 0.6931 × τL × ω0 .

(5.3.21)

Then, QL is calculated as

Now we measure this τL , and already know the resonant angular frequency of the two-mode
cavity ω0 , then we can estimate the loaded quality factor QL .
Additionally, seeing the pattern of Pr , the coupling strength of β ∗ can be determined asover-coupling (β ∗ ≥ 1) or under-coupling (β ∗ < 1) (Figure 5.28). β ∗ is estimated by)
1 ± PPinr) ,β∗ =(5.3.22)1 ∓ PPinr

where the upper double-sign corresponds is chosen for over-coupling and the down one ischosen for under-coupling. The values of Pin and Pr used to estimate β ∗ are measured in thenext step power measurement .
74

5.3.2.3

Power measurement

Now we know the correction factors of cable correction and QL and whether β ∗ is overcoupling or under-coupling. Here, the ultimate goal to estimate Q0 and Eap is accomplished.
First, using the corrected values of Pin , Pr and Eq. (5.3.22), (5.3.18), Q0 is estimated.
Next, replacing Q0 with Qt in Eq. (5.1.60),*
Ra /Qt*Eacc =Q t Pt(5.3.23)d
is acquired. As described Sec. 5.1.3, there is a relation between Eacc and Eap as
Eap ∝ Eacc ,(5.3.24)*
Eap = A Qt Pt ,(5.3.25)

then Eap is expressed aswhere A can be estimated by the numerical simulation and the values for the two-mode cavity
is listed in Table. 5.4.

Figure 5.28: The schematic diagram of Pin , Pr and Pt .
75

5.3.3ResultHere the latest result is described. Figure 5.29 shows the result of Eap and Q0 . The errors are
due to the uncertainty of the power meter and the oscilloscope utilized in our measurement.
Table 5.7 summarizes the highest achievement of Eap , and Q0 at that time when the highest
Eap was recorded. The measured Q0 for both resonant modes are about a half of the simulated
values listed in Table. 5.4 respectively. The practical values of Q0 are degenerated to someextent for the residual resistance and the outgassing.
The ratio of the resonant frequency f020 /f010 was measured as 1.9964. In order to satisfy
Table 5.7: The measured performance of the two-mode cavity. Eap indicates the highest
achievement and Q0 is the one estimated when the highest Eap was recorded.
TM010
TM020
Frequency [GHz]1.30024902
2.59587465
Q0(1.09 ± 0.06) × 108 (4.8 ± 0.3) × 107
Eap [MV/m]
11.0 ± 0.3
7.8 ± 0.3

Q0

109

TM010
TM020
108

0

2

4

6

8

10
12
E ap [MV/m]

Figure 5.29: The latest result of the vertical test for the two-mode cavity.
76

all requirements for manufacturing the two-mode cavity raised in Sec. 5.1.4, we conducted
the tuning test (next section).

5.4

Frequency Tuning Test of Two-mode Cavity

As described in Sec. 3.1.1, our two-mode cavity has two frequency tuners. Figure 5.30
indicates the tunable range of the frequency ratio of the manufactured two-mode cavity. The
blue area indicates the tunable range in the case of the use of the span tuner, and the green
one indicates in the case of the use of the piezo tuner. The red dotted line indicates the ratioof the manufactured two-mode cavity. This figure tells that we cannot reach 2.0000 in thistwo-mode cavity with the existing tuners.
In order to establish a new tuning process to realize 2.0000 as the frequency ratio, wemanufactured a model cavity made of copper (Figure 5.31). This has the same shape as theniobium two-mode cavity. Firstly, the frequency ratio of this model cavity was 1.9949, whichmeans it was impossible to realize 2.0000 with the existing tuners.
Thus, we introduced a new tuning process; shaving the inside of the model cavity (Figure5.32). We shaved a point where the electric field of the TM020 mode is relatively stronger
than that of the TM010 mode. Then, f020 can be altered largely, and the ratio can be changed.
Table 5.8 summarizes the transition of the frequency and the ratio.
As the result, the ratio became 1.9976 that is inside of the tunable range of the niobium
two-mode cavity. After the shaving process, we conducted the tuning test using the span
tuner. Figure 5.33 shows the variation of the frequency ratio against the span tuner setting.
In this test, we realized the frequency ratio of 2.0000. With this result, all requirements
for manufacturing the two-mode cavity, described in Sec. 5.1.4, were achieved. When we
manufacture the next niobium two-mode cavity, the above shaving process will be employed.

Figure 5.30: The tunable range of the frequency ratio of the manufactured two-mode cavity
and its tuners.
77

Table 5.8: The transition of the frequency ratio before and after the shaving process.
f010 [GHz] f020 [GHz] Ratio
Before shaving
1.29652.5864
1.9949After shaving
1.29502.58691.9976(a) The inside of the model cavity.

(b) The overview of the model cavity.

Figure 5.31: The model cavity made of copper.

78

Figure 5.32: Shaving the inside of the model cavity. The width is 20.0 mm, and the depth is
1.0 mm.

Figure 5.33: The variation of the frequency ratio against the span tuner.
79

80Chapter 6
Development of Feedback Control
System for Two-mode Cavity
This chapter describes the development of the feedback system for the two-mode cavity.

6.1

Source of DisturbanceIn Appendix B, superconducting cavities are modeled by a lumped-constant circuit. Here,the important equations are shown again;√
2 β∗ *1ω − ω0 iωt
Vc ≃Ra Pine(6.1.1)∗
1+β
1 + 2iQL ω0√2 β∗ *
1=
Ra Pineiωt(6.1.2)∗1+β
1 − i tan θ√
2 β∗ *=Ra Pin cos θei(ωt+θ)(6.1.3)
1 + β∗
ω − ω0tan θ = −2QL
(6.1.4)ω0
4β ∗ βt cos2 θPt =Pin(6.1.5)(1 + β ∗ )2
where θ is called as(6.1.3) and (6.1.5),

tuning angle , and Vc indicates the accelerating voltage. Using Eq.

Vc ≃

$

Ra* i(ωt+θ)Pt e
βt(6.1.6)is derived. Watching Eq. (6.1.3), it can be seen that the disturbance of Pin propagates to Vc .
Additionally, if the deference between the input RF frequency ω and the resonant frequency
ω0 fluctuate Vc too. In order to stabilize Vc , the transmitted power Pt can be used to monitor
and control. Our feedback system was developed, based on the system of cEEL accelerator
which monitors this Pt .
81

6.2

Setup of Feedback System For Two-mode Cavity

This section describes the configuration of our feedback system. The software part implemented on FPGA circuits is based on cERL accelerator. However, in our two-mode cavity
both the TM010 mode and the TM020 mode have to be controlled at the same time. The
configuration of the analog circuit is completely diﬀerent from that of cERL accelerator. The
schematic circuit for 1.3 GHz is shown in Figure 6.1. The configuration of 2.6 GHz is thesame as the 1.3 GHz system. In the following, each component is explained.
Signal Generator
The signal generator plays an important role in the functioning of the circuit. It supplies
the 1.3 GHz signal to every circuit element so that it is called the master oscillator .
In the vertical tests, its frequency is adjustable to match the two-mode cavity’s resonant
frequency. On the other hand, the frequency is fixed in this feedback test. Therefore there
occurs a diﬀerence between the frequency of the master oscillator and the resonant frequency
which reduce the generated field according to Eq. (6.1.3) - (6.1.6). This is compensated bycontrolling the input power using FPGA based feedback circuit.
Frequency Divider
The frequency divider is used to step-down the input frequency by a pre-determined factor.
In this test, 1.3 GHz is divided to generate 81.25 MHz (1/16) and 10.15625 MHz (1/128).
The former is the clock frequency for the FPGA circuit and it is also the sampling frequency
for the RF signal which will be mentioned in detail later. The latter is used to make the1.31 GHz signal (more precisely, 1.31015625 MHz) which is called the ”local oscillator (LO)”.
This is utilized to down-convert the RF signal of 1.3 GHz into 10 MHz signal which is calledthe ”intermediate frequency (IF)”1 .
LO Signal Generator
This receives two signals; one is the 1.3 GHz master oscillators and the other is the 10 MHz
signal which comes from the frequency divider. Using the two signals, the signal of 1.31
GHz is generated, whereas parasitic frequencies are also generated at the same time, forexample 1.32 GHz and 1.29 GHz. A band pass filter is used to filter the signals from thedown-converter in order to pass only the 1.31 GHz signal. In the case of the TM020 mode
signal processing, the center frequency of the band pass filter is 2.61 GHz.

1

More strictly this is 10.15625 MHz, however the notation of 10 MHz is utilized for the convenience.

