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This is to certify that the
dissertation entitled
BEAM DYNAMICS STUDIES OF FOUR-GAP

LOW-BETA SUPERCONDUCTING RESONATORS

presented by

Kihun Joh

has been accepted towards fulfillment
of the requirements for

P11- T)- degree in Physics

MffiL/F

Major proflssor

Date Sept. 30th, 1993

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BEAM DYNAMICS STUDIES OF
FOUR-GAP LOW-BETA
SUPERCONDUCTIN G RESONATORS

By

Kihun’ J oh
A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1993

ABSTRACT

BEAM DYNAMICS STUDIES OF FOUR-GAP LOW-BETA
SUPERCONDUCTING RESONATORS

BY

Kihun J oh

Four-gap superconducting resonators have been developed at Argonne for use in
the low-beta positive ion injector for ATLAS. These structures have been used success-
fully for ion velocities as low as 0.007c with q/m=0.l. It was previously observed that
at low velocities these structures can be focusing in both longitudinal and transverse
phase spaces due to an inherent alternating-phase-focusing property. Calculations
indicate that a series of three such resonators have large transverse and longitudinal

acceptances even when operated without the transverse focusing elements.

Detailed studies of the beam dynamics of these types of resonators are presented.
For this the first-order transverse and longitudinal matrices are evaluated analytically
point-by-point through the system. The phase space transformations are generalized
to accelerating systems to study acceptances in such systems. Also a new method is
presented to establish criteria for defining in a quantitative way the size and orienta-

tions of the linear acceptances in any accelerator or beam transport system.

These studies also include the effects of higher order distortions in the longitudinal
and transverse phase spaces since minimizing such aberrations is very important for

most nuclear physics applications of such accelerators.

To my mother and to the memory of my father

iii

ACKNOWLEDGEMENTS

I would like to thank the professors who instructed me at the Department of
Physics and Astronomy. I also would like to thank my colleagues, the staff, and the
faculty of the National Superconducting Cyclotron Laboratory for their support of
my education. I especially thank Dr. Al Zeller and professor Brad Sherrill for their
guidance in learning and understanding various computer codes and beam transport
systems. I thank Dr. Felix Marti for his guidance in using the code Cyclone to study

central region of the K1200 Cyclotron. I also thank the operators of the NSCL for

their help in measuring the beam emittances of the K1200 Cyclotron.

My thanks go to professor Edwin Kashy, professor Martin Berz, professor Norman
Birge, and professor J. S. Kovacs for their efforts and advice in evaluating my Ph.
D. program. I wish to give special thanks to professor Martin Berz for the useful

discussions about beam optics and for his enthusiastic instruction of his own computer

code COSY INFINITY.

I thank Dr. Richard Pardo and Dr. Kenneth Shepard at Argonne National Lab-
oratory for their help in understanding basic accelerating structures of the ATLAS

Linac. I also thank Kevin Beyer for his proofreading of this dissertation.

I want to deeply thank my thesis adviser Dr. Jerry N olen. He was always generous
in giving his busiest time for useful discussions. His creative suggestions, teaching,
and encouragements were greatly appreciated. I also thank him for considering me

for work at Argonne National Laboratory in order to write my Ph. D. dissertation.

I want to share my delight with my mother, late father, two brothers, sister-in-law,
Min Kuk, Min Cho, brother-in-law, and sister. I recognize that without their love

and support a page of this thesis would not exist.

iv

Finally I thank GOD in my heart for giving me my wife, Ki-Ok, and two children,
Minji and Minsol in the USA. They were the source of my strength and hope in

continuing my study even through the hardest times.

I am also grateful to Michigan State University and Argonne National Laboratory

for financial support during my graduate study.

Contents

LIST OF TABLES
LIST OF FIGURES

1 Introduction and Motivation

1.1 The Positive Ion Injector of ATLAS ...................
1.1.1 The Beam Optics of the PII Linac ................
1.1.2 The Resonators of the P11 Linac .................

1.2 The Beam Dynamics of Type II Resonators ..............

1.3 Goals and Motivations ..........................

2 First Order Matrix Optics

2.1 Introduction ................................
2.2 Transfer Map ...............................
2.2.1 The Multiplication of Transfer Maps ..............
2.2.2 Interpretation of Transfer Matrix ................

