A Validation Study on the Novel
Three-Dimensional Spiral Injection
Scheme with the Electron Beam for
Muon g – 2/EDM Experiment

Muhammad Abdul Rehman
Department of Accelerator Science
School of High Energy Accelerator Science
The Graduate University for Advanced Studies, SOKENDAI

A thesis submitted for the degree ofDoctor of Philosophy
2020

Supervisor:

Prof. Kazuro Furukawa (KEK/ACCL)

Reviewers:

Assoc. Prof. Hiromi Iinuma (Ibaraki University)
Prof. Takuya Kamitani (KEK/ACCL)
Assoc. Prof. Tsutomu Mibe (KEK/IPNS)
Prof. Kenichi Sasaki (KEK/IPNS)
Assis. Prof. Hiroshi Sugimoto (KEK/ACCL)
Senior Scientist Yasuhiro Kondo (JAEA)

Abstract
The muon anomalous magnetic moment (g − 2)µ is one of the most important measurements
in elementary particle physics.

The most recent measurement of (g − 2)µ at Brookhaven

National Laboratory (BNL) results in more than 3σ discrepancy compared with the equally
precise Standard Model prediction. This disparity could be indicative of new physics beyond
the standard model. To resolve this discrepancy, a new muon's g – 2/EDM (E34) experiment at
J-PARC aims to measure (g − 2)µ to a precision of 0.1 ppm and electric dipole moment (EDM)
down to the sensitivity of 1.5 × 10−21 e.cm. The key objective of measuring (g − 2)µ is to store
spin polarized µ+ beam in a magnetic field and evaluate the evolution of the spin precession
vector. In the E34 experiment, a muon beam with an emittance less than 0.5π mm mrad and a
momentum of 300 MeV/c from a muon accelerator will be injected into a compact 3-T solenoid
magnet. The muon beam will be stored on a 0.66 m diameter orbit. The solenoid magnet will
provide an unprecedented local field uniformity of 0.1 ppm (peak to peak).
A new three-dimensional spiral injection scheme has been invented to inject the beam into
the MRI-type magnet. This new injection scheme will enhance injection efficiency and overcome
technical challenges related to the small storage orbit diameter. In the spiral injection scheme,
the beam will be injected at a vertical angle into the storage magnet. The radial field of thesolenoid will decrease the vertical angle of the beam as it approaches the midplane of the magnet.
Finally, a magnetic kicker will guide the beam to the storage volume where the beam will be
stored under a weak focusing field. However, the injection of the accelerated beam into sucha small storage orbit is unprecedented; therefore, a demonstration experiment to establish thefeasibility of this new injection scheme is inevitable. The purpose of this research is to develop ascale down Spiral Injection Test Experiment (SITE) with an electron beam to demonstrate theideology behind this new injection scheme.
An electron beamline of a length of 2 m has been designed and constructed for the SITE.
The beamline consists of a triode-type thermionic electron gun with a LaB6 cathode to generate
a DC electron beam of 80 keV with a beam current of approximately 100 µA. Subsequently, amagnetic lens was placed to prevent the growth of the beam. A collimator with a diameter of 3
mm and a depth of 5 mm was placed at a distance of 0.7 m from the electron gun. The mainfunction of the collimator is to produce a differential pumping for the gas monitor, as well as to
cut the beam halo. A pair of air-core steering coils has been installed to control the transverse
position of the 80 keV electron beam. Next, three rotation quadrupole magnets were installed tocontrol the beam phase space for the spiral injection. The emittance of the electron beam after
collimation was measured by the quadrupole scan method. The emittance value in the horizontal
direction was estimated 0.61 ± 0.05 mm mrad and in the vertical direction, it was 0.41 ± 0.04
mm mrad. In order to estimate the beam transverse coupling strength a simple procedure wasdeveloped. The coupling strength of the beam after the collimator was estimated to 0.17 ± 0.6.
This small value of coupling means decoupled beam after the collimator.
A normal conducting solenoid-type storage magnet of field strength 82.5 Gauss was designed

and constructed to store the electron beam in an orbit with a diameter of 0.24 m. In the caseof SITE, the field index value n (strength of weak focusing: 0 < n < 1) of 1.65 × 10−2 wasset. This higher value of the field index relaxes the phase space requirement and vertical kicker
parameters for SITE.
A non-invasive gas monitor was used to detect the electron beam track in the storage magnet.
A charge-coupled device (CCD) camera was used to observe fluorescent light originating fromthe gas monitor. An electron beam track of four turns in the vacuum chamber of the storage
magnet was confirmed by this gas monitor, which is a good qualitative tool to efficiently verify
the electron beam track inside the storage magnet. However, a gas monitor lacks the ability
to provide quantitative information about the position of the beam and the profile in thestorage magnet. Therefore, a special type of wire scanner for determining the beam position
and measuring the profile in the solenoid magnet was designed and constructed for the SITE.
Two wire scanners were installed on the storage magnet to measure the beam profile and beam
evolution in the storage magnet.
Due to the axial symmetric field of the solenoid magnet, a strongly XY-coupled beam isrequired. To produce the required phase space for the solenoid-type storage magnet, a beamtransport line consisting of three rotatable quadrupole magnets has been designed and built
for SITE. The vertical beam size reduction by means of phase space matching and othergeometrical information have been successfully measured by the wire scanners and confirmedby the simulation. The vertical beam size had been reduced to 25.68 ± 0.31 mm with thephase matching as compared to vertical beam growth of 78.41 ± 0.83 mm without any rotatable
quadrupole magnets.
To store the beam at the center of the storage magnet, a pulsed magnetic kicker was designed
and simulated. From simulation studies, the best design parameters for the magnetic kicker havebeen determined. As explained earlier in the case of SITE, the field index value of 1.65×10−2 wasused. This value of the field index allowed to use pulse width of 50 ns for the kicker. In addition,it also provides a weak focusing trap of ±50 mm at storage region. A tracking simulation withthe best beam phase space that can achieved by the three rotatable quadrupole magnets has
been performed. And it was estimated that injection efficiency of 81% can be achieved for SITE.
The purpose of this study was to prove the ideology of 3-D spiral injection scheme withthe electron beam.

The qualitative and quantitative details of the vertical and horizontal

beam profiles and positions inside the solenoid storage magnet are one of the most important
requisites for the successful 3-D spiral injection. Therefore, a wire scanner type beam positionand profile monitor was developed for the SITE. The knowledge and experience of wire scanner
development to extract the beam profile and position for the SITE storage magnet greatly helped
to understand the beam behavior inside the solenoid storage magnet. Moreover, it paved theway for the development of a similar quantitative monitor for the successful 3-D spiral injectionin the case of E34. The method to measure the transverse beam coupling developed at the SITE
will also be implanted at E34 to estimate the coupling strength at the matching point.

Contents
List of Figures

10

List of Tables

20

1 Introduction

23

1.1

Structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

1.2

Muon's Anomalous Magnetic Dipole Moment . . . . . . . . . . . . . . . . . . . .

24

1.3

Muon's Electric Dipole Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

1.4

Historical Glimpse of Muon g – 2 Experiments . . . . . . . . . . . . . . . . . . .

27

1.5

The BNL Muon's g – 2 Experiment (E821) . . . . . . . . . . . . . . . . . . . . .

28

1.5.1

Principle of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

J-PARC Muon's g – 2/EDM Experiment . . . . . . . . . . . . . . . . . . . . . . .

31

1.6

2 Three–Dimensional Spiral Injection Scheme

35

2.1

Why Three–Dimensional Spiral Injection? . . . . . . . . . . . . . . . . . . . . . .

35

2.2

Basic idea of Spiral Injection Scheme . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.3

Vertical Kicker System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.4

Weak Focusing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.5

Beam Phase Space Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

2.5.1

Concepts in beam dynamics . . . . . . . . . . . . . . . . . . . . . . . . . .

44

2.5.2

Emittance and Twiss parameters . . . . . . . . . . . . . . . . . . . . . . .

46

2.5.3

Beam matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

2.5.4

Beam Phase Space Requirements for 3-D Spiral Injection . . . . . . . . .

49

3 Spiral Injection Test Experiment (SITE) with Electron Beam

51

3.1

Experimental Setup of SITE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.2

Electron Gun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.2.1

Emission Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

3.2.2

Electron Gun Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

3.3

Magnetic Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.4

Electric Chopper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.5

Collimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

3.6

Rotatable Quadrupole Mangets . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

7

3.6.1
3.7

3.8

Field Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Dipole Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

3.7.1

Dipole Magnet 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

3.7.2

Dipole Magnet 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Storage Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

3.8.1

75

Field Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Development of Beam Monitors for SITE's Storage Magnet
4.1

4.2

Gas Monitor

79. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.1.1

Gas Monitor Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.1.2

Optical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.1.3

Light Reflection Suppression . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.1.4

Performance of the Gas Monitor . . . . . . . . . . . . . . . . . . . . . . .

85

Wire Scanner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

4.2.1

Basic Principle and Design of Wire Scanner . . . . . . . . . . . . . . . . .

87

4.2.2

Mechanical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

4.2.3

Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

5 Beam Commissioning with DC Beam

99

5.1

Emittance Measurement By Quadrupole Scan . . . . . . . . . . . . . . . . . . . .

5.2

Emittance Measurement at SITE Beamline . . . . . . . . . . . . . . . . . . . . . 101

5.3

5.2.1

Image Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2.2

Misalignment of Quadrupole Magnets . . . . . . . . . . . . . . . . . . . . 103

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3.1

5.4

Beam Phase Space at the Collimator . . . . . . . . . . . . . . . . . . . . . 117

Coupling Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4.1

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6 Experimental Results of Phase Space Matching
6.1

99

129

Beam Phase-Space Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.1.1

Coordinates System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.1.2

A Method to Calculate the Beam Phase-Space Requirements at the
Matching Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.1.3

Errors on the Beam Phase-Space Slopes . . . . . . . . . . . . . . . . . . . 138

6.2

The Matching Beamline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.3

Three Rotatable Quadrupole Magnets Solution . . . . . . . . . . . . . . . . . . . 143
6.3.1

6.4

Beamline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.4.1

Injection without Rotatable Quadrupole Magnets . . . . . . . . . . . . . . 147

6.4.2

Injection with Normal Quadrupole Magnets . . . . . . . . . . . . . . . . . 154

6.4.3
6.5

Injection with Rotatable Quadrupole Magnets. . . . . . . . . . . . . . . 155

Discussions and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

7 Kicker System for SITE

165

7.1

Vertical Magnetic Kicker for SITE . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.2

Design of the kicker Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.3

Eddy Current Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

7.4

Kicker Power Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7.5

Injection Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.5.1

7.6

Injection Inefficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8 Conclusions

189

A Appendix

195

A.1 Wire Scanner Data Acquisition System Code (C#) . . . . . . . . . . . . . . . . . 195
B Appendix

201

B.1 The beam Injection with other settings . . . . . . . . . . . . . . . . . . . . . . . . 201
C Appendix

205

C.1 Pulsed Magnetic Kicker Tracking Code (Mathematica) . . . . . . . . . . . . . . . 205
Bibliography

211

List of Figures
1.1

Feynman diagrams for aSMµ . From left to right: first order QED contribution,
lowest order weak and hadronic contribution [3]. . . . . . . . . . . . . . . . . . .

1.2

26

The comparison between the SM model prediction value from different groups
and the latest value of the muon g – 2 measurement from E821 [7]. . . . . . . . .

26

1.3

The storage ring of Brookhaven Experiment E821 [24]. . . . . . . . . . . . . . . .

29

1.4

The outline of J-PARC Muon g – 2/EDM experiment [29]. . . . . . . . . . . . . .

31

1.5

The overview of J-PARC Muon g – 2/EDM experiment storage magnet. . . . . .

33

2.1

The concept of the conventional beam injection into the storage magnet. The
beam is injected into the storage magnet by the use of a device called the inflector.
The red dotted line shows the injected beam with the help of an inflector, while
the green thick lines show the corrected beam orbit after the kick. To avoid thebeam hit at the wall of storage magnet within the first turn, a kicker system isneeded to correct the injected beam orbit. . . . . . . . . . . . . . . . . . . . . . .

36

2.2

The layout of the three-dimensional spiral injection scheme. . . . . . . . . . . . .

37

2.3

Left: Black dashed trajectory shows the beam injection into the storage magnet
without the kick. Black thick trajectory shows the case wherein a suitable kickis applied to the beam to stop the beam at the midplane. Right: Black dashedtrajectory shows the pitch angle of the beam with respect to the vertical positionof the kick; in this case trajectory shows the beam that has not been subjected
to a kick, the trajectory represented in black thick indicate that the pitch angle
reduced to zero when the kick is applied to the beam. . . . . . . . . . . . . . . .

2.4

The radial field component B~r and stopping volume for the kick angle 60 mrad asa function of the kicker duration. . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5

38
40

The coordinate system and function of weak focusing for the charged particle.
The charged particle exhibits simple harmonic motion under the influence of the
weak focusing in the horizontal and vertical direction. . . . . . . . . . . . . . . .

41

2.6

Frenet-Serret coordinate system. . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

2.7

The beam phase-space ellipse in the horizontal plane.

Twiss parameters(β x , αx , γx ) describes the shape and orientation of the beam phase-space ellipse. .
2.8

Gaussian distribution for horizontal beam phase-space.

The solid ellipse

represents the 1σ of the Gaussian distribution. . . . . . . . . . . . . . . . . . . .
10

47
49

2.9

Left: Beam injection without phase space matching. The below plots shows thebeam cross section at the injection point. The storage magnet coordinates arereferred as global coordinates (Xg , Yg , Zg ). For the convenience, the beamphase-space at the injection point is shown in beam coordinates (X, Y, Z). It
is evident that without phase space matching beam drastically blow-up in thevertical direction. Right: Beam injection with phase space matching. Whenappropriate XY-coupling (phase space matching) was applied to the injected
beam, the vertical blow-up reduce sufficiently. . . . . . . . . . . . . . . . . . . . .

3.1

The side view of the 3D model of the three-dimensional Spiral Injection Test
Experiment (SITE). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

56

Left: Electric potential contour plots of the electron gun. Right: The electron
beam trajectories through electron gun. . . . . . . . . . . . . . . . . . . . . . . .

3.6

55

Photo of the LaB6 cathode used in electron gun. The diameter of the cathode is
2 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

54

Above: Layout of electron gun high voltage (H.V) power supply station. Below:
H.V station and electron gun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

53

Above: A Photo of SITE setup. Below: A top view photo of the straight beamline,with each component labeled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

50

57

Left: The electron gun was operated at 80 keV. H.V of electron gun was recorded
as a function of time to check the stability during the operation. Right: Currentof electron gun as a function of time. . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

58

Left: Cut view of 3-D model of SITE’ magnetic lens. Right: Axial magnetic fieldof magnetic lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.8

Schematic of electric chopper system. . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.9

Right: Mechanical structure and dimension of chopper system. Left: Potential
contours on the electrodes.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.10 Left: H.V electrode attached to the vacuum feed through and its support. Right:
Chopper’s electrodes in vacuum chamber . . . . . . . . . . . . . . . . . . . . . . .

62

3.11 Collimator used at SITE’s beamline. The diameter of the collimator is 3 mm anddepth is 5 mm. The black burn mark is due to the beam hitting on the collimator. 63
3.12 Three rotating quadruploes and their rotation system. . . . . . . . . . . . . . . .

64

3.13 The field measurement setup for the rotation quadrupole system. . . . . . . . . .

66

3.14 Above: The focusing strength of the quadrupole magnet as a function of theexcitation current. The focusing strength increase linearly with respect to theapplied current. Below: The focal length of the quadrupole magnet as a functionof the excitation current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

3.15 The magnetic field measurement of the rotation quadruple magnets system inthe longitudinal direction. Above graph shows the axial field measurement athorizontal position x = ±5, and below plot presents the axial field measurement atvertical position y = ±5. Color lines dotted lines represents the field measurementand dashed line is showing the CST simulation results. . . . . . . . . . . . . . . .
3.16 Transverse magnetic field profile of the quadrupole magnet.

68

Color dots areshowing the field measurement results and black line are CST simulation results.

69

3.17 The rectangular dipole magnet three dimensional model. . . . . . . . . . . . . . .

71

3.18 Left: The rectangular dipole magnet. The diople magnet at the SITE beamline.

71

3.19 Left: The blue solid line shows the CST simulation and red dots are correspondto the field measurement results.

Right: The percentage difference between

measured values and CST simulation. . . . . . . . . . . . . . . . . . . . . . . . .

72

3.20 The role of second dipole magnet is shown. In order to control the injected beam
angle with more freedom a second dipole magnet was added to the injection beam
pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.21 Left: The 3-D Model and dimensions of the second dipole magnet. Right: The
second dipole magnet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

3.22 Left: The field measurement and CST simulation of second bending magnet.
Right: The percentage difference between measured values and CST simulation. .

74

3.23 Left: Schematic view of storage magnet with reference trajectory and monitors.
Right: Storage magnet and vacuum chamber. . . . . . . . . . . . . . . . . . . . .

76

3.24 Above: The axial field of the storage magnet along with the vertical position of
the storage magnet. The black solid lines are the CST-simulation and color points
are the experimental field measurement. The current of the main coil was fixed to
8 A and correction coil current was changed from 0-10 A (opposite polarity to themain coil). As the correction coil’s current increases the weak focusing potential
increases. Below: The radial field of the storage magnet along with the vertical
position of the storage magnet. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1

The energy loss of the electron beam in nitrogen gas. The red point indicates theenergy loss for the 80 keV electron beam. . . . . . . . . . . . . . . . . . . . . . . .

4.2

77

80

The setup of the GM at the top of the storage magnet. The GM consists of thevalve for nitrogen gas inlet and CCD camera. The picture on the right shows
the inside view of storage magnet’s vacuum chamber and the red dotted line
corresponds as expected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.3

Configuration of the optical system. . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.4

Inside view of the storage magnet’s vacuum chamber depends upon thecharacteristics of the lens and camera. . . . . . . . . . . . . . . . . . . . . . . . .

83

4.5

Two copper sheets of height (350 mm) had been joined together to make a copper
sheet of height 700 mm. The black sheet of height 700 mm and width 1257 mm
was attached to the thin copper sheet by Araldite Rapid Epoxy. . . . . . . . . .

4.6

The procedure of cutting copper and black sheets, as well as placing the black
sheets inside the vacuum chamber. . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

83

84

Left: Vacuum chamber without the black sheet. The light reflection from the wallof the vacuum chamber can be seen very clearly. Right: Vacuum chamber withthe black sheet. The light reflection suppressed considerably after the insertionof the black sheet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

84

Above: The beam injection into the storage magnet without the black sheet.
Reflection from the wall of the vacuum chamber is the major source of background.
Below: The beam injection into the storage magnet with the black sheet. The
background decrease dramatically with black sheet insertion. . . . . . . . . . . .

4.9

86

The dimensions of the storage magnet vacuum chamber and position of the wire
scanner 1 and 2 with respect to the center of the vacuum chamber.

These

dimensions were used to calibrate the center of the wire scanners and storage
magnet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

4.10 The principle and mechanics of the wire scanner. . . . . . . . . . . . . . . . . . .

89

4.11 Above: Wire scanner 1 consists of four segmented horizontal wire to measure
the vertical beam profile and one vertical wire to measure the horizontal beamprofile. Below: Wire scanner 2 consists of only four segmented horizontal wires
to measure the vertical beam profile. . . . . . . . . . . . . . . . . . . . . . . . . .

89

4.12 The principle of the horizontal beam size measurement. The black circle shows
the beam orbit (diameter = 24 cm). The red circle presents the circular pathon which vertical wire traverse to measure the horizontal beam profile (diameter= 35 mm). The vertical wire was placed at 35 mm away from the center of thewire scanner 1 support. In order to extract the horizontal beam size the verticalwire was placed at the fixed vertical position, and wire scanner support bar was
rotated at its own axis by the rotary actuator. As vertical wire rotates it provides
the information of horizontal beam position and horizontal beam size. . . . . . .

90

4.13 Magnetically-coupled Rotary/Linear Feedthrough (MRLF) for the vertical androtary movement of the wires. On the left is the 3-D model of the MLRF [66] andto the right shows the MRLF used at the SITE. . . . . . . . . . . . . . . . . . . .

92

4.14 Above: The 3-D model of the linear actuator [67]. Below: The linear actuator forthe vertical movement of the MRLF. . . . . . . . . . . . . . . . . . . . . . . . . .

93

4.15 The 3-D model of the rotary actuator [68]. The rotary actuator used for therotation of the MRLF for the horizontal beam size measurement. . . . . . . . . .

94

4.16 Above: The velocity of the actuator was set to 1 mm/sec. To check the linearity,
data were taken at different points and fitted with a linear function. Below:
Actuator set values versus digital scale readout. To check the linearity between
the set value and readout, data was taken. The set values and readout values
agree well within the uncertainty of the digital readout system (±50µm). . . . .

95

4.17 (a) Schematic of the two-wire scanner setup. (b) Wire scanner system on the topof the storage magnet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

4.18 The control system layout for the actuator and wire scanner. . . . . . . . . . . .

97

4.19 The screenshot of the oscilloscope control program. This program was used forthe wire scanner data online plotting and data logging. . . . . . . . . . . . . . . .

98

4.20 The beam current at the wire as a function of the cathode’s heater current of theelectron gun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1

98(a) The quadrupole scan setup. The quadrupole scan had been done with all three
quadrupoles independently. (b) The cross section of the quadrupole magnet. The
bore radius of quadrupole magnet is 25 mm and effective length is 56 mm. . . . . 102

5.2

The copper plate used to observe the beam profile (via OTR). The horizontal andvertical lines were drawn on the copper plate in order to determine the conversion
factor from the pixels data of the photo to the millimeter. . . . . . . . . . . . . . 103

5.3(a) A raw picture from the CMOS-camera without applying any quadrupole
magnet. (b) The above figure converted from RGB to the Grayscale for further
analysis.

The conversion factor from pixels to mm is 10 Pixels/mm in bothtransverse planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4(a) and (b) are the beam profiles in the vertical and horizontal focus respectively.
(c) and (d) represents the beam profile data for vertical and horizontal planes. . 105

5.5

The change in RMS beam size due to the different level of the background. . . . 106

5.6

The beam position shift due to the quadrupole magnet misalignment error. The
black points are weighted average of data points. And red line is the linear fit to
the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.7(a), (c) and (e) shows the vertical beam size square as a function of the quadupole
magnet 1, 2 and 3 respectively. Whereas (b), (d) and (f) shows the horizontal
beam size square as a function of the quadupole magnet 1, 2 and 3 respectively.
The color points in each picture are correspond to the repeated measurement andblack points are the weighted mean of the repeated measurement. . . . . . . . . . 109

5.8

The beam images with different strength of vertical focused quadrupole magnet
1. False colors were added to the pictures for the better visibility. . . . . . . . . . 110

5.9

The beam images with different strength of horizontal focused quadrupole magnet
1. False colors were added to the pictures for the better visibility. . . . . . . . . . 111

5.10 The beam size squared as function of quadrupole strength. (a) and (b) shows
the vertical and horizontal scan for the quadupole 1 respectively. (c) and (d)shows the vertical and horizontal scan for the quadrupole 2. (e) and (f) shows thevertical and horizontal scan for the quadrupole 3. . . . . . . . . . . . . . . . . . . 112
5.11 the comparison of the emittance values between experimantal results and CST-PS
simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.12 The βx and βy at Q1,Q2, Q3 and their transformation to the collimator position. 119
5.13 The αx and αy at Q1, Q2, Q3 and their transformation to the collimator position. 120
5.14 The γx and γy at Q1,Q2, Q3 and their transformation to the collimator position. 121
5.15 The phase space distribution at the collimator. This phase space satisfy theemittance and Twiss parameters given in the Table 5.9. . . . . . . . . . . . . . . 122
5.16 The layout of the beamline for the 4-D beam matrix reconstruction. . . . . . . . 126
5.17 (a) The gray-scale (setting#14) picture. (b) The 3-D histogram of the above
picture. The background was estimated as a 2D slope picture data. The red patchshows the background estimation. (c) The green highlighted region indicts thearea selected for the RMS beam size measurement after the background estimation.127
5.18 The beam images with different settings. False colors were added to the pictures
for the better visibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.1

Left: A horizontal flat beam was assumed inside the storage magnet and reverse
track it to find the required beam slopes at the injection point. Right: The
beam frame is used to describe the beam shape at the injection point. X andY represents the horizontal and vertical planes respectively, and Z is the beam
motion direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.2

The slopes of the beam correlations at the matching point Yg = −600 mm. The
beam correlations plots are shown in the beam coordinates. The dots represents
the data points obtained from the reverse tracking at the matching point and solidline (red line) are the linear fit to these points. . . . . . . . . . . . . . . . . . . . 131

6.3

The input beam phase-space distribution. The the values of emittance and Twiss
parameter for these distributions are given in Table 6.2. . . . . . . . . . . . . . . 136

6.4

The coupled beam phase-space at the matching point. The line (blue) are therequired beam slopes at the injection point. . . . . . . . . . . . . . . . . . . . . . 136

6.5(a) The beam injection into the storage magnet without phase space matching,
the phase-space distribution and the emittance and Twiss parameters are given in
Fig. 6.3 and Table 6.2 respectively. (b) The beam injection into the storage magnet
with phase-space matching, the phase-space distribution and the emittance andTwiss parameters are given in Fig. 6.4 and Table 6.3, 6.4 respectively. . . . . . . 137

6.6

The slopes of the beam phase-space at the matching point. Error of 5%, 10%,15%, and 20% was introduced to each correlation in order to find their effect on
the vertical beam size in the storage magnet. . . . . . . . . . . . . . . . . . . . . 138

6.7(a) The beam injection into the storage magnet with 0% error on the axy slope,
while other slopes (axx0 , ayy0 ) were kept at the original values. Where (b), (c),and (d) shows the beam injection into the storage magnet with 5%, 10%, 15%,
error on slope axy slope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.8

The vertical beam size growth at the kick point due to the change in the slope
values. The biggest growth in the beam size was observed due to the change in
X − Y correlation slope. And the (X, X 0 ) correlation has the least effect on the
beam growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.9

The evolution of the beta function and beam size through the array of six
quadrupole magnets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.10 Left: The 3-D plot of tracking simulation of the beam injection with appropriate
coupling as mentioned in the Table 6.3 and 6.4. Right: The histogram of thevertical beam size as a function of the vertical beam position. . . . . . . . . . . . 142
6.11 (a) The layout of the beamline, beamline composed of three rotatable quadrupole
magnets, a rectangular bending magnet and drift space. The rotatable quadrupole
magnets were placed in the downstream direction of the collimator.

Thedimensions and mutual distances among magnets are also shown. (b) The photoof the experimental setup of the matching beamline. . . . . . . . . . . . . . . . . 145
6.12 The simulated beam phase-space with the three rotation quadrupole magnets.
The solid blue line shows the ideal slopes for the beam phase-space matching.
The beamline setting is described in the Table 6.7. . . . . . . . . . . . . . . . . . 146
6.13 The beam profile of the injected beam. The false colors were added to the picture
for better visualization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.14 Above: The simulation of the beam phase-space distribution without applying
quadrupole magnets.

Below: The injected beam profile at the beam profile

monitor. The false colors were added for the better visual effects. . . . . . . . . . 150
6.15 Above: The gas monitor results of the beam injection into the storage magnet
without rotation quadrupole magnets. The false colors were added to the photofor better visualization. It is evident from the gas monitor results that the beam
blow-up in vertical direction as it traverses through the storage magnet. . . . . . 151
6.16 (a) The tracking simulation of the beam injection ( CST-PS) into the storage
magnet without phase-space matching. The black open circle shows the points at
which the beam sizes were measured. (b) The vertical beam size was measured
by using a wire scanner. The black points are the wire scanner 1 data. The green
histogram is a simulation result of CST-PS. The orange line is a global fit toexperimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.17 (a) The tracking simulation of the beam injection (CST-PS) into the storage
magnet without phase-space matching. The black open circle shows the points at
which the beam sizes were measured. (b) The black points are the wire scanner
2 data. The green histogram is a simulation result of CST-PS. The orange line is
the fit to experimental data. The wire scanner 2 is located at 180 Degree away
(cyclotron period) from the wire scanner 1 position. . . . . . . . . . . . . . . . . 153
6.18 (a) The beam phase-space at the injection point with the Twiss parameters aslisted in the Table 6.3 , but coupling parameters were set to zero. (b) Particle
Tracking with the beam phase-space as mentioned in the in (a) as the beam
traverse through the storage magnet beam blow-up drastically in the verticaldirection. It signifies that the xy-coupling to the beam cross section is the mostimportant parameter to controls the vertical beam spread in the solenoidal-type
storage magnet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.19 Above: The gas monitor results of the beam injection into the storage magnet
with rotation quadrupole magnets set up as mentioned in Table 6.7. Below: The
horizontal projection of the above picture. The three turns can be visualized easily
from this data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.20 (a) The tracking simulation of the beam injection ( CST-PS) into the storage
magnet without phase-space matching. The black open circle shows the points
at which the beam sizes were measured. (b) The vertical beam size is measured
by using a wire scanner. The black points are the wire scanner 1 data. The redhistogram is a simulation result of CST-PS. The orange line is the global fit toexperimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.21 (a) The tracking simulation of the beam injection ( CST-PS) into the storage
magnet without phase-space matching. The black open circle shows the points at
which the beam sizes were measured. (b) The vertical beam size was measured
by using a wire scanner. The black points are the wire scanner 2 data. The
red histogram is a simulation result of CST-PS. The orange line is the fit toexperimental data.