82

Figure 6.1: The feedback system developed for the two-mode cavity.
83Mixer
The mixer receives two inputs; one is the pickup signal of 1.3 GHz which comes from thetwo-mode cavity and the other is the LO of 1.31 GHz. In the mixer, two signals are mixed,
i.e. multiplied mathematically to generate the intermediate frequency (IF) signal;
SLO × SRF = ALO sin(2πfLO t + φLO ) × ARF sin(2πfRF t + φRF ),(6.2.1)

where ALO is the amplitude of the LO signal, fLO is the frequency of the LO, which is 1.31
GHz. φRF is the phase of the LO signals. The notations for the 1.3 GHz RF signals aredefined similarly. In the mixer, ALO is set high enough, that is ALO is saturated. This
enables the IF amplitude AIF to be proportional to ARF . Therefore the IF signal is generated
by using sinusoidal product-sum formulae as
SLO × SRF = AIF cos(2πfIF t + φIF )AIF = aARF
fIF = fLO − fRF = 1.31 GHz − 1.30 GHz = 10 MHz
φIF = φLO − φRF ,(6.2.2)(6.2.3)(6.2.4)(6.2.5)where a is a proportional coeﬃcient to convert ARF to AIF , and is assumed to be a constant.
The signal of the frequency fLO + fRF , generated as a by-product, is cut oﬀ by the filter
equipped with the mixer. Current FPGAs cannot process high frequency signal of O(1 GHz).
Therefore it is the common practice to convert a GHz signal to a MHz signal.
FPGA
This is the key for the feedback. It receives only one signal as an input; down-converted IF
signal SIF = AIF sin(2πfIF t+φIF ). Now we would like to measure both the amplitude AIF andthe phase φIF , however we can only measure the value of the magnitude of SIF at the timet. Therefore the sampling frequency fs plays an active role. Here the idea of IQ sampling is
introduced, based on [35]. Now we would like to know A and φ. For this we transform themto I and Q as;
I = A cos φ
Q = A sin φ*
A = I 2 + Q2
% &Q
φ = atan.
I(6.2.6)(6.2.7)(6.2.8)(6.2.9)Figure 6.2 shows the relation between A, φ and I, Q. Now what we can do with one ADC isto measure xt , the sampled IF signal SIF at the time t. The data is acquired at 80 MHz as
mentioned before. Therefore the sampling constant m is calculated as
80 MHz10 MHz
= 8.

m=

84

(6.2.10)(6.2.11)

This situation is called Oversampling , therefore the acquisition phase resolution ∆φ is
360 deg./8 = 45 deg.. The value of x is expressed by using I and Q as
x1 = I cos(∆φ) + Q sin(∆φ).

(6.2.12)x2 = I cos(2∆φ) + Q sin(2∆φ)...
xn = I cos(n∆φ) + Q sin(n∆φ).

(6.2.13)(6.2.14)(6.2.15)

Then n samples are taken,

using the notation of a1 , ..., an and b1 , ..., bn instead of cos(∆φ), ..., cos(n∆φ) and sin(∆φ), ...,..., sin(n∆φ) thena1 I + b 1 Q − x 1 = 0...
an I + bn Q − xn = 0.

(6.2.16)(6.2.17)(6.2.18)

Multiplying ai (or bi ) and having sums from i to n, then
n
+

I
I

a2i + Q

i=1
n
+
i=1

n
+

ai b i −i=1n+ai bi + Q

i=1

b2i −

n
+

a i xi = 0(6.2.19)bi xi = 0.

(6.2.20)

i=1

n
+
i=1

Figure 6.2: The definition of IQ.
85

Here new notations are introduced as
s1 =

n
+

xi b i(6.2.21)xi a i(6.2.22)a2i(6.2.23)

ai bi(6.2.24)b2i(6.2.25)

α11 I + α12 Q = s1
α12 I + α22 Q = s2 .

(6.2.26)(6.2.27)

s2 =

i=1
n
+
i=1

α11 =
α12 = α21 =
α22 =

n
+
i=1
n
+
i=1
n
+
i=1

then

Now n = 8 and ∆φ = 45 deg. thenn+sin(i∆φ) cos(i∆φ) = α12 = α21 = 0(6.2.28)

i=1

n
+

i=1
n
+

sin2 (i∆φ) = α22 =n=42(6.2.29)

cos2 (i∆φ) = α11 =n=42(6.2.30)i=1

are establish. Using these equations with Eq. (6.2.26), (6.2.27),
2s1
n
2s2
Q=
n
I=

(6.2.31)(6.2.32)

are derived. Substituting Eq. (6.2.21), (6.2.22) into the above equations,n2+I=xi sin(i∆φ)
n i=1(6.2.33)

Q=(6.2.34)n2+
xi cos(i∆φ)
n i=1
86

are acquired. From these equations, I and Q can be calculated using the 8 continuous ADC
values.
After calculating I and Q from ADC, I and Q are compared with the set values. With
its diﬀerence, PI control process is conducted.
P denotes proportional control
which outputs the proportional signal against the input. And I denotes integral control
which outputs the time-integral of the input. These control strength can be adjusted tominimize the fluctuation of I and Q. After the PI control, the new digital signal are processed
at a low pass filter (LPF), then converted into new I and Q. In our FPGA circuit, additional
LPF called monitor filter was installed in front of the PI control in order to reduce the
ADC noise. This cut-oﬀ frequency was set at 5 kHz.

6.3

Perfomance Test

We did the performance tests in two ways;
1. Single feedback
2. Dual feedback
In this order the detailed results will be explained.

6.3.1

Single feedback

Here the feedback system functioned only for the TM020 mode. The resonant frequency
was 2.59547 GHz. First the stability with no feedback (integral control coeﬃcient KI andproportional control coeﬃcient KP equal to zero) was measured (Figure 6.3). Next KI and
KP were increased to see the stabilities (Figure 6.4, 6.5). Table 6.1 summarizes the results ofsingle feedback test, showing the stability of amplitude and phase of the transmitted powerPt .

Table 6.1: The result of the single feedback.
Feedback Constants Stability of Amplitude Stability of Phase
KI=0, KP=0(2.4 ± 0.6) × 104 ppm(21 ± 5) mdeg.
2
KI=0, KP=200(6 ± 1) × 10 ppm(1.4 ± 0.2) mdeg.
KI=2000, KP=200 (2.2 ± 0.4) × 102 ppm (0.52 ± 0.07) mdeg.
87

×10

3

Measurement Time T : 40 ms

Averaged Stability : (45378 ± 6072) ppm

KI=0, KP=0

The Best Stability : 21890 ppm

w/o Monitor Filter

The Worst Stability : 60109 ppm

100

80

Stability of the Amplitude [ppm]

Stability of the Amplitude [ppm]

120

60

40

100

×10

3

90

Measurement Time T : 40 ms

Averaged Stability : (24373 ± 6014) ppm

80

KI=0, KP=0

The Best Stability : 1862 ppm

70

w/ Monitor Filter

The Worst Stability : 53905 ppm

60
50
40
30
20

20

10
0

1

2

3

4

5

6

7

8

9

0

10

1

2

3

4

5

6

7

140
Averaged Stability : (35 ± 8) mdeg.

Measurement Time T : 40 ms

120
100

KI=0, KP=0

The Best Stability : 1 mdeg.

w/o Monitor Filter

The Worst Stability : 67 mdeg.

80
60

8

9

10

Measurement Number

Stability of the Phase [mdeg.]

Stability of the Phase [mdeg.]

Measurement Number

90
80

Averaged Stability : (21 ± 5) mdeg.

Measurement Time T : 40 ms

70
60

KI=0, KP=0

The Best Stability : 2 mdeg.

w/ Monitor Filter

The Worst Stability : 45 mdeg.

50
40
30

40
20
20
01

10
2

3

4

5

6

7

8

9

10

Measurement Number(a) Without monitor filter

01

2

3

4

5

6

7

8

(b) With monitor filter

Figure 6.3: The single feedback for the TM020 mode with KI=0, KP=0.

88

9

10

Measurement Number

Measurement Time T : 40 ms

Averaged Stability : (698 ± 153) ppm

KI=0, KP=200

The Best Stability : 150 ppm

w/o Monitor Filter

The Worst Stability : 1337 ppm

2000

Stability of the Amplitude [ppm]

Stability of the Amplitude [ppm]

2500

1500

1000

500

2000
1800

Measurement Time T : 40 ms

Averaged Stability : (638 ± 100) ppm

1600

KI=0, KP=200

The Best Stability : 260 ppm

1400

w/ Monitor Filter

The Worst Stability : 998 ppm

1200
1000
800
600
400
200

0

1

2

3

4

5

6

7

8

9

0

10

1

2

3

4

5

6

7

7
Averaged Stability : (1.6 ± 0.3) mdeg.

Measurement Time T : 40 ms

6
5

KI=0, KP=200

The Best Stability : 0 mdeg.

w/o Monitor Filter

The Worst Stability : 3.3 mdeg.

4
3
2

9

10

4
3.5

Averaged Stability : (1.4 ± 0.2) mdeg.

Measurement Time T : 40 ms
KI=0, KP=200

The Best Stability : 0.5 mdeg.

w/ Monitor Filter

The Worst Stability : 2.2 mdeg.

3
2.5
2
1.5
1

1
01

8

Measurement Number

Stability of the Phase [mdeg.]

Stability of the Phase [mdeg.]