2.3 Phase Space Transformation .......................
2.3.1 2-dimensional Phase Space Ellipse ................
2.3.2 N-dimensional Phase Space Ellipsoid for an Axially Symmetric
Beam ...............................

2.3.3 Envelope of Phase Space Ellipsoid ................
2.3.4 Example of n=4 Case .......................

3 The Equations of Motion of Linac Components

3.1 Introduction ................................
3.2 Equations of Motion ...........................
3.2.1 RF Resonator ...........................
3.2.2 Magnetic Solenoid ........................
3.2.3 Drift Space ............................

vi

ix

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18
18
19
20
21
22
22

25
28
29

33
33
34
36
37

3.3 Linear Transfer Map ...........................

3.3.1 Transfer Map of a Drift Space ..................
3.3.2 Transfer Map of a RF Resonator ................
3.3.3 Transfer Map of a Magnetic Solenoid ..............
3.4 Implementation by Computer Programs ................
3.4.1 Numerical Integration ......................
3.4.2 Model Solenoid ..........................

Application to Resonator Study

4.1 Introduction ................................
4.2 Focusing Power ..............................
4.2.1 Transverse Case ..........................
4.2.2 Longitudinal Case .........................
4.3 Focusing Structures of II and I2 type Resonators ...........
4.3.1 Transverse Focusing .......................
4.3.2 Longitudinal Focusing ......................
4.3.3 Alternating Phase Focusing ...................
4.4 The Limits of RF Phase Offset of Resonators ..............
4.5 Parameters on Focusing .........................
4.5.1 Injection Energy .........................
4.5.2 RF Phase Offset .........................
4.6 Initial Conditions for Focusing .....................
4.7 Conclusions ................................

Applications to Acceptance Study

5.1 Introduction ................................
5.2 Geometrical Acceptance .........................
5.2.1 Transverse Acceptance ......................
5.2.2 Longitudinal Acceptance .....................
5.2.3 Fitting of Geometrical Acceptance ...............
5.3 Linear Acceptance ............................
5.3.1 Scaling of Nonlinearity by Linear Transformation .......
5.3.2 Scaling of Nonlinearity by RMS Emittance ...........
5.4 Application to an Alternating-Phase-Focusing System .........
5.4.1 Geometrical Acceptances on Incident Velocities ........
5.4.2 Estimation of the Linear Acceptances ..............
5.5 Nonlinearity of a System .........................

vii

39
40
42
48
50

54
54
55
56
58
61
61
62
62
64
65
65
66
67
69

85
85
86
87
88
89
89
89
90
92
92
93
95

5.6 Application to the PI1 Linac ....................... 97

5.6.1 Incident Velocity on Geometrical Acceptances ......... 97

5.6.2 Fitting of Solenoid Strength ................... 97

5.6.3 The Selection of Nonlinear Optical Elements .......... 98

5.7 Analysis of the Two Focusing Systems ................. 99
5.8 Conclusions ................................ 101

6 Summary and Discussions 118
A Electromagnetic Fields in an RF Resonator 121
B The System in a Cylindrically Symmetric Magnetic Field 129
8.1 Magnetic Field in Magnetic Solenoid .................. 129
B.2 Angular momentum conservation .................... 131
B.3 Rotation of a Particle in Magnetic Field ................ 132
BA Decoupling of Equations of Motion ................... 132
3.5 Rotation of Transfer Map ........................ 134

C Calculation of Fringe Field 136
C.1 Length of Fringe Field Tail ........................ 136
C.2 Transfer Map of a Solenoid ........................ 137
LIST OF REFERENCES 139

viii

List of Tables

1.1 Accelerating structures of the PII resonators ............... 17

4.1 Focusin powers of each accelerating gap of 11 and 12 type resonators
for “At 2 with the rf settings and initial velocities shown in Figure 4.3
(II) and Figure 4.4 (I2). ......................... 70

5.1 Focusing structures of the resonator focusing system for 40Ar12+ beam

with incident velocity 0.0085c ....................... 103
5.2 Focusing structures of the resonator focusing system for 238U24+ beam
with incident velocity 0.0070c ....................... 104

ix

List of Figures

1.1

1.2
1.3
1.4

1.5

1.6

1.7

1.8

The layout and main components of the positive ion injector (PII) of

ATLAS. ..................................
The configuration of the PII linac .....................
The resonators of the P11 linac. .....................