The wire scanner 2 is located at 180 Degree away (halfcyclotron period) from the wire scanner 1 position. . . . . . . . . . . . . . . . . . 159
6.22 (a) The 3-D plot of the 2-D projection of the beam profile in the storage magnet.
The positive X-axis corresponds to the wire scanner 1 location and negative show
the wire scanner 2 location. The black histogram is the CST-PS simulation andgrey plot show the experimental data. (b) The zoom up view of the wire scanner
1 location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.23 The horizontal beam profile at different vertical positions. Average horizontal
beam profile form first turn to the second turn was 0.65 mm (1σ) . . . . . . . . . 161

6.24 The comparison of the vertical beam sizes growth in the storage magnet withoutphase-space matching, with appropriate combination of rotatable quadrupole
magnets and simulated perfect matching. . . . . . . . . . . . . . . . . . . . . . . 163
6.25 The comparison of the pitch angle between without phase space matching andwith appropriate combination of rotatable quadrupole magnets. . . . . . . . . . . 164
7.1(a) The concept of the vertical kicker. The beam enters in the storage magnet at
40 Degree and radial field of the storage magnet reduce the injection to 4-5 Degree
as it approaches the boundary of the weak focusing field. At Yg = −100 mm a
vertical kick applied to the beam to gradually reduce the pitch angle of the beam
to the zero. (b) The pitch angle of the injected beam. And its reduction due tothe radial field of the storage magnet and kicker field. . . . . . . . . . . . . . . . 166

7.2

The character of the symmetrical kicker. The orange line shows the reduction in
the pitch angle of the beam from the injection point to the kick point. The blue
curve shows the kick due to the symmetrical kicker, the pitch angle of the beam
reduces smoothly in the of the symmetrical kicker. After applying kick the pitch
angle reduces to the zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.3

The symmetrical kick and separatrix. The blue curve is the symmetrical kick.
The closed circles shows the stable storage orbit. If the particles lands on theclosed circle after the kick the will get store. Th red circles shows the storage
orbit for Yg = ±30 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7.4

The character of the localized kicker. The orange line shows the reduction in thepitch angle of the beam from the injection point to the kick point. The black
curve shows the kick due to the localized kicker, the pitch angle reduces to zeroin 10 steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.5

The localized kick and separatrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.6

The oscillation of the pitch angle in the weak focusing field. . . . . . . . . . . . . 172

7.7

The oscillation of the beam in the vertical direction in the weak focusing field. . 172

7.8

Above: The design of the symmetrical kicker. Below: The radial magnetic fieldof the symmetrical kicker coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.9

Above: The design of the localized kicker. Below: The kicker field of the localized
kicker coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7.10 The vacuum chamber and kicker symmetrical kicker coils. . . . . . . . . . . . . . 177
7.11 The average ohmic losses in the wall of the vacuum chamber due to thesymmetrical kicker coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.12 The vacuum chamber and kicker localized kicker coils. . . . . . . . . . . . . . . . 178
7.13 The average ohmic losses in the wall of the vacuum chamber due to the localized
kicker coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.14 The power loss in the wall of the vacuum chamber due to the symmetric andlocalized kicker coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.15 The magnetic field reduction in the case of symmetrical kicker due to eddy current
flow in the wall of the vacuum chamber. . . . . . . . . . . . . . . . . . . . . . . . 179
7.16 The magnetic field reduction in the case of localized kicker due to eddy current
flow in the wall of the vacuum chamber. . . . . . . . . . . . . . . . . . . . . . . . 180
7.17 (a) The circuit diagram of the kicker power supply. (b) The current through kicker
coils. (d) The voltage across kicker coils. . . . . . . . . . . . . . . . . . . . . . . . 181
7.18 The beam vertical phase-space at the kick point.(a) The vertical phase space
acceptance of the kicker. (b) The vertical phase-space at the kick point withphase-space matching. (c) The same vertical phase-space as mentioned in (b)except flipped it’s sign to the focusing. This case has been simulated in order toshow the injection efficiency in the ideal beam phase-space at the kick point. (d)The vertical phase-space at the kick point without any phase-space matching. . . 183
7.19 The kicker tracking with the phase-space matching (XY-coupled) to estimate theinjection efficiency. Th grey lines shows the separatrix for the field index 1.65 ×
10−2 . The closed contours presents the stable region and open hyperbolas shows
the un-stable region. The red closed lines are presents the stored beam, whereas,
open line shows the lost particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.20 The kicker tracking with flipped XY-coupled phase space.

The closed blue

contours represents the stored beam. In this case 100% beam get stored. . . . . . 185
7.21 The tracking of the beam in transient field of the kicker with the energy spreadof 4E/E = 6 × 10−4 . In this scenario, the injection efficiency reduce to 67%. . . 187
A.1 The screenshot of the DPO–7104 oscilloscope control program. . . . . . . . . . . 195
B.1 The beam phase space with setting others than best run are shown. In (a) and(b) (X, Y ), and (X, X 0 ) were matched preferentially while (Y, Y 0 ) was ignored.

201

B.2 The wire scanner a and 2 results with beam phase–space setting as shown in Fig.B.1.202
B.3 The comparison of the vertical beam sizes growth in the storage magnet with thephase space settings as shown in Fig. B.1. . . . . . . . . . . . . . . . . . . . . . . 203

List of Tables
3.1

Comparison of Parameters Between E34 and SITE. . . . . . . . . . . . . . . . . .

52

3.2

Electron gun specifications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.3

Electric Chopper System Parameters. . . . . . . . . . . . . . . . . . . . . . . . .

61

3.4

Rotating Quadrupole Magnet Parameters. . . . . . . . . . . . . . . . . . . . . . .

64

3.5

Dipole Magnet 1 Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

3.6

Dipole Magnet 2 Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.7

Storage Magnet Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

4.1

Wire Scanner Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.1

The Quadrupole Magnets Vertical Misalignment. . . . . . . . . . . . . . . . . . . 106

5.2

The Quadrupole Magnets Horizontal Misalignment. . . . . . . . . . . . . . . . . . 106

5.3

The beam size square (Weighted Mean) and quadrupole magnet strength values
for Q1-scan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.4

The beam size square (Weighted Mean) and quadrupole magnet strength values
for Q2-scan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.5

The beam size square (Weighted Mean) and quadrupole magnet strength values
for Q3-scan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.6

The transverse emittance values form Q1, Q2, and Q3 scans. . . . . . . . . . . . 116

5.7

The values of Twiss parameters from Q1, Q2, and Q3 scans at the quadrupole
magnet location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.8

Twiss parameters transformation from Q1,Q2 and Q3 to the Collimator . . . . . 118

5.9

The weighted mean of Emittance and Twiss parameters at the Collimator . . . . 118

5.10 The drift spaces of the beamline. . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.11 The beamline setting and respective values of σ1,11 , σ1,33 , σ1,13 . . . . . . . . . . . 126
5.12 The values of the beam matrix elements at the reconstruction point. . . . . . . . 128
5.13 The Coupling Strength for the SITE beamline. . . . . . . . . . . . . . . . . . . . 128
6.1

The beam slopes at the matching point. . . . . . . . . . . . . . . . . . . . . . . . 132

6.2

The initial beam phase space parameters. . . . . . . . . . . . . . . . . . . . . . . 135

6.3

The beam phase-space parameters at the matching point. . . . . . . . . . . . . . 135

6.4

Coupling Parameters for the Matched Beam Phase Space. . . . . . . . . . . . . . 135

6.5

Matching beamline quadrupole magnets parameters . . . . . . . . . . . . . . . . 143
20

6.6

The drift spaces for the beamline. . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.7

The Three Rotatable Quadrupole Magnets Matching Beamline Parameters. . . . 146

6.8

The Comparison of the Beam Slopes between the Simulation and Measured fromthe Beam Profile Monitor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.9

The Vertical Beam Sizes as a function of the Vertical Position of the Storage Magnet162

6.10 The Pitch Angel as a Function of the Vertical Position of the Storage Magnet . . 164
7.1

The beam position and pitch angle at kick point. . . . . . . . . . . . . . . . . . . 168

7.2

Symmetrical Kicker Coil Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.3

Localized Kicker Coil Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.4

Power loss due to the Eddy Current. . . . . . . . . . . . . . . . . . . . . . . . . . 176

7.5

Kicker Power supply Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.6

Injection Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Chapter 1

Introduction
Ever since the discovery of the muon [1], it has been observed to play a vital role inunderstanding our universe. The anomalous magnetic dipole moment of the muon is one of
the intriguing measurements that can extend our knowledge of physics beyond the Standard
Model. The most recent measurement of the muon's g – 2 at BNL(E821) was achieved with adiscrepancy of more than 3.5 σ as compared with the equally precise Standard Model prediction.
This unresolved discrepancy could be indicative of the establishment of new physics beyond
the Standard Model. Therefore, a new J-PARC (Japan Proton Accelerator Research Complex)
muon's g – 2/EDM (E34) is under preparation to resolve this fascinating discrepancy. The E34
experiment will employ a completely new approach to measure the muon's g – 2. The final goalof the E34 is to measure the muon's g – 2 with a precision of 0.1 ppm and the electron dipole
moment (EDM) down to the value of 10−21 e.cm.
To measure the muon's g – 2 a low-emittance (0.5 π mm mrad) polarized muon beam will be
stored in a precise magnetic field to determine the evolution of the spin precession vector withrespect to time. In the E34 experiment, the muon beam with a momentum of 300 MeV/c will be
injected into a 3-T magnetic resonance imaging (MRI)-type solenoid magnet to store the muon
beam on a 0.66 m diameter orbit. The MRI-type storage magnet will provide an unprecedented
local field uniformity of 0.1 ppm. A new three-dimensional spiral injection scheme has been
invented to inject the beam into the MRI-type magnet. This new injection scheme will enhance
injection efficiency and overcome technical challenges that are related to the small storage orbit
diameters.
The three-dimensional spiral injection scheme is an unprecedented injection idea; therefore,a demonstration experiment to establish the feasibility of this new injection scheme is inevitable.
Herein, I describe the demonstration of a novel spiral injection scheme with the electron beam
for the E34 experiment.
The Spiral Injection Test Experiment (SITE) consisting of a 2 m long straight beamline withenergy of 80 keV and a solenoid storage magnet to store the electron beam has been designed
and built. The requirement of the XY-coupled beam for the solenoid-type storage magnet is23

1.1. STRUCTURE OF THIS THESIS
confirmed, and the beam monitor for the beam position and profile in the storage magnet toconfirm appropriate matching is also discussed. The kicker system to store the electron beamat the midplane of the storage magnet has also been developed. Other important aspects of the
spiral injection scheme will also be illustrated in this thesis.

1.1

Structure of this thesis

This thesis is structured into eight chapters. A brief description of each chapter is as follows:• The following sections of this chapter will briefly describe the physics background of the
Muon's g – 2/EDM properties. The principle of the Muon's g – 2/EDM experiments andcurrent status will also be discussed.
• Chapter 2 describes the basic idea and requirements of a spiral injection scheme. The
concept of a kicker system to guide the beam to the storage region and the weak focusing
system for the beam confinement has been described and discussed along with the phasespace requirement for the solenoid-type magnet.
• The description of the SITE with the detail of each component is discussed in Chapter 3.
• Chapter 4 explains the development of the beam monitors for the solenoid-type storage
magnet. Two kinds of monitors have been built for beam monitoring in the storage magnet,
namely: a non-invasive gas monitor for qualitative monitoring and a wire scanner for
quantitative monitoring.
• Chapter 5 presents DC beam commissioning at SITE. The emittance of the beam at theentrance of the matching beamline has been measured by the quadrupole scan is describedin detail.
• Chapter 6 explains the experimental results of phase space matching and its effect onvertical beam size reduction.
• To stop the beam at the midplane of the storage magnet, a kicker system is needed. A
detailed simulation study of the kicker and its development is described in chapter 7.
• Chapter 8 provides the conclusion and future outlooks.

1.2

Muon's Anomalous Magnetic Dipole Moment

The Muon possesses an intrinsic magnetic field and spin angular momentum. When muons
are placed in an external magnetic field, their internal magnet tends to align (rotate) itself
with the external magnetic field. The strength of the internal magnet and the rate of the24CHAPTER 1. INTRODUCTION
magnet's gyration determine the muon's gyro-magnetic ratio. For muons, the relation between
the magnetic dipole moment and the spin is as follows:µ~ µ = gµ

e
~s
2mµ

(1.1)

Where, µ~ µ is the magnetic dipole moment (MDM), g is the gyromagnetic ratio, e is theelementary charge on muon, mµ is the mass of muon, and ~s is the spin of muon. In Dirac's
quantum theory the value of g is equal to 2 for all fundamental particles like muons [2].
gµ = 2(1 + aµ ),aµ =gµ − 22(1.2)

The aµ is known as an anomaly or anomalous magnetic dipole moment of the muon. In
the framework of the Standard Model (SM) of particles of physics, the small finite value of aµappears due to the quantum fluctuations or the exchange of virtual particles. The SM value canbe shown as a sum of three gauge sectors [3]eakaSM= aQED+ aW+ ahad
µ
µ
µ
µ

(1.3)

Figure 1.1 shows the Feynman diagram for the contributions from QED, weak and hadronic
sectors [3].

=The most recent prediction of QED and weak contributions are: aQEDµ

eak
(11658471.90 ± 0.01) × 10−10 [4] and aW= (15.36 ± 0.1) × 10−10 [5], respectively. Theµ

results
hadronic contribution was recently reported in [6, 7]. The total SM prediction for aSMµ
inaSM= (11659181.08 ± 3.78) × 10−10
µ

(1.4)

The Muon g–2 experiment (E821) at Brookhaven National Lab (BNL) [8] concluded aExp
asµfollows:aExp= (11659209.10 ± 6.33) × 10−10
µ

(1.5)

The deviation from SM prediction is:
4aµ = aExp
− aSM
= (28.02 ± 7.37) × 10−10
µ
µ

(1.6)

This difference leads to the 3.8σ discrepancy between aµ value from SM prediction and BNL g–2 experiment measurement.
Even though the SM is theoretically successful, its inability to include gravity and lack of
explanation for dark matter has led many scientists to believe that it is incomplete. The aµ could
be a signal of New Physics (NP) beyond the SM. One of the theoretical schemes for NP would
be the supersymmetric (SUSY) expansion of the SM. SUSY implements a symmetric mapping
between fermions (spin 12 ) and bosons (integer spin) by changing the spin by 1/2 units. The
SUSY scenario can be an explanation for the muon anomaly. For a SUSY mass equal to Λ, thecontribution to aµ is given by [9, 10]:
SY ∼aSU
= sign(µ)130 × 10−11 tan βµ

100 GeV
Λ

2

(1.7)

Where, the factor tan β is the ratio of the vacuum expectation values of the two Higgs fields,
the sign(µ) is the SUSY µ parameter.
25

1.2. MUON'S ANOMALOUS MAGNETIC DIPOLE MOMENT

Figure 1.1: Feynman diagrams for aSMµ . From left to right: first order QED contribution, lowest
order weak and hadronic contribution [3].

Figure 1.2: The comparison between the SM model prediction value from different groups andthe latest value of the muon g – 2 measurement from E821 [7].

26CHAPTER 1. INTRODUCTION

1.3

Muon's Electric Dipole Moment

The finite value of the permanent EDM of elementary particles violates the parity (P)and time-reversal (T) symmetries under the CPT theorem, where C represents the charge
conjugation.

According to the SM, the EDMs of elementary particles is far beyond theexperimental sensitivity. However, in extensions of the SM, obtaining larger EDMs of elementary
particles is possible. Hence, any non-zero value of EDM could be an indicator of new physics [11].
The EDM of the muon can be shown as follows:d~µ = ηµ

e
~s
2mµ c

(1.8)

Where, ηµ is the size of the EDM, e and m are the charge and mass of the muon, and ~sis the muon spin. The muon’s EDM and aµ can be derived from similar operators; therefore,aµ discrepancy from the SM can be an evidence of a non-zero EDM [12]. This is one of the
motivation to search for the muon's EDM simultaneously with the aµ measurement. The E821
provided the most stringent constraint on the EDM value [13]:d~µ ≤ 1.8 × 10−19 e.cm

1.4

(1.9)

Historical Glimpse of Muon g – 2 Experiments

The first aµ measurement was performed in 1957 by Garwin, et al., and they reported a
final value of gµ = 2(1.00113) with the precision of 0.007% [14]. This measurement agrees wellwith Schwinger’s first order QED calculation [15]. After Garwin, et al., there was motivation
to test the QED more stringently to observe the failure of QED. Therefore, a new measurementof aµ with the precision of 5164 ppm was implemented at the CERN [16, 17]. The first CERN
experiment concluded on the following aµ value with the precision of 5164 ppm:
aµ = (1162 ± 5) × 10−6

(1.10)

The first CERN-I experiment was performed with a non-relativistic muon beam at a rest
muon lifetime of 2.2 µs. Therefore, a measurement period to a few lifetimes was the limitationof the CERN-I experiment.
The next experiment at CERN was conducted with a storage ring whose diameter was 5 m.
The magnetic field was 1.711 T, P~µ = 1.27 GeV/c, and γµ = 12.06. A weak magnetic focusing
was used to confine the beam in the storage magnet [18]. The CERN-II experiment concluded
on a aµ value with a precision of 266 ppm:
aµ = (11661 ± 3.1) × 10−7

(1.11)The storage ring of the CERN-II experiment contained a pion production target inside thering. Therefore, the background from the pion production target installed inside the storage
ring deteriorated the performance of the detector, and limited the sensitivity of the CERN-II
experiment.
27

1.5. THE BNL MUON'S G – 2 EXPERIMENT (E821)
The CERN-III [19] experiment was performed with the following four improvements over the
CERN-II experiment:• Direct injection of a pion beam into the storage ring, which reduces the background forthe detector• Improved injection: Use of a device called an inflector for the beam injection to cancel thefringe field of the storage magnet• Longer muon lifetime 64.4 µs with the use of a magic momentum γ = 29.3, which results
in an increased measurement time
• The more homogeneous magnetic field as compared to CERN-2
The CERN-III concluded on a aµ value with the precision of 7.3 ppm
aµ = (1165924 ± 8.5) × 10−9

1.5

(1.12)The BNL Muon's g – 2 Experiment (E821)

To further enhance the precision of aµ compared to the CERN—III, the muon g – 2
experiment at Brookhaven National Laboratory (BNL) (E821) was started in 1984. E821 aimed
to measure the muon anomaly up to a precision of 0.54 ppm, where the electroweak contribution
becomes measurable. This level of precise measurement of the muon anomaly is also a stringent
constraint for new physics.
The BNL g – 2 experiment used the same experimental techniques as in CERN-III. However,
several ingenious techniques have been introduced to enhance the precision of aµ . The 24 GeV/cproton beam from the BNL Alternating Gradient Synchrotron (AGS) strikes at the pion
production target.

This secondary pion traverses through the 80 m long pion-muon-decay

channel. The tertiary-polarized muon beam from pion decay is directly injected into the storage
ring of 711 cm.
A superconducting inflector [20] was used to inject the beam into the storage magnet. The
inflector cancels the fringe field of the storage magnet. The storage ring [21] was a continuous
superconducting magnet of field strength 1.45 T. Electric quadrupoles were used to confine themuon beam into the storage magnet. A pulsed kicker system [22] was also placed in the storage
ring to prevent the beam from hitting the wall of the inflector within the cyclotron period of 149 s.
Figure 1.3 shows the storage ring of the BNL g – 2 experiment.
The decay electrons were detected by the calorimeters to measure the precession frequency.
The magnetic field was measured with nuclear magnetic resonance probes [23]. The BNL g – 2
experiment concluded on a final value of aµ after the series of measurement with the precisionof 0.54 ppm [8]aµ = (116592089 ± 54 ± 33 ± 63) × 10−11(1.13)

Where, the first error is statistical, the second is systematic, and the third is the total error.
28

CHAPTER 1. INTRODUCTION

Figure 1.3: The storage ring of Brookhaven Experiment E821 [24].

1.5.1

Principle of Measurement

The measurement principle of the anomalous magnetic moment of muon in the BNL g – 2
experiment was the same as in CERN-III. The cyclotron frequency for the muons circulating inthe storage ring in the presence of the magnetic and electric field is given by:
"
!#
~ ×E
~
γ
B
e β~
− 2
ω
~c = −m γ
γ −1c(1.14)~ E~ are the magnetic andWhere, e is the elementary charge, m is the mass of particle,Belectric fields, respectively. β and γ are relativistic factors given as:
~v
β~ =
c

,

1
~γ = p1 − β2(1.15)The spin rotation frequency was first derived by the [25]. Here, I will present the version given
in [26]eω~s = −
m

"

! #






~
g−2 1 ~
g−2
γ
g
−
2
1
E
~B
~ β~ −
+
B−
+
β.
+
β~ ×
β~
2
γ
2
γ+1
2
γ+1
c
!#
"
!
~
~
e η E
γ
E
~
−
−
β~ ×
β~ + β~ × B
m 2 c
γ+1
c
(1.16)The upper term in Eq. (1.16) corresponds to the precession frequency due to the g factor,
while the lower term correlates with the spin rotation due to the EDM. If we consider the value29

1.5. THE BNL MUON'S G – 2 EXPERIMENT (E821)
of η to be negligibly small, then the lower term will vanish. The difference between spin rotation
frequency and cyclotron frequency (~ωaµ = ω~s − ω~ c ) is called the anomalous spin precession
frequency (~ωaµ ). Further, by defining aµ ≡ g−2
2 Eq. (1.16) will become" ~ ~#
e
γ
1
β×E
~
~
~
~
ω
~ aµ = −aµ B − aµ(β.B)β − aµ − 2
m
γ+1
γ −1
c

(1.17)The BNL g – 2 experiment selected a so called magic γmagic ≈ 29.3 at γmagic ; consequently,
the third term in Eq. (1.17) vanishes along with the second term in Eq. (1.17) if the muon
only lies in a plane perpendicular to the magnetic field of the storage magnet. Subsequently,
Eq. (1.17) is simplified to the following form:ω~ aµ = −

e
~
aµ B
m

(1.18)In experiment, ω~ aµ is measured using the decay positron signals, i.e.,µ+ → e+ + ν µ + νe

(1.19)~ is measured by observing the Larmor frequency of stationary protons
The magnetic field B(~ωp ) with the NMR probes. In BNL g – 2 experiment magnetic field was determined with pulsed
nuclear magnetic resonance (NMR).
~
eB
2mp

ω
~ p = gp!
(1.20)~ becomes:
Where, gp is the gyromagnetic factor of the proton. On rearranging Eq. (1.20), the B~p~ = 2mp ωBegp(1.21)~ from (1.21) into (1.18) and solve for aµ :Put the value of B
gp mµ ω~a
2 mp ω
~p(1.22)~a
gp mµ ω(1 + aµ )gµ mp ω~p(1.23)

gµ mp
gp mµ

(1.24)aµ =Equation (1.22) can also be written as:aµ =Let’s define
λ≡

R≡

ω
~a
ω
~p

Hence aµ will become:
aµ =

R
λ−R

(1.25)

Here, λ = 3.183345137 ± 85 is the muon-to-proton magnetic moment ratio. This value
was measured by Muonium (the µ+ e− atom) hyperfine structure experiment [27].

A newhigh precision measurement of Muonium hyperfine structure experiment is under preparation at
J-PARC [28].
30CHAPTER 1. INTRODUCTION

Figure 1.4: The outline of J-PARC Muon g – 2/EDM experiment [29].

1.6

J-PARC Muon's g – 2/EDM Experiment

The J-PARC muon g – 2/EDM experiment (E34) intends to measure aµ with the statisticaluncertainty of 450 ppb and systematic less than 70 ppb. The EDM is probed down to the valueof 1.5 × 10−21 e.cm [29]. The overview of E34 is shown in Fig. 1.4.
The E34 will employ many new and novel approaches to reach the goal sensitivity of aµ anddµ . The salient features of E34 are as follows:• Low emittance: In E34, a primary proton beam of energy 3 GeV will hit the graphitetarget. The proton beam collision at the graphite target will produce π + (pions). These
pions decay near the surface of the target and produce surface muon beam of momentum
29.8 MeV/c and approximately 100% polarization.
π + → µ+ + νµ(1.26)

Since the surface muon beam is tertiary, it comprises a very large intrinsic emittance. A
surface muon beam transport line has been designed to transfer the muon beam to thecooling station [30]. To reduce the transverse emittance, the surface muon beam will hiton the laser-ablated silica aerogel. The majority of the muons will diffuse from the aerogeltarget into the vacuum region as the muonium atoms (Mu:µ+ e− ) [31, 32]. Subsequently,
Mu will be ionized by the laser to remove the e+ . Hence, a low emittance or thermal muon
beam will be developed in this way.
• Muon reacceleration: The thermal muons will be accelerated by the series of acceleration
method, to the momentum of 300 MeV/c. Recently, first stage muon acceleration by RFQ
has been successfully demonstrated [33].
• Compact storage magnet: A compact MRI-type solenoid magnet of field strength 3 T will
be used as a storage magnet for the E34. The choice of solenoid magnet has an advantage
31

1.6. J-PARC MUON'S G – 2/EDM EXPERIMENTof an unprecedented local field uniformity of 0.1 ppm at the storage plane. This storage
magnet has a small fiducial volume. The diameter of the stored beam will be 0.66 m.
Therefore, the conventional injection scheme is not applicable. A novel three-dimensional
spiral injection scheme has been invented to inject the beam into the compact storage
magnet [34]. Chapter 2 will describe this injection scheme in detail. Due to the injection of
the low emittance beam into the storage magnet, a very weak magnetic focusing (n ∼ 10−4 )is sufficient to confine the muon beam within a storage orbit. The storage magnet is shownin Fig 1.5.
As described earlier, a low emittance beam can be injected into the compact storage magnet
and a very weak magnetic focusing (n ∼ 10−4 ) can confine the muon beam. Therefore, no strong
electric focusing is necessary in the storage magnet.
ω~ =ω
~a + ω
~η

"

~ ~
η
e
β×E
~ − aµ − 1
aµ B
+
ω
~ =−
m
γ2 − 1c
2(1.27)~~+Eβ~ × Bc!#(1.28)

Since in E34 no electric field is present, the second term vanishes, and no magic momentumis needed; this is in contrast to the E821.
ω~ =−e h ~ η ~ ~ iaµ B + (β × B)m
2

(1.29)The precession vectors, ωa and ωη , are orthogonal to each other. Therefore, E34 is able tomeasure g – 2 and EDM simultaneously. The principle of aµ extraction is the same as mentioned
in Section 1.5.1.

32

CHAPTER 1. INTRODUCTION

Figure 1.5: The overview of J-PARC Muon g – 2/EDM experiment storage magnet.

33

1.6. J-PARC MUON'S G – 2/EDM EXPERIMENT

34

Chapter 2

Three–Dimensional Spiral
Injection Scheme
In this chapter, the need for the three-dimensional spiral injection scheme will be described. The
basic concept of the spiral injection and essential techniques to realize this injection scheme will
also be explained.

2.1

Why Three–Dimensional Spiral Injection?

Conventional accelerator injection systems contain an inflector magnet for the beam injection
into the storage ring. The inflector magnets are essential components of the injection system tominimize the interference from the storage magnet field to the injected beam bunches. The BNL
muon g – 2 experiment used the superconducting inflector [20] to cancel the storage magnet of
filed 1.7 T to avoid interference of the storage magnet field to the injected beam.
After the beam injection into the storage magnet, a kicker magnet is essential to correct the
injected beam orbit to avoid the beam hit at the wall of the inflector within one cyclotron period.
The cyclotron frequency (~ωc ) and time-period (T ) in the presence of only the magnetic field canbe calculated as follows:
ω
~c =

~
qB
mµ γ

T =

,

2π
ωc

(2.1)

~ is the magnetic field and q, m, γ are charge, mass, and Lorentz factor of the charge
Where, Bparticle, respectively. In the case of the BNL g – 2 experiment γ = 29.2, magnetic field of 1.7 T
results in T = 149 ns. The required kick angle to correct the injected beam can be calculated asfollows:
θ=

s
r

(2.2)

Where, θ is the kick angle, s is the required displacement, and r is the radius of the storage
ring. In E821 the radius of the stored beam was 7.11 m and the required displacement to avoid the35

2.1. WHY THREE–DIMENSIONAL SPIRAL INJECTION?
beam hit at the wall of the inflector was 0.077 m, resulting in the required kick angle of 10 mrad.
Hence, in E821 the injected muon beam was kicked by angle 10 mrad within 140 ns [22]. The
concept of the conventional injection scheme is shown in Fig. 2.1.

Figure 2.1: The concept of the conventional beam injection into the storage magnet. The beam
is injected into the storage magnet by the use of a device called the inflector. The red dottedline shows the injected beam with the help of an inflector, while the green thick lines show thecorrected beam orbit after the kick. To avoid the beam hit at the wall of storage magnet within
the first turn, a kicker system is needed to correct the injected beam orbit.
As explained in Section 1.5, the J–PARC Muon g – 2/EDM experiment (E34) will use acompact solenoid magnet of strength 3 T. This compact storage magnet will provide the local
field uniformity of 0.1 ppm, which is essential to reach the ultimate sensitivity goal of 0.14 ppm
for aµ . The diameter of the storage orbit will be 0.66 m. The 3 T magnetic field is too high to
be canceled by the inflector or any other existent technology.
In E34, the value of γ = 3 and field strength of 3 T results in the cyclotron period of 7.4 ns.
In the case of E34, the radius of the stored beam orbit is 0.33 m, and according to Eq. (2.2), therequired kick angle to correct the beam orbit will become 233 mrad within 7.4 ns. The 233 mradkick within 7.4 ns is currently impossible to be achieved using any available technology. Hence,36

CHAPTER 2. THREE–DIMENSIONAL SPIRAL INJECTION SCHEME
one can conclude that conventional injection scheme is not applicable in the case of E34 because:• The 3 T magnetic field cannot be canceled by the inflector• The 233 mrad kick angle within 7.4 ns rapidly expands to be achieved by current technology
A novel three-dimensional spiral injection scheme has been invented to overcome the earlier
stated challenges and to achieve high injection and storage efficiency. The next section will
describe the basic concept of the newly invented injection scheme.

2.2

Basic idea of Spiral Injection Scheme

In the three-dimensional spiral injection scheme, the beam will be injected at a vertical angle
into the solenoid magnet. The radial magnetic field of the solenoid will reduce the injection
angle to a smaller value as it approaches to the midplane. To stop the vertical motion, akicker will apply a force in the opposite direction of the vertical beam motion, and the beam
will completely stop at the midplane of the storage magnet. The duration of the kick could
be in several cyclotron periods. To confine the vertical motion of the beam in the storage
volume, a very weak magnetic focusing will be applied. Figure 2.2 shows the basic layout of
the three-dimensional spiral injection scheme. In Fig 2.2 (Xg , Yg , Zg ) are the storage magnet
coordinates. The detailed basic concept can be found at [34].

Figure 2.2: The layout of the three-dimensional spiral injection scheme.