Measurement Number

0.5
2

3

4

5

6

7

8

9

10

Measurement Number(a) Without monitor filter

01

2

3

4

5

6

7

8

(b) With monitor filter

Figure 6.4: The single feedback for the TM020 mode with KI=0, KP=200.

89

9

10

Measurement Number

900

Measurement Time T : 40 ms

Averaged Stability : (277 ± 62) ppm

800

KI=2000, KP=200

The Best Stability : 87 ppm

700

Stability of the Amplitude [ppm]

Stability of the Amplitude [ppm]1000

The Worst Stability : 578 ppm

w/o Monitor Filter

600
500
400
300

800
Measurement Time T : 40 ms

Averaged Stability : (224 ± 36) ppm

KI=2000, KP=200

The Best Stability : 113 ppm

w/ Monitor Filter

The Worst Stability : 423 ppm

700
600
500
400
300
200

200
100

100
0

1

2

3

4

5

6

7

8

9

0

10

1

2

3

4

5

6

7

2.4
2.2

Averaged Stability : (0.67 ± 0.15) mdeg.

Measurement Time T : 40 ms

2

KI=2000, KP=200

The Best Stability : 0.12 mdeg.

w/o Monitor Filter

The Worst Stability : 1.25 mdeg.

1.8
1.6
1.4
1.2
1
0.8
0.6

9

10

1.6
1.4

Averaged Stability : (0.52 ± 0.07) mdeg.

Measurement Time T : 40 ms
KI=2000, KP=200

The Best Stability : 0.24 mdeg.

w/ Monitor Filter

The Worst Stability : 0.86 mdeg.

1.2
1
0.8
0.6
0.4

0.4

0.2

0.2
01

8

Measurement Number

Stability of the Phase [mdeg.]

Stability of the Phase [mdeg.]

Measurement Number

2

3

4

5

6

7

8

9

10

Measurement Number(a) Without monitor filter

01

2

3

4

5

6

7

8

(b) With monitor filter

Figure 6.5: The single feedback for the TM020 mode with KI=2000, KP=200.

90

9

10

Measurement Number

6.3.2

Dual Feedback

In the dual feedback test, the feedback system functioned for both the TM010 mode and theTM020 mode. KI and KP were set to 5000 and 500 respectively where the deviation of theamplitude and phase were minimized. In this time, the monitor filter and the duration time
for the measurement were changed. Figure 6.8 - 6.7 show each result. Table 6.2 and 6.3
summarize the results of dual feedback tests.
Table 6.2: The result of dual feedback for the TM010 mode.
Measurement Time Monitor Filter Stability of Amplitude Stability of Phase
T =40 ms
w/o
(2.4 ± 0.5) × 102 ppm(0.9 ± 0.2) mdeg.
w/(2 ± 1) × 10 ppm(0.6 ± 0.1) mdeg.
2
T =50 µs
w/o
(3.4 ± 0.9) × 10 ppm(0.5 ± 0.1) mdeg.
w/(2.4 ± 0.6) × 10 ppm (0.04 ± 0.01) mdeg.

Table 6.3: The result of dual feedback for the TM020 mode.
Measurement Time Monitor Filter Stability of Amplitude Stability of Phase
T =40 ms
w/o
(1.0 ± 0.3) × 102 ppm (0.21 ± 0.05) mdeg.
w/(5 ± 2) × 10 ppm(0.06 ± 0.01) mdeg.
T =50 µsw/o
(9 ± 4) × 10 ppm(0.23 ± 0.08) mdeg.
w/(9 ± 8) ppm(0.03 ± 0.01) mdeg.

91

900

Measurement Time T : 40 ms

Averaged Stability : (240 ± 54) ppm

800

KI=5000, KP=500

The Best Stability : 3.69 ppm

700

w/o Monitor Filter

The Worst Stability : 512 ppm

Stability of the Amplitude [ppm]

Stability of the Amplitude [ppm]

1000

600
500
400
300

200
180

Measurement Time T : 40 ms

Averaged Stability : (22 ± 13) ppm

160

KI=5000, KP=500

The Best Stability : 1.35 ppm

140

w/ Monitor Filter

The Worst Stability : 103 ppm

120
100
80
60

200

40

100

20

0

1

2

3

4

5

6

7

8

9

0

10

1

2

3

4

5

6

7

4
3.5

Averaged Stability : (0.9 ± 0.2) mdeg.

Measurement Time T : 40 ms
KI=5000, KP=500

The Best Stability : 0.1 mdeg

w/o Monitor Filter

The Worst Stability : 1.9 mdeg

3
2.5
2
1.5

8

9

10

Measurement Number

Stability of the Phase [mdeg.]

Stability of the Phase [mdeg.]

Measurement Number

2
Averaged Stability : (0.6 ± 0.1) mdeg.

1.8

Measurement Time T : 40 ms

1.6

KI=5000, KP=500

The Best Stability : 0.2 mdeg

w/ Monitor Filter

The Worst Stability : 1.0 mdeg

1.4
1.2
1
0.8
0.6

1
0.4
0.5
01

0.2
2

3

4

5

6

7

8

9

10

Measurement Number(a) Without monitor filter

01

2

3

4

5

6

7

8

9

(b) With monitor filter

Figure 6.6: The dual feedback for the TM010 mode with KI=5000, KP=500 for T = 40 ms.

92

10Measurement Number

1600

Measurement Time T : 50 µs

Averaged Stability : (344 ± 91) ppm

KI=5000, KP=500

The Best Stability : 8.6 ppm

w/o Monitor Filter

The Worst Stability : 890 ppm

1400
1200

Stability of the Amplitude [ppm]

Stability of the Amplitude [ppm]

1800

1000
800
600
400

1

90

Measurement Time T : 50 µs

Averaged Stability : (24 ± 6) ppm

80

KI=5000, KP=500

The Best Stability : 0.04 ppm

70

w/ Monitor Filter

The Worst Stability : 52 ppm

60
50
40
30
20

200
0

100

10
2

3

4

5

6

7

8

9

0

10

1

2

3

4

5

6

7

2.5
Measurement Time T : 50 µs

2

Averaged Stability : (0.5 ± 0.1) mdeg.

KI=5000, KP=500

The Best Stability : 0.04 mdeg

w/o Monitor Filter

The Worst Stability : 1.2 mdeg

1.5

1

8

9

10

Measurement Number

Stability of the Phase [mdeg.]

Stability of the Phase [mdeg.]

Measurement Number

0.2
0.18

Measurement Time T : 50 µs

0.16KI=5000, KP=500

The Best Stability : 0.001 mdeg

w/ Monitor Filter

The Worst Stability : 0.10 mdeg

0.14

Averaged Stability : (0.04 ± 0.01) mdeg.

0.12
0.1
0.08
0.06
0.04

0.5

0.02
01

2

3

4

5

6

7

8

9

10

Measurement Number(a) Without monitor filter

01

2

3

4

5

6

7

8

9

(b) With monitor filter

Figure 6.7: The dual feedback for the TM010 mode with KI=5000, KP=500 for T = 50 µs.

93

10Measurement Number

Measurement Time T : 40 ms

Averaged Stability : (100 ± 25) ppm

KI=5000, KP=500

The Best Stability : 0.23 ppm

w/o Monitor Filter

The Worst Stability : 293 ppm

500

400

Stability of the Amplitude [ppm]

Stability of the Amplitude [ppm]

600

300

200

100

0

1

600
Measurement Time T : 40 ms

Averaged Stability : (52 ± 22) ppm

KI=5000, KP=500

The Best Stability : 0.04 ppm

w/ Monitor Filter

The Worst Stability : 309 ppm

500

400

300

200

100

2

3

4

5

6

7

8

9

0

10

1

2

3

4

5

6

7

1
Averaged Stability : (0.21 ± 0.05) mdeg.

0.9

Measurement Time T : 40 ms

0.8

KI=5000, KP=500

The Best Stability : 0.045 mdeg

w/o Monitor Filter

The Worst Stability : 0.43 mdeg

0.7
0.6
0.5
0.4

8

9

10

Measurement Number

Stability of the Phase [mdeg.]

Stability of the Phase [mdeg.]

Measurement Number

0.25
Averaged Stability : (0.06 ± 0.01) mdeg.

Measurement Time T : 40 ms

0.2

KI=5000, KP=500

The Best Stability : 0.04 mdeg

w/ Monitor Filter

The Worst Stability : 0.14 mdeg

0.15

0.1

0.3
0.2

0.05

0.1
01

2

3

4

5

6

7

8

9

10

Measurement Number(a) Without monitor filter

0

1

2

3

4

5

6

7

8
9
10
Measurement Number(b) With monitor filter

Figure 6.8: The dual feedback for the TM020 mode with KI=5000, KP=500 for T = 40 ms.