The curve of energy gain versus incident rf phase angle of type I1
resonator. The dotted line is the typical energy gain curve of a high
beta resonator (not scaled). .......................

The actions of drift space on the transverse (left) and the longitudinal
(right) phase spaces. The initial phase space ellipses are upright. . . .
The transverse phase space transformations of ”'OAr12+ with incident
velocity 0.0085c through type II resonator. Starting from the top left
the positions of plots are at the entrance of gapl, at the center of gapl,
and at the exit of gapl or at the entrance of gap2 etc. The amplitude
of electric field gradient and initial rf phase angle were 4.5 MV/m and
—10 degrees respectively. The initial emittance of the beam was 50
mm-mr (area 501r mm-mr) without longitudinal emittance. ......
The longitudinal phase space transformations of 40Ar12+ with incident
velocity 0.0085c through type 11 resonator. Starting from the top left
the positions of plots are at the entrance of gapl, at the center of gapl,
and at the exit of gapl or at the entrance of gap2 etc. The amplitude
of electric field gradient and initial rf phase angle were 4.5 MV/m and
—10 degrees respectively. The initial emittance of the beam was 5 keV
-nsec (area 51r keV-nsec) without transverse emittance. ........
The growth of longitudinal phase space of 4OAr12+ with incident veloc-
ity 0.0085c through type II resonator. Starting from the top left the
positions of plots are at the entrance of gapl, at the center of gapl,
and at the exit of gapl or at the entrance of gap2 etc. The amplitude
of electric field gradient and initial rf phase angle were 4.5 MV/m and
—10 degrees respectively. The initial emittance of the beam was 50
mm-mr (area 501r mm-mr) without longitudinal emittance. ......

10

11

12

13

1.9

1.10

1.11

2.1

3.1

4.1

4.2

4.3

4.4

4.5

The transverse phase space transformations of 40Ar12+ with incident

velocity 0.0085c through type I1 resonator. Starting from the top left
the positions of plots are at the entrance of gapl, at the center of gapl,
and at the exit of gapl or at the entrance of gap2 etc. The amplitude
of electric field gradient and initial rf phase angle were 4.5 MV/m and
—10 degrees respectively. The initial emittance of the beam was 50
mm-mr (area 50w mm-mr) with longitudinal emittance 5 keV-nsec. . . 14

The longitudinal phase space transformations of 40Ar12+ with incident
velocity 0.0085c through type II resonator. Starting from the top left
the positions of plots are at the entrance of gapl, at the center of gapl,
and at the exit of gapl or at the entrance of gap2 etc. The amplitude
of electric field gradient and initial rf phase angle were 4.5 MV/m and
—10 degrees respectively. The initial emittance of the beam was 50
mm-mr (area 501r mm-mr) with longitudinal emittance 5 keV-nsec. . . 15

The transverse (upper) and longitudinal (lower) focusing properties of

type 11 resonator for 40Ar12+ beam with incident velocity 0.0085c. The
amplitude of electric field gradient and initial rf phase angle were 4.5
MV/ m and —10 degrees respectively. .................. 16

A two dimensional beam phase space ellipse. The equation of the

ellipse is given by 73:2 + Zazz’ + 313,2 = e. The area of the phase space

ellipse is re. The c is called beam emittance ............... 32
The normalized magnetic field profile on axes of the model solenoid
with half aperture 1.5 cm and body length 10 cm ............ 53

The point-to-parallel system(left) and the parallel-to-point system(right).
P1 and P2 are two different principal planes. .............. 71

The focusing and defocusing actions of a lens. (:c’lz) or -(Aw|At) is
positive for ray 1 to give net defocusing action and negative for ray 2
to give net focusing action ........................ 72

The focusing power at each cell of an 11 type resonator for “mArl'”
beam with initial value of B = 0.0085 (bottom). The solid line is for
transverse direction and dotted line is for longitudinal direction. The
top shows the B profile along the resonator axis. The middle part is
an axial electric field profile of the I1 type resonator which has a four
gap structure. ............................... 73

The focusing power at each cell of an I2 type resonator for 40Ar12+
beam with initial value of 3 = 0.0164 (bottom). The solid line is for
transverse direction and dotted line is for longitudinal direction. The
top shows the ,3 profile along the resonator axis. The middle part is
an axial electric field profile of the I2 type resonator which has a four
gap structure. ............................... 74

The stability region of the rf phase offset of 11 (up) and 12 type (down)

resonators as a function of initial ,6 for “Ari” beam .......... 75

xi

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

5.1

5.2

5.3

5.4

5.5

5.6

5.7

The stability region of the rf phase offset of 11 (up) and I2 type (down)

resonators as a function of initial 3 for 238U2'” beam. .........
The plots of (z’lm) and -—(Aw|At) vs. initial beta B.- for an 11 type

resonator. The solid line is for 40Ar12+ and the dotted line is for 238U2”.