2.3

Vertical Kicker System

As explained earlier, the beam is injected into the storage magnet at a vertical angle;
therefore, the beam has a component parallel to the axial magnetic field of the solenoid magnet
and starts spiraling through the storage magnet. The radial fringe field of the solenoid storage
magnet will reduce the injection angle of the beam to some extent.
37

2.3. VERTICAL KICKER SYSTEM
To store the beam at the center of the storage magnet, the pitch angle of the beam must be
reduced to almost zero as it reaches the storage region. A pulsed magnetic kicker will be usedP

to decrease the pitch angle ψ = |Py| of the beam to zero, where Py is the beam momentum inthe vertical direction and | P | is the total momentum of the particle. Figure 2.3 illustrates thecharacter of the magnetic kicker. The black dashed trajectory shows the motion of the beam
inside the storage magnet without a kicker; it is observed that the injection angle of the beam
reduced due to the radial fringe field but does not decrease to zero. The beam passed throughthe middle region of the storage magnet without getting stopped. When the appropriate kick isapplied to the beam, as shown in the black thick trajectory, the pitch angle is reduced to zero.
The curved region in Fig. 2.3 (right) depicts the weak focusing region.

Figure 2.3: Left: Black dashed trajectory shows the beam injection into the storage magnet
without the kick. Black thick trajectory shows the case wherein a suitable kick is applied to thebeam to stop the beam at the midplane. Right: Black dashed trajectory shows the pitch angleof the beam with respect to the vertical position of the kick; in this case trajectory shows thebeam that has not been subjected to a kick, the trajectory represented in black thick indicatethat the pitch angle reduced to zero when the kick is applied to the beam.
The pulsed magnetic kicker has to produce the radial field, which will be perpendicular tothe vertical component of the momentum of the beam. The required radial magnetic field anda stopping volume [35] (region needed to stop the beam) may be calculated as follows:~
F~ = q~v × B,ax

F~ = m~a,





⇒

~
m~a = q~v × B

B z vy − B y v z






q 




 ay  = B x vz − B z vx 

 m

az
B y v x − B x vy
qvL BR (R, y) sin(ωt)
mpWhere, vL = vx2 + vz2 , integrating Eq. (2.5) with respect to time gives:ay =vy = vy0 +

q
m

Z t

vL BR (R, y) sin(ωt)dt

0

38

(2.3)

(2.4)

(2.5)

(2.6)

CHAPTER 2. THREE–DIMENSIONAL SPIRAL INJECTION SCHEME
In order to simplify above equation let’s assume small angle approximation in that case vL
will remain constant and can be taken out of integral in Eq. (2.6)
vy = vy0 +

q
vL
m

Z t

BR (R, y) sin(ωt)dt

(2.7)

0

For half sine shape t = T2 = ωπ . Solving Eq. (2.6) for BR (R, y) gives:
BR (R, y) = −mvy0 ω
2qvL(2.8)

If we consider θkick = vy0 /vL and ω = 2π
T , where T is the time period, then Eq. (2.8) will
become:BR (R, y) = −πm θkick
q T

(2.9)

The total kicker field for the half sine shape can written as follows:
Bkick = BR . sin



2πt
Tkick



(2.10)

Where, Tkick is the kick duration. Integrating Eq. (2.6) can give the volume to stop the beam
as follows:
y0 = −

Z t0 + ωπvy (t)dt =

t0

c T
Py
mγ 4 0(2.11)

Where Py0 is the vertical momentum, m, γ and c are the mass, Lorentz factor and speedof light. Figure 2.4 shows the behavior of Eq. (2.9) and (2.11) for the 60 mrad kick angle. As
the time period increases, the required strength of kick field decreases and stopping volume
increases. These analytical formulas represent only rough estimation. To calculate the realistic
field for the kick system and determine the behavior of charge particle through the kicker field,
a numerical simulation is necessary. The kicker coil has been designed in the CST [36], anda custom Runge-Kutta 4th Order Method (RK–4 ) program in Mathematica [37] has also beendeveloped to track charged particles in the transient field of the kicker. In Chapter 7, the detailed
kicker study for SITE will be described.

39

2.4. WEAK FOCUSING SYSTEM

Figure 2.4: The radial field component B~r and stopping volume for the kick angle 60 mrad as afunction of the kicker duration.

2.4

Weak Focusing System

The kicked beam will be stopped at the midplane of storage magnet. To confine the stored
beam orbit into the required vertical and horizontal amplitudes, a weak magnetic focusing will
be applied [38, 39]. In this section, we will derive the equation of motion for the charged particle
under the weak focusing field. Figure 2.5 shows the coordinate system and the function of the
weak focusing field. Under the weak focusing field, the charged particle exhibits simple harmonic
motion. The first complete framework of weak focusing was developed by D. W. Kerst et al. [40].
In this text the version given in [41] will be used.
Let us consider a local coordinate system (x̂, ŷ, ŝ), that moves along the ideal particle. The
restoring force for the orbit other than the ideal orbit can be shown as Lorentz force andcentripetal force, i.e.,
Fy = qβcBx − γmÿ

(2.12)

Fx = −qβcBy + γmẍ(2.13)

Where, Fx , Fy and Bx , By are the components of the force and magnetic field vectors in
the horizontal and vertical directions, respectively. β and γ are relativistic factors, m is the rest
mass of particle and ẍ, ÿ are the accelerations in the horizontal and vertical directions, and c isthe speed of light. Then, the position vector is:~r = Rx̂ + y ŷ40

(2.14)

CHAPTER 2. THREE–DIMENSIONAL SPIRAL INJECTION SCHEME

Figure 2.5: The coordinate system and function of weak focusing for the charged particle. The
charged particle exhibits simple harmonic motion under the influence of the weak focusing in
the horizontal and vertical direction.
While the corresponding velocity and acceleration is given by:~v = ~r˙ = Ṙx̂ + Rx̂˙ + ẏ ŷ = Ṙx̂ + Rθ̇ŝ + ẏ ŷ(2.15)~a = ~v˙ = R̈x̂ + (2Ṙθ̇ + Rθ̈)ŝ + Rθ̇ŝ˙ + ÿ ŷ(2.16)~v
If ŝ˙ = −θ̇x̂ = − Rx̂ and inserting it in Eq. (2.16) gives:


v2
2ẋv
~a = ẍ −x̂ +ŝ + ÿ ŷRR(2.17)

Furthermore, on inserting ÿ and ẍ in Eq. (2.12) and (2.13) and solving for ÿ and ẍ gives:
qβcBx
=0
γm

(2.18)

v2
qβcBx
+
=0
R
γm

(2.19)ÿ −ẍ −Changing the derivative from time to θ gives:
t=

R
θ
βc

→

d
βc d
=
dt
R dθ

(2.20)

If p = γmβc, and writing Eq. (2.18) and (2.19) in terms of θ gives:d2 y qBx 2
−
R =0
dθ2
p
d2 x
+
dθ2




qBy
R−1 R=0p
41(2.21)(2.22)

2.4. WEAK FOCUSING SYSTEM
Since we know that magnetic rigidity is p = qB0 ρ, and from Fig. 2.5 R = ρ 1 + xρ , thenEq. (2.22) will become:





By
d2 x
x
+
1
+
−
1
R=0
dθ2
B0
ρ

The magnetic guide field for the off-orbits will become:


1 ∂By
By = B0 1 +x + ...
B0 ∂x(2.23)(2.24)

Ignoring the second order and higher terms in Eq. (2.24) and define a quantity “n”
n=

ρ ∂By
B0 ∂x

(2.25)

where, “n” is known as the field index, and insert Eq. (2.24) with the definition of field index in
Eq. (2.23):

d2 x+ (1 − n)x = 0dθ2(2.26)

A similar procedure for Eq. (2.18) will result in:d2 y+ ny = 0dθ2(2.27)

Equation (2.26) and (2.27) describe the weak focusing. To have stable motion in the verticaland horizontal planes, the stability criterion is as follows:
0<n<1(2.28)

Where, n < 1 is for the horizontal stability and n > 0 corresponds to the vertical stability. Theseequations are in the form of a simple harmonic oscillator, and their solutions are well known.
√√x(θ) = A cos(θ 1 − n) + B sin(θ 1 − n)(2.29)y(θ) = C cos(θn) + D sin(θn)(2.30)and derivativeẋ(θ) =√√√

1 − n −A sin(θ 1 − n) + B cos(θ 1 − n)(2.31)ẏ(θ) = n [−C sin(θn) + D cos(θn)](2.32)

To find constants A and B, consider the following initial condition.
√1−nx0 = x(θ = 0) = A,
ẋ0 =
B
ρ

A = x0 ,

B=√

ρ
ẋ0
1−n

(2.33)(2.34)

Similarly, for the vertical direction, the results will be:
C = y0 ,ρ
D = √ ẏ0n42(2.35)

CHAPTER 2. THREE–DIMENSIONAL SPIRAL INJECTION SCHEME
Insert Eq. (2.34) and (2.35) into x(θ), ẋ(θ), y(θ), ẏ(θ)√√ρ
x(θ) = x0 cos(θ 1 − n) + √ẋ0 sin(θ 1 − n)1−n√ẋ(θ) = −√√1−nx0 sin(θ 1 − n) + ẋ0 cos(θ 1 − n)ρρy(θ) = y0 cos(θn) + √ ẏ0 sin(θn)
n(2.36)(2.37)(2.38)√ẏ(θ) = −nx0 sin(θn) + ẏ0 cos(θn)ρ(2.39)Taking θ = ρs , ρ is the ideal beam orbit, and defining betatron phases:
φx (s) ≡s√
1 − n,
ρφy (s) ≡s√nρ

(2.40)

Then, the solutions can be written in matrix form as follows:


√ρ
sin(φ
(s))
cos(φx (s))
x1−nMH =  √
− 1−n
sin(φ
(s))
cos(φ
(s))
x
x
ρ

MV = 

cos(φy (s))ρ
n sin(φy (s))

− nρ sin(φy (s))

cos(φy (s))



(2.41)(2.42)The transformation for horizontal and vertical directions will be as follows:
X = MH X0(2.43)

Y = MV Y0(2.44)The number of oscillation performed by a charged particle in horizontal or vertical plane is
known as a ’tune’. The horizontal and vertical tunes can be calculated as follows:νx =√1 − n,

νy =√n(2.45)The realistic design of coils to obtain the required weak focusing field will be discussed in
the next chapter. For the J–PARC g – 2 experiment, the field index is set at n = 1.5 × 10−4 .
This small value of “n” was chosen to avoid the contribution to aµ precession from the weakmagnetic focusing. In the case of SITE, we set the field index value to 1.65 × 10−2 . The purposeof SITE is to prove the feasibility of this new idea, not to obtain any physics results. Therefore,
this large value of “n” was chosen for the ease of the experiment.

2.5

Beam Phase Space Requirements

In this section, the basic concepts in transverse beam dynamics and related parameters will
be described briefly. For a detailed review on beam dynamics see [42], [43], [44], and [45]. In
the later part of this section, the beam phase-space requirements for the 3-D spiral injection will
be described.
43

2.5. BEAM PHASE SPACE REQUIREMENTS

2.5.1

Concepts in beam dynamics

In accelerator physics, the motion of each particle is described by using the coordinates along
the design trajectory s of a reference particle. This coordinate system is know as Frenet-Serret,as shown in Fig 2.6. The 6-dimensional phase space vector in Frenet-Serret coordinate system
is given by:


x





 0 
 x 




 y 

X(s) =  0 ,
 y 




 z 


δ

(2.46)

where x and y are the transverse displacements from the design trajectory s, z is longitudinaldy0
displacement from the reference particle. The coordinates x0 = dx
ds , y = ds represent thehorizontal and vertical slopes with respect to the reference trajectory, δ = ∆p
ps is the momentum
deviation from the reference momentum. In some instances it is more convenient to use thetransverse momenta px , py instead of slopes. The relationship between them can be describedp

as following: x0 = ppxs , y 0 = pys , where ps denotes the particle momentum along the reference
trajectory. The later convention is used in this thesis.

Figure 2.6: Frenet-Serret coordinate system.

44

CHAPTER 2. THREE–DIMENSIONAL SPIRAL INJECTION SCHEME
The motion of charged particles through an electromagnetic field (beamline), is govern by
the Lorentz force:~ + ~v × B),~
F~ = q(E(2.47)~ and B
~ are the electric and magnetic
where q is the charge of the particle, ~v its velocity, and E
field vectors respectively. Electric fields are used to accelerate the particles, while magnetic fields
are used to deflect and to focus the beam. In this text, we will concentrate only on the effect ofmagnetic fields on the particle motion.
If the beamline consists only of drift spaces, dipole, (skew) quadrupole magnets, the magnetic
fields depend linearly on the deviation of the particle from the reference path. If we ignore themomentum deviation (δ = 0) and longitudinal motion , then the equations of horizontal andvertical motion can be written as:
x00 (s) + kx · x(s) = 0,

y 00 (s) + ky · y(s) = 0,

(2.48)

where kx and ky are
kx =

1 dB
1
−
,
2
ρ
B · ρ dx

, ky =1 dB
B · ρ dx

(2.49)The parameter ρ is the bending radius of the reference trajectory at the position s and
1 dB
B·ρ dxshows focusing strength of the quadrupole magnet. Eq. (2.48) is known as Mathieu-Hillequations. The detailed derivation of Eq. (2.48) can be found at [42].
The differential equation of the motion of a particle through the individual components(drift space, quadrupole, and bending magnets, etc.) can be established explicitly. Therefore,
the solutions of the equations of motion with linear approximation can be shown as:
X(s) = M · X(s0 ),


x





m11


 
 0   x   m21

=

 
 y   m31

 
y0
m41

m12

m13

m22

m23

m32

m33

m42

m43

m14

(2.50)


x0





 0 
m24   x0 




m34   y0 


0
m44
y0

(2.51)The 4×4 matrix M shows the transfer matrix that describes the transportation of the particle
phase-space from the entrance to the exit of a beamline. When a particle passes through abeamline composed of different components, the final phase space can be shown as the productof the individual transfer matrices:X(s) = Mn · Mn−1 · ..... · R1 · X(s0 ),(2.52)

where X(s) and X(s0 ) are phase space vectors at the initial and final position respectively,
and M1 , M2 , ...Mn represents the transfer matrix of individual components of the beamline from45

2.5. BEAM PHASE SPACE REQUIREMENTS
s0 to s. In the case of completely decoupled motion, the off-diagonal elements in Eq. (2.51) will
become zero. The transformation matrices of each beamline component (drift spaces, bending
magnet, quadrupole magnet) will be described in the forthcoming Chapter.

2.5.2

Emittance and Twiss parameters

The Mathieu-Hill equation (Eq. (2.48)) is similar to the simple harmonic oscillator, therefore,
the general solution to Eq. (2.48) and it’s derivatives can be written as [42]q(s) =q
q βq (s) · cos(φq (s) + φq0 )(2.53)√

(αq (s) · cos(φq (s) + φq0 ) + sin(φq (s) + φq0 )(2.54)
βwhere q stands for the one of the transverse coordinate x or y and , φq0 are constants. The
q (s) =
0phase function φq (s) is given by
φq (s) =Z sds̄

s0 βq (s̄)

+ φq0(2.55)

Eq. (2.53) and (2.54) can be a solution of Eq. (2.48), if the following relation satisfied:1
1
βq (s)βq00 (s) − βq02 + Kq (s)βq2 (s) = 1
2
4
Combined Eq. (2.53) and (2.54) in terms of invariant of motion 0γ(s)q(s)2 + 2α(s)q(s)q (s) + β(s)q 0 (s)2 = (2.56)(2.57)

2)
where α(s) = − 21 β 0 (s) and γ = (1+α(s). Eq. (2.57) is the equation of an ellipse. The ellipseβ(s)parameters β, α and γ, are called Twiss parameters.
The single particle, whose motion is governed by Eq. (2.48), traverse along the contour of anellipse given by Eq. (2.57). The shape and orientation of an ellipse change as it moves throughthe accelerator, but its area remains constant.
So far, the motion of a single particle within phase space was described. Now let’s consider
ensemble of particles, whose trajectories fill a certain area in the horizontal phase space (x, x0 ).
As shown above, an ensemble of particles always remains inside of a phase space ellipse, whilemoving along the accelerator. Only the shape and orientation of this ellipse changes while its
area remains constant. The area A of this ellipse, enclosing a certain fraction of particles inphase space, divided by π, is called horizontal emittance x
x =

A
π

(2.58)

Figure 2.7 denotes the horizontal phase space ellipse and also shows the elaborate therelationship between the ellipse and Twiss parameters.

And horizontal phase-space ellipse

equation can be shown as below:
γx x2 + 2αx xx0 + βx (s)x02 = 
46(2.59)

CHAPTER 2. THREE–DIMENSIONAL SPIRAL INJECTION SCHEME

Figure 2.7: The beam phase-space ellipse in the horizontal plane. Twiss parameters (β x , αx , γx )describes the shape and orientation of the beam phase-space ellipse.

472.5. BEAM PHASE SPACE REQUIREMENTS

2.5.3

Beam matrix

Let’s write the equation of the horizontal phase space ellipse by introducing the symmetrictwo-dimensional beam matrix:



x0

x




σ11

σ12

σ21

σ22

−1 




x
x0


=1

(2.60)

Since, σ21 = σ12 , this can be written as00

σ22 x2 + 2σ12 xx + σ11 x 2 = detσ,(2.61)

Comparing Eq. (2.59) and Eq. (2.61) will results in the Twiss parameters, the emittance, andthe beam matrix:


σ=

σ11

σ12

σ21

x =

σ22
√





 = x 

detσ =

βx
−αx

−αx
γx


(2.62)

q
2 .
σ11 σ22 − σ12(2.63)

This analogy can be generalized to n-dimensional phase space. The emittance expresses thearea (or volume) of the phase space ellipse consist of all or a certain portion of particles of thebeams. The 4 × 4 beam matrix can be written as below:




hxxi

hxx0 ihxyi

hxy 0 i
 0
 hx xiσ= hyxi
hy 0 xi

hx0 x0 i

hx0 yihyx0 ihyyi

hy 0 x0 i

hy 0 yi hx0 y 0 i σ =  xx0T
σxy
hyy i 
hy 0 y 0 i

σxy
σyy


.

(2.64)

Where, hxxi, hyyi are RMS beam sizes. The most common choice to describe the distributionof particles is Gaussian distribution. If we consider a two-dimensional Gaussian phase-space
distribution in the horizontal plane and only consider the 2-D emittance defined in Eq. (2.63),
the ellipse calculated by the second moments include about 1σ of the particles in the beam.
Figure 2.8 represents the Gaussian distribution for horizontal phase-space. The solid ellipse
denotes the 1σ of the Gaussian distribution.
In Chapter 5, a technique known as a quadrupole scan will be introduced in detail to measure
the emittance of the beam.

48

CHAPTER 2. THREE–DIMENSIONAL SPIRAL INJECTION SCHEME

Figure 2.8: Gaussian distribution for horizontal beam phase-space. The solid ellipse represents
the 1σ of the Gaussian distribution.

2.5.4

Beam Phase Space Requirements for 3-D Spiral Injection

Due to the axial symmetric field of the solenoid magnet a strongly XY -coupled beam isneeded for the successful 3-D spiral injection scheme. Moreover, the radial fringe field of thesolenoid magnet causes the vertical blow-up of the injected beam, as the beam moves throughthe storage magnet. The particles at different vertical positions sense different radial fieldand eventually takes different vertical paths. Therefore, the beam phase-space matching is
indispensable for 3-D spiral injection.
In order to overcome the vertical blow-up of the beam, a matched beam (or appropriate
XY -coupled) at the injection point is required. Here, XY -Coupling is referred to the rotationof the principle axes between horizontal (X, X 0 ) and vertical (Y, Y 0 ) planes. The detailed
description/calculations and results of the beam phase-space matching will be discussed in
Chapter 6.
Figure 2.9 (left) represents the beam injection without phase space matching and the below
plot shows the beam cross-section at the injection point.

In Fig. 2.9 the storage magnet

coordinates will be referred as global coordinates (Xg , Yg , Zg ). For the convenience, the beamphase-space at the injection point is shown in beam coordinates (X, Y, Z). It is evident thatwithout phase space matching beam drastically blow-up in the vertical direction. Figure 2.9(right) shows the beam injection with appropriate phase space matching. When appropriate
XY-coupling (phase space matching) was applied to the injected beam, the vertical blow-up
reduced sufficiently.
49

2.5. BEAM PHASE SPACE REQUIREMENTS

Figure 2.9: Left: Beam injection without phase space matching. The below plots shows thebeam cross section at the injection point. The storage magnet coordinates are referred as global
coordinates (Xg , Yg , Zg ). For the convenience, the beam phase-space at the injection point isshown in beam coordinates (X, Y, Z). It is evident that without phase space matching beam
drastically blow-up in the vertical direction. Right: Beam injection with phase space matching.
When appropriate XY-coupling (phase space matching) was applied to the injected beam, thevertical blow-up reduce sufficiently.

50

Chapter 3

Spiral Injection Test Experiment(SITE) with Electron Beam
The three-dimensional spiral injection scheme is an unproven idea, and no real precedent
exists for this new injection scheme.

Therefore, a demonstration experiment to prove thefeasibility of this unprecedented injection scheme is inevitable. A scale down Spiral Injection
Test Experiment (SITE) via the use of an electron beam was developed at the KEK Tsukuba
campus. The following are the salient features of SITE:• SITE used a scale-down solenoid storage magnet. The weak focusing and magnetic kicker
are included.
• SITE provides the freedom to experiment with the electron beam without any stringent
time/cost constraint of the muon beam time availability.
• SITE provided an opportunity to develop the beam monitors for the solenoid storage
magnet.
• The beam phase space matching for the solenoid-type storage magnet is challenging andis also tested at SITE.
In this chapter, the SITE experimental setup will be described in detail.

3.1

Experimental Setup of SITE

The SITE setup consists of a 2 m long beamline and a solenoid storage magnet to store theelectron beam. The beamline consists of following components:• Electron Gun: A triode-type thermionic electron gun with a LaB6 cathode was used togenerate the DC electron beam. In Section 3.2, the electron gun details will be discussed.
51

3.1. EXPERIMENTAL SETUP OF SITE
• Magnetic Lens: After the electron gun, a magnetic lens is placed to focus the low energy
beam. A pair of steering coils is also placed to control the transverse position of the beam.
The brief detail of the magnetic lens is shown in Section 3.3.
• Electric Chopper: An electric chopper system is placed after the magnetic lens to generate
the pulsed beam. In Section 3.4, the electric chopper system will discussed.
• Collimator: A collimator was placed after the electric chopper. The collimator served as
the beam dump for the chopper system and was also used to create the differential vacuum
pumping system for the gas monitor in the storage magnet. The collimator is describedin 3.5.
• Rotating quadrupole magnets: Three rotating quadrupole magnets has been placed after
the collimator. These magnets are used to produce the required phase space for the storage
magnet. The design and field measurement is discussed in Section 3.6.
• Dipole Magnet: A bending magnet (BM1) was placed on the straight beamline to deflect
the beam at 40 Degree toward the storage magnet. A second bending magnet (BM2) was
also placed near the injection point to control the injection angle slightly. In Section 3.7
dipole magnets will be discussed in detail.
• Storage Magnet: The dipole magnet will guide the beam to the normal conducting solenoid
magnet. The storage magnet has been designed to store the electron beam into the orbitof 0.24 m diameter. Details of the storage magnets are discussed in 3.8.
A pulsed magnetic kicker was also developed to store the beam at the center of the storage
magnet. The kicker system for the SITE will be discussed in chapter 7 of this thesis. The layoutof the SITE experimental setup with each component is shown in Fig. 3.1. A comparison ofparameters between SITE and E34 is given in Table 3.1. Some progress reports of SITE can befound at [46, 47, 35]. The SITE experimental setup photo is shown in Fig. 3.2.
Table 3.1: Comparison of Parameters Between E34 and SITE.
Parameters

E34

SITE

Particle Specieµ+e−Magnetic field strength

3T

82.5 Gauss

Momentum

300 MeV/c

297 keV/c

Cyclotron Period

7.4 ns5 ns

Storage orbit diameter

0.66 m0.24 mXY -Coupling

Yes

Yes

52

CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM

Figure 3.1: The side view of the 3D model of the three-dimensional Spiral Injection Test
Experiment (SITE).

3.2

Electron Gun

A thermionic electron gun with a LaB6 cathode of 2 mm diameter was used to produce theelectron beam. This electron gun was previously used for the production high brightness X-ray
experiment [48] at KEK. The electron gun was operated at 80 keV and the maximum achievablebeam current was 100 mA. From a radiation and thermal safety point of view, the electron gun
was operated at 0.5 mA. An isolation transformer of a maximum capacity of 200 kV was used totransfer electrical power from the AC source to a high-voltage power supply electron gun. The
specifications of the electron gun are given in Table 3.2. The experimental layout of the electron
gun high voltage (H.V) station is shown 5.1.
Table 3.2: Electron gun specifications.
Parameters

ValuesExtraction Voltage

80 kV

Maximum current

100 mA

Grid type

Aperture

Cathode material

LaB6

Cathode Diameter

2.0 mmCathode Grid Voltage

2 keV

The electron gun consists of a LaB6 cathode of diameter 2.0 mm. The LaB6 cathode waschosen due to its low work function (2.70 eV), modest vacuum requirement, and a 50-fold greater
service life than that of tungsten cathode. Figure 3.4 shows the photo of the LaB6 cathode. After
53

3.2. ELECTRON GUN

Figure 3.2: Above: A Photo of SITE setup. Below: A top view photo of the straight beamline,with each component labeled.

54CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM

Figure 3.3: Above: Layout of electron gun high voltage (H.V) power supply station. Below:
H.V station and electron gun.

55

3.2. ELECTRON GUN
the cathode, a Wehnelt electrode was placed to focus the emitted electrons. An aperture grid
was placed after the Wehnelt to control the beam current from the electron gun. Afterward, theaperture grid anode was placed to extract the emitted electrons. The 3-D simulation of electron
gun has been done in the CST-PS [36]. The potential contours and electron beam trajectories
are shown in Fig. 3.5. The emittance of the beam at the exit of the electron gun (Anode) in
CST-PS [36] simulation was 1.45 mm mrad in both transverse planes.

Figure 3.4: Photo of the LaB6 cathode used in electron gun. The diameter of the cathode is
2 mm

56

CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM

Figure 3.5: Left: Electric potential contour plots of the electron gun. Right: The electron beam
trajectories through electron gun.

3.2.1

Emission Process

In the thermionic emission process, a piece of metal or appropriate conducting material(cathode: in our case LaB6 ) is heated to a high temperature. If the applied temperature is highenough, a part of electrons get kinetic energy equal or greater than the work function (φ) of thecathode material, then electrons will escape through the surface of the cathode.
In thermionic emission, the maximum number of emitted particles depends on thetemperature of the surface. There are two regimes for the thermionic emission.
• In the temperature-limited model, the emission current depends on temperature of thecathode, while in the case of the temperature-limited regime, Richardson’s law [49] gives
the current density of emitted electrons.
J(x) = AR T exp
2



−φ
kT



(3.1)

Where, AR is Richardson’s constant with a value 1.2 × 106 mA
2 k2 , φ is the work function of
the metal, T is the applied temperature, and k is the Boltzmann constant.
• In the case of the space-charge regime, the cathode temperature is set at its maximum,
and the emission of particles depends on the extraction voltage only. The current density
in the case of space-charge limited regime is given by Childs-Langmuir law [50]r ! 3
40 2e VA2
J(x) =
9
m
b2

(3.2)

Where, J(x) is the current density, VA is the applied voltage, 0 is the permittivity of freespace, b is the distance from emitting surface, m and e are the mass and charge of theelectron.
During beam time of SITE, the electron gun was operated in a temperature-limited regime,
providing the freedom to control the beam current at a fixed applied voltage.
57

3.2. ELECTRON GUN

3.2.2

Electron Gun Stability

During the beam operation of the electron gun H.V and current in H.V circuit, were saved
on logger [51], to check the stability of the electron gun. The beam current of the electron gun
can be estimated by subtracting the current in the H.V circuit while the heater of cathode is
OFF from the total current flows in the H.V circuit while heater of the cathode is ON
Ibeam = It − Ibg(3.3)

Where, Ibeam is the beam current, It is the total current flows in the H.V circuit while heater
is On. And Ibg is the current in the H.V circuit while heater was off. Figure 3.6 shows therecorded H.V and current data. The current on the right in Fig. 3.6 decreases rapidly after
the electron gun is turned on, one of the possibility of this rapid decrease is due to the spoiled
vacuum caused by the out-gassing of the cathode. After the operation of 30 minutes the vacuum
becomes stable and current also settles down. The electron gun was operated at 80 keV, voltage
stability was 0.008% (80000 ± 5.29 V). The beam current stability was 0.94% (1.2 ± 0.01 mA).
This voltage and current stability is enough small to not cause any significant beam instability
and acceptable for the SITE.

Figure 3.6: Left: The electron gun was operated at 80 keV. H.V of electron gun was recorded asa function of time to check the stability during the operation. Right: Current of electron gun as
a function of time.

58

CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM

3.3

Magnetic Lens

A magnetic lens is a beam optics manipulation component that is used to focus the beam inboth transverse directions. In SITE, a magnetic lens was placed after the electron gun. Magnetic
lens consists of a coil of copper wires inside the iron pole pieces. A current through the coils
produces a magnetic field in the bore of the pole pieces. The rotationally symmetric magnetic
field is inhomogeneous in such a way that it is weak in the center of the gap and becomes stronger
close to the bore. Electrons close to the center are less strongly deflected than those passing thelens far from the axis. The overall effect is that the beam is focused on a spot. The transfer
matrix of a magnetic lens can be written as a product of rotation matrix in and focusing matrix
in both transverse directions:Msolenoid = Mrotation · Mf ocusingcos(kL)0sin(kL)

(3.4)

0








0
− cos(kL)0sin(kL) 
Mrotation =  − sin(kL)0cos(kL)00− sin(kL)0cos(kL)sin(kL)cos(kL)
0
0
k




 − sin(kL) cos(kL)00Mfocusing = sin(kL) 00
cos(kL)k00− sin(kL) cos(kL)

(3.5)

(3.6)

B
Where, k is a focusing strength (k = 2Bρ
) and L is a length of magnetic lens. Hence,magnetic lens focus provides the focusing in both transverse planes simultaneously [52, 53] witha rotation of transverse planes by an angle of kL. Figure 3.7 (left) shows the cut view of 3-D
model of SITE magnetic lens, while the axial field of the magnetic lens is shown on the right.

59

3.4. ELECTRIC CHOPPER

Figure 3.7: Left: Cut view of 3-D model of SITE’ magnetic lens. Right: Axial magnetic field ofmagnetic lens.