94

Measurement Time T : 50 µs

Averaged Stability : (94 ± 43) ppm

KI=5000, KP=500

The Best Stability : 0.0 ppm

w/o Monitor Filter

The Worst Stability : 434 ppm

Stability of the Amplitude [ppm]

Stability of the Amplitude [ppm]

800
700
600
500
400
300

140

Measurement Time T : 50 µs

Averaged Stability : ( 9 ± 8) ppm

120

KI=5000, KP=500

The Best Stability : 0.0 ppm

100

w/ Monitor Filter

The Worst Stability : 76 ppm

80
60

200

40

100

20

0

1

2

3

4

5

6

7

8

9

0

10

1

2

3

4

5

6

7

1.4
Measurement Time T : 50 µs

Averaged Stability : (0.23 ± 0.08) mdeg.

1.2
1KI=5000, KP=500

The Best Stability : 0.0 mdeg.

w/o Monitor Filter

The Worst Stability : 0.72 mdeg.

0.8
0.6
0.4

9

10

0.16
0.14

Measurement Time T : 50 µs

Averaged Stability : (0.03 ± 0.01) mdeg.

KI=5000, KP=500

The Best Stability : 0.0 mdeg

w/ Monitor Filter

The Worst Stability : 0.1 mdeg

0.12
0.1
0.08
0.06
0.04

0.2
01

8

Measurement Number

Stability of the Phase [mdeg.]

Stability of the Phase [mdeg.]

Measurement Number

0.02
2

3

4

5

6

7

8

9

10

Measurement Number(a) Without monitor filter

01

2

3

4

5

6

7

8

9

(b) With monitor filter

Figure 6.9: The dual feedback for the TM020 mode with KI=5000, KP=500 for T = 50 µs.

95

10Measurement Number

6.3.3

Consideration of Ultimate Control Stability

Up to this point, the stabilities are measured at the ADCs in the form of the power in theintermediate frequency;
SIF = AIF cos(2πfIF t + φIF )AIF = aARF
fIF = fLO − fRF
φIF = φLO − φRF .

(6.3.1)(6.3.2)(6.3.3)(6.3.4)

The form that should be compared with the requirement given by GPT simulations is Eap =E0 cos(2πfRF t + φRF ). In order to convert the stabilities for AIF and φIF into the stabilities
for E0 and φRF , some steps are made;• ∆AIF /AIF → ∆ARF /ARF → ∆E0 /E0• ∆φIF → ∆φRF → ∆φap .
For the amplitude, under the transformation from ∆AIF /AIF to ∆ARF /ARF , the stability does
not change because the sinusoidal product-sum transformation in the mixer is performed ina saturated range. The next transformation from ∆ARF /ARF to ∆E0 /E0 uses the following
relation;∆E0
1 ∆ARF=E0
2 ARF(6.3.5)from the relation of Eq. (5.3.25).
For the phase, ∆φIF includes both ∆φLO and ∆φRF . Therefore the relation ∆φIF ≥ ∆φRFholds. Since ∆φap = ∆φRF , ∆φIF is a conservative estimate of the stability of phase ∆φap .
From the above consideration, the best results of the dual feedback test are summarizedin Table 6.4.

Table 6.4: The result of dual feedback for Eap . The measurement time T is 50 µs and themonitor filter is turned on.
Resonant Mode Stability of Amplitude Stability of Phase
TM010(12 ± 3) ppm(0.04 ± 0.01) mdeg.
TM020(5 ± 4) ppm(0.03 ± 0.01) mdeg.
96Chapter 7
Ultimate Performance ofSuperconducting-Radio-Frequency
Transmission Electron Microscope
In previous chapters, the development of the essential technologies for the superconducting
radio frequency transmission electron Microscope (SRF-TEM) was discussed. In this chapter,the performance of each technology is summarized. Then the ultimate performance of the
300 kV SRF-TEM is estimated. After that, the future perspectives for the SRF-TEM areconsidered. Finally, the achievements in this study is summarized.

7.1

Performance Summary of Essential Technology for
Superconducting Radio Frequency Transmission Electron MicroscopeThis section describes the performance of each essential technology for SRF-TEM. The technologies are;
1. The two-mode cavity which realizes low energy dispersion required for the SRF-TEM,
2. The RF feedback system which enables the two-mode cavity to generate stable accelerating fields.

7.1.1

Performance of the Two-mode Cavity

We manufactured two two-mode cavities and performed performance tests twice for each
two-mode cavity. The most important performance index determining the spatial resolutionof the 300 kV SRF-TEM prototype is the accelerating field Eap . Table 7.1 summarizes Eapof both the TM010 mode and the TM020 mode, achieved in the last performance test of thesecond two-mode cavity. The requirements for Eap which is calculated in Chapter 4 is also
97

Table 7.1: The comparison of the requirement and the achievement of Eap . The requirements
is in the case of 300 kV SRF-TEM. The values of the achievement are measured in the last
performance test of two-mode cavity manufactured as the second prototype.
Requirement
Achievement
TM010 7.46 MV/m (11.0 ± 0.3) MV/m
TM020 7.20 MV/m (7.8 ± 0.3) MV/m
Table 7.2: The result of the accelerating field feedback.
Requirement
Achievement
TM010 Amplitude
0.0040 %(0.0012 ± 0.0003) %Phase
150 mdeg.
(0.04 ± 0.01) mdeg.
TM020 Amplitude
0.0140 %(0.0005 ± 0.0004) %Phase
55 mdeg.
(0.03 ± 0.01) mdeg.

shown. Table 7.1 indicates that the measured Eap satisfies the requirements for both theTM010 mode and the TM020 mode.

7.1.2

Performance of RF Feedback System

The RF feedback system for the two-mode cavity is necessary in order to stabilize the highaccelerating fields for minimizing the energy dispersion. Table 7.2 compares the requirements
for the stable drive and the achievement of the performance test of the RF feedback system.
The requirements are determined in order to achieve the energy dispersion of 4.0 × 10−5
(Chapter 4). The achievement values are the best ones recored in the performance tests.
The measurement time is 50 µs, that is four times the typical image acquisition time, andthe monitor filter is used. Table 7.2 indicates that the developed feedback system satisfies
the requirement for the energy dispersion of 4.0 × 10−5 .
Based on the result of the RF feedback test, we can estimate the energy dispersion from
Table 7.2 and the relation described in Figure 4.11 - 4.14. The expected value of energy
dispersion is 1.6 × 10−5 , a two times better figure than the requirement.

7.2

Estimation of Ultimate Performance of 300 kV Superconducting Radio Frequency Transmission Electron Microscope

As described in the previous section, the energy dispersion is estimated as 1.6 × 10−5 fromthe performance tests for the 300 kV SRF-TEM. This corresponds to the energy spread of 5eV for the 300 kV SRF-TEM system employing the developed technologies of the two-mode98

cavity and the RF feedback system. Using Eq. (3.3.4) and the lens parameters of H-9000
NAR in Table 7.3, ∆′′ = 36 nm is obtained. Figure 7.1 shows a LTF with this value and 15
nm defocus. The spatial resolution reaches the best value of 333 pm.

Table 7.3: Parameters of Hitachi 300 kV TEM H-9000 NAR. The values of σ(x) and β are
not published, then values of Hitachi 1.2 MV FE-TEM are used.
C3
0.7 µ mCc
1.4 mm∆J/J 1.0 × 10−5
σ(x)
10 pm
β
0.05 mdeg.

1
0.9

E ps: Phase Contrast Transfer Function sinΧ0.8

E fs : Correction Term for Energy Dispersion

0.7

E is : Correction Term for Image Spread

0.6

E sc: Correction Term for Incident Angle

0.5

Modified Phase Contrast Transfer Function

0.4
0.3
0.2
0.1
0
0

1

2

3

4

5

6

7
8
9
10
-1
Spatial Frequency [nm ]Figure 7.1: The lens transfer function of 300 kV SRF-TEM in the case of ∆′′ = 36 nm and∆f = 15 nm. The umax is 3.0 nm−1 , and the resulting spatial resolution is 333 pm.
99

7.3

Future Perspective

This thesis reports the overall system design and the establishment of the essential technologies for the SRF-TEM. This section describes the future perspectives to develop and study
the first prototype of the 300 kV SRF-TEM, the 3 MV and 10 MV machines.

7.3.1

Fabrication of Cryostat for Two-mode Cavity

The acceleration voltage of the first prototype is 300 kV. Our photocathode gun can accelerate
electrons with up to 100 kV. In this situation, we accelerate electrons with 60 kV with this DC
gun, and then other 240 kV is given by the two-mode cavity. The accelerating performanceof these technologies are already confirmed.
In order to cool the developed two-mode cavity to 4.2 K, we need a cryostat installed into
the 300 kV SRF-TEM prototype. The designed performances are summarized in Table 7.4.
It is noted that accelerators are usually constructed horizontally, while our SRF-TEM will beassembled vertically. Therefore, the countermeasure against the terrestrial magnetism will
be modified compared with the conventional ones.
Table 7.4: The designed performance of the cryostat installed into the 300 kV SRF-TEM
prototype.
Operating Temperature
Vacuum Level for Two-mode Cavity
Dissipating Power of Two-mode Cavity
Heat Intrusion
Volume of Liquid Helium
Volume of Liquid Nitrogen
Frequency Tuner (Mechanical)
Magnetic Shield

7.3.2

4.2 K
1.0 × 10−7 Pa
17 W
5W
100 ℓ
10 ℓ
± 2 MHz (TM010 mode), ± 4 MHz (TM020 mode)
up to 20 mG

Performance Evaluation of 300 kV SRF-TEM Prototype

After Assembling the 300 kV SRF-TEM prototype, we will conduct the performance evaluation test. This is composed of three processes;
1. Measurement of the energy spread of the beam
2. Measurement of the spatial resolution using an amorphous specimen
3. Observation of each kind of specimens.