The plots of (:c'lz) and —(Aw|At) vs. initial beta B,- for an I2 type

resonator. The solid line is for 40Ar12+ and the dotted line is for 23f'l‘Uz‘H".

The plots of (m’lz) and ~—(Aw|At) vs. rf phase offset (to for an 11 type

resonator. The solid line is for 4”Arl2+ and the dashed line is for 238U2”.

The plots of (z’lm) and ——(Aw|At) vs. rf phase offset (1)0 for an I2 type

resonator. The solid line is for 4°Ar12+ with ,3,- = 0.0164 and the dashed
line is for 238U24+ with ,6,- = 0.0164. ...................

The plot for initial beta ,8.- vs. final beta B, of an 11 type resonator.

The solid line is for "OArl2+ and the dashed line for 238U2‘1+ beam. . .
The plot for initial beta ,8,- vs. final beta ,3; of I2 type resonator. The

solid line is for 40Ar12+ and dashed line for ”BUZZ” beam ........

The transverse raytracing plots for 40Ar12+ (up) and 23”U2” (down)
beams with parallel rays at the beginning. ...............

The longitudinal raytracing plots for 40Ar12+ (up) and 238U?” (down)
beams with parallel rays (AW 2 0) at the beginning. .........

The geometrical transverse (upper) and longitudinal (lower) accep-
tances as a functions of incident velocities. ...............

The beam envelopes of beam radius (upper) and deviation of time of
flight (lower) starting from maximum emittance .............

The phase space transformations of the transverse linear acceptance for
a 2M’Uz‘” beam with an incident velocity of 0.007c and with a deviation

of 1%. ...................................

The phase space transformations of the transverse linear acceptance
for a 238U24+ beam with an incident velocity 0.007c with a deviation

of 5%. ...................................

The phase 5 ace transformations of the longitudinal linear acceptance
for a 238U24 beam with an incident velocity 0.007c ...........

The relationship between scalings of nonlinearity by linear transforma-
tion and by rms emittance for a 238U24+ beam with an incident velocity
0.007c. The initial geometrical transverse acceptance was 140.49 mm
- mr without longitudinal phase space. .................

The relationship between scalings of nonlinearity by linear transforma-
tion and by rms emittance for a 238U24+ beam with an incident velocity
0.007c. The initial geometrical longitudinal acceptance was 51.07 keV
- nsec without transverse phase space ...................

xii

76

80

81

82

83

84

105

106

107

108

109

110

111

5.8

5.9

5.10

5.11

5.12

5.13

A.1

The beam envelopes of beam radius (upper) and deviation of time
of flight (lower) starting with a maximum emittance in the system
using solenoids for the transverse focusing. The incident velocity of
the 238U24+ beam was 0.0085c .......................

The phase space transformations of the transverse linear acceptance in
the system using solenoids for the transverse focusing. The incident

velocity of the 238U24+ beam was 0.0085c .................

The phase space transformations of the longitudinal linear acceptance
in the system using solenoids for the transverse focusing. The incident

velocity of the 238U2'” beam was 0.0085c .................

the relationship between scalings of nonlinearity by linear transfor-
mation and by rms emittance for a 238U24+ beam with an incident
velocity 0.0085c in the system using solenoids for transverse focusing.
The initial geometrical transverse acceptance was 263 mm-mr without
longitudinal phase space ..........................

The relationship between scalings of nonlinearity by linear transforma-
tion and by rms emittance for a 238U?” beam with an incident velocity
0.0085c in the system using solenoids for transverse focusing. The ini—
tial geometrical longitudinal acceptance was 117.4 keV~nsec without
transverse phase space ...........................

The effective beta-functions of the transverse (upper) and the longitu-

dinal (lower) phase spaces for 238U2"+ beam with an incident velocity
0.0085c in the system using solenoids for the transverse focusing. The
dotted lines are for Twiss parameters from the geometrical acceptances

and the solid line is for Twiss parameters from the linear acceptances.