3.4

Electric Chopper

An electric chopper system has been developed to generate the pulsed electron beam for the
SITE. Fast response, simple structure, and low cost make the electric chopper system the best
choice for the generation of pulse beam for the low energy beam.
The electric chopper system consists of a pair of the electrode and a beam dump. One of
the electrodes is attached to the positive DC H.V power supply, which permanently deflects
the electron beam to beam dump location (in our case collimator: section 3.5). The otherelectrode was connected to the pulse power supply, during the active time of the pulse power
supply the beam passes through the collimator and otherwise remain deflected and stopped at
the collimator. Figure 3.8 shows the schematic and principle of the electric chopper system.
The required voltage for the electric chopper system depends on the electron beam energy,
electrode spacing, length of the electrodes, and distance to the beam dump location with respect
to the position of the electrode. The required deflection angle θ can be calculated as following:
θ=

qVd l
(γmc2 β 2 )d(3.7)

Where, q is charge of elementary particle, l is the length of electrodes, Vd is the required
deflection voltage, d is the gap between the electrodes, m is the rest mass of electron, c is the
speed of light, γ and β are the relativistic factor of the beam.
y=(γ − 1)Vd ll(D + ),(γβ 2 )dVacc
2

(3.8)

where, y is a distance on which the beam deflects, Vacc is the accelerating voltage of thebeam, D is a distance of beam dump (collimator) from the electrodes end, and l is the length ofelectrodes.
The parameters of the SITE electric chopper system are given in the Table 3.3. A simulationof the chopper system had been carried in CST-ES [36]. Figure 3.9 (left) shows the 3-D model60

CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM
of the chopper system and the picture on the right depicts the potential contours of the chopper
system in CST simulation. Figure 3.9 (left) presents the support structure and feed throughconnection to the H.V plate, while on the right is the cross section of the electrodes assemblyin the vacuum chamber. The ceramic feedthrough is a standard 5 kV single-ended coaxial MHV
connector.

Figure 3.8: Schematic of electric chopper system.

Table 3.3: Electric Chopper System Parameters.
Parameter

Value

Chopper Deflecting Voltage

1.6 kVElectrode Length

35 mm

Aperture (Width × Height)

20 mm × 10 mmDeflection Angle

0.04 radPulse Width

2 ns

Frequency

50 Hz

Capacitance

50 pF

61

3.4. ELECTRIC CHOPPER

Figure 3.9: Right: Mechanical structure and dimension of chopper system. Left: Potential
contours on the electrodes.

Figure 3.10: Left: H.V electrode attached to the vacuum feed through and its support. Right:
Chopper’s electrodes in vacuum chamber
62

CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM

3.5

Collimator

A collimator of diameter 3 mm and depth of 5 mm was placed after the electric chopper
system. The following are the functions of the collimator:• It serves as beam dump for the electric chopper system• It was also used to create the differential vacuum pumping system for the gas monitor
of the storage magnet. The differential pumping is a method wherein the large pressure
difference is maintained between different regions of a single vacuum system. The pressure
difference is maintained by a collimator and extra pumping on the high vacuum region
part. In the straight section of the beamline, a high vacuum was the requirement of theelectron gun operation (1.5 × 10−6 mbar), the medium vacuum (1.5 × 10−3 mbar) with theinsertion of N2 gas was maintained in the storage magnet.
• The collimator also served to remove the beam halo from the electron gun.
Figure 3.11 shows the collimator used at SITE. The black burn mark is due to the beam hitting
the collimator.

Figure 3.11: Collimator used at SITE’s beamline. The diameter of the collimator is 3 mm anddepth is 5 mm. The black burn mark is due to the beam hitting on the collimator.

63

3.6. ROTATABLE QUADRUPOLE MANGETS

3.6

Rotatable Quadrupole Mangets

As described in Chapter 2 an XY -coupled beam is key to the successful spiral injection.
To produce the required XY -coupled beam for the phase space matching, quadrupole magnets
with arbitrary rotation angles are required. For SITE, three rotating quadrupole magnets havebeen designed and manufactured. These rotating quadrupole magnets can rotate ±45 Degree
around the axial field direction. The rotating quadrupole magnets system consists of a baseplate to support the quadrupole and rotation system. The quadrupole magnets were enclosed
in a circular ring, and the ring was placed in a support system such that we can rotate it about±45 Degree manually. Figure 3.12 shows the rotating quadrupole magnets and support system
for the rotation. Table 3.4 summarizes the rotating quadrupole magnet parameters.

Figure 3.12: Three rotating quadruploes and their rotation system.

Table 3.4: Rotating Quadrupole Magnet Parameters.
Parameter

Value

Rotation Angle±45 Degree

Number of Quads

3

Ampere-Turns per pole (NI)50 A tNumber of Turns

25

Field Gradient at 2 A

0.19 T/m

Bore Radius

25 mm

Pole Length

35 mm

Effective Length

56 mm

64

CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM
The motion of particle through arbitrary angle quadrupole magnet can be found by thefollowing transfer matrix:
MRot.Quad = R(α)−1 · MQuarupole · R(α)(3.9)

Where, MRot.Quad is the rotation quadrupole magnet transfer matrix, R(α) is rotation matrix
with angle α, and MQuarupole is the normal quadrupole magnet transfer matrix. For verticaland horizontal direction, quadrupole magnets can be written as follows:

cosh (klq ) √ k sinh √klq
MQV = 


0

0
√ 
cos
klq

√ 
 √
 − k sinklq

MQH = 

0


0

√
sinh( klq )
√
k


√

cosh

0

0
√
sin( klq )
√
 k 

cos

√







√


sin( klq )
√

klq
cos
k
√ 
√  
√− k sin
klq
cos
klq

klq

0



0

0
√

0

0

0

0
√

klq


klq
cosh
√ 
√
k sinhklq

00(3.10)






0

√

sinh( klq )

√

k
√
 
klq
cosh

(3.11)

Where, MQV and MQH are the transfer matrix of the quadrupole magnet in the vertical andhorizontal direction. Where, k[m−2 ] is the focusing strength of the quadrupole, magnet which
depends on the field gradient g[T/m],k[m−2 ] = 0.3

g[T/m],
P [GeV/c]g[T/m] =2µ0 nI
,
R2

f=

1
klq

(3.12)

where, P (GeV/c) is the momentum, µ0 is the permeability of free space and it’s value is
4π × 10−7 H/m, n is number of turns, I is the excitation current, R is the bore radius of thequadrupole magnet, f is the focal length of the quadrupole magnet, lq is the effective length of
the quadrupole magnet and given as follows:
R +∞
lq =

−∞

~g (z)dz
g0,

lq ≈ l + R(3.13)The rotation matrix can be written as:cos(α)00cos(α)
R= − sin(α)
00− sin(α)

65sin(α)0sin(α) 
cos(α)00cos(α)
0(3.14)

3.6. ROTATABLE QUADRUPOLE MANGETS

3.6.1

Field Measurement

The field measurement setup for the rotation quadrupole magnet is shown in Fig. 3.13. The
field measurement setup consists of the following components• Hall probe• An XY Z stage to adjust the position of hall probe• A linear actuator to sweep hall probe
In order to measure the magnetic field, the hall probe [54] had been attached to the aluminum
bar and mounted on the XY Z stage. The hall probe [54] and XY Z-stage setup were mounted
on the linear guide to sweep hall probe in the desired direction.
Figure 3.14 is showing the focusing strength of the quadrupole magnet as a function of theexcitation current. The focusing strength increases linearly with respect to the applied current.
Below: The focal length of the quadrupole magnet as a function of the excitation current.
Figure 3.15 presents the magnetic field measurement of the rotation quadruple magnets
system in the longitudinal direction. The above graph shows the axial field measurement at ahorizontal position x = ±5, and the below plot presents the axial field measurement at a vertical
position of y = ±5. The colored dotted lines present the field measurement and the solid blackline is showing the CST simulation results.
Figure 3.16 shows the transverse magnetic field profile of the quadrupole magnet. The color
dots are showing the field measurement results and the black line are CST simulation results.
The simulation and experimental results are in good agreement with each other.

Figure 3.13: The field measurement setup for the rotation quadrupole system.

66

CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM

Figure 3.14: Above: The focusing strength of the quadrupole magnet as a function of theexcitation current. The focusing strength increase linearly with respect to the applied current.
Below: The focal length of the quadrupole magnet as a function of the excitation current.

673.6. ROTATABLE QUADRUPOLE MANGETS

Figure 3.15: The magnetic field measurement of the rotation quadruple magnets system in thelongitudinal direction. Above graph shows the axial field measurement at horizontal position
x = ±5, and below plot presents the axial field measurement at vertical position y = ±5. Colorlines dotted lines represents the field measurement and dashed line is showing the CST simulation
results.

68

CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM

Figure 3.16: Transverse magnetic field profile of the quadrupole magnet. Color dots are showing
the field measurement results and black line are CST simulation results.

69

3.7. DIPOLE MAGNET

3.7

Dipole Magnet

In order to inject the electron beam into the storage magnet, a rectangular dipole magnet
has been built. A second dipole magnet also has been mounted on the injection pipe to slightly
control the injection angle of the beam. The forthcoming sections will describe the details of thedipole magnet.

3.7.1

Dipole Magnet 1

A rectangular pole dipole magnet was designed for the SITE. The rectangular dipole magnet
simply deflects the beam in the bending direction, whereas in the non-deflecting plane it has afocusing effect. The transfer matrix of the rectangular dipole magnet [42] is given as follows:
Mrectangular = Medge · Mdipole · Medge(3.15)For a deflection angle θ, α = θ2 . The transformation matrix in deflecting plane is given as:MH = 

10

cos(θ)

ρ sin(θ)



1

0



− tan(α)ρ

1− sin(θ)ρ

cos(θ)

− tan(α)ρ

1MH = 

1

ρ sin(θ)



0

1



(3.16)(3.17)

Whereas transformation matrix of rectangular bending magnet in the non deflecting plane isgiven as:
MV = 

1− tan(α)ρ

MV = 

0



1



1
0

ρθ
0



1

0





− tan(α)ρ

11 − θ tan( θ2 )ρθ− ρ1 tan( θ2 )(2 − θ tan( θ2 ))1 − θ tan( θ2 )(3.18)



(3.19)

Where, ρ is the bending curvature. The bending curvature can be calculated from magnetic
rigidity formulaBρ =10P (GeV/c)
2.9979(3.20)The N I (Ampere-Turns per pole) and deflection angle are given as follows:
NI =

Bg
,
µ0

θ=

B0 lef f ectiveBρ(3.21)

where lef f ective is effective length and can be calculated as follow
R +∞lef f ective =−∞B(z)dzB0,

lef f ective ≈ l + gap(3.22)The magnet design and geometry result on pole length of 46 mm, with a gap of 18 mm
and yoke size of 103 mm × 140 mm. The dipole magnet consists of two coils with 30 windings70

CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM

Figure 3.17: The rectangular dipole magnet three dimensional model.

Figure 3.18: Left: The rectangular dipole magnet. The diople magnet at the SITE beamline.
713.7. DIPOLE MAGNET
Table 3.5: Dipole Magnet 1 Parameters.
Parameter

Value

Ampere-Turns per pole

90 A tNumber of Turns

30

Pole Gap

18 mm

Pole Length

56 mm

Effective Length

78 mmPole Shape

RectangularBρ

9.90 × 10−4 T.meach. Figure 3.17 shows the different views of the 3-D model of the dipole magnet. Table 3.7
summarize the parameters of the dipole magnet.
The magnetic field of the dipole magnet has been measured by the hall probe [54]. Figure 3.19
left shows the magnetic field measurement and CST simulation. Figure 3.19 right is a percentage
difference between the simulation and the experiment. The difference in good field region is lessthan 0.4%, but at the edges, the difference becomes 15% due to hall probe alignment error. Hall
probe was aligned with bubble gauge which has error of 10%. The simulation and experimental
results are in agreement within the uncertainty.

Figure 3.19: Left: The blue solid line shows the CST simulation and red dots are correspond to
the field measurement results. Right: The percentage difference between measured values andCST simulation.

72

CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM

3.7.2

Dipole Magnet 2

A second dipole magnet has been placed at the injection pipe. The second dipole magnet
increase injection angle acceptance into the storage magnet. Figure 3.20 shows the function of
second dipole magnet. Figure 3.21 presents the 3-D model and dimensions of the second dipole
magnet. Parameters of the second dipole magnet are shown in the Table 3.6. Field measurement
results with CST-EM simulation comparison are shown in the Fig. 3.22.
Table 3.6: Dipole Magnet 2 Parameters.
Parameter

Value

Ampere-Turns

100 A tNumber of Turns

100

Pole Gap

49.5 mmPole Length

18 mm

Effective Length

78.5 mmBρ

9.90 × 10−4 T.m

Figure 3.20: The role of second dipole magnet is shown. In order to control the injected beam
angle with more freedom a second dipole magnet was added to the injection beam pipe.

73

3.7. DIPOLE MAGNET

Figure 3.21: Left: The 3-D Model and dimensions of the second dipole magnet. Right: The
second dipole magnet.

Figure 3.22: Left: The field measurement and CST simulation of second bending magnet. Right:
The percentage difference between measured values and CST simulation.

74

CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM

3.8

Storage Magnet

The storage magnet is a solenoid electromagnet placed inside a cylindrical iron yoke of height
800 mm and a diameter of 600 mm, the thickness of iron yoke is 5 mm. The top and bottom
sides of the iron yoke are closed by a 20 mm thick iron plate to avoid field leakage from thestorage magnet. The iron poles of 120 mm height and thickness 20 mm are placed at the centerof the top and bottom iron plates. Both upper and lower iron plates have four viewing windows.
Upper windows are used to install CCD-cameras and feed-throughs for wire scanner. The holes
on lower iron yoke were used for gas valve, CCD-camera and also to inject the electron beam.
The main coil of the storage magnet has 540 turns. A correction coil of 80 turns has been placedat the center of the main coils to produce weak focusing potential. Figure 3.23 left shows theschematic of the storage magnet, and right is a solenoid magnet and vacuum chamber.
Table 3.7: Storage Magnet Parameters.

3.8.1

Parameter

Value

Iron Yoke Height

800 mmMain Coil Turns

540

Correction Coil Turns

80

Field Strength82.5 Gauss

Field Index (n)1.65 × 10−2

Field Measurement

The magnetic field of the storage magnet has been measured by the ALPHA LAB INC. [54] hallprobe, with an accuracy of 1%. Figure 3.24 above shows the axial field of the storage magnet
along with the vertical position of the storage magnet and below is the radial field of the storage
magnet along with the vertical position of the storage magnet. Above: The axial field of thestorage magnet along with the vertical position of the storage magnet. The black solid lines are
the CST-simulation and color points are the experimental field measurement. The current of
the main coil was fixed to 8 A and correction coil current was changed from 0-10 A (opposite
polarity to the main coil). As the correction coil’s current increases the weak focusing potential
increases. Below: The radial field of the storage magnet along with the vertical position of thestorage magnet.

75

3.8. STORAGE MAGNET

Figure 3.23: Left: Schematic view of storage magnet with reference trajectory and monitors.
Right: Storage magnet and vacuum chamber.

76

CHAPTER 3. SPIRAL INJECTION TEST EXPERIMENT (SITE) WITH ELECTRON
BEAM

Figure 3.24: Above: The axial field of the storage magnet along with the vertical position of
the storage magnet. The black solid lines are the CST-simulation and color points are theexperimental field measurement. The current of the main coil was fixed to 8 A and correction
coil current was changed from 0-10 A (opposite polarity to the main coil). As the correction
coil’s current increases the weak focusing potential increases. Below: The radial field of thestorage magnet along with the vertical position of the storage magnet.

77

3.8. STORAGE MAGNET

78

Chapter 4

Development of Beam Monitors
for SITE's Storage Magnet
Beam monitoring inside the solenoid-type storage magnet is very challenging due to themechanical constraints of the solenoid magnet. Two kinds of beam monitors have developed andtested at the SITE
• Gas Monitor: A Gas Monitor (GM) for non-invasive beam monitoring inside the storage
magnet is developed. The GM is a robust approach to qualitatively monitor beams inside
the storage magnet. It provides information about the beam orbit position, the number ofturns, and so on.
• Wire Scanner: The GM is a quick way to monitor the beam inside the storage magnet
but it lacks the important quantitative information about the beam position and profile
inside the storage magnet. Therefore, a unique wire scanner system has been designed for
qualitative beam monitoring inside the solenoid storage magnet.
In this chapter, the detailed principle and development of the beam monitor for the SITE
storage magnet will be described.

4.1

Gas Monitor

When the electron beam passes through nitrogen gas, it excites the nitrogen gas molecules.
Nitrogen gas de-excitation results in a light emission in the blue wavelength range 390 nm <λ < 470 nm [55]. This kind of non-invasive monitor in literature is known as beam-induced
fluorescence, gas monitor, or fluorescence Monitor [56, 57]. In this text, we will call it Gas
Monitor (GM). The light emission process from the excited nitrogen gas molecule can be written
as below:N2∗ → N2 + hν
79

(4.1)

4.1. GAS MONITOR
Where, N2∗ shows the excited nitrogen gas molecule, and N2 presents the de-excited nitrogengas molecule after emitting a photon. The photons yield emitted from the gas and beaminteraction is given as:

Yγ = NedE
× 4x × ρ × Ω
dx

(4.2)

Where, Ne is the number of electrons (corresponding to the beam current), dE
dx is the deposit
energy in the material[58, 59], 4x is the beam travel distance in the gas , ρ is density of gas,and Ω depends on the optical system. The dE
dx , which is the energy loss by ionization [3] can becalculated from the Bathe-Bloch formula as given below:
1 2me c2 β 2 γ 2 Tmax
δ(βγ)
dE
2
2Z 1
−β −
= Kz
ln
dx
A β2 2I2
2(4.3)

The energy loss for the 80 keV electron beam in nitrogen gas is equal to 4.23 M eV cm2 /g.
Fig. 4.1 shows the energy loss of the electron beam in the nitrogen gas.The ρ = 0.011P (ρ =1
Rspe. T ).

A beam current of 100 µA, Ω = 8.33 × 10−5 , and a gas pressure of 0.015 Pa (1.5 ×

10−4 mbar) are assumed. By inserting these values in Eq. (4.2), the number of photons will be:

Yγ = 2.90 × 106(4.4)

Figure 4.1: The energy loss of the electron beam in nitrogen gas. The red point indicates theenergy loss for the 80 keV electron beam.

80

CHAPTER 4. DEVELOPMENT OF BEAM MONITORS FOR SITE'S STORAGE
MAGNET

4.1.1

Gas Monitor Setup

The electron beam leaves the high vacuum region of a pressure of 1.5 × 10−6 mbar through a
3 mm diameter and 5 mm thick stainless steel collimator, into a nitrogen gas dominated medium
vacuum region of a pressure of 1.5 × 10−3 mbar. This medium vacuum region consists of abending section and a storage magnet. This technique of maintaining two different vacuums
by a small diameter collimator is known as the differential vacuum pumping. The GM setup
comprises a valve for the nitrogen gas inlet [60], a CCD camera to capture photons from theviewing port of nitrogen gas volume along the electron beam path. Figure 4.2 shows the setup
of a GM for the SITE.

Figure 4.2: The setup of the GM at the top of the storage magnet. The GM consists of thevalve for nitrogen gas inlet and CCD camera. The picture on the right shows the inside view ofstorage magnet’s vacuum chamber and the red dotted line corresponds as expected.

4.1.2

Optical System

The light from the nitrogen gas de-excitation was collected by a Hamamatsu 5985 B/W
Chilled CCD camera [61]. The lens of F-number 1.8 and focal length 4.8 mm was used [62].
Figure 4.3 shows the configuration of the optical system. The optical system depends on thecharacteristic of the lens and camera. In Fig. 4.3 G is the object size, which is the diameter ofthe vacuum chamber, i.e., 400 mm in our case. The g is the minimum object distance and it is
the characteristic of the lens, which in case our lens has a value is 0.3 m. B is the image size on
81

4.1. GAS MONITOR
00

the camera lens. In the case of the Hamamatsu C5985 CCD camera, the sensor size is 1/2 (6.40mm). f is the focal length of the lens, which can be calculated by the following formula:
0

g
B +1

f = G

(4.5)

The focal length of a lens for a given system depends upon three factors• Sensor size of the camera• Minimum object distance• Object size
The depth of the storage magnet’s vacuum chamber is 600 mm. The lens of focal length
4.8 mm and camera specification restrict the view up to 400 mm from the bottom of the vacuum
chamber. The possible inside view of the storage magnet vacuum chamber is shown in Fig. 4.4.
The gray and blue areas in Fig. 4.4 represent the area of storage magnet, which can be capturedby the use optical system.

Figure 4.3: Configuration of the optical system.

4.1.3

Light Reflection Suppression

The light reflection from the vacuum chamber’s wall and the bottom plate is a big source of
background light in the CCD-camera data. The wall of the vacuum chamber was covered witha black sheet to avoid light reflection from the chamber wall. A Light Exclusion Super Black
Sheet from SHIBUYA OPTICAL CO., LTD [63] was used to blacken the walls and the bottom
plate of the vacuum chamber. As a precaution, the black sheet was not attached directly to thewall of the storage magnet. First, we pasted it on the thin copper sheet (0.5 mm) by Araldite
Rapid epoxy [64] and then inserted it inside the storage magnet.
82

CHAPTER 4. DEVELOPMENT OF BEAM MONITORS FOR SITE'S STORAGE
MAGNET

Figure 4.4: Inside view of the storage magnet’s vacuum chamber depends upon the characteristicsof the lens and camera.

Figure 4.5 shows the black sheet pasted on to thin copper sheet using Araldite Rapid epoxy.
Figure 4.6 shows the process of cutting the copper and black sheets and inserting them into thevacuum chamber.

Figure 4.5: Two copper sheets of height (350 mm) had been joined together to make a copper
sheet of height 700 mm. The black sheet of height 700 mm and width 1257 mm was attachedto the thin copper sheet by Araldite Rapid Epoxy.

83

4.1. GAS MONITOR

Figure 4.6: The procedure of cutting copper and black sheets, as well as placing the black sheets
inside the vacuum chamber.

Figure 4.7: Left: Vacuum chamber without the black sheet. The light reflection from the wallof the vacuum chamber can be seen very clearly. Right: Vacuum chamber with the black sheet.
The light reflection suppressed considerably after the insertion of the black sheet.
84CHAPTER 4. DEVELOPMENT OF BEAM MONITORS FOR SITE'S STORAGE
MAGNET

4.1.4

Performance of the Gas Monitor

The major concern after inserting the black sheet was maintaining the vacuum level of thevacuum chamber. After the black sheet insertion. It was found that, the black sheet did notcause any significant out-gassing in the vacuum chamber. And the vacuum pressure reached thedesired value of the vacuum 7.2 × 10−5 mbar within 2 hours.
Figure 4.8 shows the difference between the beam injection with and without the black sheet.
In Fig. 4.8 (above) the electron beam enter through the injection hole and takes a spiral path
into the storage magnet, the light reflection at the bottom plate of the storage magnet’s vacuum
chamber can be seen very clearly. This light reflection seems to suppers the original beam signal
from the Gas monitor. Figure 4.8 (below) shows the instance of the electron beam injection
into the storage magnet with the black sheet insertion at walls of the vacuum chamber. The
background light from the reflective surface of the vacuum chamber completely vanishes with
the insertion of the black sheet.

854.1. GAS MONITOR

Figure 4.8: Above: The beam injection into the storage magnet without the black sheet.
Reflection from the wall of the vacuum chamber is the major source of background. Below:
The beam injection into the storage magnet with the black sheet. The background decrease
dramatically with black sheet insertion.

86

CHAPTER 4. DEVELOPMENT OF BEAM MONITORS FOR SITE'S STORAGE
MAGNET

4.2

Wire Scanner

The information of the beam vertical and horizontal profiles inside the storage magnet
is crucial to confirm the appropriate phase space matching.

Therefore, quantitative beam

monitoring inside the storage magnet is one of the keys to the successful three-dimensional
spiral injection scheme.
The wire scanners are extensively used in the accelerator complex for the transverse beamprofile measurement; however, it is the first instance of using such a kind of monitor forthe solenoid storage magnet. The mechanical constraints of the solenoid magnet make beam
monitoring very challenging. Two-wire scanners of a stroke length of 500 mm were designed anddeveloped to measure the beam profile, position, and evolution in the storage magnet [65]. The
forthcoming sections will describe the details of the wire scanner.

4.2.1

Basic Principle and Design of Wire ScannerIn the wire scanner-type monitors, the beam hits the wire and deposits energy into the wireand charges converted to a corresponding current. The voltage across a resistor is proportional
to the instantaneous current absorbed in the wire. To measure the transverse beam profile,wires were moved perpendicularly with respect to the beam position, and the beam current wasmeasured at each position.
Two-wire scanners were designed and built for the SITE. Herein, I will refer to them as“wire scanner 1” and “wire scanner 2”. The wire scanner 1 was installed at 2 cm away from theinjection point and wire scanner 2 was placed at 180 Degrees (half cyclotron period) away fromwire scanner 1 for the beam profile measurement at the half cyclotron period. Wire scanner 1
consists of four horizontal wires and one vertical wire. The horizontal wire was divided into four
segments to measure the radial offset of the beam. The horizontal wires were moved verticallyto measure the vertical beam profile. One vertical wire is also installed on the wire scanner 1 tomeasure the horizontal beam profile. To measure the horizontal beam profile, the vertical wire
was rotated around the axis of the supporting bar of wire scanner 1. The support bar of thewire scanner was placed at the 12 cm away from the center of the storage magnet.
Figure 4.9 shows the dimensions of the storage magnet vacuum chamber and position of thewire scanner 1 and 2 with respect to the center of the vacuum chamber. These dimensions wereused to calibrate the center of the wire scanners and storage magnet. In the case of the SITE,
a copper wire of 1 mm diameter was used to measure the horizontal and vertical beam profile.
Figure 4.10 presents the concept and mechanics of the wire scanner. Figure 4.11 shows the 3-Dmodel and the actual build of the wire scanner.

874.2. WIRE SCANNER

Figure 4.9: The dimensions of the storage magnet vacuum chamber and position of the wire
scanner 1 and 2 with respect to the center of the vacuum chamber. These dimensions were used
to calibrate the center of the wire scanners and storage magnet.

88CHAPTER 4. DEVELOPMENT OF BEAM MONITORS FOR SITE'S STORAGE
MAGNET

Figure 4.10: The principle and mechanics of the wire scanner.

Figure 4.11: Above: Wire scanner 1 consists of four segmented horizontal wire to measure thevertical beam profile and one vertical wire to measure the horizontal beam profile. Below: Wirescanner 2 consists of only four segmented horizontal wires to measure the vertical beam profile.

894.2. WIRE SCANNER

Figure 4.12: The principle of the horizontal beam size measurement. The black circle shows
the beam orbit (diameter = 24 cm). The red circle presents the circular path on which verticalwire traverse to measure the horizontal beam profile (diameter = 35 mm). The vertical wire
was placed at 35 mm away from the center of the wire scanner 1 support. In order to extract
the horizontal beam size the vertical wire was placed at the fixed vertical position, and wire
scanner support bar was rotated at its own axis by the rotary actuator. As vertical wire rotates
it provides the information of horizontal beam position and horizontal beam size.
The detailed principle of the horizontal beam profile measurement is shown in the Fig. 4.12.
The black circle shows the beam orbit (diameter = 24 cm). The red circle presents the circularpath on which vertical wire traverse to measure the horizontal beam profile (diameter = 35 mm).
The vertical wire was placed at 35 mm away from the center of the wire scanner 1 support. In
order to extract the horizontal beam size the vertical wire was placed at the fixed vertical
position, and wire scanner support bar was rotated at its own axis by the rotary actuator. As
vertical wire rotates it provides the information of horizontal beam position and horizontal beam
size. The radial position R of the vertical wire can be computed as follows:
R2 = (r0 + r1 cos(θ))2 + r12 sin(θ)
R2 = r02 + 2r0 r1 cos(θ) + r12(4.6)

R = r0 + r1 cos(θ)
where r0 is the beam orbit radius and equal to 120 mm, r1 is a distance of the vertical wirefrom the beam orbit and equal to the 35 mm. The θ is the angular distance and given as follows:
θ=

n
X

ωti

(4.7)

i=1

Where, ω is the angular velocity of the rotary actuator and equal to the 0.5 Degree/sec. The
90

CHAPTER 4. DEVELOPMENT OF BEAM MONITORS FOR SITE'S STORAGE
MAGNETsummation is over the required range of distance, and ti is the position of the ith step.

4.2.2

Mechanical Design

The mechanical design of the wires scanners composed of following items:• Magnetically-coupled Rotary/Linear Vacuum Feedthrough (MRLF) [66]
• A linear actuator for the vertical motion [67]• A rotary actuator for the rotation of feedthrough [68]• T-slot aluminum frame• Electric feedthrough to obtain the current signal of the wire out of the storage magnet
The copper wires of 1 mm diameter were supported by the holes of the GFRP (Glass Fiber
Reinforced Epoxy). The wires through the GFRP holes go to the electric feedthrough. TheGFRP support for the copper wires was used to keep the copper wires isolated from the rest of
the system (to prevent current leakage). The outside of the GFRP was enclosed with metal toprevent charge up due to the beam hit.
The copper wire and its support box were attached to the hollow copper pipe. The hollow
support bar (made of copper) was used to connect copper wires with the electric feedthrough.
The support bar was attached to the magnetically-coupled rotary/linear vacuum feedthrough(MRLF) [66] to move wires in the desired directions. Figure 4.13 shows the MRLF used at SITE.
The MRLF was attached to a driving system to move the MRLF precisely and remotely during
the beamtime.
The deriving system was composed of a linear actuator [67] for the vertical beam profile
measurement and a rotary actuator for the horizontal beam profile measurement [68]. Figure 4.14
and 4.15 show the linear and rotary actuator used for the SITE’s wire scanner. The schematic
view of the wire scanner system is shown in Fig. 4.17. Table 4.1 shows the parameters of the wire
scanner. Figure 4.16 shows the linearity between actuator set values and digital scale readout.
We confirmed the linearity within the uncertainty of the digital scale readout ±0.5µm.
Table 4.1: Wire Scanner Parameters
Parameter

Value

Number of wire scanners

2

Stroke Length

500 mmPosition Repeatability

0.05 mm

Resolution

1 mmWire Diameter

1 mm

Beam Intensity

20-100 µA

91

4.2. WIRE SCANNER

Figure 4.13: Magnetically-coupled Rotary/Linear Feedthrough (MRLF) for the vertical androtary movement of the wires. On the left is the 3-D model of the MLRF [66] and to the rightshows the MRLF used at the SITE.