100

The first one indicates the direct measurement of the energy spread ∆T using the device
called Electron Energy Loss Spectrometer (EELS) . Originally, this device is utilized
to measure the energy loss at specimens in order to study the characteristics of specimens
in TEM observations. In our measurement, we wont’t set any specimens, then the genuineenergy spread of the beam can be estimated.quantitatively
The second one is the common method to estimate the spatial resolution. The specimen composed of amorphous carbon and gold crystal is utilized (Figure 7.2). Carbons
are distributed with a random spacing, then observing the specimen in diﬀraction mode, adiﬀraction pattern corresponding to each defocus ∆f is acquired, and the reciprocal space
distance u is estimated. umax , the maximum value of u becomes the inverse of the spatial
resolution ∆r of the machine1 .
In order to study the performance of the prototype quantitatively, the first and the second
measurements are conducted. Additionally, the third test is conducted as the investigation
for the practical uses.
With the above, the performance of the 300 kV SRF-TEM prototype will be confirmed.

Figure 7.2: The schematic figure of the specimen utilized in the measurement of the spatial
resolution ∆r. The red circle indicates a gold atom, and the black one indicates a carbon
atom. Here, the lattice constant of golds is 230 pm.

7.3.3

Future SRF-TEM with Higher Accelerating Voltage than
300 kVThe 300 kV SRF-TEM is the first prototype for the proof-of-principe of the SRF technology
application into TEMs. As described in Sec. 3.3.1, the great advantage of the SRF-TEM
1

To estimate each u, the lattice constant of gold crystal is used to know the scaling factor of u.

101

appears when the accelerating voltage exceeds 3 MV that is the world’t highest value at thepresent moment. Here, tasks for the developments of the future 3 MV and 10 MV SRF-TEMs
are summarized.
Magnetic Field of Lens
The magnetic field of the objective lens of a conventional TEM reaches about 2 T. In thecase of the higher voltage than 3 MV, the conventional TEM technology cannot handle thissituation. The established technology of superconducting magnet utilized in high energy
accelerators is expected to realize the strong magnetic field needed for the high energy SRFTEMs. NbTi is a typical material for superconducting magnet coil. In the Large Hadron
Collider (LHC), the dipole magnets with the magnetic field of 8.3 T have been operated[36].
The high energy SRF-TEMs can be realized by replacing the conventional normal conducting
magnets with superconducting magnets as well as the accelerating part.
Optimization of Two-mode Cavity
The current two-mode cavity was designed for the acceleration from 60 kV to 300 KV. The
gap length of the cavity is too narrow to eﬃciently accelerate electrons with the energy higher
than 300 kV. It is expected that a new design of two-mode cavity with the gap length larger
than the current prototype realize a more eﬀective acceleration, and a downsizing of the highenergy SRF-TEM.
Requirement for RF Feedback Control
Currently we consider the 4 K operation for the 300 kV SRF-TEM. However, the dissipating
power becomes larger for the high energy SRF-TEMs of 3 MV and 10 MV. In order tosuppress this, the 2 K operation is eﬀective. However, the RF control is further more diﬃcult
for large quality factor Q0 as seen in Eq. (6.1.3) - (6.1.4). In practical development of highenergy SRF-TEMs, it has to be studied whether the 2 K operation or the 4 K operation isappropriate.

7.4

Achievement in This Thesis

We aim to develop a SRF-TEM, the world’s first TEM machine employing superconducting
RF technologies. This technology is expected to enable TEMs of higher beam energy above
3 MeV, the highest value of the present TEMs. The key is the two-mode accelerating cavity.
Very low energy dispersion is required for TEM in order to achieve good spatial resolution.
This is not achievable by the conventional RF cavities used for high energy accelerators, but it
is possible with the two-mode cavity as proven by the numerical simulation and demonstrated
by the tests of prototype cavities. This new type of acceleration scheme is realized by thetwo-mode cavity of original design employing the superconducting technology. Based on
the electric field distribution calculated for the configuration of the prototype two-mode102

cavity, an optimization of the overall design of 300 kV SRF-TEM and 3 MV SRF-TEM was
performed. The values of electric field and its error margin were used respectively to define
the requirements for the two-mode cavity construction and the RF feedback system.
Based on the experiences and knowledge that have been cultivated for high energy accelerators, we established the process to fabricate the two-mode cavity, and conducted theperformance tests. This confirms that the developed two-mode cavity satisfies the requirements set by the design optimization of the 300 kV SRF-TEM.
In order to minimize the energy dispersion of electron beam arising from fluctuation of
the electric field and the waveform, the operation of the two-mode cavity has to be stabilized. We introduced a RF feedback system. Based on the technology of cERL accelerator,
we established the special feedback system dedicated to SRF-TEM, and conducted the performance tests. We confirmed that the developed system satisfies the requirements set by
the overall design optimization.
From the above, we established the fundamental technologies for SRF-TEM and pioneered
the application of superconducting RF technologies to TEMs with higher energy and good
spatial resolution allowing observation of thicker objects than ever, opening a new powerfulviews for researches in variety of scientific field.

103

104

Appendix A
Imaging Theory of High-Resolution
Transmission Electron MicroscopeThis part is a supplement for 2.3 to explain the imaging theory in more detail, then there
are some overlaps with 2.3.

A.1

Phase Change by Electrostatic Potential

Phase Contrast Theory uses a wave function to consider a phase change of electron wave by
a nucleus. This means that image contrast is generated for to see how the phase of incident
electrons’ wave functions is altered by nuclei’s electrostatic potential.
Generally a refraction index n is defined by
λ1
,
λ2

n=

(A.1.1)

where λ1 is a de Broglie wave length of an electron propagating in a space free from any
electrostatic potentials of nuclei and λ2 is one in a space in potential V . Using λ = h/p,
Each λ is
λ1

=√

hc
2T me c2 +T 2

(A.1.2)λ2 = √hc
2(T +eV )me c2 +(T +eV )2(A.1.3)where me c2 is a rest mass of electron equal to 511 keV, e is elementary charge, equal to1.60 × 10−19 C. The dependence of de Broglie wave length of a electron on accelerating energy
T is illustrated in Figure A.1.
Substituting Eq.(A.1.2), Eq.(A.1.3) into Eq.(A.1.1) and ignore the second order of V
(V ≪ E, me c2 is reasonable here), relativistic refraction index nr is followed as
nr ∼
=1+

V (me c2 + T ),E(2me c2 + T )105(A.1.4)

where E is an accelerating voltage of TEM and T = eE. Phase change η can be written asη=2π(n − 1)∆z
λ(A.1.5)

Substituting the equation of relativistic refraction index Eq.(A.1.4), we get relativistic phasechange ηr as
2π me c2 + Tηr =V (r)∆z.
λE 2me c2 + T(A.1.6)

Here a new characteristic ”interaction coeﬃcient” σr is introduced as σr = 2π(me c2 +T )/λE(2me c2 + T ), then the wave function for a relativistic phase change satisfies
ψr (x, y) = exp {iσr V (r)∆z}.

(A.1.7)

It is usual for TEMs to take the accelerating energy from O(100 kV) to O(1 MV) therefore
T ∼ O(me c2 ). Therefore non-relativistic interaction coeﬃcient σc comes to π/λE in norelativistic energy region. A wave function for a non-relativistic phase change can be definedasψc (x, y) = exp {iσc V (r)∆z}.

(A.1.8)de Broglie wave length λ [pm]An interaction coeﬃcient σr decides how much an electrostatic potential in a specimen contributes to a phase change and in higher energy region, it gets smaller and comes closer to

102

10

1
10−1
10−2
10−3
10−4 −1
10

1

10

3

102
10
Accelerating Energy T [MeV]

Figure A.1: de Broglie wave length varying along with the accelerating energy T .
106

2πe/hc (see Figure A.2). For example, σ100 kV , the interaction coeﬃcient in the accelerating
voltage of 100 kV can be compared with one in 10 MV, σ10 MV . Then,
σ10 MV8.13 × 10−13 m−1 eV−1=
≃ 0.55.
σ100 kV
1.48 × 10−12 m−1 eV−1

(A.1.9)This means that TEMs with 10 MV accelerating voltage has a half of ability to transfer
electrostatic potentials in a specimen to a phase change compared with that of 100 kV TEM.
Incoming electrons pass through a specimen feeling a potential V (r), then coming to an
objective lens. ηnet , the net path diﬀerence generated by potentials V (r) in a specimen is
calculated as!
!
ηnet = σr V (r)dz = σr V (x, y, z)dz = σr Vp (x, y),(A.1.10)where Vp (x, y) is called as a projection potential which is integrated all potentials belong
beam axis. Here ηnet ≪ 1, then the phase wave function ψi can be approximated likeψi = exp(iσVp )= exp(iηnet )= 1 + iηnet .