Cylindrical pill box .............................

xiii

112

113

114

115

116

117

128

Chapter 1

Introduction and Motivation

1.1 The Positive Ion Injector of ATLAS

The positive ion injector (PII) for the ATLAS heavy ion accelerator was designed to
inject ions as heavy as uranium into the existing ATLAS heavy ion linac [Pardo 87b],
[Bollinger 93]. The basic components of PII consist of an electron cyclotron resonance

(ECR) ion source and superconducting linac as shown in Figure 1.1.

The beam from the ECR positive ion source is magnetically analyzed (analyzer
1), bunched (buncher 1), and accelerated by the 350kV accelerator tube. The beam
is passed through object slits and then further analyzed at analyzer 2. After the next
90° bend the beam is chopped to remove the tails of unbunched beam, and finally

the beam is rebunched at buncher 2 and injected into the linac.

The linac is able to accelerate ions from a velocity of 0.008c to 0.04c. The config-
uration of the PII linac as shown in Figure 1.2 is an array of four types of supercon-

ducting resonators and superconducting focusing solenoids.

1.1.1 The Beam Optics of the PII Linac

The basic concept of the P11 linac beam optics is to use an alternating array of various
types of superconducting resonators and superconducting focusing solenoids as shown

1

in Figure 1.2. Longitudinal focusing of the beam is done by the independently-phased
superconducting resonators (which are generally transverse defocusing). Transverse
focusing of the beam is done by superconducting solenoids located between the res-
onators. Also, the field strengths of the focusing solenoids are adjusted to make
beam waists at the center of each resonator in order to minimize beam envelopes at
the resonators. This reduces nonlinear distortions of transverse phase space due to
radial defocusing forces in the resonators and the coupling of radial and longitudi-
nal phase spaces. Therefore, the frequent alternation of longitudinal and transverse
focusings would maintain a beam stable through the linac without serious degrad-
ing of the beam quality. As a consequence of having minimum beam envelopes at
the resonators the beam envelopes at the focusing solenoids are maximized. Possible
nonlinear effects from the solenoids in the P11 linac were not investigated before this

work.

1.1.2 The Resonators of the P11 Linac

The resonators of the P11 linac [Shepard 87], [Shepard 89] consist of four different
types of accelerating structures (type Il-I4) to span a wide range of velocities from
0.008c to 0.04c. To accept very low velocity heavy ions the resonators have short

accelerating gaps with a four-gap interdigital structure and very low rf frequencies of

48.5 MHz (II-I3) and 72.75 MHz (14).

1.2 The Beam Dynamics of Type I1 Resonators

The most interesting aspects of beam dynamics are in the type II resonator mainly
because of its rapid velocity change starting from very low velocities. In usual oper-
ating cases the beam energy is more than doubled in the first resonator. Figure 4.3

in Chapter 4 shows a very rapid velocity change of an 40Ar12+ beam through the

four-gaps of 11 under the given axial electric field profiles. One aspect of this low
velocity effect in type I1 is its unusual pattern of energy gain vs initial rf phase
angles [Bollinger 92]. Unlike usual cosine like patterns of high velocity accelerating

structures it shows its own energy gain curve as shown in Figure 1.4.

Another aspect of the low velocity effect in type 11 is its very strong transverse
focusing properties even for heavy ion beams like uranium which have low incident
energies. This strong transverse focusing of type I1 was realized from raytracing stud-
ies [Pardo 87a]. The focal length of type 11 for relatively light beam like ”Arr” is
short enough to form a focal point inside the resonator as shown in Figure 1.11. This
means that type 11 might act as a focusing element simultaneously both for longitu-
dinal and for transverse directions. This kind of triple focusing was studied already in
a heavy ion cryogenic linac [Chambers 76]. This rather surprising result implies that
type 11 resonators might have alternating-phase-focusing properties because a sim-
ple accelerating gap does not have both transverse and longitudinal focusing actions
simultaneously. Also, it turned out that even type 12 resonators have nonnegligi-
ble transverse focusing strengths and alternating-phase-focusing properties similar to

type 11 resonators.