92

CHAPTER 4. DEVELOPMENT OF BEAM MONITORS FOR SITE'S STORAGE
MAGNET

Figure 4.14: Above: The 3-D model of the linear actuator [67]. Below: The linear actuator forthe vertical movement of the MRLF.

93

4.2. WIRE SCANNER

Figure 4.15: The 3-D model of the rotary actuator [68]. The rotary actuator used for the rotationof the MRLF for the horizontal beam size measurement.

94

CHAPTER 4. DEVELOPMENT OF BEAM MONITORS FOR SITE'S STORAGE
MAGNET

Figure 4.16: Above: The velocity of the actuator was set to 1 mm/sec. To check the linearity,
data were taken at different points and fitted with a linear function. Below: Actuator set values
versus digital scale readout. To check the linearity between the set value and readout, data wastaken. The set values and readout values agree well within the uncertainty of the digital readout
system (±50µm).

954.2. WIRE SCANNER

Figure 4.17: (a) Schematic of the two-wire scanner setup. (b) Wire scanner system on the topof the storage magnet.

96

CHAPTER 4. DEVELOPMENT OF BEAM MONITORS FOR SITE'S STORAGE
MAGNET

4.2.3

Data Acquisition System

The signal from the wires was taken out of the vacuum chamber via vacuum feedthroughs
and terminated into the 1 MΩ terminal of an oscilloscope. The Tektronix DPO 7104 (1 GHz)
oscilloscope was used [69]. A GUI1 C# program was developed in Microsoft Visual Studio [70]to log the data from the wire scanner. The vertical and rotary actuators were controlled by theirrespective programs provided by the vendor. This program also features online plotting during
the beam test. The control system layout of the for the actuator and oscilloscope is shown in
Fig. 4.18. Figure 4.19 is the screenshot of the oscilloscope control program. Figure 4.20 presents
the beam current at the wire as a function of the cathode’s heater current of the electron gun.
During the beam operation electron gun was operated in temperature limited regime, therefore
the beam current increase as a function of the cathode’s heater current. The electron gun was
operated at the cathode’s heater current in the range 0.72-0.80 A due to radiation and thermal
safety point of view. Under this region a good linearity between cathode’s heater current andthe beam current at the wire can be seen from the Fig. 4.20.

Figure 4.18: The control system layout for the actuator and wire scanner.

1 Graphical User Interface97

4.2. WIRE SCANNER

Figure 4.19: The screenshot of the oscilloscope control program. This program was used for thewire scanner data online plotting and data logging.

Figure 4.20: The beam current at the wire as a function of the cathode’s heater current of theelectron gun.

98

Chapter 5

Beam Commissioning with DC
BeamThis chapter will describe the basic beam studies conducted at SITE.

5.1

Emittance Measurement By Quadrupole Scan

The charged particle emittance is a measure of beam size and beam divergence angle. Hence,
it is necessary to measure this figure of merit to describe charged particle beam quality. In this
Chapter, a well-known quadrupole scan technique will be described to measure the emittance ofthe beam.
In the quadrupole scan technique, the beam size is changed by varying the strength of thequadrupole magnet [71]. To determine beam emittance at the point P0 , a beamline with one
quadrupole magnet and a one screen monitor 1 at the location P1 is needed. The transformationof sigma matrix is as follows:
σ1 = M σ0 M T(5.1)

Where, σ0 is the beam matrix at the location P0 and σ1 corresponds to the location P1 and
M is a transport line matrix from point P0 to P1 . The full 4 × 4 beam matrix (σ) is given as:


σ11



 σ21
σ=

 σ31

σ41

σ12

σ13

σ22

σ23

σ32

σ33

σ42

σ43

σ14



 

σ24 
σ
 =  xx

T
σ34 
σxy

σ44

σxy
σyy




(5.2)

T
The terms σxy and σxy
represents the inter plane coupling. In Section 5.4 inter plane terms

will be treated in detail. In this section, I will treat horizontal and vertical planes completely
1 In the case of SITE, OTR (Optical Transition Radiation) from a 0.1 mm thick sheet of copper was used toobserve the transverse beam size.

99

5.1. EMITTANCE MEASUREMENT BY QUADRUPOLE SCAN
T
independent. Therefore, σxy and σxy
will vanish. Hence, the horizontal and vertical terms in

Eq. (5.1) can be written independently as follows:
σxx = hx2 i

hxx0 ihx0 xihx02 i,σyy = hy 2 ihyy 0 i

hy 0 yihy 02 i

(5.3)

In Eq. (5.1), M is the transport line matrix and M T is the transpose of M . In the case of
one quadruple and drift space, M will be:
M = D.QF(5.4)

Where, QF is the transfer matrix of the quadrupole in the focusing direction and D is thedrift space transfer matrix.
√ 
klq
cos
√ 
QF =  √klq− k sin√sin( klq )
√
 k 



cos

√

klq





1

d

0

1

√



,D = 


(5.5)



The M will become:


M =

m11

m12

m21

m22





=

cos

√

√ 
√
klq − d k sin
klq
√ 
√
− k sin
klq


d cos

√
sin( klq )
√
+
 k

klq
√
klq
cos


 (5.6)

Where, k[m−2 ] is the strength of QF , lq [m] is the effective length of QF and d is a distancefrom QF to the screen monitor. Then substituting Eq. (5.6) and (5.2) in Eq. (5.1) will results
σ1,11 as follows:
σ1,11 = m211 σ0,11 + 2m12 m11 σ0,12 + m212 σ0,22(5.7)

Where, m11 and m12 are the matrix elements, as shown in Eq. (5.6). Rewriting Eq. (5.7) interms of the matrix elements m11 and m12 gives:√ √ 2

√ 
√ 
√
√
σ1,11 = cos
klq − d k sin
klq
σ0,11 + 2 cos
klq − 2d k sin
klq







2
√
√
√  sin√  sinklq
klq
d cos σ0,12 + d cos
 σ0,22
√
√
klq +
klq +
k
k

(5.8)

The emittance and Twiss parameters can be measured at the location P0 by measuring thebeam size at P1 (σ1,11 : Beam size squared) as a function of the quadrupole magnet strengthand then fitting it with Eq. (5.8). The σ0,11 , σ0,12 , and σ0,22 will be extracted from the fitting
function and then the corresponding emittance (0 ) and Twiss (β0 , α0 , γ0 ) parameters at thelocation P0 can be computed as follows:
0 =

q

2
σ0,11 σ0,22 − σ0,12

(5.9)

σ0,11
0

(5.10)β0 =100

CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM

σ0,12
0

(5.11)

σ0,22
0(5.12)

α0 = −

γ0 =The error propagation for emittance is as follows:s2 σ 2 + δσ 2 σ 2
1 4 | δσ12 σ12 |2 −δσ221111 22δ = ±
2
2
σ11 σ22 − σ12(5.13)

Where, δ is the emittance error. δσ11 , δσ12 , and δσ22 represent the errors on the beam matrix
element.

5.2

Emittance Measurement at SITE Beamline

Three quadrupole magnets had been placed after the collimator for the phase space matching.
These quadrupole magnets also have been used to measure the emittance of the beam after thecollimator. The beam profile monitor consists of a copper plate and CMOS-camera. Whenthe electron beam hits the copper plate placed at 45 Degree with respect to the beam, opticaltransition radiation (OTR) arises at the surface of the copper plate due to the motion of acharged particle beam from the vacuum to the metal. The OTR from the copper was observedby the CMOS-camera [72]. The beam profile monitor setup is shown in Fig. 5.1 (a). The
distance of the quadrupole magnets from the beam profile monitor are also shown in Fig. 5.1(a). Figure 5.1 (b) represents the cross section of the quadrupole magnet. The bore radius ofquadrupole magnets is 25 mm and effective length is 56 mm.

5.2.1

Image Analysis

The beam images from the CMOS-camera as a function of the quadruple strength were savedon PC. A Mathematica [37] program has been developed to extract the beam size informationfrom the picture data. The following steps have been done to analyze the beam profile from theimages:• Convert RGB data to Grayscale• Measure beam profiles across vertical or horizontal pixels• Remove background from beam profile data
• Measure RMS size
Figure 5.3 (a) shows a raw picture of beam profile from copper plate taken by the
CMOS-camera without applying any quadrupole magnet. Figure 5.3 (b) is the conversion ofFig. 5.3 (a) RGB to the Grayscale for further analysis. Figure 5.2 shows the copper plateused to observe the beam profile (via OTR). The horizontal and vertical lines were drawn on
101

5.2. EMITTANCE MEASUREMENT AT SITE BEAMLINE

Figure 5.1: (a) The quadrupole scan setup. The quadrupole scan had been done with all three
quadrupoles independently. (b) The cross section of the quadrupole magnet. The bore radiusof quadrupole magnet is 25 mm and effective length is 56 mm.
the copper plate in order to determine the conversion factor from the pixels data of the phototo the millimeter. The conversion factor from pixels to mm can be obtained by dividing thenumber of pixels into horizontal and vertical directions to the known distance in millimeters.
The conversion factor from pixels to mm was 10±0.2 Pixels/mm in both transverse planes.
Figure 5.4 is an illustration of the analysis process. Figure 5.4 (a) and (b) represents thebeam images with vertical and horizontal focus respectively. And Fig. 5.4 (c) and (d) are the slice(contains 10 pixels) of horizontal and vertical projections to obtain a transverse beam profile.
The apparent tilted beam profile in Fig. 5.4 (d) is due to the rotation of (±5) Degree of screen
monitor itself. For coupling measurement (section 5.4) we corrected the screen monitor tilt to±1 Degree.
It is necessary to remove the background from the beam profile data in order to measure theRMS beam size. The background was estimated from the slope fit. The green highlighted curvein Fig. 5.4 (c) and (d) indicted the area selected for the RMS beam size measurement. Figure 5.5
shows the change in RMS beam size due to the different level of the background. This kind ofchange will cause ±0.004 [mm] and ±0.005 [mm] of systematic error to vertical and horizontal
beam sizes respectively.

102

CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM

Figure 5.2: The copper plate used to observe the beam profile (via OTR). The horizontal andvertical lines were drawn on the copper plate in order to determine the conversion factor fromthe pixels data of the photo to the millimeter.

5.2.2

Misalignment of Quadrupole Magnets

The quadrupole magnet had been aligned on the beamline by Laser Survey meter. In orderto know the misalignment of the quadrupole magnets precisely with the beam, a so-called
beam-based alignment method can be used. In the beam based alignment of the quadrupole
magnet method, the position shift of the transverse beam profile as a function of the quadrupole
magnets strength was obtained from the beam profile monitor. And the misalignment of thequadrupole magnets can be estimated from the information of the beam shift on the screen
monitor as a function of the quadrupole magnet strength.
Due to the quadrupole magnet misalignment an error dipole field produced. The angularkick to the beam due to the quadrupole magnet misalignment of distance 4d is given as [73]:

θ=

gl
4d
4d=
Bρ
f

(5.14)

Where, g is the quadrupole magnet field gradient, Bρ is the magnetic rigidity, f is focal
length and l is effective length of the quadrupole magnet. If one knows the effective angular kickdue to the misalignment then the Eq. (5.14) can be written as:

4d =

θBρ
= θf
gl

(5.15)

Figure 5.6 shows the beam position shift due to the quadrupole magnet misalignment error.
Each measurement was repeated 10 times. The black points represent the weighted average ofdata points. And the red line is the linear fit to the data. Table 5.1 and 5.2 are showing theeffective kick and misalignment of Q1-Q3 in vertical and horizontal direction respectively.
From this study we found that quadrupole magnet are miss aligned within ±1 mm. And thissmall misalignment does not cause any significant effect on beam. Furthermore, horizontal and103

5.2. EMITTANCE MEASUREMENT AT SITE BEAMLINE

Figure 5.3: (a) A raw picture from the CMOS-camera without applying any quadrupole magnet.
(b) The above figure converted from RGB to the Grayscale for further analysis. The conversion
factor from pixels to mm is 10 Pixels/mm in both transverse planes.

104

CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM

Figure 5.4: (a) and (b) are the beam profiles in the vertical and horizontal focus respectively.
(c) and (d) represents the beam profile data for vertical and horizontal planes.

1055.2. EMITTANCE MEASUREMENT AT SITE BEAMLINE

Figure 5.5: The change in RMS beam size due to the different level of the background.
vertical steering coils had been placed after the quadrupole magnets to compensate the effect of
the quadrupole magnets misalignment.
Table 5.1: The Quadrupole Magnets Vertical Misalignment.
Quadrupole Magnet

θ [mrad]

4d [mm]

Q1

2.25

0.38

Q2

5.86

0.94

Q3

5.13

0.78

Table 5.2: The Quadrupole Magnets Horizontal Misalignment.
Quadrupole Magnet

θ [mrad]

4d [mm]

Q1

4.74

0.80

Q2

0.11

0.018

Q3

3.67

0.56

106

CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM

Figure 5.6: The beam position shift due to the quadrupole magnet misalignment error. The
black points are weighted average of data points. And red line is the linear fit to the data.

1075.3. RESULTS

5.3

Results

The emittance measurement at SITE was performed by all three quadrupoles magnets.
Figure 5.7 (a), (c) and (e) shows the vertical beam size square as a function of the quadrupole
magnet 1, 2 and 3 respectively. Whereas, (b), (d) and (f) shows the horizontal beam size square
as a function of the quadupole magnet 1, 2 and 3 respectively. The color points in each picture
are correspond to the repeated measurement and black points are the weighted mean of therepeated measurement. Figure 5.8 and 5.9 shows the beam images with different strengthsof vertical and horizontal focused quadrupole magnet 1 respectively. The quadrupole magnet
strength corresponding to the each picture is given in Table 5.3.
The errors in Fig. 5.7 are statistical errors calculated from the errors of the rms beam size
measurements. The transverse beam sizes were measured 12 times for Q1 (both transverse
planes) and 6 time for the Q2 and Q3 scans. The weighted mean of each beam size can becalculated as follows:
ˆ =
hxi

i=n
X
xi /σ 2
i

1/σi2
i=1

(5.16)

ˆ > is the weighted mean and
Where, (x1 , x2 , ...xn ) are correspond to the beam sizes and < x
σi is the error of each value. The error of the weighted mean (σ) is given as follows:
s
σ=

1
Pi=n

2
i=1 1/σi(5.17)

Figure 5.10 shows the weighted mean of beam size squared as function of quadrupole strength.
Figure 5.10 (a) and (b) shows the vertical and horizontal scan for the quadupole 1 respectively.
(c) and (d) shows the vertical and horizontal scan for the quadupole 2. Figure 5.10 (e) and(f) shows the vertical and horizontal scan for the quadupole 3. The data were fitted with theEq. (5.8). Table 5.3, 5.4 and 5.5 shows the quadrupole magnets strength and beam size square(weighted mean) for Q1, Q2 and Q3 scans respectively.
Matrices (5.18), (5.19) and (5.20) represents the sigma matrices at Q1, Q2 and Q3
respectively. Table 5.6 shows the measured emittance values, calculated by using Eq.(5.9) for
Q1, Q2 and Q3, and the comparison between the thick lens and thin lens approximation alsopresented. Table 5.7 shows the Twiss parameters, calculated by using Eq.(5.10), (5.11) (5.12) for
thick lens results. The emittacne calculated from the CST simulation was 0.5 mm mrad in bothtransverse planes. The apparent difference in measured value (Q-scan) of emittance from thesimulation is due to the the cathode degradation overtime. Figure 5.11 details the comparisonof the emittance values between experimantal results and CST-PS simulation.

108

CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM

Figure 5.7: (a), (c) and (e) shows the vertical beam size square as a function of the quadupole
magnet 1, 2 and 3 respectively. Whereas (b), (d) and (f) shows the horizontal beam size square
as a function of the quadupole magnet 1, 2 and 3 respectively. The color points in each picture
are correspond to the repeated measurement and black points are the weighted mean of therepeated measurement.

109

5.3. RESULTS

Figure 5.8: The beam images with different strength of vertical focused quadrupole magnet 1.
False colors were added to the pictures for the better visibility.

110CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM

Figure 5.9: The beam images with different strength of horizontal focused quadrupole magnet
1. False colors were added to the pictures for the better visibility.

111

5.3. RESULTS

Figure 5.10: The beam size squared as function of quadrupole strength. (a) and (b) shows thevertical and horizontal scan for the quadupole 1 respectively. (c) and (d) shows the vertical andhorizontal scan for the quadrupole 2. (e) and (f) shows the vertical and horizontal scan for thequadrupole 3.

112

CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM
Table 5.3: The beam size square (Weighted Mean) and quadrupole magnet strength values forQ1-scan.
k (m−2 )Vertical (mm2 )Horizontal (mm2 )5.75

3.29 ± 0.10

3.24 ± 0.10

11.51

2.36 ± 0.07

2.57 ± 0.07

17.27

2.04 ± 0.06

2.08 ± 0.06

23.03

1.65 ± 0.06

1.72 ± 0.05

28.78

1.21 ± 0.04

1.39 ± 0.05

34.54

0.92 ± 0.04

1.06 ± 0.05

40.30

0.697 ± 0.027

0.80 ± 0.04

46.06

0.398 ± 0.018

0.536 ± 0.025

51.81

0.250 ± 0.014

0.333 ± 0.017

57.57

0.145 ± 0.009

0.209 ± 0.016

63.33

0.082 ± 0.006

0.151 ± 0.011

69.09

0.059 ± 0.006

0.119 ± 0.011

74.84

0.064 ± 0.008

0.130 ± 0.016

80.60

0.120 ± 0.008

0.180 ± 0.017

86.36

0.199 ± 0.014

0.262 ± 0.021

92.12

0.366 ± 0.023

0.391 ± 0.026

97.87

0.589 ± 0.033

0.59 ± 0.04

105.550.78 ± 0.04

0.90 ± 0.05

113

5.3. RESULTS
Table 5.4: The beam size square (Weighted Mean) and quadrupole magnet strength values forQ2-scan.
k (m−2 )Vertical (mm2 )Horizontal (mm2 )5.75

2.75 ± 0.07

3.26 ± 0.12

11.51

2.32 ± 0.06

2.52 ± 0.10

17.27

1.77 ± 0.06

2.32 ± 0.09

23.03

1.43 ± 0.05

1.73 ± 0.06

28.78

1.12 ± 0.04

1.27 ± 0.05

34.54

0.91 ± 0.04

1.12 ± 0.05

40.30

0.631 ± 0.030

0.78 ± 0.04

46.06

0.464 ± 0.024

0.60 ± 0.04

51.81

0.295 ± 0.021

0.417 ± 0.025

57.57

0.208 ± 0.015

0.244 ± 0.016

63.33

0.131 ± 0.012

0.151 ± 0.011

69.09

0.098 ± 0.011

0.091 ± 0.009

74.84

0.095 ± 0.012

0.055 ± 0.007

80.60

0.115 ± 0.013

0.054 ± 0.004

86.36

0.178 ± 0.020

0.075 ± 0.007

92.12

0.257 ± 0.024

0.131 ± 0.011

97.87

0.352 ± 0.024

0.193 ± 0.015

105.550.53 ± 0.04

0.372 ± 0.026

111.220.71 ± 0.05

0.53 ± 0.04

114

CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM
Table 5.5: The beam size square (Weighted Mean) and quadrupole magnet strength values forQ3-scan.
k (m−2 )Vertical (mm2 )Horizontal (mm2 )5.75

2.30 ± 0.08

2.92 ± 0.08

11.51

1.96 ± 0.07

2.59 ± 0.07

17.27

1.41 ± 0.06

2.03 ± 0.06

23.03

1.23 ± 0.05

1.85 ± 0.05

28.78

0.93 ± 0.04

1.54 ± 0.05

34.54

0.79 ± 0.04

1.22 ± 0.05

40.30

0.569 ± 0.028

0.96 ± 0.04

46.06

0.436 ± 0.024

0.96 ± 0.04

51.81

0.274 ± 0.020

0.512 ± 0.024

57.57

0.168 ± 0.012

0.385 ± 0.020

63.33

0.105 ± 0.009

0.260 ± 0.016

69.09

0.067 ± 0.007

0.156 ± 0.012

74.84

0.045 ± 0.005

0.102 ± 0.010

80.60

0.039 ± 0.004

0.074 ± 0.009

86.36

0.051 ± 0.005

0.064 ± 0.010

92.12

0.077 ± 0.009

0.068 ± 0.010

97.87

0.134 ± 0.010

0.108 ± 0.011

103.530.200 ± 0.017

0.171 ± 0.015

109.390.306 ± 0.021

0.268 ± 0.023

115.150.390 ± 0.03

0.379 ± 0.028

120.900.59 ± 0.04

0.52 ± 0.04

115

5.3. RESULTS



0.79



 1.61σQ1 = 

 0

0

0.96


 1.65σQ2 = 

 0

0

1.17


 1.98σQ3 = 

 0

0

0

3.71



0
0 


0.78 1.55 
1.55 3.3

0
0


0
0 


1.19 1.85 
1.85 3.03

0
0


0
0 


1.21 1.98 
1.98 3.4

0
0
1.65
3.19
0
0
1.98
3.69
0
0

0



1.61

(5.18)(5.19)(5.20)

Table 5.6: The transverse emittance values form Q1, Q2, and Q3 scans.
x [mm mrad]

y [mm mrad]

Q1

0.59 ± 0.07

0.39 ± 0.06

Q2

0.58 ± 0.09

0.43 ± 0.07

Q3

0.62 ± 0.09

0.44 ± 0.08

Table 5.7: The values of Twiss parameters from Q1, Q2, and Q3 scans at the quadrupole magnet
location.
βx [m]

βy [m]αxαy

γx [m rad]γy [m rad]Q1

1.35 ± 0.15

2.0 ± 0.18

−2.73 ± 0.30

−4.0 ± 0.6

6.3 ± 0.7

8.5 ± 0.35

Q2

1.65 ± 0.22

2.60 ± 0.18

−2.85 ± 0.40

−4.3 ± 0.7

5.6 ± 0.9

9.0 ± 1.4

Q3

1.89 ± 0.25

2.75 ± 0.16

−3.2 ± 0.25

−4.5 ± 0.9

6.5 ± 1.2

8.10 ± 1.21

116

CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM

Figure 5.11: the comparison of the emittance values between experimantal results and CST-PS
simulation.

5.3.1

Beam Phase Space at the Collimator

The quadrupole scan gives the Twiss parameters and emittance at the quadrupole location.
In the transport line design, we start from the collimator position. Therefore, it is necessary tofind the beam phase space shape (Twiss parameters) and emittance at the collimator location.
The beam matrix transform like
σ1 = M σ0 M T(5.21)

where M is the transport line matrix and M T is its transpose. In our case, the beam matrix
is known at the quadrupole location and in order to find the beam matrix at the collimator
location, we have to take the inverse of the above equation.
σ0 = M −1 σ1 (M T )−1(5.22)

If the elements of sigma matrix are known at the collimator then the Twiss parameters can becalculated by the Eq. (5.22). In the case of linear beam dynamics the beam emittance remains
constant but Twiss parameters transform along the beamline. Another relation to transform
Twiss parameter through drift [71] is as following

 
β(l)
1 −2l

 

 
 α(l)  =  0
1

 
γ(l)
0
0
117

l2



β0







−l   α0 


1
γ0

(5.23)

5.3. RESULTS
above relation gives the Twiss parameters at forward location in. In order to find Twiss
parameters at the collimator location, we have to take inverse of the above relation


 

β
1 2l l2
β0 (l)


 



 

 α0 (l)  =  0 1 l   α 


 

γ
0 0 1
γ0 (l)(5.24)

2The β and α represent beam size and tilt respectively. And γ = 1+αdepends upon β and
βα and remain constant in linear beam transport line.
The Twiss parameters from the quadruple magnet 1 to 3 scan had been transformed to thecollimator location by using Eq. (5.24). Table 5.8 shows the transformed Twiss parameters at thecollimator. Figure 5.12, 5.13 and 5.14 shows the transformation of β, α and γ from quadrupole
magnet location to the collimator. The trend of transformation follows the Eq. (5.24).
Table 5.9 shows the weighted mean of emittance and Twiss parameter values at the collimator
position.
Figure 5.15 shows the phase space distribution at the collimator point. Based on Fig. 5.3,
Gaussian distribution was used to define the beam phase-space at the collimator location. (6.11).
Table 5.8: Twiss parameters transformation from Q1,Q2 and Q3 to the Collimator
βx [m]

βy [m]αxαy

γx [mrad]γy [mrad]Q1-Colli.

1.33 ± 0.15

1.98 ± 0.18

−2.70 ± 0.31

−3.9 ± 0.6

6.3 ± 0.7

8.50 ± 1.35

Q2-Colli

1.27 ± 0.19

2.00 ± 0.18

−2.41 ± 0.41

−3.6 ± 0.7

5.6 ± 1.1

9.0 ± 1.4

Q3-Colli

1.10 ± 0.25

1.45 ± 0.16

−2.01 ± 0.25

−3.5 ± 0.9

4.99 ± 0.9

8.10 ± 1.21

Table 5.9: The weighted mean of Emittance and Twiss parameters at the Collimator
Parameters

Horizontal

Vertical [mm mrad]0.61 ± 0.05

0.41 ± 0.04

β [m]1.26 ± 0.10

1.78 ± 0.09

α−2.30 ± 0.17

−3.69 ± 0.38

γ [mrad]5.01 ± 0.49

8.4 ± 0.75



0.76871.4057
 1.4057
σInitial = 


0

0

3.0474
0
0

118

0

0








0.7322 1.5177 

1.5177 3.3738
00(5.25)

CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM

Figure 5.12: The βx and βy at Q1,Q2, Q3 and their transformation to the collimator position.

119

5.3. RESULTS

Figure 5.13: The αx and αy at Q1, Q2, Q3 and their transformation to the collimator position.

120

CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM

Figure 5.14: The γx and γy at Q1,Q2, Q3 and their transformation to the collimator position.

121

5.4. COUPLING MEASUREMENT

Figure 5.15: The phase space distribution at the collimator. This phase space satisfy theemittance and Twiss parameters given in the Table 5.9.

5.4

Coupling Measurement

If one knows the full beam matrix, then the beam coupling can be estimated. At the SITE
beamline a weak magnetic lens (solenoid) was placed after the electron gun to focus the electron
beam to the collimator. The magnetic lens is a coupled linear optics element. Therefor, it isnecessary to estimate the beam coupling caused by this weak magnetic lens. If one does notassume a zero coupling term in the beam matrix. Then, the transformation of the beam matrix
from point p0 (σ0 ) to p1 (σ1 ) will be as follows:
σ1 = M σ 0 M T ,



hxxi

hxx0 ihxyi

hxy 0 i
 0
 hx xi
σ0 = 
 hyxi
hy 0 xi

hx0 x0 i

hx0 yihyx0 ihyyi

hy 0 x0 i

hy 0 yi
hx0 y 0 i ,
hyy 0 i 
hy 0 y 0 i
σ0 = 

σ0,XX

σ0,XY

T
σ0,XY

σ0,Y Y (5.26)

Where, M is transport line matrix:


m11



 m21
M =

 m31

m41

m12

m13

m22

m23

m32

m33

m42

m43

m14





m24 


m34 

m44

(5.27)

If we set the beamline between p0 and p1 without transverse coupling (only normal quadrupole122

CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM
magnets) element, the off-diagonal terms become zero. And Eq.(5.27) will become:
m11 m1200 m21 m22
0
0 


M =

 0
0
m33 m34 
00m43 m44(5.28)The beam transverse sizes and x − y terms will become:hx2 i = σ1,11 = m211 σ0,11 + 2m11 m12 σ0,12 + m212 σ0,22(5.29)

hy 2 i = σ1,33 = m233 σ0,33 + 2m33 m34 σ0,34 + m234 σ0,44(5.30)

hxyi = σ1,13 = m11 m33 σ0,13 + m12 m33 σ0,14 + m11 m34 σ0,23 + m12 m34 σ0,24(5.31)The hxyi term can be calculated as a covariance between x and y as follows:nhxyi = σ1,13 =1X
hxi yi in i=1(5.32)

If the beam is decoupled then the projected rms emittances are the square root of thedeterminant σxx and σyy , and can be shown as x and y for horizontal and vertical planes
respectively.
µ =qhµµihµ0 µ0 i − hµµ0 i2(5.33)

Where, µ refers to either x or y. If inter plane coupling is non-zero then the eigen emittances
1 and 2 can be defined as follows [74, 75, 76]:
1
1 =
2

r

1
2 =
2

r

−tr [(σ0 J)2 ] +q

tr2 [(σ0 J)2 ] − 16 | σ0 |(5.34)−tr [(σ0 J)2 ] −q

tr2 [(σ0 J)2 ] − 16 | σ0 |(5.35)

Where, J is four dimensional skew-symmetric matrix given as follows:




0

1

0

0



 −1
J =

 0

0

0

0

0

0



0 


1 

1

0 −1

(5.36)The product of the eigen emittances cannot be greater than the product of the project rms
emittances and four-dimensional beam rms emittance can be computed as follows:1 2 =p
| σ 0 | ≤ x y123(5.37)

5.4. COUPLING MEASUREMENT
Where, σ0 is beam matrix at the desired reconstruction point. In order to measure interplane coupling a parameter 't' can be defined as follows:

t≡

x y
− 1 ≥ 0,
1 2

(5.38)t equal to zero correspond to the no interplane correlation and t > 0 represents thetransversely coupled beam.