(A.1.11)(A.1.12)(A.1.13)

The image intensity I can be determine by I = ψi∗ ψi , however ηnet ≪ 1 works here then the

Relativistic Interaction Coefficient σ r [m-1eV-1]

−12

5 ×10
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
10−2

10−1

3

102
10
10
Accelerating Energy T [MeV]1

Figure A.2: Variation of Interaction Coeﬃcient σ When Accelerating Voltage Is Changed.

107higher order larger including the second order of ηnet should be ignored, thenI = ψi∗ ψi= (1 − iηnet )(1 + iηnet )= 1,(A.1.14)(A.1.15)(A.1.16)

which means the image has no contrast.

A.2

Phase Shift at Objective Lens

Here the eﬀect of the phase shift by the objective lens shall be considered then it can be
explained that in practice TEMs can get a contrast images. The wave function that corresponds to the phase shift by the objective lens is called Lens Transfer Function (LTF) ,and is expressed as
LTF = exp(−iχps ) exp(χfs ) exp(χis ) exp(χsc )ππ
χps = C3 λ3 u4 + C5 λ5 u6 + π∆f λu223
π2 2 2 4
χfs = − λ ∆ u2
χis = −2π 2 σ 2 (x)u2"#2
χsc = −π 2 u20 (C3 λ2 u2 + ∆f )λu ,(A.2.1)(A.2.2)(A.2.3)(A.2.4)(A.2.5)where the notations were explained before in 2.3.
The phase wave function ψi in Eq. (A.1.13) is modified as
ψi = 1 + iσVp ⊗ F̂ [LTF]= 1 + iσVp ⊗ F̂ [exp(−iχps + χfs + χis + χsc )]= 1 + iσVp ⊗ F̂ [exp(−iχps + χR )] ,(A.2.6)(A.2.7)(A.2.8)where ⊗ represents convolution, F̂ stands for inverse Fourier transformation and replacedχfs + χis + χsc with χR . Then the image intensity I becomes
4
54
5
I = 1 + iσVp ⊗ F̂ [exp(−iχps + χR )]1 − iσVp ⊗ F̂ [exp(+iχps + χR )](A.2.9)= 1 − iσVp ⊗ F̂ [exp(iχps + χR )] + iσVp ⊗ F̂ [exp(−iχps + χR )](A.2.10)= 1 + 2σVp ⊗ F̂ [sin(χps ) · exp(χfs + χis + χsc )] .

(A.2.12)= 1 + 2σVp ⊗ F̂ [sin(χps ) · exp(χR )]108(A.2.11)

Appendix B
Response Calculation ofSuperconducting Cavity by
Lumped-constant Circuit Modeling
It is useful that superconducting cavities are modeled as lumped-constant circuit in order toconsider the operation. This description is based on [37].

B.1

Relation Between Cavity and Coupler

Figure B.1 shows the lumped-constant circuit model of the two-mode cavity and the powertransmission lines. The left part represents the power source including the reflection power.
The right part represents the transmitted power. The resistance Rc , the coil Lc and thecapacitor Cc represents the two-mode cavity. The Voltage Vin and the current Iin travel fromthe RF source G to the cavity, and the Voltage Vr and the current Ir comes back to the sourceof G, reflected at the cavity. z0 represents the characteristic impedance of the transmittedlines. The Voltage Vt , the current It and the power Pt travel from cavity to an external
circuit. Iin and Ic1 indicates currents flowing at a voltage transformer that represents the

Figure B.1: The lumped-constant circuit model of the two-mode cavity.

109

input power coupler whose coupling ratio is 1 : κin . Similarly, It and Ic2 indicates currents
flowing at a voltage transformer that represents the output power coupler whose coupling
ratio is 1 : κt . The Voltage of Vg represents the voltage of the power source, and the voltageof Vc represents the cavity voltage.
One relation is established among the input voltage Vin , the reflection voltage Vr and thepower source voltage Vg as
Vg = Vin + Vr .

(B.1.1)

Vin = z0 Iin
Vr = z 0 I r
Ig = Iin − Ir

(B.1.2)(B.1.3)(B.1.4)Vin − Vr = z0 Ig(B.1.5)Additionally,

are established. Therefore,

is derived. Using Vin and Vr , the cavity voltage Vc and the coupling current Ic1 can be
expressed as
Vc = κin Vg = κin (Vin + Vr )
Ig
1
Ic1 =
=
(Vin − Vr ).
κinκin z0(B.1.6)(B.1.7)

Using the above equations,
Vc
2Vin
Ic1 = − 2 +κin z0 κin z0
Vc
Vr =
− Vin
κin

(B.1.8)(B.1.9)are established. Similarly, the voltage Vt and the current Ic2 are expressed as
V t = z0 I t
V c = κ t Vt
It
Ic2 = − ,κt(B.1.10)(B.1.11)(B.1.12)then,
Vc
Ic2 = − 2
κt z 0(B.1.13)(B.1.14)110

is derived. Each power has a relation with its corresponding voltage as
1|Vin |22z0
1
Pr =|Vr |22z0
1
Pt =|Vt |22z0
1
Pc =|Vc |2 .
2Rc

Pin =

B.2

(B.1.15)(B.1.16)(B.1.17)(B.1.18)

Diﬀerential Equation for Superconducting Cavity
and General Solution

Considering Kirchhoﬀ’s circuit law for the cavity,!
d1
1
C c Vc ++Vc dt = Ic1 + Ic2
dt
Rc Lc

(B.2.1)is established. Substituting Eq. (B.1.8) and (B.1.13) into the above equation and taking atemporal diﬀerentiation,
%
&
d2
d
2Vin
∗ d
2
Vc + 2λ Vc + ω0 Vc =(B.2.2)dt2
dt
dt κin z0 Cc
1
1
1
2λ∗ =
+ 2
(B.2.3)2
Rc Cc κin z0 Cc κt z0 Cc
1
ω02 =(B.2.4)
Lc Cc
are derived, where ω0 is the resonant angular frequency of the superconducting cavity. As
explained in Sec. 5.1.3, the quality factor Q0 is written as
Q0 =

ω0 U
,
Pc

(B.2.5)and the stored energy U is described as
1
U = Cc Vc2
21= Lc Ic2 ,2then substituting Eq. (B.1.18) in place of Pc , the quality factor is expressed as
2
Cc
Q0 =
Rc .
Lc
111

(B.2.6)(B.2.7)(B.2.8)

The shunt impedance of the cavity Ra is defined as
Vc.
Pc(B.2.9)Ra = 2Rc .

(B.2.10)Ra =Substituting Eq. (B.1.18), Ra is written as

Additionally, new parameters are introduced as
Rc
κ2in z0Rcβt = 2 .
κt z 0βin =(B.2.11)(B.2.12)

Theses are determined by the coupling strength of κin and κt . The loaded quality factor QL
is expressed as
1
1=(1 + βin + βt )
QL
Q0
1
=
(1 + β ∗ ) .
Q0(B.2.13)(B.2.14)

Then,
1
1
1
+ 2
+ 2
Rc Cc κin z0 Cc κt z0 Cc
%
&
1
Rc
Rc
=
1+ 2 + 2
Rc C cκin z0 κt z0
ω0=(1 + βin + βt )Q0ω0=(1 + β ∗ )
Q0
ω0
=
.
QL

2λ∗ =

Eq. (B.1.9) and (B.2.2) are expressed using β ∗ and Ra as22β ∗ z0Vr =Vc − Vin
Ra
√ ∗2
d2
d
2
β
Ra d
Vc + 2λ∗ Vc + ω02 Vc = 2λ∗
Vin .
2
∗
dt
dt
1 + β 2z0 dt(B.2.15)(B.2.16)(B.2.17)(B.2.18)(B.2.19)(B.2.20)(B.2.21)Eq. (B.2.21) is rewritten as a general form liked2ddVc + 2λ∗ Vc + ω02 Vc = f (t)
2
dt
dt
dt
112

(B.2.22)

and its general solution Vf is
V f = C+ e

−λ+ t

+ C− e−λ− t)λ± = λ ± λ∗ 2 − ω02

1+
λ+ − λ−

!