A final aspect of the low velocity effect is the possibility of distortions of transverse
and longitudinal phase spaces coming from higher order radial forces and coupling of
longitudinal and transverse phase spaces when a beam envelope is large. Figure 1.5
provides information about the sign conventions of transverse (left) and longitudinal
(right) phase spaces in the raytracing calculations. The details will be presented in

Chapter 3.

Figures 1.6 - 1.10 show transformations of phase spaces with very strong focusing
properties both for transverse and for longitudinal directions: The new growth of the

longitudinal phase spaces from the transverse phase space due to off axial fields appear

clearly in Figure 1.8. Also, the distortions of the longitudinal phase space on axes
are shown with asymmetric distributions both in time and energy structures in Fig-
ure 1.7. The effects of coupling between the transverse and longitudinal coordinates

are well represented in Figure 1.9 and Figure 1.10.

1.3 Goals and Motivations

Despite the unusual behaviors of a beam in the very low velocity region of PII they
were understood in rather qualitative ways. Since the beam is very slow it is not
easy to find any analytical solutions in closed forms or even in Taylor series forms
of its equations of motion. The brute force method of raytracing does not provide
much quantitative and systematic informations about beam dynamic properties of a

system.

One of the goals in this thesis is to understand the beam dynamics of P11 and also
each of its resonators in a more detailed and quantitative way than what was previ-
ously known. It might present new possibilities to make use of inherent alternating-
phase-focusing properties of the P11 resonators [Nolen 93] without using any trans-
verse focusing elements at the beginning stage of PII. To study each resonator would

be useful because of its potential applications for ions with velocities less than 0.007c

and q/m less than 0.1 [Nolen 93].

The other goal of this thesis is to extend the use of matrix optics to rf resonators
with very low velocity ion beams and generalize phase space transformations to sys-
tems with acceleration [Joh 93a]. There have been several ion optical matrix optics
codes for If cavities based on analytical modelling of rf fields [Ben-Zvi 79], [Lu 87]
or based on thin lens approximations [Brown 79], [Servranckx 85], [Berz 93]. Also,

the matrix optics formalism has been extended and developed at SLAC [Brown 67],

[Brown 81] to the linear phase space transformations of systems without acceleration.

Even though the PI1 linac consists of only three kinds of optical elements namely
rf resonators, magnetic solenoids and drift spaces it is the most complex system in the
sense that the time dependent accelerating field exists in the rf resonators. Also, due
to rapid changes in velocity during the acceleration some kinds of approximations,
like thin lens for a resonator, are no longer valid. So we will test another way to use
matrix optics for rf cavities to see the possible use of matrix optics for studies of rf

resonators and phase space matchings of a linac system etc.

We have been mainly interested in the low velocity region in PII because the
nonlinearity of a beam might not be significant after gaining some energy from ac-
celerations if the initial phase spaces are well matched. Also, because of the smaller
aperture size of type 11 resonators and the reduction of transverse emittance with
acceleration, we expect that the acceptances of the P11 linac will be mainly deter-
mined by the first Il resonator or the first few elements of P11. So it might be useful
to study the beginning part of the system to understand the beam dynamics of the

Whole system. This has been one of the motivations for this thesis.

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Figure 1.5: The actions of drift space on the transverse (left) and the longitudinal
(right) phase spaces. The initial phase space ellipses are upright.

11

”Arm. s,=-o.ooss s/A=33.ss(keV/A)
11: E0 =- 4.6MV/m. ¢,, = -10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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—4-20 2 4 —4—20 2 4 ~4-20 2 4

Figure 1.6: The transverse phase space transformations of ""Arl2+ with incident ve-

locity 0.0085c through type 11 resonator. Starting from the top left the positions of
plots are at the entrance of gapl, at the center of gapl, and at the exit of gapl or
at the entrance of gap2 etc. The amplitude of electric field gradient and initial rf
phase angle were 4.5 MV/m and -10 degrees respectively. The initial emittance of
the beam was 50 mm-mr (area 501r mm-mr) without longitudinal emittance.

12

“Arm. 5,-0.0055 s/A=53.ss(keV/A)
11: s. - 4.5IN/m. ¢, =- -10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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_ blllllllllllll\JlF _ :llll lllllllllllll— _ lllllllllllll 11 ‘
-1—O.5 0 0.5 1 -—1—O.5 0 0.5 1 -1—0.5 0 0.5 1

Figure 1.7: The longitudinal phase space transformations of "’Ar'z+ with incident
velocity 0.0085c through type II resonator. Starting from the top left the positions
of plots are at the entrance of gapl, at the center of gapl, and at the exit of gapl
or at the entrance of gap2 etc. The amplitude of electric field gradient and initial rf
phase angle were 4.5 MV/m and —10 degrees respectively. The initial emittance of
the beam was 5 keV -nsec (area 51r keV-nsec) without transverse emittance.