5.4.1

Results

In the case of SITE, three quadrupole magnets were used to measure the coupling of thebeam. Figure 5.16 shows the beamline layout for the coupling measurement. The following
beamline transfer matrices were assumed for the coupling measurement:
M = L4 .Q3V .L3 .Q2H .L2 .Q1V .L1(5.39)

M = L4 .Q3H .L3 .Q2V .L2 .Q1H .L1(5.40)

Where, Q2V and Q1H are vertical and horizontal focus normal quadruple magnets
respectively. The L1 − L4 are drift space as given in Table 5.10. At least four measurements(i.e., 12 equations) are necessary for this evaluation. Although, more measurements will confine
the problem more tightly. Therefore, the beam profile measurement with 20 different beamline
settings was carried out. First 10 beam profile measurements were carried out with beamline
setting as mentioned in (5.39) and later 10 were taken with (5.40).
Figure 5.18 shows the beam profiles with the 10 different beamline settings correspond tothe (5.40). False colors were added to the Fig. 5.18 for better visibility. Table 5.11 shows thetransverse beam profile and coupling term at the profile monitor for 20 different settings. In
Table 5.11, k1 , k2 and k3 represents the focusing strength of Q1H , Q2V and Q3H respectively.
The RMS values of the beam transverse size and X-Y term were used for this measurement.
Figure 5.17 (a) shows the gray-scale (setting#4) picture. Figure 5.17 (b) is 3-D histogram of theabove picture. It is necessary to remove the background from the beam profile data in order tomeasure the RMS beam size. The background was estimated (BE) as a 2-D slope of the picture
data by using the following function:
BE = c + m1 x + m2 y

(5.41)

Where, c is a constant and m1 and m2 are slopes in the horizontal and vertical directions.
The red inclined plane in Fig 5.17 (b) shows the background estimation. The green highlighted
region in Fig. 5.17 (c) indicts the area selected for the RMS beam size measurement after the
background subtraction. The conversion factor from pixels to mm is 10 Pixels/mm in bothtransverse planes.
124CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM
The aim is to find the coefficients (beam matrix element) of Eq. (5.29), (5.30) and (5.31)
in a least-squares sense which best fits all the measured data. For 20 measurement, consider asystem of n = 20 equations with 3 variables for horizontal plane:
 

1
m1 m1
σ1,11
  11 11 2   2 2
σ1,11  m11 m11
  3   3 3
σ1,11  = m11 m11
 ..
 ..   .  
.
 

20 20
20
m11 m11σ1,112m111 m112

m112 m112

2m211 m212
m212 m212  σ0,11




m312 m312  σ0,12 


..


σ
.
0,22


2m311 m312
..
.
20
2m20
11 m12(5.42)

20m20
12 m12The method of least squares was used to find the solution of over determined set of equations
Eq. (5.42). Let’s say transformation matrix A, beam size matrix at the screen monitor Σ1x andcoefficient matrix Σ0x , then the Eq. (5.42) can be written as follows:Σ1x = AΣ0x

(5.43)

Σ0,x = (AT A)−1 AT Σ1,x ,(5.44)

Σ0,y = (AT A)−1 AT Σ1,y .

(5.45)

similarly for vertical plane

For interplane correlations terms the system of n = 20 equations with 4 unknowns can be
written as follows:

 
1
m1 m1
σ1,13

  11 33
 2   2 2
σ1,13  m11 m33  3   3 3
σ1,13  = m11 m33 ..
 ..   .  
.

 
20
20 20
σ1,13
m11 m33

m112 m133

m111 m134

m212 m233

m211 m234

m312 m333..
.

m311 m334
..
.

20
m20
12 m33

20m20
11 m34m112 m134





 σ
0,13


m212 m234  




σ
0,14

3
3


m12 m34   σ 
  0,23  σ0,2420m20
12 m34(5.46)

And similarly Eq. (5.44) and (5.47) the solution of (5.46) can be written as follows:
Σ0,xy = (AT A)−1 AT Σ1,xy .

(5.47)A Mathematica program has been developed to solve these system of equation. Table 5.12
shows values of the elements of σ0 matrix at reconstruction point. The coupling strength for
the SITE beamline is shown in Table 5.13. The value of 1 2 = 0.64 was obtained. And x yresults in 0.75. Which corresponds to a coupling strength of t = 0.17. In the case of the strongly
coupled beam (skew quadrupole magnet) the value of t will be in the range of 2.8. Moreover, inthe case of SITE, the required coupling strength (forth coming Chapter) at the matching pointis 6.24. Thus, the coupling strength of t = 0.17 represents the insignificant beam coupling at thecollimator location. Hence, the emittance and Twiss parameters measured from the Quad-scanwill be used for the further studies.
125

5.4. COUPLING MEASUREMENT

Figure 5.16: The layout of the beamline for the 4-D beam matrix reconstruction.

Table 5.10: The drift spaces of the beamline.
Parameters

Length [mm]

L1

2.35

L2

12.25

L3

22.45

L4

354.45

Table 5.11: The beamline setting and respective values of σ1,11 , σ1,33 , σ1,13 .
Setting

k1 (m−2 )k2 (m−2 )k3 (m−2 )σ1,11 (mm2 )σ1,33 (mm2 )σ1,13 (mm2 )1-38.5723.10-21.265.95 ± 0.08

1.180 ± 0.016

0.00035 ± 0.00028

2-63.7044.37-21.26

6.31 ± 0.08

0.681 ± 0.009

0.0014 ± 0.0006

3-63.7021.17-23.2011.60 ± 0.15

0.1936 ± 0.0018

0.0656 ± 0.0032

4-82.9455.97-32.869.74 ± 0.11

0.2136 ± 0.0021

0.0346 ± 0.0024

5

-0.00

52.10

-32.86

1.138 ± 0.014

5.76 ± 0.11

0.081 ± 0.005

6-25.03

69.50

-9.57

0.686 ± 0.008

6.63 ± 0.14

0.067 ± 0.005

7-25.03

84.97

-9.57

0.2477 ± 0.0032

7.77 ± 0.20

0.206 ± 0.008

8-25.03102.37-9.57

0.156 ± 0.006

21.1 ± 1.0

0.749 ± 0.009

9-55.97102.37-40.6

1.405 ± 0.014

2.738 ± 0.027

0.041 ± 0.002

10-55.97102.37-67.66

3.40 ± 0.05

1.546 ± 0.020

0.0315 ± 0.0025

11

19.23

-36.63

1.83

4.36 ± 0.06

2.642 ± 0.035

0.0100 ± 0.0012

12

19.23

-18.2720.3

6.60 ± 0.08

0.617 ± 0.007

0.0008 ± 0.0004

13

19.23

-98.5013.43

10.52 ± 0.13

0.0935 ± 0.0009

0.110 ± 0.004

14

19.23

-158.43139.2

1.285 ± 0.014

0.453 ± 0.005

0.108 ± 0.004

15

0.096

-158.43139.2

2.021 ± 0.023

0.22662 ± 0.0026

0.313 ± 0.006

16

40.50

-156.50

116.961.424 ± 0.014

0.874 ± 0.009

0.014 ± 0.0015

17

30.83

-162.30172.06

0.0922 ± 0.0012

2.196 ± 0.033

0.063 ± 0.005

18

48.23

-65.6334.70

1.559 ± 0.018

5.98 ± 0.11

0.0255 ± 0.0025

19

81.10

-79.1773.46

0.0655 ± 0.0005

12.22 ± 0.10

0.307 ± 0.004

20

81.10

-189.37

73.46

3.69 ± 0.04

0.1132 ± 0.0011

0.112 ± 0.004

126

CHAPTER 5. BEAM COMMISSIONING WITH DC BEAM

Figure 5.17: (a) The gray-scale (setting#14) picture. (b) The 3-D histogram of the above
picture. The background was estimated as a 2D slope picture data. The red patch shows the
background estimation. (c) The green highlighted region indicts the area selected for the RMS
beam size measurement after the background estimation.

127

5.4. COUPLING MEASUREMENT

Figure 5.18: The beam images with different settings. False colors were added to the pictures
for the better visibility.
Table 5.12: The values of the beam matrix elements at the reconstruction point.
Parameters

ValuesParameters

Valuesσ0,11 [mm2 ]0.444 ± 0.016

σ0,22 [mrad2 ]5.36 ± 0.05

σ0,33 [mm2 ]0.764 ± 0.023

σ0,44 [mrad2 ]0.408 ± 0.06

σ0,12 [mm mrad]

1.246 ± 0.022

σ0,34 [mm mrad]1.560 ± 0.035

σ0,13 [mm2 ]−0.1011 ± 0.0011

σ0,14 [mm mrad]−0.0290 ± 0.0029

σ0,23 [mm mrad]−0.2228 ± 0.0028

σ0,24 [mrad2 ]0.570 ± 0.008

Table 5.13: The Coupling Strength for the SITE beamline.
Parameters

Valuesx y [(mm mrad)2 ]p
1 2 = | σ0 | [(mm mrad)2 ]0.75 ± 0.1
0.64 ± 0.3

t0.17 ± 0.6
128Chapter 6

Experimental Results of Phase
Space Matching
One of the most important aspects of the 3-D spiral injection scheme is to produce therequired beam phase-space for the injection and confirm its effects on the reductions of verticalbeam size. In this chapter, the details of the beam phase-space matching requirements, beamline,
and experimental results of the beam phase-space matching on the vertical beam size in thestorage magnet will be described and discussed.

6.1

Beam Phase-Space Requirements

6.1.1

Coordinates System

Here, we will distinguish between the global and local coordinates system. The beam injection
and 3-D spiral plots will be shown in the storage magnet or global coordinate system. Whereas,
the beam phase-space at the injection point will be described in the beam coordinate system
for the sake of convenience. In Fig. 6.1 (Xg , Yg , Zg ) denotes the global (solenoid) coordinate
system and (X, Y, Z) is the beam coordinate (beam frame), that is used to describe the 4-D
beam phase-space at the matching point. In Fig. 6.1 X and Y represents the horizontal andvertical planes respectively, and Z is the beam motion direction. The X 0 and Y 0 are defined asfollows:
X0 =

PX
,
P

Y0 =

PY
,
P

P =

q
2 + P2 + P2
PX
Y
Z

(6.1)

where, PX , PY represents the horizontal and vertical momentum components respectively,
and P is total momentum.

1296.1. BEAM PHASE-SPACE REQUIREMENTS

Figure 6.1: Left: A horizontal flat beam was assumed inside the storage magnet and reverse
track it to find the required beam slopes at the injection point. Right: The beam frame is used todescribe the beam shape at the injection point. X and Y represents the horizontal and verticalplanes respectively, and Z is the beam motion direction.

130

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING

Figure 6.2: The slopes of the beam correlations at the matching point Yg = −600 mm. The
beam correlations plots are shown in the beam coordinates. The dots represents the data points
obtained from the reverse tracking at the matching point and solid line (red line) are the linear
fit to these points.

131

6.1. BEAM PHASE-SPACE REQUIREMENTS

6.1.2

A Method to Calculate the Beam Phase-Space Requirements at
the Matching Point

Due to the axial symmetric field of the solenoid magnet a strongly XY -coupled beam isneeded for the successful 3-D spiral injection scheme. Here, XY -coupled will be referred to therotation of the principal axes of horizontal (X, X 0 ) and vertical (Y, Y 0 ) planes.
In order to find the required beam phase-space at the injection point, we assumed a horizontal
particle distribution at the boundary of weak focusing field (Yg = −110 mm)) from the center ofthe storage magnet, and reverse track it to the matching point (outside of the storage magnet(Yg = −600 mm). The matching point was chosen outside of the storage magnet as shownin Fig. 6.1, because the beam optics elements (rotatable quadrupole magnets) for the beamphase-space matching cannot be placed inside of the storage magnet.
At the matching point (Yg = −600 mm), the beam phase-space from the reverse tracking
in beam coordinate system was calculated. Figure 6.2 presents the beam phase-space at thematching point. The solid dots (black points) in Fig. 6.2 are the data points obtained from thereverse tracking and solid line (red line) is the linear fit to these points. The axy , axx0 , ayy0 ,and ax0 y0 are slopes of the (X, Y ), (X, X 0 ), (Y, Y 0 ), and (X 0 , Y 0 ) correlations respectively.
The values of these slopes are shown in Table 6.1. The beam phase-space at the matching pointshows the strong coupling between horizontal and vertical planes.
Table 6.1: The beam slopes at the matching point.
Correlation

Slope

Value

X, Y

axy

−0.40

X, X 0axx0 [mm/mrad]

−0.95

0

ayy0 [mm/mrad]−1.70

ax0 y0

−0.71

Y, Y

X 0, Y 0

After finding the required beam phase-space slopes at the matching point as shown in Fig. 6.2,
the next target is to find the Twiss and coupling parameters for the distribution of particles at
the matching point.
Here, we will describe a transfer matrix method to compute the coupling of the beam at
the matching point. Let’s assume a matrix M which consists of rotation and Twiss parametersmatrix [44, 45, 77]:

−1
M = Uout· D · Uin ,(6.2)

where, Uin and Uout are input (initial) and output (matching point) rotation matrix, which132

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING
satisfy the symplectic 1 condition. In the case of, uncoupled input beam phase-space Uin will
be equal to identity matrix. In the case of SITE, the input beam phase-space has no coupling,
therefore, Eq. (6.2) can be written as follow:
M = U −1 · D(6.4)

The 4 × 4 U matrix is given as follow:


0

−R4

µ

R3

R2

µ

R4

0

µ



 0
Uout = 

 R1

R3



R2



−R1 


0 

µ

(6.5)

where, R1 , R2 , R3 , R4 are known as coupling parameters (unit less) and non-zerovalue of these parameters shows the strength of the coupling. The quantity µ is given aspµ =1 − (R1 R4 − R2 R3 ). The matrix D is a diagonal matrix consist of beam input andoutput Twiss parameters. The matrix D can be written as following:

D=



DX

0

0

DY

(6.6)



Where, DX in Eq. (6.6) is given as follow:
DX = 

βx

αout
x

√ out

−1

0

√ 1outpβxφX = 


0


p
in
βx

√1 in
βx
αin
x


φX 




βxout



√ in
βx

cos(φx )sin(φx )

sin(φx )cos(φx )



(6.7)

(6.8)

Where, φX is phase advance and correspond to identity in our case. Similarly for Dy . The
full form of 4 × 4 D matrix can be written as follows:



q out
βx
in
βx


in

αout
 p αx√− p x√
in
out
in
out

βx
βx
βx
βx

D=

0



0

q

0

0

0



in
βx
out
βx

0

0

βyout
βyin

0












r

0
0

in

p αy√
βyin

αout

− p iny√ out
out

βy

βy

βy

r

βyin
βyout

(6.9)

An iterative procedure has been adopted to calculate the coupling parameters.
1 A matrix U is symlectic if it satisfy the following condition [78]:
U T ΩU = Ω,

0



1

0

0

−1 0

0 0

0

0

0



1

0

−1

0

Ω=

133

0

(6.3)

6.1. BEAM PHASE-SPACE REQUIREMENTS
By changing the R1 − R4 values, we can change the rotation matrix U and the slope of the
input beam. We employed an iterative procedure to control (change: R1 − R4 ) the correlation
slopes of (X, Y ), (X, X 0 ), (Y, Y 0 ) and (X 0 , Y 0 ) until the slope of the input distributions
coincide with the axy , axx0 , ayy0 , and ax0 y0 respectively. In beginning of the iteration procedure
the following Twiss parameters were assumed as a starting point:

βxout = βxin

αxout = axx0 × βxout

βyout = βyin

αyout = ayy0 × βyout(6.10)

These Twiss parameters also has been adjusted to match the slopes of input beam phase-space
distribution with the required slope values.
The input beam phase-space distribution as shown in Fig. 6.3 was defined with the emittanceand Twiss parameters values, measured by the Q-scan, as described in the Section 5.3.1 ofChapter 5.

The output beam phase-space after getting the required Twiss and coupling

parameters is shown in Fig. 6.4, the blue lines in Fig. 6.4 are the slopes of the correlation((X, Y ), (X, X 0 ), (Y, Y 0 ), (X 0 , Y 0 )) as shown in Table 6.1 and Fig. 6.2.
The input beam matrix (collimator location) of phase space distribution as shown in Fig. 6.3
can be written as σinput :

0.76871.405700
 1.4057
σinput = 


0

0

3.0474000.732201.5177
1.5177 
3.37380(6.11)The beam matrix at the matching point σoutput after applying iterative procedure will become
as following (σoutput = M · σinput · M T , where M is as mentioned in Eq. (6.4) after the iteration
procedure):−6.8671−2.8854
 −6.8671
σoutput =  −2.8854

4.58116.4397
−3.7417 
1.3850 −2.0953 −2.0953 4.2997

2.6629−3.74174.5811

7.59842.6629(6.12)

Note, that in the case of the coupled beam off-diagonal terms appear. And coupling strengthparameter t (Eq.(5.38)) is equal to 6.24, which signify a strongly XY -coupled beam.
Table 6.2 shows the beam emittance and Twiss parameters for the input beam phase-space
distribution and Table 6.3 presents the Twiss parameters of the matched beam. Table 6.4
details the values of the coupling parameters (R1 , R2 , R3 , R4 ) for the matched beam. Figure 6.5(a) shows the beam injection into the storage magnet without phase space matching, thephase-space distribution and the emittance and Twiss parameters are given in Fig. 6.3 andTable 6.2 respectively. Whereas, Fig. 6.5 (b) presents the beam injection into the storage magnet
with phase-space matching, the phase-space distribution and the emittance and Twiss parameters
134

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING
are given in Fig. 6.4 and Table 6.3, 6.4 respectively. It can be seen clearly that the beam verticalblow-up reduce dramatically with appropriate matched beam.
Table 6.2: The initial beam phase space parameters.
Parameters

Horizontal

Vertical [mm mrad]0.61 ± 0.05

0.41 ± 0.04

βin [m]1.26 ± 0.10

1.78 ± 0.09

αin

−2.30 ± 0.17

−3.69 ± 0.38

γin [m rad]5.02 ± 0.49

8.4 ± 0.75
0R1 , R2 , R3 , R4p
| σinput |0.25

Table 6.3: The beam phase-space parameters at the matching point.
Parameters

Horizontal

Vertical

βout [m]

3

3

αout
p
| σoutput |

3

5.2
0.25
6.24

t

Table 6.4: Coupling Parameters for the Matched Beam Phase Space.
Parameters

ValuesR1−0.50

R2 [m]−0.60

R3 [1/m]

−0.99

R4

−1.09

135

6.1. BEAM PHASE-SPACE REQUIREMENTS

Figure 6.3: The input beam phase-space distribution. The the values of emittance and Twiss
parameter for these distributions are given in Table 6.2.

Figure 6.4: The coupled beam phase-space at the matching point. The line (blue) are therequired beam slopes at the injection point.

136

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING

Figure 6.5: (a) The beam injection into the storage magnet without phase space matching,
the phase-space distribution and the emittance and Twiss parameters are given in Fig. 6.3
and Table 6.2 respectively. (b) The beam injection into the storage magnet with phase-space
matching, the phase-space distribution and the emittance and Twiss parameters are given in
Fig. 6.4 and Table 6.3, 6.4 respectively.

137

6.1. BEAM PHASE-SPACE REQUIREMENTS

6.1.3

Errors on the Beam Phase-Space Slopes

In this section, the effect of errors on the slopes of beam phase-space correlation will be
described. A pseudo error was introduced to each slope (axy , axx0 , ayy0 , ax0 y0 ) independently
and its effect on the vertical beam size growth was checked by tracking simulation. From thisstudy, we found that which beam phase-space correlation is most and least sensitive to the beamvertical blow-up.
Figure 6.6 shows the slopes of the beam phase-space at the matching point. The blue lines
in Fig. 6.6 shows the slopes without any error while the other colors represent 5% (black), 10%(red), 15% (green), and 20% (orange) errors to the original slope value. In order to reveal theeffect of each correlation on the vertical beam size growth in the storage magnet, error to eachslope of correlation was applied independently, while other slopes were kept at their originalvalues.
Figure 6.7 shows the tracking simulation of the beam injection into the storage magnet
with the error on (X, Y ) correlation, while the other slopes were kept at their original value.
Figure 6.7 (a) shows the tracking with 0% error on the axy slope, whereas (b), (c), and (d)shows the beam injection into the storage magnet with 5%, 10%, 15%, error on slope of axy .
The vertical beam size growth as a function of the error on axy slope can be seen very clearly.

Figure 6.6: The slopes of the beam phase-space at the matching point. Error of 5%, 10%, 15%,
and 20% was introduced to each correlation in order to find their effect on the vertical beam sizein the storage magnet.

138

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING
Figure 6.8 present the summary of the vertical beam size growth at the kick point due to thechange in the slopes of axy , axx0 and ayy0 . The biggest growth in the beam size was observed
due to the change in (X, Y ) correlation slope (axy ). In the case of 20% change in axy slope beam
size grows to the 5.06 mm, whereas, it grows to the 3.20 mm and 2.15 mm in the case of 20%change in ayy0 , and axx0 respectively.
The radial fringe field of the solenoid magnet causes the vertical blow-up of the injected beam.
As the beam moves through the storage magnet, the particles at different vertical positions
sense different radial field and eventually takes different vertical paths. Therefore, the (X, Y )correlation slope (axy ) is most sensitive to the vertical blow-up of the beam. In the injection
region of the storage magnet, there is no vertical focusing force is present, therefore, the error
on ayy0 slope also have effect on vertical blow-up. Whereas, in the horizontal direction solenoid
magnet has intrinsic focusing, therefore, the error on axx0 slope does not impose any significant
effect on the vertical beam size growth.
For the field index value of 1.65 × 10−2 the vertical beam size should be around 8 mm inorder to have sufficient injection efficiency (forth coming section). This study shows that, If we
control the slope of axy with the 10% error and ayy0 and axx0 with slopes error of 20% than goalof 8 mm beam size can be achieved.

1396.1. BEAM PHASE-SPACE REQUIREMENTS

Figure 6.7: (a) The beam injection into the storage magnet with 0% error on the axy slope,
while other slopes (axx0 , ayy0 ) were kept at the original values. Where (b), (c), and (d) shows
the beam injection into the storage magnet with 5%, 10%, 15%, error on slope axy slope.

140

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING

Figure 6.8: The vertical beam size growth at the kick point due to the change in the slope values.
The biggest growth in the beam size was observed due to the change in X − Y correlation slope.
And the (X, X 0 ) correlation has the least effect on the beam growth.

6.2

The Matching Beamline

The SAD 2 [79] program was utilized to calculate the beamline optics for the phase space
matching. Ideally, four rotatable quadrupole magnets are enough to generate the required
beam phase-space. In the case of four rotatable quadrupole magnets, one will have 8 freeparameters, to control the 8 parameters of the beam optics (αx,y , βx,y and R1 , R2 , R3 , R4 ). In
the case of four quadrupole magnets required strength for the rotatable quadrupole magnets
and drift space becomes stringent. Therefore, at least six quadrupole magnets are required togenerate phase-space matching with the realistic parameters of the beamline. Figure. 6.9 show
the evolution of the beta function and beam size through the optics element. Table 6.5 represents
the required rotation angles and the strength of the beamline quadrupole magnets. The positive
values of the k (quadrupole magnet strength) is corresponds to the horizontal focus and negative
shows the vertical focus. Figure 6.10 (left) 3-D plot of tracking simulation of the beam injection
with appropriate coupling as mentioned in the Table 6.3 and 6.4. Figure 6.10 (right) represents
the histogram of the vertical beam size as a function of the vertical beam position. In the case,of the perfect matching vertical beam size still grows to the 5.3 mm at kick point due to thefinite size of beam phase-space area at the injection point.

2 Strategic Accelerator Design program developed at the KEK.

141

6.2. THE MATCHING BEAMLINE

Figure 6.9: The evolution of the beta function and beam size through the array of six quadrupole
magnets.

Figure 6.10: Left: The 3-D plot of tracking simulation of the beam injection with appropriate
coupling as mentioned in the Table 6.3 and 6.4. Right: The histogram of the vertical beam size
as a function of the vertical beam position.

142

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING
Table 6.5: Matching beamline quadrupole magnets parameters

6.3

Parameters

Rotation Angle (Radian)K[m−2 ]

Q1

0.22602.58

Q20.4758−2.10

Q31.65871.10

Q4−0.3314−0.83

Q50.74940.86

Q61.8395

1.83

Three Rotatable Quadrupole Magnets Solution

As mentioned in the previous section six quadrupole magnets are needed to perfectly matchthe initial uncoupled beam to the 3-D spiral injection requirement. Due to the limited space at
the SITE beamline, it was not possible to construct a beamline consist of six rotation quadrupole
magnets for phase-space matching.
In the case of SITE, the value of the field index is n = 1.65 × 10−2 which provides weakfocusing in the region of Yg = ±50 mm. From the kicker simulation3 it was determined that if
the vertical beam size becomes 8 mm at kick point, then nearly 80% of the beam can be stored.
The field index value of n = 1.65 × 10−2 provides a big region to store the beam, hence it relaxes
the beam phase-space matching requirement.
From the tracking simulation, it was found that the appropriate combination of threerotatable quadrupole magnets can reduce the vertical beam size up to the 8.06 mm at the kickpoint. Therefore, three rotation quadrupole magnets were designed and manufactured for the
SITE.

6.3.1

Beamline

The matching beamline consists of three rotatable quadrupole magnets, a bending magnet, anddrift spaces. Figure 6.11 (a) shows the dimensions of the matching beamline and (b) presents
the photo of the experimental setup. These three rotatable quadrupole magnets can be rotatedat an arbitrary angle around the beam. The transfer matrix of the beam transport line can be
written as follow:
M = L5 · Bvf · L4 · Q3 · L3 · Q2 · L2 · Q1 · L1

(6.13)

where, M is a transport line matrix, D represents the drift space, Q1 , Q2 , and Q3 denotes
the quadrupole magnets. In the SITE beamline, a rectangular pole dipole magnet was utilized to
inject the beam into the storage magnet. The rectangular bending magnet acts as a drift space
3 The electron beam tracking in the transient field of the kicker coil.

discussed in detail.

143

In the forthcoming Chapter, it will be6.3. THREE ROTATABLE QUADRUPOLE MAGNETS SOLUTION
in the deflection plane, in the non-deflecting plane it has a finite focusing effect. In the case of
40 Degree injection angle the SITE’s bending magnet has focusing strength of kb [m−2 ] = 21.88
in the non-deflecting plane. This effect was also included in the transport line program.
A simple beamline program had been developed to set the slopes of the input beamphase-space distribution to the required slopes by changing the focusing strength and rotation
angle of the quadrupole magnets. The slope of (X, Y ) slope was preferentially adjusted to therequired slope and ayy0 partially adjusted to the required slope value. And we neglected the axx0
slope as mention in the section 6.1.3.
Table. 6.6 lists beamline drift spaces and Table 6.7 shows the experimental set values ofquadrupole magnet strengths and rotation angles. The positive value of k shows the horizontal
focus while negative values correspond to the vertical focus. These beamline settings wereobtained from the program as described above.
As explained in Section 6.1.3, when the beam traverse through the storage magnet, theparticles at different vertical positions sense different radial field and eventually takes different
vertical paths. Therefore, the (X, Y ) correlation slope (axy ) is most sensitive to the verticalblow-up of the beam. In the injection region of the storage magnet, there is no vertical focusing
force is present, therefore, the error on ayy0 slope also have a considerable effect on verticalblow-up. Whereas in the horizontal direction solenoid magnet has intrinsic focusing, therefore,
the error on axx0 slope does not impose any significant effect on the vertical beam growth.
The simulated beam phase-space with the three rotation quadrupole magnets settings as
mentioned in the Table 6.7 are shown in the Fig. 6.12. Figure 6.13 shows the beam profile of theinjected beam. False colors were added to the picture for better visualization. The white line in
Fig. 6.13 is the fitted slope to the beam profile. For the reasons explained above the slopes of(X, Y ) and (Y, Y 0 ) correlations were preferentially matched with the ideal slopes.
Table 6.6: The drift spaces for the beamline.
Parameters

Length [mm]

L1

2.35

L2

12.25

L3

22.45

L4

28.25

L5

210.3

144

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING

Figure 6.11: (a) The layout of the beamline, beamline composed of three rotatable quadrupole
magnets, a rectangular bending magnet and drift space. The rotatable quadrupole magnets wereplaced in the downstream direction of the collimator. The dimensions and mutual distancesamong magnets are also shown. (b) The photo of the experimental setup of the matching
beamline.

145

6.3. THREE ROTATABLE QUADRUPOLE MAGNETS SOLUTION
Table 6.7: The Three Rotatable Quadrupole Magnets Matching Beamline Parameters.
Focusing

Quadrupole

Rotation Angle (Degree)k[m−2 ]

H

Q1

20.0

53.17

V

Q2

26.0

−67.76V

Q3

0.0−24.14

Figure 6.12: The simulated beam phase-space with the three rotation quadrupole magnets. The
solid blue line shows the ideal slopes for the beam phase-space matching. The beamline setting
is described in the Table 6.7.

146

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING

Figure 6.13: The beam profile of the injected beam. The false colors were added to the picture
for better visualization.

6.4

Results

In this section, firstly the beam injection into the storage magnet without rotation quadrupole
magnets will be described and in the later part, the beam injection with rotatable quadrupole
magnets will be described. The results of the gas monitor and wire scanner with different
injection are presented and discussed to examine the effect of the beam phase-space matchingon the vertical beam size growth. Here, only the results of the beam injection into the storage
magnet without any phase space matching and with the best setting that was achieved withthe three rotation quadrupole magnets are presented. The wire scanner results with some othersettings of the three rotation quadrupole magnets can be seen in Appendix B.