′

−∞

′

{λ+ e−λ+ (t−t ) − λ− e−λ− (t−t ) }f (t′ )dt′∗

(B.2.23)(B.2.24)

where C± are the constants determined by the initial conditions. In the case of superconducting cavities, λ∗ ≪ ω0 , then the above equations are rewritten as
!
1
′
′
−λ+ t
−λ− t
V f ≃ C+ e
+ C− e+{e−λ+ (t−t ) + e−λ− (t−t ) }f (t′ )dt′
(B.2.25)2 −∞λ± = λ∗ ± iω0 .
(B.2.26)

B.3

Stationary State of Superconducting Cavity

To satisfy Eq. (B.1.15), the input voltage is defined as*
Vin = 2z0 Pin eiωt

(B.3.1)

where ω is the angular frequency of the signal generator, and ω − ω0 is finite. In order toacquire the stationary solution for Eq. (B.2.25), setting C± = 0, and substituting Eq. (B.3.1)
in place of f (t′ ) in Eq. (B.2.25),√! 0
12 β∗ *∗ −λ+ (t−t′ )∗ −λ− (t−t′ )iωt′ ′
Vc ≃
R
P
×
λ
e
+
λ
e
e
dt
(B.3.2)ain
1 + β∗−∞
is derived. Setting the integrand G± (t) as!
′′±
G (t) =
λ∗ e−λ± (t−t ) eiωt dt′ ,(B.3.3)−∞and substituting Eq. (B.2.19) in place of λ∗ ,λ∗eiωt
λ∗ + i(ω ± ω0 )1
ω ± ω0 iωt=e .
1 + 2iQL ω0

G± (t) =113(B.3.4)(B.3.5)

In the case of Superconducting cavities, QL ≫ 1 and ω ≃ ω0 , then |G+ (t)| ≪ |G− (t)| isestablished. Therefore, ignoring G+ (t), Eq. (B.3.2) is rewritten as√
2 β∗ *1ω − ω0 iωt
Vc ≃Ra Pine(B.3.6)∗1+β
1 + 2iQL ω0√2 β∗ *
1=
Ra Pineiωt(B.3.7)∗1+β
1 − i tan θ√
2 β∗ *=Ra Pin cos θei(ωt+θ)(B.3.8)1 + β∗
ω − ω0tan θ = −2QL.
(B.3.9)ω0This θ is called as tuning angle .
Substituting Eq. (B.3.8) in place of Vc and Eq. (B.3.1) in place of Vin in Eq. (B.2.20)
respectively, the reflection voltage Vr in stationary states is calculated as
%
&
*
2β ∗
iθ
Vr = 2z0 Pincos θe − 1 eiωt
(B.3.10)
1 + β∗*β ∗ ei2θ − 1 iωt= 2z0 Pine .
(B.3.11)
1 + β∗
Using Eq. (B.1.16) and (B.3.11), the reflection power Pr is calculated as- ∗ i2θ
- β e − 1 -2
- Pin
Pr = -1 + β∗ 1 − 2β ∗ cos 2θ + β ∗ 2=
Pin .
(1 + β ∗ )2

(B.3.12)(B.3.13)

When ω = ω0 (θ = 0),%Pr =1 − β∗
1 + β∗&2

Pin(B.3.14)is established, and when ω ̸= ω0 , Pr gets the minimum value
Pr = sin2 θPin

(B.3.15)at β ∗ = 1. Our two-mode cavity was manufactured to satisfy β ∗ = 1 in order to minimize
the reflection power. Similar, from Eq. (B.1.18), (B.2.10) and (B.3.8), the dissipating power
Pc in stationary states is calculated asPc =4β ∗ cos2 θPin .
(1 + β ∗ )2

(B.3.16)

From Eq. (B.1.11), (B.1.17) and (B.3.8), the transmitted power Pt is calculated asPt =4β ∗ βt cos2 θPin .
(1 + β ∗ )2
114

(B.3.17)

B.4

Response to Pulsed Power

Here, it is considered that the input power is pulsed. Owing to monitor the response of thecavity to the pulsed input power, β ∗ , QL , βin and βt can be estimated. The indices of βinand βt can be considered stable, then the transition of Q0 and Vc can be estimated with only
once measurement of the pulsed input power.
Now, considering the situation of ω = ω0 . when t = 0, the switch of the input powerturns on, and when t = toﬀ , the switch turns oﬀ. The amplitude of the input power isconsidered constant (Figure B.2). Given the initial condition of Vc = 0, dVc /dt = 0 at t = 0,
Eq. (B.2.25) becomes√.
2 β∗ *∗ /Vc (t) ≃Ra Pin 1 − eλ t eiω0 t(0 ≤ t < toﬀ ) .
(B.4.1)∗
1+βFrom Eq. (B.2.20), (B.3.1) and (B.4.1),
%
&
*
2β ∗
−λ∗ t
Vr (t) = 2z0 Pin(1 − e) − 1 eiω0 t
∗
1+β
∗
*
1 − β ∗ + 2β ∗ e−λ t iω0 t= − 2z0 Pine(0 ≤ t < toﬀ ) .
1 + β∗

Figure B.2: Power responses to the pulsed input power.

115(B.4.2)(B.4.3)

are derived. From the above equation and Eq. (B.1.16),%∗ &2
1 − β ∗ + 2β ∗ e−λ tPr =Pin(0 ≤ t < toﬀ ) .
1 + β∗(B.4.4)is derived. Similarly, from Eq. (B.1.17), (B.1.18) and (B.4.1),∗

4β ∗ βt (1 − e−λ t )2Pin
(0 ≤ t < toﬀ )(1 + β ∗ )2∗4β ∗ (1 − e−λ t )2
Pc =Pin
(0 ≤ t < toﬀ )(1 + β ∗ )2Pt =

(B.4.5)(B.4.6)are derived. Next, the case is considered that the switch turns oﬀ at t = toﬀ . In this case,the amplitude of Vc at t = toﬀ equals to the magnitude of Eq. (B.3.8), then√
2 β∗ *∗Vc ≃Ra Pin e−λ (t−toﬀ) ei(ωt+θ)(B.4.7)∗1+β(B.4.8)is established. From Eq. (B.2.20), (B.4.7) and substituting Vin = 0,*2β ∗ −λ∗ (t−toﬀ ) iω0 t
Vr = 2z0 Pinee(t ≥ toﬀ )1 + β∗(B.4.9)(t ≥ toﬀ )(B.4.10)is derived. From Eq. (B.1.16) and (B.4.9),
%
&2
2β ∗
∗
Pr (t) =e−2λ (t−toﬀ) Pin∗
1+β

is derived. Similarly, from Eq. (B.1.11), (B.1.17), (B.1.18) and (B.4.7),
4β ∗ βt −2λ∗ (t−toﬀ )ePin(1 + β ∗ )2
4β ∗
∗
Pc =
e−2λ (t−toﬀ ) Pin∗
2(1 + β )Pt =(t ≥ toﬀ )(B.4.11)(t ≥ toﬀ )(B.4.12)are derived. Figure B.2) shows the various patterns of Vr whose amplitude at t = toﬀ isdepend on β ∗ . The situation of beta∗ > 1 is called over-coupling and β ∗ < 1 is called
under-coupling . Using the information of the Pr patterns, the coupling strength of β ∗ canbe determined.
Moreover, measuring the decay slope of Pt , the loaded quality factor Q0 can be determined. From Eq. (B.4.11), the decay time t − toﬀ = τL elapsed to reach a half value of thepeak is expressed as1∗e−2λ τL = .
2(B.4.13)

QL ≃ 0.6931 × τL × ω0 .

(B.4.14)

Then, QL is calculated as

116

B.5

Estimate Q0 and Vc

Now, we know the coupling strength of β ∗ (overcooling or under-coupling) and the value of
QL . In order to estimate the value of Q0 , it is necessary to estimate β ∗ . As described in Sec.
B.3, the reflected power Pr in stationary states are given as%&2
1 − β∗Pr =Pin(B.5.1)
1 + β∗
which is already shown as Eq. (B.3.14). Solving this equation in terms of β ∗ ,)1 ± PPinr)β∗ =1 ∓ PPinr(B.5.2)is derived, where the upper double-sign corresponds is chosen for β ∗ ≥ 1 (over-coupling) andthe down one is chosen for β ∗ < 1 (under-coupling). Then, from Eq. (B.2.14), the value of
Q0 is estimated. It should be noted that, Q0 is always changed because the dissipating powerin the cavity Pc gets larger at a hight accelerating field. On the other hand, Qt and Qe arestatic in the process of performance tests because U/Pt and U/Pe are determined only by
the shape of the cavity.
The amplitude of the accelerating voltage |Vc | is estimated by Eq. (B.3.8) as√
2 β∗ *|Vc | =Ra Pin .
(B.5.3)
1 + β∗
Using Eq. (B.4.11), Eq. (B.5.3) is expressed as*|Vc | = Ra Pt .

As Eq. (5.1.60), the accelerating field Eacc is expressed as*
Ra /Q0*Eacc =Qt Pt .
d

(B.5.4)(B.5.5)

There is a relation between Eacc and Eap as

Eap ∝ Eacc ,(B.5.6)*
Eap = A Qt Pt ,(B.5.7)then Eap is expressed aswhere A is a constant and can be estimated by the numerical simulation.