 

50
40'
30—
20
10

0

—O....2-0100102

50
40
30
20
10

0

—0.2-0.1 0 0.1 0.2

50
40
30
20
10

O

—0.2-0.1 0 0.1 0.2

Figure 1.8: The growth of longitudinal phase space of ""Ar

“Ax-"t p,=0.0005 E/A=33.66(keV/A)

13

11: E0 8 4.5MV/rn. $0 = -10

 

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with incident velocity

0.0085c through type I1 resonator. Starting from the top left the positions of plots
are at the entrance of gapl, at the center of gapl, and at the exit of gapl or at the
entrance of gap2 etc. The amplitude of electric field gradient and initial rf phase angle
were 4.5 MV/m and —10 degrees respectively. The initial emittance of the beam was

50 mm-mr (area 501r mm-mr) without longitudinal emittance.

 

14

”Arm. p,=0.0085 E/A=33.66(keV/A)
II: a, - 4.5IN/m. ¢, =- -10

 

 

 

  

 

 

 

 

 

 

 

 

 

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—4-20 2 4 —4—20 2 4 -4-20 2 4

Figure 1.9: The transverse phase space transformations of “’Ar'“ with incident ve-

locity 0.0085c through type 11 resonator. Starting from the top left the positions of
plots are at the entrance of gapl, at the center of gapl, and at the exit of gapl or
at the entrance of gap2 etc. The amplitude of electric field gradient and initial rf
phase angle were 4.5 MV/ m and —10 degrees respectively. The initial emittance of
the beam was 50 mm-mr (area 501r mm-mr) with longitudinal emittance 5 keV-nsec.

 

“Arm. p.=o.ooas E/A=33.65(keV/A)
11: so .. 4.5MV/m. II. = —10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Figure 1.10: The longitudinal phase space transformations of “Al-12+ with incident
velocity 0.0085c through type 11 resonator. Starting from the top left the positions
of plots are at the entrance of gapl, at the center of gapl, and at the exit of ‘ gapl
or at the entrance of gap2 etc. The amplitude of electric field gradient and initial rf
phase angle were 4.5 MV/ m and —10 degrees respectively. The initial emittance of
the beam was 50 mm-mr (area 501r mm-mr) with longitudinal emittance 5 kchnsec.

16

“Arm. p,=o.ooas

11: E. = 4.5 (IN/m). so = -10

o A
-5 \

 

10

 

I!

X(mm)
OI

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0.1 0.2 0.3 0.4 0.5 20!!)
’1? 0.4
s n
E 0.2 /\
/—-N
0.0
\_—’
—0.4
0 0.1 0.2 0.3 0.4 0.5 20!!)

Figure 1.11: The transverse (upper) and longitudinal (lower) focusing properties of
type 11 resonator for 4"Ar'z'l" beam with incident velocity 0.0085c. The amplitude

of electric field gradient and initial rf phase angle were 4.5 MV / m and —10 degrees

respectively.

 

17

Table 1.1: Accelerating structures of the P11 resonators.

 

 

 

 

 

RF Active Nominal Number
Type No. of Frequency fl Length Field of
Gaps (MHz) (cm) (MV/m) Units
I, 4 48.5 0.008 10 4.5 1
12 4 48.5 0.015 16 3.0 2
13 4 48.5 0.024 25 3.0 5
I, 4 72.75 0.036 25 3.0 10

 

 

 

 

 

 

 

 

 

Chapter 2

First Order Matrix Optics

2. 1 Introduction

Matrix optics is a way to represent solutions of differential equations of motion of a
particle or ray in a compact matrix form called a transfer matrix or transfer map. The
transfer map relates initial coordinates of a particle or ray to its final coordinates. The
transfer map is called lst order if it contains lst order solutions only and nth order if
it contains solutions up to nth order. It is equivalent to a Taylor series approximation

of the corresponding order.