6.4.1

Injection without Rotatable Quadrupole Magnets

In the first step, the beam was injected into the storage magnet while rotatable quadrupole
magnets were off. This experiment was performed, in order to reveal the effect of a complete
beam mismatch on the vertical beam size growth. Figure 6.14 (above) shows a simulation of
the beam phase-space distribution without applying rotatable quadrupole magnets and Fig. 6.14
(below) is the injected beam profile at the beam profile monitor.
Gas Monitor Results
As reported in Chapter 4, in gas monitor the electron beam passes through nitrogen gas, it
excites the nitrogen gas molecules. Nitrogen gas de-excitation results in a light emission in theblue wavelength range 390 nm < λ < 470 nm [55]. The light from the nitrogen gas de-excitation
1476.4. RESULTS
was collected by a CCD camera [61]. Figure 6.15 (above) is the gas monitor results of the beam
injection into the storage magnet without rotatable quadrupole magnets. The false colors wereadded to the photo for better visualization. Figure 6.15 (below) shows the horizontal projectionof the above picture. As expected, the beam blow-up in the vertical direction as it traverses
through the storage magnet without phase-space matching, this effect can be easily visualized
qualitatively from the gas monitor result as shown in Fig. 6.15. The gas monitor provides a quick
and efficient way of beam monitoring inside the storage magnet, but the lack the quantitative
information of the beam size inside the storage magnet. Therefore, a wire scanner was utilized
to extract the qualitative information of the beam position and size in the storage magnet.
Wire Scanner Results
In order to measure the beam profile and position information inside the storage magnet, a new
type of wire scanner was employed. Please refer to Section 4.2 of Chapter 4, for the detailed
principle and structure of the wire scanner system.
The signal from the wires was taken out of the vacuum chamber via electrical vacuum
feedthroughs and terminated into the 1 MΩ terminal of an oscilloscope [69]. The data (voltages)
were collected using a custom made GUI C# program. Please refer to Appendix A for detail of
C# code. The beam current can be calculated by using Ohm’s law: V = IR, where V is thevoltage collected from the oscilloscope, R is the resistance, and equal to 1 MΩ, I is the beam
current. During the beam test, the beam current at the Faraday cup was 40 µA.
Figure. 6.16 (a) shows the CST-PS [36] tracking simulation of the beam injection into thestorage magnet with the beam phase-space as given in Fig. 6.14 (No phase-space matching).
The beam current was set 40 µA for the CST-PS simulation. The black open circles show thepoints at which the beam sizes were measured by using the wire scanner 1. For the convenience,
the wire scanner results and simulation are shown in global coordinates (Xg , Yg , Zg ). Here, Yg
shows the storage magnet (solenoid) axis.
Figure 6.16 (b) details the data of wire scanner 1. The horizontal axis (Yg ) in Fig. 6.16 (b)shows the vertical position of the wire scanner in the storage magnet, and the vertical axis
presents the beam current measured along the storage magnet axis. The peaks position in
Fig. 6.16 (b) illustrate the beam signal at the different vertical position, and the width of thepeak shows the vertical beam size along the storage magnet axis. The distance between thesepeaks shows the spacing between electron beam orbits, here we refer it as pitch distance. The
curves in wire scanner data seem to follow the Gaussian distribution. Therefore, we fit theexperimental data with the Gaussian function. The multi-peak Gaussian function used to fit theexperimental data globally can be written as follows:

f (x) = w +

n
X

1
p

i=1

e
2

2πσi

−(xi −µi )2
2σ 2
i

,

(6.14)

where w is the estimation of the baseline and n corresponds to the number of peaks. The
148

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING
black points in Fig. 6.16 (b) are the wire scanner 1 data, the green histogram illustrates thesimulation results of vertical beam size. The orange line is a global fit to the experimental data.
The dashed blue line corresponds to the local peak fitting.
Figure 6.17 reports the wire scanner 2 data. The wire scanner 2 was located at 180 Degree
away (half cyclotron period) from the wire scanner 1 position. Figure 6.17 (a) shows thesimulation and the black open circle illustrates the points at which vertical beam size wasmeasured. Figure 6.17 (b) presents the wire scanner 2 data, black points illustrate wire scanner
data and the green histogram presents the simulation results of vertical beam size, and orange
and blue curves are global and local fitting to the data. The simulation and experimental data
are in good agreement with each other. And no significant discrepancy was found
The vertical beam size at the injection point (Yg = −345.46 mm) was 1.4±0.02 mm. Whenthe beam reaches near to the kick point (Yg = -59.57 mm), the vertical beam size increase up
to 31.25±0.21 mm. This large value of vertical beam size at the kick point will greatly reducethe injection efficiency. This drastic beam blow-up in the storage magnet occurred due to beamphase-space miss-matching at the injection point. These results of the wire scanner indicate theneed for the beam phase-space matching, to avoid the beam blow-up in the vertical direction.

149

6.4. RESULTS

Figure 6.14: Above: The simulation of the beam phase-space distribution without applying
quadrupole magnets. Below: The injected beam profile at the beam profile monitor. The false
colors were added for the better visual effects.

150CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING

Figure 6.15: Above: The gas monitor results of the beam injection into the storage magnet
without rotation quadrupole magnets. The false colors were added to the photo for better
visualization. It is evident from the gas monitor results that the beam blow-up in verticaldirection as it traverses through the storage magnet.

1516.4. RESULTS

Figure 6.16: (a) The tracking simulation of the beam injection ( CST-PS) into the storage
magnet without phase-space matching. The black open circle shows the points at which thebeam sizes were measured. (b) The vertical beam size was measured by using a wire scanner.
The black points are the wire scanner 1 data. The green histogram is a simulation result of
CST-PS. The orange line is a global fit to experimental data.

152

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING

Figure 6.17: (a) The tracking simulation of the beam injection (CST-PS) into the storage magnet
without phase-space matching. The black open circle shows the points at which the beam sizes
were measured. (b) The black points are the wire scanner 2 data. The green histogram is asimulation result of CST-PS. The orange line is the fit to experimental data. The wire scanner
2 is located at 180 Degree away (cyclotron period) from the wire scanner 1 position.

1536.4. RESULTS

6.4.2

Injection with Normal Quadrupole Magnets

In this case, the beam phase only with the Twiss parameters as mentioned in the Table 6.3
assumed and coupling parameters were set to zero, in order to reveal the effect of only Twiss
parameter matching on the beam vertical size in the storage magnet. Figure 6.18 (a) shows
the beam phase–space at the injection point. Figure 6.18 (b) illustrate the tracking in CST-PS
with the beam phase space as mentioned in the Figure 6.18 (a), as the beam traverse throughthe storage magnet beam blow-up drastically in the vertical direction. It shows that the mostxy-coupling to beam cross section is the most important parameter to controls the vertical beam
spread in the solenoidal-type storage magnet.

Figure 6.18: (a) The beam phase-space at the injection point with the Twiss parameters aslisted in the Table 6.3 , but coupling parameters were set to zero. (b) Particle Tracking withthe beam phase-space as mentioned in the in (a) as the beam traverse through the storage
magnet beam blow-up drastically in the vertical direction. It signifies that the xy-coupling to
the beam cross section is the most important parameter to controls the vertical beam spread in
the solenoidal-type storage magnet.

154

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING

6.4.3

Injection with Rotatable Quadrupole Magnets

In this section, the effect of the rotatable quadrupole magnets on the beam size growth in
the storage magnet will be described.
Gas Monitor Results
The gas monitor result of the beam injection into the storage magnet with rotation quadrupole
magnet setup as given in Table 6.7 is shown in Fig. 6.19 (above). The false colors were added tothe photo for better visualization. The beam enters into the storage magnet from the injection
hole and starts taking a spiral path into the storage magnet. Figure 6.19 (below) presents thehorizontal projection of the above picture at a position highlighted in a thin line in Fig. 6.19
(above). The three turns can be visualized easily from this data. As anticipated from thesimulation, the vertical beam blow-up significantly decreases with the appropriate setting of
three rotatable quadrupole magnets.
Wire Scanner Results
Figure 6.20 (a) shows the beam tracking simulation with the beam phase-space as shown in theFig. 6.12, the black open circle presents the positions where beam size was measured by the wire
scanner. Figure 6.20 (b) shows the vertical beam size measured by using a wire scanner withthe rotatable quadrupole magnets set up. Note that the vertical axis is the log scale. The black
points are the wire scanner 1 data and the red histogram is a simulation result of CST-PS. Theorange line is the multi-peak Gaussian fit to experimental data and dashed blue lines correspondto the local fit to each peak.
The wire scanner 2 simulation and data is shown in Figure 6.21. The wire scanner 2 is locatedat 180 Degree away (half cyclotron period) from the wire scanner 1 position. Figure 6.20 (b)represents the vertical beam size measured by using a wire scanner 2. The red histogram isa simulation result of CST-PS. The orange line is the fit to experimental data. As the beam
traverse through the storage magnet, the amplitude of the peak decrease but the area underR
the curve (A = Idy, A is the area and I is the beam current) remains the same. In the wire
scanner 1 data, the area under the first peak (injection point), second peak (first turns), andR
third peak (second turn) is A = Idy = 41.91 A mm, 40.31 Amm, and 39.69 Amm respectively.
From injection point to second turn the area under the curve changes 5.3%. This decrease could
be due to some lost particles as the beam moves through the storage magnet.
The wire scanner results clearly show that the vertical beam blow-up reduced dramaticallywith the rotatable quadrupole magnets. The vertical beam size at the injection point (Yg= –344.30 mm) was 2.51±0.02 mm. When the beam reaches near to the kick point (Yg =–82.61 mm), the vertical beam size becomes to 8.06±0.21 mm. This value of vertical beam sizeat the kick point will provide 81% of the injection efficiency. This reduction in vertical beam
size blow-up confirms our beam phase-space matching strategy.
155

6.4. RESULTS
The horizontal beam profile also has been measured by rotating vertical wire on the wire
scanner 1. Please refer to the Section 4.2.1 of Chapter 4 for the detailed mechanism of horizontal
beam size measurement monitor. Since the only one rotary actuator was available, therefore,horizontal beam profile had been measured only at the wire scanner 1 position. In order toextract the horizontal beam size, the vertical wire was placed at the fixed vertical position, andthe wire scanner support bar was rotated at its own axis by the rotary actuator. As verticalwire rotates it provides the information of horizontal beam position and horizontal beam size.
Figure 6.22 (a) presents the 2-D projection of the beam profile in the storage magnet. The
positive Xg -axis corresponds to the wire scanner 1 location and negative show the wire scanner
2 location. The black histogram is the CST-PS simulation and gray color show the experimental
data. Figure 6.22 (b) is the zoom up view of the wire scanner 1 location. The horizontal beamprofile at different vertical positions is shown in Fig. 6.23. The black points are the experimental
data, the red histogram is the simulation, and the orange line is a fit to the experimental data. As
explained earlier solenoid-type storage magnet has an intrinsic focusing effect in the horizontal
direction, therefore, a horizontal beam does not grow in the storage magnet as the beam traverse
through the storage magnet. The average horizontal beam profile form first turns to the second
turn was 0.65 mm (1σ).

156

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING

Figure 6.19: Above: The gas monitor results of the beam injection into the storage magnet withrotation quadrupole magnets set up as mentioned in Table 6.7. Below: The horizontal projectionof the above picture. The three turns can be visualized easily from this data.

157

6.4. RESULTS

Figure 6.20: (a) The tracking simulation of the beam injection ( CST-PS) into the storage
magnet without phase-space matching. The black open circle shows the points at which thebeam sizes were measured. (b) The vertical beam size is measured by using a wire scanner. The
black points are the wire scanner 1 data. The red histogram is a simulation result of CST-PS.
The orange line is the global fit to experimental data.
158

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING

Figure 6.21: (a) The tracking simulation of the beam injection ( CST-PS) into the storage magnet
without phase-space matching. The black open circle shows the points at which the beam sizes
were measured. (b) The vertical beam size was measured by using a wire scanner. The black
points are the wire scanner 2 data. The red histogram is a simulation result of CST-PS. Theorange line is the fit to experimental data. The wire scanner 2 is located at 180 Degree away
(half cyclotron period) from the wire scanner 1 position.
159

6.4. RESULTS

Figure 6.22: (a) The 3-D plot of the 2-D projection of the beam profile in the storage magnet. The
positive X-axis corresponds to the wire scanner 1 location and negative show the wire scanner
2 location. The black histogram is the CST-PS simulation and grey plot show the experimental
data. (b) The zoom up view of the wire scanner 1 location.

160

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING

Figure 6.23: The horizontal beam profile at different vertical positions. Average horizontal beamprofile form first turn to the second turn was 0.65 mm (1σ).

161

6.5. DISCUSSIONS AND SUMMARY

6.5

Discussions and Summary

The dramatic reduction in the vertical beam size was observed with the appropriate settingof three rotatable quadrupole magnets. Without any beam phase-space matching, the beam
size grows to 31.25 ± 0.21 mm (FWHM) at the kick point of the storage magnet. The verticalbeam blow-up was reduced to 8.06 ± 0.21 mm (FWHM) at the kick point with the appropriate
combination of the three rotation quadrupole magnets. The horizontal beam size does not growas beam moves through the storage magnet. The horizontal beam size was estimated 0.65 mm(1σ). In conclusion vertical beam size reduce to 3× with the three rotatable quadrupole magnets
as compared to the beam injection without any rotatable magnets. The charged particle tracking
simulation in CST-PS and experimental results are in good agreement with each other.
Figure 6.24 shows the comparison of the beam vertical beam size growth as a function of theaxial position of the storage magnet among different cases. The comparison between numericalvalues is given in the Table 6.9. Even in the case of the perfect matching the vertical beam
size grows to the 6.3 mm (FWHM), whereas in the case of the partial matching vertical beamincreases up to 8.06 ± 0.21 mm. In the case of the SITE, we choose the high value of field index
1.65 × 10−2 , this field index provides the focusing in the region of Y = ±50 mm. Therefore,
partial matching of the beam phase-space is enough to store nearly 81% of the beam. In thenext chapter, this topic will be discussed in detail.
Table 6.8 provides the comparison between ideal slope and measured slope of the X − Y .
Table 6.8: The Comparison of the Beam Slopes between the Simulation and Measured from theBeam Profile Monitor.
Correlation

Ideal SlopeNo quads (Slope)With Quads (Slope)

X-Y

0.40

0

0.35

Table 6.9: The Vertical Beam Sizes as a function of the Vertical Position of the Storage Magnet
Without Quads

With rotation Quads

Y0 [mm]

FWHM [mm]Y0 [mm]

FWHM [mm]-345.46

1.4 ± 0.02

-344.30

2.51 ± 0.04

-134.34

10.62 ± 0.23

-141.34

2.71 ± 0.03

-59.5731.25 ± 0.21

-82.618.06 ± 0.21

21.36

78.41 ± 0.83

-39.4918.96 ± 0.14

86.558122.45 ± 1.11

-3.47

25.68 ± 0.31

49.22

40.39 ± 0.81

In order to completely stop the vertical motion of the beam in the storage magnet, a verticalkick is needed. To kick the beam appropriately the pitch angle at the kick point must reduce to162

CHAPTER 6. EXPERIMENTAL RESULTS OF PHASE SPACE MATCHING

Figure 6.24: The comparison of the vertical beam sizes growth in the storage magnet withoutphase-space matching, with appropriate combination of rotatable quadrupole magnets andsimulated perfect matching.
5-3 Degree. The small pitch angle reduces the high power and fast timing constraints of the kicker
power supply. As shown in Fig. 6.24 with partially matched beam pitch angle at Y = −82.6mm reduces to 3.37 Degree. Hence in the case of the partially matched beam injection, we cankick the beam appropriately. The following formula can be used to compute the pitch angle ofthe beam:ψ ≈ tan−1 (p)
πD(6.15)

where p is the pitch distance between two consecutive turns and D is the diameter of the spiraltrajectory. Figure 6.25 shows the pitch angle of the beam as it traverses through the storage
magnet. The comparison between the numerical values of the pitch angle between unmatchedand partially matched beam is given in the Table 6.10. Note that the pitch angle at the kickpoint can be controlled by the injection angle.

1636.5. DISCUSSIONS AND SUMMARY

Table 6.10: The Pitch Angel as a Function of the Vertical Position of the Storage Magnet
Without Quads

With rotation Quads

Y0 [mm]ψ [Degree]

Y0 [mm]ψ [Degree]-342.1219.47-344.30

15.06-75.538.43-141.34

4.45

36.34

13.73

-82.6

3.27

Figure 6.25: The comparison of the pitch angle between without phase space matching and withappropriate combination of rotatable quadrupole magnets.

164Chapter 7

Kicker System for SITE
To store the beam at the center of the storage magnet, the vertical spiral motion of the beam
must be completely stop, as it reach to the storage plane. Or in other words the pitch angleof the beam must reduce to around 0° as it reach to the storage plane. The remaining small
pitch angle (in the range of few mrad) of the beam will be compensated by the designed weakfocusing field at the storage plane. A pulsed vertical kicker will be utilized to stop the verticalmotion of the beam at center of the storage magnet. The vertical kicker will generate the pulsed
radial field of magnitude of few gauss and several cyclotron periods, to gradually reduced thepitch angle the beam. Figure 7.1 presents the concept of the kicker. The black dashed trajectoryshows the motion of the beam inside the storage magnet without a kicker; it is observed that the
injection angle of the beam reduced due to the radial fringe field but does not decrease to zero.
The beam passed through the middle region of the storage magnet without getting stopped.
When the appropriate kick is applied to the beam, as shown in the black solid trajectory, thepitch angle is reduced to zero.
In this Chapter, the following studies for the SITE kicker will be described• Basic concept and design for the SITE kicker• Tracking studies to determine the kicker requirements• Eddy current consideration for kicker system

165

Figure 7.1: (a) The concept of the vertical kicker. The beam enters in the storage magnet at 40
Degree and radial field of the storage magnet reduce the injection to 4-5 Degree as it approaches
the boundary of the weak focusing field. At Yg = −100 mm a vertical kick applied to the beam
to gradually reduce the pitch angle of the beam to the zero. (b) The pitch angle of the injected
beam. And its reduction due to the radial field of the storage magnet and kicker field.

166

CHAPTER 7. KICKER SYSTEM FOR SITE

7.1

Vertical Magnetic Kicker for SITE

In the case SITE following two kinds of kickers were considered• Symmetrical Kicker• Localized Kicker
In symmetrical type kicker, the kick field will be applied to the beam throughout the regionof the kick point to the stop position (storage plane). The pitch angle reduces continuously inthe case of the symmetric kick. Figure 7.2 shows the symmetrical kick. The beam is injected into
the storage magnet at 0.7 rad. The orange line in Fig.7.2 shows the reduction of the injection
angle (pitch angle) due to the radial field of the storage magnet. The pitch angle reduces to
51.14 mrad at vertical position of −98.70 mm. At this point, the kick will be applied to thebeam. The motion of the particles in the time-varying kicker field was simulated by the 4-thorder Runge-Kuta method in Mathematica [37]. The half-sine type pulse shape of width 50 ns
was considered for this simulation. The longer pulse width decrease stringent requirement of thepower supply. The blue curve in Fig. 7.2 shows the decrease in the pitch angle to 0 rad as itreaches the storage plane, due to the symmetrical kick.
In the case of the SITE, the value of the field index (n: Weak Focusing Strength) is set to1.65 × 10−2 . The corresponding speratrix1 for this field index value are plotted in the Fig. 7.3.
The closed circles show the stable regions and hyperbolic curves show the unstable regions. As
far as the particles arrive in the closed contours they will trap in the weak focusing field andperform the simple harmonic motion. As described in the last chapter the beam size grows to
the 25.68 mm. Therefore, the weak focusing trap is big enough to store 81% of the beam. The
red contours in Fig. 7.3 show the stable storage orbit of Yg = ± 30 mm.
In a localized type kicker, the kick field will be applied to the beam only in the limited regionof the kick point to the stop position (storage plane). The pitch angle reduces in steps in the caseof the localized kick. Figure 7.4 shows the character of the localized kick. The pulse width is
50 ns and one cyclotron period is 5 ns, hence beam has to pass ×10 through the localized kicker
before pitch angle reduced to the zero at the storage plane. The orange line in Fig.7.4 shows the
reduction of the injection angle (pitch angle) due to the radial field of the storage magnet. The
pitch angle reduces to 57.14 mrad at vertical position of −101.60 mm. The black step-curve in
Fig. 7.4 shows the decrease in the pitch angle to 0 rad as it reaches the storage plane, due to the
localized kicker. Figure 7.5 represents the localized kick and separatrix. The decrease in steps
is due to the limited region of the kicker. Table 7.1 shows the beam position and pitch angle at
the kick point for the symmetrical and localized kicker.
The oscillation of the stored beam (vertical direction and pitch angel) in the weak focusing
potential is shown in Fig. 7.8 and Fig.7.9. The horizontal and vertical tune of the beam are as
1 A phase-curve which connects the stable and unstable regions [80]167

7.1. VERTICAL MAGNETIC KICKER FOR SITE
following:νx =√1 − n = 0.9919,

νy =

√

n = 0.1284(7.1)

Table 7.1: The beam position and pitch angle at kick point.
Parameters

Symmetrical kicker

Localized kicker

Injection Angle [rad]

0.70

0.70

Kick Position [mm]-98.70-101.60

Pitch Angle at kick point [mrad]

51.14

57.59

Kick duration [nsec]

50

50

Field Index (n)1.65 × 10−2

1.65 × 10−2

Figure 7.2: The character of the symmetrical kicker. The orange line shows the reduction in thepitch angle of the beam from the injection point to the kick point. The blue curve shows the
kick due to the symmetrical kicker, the pitch angle of the beam reduces smoothly in the of thesymmetrical kicker. After applying kick the pitch angle reduces to the zero.

168

CHAPTER 7. KICKER SYSTEM FOR SITE

Figure 7.3: The symmetrical kick and separatrix. The blue curve is the symmetrical kick. The
closed circles shows the stable storage orbit. If the particles lands on the closed circle after the
kick the will get store. Th red circles shows the storage orbit for Yg = ±30 mm.

169

7.1. VERTICAL MAGNETIC KICKER FOR SITE

Figure 7.4: The character of the localized kicker. The orange line shows the reduction in thepitch angle of the beam from the injection point to the kick point. The black curve shows the
kick due to the localized kicker, the pitch angle reduces to zero in 10 steps.

170

CHAPTER 7. KICKER SYSTEM FOR SITE

Figure 7.5: The localized kick and separatrix.

1717.1. VERTICAL MAGNETIC KICKER FOR SITE

Figure 7.6: The oscillation of the pitch angle in the weak focusing field.

Figure 7.7: The oscillation of the beam in the vertical direction in the weak focusing field.

172

CHAPTER 7. KICKER SYSTEM FOR SITE

7.2

Design of the kicker Coils

A pair of circular coils with the current in the opposite directions can generate the radial field.
This kind of coil configuration can be used to produce symmetrical kick for the beam. Figure 7.8
above shows the symmetrical kicker coil design. The diameter of these coils is 220 mm and theirmutual distance is 240 mm. The radial magnetic field profile at the radius 120 mm along the axial
direction is shown in Fig. 7.8. The total current required to produce the appropriate symmetricalkick is 15 A. Table 7.2 shows the parameters for the symmetrical kicker coils. The inductanceof each coil is 0.67 µH. If they are connected in parallel then total inductance of the coils will
become LT = 0.34 µH.
LT =

L1 L2
L1 + L2(7.2)

Table 7.2: Symmetrical Kicker Coil Parameters.
Parameters

ValuesDiameter [mm]220Mutual Distance [mm]240

Inductane of each coil [µH]0.67

Current [A]15

A dipole coil can serves as a localized kicker for 3-D spiral injection. Figure 7.9 represents the
localized kicker coil design. The blue circles in Fig. 7.9 represents the beam trajectory. Figure 7.9
below show the localized kicker field at the Yg =0 mm. The field direction is perpendicular tobeam motion. Hence, this field can be used to produce a kicker field. The geometrical andelectrical parameters of the localized kicker are shown in Table 7.3. The kicker coils were designed
in CST-MS [36].
Table 7.3: Localized Kicker Coil Parameters.
Parameters

ValuesWidth [mm]62Gap [mm]20

Height [mm]175

Inductane of each coil [µH]0.44

Current [A]

16

173

7.2. DESIGN OF THE KICKER COILS

Figure 7.8: Above: The design of the symmetrical kicker. Below: The radial magnetic field of
the symmetrical kicker coils.

174

CHAPTER 7. KICKER SYSTEM FOR SITE

Figure 7.9: Above: The design of the localized kicker. Below: The kicker field of the localized
kicker coils.

175

7.3. EDDY CURRENT CONSIDERATIONS

7.3

Eddy Current Considerations

According to Faraday’s law of induction the change in the magnetic field produce loops ofelectric current or Eddy current in the conductor. The kicker coils are the transient device andproduce eddy current in the wall of the vacuum chamber. The eddy current flow in the wallof the vacuum chamber can reduce the magnitude of the kicker field. In addition, it can also
cause a thermal load for the vacuum chamber. In order to estimate the Eddy current effect, asimulation in the CST-LT [36] was conducted.
The vacuum chamber is made of SUS316. And the conductivity of SUS316 is 1.334×106 S/m.
Figure 7.10 shows a vacuum chamber with the symmetrical kicker coils model. Figure 7.11 shows
the contours plot of the average ohmic loss in the wall of the vacuum chamber due to Eddy
current. The ohmic loss follows the shape of the current pulse. The maximum power loss in thecase of the symmetrical kicker at the wall vacuum chamber is 1.1 Watt. Figure 7.12 shows thevacuum chamber with localized kicker and Fig. 7.13 shows the average ohmic loss in the case of
the localized kicker. The maximum power loss in the case of the localized kicker is 0.49 Watt.
Figure 7.14 represents the power losses as a function of the kicker duration for symmetrical(Blue) and localized (Black) kicker coil. Note that the power loss is small in the case of thelocalized kicker due to the smaller and confined volume of the localized kicker coil. Whereas in
the case of symmetrical kicker large area is exposed to the surface of the vacuum chamber andhence results in more power loss. Figure 7.15 and 7.16 show the decrease in the magnetic field
due to the eddy current in the symmetric and localized kicker coils respectively. In the case ofsymmetric kicker field strength reduce nearly 30% and in the case of localized kicker reduction
is only 4%.
The power loss of 1.1 Watt and 0.4 Watt can not cause any significant thermal load. Therefore,
this kicker field reduction due to the Eddy current can be easily compensated by the increase ofcurrent from power.
Table 7.4: Power loss due to the Eddy Current.
Parameters

Symmetrical kicker

Localized kicker

Power Loss [Watt]

1.10

0.49

Field reduction

30%

4%

176

CHAPTER 7. KICKER SYSTEM FOR SITE

Figure 7.10: The vacuum chamber and kicker symmetrical kicker coils.

Figure 7.11: The average ohmic losses in the wall of the vacuum chamber due to the symmetricalkicker coils.

1777.3. EDDY CURRENT CONSIDERATIONS

Figure 7.12: The vacuum chamber and kicker localized kicker coils.

Figure 7.13: The average ohmic losses in the wall of the vacuum chamber due to the localized
kicker coils.

178

CHAPTER 7. KICKER SYSTEM FOR SITE

Figure 7.14: The power loss in the wall of the vacuum chamber due to the symmetric andlocalized kicker coils.

Figure 7.15: The magnetic field reduction in the case of symmetrical kicker due to eddy current
flow in the wall of the vacuum chamber.

179

7.4. KICKER POWER SUPPLY

Figure 7.16: The magnetic field reduction in the case of localized kicker due to eddy current flowin the wall of the vacuum chamber.

7.4

Kicker Power Supply

The kicker power supply was designed and built-in collaboration with the private company’Pulsed Power Japan Laboratory Ltd’ [81]. Table 7.5 represents the parameters of the kicker
power supply.

The required voltages and capacitance for the kicker power supply can becomputed as follow:
C=


T 2
π
L
r

(7.3)

L
C

(7.4)

√
T = π LC

(7.5)

V =I

The MicroCap [82] software was used to design the kicker power supply. Figure 7.17 (a)shows circuit diagram of the kicker power supply. Figure 7.17 (b) and (c) represents the current
through kicker coils and voltage across kicker coils respectively.

180

CHAPTER 7. KICKER SYSTEM FOR SITE

Figure 7.17: (a) The circuit diagram of the kicker power supply. (b) The current through kicker
coils. (d) The voltage across kicker coils.

1817.5. INJECTION EFFICIENCY
Table 7.5: Kicker Power supply Parameters.

7.5

Parameters

ValuesInductance [ µH]

1 (including Cables)

Capacitance [nF]

0.2533High Voltage [kV]5

Current [A]20Frequency [Hz]50

Pulse Width (half sine) [nsec]

50

Injection Efficiency

In order to determine the injection efficiency multi-particle tracking had been performed
with different vertical initial phase-space conditions at the kick point. Figure 7.18 shows thebeam vertical phase-space with different conditions at the kick point. In Fig. 7.18 (a) shows thevertical phase-space acceptance of the kicker in the case of n = 1.65 × 10−2 , (b) is the verticalphase-space at the kick point with phase-space matching that can be achieved with the threerotatable quadrupole magnets, (c) presents the same vertical phase-space as mentioned in (b)except flipped it to the focusing, and (d) shows the vertical phase-space at the kick point without
any phase-space matching.
The multi-particle simulation with 100 samples in the transient field of the kicker and weakfocusing index n = 1.65 × 10−2 was carried out. In the case of, no phase space matching (noXY-coupling) vertical beam phase-space completely mismatch with the acceptance of the kicker
and only a few stray particles get trapped in the weak focusing potential. In this case, theinjection efficiency is only 12%.
The red points and ellipse in Fig. 7.18 presents the vertical phase-space at kick point point,
achieved by three rotation quadrupole magnets. In this case, the vertical phase-space partiallymatched with the kicker acceptance. Figure 7.19 shows the kicker tracking with this verticalphase-space (XY-coupled) to estimate the injection efficiency. The closed red contours presents
the stored beam and open hyperbolas are lost particles. In this case injection efficiency is 81%.
The blue points in the Fig. 7.18 presents the ideal vertical phase-space at the kick point.
It is the same vertical phase-space as mentioned in (b) except flipped it’s sign to the focusing.
This situation has been simulated to show the injection efficiency in the case of the ideal case.
Figure 7.20 shows the kicker tracking with flipped XY-coupled phase space to estimate theinjection efficiency. The closed blue contours represents the stored beam. In this case injection
efficiency is 100%.
The injection efficiency with different vertical phase spaces is summarized in Table 7.6.

182

CHAPTER 7. KICKER SYSTEM FOR SITE
Table 7.6: Injection Efficiency
ID

Condition

Stored/Total

Injection Efficiency(b)

XY-coupled

81/10081%(c)Flipped XY -coupling (Ideal)

100/100

100%(d)

No XY -coupling

12/10012%

Figure 7.18: The beam vertical phase-space at the kick point.(a) The vertical phase space
acceptance of the kicker. (b) The vertical phase-space at the kick point with phase-space
matching. (c) The same vertical phase-space as mentioned in (b) except flipped it’s sign to
the focusing. This case has been simulated in order to show the injection efficiency in the ideal
beam phase-space at the kick point. (d) The vertical phase-space at the kick point without any
phase-space matching.

183

7.5. INJECTION EFFICIENCY

Figure 7.19: The kicker tracking with the phase-space matching (XY-coupled) to estimate theinjection efficiency. Th grey lines shows the separatrix for the field index 1.65 × 10−2 . The closed
contours presents the stable region and open hyperbolas shows the un-stable region. The redclosed lines are presents the stored beam, whereas, open line shows the lost particles.

184

CHAPTER 7. KICKER SYSTEM FOR SITE

Figure 7.20: The kicker tracking with flipped XY-coupled phase space. The closed blue contours
represents the stored beam. In this case 100% beam get stored.