117

118

Bibliography[1] A. Tonomura, et al., Observation of individual vortices trapped along columnar defects
in high-temperature superconductors , Nature, vol412, 620-622, (2001)[2] X. Z. Yu, et al., Real-space observation of a two-dimensional skyrmion crystal , Nature,
465, 901, (2010)[3] F. N. Rybakov, et al., New spiral state and skyrmion lattice in 3D model of chiralmagnets , New J. Phys. 18, 045002 (2016)[4] T. Akashi, et al., Aberration corrected 1.2-MV cold field-emission transmission electron
microscope with a sub-50-pm resolution”, Applied Physics Letters, 106, 074101, (2015)[5] 大木正路,

高電圧工学 , 槇書店, (1982)[6] T. Akashi, et al., Aberration corrected 1.2-MV cold field-emission transmission electron
microscope with a sub-50-pm resolution (Supplementary material) , Applied Physics
Letters 106, 074101 (2015)[7] Nakamura et al., Expandable Megakaryocyte Cell Lines Enable Clinically Applicable
Generation of Platelets from Human Induced Pluripotent Stem Cells , Cell Stem Cell,
14, 1-14, (2014)[8] M. Haider et al., Electron Microscopy Image Enhanced , Nature 392, 768, (1998）[9] M. Kuwahara, et al., 30-kV spin-polarized transmission electron microscope with GaAsGaAsP strained superlattice photocathode , Appl. Phys. Lett. 101, 033102, (2012)[10] Ray F. Egerton, Physical Principles of Electron Microscopy -An Introduction to TEM,
SEM, and AEM-”, Springer, (2005)[11] N. Yamamoto, et al., High brightness and high polarization electron source using transmission photocathode with GaAs-GaAsP superlattice layers”, JORNAL OF APPLIEDPHYSICS, 103, 064905, (2008)[12] S. Pastuszka, et al., Preparation and performance of transmission-mode GaAs photocathodes as sources for cold dc electron beams”, JORNAL OF APPLIED PHYSICS,
88, 6788, (2000)
119[13] L. Reimer, H. Kohl, Transmission Electron Microscopy -Physics of Image FormationFifth Edition”, Springer, (2008)[14] 田中信夫, 電子線ナノイメージング-高分解能 TEM と STEM による可視化-”, 内田老鶴圃, (2009)[15] M. Haider, et al., Information Transfer in a TEM Corrected for Spherical and Chromatic Aberration”, Microscopy and Microanalysis, 16, 393-408, (2010)[16] PDG,

Particle Physics Booklet”, (2014)[17] 松葉俊哉, 本田洋介, 拡散モデルによるカソードの時間応答の計算 , KEK cERL internalreport, (2010)[18] B. W. Reed et al. , Electron Optical Requirements for Sub-Nanosecond Microscopy ,Microscopy and Microanalysis, Volume 11, Supplement S02, 1480-1481, (2005)[19] General Particle Tracer,
Physics, (2012)

General Particle Tracer User Manual Version 2.82 , Pulser[20] William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery,
Numerical Recipes in C++ Second Edition , Cambridge University Press, (2002)[21] J. Bardeen et al. ,108:1175(1957)[22] 渡辺正,

Microscopic Theory of Superconductivity , Phys. Rev.

元素大百科事典, 朝倉書店, (2007)[23] ESPI Metals, Technical Data Niobium , http://www.espimetals.com/index.php/technicaldata/
170-niobium, (2012)[24] W. M. Haynes et al. ,CRC handbook of Chemistry and Physics , CRC Press, (2012)[25] Hasan Padamsee et al. ,RF Superconductivity for Accelerators , (1998)[26] 齋藤健治, 東京大学大学院理学系研究科物理学専攻集中講義ノート, (2011)[27] K. Halbach and R. F. Holsinger, SUPERFISH-A COMPUTER PROGRAM FOR
EVALUATION OF RF CAVITIES WITH CYLINDRICAL SYMMETRY , Particle
Accelerators, Vol. 7, pp. 213 - 222, (1976)[28] P. Kneisel,(1980)

Surface Preparation of Niobium , Proceedings of SRF Workshop 1980,[29] Y. Morita et al. , ILC 超伝導加速空洞のための電解研磨法における研磨量の電極間距離依存に関する研究 , Proceedings of the 5th Annual Meeting of Particle Accelerator
Society of Japan and the 33rd Linear Accelerator Meeting in Japan, (2008)120

[30] M. L. Kinter et al., Chemical Polish for Niobium Microwave Structures , J. Appl.
Phys. 41, 828, (1970)[31] 日本工業規格,

B9920 クリーンルームの空気清浄度の評価方法 , (2002)[32] 日本工業規格,

K0552 超純水の電気伝導率試験方法 , (1994)[33] 国立天文台編纂,

理科年表平成 5 年 , 丸善株式会社, (1992)[34] HP,

HP 5350B/5351B/5352B マイクロ波周波数カウンタ 操作ガイド , (1988)[35] M. Omet, Doctoral Thesis, Digital Low Level RF Control Techniques and Procedures
Towards the International Linear Collider , (2014)[36] L. Bottura ,et al., Superconducting Magnets for Particle Accelerators , IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 63, NO. 2, APRIL 2016, 751-776, (2016)[37] 中里俊晴, 博士論文 超伝導 500MHz 加速空洞の研究 , (1981)

121

122

Acknowledgement
This thesis would not have been accomplished without the supports of many people. I will
express my gratitude to them.
First of all, I would like to express my deepest gratitude to my supervisor, Prof. Satoru
Yamashita. He gave the opportunity to participate in this study, which became a desirable
experience for me. He gave me a lot of advices and guidances through many discussions
continuously, which always encouraged me to accomplish this study.
This study is carried out in High Energy Accelerator Research Organization (KEK). I
would like to thank all members of the KEK research team.
I would like to express my special gratitude to Dr. Yukihide Kamiya. He gave me a lot ofadvices about the entire study, the presentations of the conferences and papers. Additionally,he spent an enormous amount of time to equip me the knowledge of accelerators. I would
like to express my appreciation to Dr. Atsushi Enomoto. He gave me the knowledge about
accelerators and shared his valuable experiences with Dr. Yukihide Kamiya. He also created
a comfortable environment for me to study in KEK.
I would like to express my best gratitude to Prof. Takaaki Furuya for the development of
the superconducting cavities. His knowledge and experience created in the longtime researchin KEK led us to the success of the production of the unprecedented cavity. As well as
the manufacturing the cavity, he gave me a technique of the vertical test of superconducting
cavities. It was memorable that he and I carried out the performance tests late into the night.
I would like to express my appreciation to Mr. Yoshisato Funahashi. He supported me at the
every moment of manufacturing cavities, including EBW and polishing. I would like to thankMr. Motoaki Sawabe for the support of the cavity polishing. With his extensive knowledge
and experience, we performed the polishing of the complicated-shape cavity successfully. I
would express my appreciation to Prof. Hiroshi Sakai and Dr. Michiru NIshiwaki for the
support of the vertical tests of the cavities. Moreover, they helped me to carry out the study
by teaching me about superconducting cavities.
I would like to express my special thanks to Prof. Shinichiro Michizono for the development of the feedback system. With his expertise of the feedback system, we realized thestable control system for the superconducting cavity. I also would like to express my thanks
to Dr. Qiu Feng. He helped us to perform the feedback tests for a long time. With his
dedicated cooperation, we measured the stability of our feedback system.
I would like to thanks to Dr. Masahiro Yamamoto. He is a expert of photocathode gun,and gave me a lot of guidance for the electron gun and the beam dynamic simulation. I

123would also express my gratitude to Prof. Masao Kuriki of Hiroshima University, who taughtme the implementation of the beam dynamics simulation.
I would like to express my special thanks to Mr. Yoichi Ose of Hitachi High-Technologies
Corporation. He gave us a plenty of expertise of electron microscopy. In addition, I would liketo thank Mr. Shigeto Isakozawa of Hitachi High-Technologies Corporation and Dr. Takeshi
Kawasaki of Hitachi for their generous supports.
I would like to express my gratitude to Prof. Yukinori Kobayashi, Dr. Jacqueline Yan,
Prof. Takako Miura, Dr. Jin Xiuguang, Prof. Tsukasa Miyajima of KEK, and Dr. Koji
Kimono of National Institute for Materials Science.
I would like to express my thanks to all members of the Yamashita group of ICEPP in
the university of Tokyo: Dr. Tomohiko Tanabe, Dr. Masakazu Kurata, Mr. Ryo Katayama,
Mr. Takahito Yamada, Ms. Sei Ieki, Mr. Naoki Nagakura, Mr. Shohei Fujikura and Mr. Yu
Kato. I would also express my gratitude to the old members of the Yamashita group: Prof.
Kenji Mishima, Dr. Taikan Suehara, Dr. Takashi Kubota, Dr. Yuichi Morita, Dr. Hidetoshi
Otono, Dr. Hideyuki Oide, Mr. Harumichi Yokoyama, Mr. Takuaki Mori and Mr. Kento
Kasuya. I am honored to have been a colleague of them.
I would like to express my deepest gratitude to ICEPP secretaries during my belonging:
Ms. Masako Shiota, Ms. Yoko Takemoto, Ms. Kayo Yamaura, Ms. Yoshie Tezuka, Ms.
Akiko Miyazono, Ms. Ritsuko Ambiru, Ms. Megumi Suzuki. With their generous supports,
I accomplished this study.
Finally, I would like to thank my mother Sakae and my father Toru, who have given mea plenty of opportunities and encouraged me all the time.

124