185

7.6. SUMMARY

7.5.1

Injection Inefficiencies

The most probable reasons for the injection inefficiencies are following:• Vertical beam size at the kick point• Energy spread• The beam pulse length
The major reason for the injection inefficiency is the bigger vertical beam size at the kickthan the acceptance of the the weak focusing index. As shown in the the previous section, in thecase of no phase-space matching beam size grows to the 31 mm at the kick point and injection
efficiency is only 12%. In the case of phase-space matching with three rotatable quadrupole
magnets the vertical beam size reduces to the 8.06 mm at the kick point and approximately 80%
beam gets stored in this case.
The other cause of the injection inefficiency could be the beam energy spread. The energy
spread (4E) is associated to the accelerating voltage of the electron gun. The better stability ofaccelerating voltage results in the lower energy spread. For the SITE’s electron gun the energy
spread from the accelerating voltage stability was estimated about 0.01% (4E/E = 6 × 10−5 ).
This small value of the energy spread does not cause any effect on the injection efficiency. In
order to reveal the effect of the energy spread on the injection efficiency a factor of large energy
spread was assumed (4E/E = 6×10−4 ). In this scenario, the injection efficiency reduce to 67%.
Figure 7.21 presents the tracking of the beam in transient field of the kicker with the energy
spread of 4E/E = 6 × 10−4 .
The beam of 2 ns pulse length will be generated by the electric chopper system. In order toshow the effect of beam pulse length on the injection inefficiency, tracking simulation with 2 ns.
The kicker duration is 50 ns. Because kicker duration is enough longer as compared to the beampulse length of 2 ns, therefore, it has no effect on the injection efficiency.

7.6

Summary

The symmetrical and localized types of kickers are designed for the SITE. The transient
tracking simulation had been done by the custom made program (4th order Runge-Kutta
method) to simulate the characteristics of the beam in the kicker field. The beam acceptanceand storage volume was also computed by the tracking simulation.
The realistic design of both types of kickers coils was presented. In the case of symmetricalkick an Anti-Helmholtz type kicker will serve the purpose, whereas in the case of the localized
kicker a simple dipole coil can produce the required kick field.
The power losses due to Eddy current in the wall of the vacuum chamber was also estimated.
It was found that the losses due to the Eddy current is minor and can be compensated by simply
increasing the current of the kicker power supply.
186

CHAPTER 7. KICKER SYSTEM FOR SITE

Figure 7.21: The tracking of the beam in transient field of the kicker with the energy spread of
4E/E = 6 × 10−4 . In this scenario, the injection efficiency reduce to 67%.

187

7.6. SUMMARY
The kicker power supply had been designed and manufactured in collaboration with a privatecompany. The test of the kicker power supply also has been performed.

188

Chapter 8

Conclusions
The muon anomalous magnetic moment (g − 2)µ is one of the most important measurements
in elementary particle physics.

The most recent measurement of (g − 2)µ at Brookhaven

National Laboratory (BNL) results in more than 3σ discrepancy compared with the equally
precise Standard Model prediction. This disparity could be indicative of new physics beyond
the standard model. To resolve this discrepancy, a new muon’s g – 2/EDM (E34) experiment at
J-PARC aims to measure (g − 2)µ to a precision of 0.1 ppm and electric dipole moment (EDM)
down to the sensitivity of 1.5 × 10−21 e.cm. The key objective of measuring (g − 2)µ is to store
spin polarized µ+ beam in a magnetic field and evaluate the evolution of the spin precession
vector. In the E34 experiment, a muon beam with an emittance less than 0.5π mm mrad and a
momentum of 300 MeV/c from a muon accelerator will be injected into a compact 3-T solenoid
magnet. The muon beam will be stored on a 0.66 m diameter orbit. The solenoid magnet will
provide an unprecedented local field uniformity of 0.1 ppm (peak to peak).
A new three-dimensional spiral injection scheme has been invented to inject the beam into
a compact solenoid magnet. This new injection scheme will enhance injection efficiency andovercome technical challenges related to the small storage orbit diameter. In the spiral injection
scheme, the beam will be injected at a vertical angle into the storage magnet. The radial
component of the fringe field of the solenoid magnet will decrease the vertical angle of the beam
as it approaches the midplane of the magnet. Finally, a magnetic pulsed kicker will guide thebeam to the storage volume where the beam will be stored under a weak focusing field. However,
the injection of the accelerated beam into such a small storage orbit is unprecedented; therefore,a demonstration experiment to establish the feasibility of this new injection scheme is inevitable.
The purpose of this research was to develop a scale down Spiral Injection Test Experiment(SITE) with an electron beam to demonstrate the ideology behind this new injection scheme.
An electron beamline of a length of 2 m has been designed and constructed for the SITE. The
beamline consists of a triode-type thermionic electron gun with a LaB6 cathode to generate a DC
electron beam of 80 keV with a beam current of approximately 100 µA. Subsequently, a magnetic
lens was placed to prevent the growth of the beam. A collimator with a diameter of 3 mm and189

a length of 5 mm was placed at a distance of 0.7 m from the exit of the electron gun. A pair of
air-core steering coils has been installed to control the transverse position of the 80 keV electron
beam. Next, three rotation quadrupole magnets were installed to control the beam phase space
for the spiral injection. An electron beam in a straight beam line was successfully commissioned
and confirmed at several locations via fluorescent screen monitors. The emittance of the electron
beam after collimation was measured by the quadrupole scan method. The emittance value inthe horizontal direction was estimated 0.61 ± 0.05 mm mrad and in the vertical direction, it
was 0.41 ± 0.04 mm mrad. In order to estimate the beam transverse coupling strength a simple
procedure was developed. The coupling strength of the beam after the collimator was estimatedto 0.17 ± 0.6. This small value of coupling means decoupled beam after the collimator.
A normal conducting solenoid-type storage magnet of field strength 82.5 Gauss was designed
and constructed to store the electron beam in an orbit with a diameter of 0.24 m. In the caseof SITE, the field index value n (strength of weak focusing: 0 < n < 1) of 1.65 × 10−2 wasset. This higher value of the field index relaxes the phase space requirement and vertical kicker
parameters for SITE.
A non-invasive gas monitor was used to detect the electron beam track in the storage magnet.
A charge-coupled device (CCD) camera was used to observe fluorescent light originating fromthe gas monitor. An electron beam track of four turns in the vacuum chamber of the storage
magnet was confirmed by this gas monitor, which is a good qualitative tool to efficiently verify
the electron beam track inside the storage magnet. However, a gas monitor lacks the ability
to provide quantitative information about the position of the beam and the profile in thestorage magnet. Therefore, a special type of a wire scanner for determining the beam position
and measuring the profile in the solenoid magnet was designed, constructed and successfullycommissioned for SITE. Two wire scanners were installed on the storage magnet to measure thebeam profile and beam evolution in the storage magnet.
Due to the axial symmetric field of the solenoid magnet, a strongly XY-coupled beam isrequired. To produce the required phase space for the solenoid-type storage magnet, a beamtransport line consisting of three rotatable quadrupole magnets has been designed and built
for SITE. The vertical beam size reduction by means of phase space matching and othergeometrical information have been successfully measured by the wire scanners and confirmedby the simulation. The vertical beam size had been reduced to 25.68 ± 0.31 mm with thephase matching as compared to vertical beam growth of 78.41 ± 0.83 mm without any rotatable
quadrupole magnets.
To store the beam at the center of the storage magnet, a pulsed magnetic kicker was designed
and simulated. From simulation studies, the best design parameters for the magnetic kicker havebeen determined. As explained earlier in the case of SITE, the field index value of 1.65×10−2 wasused. This value of the field index allowed to use pulse width of 50 ns for the kicker. In addition,it also provides a weak focusing trap of ±50 mm at storage region. A tracking simulation withthe best beam phase space that can achieved by the three rotatable quadrupole magnets has
190

CHAPTER 8. CONCLUSIONS
been performed. And it was estimated that injection efficiency of 81% can be achieved for SITE.
The purpose of this study was to prove the ideology of 3-D spiral injection scheme withthe electron beam.

The qualitative and quantitative details of the vertical and horizontal

beam profiles and positions inside the solenoid storage magnet are one of the most important
requisites for the successful 3-D spiral injection. Therefore, a wire scanner type beam positionand profile monitor was developed for the SITE. The knowledge and experience of wire scanner
development to extract the beam profile and position for the SITE storage magnet greatly helped
to understand the beam behavior inside the solenoid storage magnet. Moreover, it paved theway for the development of a similar quantitative monitor for the successful 3-D spiral injectionin the case of E34. The method to measure the transverse beam coupling developed at the SITE
will also be implanted at E34 to estimate the coupling strength at the matching point.
Hence, evidence from this study concludes that the novel 3-D spiral injection scheme can beapplied to the E34 storage magnet with the appropriate phase space matching and a quantitative
monitor for the solenoid type storage magnet.

191

Acknowledgment
There are so many people I wish to thank who have in one way or another made this thesis
possible. I would like to express my sincere gratitude to my supervisor Prof. Kazuro Furukawa
for his constant support, guidance, and motivation academically and otherwise. The door toProf. Kazuro Furukawa’s office was always open whenever I ran into a trouble spot.
I would like to express my deepest gratitude to Prof. Hiromi Iinuma for her patient guidance,
enthusiastic encouragement and useful critiques of my research work.
I wish to give special thanks to my Co-supervisor Prof. Tsutomu Mibe for his incredible
support and encouragement.
I am much obliged to Prof. Satoshi Ohsawa for his support academically and in my daily
life. His teaching and support were indispensable for the completion of this thesis.
I would like to thank Prof. Hisayoshi Nakayama. He has been supportive since the days I
began my work at the experiment.
I am indebted to Mr. Hiromi Hisamatsu for his kind help during the experiment and forproviding us many crucial vacuum components. I would also like to thank Prof. Tohru Honda
for his special interest and care about my research project.
I am grateful to Prof.

Naohito Saito for giving me many constructive comments andsuggestions during weekly meetings. I would also like to thank all Saito’s lab members: Mr.
Ce Zhang, Mr. Taha Youssef, Mr. Hiromasa Yasuda, Mr. Takahiro Ushizawa, Dr. Yutaro Sato.
I am thankful to Mr. Hirohiko Someya for kindly letting me use his laboratory for coilwinding. I am much obliged to Prof. Toshiyasu Higo and Prof. Izumi Mochizuki for theirgenerous support for this project.
I am greatly indebted to KEK LINAC (e− /e+ ) staff for their support and help for SITE.
I gratefully acknowledge the support of the SOKENDAI office Ms. Harumi Michikawa, Ms.
Noriko Yamamura, Ms. Minami Ono, and Ms. Kimura.
I am grateful to Dr. Sigit B Wibowo, Dr. Emy Mulyani, Dr. Na Liu, Mr. Di Wang, and Mr.
Du Baiting for their friendship and being awesome office mates.
Finally, I must express my gratitude to my mother and siblings for their prayers and countless
love for me.

Appendix A

Appendix
A.1

Wire Scanner Data Acquisition System Code (C#)

A GUI program in C# was developed for data acquisition and online plotting of wire scannerdata. Figure A.1 is the screenshot of the control program.

Figure A.1: The screenshot of the DPO–7104 oscilloscope control program.

Listing A.1: C# code for Tektronix DPO–7104 Oscilloscope
1

using System;

2

using System. Collections . Generic ;3

using System.ComponentModel;

4

using System.Data;

5

using System.Drawing;

6

using System.Linq;

7

using System.Text;

195

8

using System.Threading.Tasks;

9

using System.Windows.Forms;

10

using TekVISANet;11

using TekInstrMgr;

12using VISAInstallCheckLib ;13

using System. Collections ;14

using System.Diagnostics ;15

using System.IO;

16
17

namespace DPO7104 Control WinForm

18

{

19

public partial class Form1 : Form

20

{

21

bool flag ;22

private const string OSCILLOSCOPE = "TCPIP::169.254.138.160::INSTR";

23

public Form1()

24

{

25

InitializeComponent () ;

26

}

27

private void Button1 Click( object sender , EventArgs e)

28{
string response ;29
30

ArrayList resources = new ArrayList();31

VISA tekVISA = new VISA();32

tekVISA.FindResources("?*", out resources ) ;33

Console.ForegroundColor = ConsoleColor.Green;

34

tekVISA.Open(resources [0]. ToString()) ;35

tekVISA.Write("*IDN?");36bool status = tekVISA.Read(out response);37

if ( status )

38

{

39

MessageBox.Show("Instrument *IDN? query returned:

40\n" + response.ToString() , "Device Connection
Test");

41

}

42
43

}

44

private void Button2 Click( object sender , EventArgs e)

45

{

46

DataTable dt1;47

dt1 = new DataTable();48

dt1.Columns.Add("#ID");49

dt1.Columns.Add("time (ms)");50

dt1.Columns.Add("Ch1_V_Mean");51

dt1.Columns.Add("Ch2_V_Meam");52

dt1.Columns.Add("Ch3_V_Meam");53

dt1.Columns.Add("Ch4_V_Meam");54

dataGridView1.DataSource = dt1;

55
56

DataRow r1;

57
58

Stopwatch s = new Stopwatch();59

s . Start () ;60

double v = 0.001;

61

double t ;62

double sf = 0;63
64

DateTime now = DateTime.Now;

65
66

double num = 0;67

flag = true;

68

VISA tekVISA = new VISA();

69
70

ArrayList resources = new ArrayList();71

tekVISA.FindResources("?*", out resources ) ;7273bool opened = false ;74

if ( resources .Contains(OSCILLOSCOPE))
opened = tekVISA.Open(OSCILLOSCOPE);

75
76
77

if (opened)

78

{

79

while ( flag ==true)

80

{

81

t = (s. ElapsedMilliseconds ) ;82

sf = v ∗ (t / 1000);

8384

r1 = dt1.NewRow();85//id of measurement86

r1 [0] = num.ToString();87//time88

r1 [1] = (s. ElapsedMilliseconds ) .ToString() ;89
90

if (checkBox1.Checked)

91

{

92

tekVISA.Write("MEASUREMENT:MEAS1:TYPE MEAN");93

tekVISA.Write("MEASUREMENT:MEAS1:SOURCE CH1");

94

tekVISA.Write("MEASUREMENT:MEAS1:STATE ON");95///write command to read RMS

96

tekVISA.Write("MEASUREMENT:MEAS1:VALUE?");97

string response = "";98///read data

99

tekVISA.Read(out response);

100

double rv1 = Convert.ToDouble(response);101

r1 [2] = response.ToString() ;102

if (rv1 < 99.0000E+36)

103

{

104

this . chart1 . Series ["Ch1"].Points .AddXY((v105∗ (t / 1000))∗1000, response) ;Application .DoEvents();

106

}

107
108

}

109
110

if (checkBox2.Checked)

111

{

112

tekVISA.Write("MEASUREMENT:MEAS2:TYPE MEAN");113

tekVISA.Write("MEASUREMENT:MEAS2:SOURCE1 CH2");

114

tekVISA.Write("MEASUREMENT:MEAS2:STATE ON");115///write command to read RMS

116

tekVISA.Write("MEASUREMENT:MEAS2:VALUE?");117

string response1 = "";118///read data

119

tekVISA.Read(out response1);

120
121

r1 [3] = response1.ToString() ;122

double rv2 = Convert.ToDouble(response1);

123
124

if (rv2 < 99.0000E+36)

125

{

126

this . chart1 . Series ["Ch2"].Points .AddXY((v127∗ (t / 1000)) ∗ 1000, response1) ;
}

128
129

}

130

if (checkBox3.Checked)

131

{

132

tekVISA.Write("MEASUREMENT:MEAS3:TYPE

133

MEAN");

134

tekVISA.Write("MEASUREMENT:MEAS3:SOURCE1

135

CH3");

136

tekVISA.Write("MEASUREMENT:MEAS3:STATE

137

ON");

138

///write command to read RMS

139

tekVISA.Write("MEASUREMENT:MEAS3:VALUE?");140

string response2 = "";141///read data

142

tekVISA.Read(out response2);

143//textBox1.Text = response2.ToString() ;144

r1 [4] = response2.ToString() ;145

double rv1 = Convert.ToDouble(response2);

146
147

if (rv1 < 99.0000E+36)

148{
this . chart1 . Series ["Ch3"].Points .AddXY((v149∗ (t / 1000)) ∗ 1000, response2) ;

150

}

151
152

}

153

if (checkBox4.Checked)

154

{

155

tekVISA.Write("MEASUREMENT:MEAS4:TYPE

156

MEAN");

157

tekVISA.Write("MEASUREMENT:MEAS4:SOURCE1

158

CH4");

159

tekVISA.Write("MEASUREMENT:MEAS4:STATE

160

ON");

161

///write command to read RMS

162

tekVISA.Write("MEASUREMENT:MEAS4:VALUE?");163

string response3 = "";164///read data

165

tekVISA.Read(out response3);166// textBox1.Text = response3.ToString() ;167

double rv = Convert.ToDouble(response3);

168
169

if (rv < 99.0000E+36)170{
r1 [5] = response3.ToString() ;

171
172
173

this . chart1 . Series ["Ch4"].Points .AddXY((v174∗ (t / 1000)) ∗ 1000, response3) ;
}

175

}

176
177
178

num++;

179

dt1.Rows.Add(r1);

180

Application .DoEvents();}181

s .Stop();

182

}

183
184

}

185
186

private void Button3 Click( object sender , EventArgs e)

187{
flag = false ;

188
189

}

190
191

private void Button5 Click( object sender , EventArgs e)

192

{

193

SaveFileDialog savechart = new SaveFileDialog();

194savechart . Filter = "JPeg Image|*.jpg|Bitmap Image|195*.bmp|Gif Image|*.gif";196

savechart . Title = "Save the Chart";197

savechart .ShowDialog();198

if ( savechart .FileName != "")

199

{

200

System.IO.FileStream fs =201

(System.IO.FileStream)savechart .OpenFile();202

this . chart1 .SaveImage(fs,203

System.Drawing.Imaging.ImageFormat.Jpeg);

204

string path =205

System.IO.Path.GetFullPath(savechart .FileName);
MessageBox.Show("Polt Saved" + path);

206

}

207

}

208
209
210

private void Button6 Click( object sender , EventArgs e)

211

{

212

chart1 . Series ["Chanel1"].Points. Clear () ;213

chart1 . Series ["Chanel2"].Points. Clear () ;214

chart1 . Series ["Chanel3"].Points. Clear () ;chart1 . Series ["Chanel4"].Points. Clear () ;

215

}

216
217
218

private void Button4 Click( object sender , EventArgs e)

219

{

220

double num;221

num = double.Parse(textBox2.Text);

222// textBox5.Text = ”Waveform format: BYTE”;

223

TextWriter writer = new StreamWriter

224(@"C:\\Users\\M. A. Rehmam\\Desktop\\Tek_data\

225\tek_meas_data_ws1_"+ num.ToString() + ".txt");

226
227
228

for ( int i = 0; i < dataGridView1.Rows.Count − 1; i+

229

+)

230

{

231

for ( int j = 0; j < dataGridView1.Columns.Count;

232

j++)

233

{

234

writer .Write("\t" +235

dataGridView1.Rows[i ]. Cells [ j ]. Value.ToString()

236+ "\t");

237

}

238

writer .WriteLine("");
}

239
240

writer . Close() ;241

MessageBox.Show("Data Exported");
}

242

}

243
244

}

Appendix B

Appendix
B.1

The beam Injection with other settings

In this section, the wire scanner results with settings other than the best run, with the threerotation quadrupole magnets are shown. In Fig. B.1 (a) and (b) (X, Y ), and (X, X 0 ) werematched preferentially while (Y, Y 0 ) was ignored. Figure B.1 shows the wire scanner 1 and 2
results with beam phase–space setting as shown in Fig.B.1. Figure B.3 presents the comparisonof the vertical beam sizes growth in the storage magnet with the phase space settings as shownin Fig. B.1.

Figure B.1: The beam phase space with setting others than best run are shown. In (a) and (b)(X, Y ), and (X, X 0 ) were matched preferentially while (Y, Y 0 ) was ignored.
201

Figure B.2: The wire scanner a and 2 results with beam phase–space setting as shown in Fig.B.1.

Figure B.3: The comparison of the vertical beam sizes growth in the storage magnet with thephase space settings as shown in Fig. B.1.

Appendix C

Appendix
C.1

PulsedMagnetic

KickerTracking

Code(Mathematica)
This is the code developed for the particle tracking in the time varying field of the kicker.
Listing C.1: Example code
1(∗Time Doamin Tracking∗)

2

Bz = Import[NotebookDirectory[] <> "6bz.txt", "Table"];3

Bx = Import[NotebookDirectory[] <> "6bx.txt", "Table"];4

BzD = Dimensions[Bz][[1]];5BxD = Dimensions[Bx][[1]];6

oscil = 0; t11 = 100;

7
8
9
10

kick = Import[NotebookDirectory[] <> "currentpath_inner coil_10A.txt","Table"];z1 = Table[i , {i , −300, 300, 1}];

11
12

Tf = t11 10ˆ−9; dt =13

1 10ˆ−12; ttime = (Tf/dt); runT = (Tf/dt)/2 + oscil;

14

ParallelEvaluate [15
16

MultiPart [ xi0 , yi0, zi0 , \[Theta]0 ] :=Module[{xi = xi0, yi = yi0, zi = zi0, \[Theta] = \[Theta]0},17z = Table[i , {i , −350, 350, 1}];

18

BTa =

19
20
21
22

Table[{z [[ i ]], 0.979 Bz[[ i , 2]], −0.979 Bx[[i , 2]]}, {i , 1,
BzD − 1}];
br = Table[{z1[[ i ]], 1.98 kick [[ i , 2]]}, {i , 1,
Dimensions[kick ][[1]] − 1}];

23(∗RK−4 Method half sine shape pitch angle \[Theta]=24

2.12 Degree∗)25
26

q = 1.6 10ˆ−19; k = 80; c = (3 10ˆ8) 10ˆ3;

27

205

28

Tf = t11 10ˆ−9; dt = 1 10ˆ−12; ttime = (Tf/dt);

29

Br = (0.571 10ˆ−4);(∗\[Theta]=0.5Degree;∗)30

er = 511;

31

et = k + er; \[Gamma] = et/er; \[Beta] =32

Sqrt[1 − (1/\[Gamma]ˆ2)] // N; m = 9.1 10ˆ−31 \[Gamma];

33

vt = −\[Beta] c // N;

34

vxi = vt Cos[\[Theta ]];35

vyi = vt 0.;36

vzi = vt Sin [\[ Theta ]];37

t = t11 10ˆ−9;38mat = {};

39
40

nnn = (Tf/dt)/2;

41

Clear [K, l , M, n, o, P];

42{ While[nnn >= 0 \[And] zi < 300 \[And] zi > −300, t = t − dt;

4344

k1 = q /m (vyi BTa[[Position [BTa, Round[zi ]][[1,vzi( If [nnn >= 0,
br [[ Position [ br , Round[zi ]][[1,451]], 2]] −1]], 2]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],460] + BTa[[Position[BTa, Round[zi ]][[1,471]], 3]] Cos[48

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) ) dt ; l1 = vxi dt ;49

k2 = q /m ((vyi + k1/2) BTa[[Position[BTa, Round[zi ]][[1,502]] − (vzi + k1/

512) ( If [nnn >= 0,
br [[ Position [ br , Round[zi ]][[1,

52

1]],

1]], 2]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],

53

0] +

54

BTa[[Position [BTa, Round[zi ]][[1,551]], 3]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) ) dt ;5657

l2 = (vxi + k1/2) dt;

58

k3 = q /m ((vyi + k2/2) BTa[[Position[BTa, Round[zi ]][[1,592]] − (vzi + k2/

602) ( If [nnn >= 0,
br [[ Position [ br , Round[zi ]][[1,

61

1]],

1]], 2]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],620] + BTa[[Position[BTa, Round[zi ]][[1,631]], 3]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) ) dt ;6465

l3 = (vxi + k2/2) dt;

66

k4 = q /m ((vyi + k3) BTa[[Position [BTa, Round[zi ]][[1,672]] − (vzi +68k3) ( If [nnn >= 0,
br [[ Position [ br , Round[zi ]][[1,

69

1]],

1]], 2]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],700] + BTa[[Position[BTa, Round[zi ]][[1,711]], 3]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) ) dt ; l4 = (vxi + k3) dt;

72
73

K = 1/6 (k1 + 2 k2 + 2 k3 + k4);74

l = 1/6 (l1 + 2 l2 + 2 l3 + l4);75

vxf = vxi + K ; xf = xi + l;

76

m1 =

77

q /m (vzi ( If [nnn >= 0,
br [[ Position [ br , Round[zi ]][[1,781]], 2]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],79800] + BTa[[Position[BTa, Round[zi ]][[1,81ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) −82

vxi BTa[[Position [BTa, Round[zi ]][[1,83n1 = vyi dt ;84

m2 = q /m ((vzi + m1/

1]], 3]] Cos[1]], 2]]) dt ;2) ( If [nnn >= 0,85

br [[ Position [ br , Round[zi ]][[1,861]], 2]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],87880] + BTa[[Position[BTa, Round[zi ]][[1,89ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) − (vxi + m1/2) BTa[[
Position [BTa, Round[zi ]][[1,9091n2 = (vyi + m1/2) dt;

92

m3 = q /m ((vzi + m2/

1]], 3]] Cos[1]], 2]]) dt ;2) ( If [nnn >= 0,93

br [[ Position [ br , Round[zi ]][[1,941]], 2]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],950] + BTa[[Position[BTa, Round[zi ]][[1,961]], 3]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) − (vxi + m2/2) BTa[[

97

Position [BTa, Round[zi ]][[1,9899n3 = (vyi + m2/2) dt;

100

m4 = q /m ((vzi +1]], 2]]) dt ;m3) (If [nnn >= 0,101

br [[ Position [ br , Round[zi ]][[1,1021]], 2]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],

103
104

0] + BTa[[Position[BTa, Round[zi ]][[1,105ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) − (vxi + m3) BTa[[
Position [BTa, Round[zi ]][[1,1061]], 3]] Cos[1]], 2]]) dt ;107

n4 = (vyi + m3) dt; M = 1/6 (m1 + 2 m2 + 2 m3 + m4);108

n = 1/6 (n1 + 2 n2 + 2 n3 + n4);109

vyf = vyi + M; yf = yi + n;

110

o1 =

111
112
113
114
115
116
117
118

q /m (vxi ( If [nnn >= 0,
br [[ Position [ br , Round[zi ]][[1,1]], 2]] Sin [ArcSin[ yi /Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],0] + BTa[[Position[BTa, Round[zi ]][[1,1]], 3]]

ArcSin[ yi /Sqrt[ xi ˆ2 + yi ˆ2]]]) −vyi ( If [nnn >= 0,
br [[ Position [ br , Round[zi ]][[1,1]], 2]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],1190] + BTa[[Position[BTa, Round[zi ]][[1,120ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) ) dt ; p1 = vzi dt ;

121
122
123
124
125

Sin [

1]], 3]] Cos[o2 = q /m ((vxi + o1/
2) ( If [nnn >= 0,
br [[ Position [ br , Round[zi ]][[1,1]], 2]] Sin [ArcSin[ yi /Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],0] + BTa[[Position[BTa, Round[zi ]][[1,1]], 3]]

Sin [

126
127
128
129
130
131

ArcSin[ yi /Sqrt[ xi ˆ2 + yi ˆ2]]]) − (vyi + o1/
2) ( If [nnn >= 0,
br [[ Position [ br , Round[zi ]][[1,0] + BTa[[Position[BTa, Round[zi ]][[1,p2 = (vzi + o1/2) dt;

133

o3 = q /m ((vxi + o2/

135
136

1]], 3]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) ) dt ;

132

134

1]], 2]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],2) ( If [nnn >= 0,
br [[ Position [ br , Round[zi ]][[1,1]], 2]] Sin [ArcSin[ yi /Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],137

0] + BTa[[Position[BTa, Round[zi ]][[1,138ArcSin[ yi /Sqrt[ xi ˆ2 + yi ˆ2]]]) − (vyi + o2/

139
140
141

br [[ Position [ br , Round[zi ]][[1,1]], 2]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],0] + BTa[[Position[BTa, Round[zi ]][[1,143

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) ) dt ;144

p3 = (vzi + o2/2) dt;

145

o4 = q /m ((vxi +

147
148
149
150
151
152
153
154
155

br [[ Position [ br , Round[zi ]][[1,1]], 2]] Sin [ArcSin[ yi /Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],0] + BTa[[Position[BTa, Round[zi ]][[1,1]], 3]]Sin [ArcSin[ yi /Sqrt[ xi ˆ2 + yi ˆ2]]]) − (vyi +o3) ( If [nnn >= 0,
br [[ Position [ br , Round[zi ]][[1,1]], 2]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf ],0] + BTa[[Position[BTa, Round[zi ]][[1,1]], 3]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) ) dt ; p4 = (vzi + o3) dt;
o = 1/6 (o1 + 2 o2 + 2 o3 + o4);

157

P = 1/6 (p1 + 2 p2 + 2 p3 + p4);

158

vzf = vzi + o; zf = (zi + P);

159

AppendTo[mat, {t , xf , yf , zf , vxf , vyf , vzf , Sqrt[

161

vxfˆ2 + vyfˆ2 + vzfˆ2],

162

Sqrt [( BTa[[Position [BTa, Round[zi ]][[1,1631]], 3]] Cos[o3) ( If [nnn >= 0,

156

160

Sin [

2) ( If [nnn >= 0,

142

146

1]], 3]]1]], 3]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) ˆ2 + (BTa[[

164

Position [BTa, Round[zi ]][[1,165ArcSin[ yi /Sqrt[ xi ˆ2 + yi ˆ2]]]) ˆ2],1]], 3]]166

If [nnn >= 0,167

Sqrt [( br [[ Position [ br , Round[zi ]][[1,168ArcSin[ yi /Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [169

t (2 \[Pi ]) /Tf])ˆ2 + (br [[Sin [1]], 2]] Sin [170

Position [ br , Round[zi ]][[1,171ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]] Sin [ t (2 \[Pi ]) /Tf]) ˆ2],1]], 2]] Cos[

1720], BTa[[Position [BTa, Round[zi ]][[1,1732]], ((BTa[[Position [BTa, Round[zi ]][[1,

174

1]],
1]], 3]] Cos[

ArcCos[xi/Sqrt[ xi ˆ2 + yi ˆ2]]]) + (BTa[[175

Position [BTa, Round[zi ]][[1,176ArcSin[ yi /Sqrt[ xi ˆ2 + yi ˆ2]]]) ) }]; vxi = vxf; vyi = vyf;

177

vzi = vzf; xi = xf; yi = yf;

178

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