The advantages of the matrix method compared with raytracing calculations are:
First, it represents a system by the transfer map not by an individual particle as in
raytracing, where it is necessary to integrate equations repeatedly for different initial
conditions. Second, because of its explicit representation of a system, optimization or
fitting is possible with an appropriate optimizer or even from a simple formula. For
example, in a system consisting of a magnetic solenoid with two drift spaces before
and after the solenoid, the magnetic field strength for an image condition can be

found easily by an optimizer once the first order transfer map is known.

The other advantage of the matrix method itself is its extension to the transfor-

mation of the phase space ellipse of a bunch of particles or beam. In accelerator

18

19

physics the trace of a beam is usually more important than the trace of an individual
particle, because a beam consists of numerous individual particles. The trace of a
beam through a system is well described up to first order by the transformation of
a phase space ellipse which can be calculated from a first order transfer map of the

system.

First order matrix theory has been developed i.e. for transformation of particle
coordinates and phase space ellipsoids when there is no acceleration (damping) in a
system [Brown 73],[Brown 81], where the determinant of the transfer map is equal to
one. It has been recently generalized to an accelerating system for betatron function

parameterization in two dimensional decoupled phase space [Douglas 88].

This chapter briefly reviews the basic equations of first order matrix optics. Then
we independently generalize first order phase space transformation based on the ge-
ometric characteristics of the phase ellipse for systems with acceleration, i.e. where

the determinant of transfer map is not equal to one.

2.2 Transfer Map

The trajectory of an arbitrary particle in a beam is described by deviations from
a reference trajectory in a given phase space. We assume that the solution of the
equation of motion of these coordinate deviations of an arbitrary particle is given
by a Taylor expansion. If the initial deviation is not large the first few terms in the

Taylor expansion would be enough to describe the trajectory.

For a simple case we consider the two dimensional phase space of (3,3’). The

final phase space coordinates, at distance 2 in the system, can be written by a Taylor

20

expansion in terms of the initial coordinates:

we) ——- :(xlxaxSImaxs , (2.1)
II)

x’(z) = :(z'lx:,x::)x:,z:: . (2.2)
I J

In the expansion the subscript 0 means initial deviation and the sum of 13+ j determines

the order of solution. The expansion coefficient is an abbreviation of the Taylor

 

coefficients:
. . 1 614-153
r I1
, = . . . 2.3
(3'3030) (I: + j)! 8123061236 ( )
If we consider the first order solution it can be expressed in matrix form:

a: (:clm) (:clzc’) 2:0
2 . (2.4)

3’ (m'lx) (3’13’) 236

Then the large parenthesis which contains Taylor expansion coefficients is called the
transfer map or transfer matrix of the system. So if we apply the transfer map
to the initial coordinates we get the final coordinates of the particle. The transfer
map for higher order calculations can be derived from higher order Taylor expansion

coefficients.

2.2.1 The Multiplication of Transfer Maps

Sometimes it is useful to build a whole system for several subsystems such as focusing
elements, bending elements, and drifts. In this case we need multiplication of transfer

maps. We consider a system consisting of two subsystems.

Let the transfer map from the point II to j be M (i —’ j) and F(k) be the coordinate

vector at the point 1:. Then we get:

5(2) 2 M(1 _. 2)F(1) , (2.5)

21

17(3) :2 M(1—9 3)F(1) . (2.6)
But 5(3) can be rewritten by using equation (2.5):

5(3) 2 fil(2—+3)F(2)

= M(Z —2 3)]ll(1 ——2 2)17(l) . (2.7)
By comparing equations (2.6) and (2.7) we obtain:
M(1 .4 3) = M(2 _. 3)M(1 ... 2) . (2.8)

Similarly, transfer maps of any number of subsystems can be multiplied in inverse
order to get the transfer map of the whole system [Berz 92]. If the transfer maps
are first order then their multiplications comply with matrix algebra. This would be

extended to higher order transfer maps represented in the form of a matrix.

2.2.2 Interpretation of Transfer Matrix

If a: and :c' in equation (2.4) represent the position and inclined angle of a ray,

a transfer matrix element has its own physical meaning for a system [Brown 67],

[Wollnik 87a].

1. If (mlm) = 0 then we get from the first row of equation (2.4):
a: : (zlm’)m6 .
This means that final coordinate a: does not depend on initial coordinate 2:0.
So all particles or rays with the same inclined angle relative to the optic axis
are focused to the same point. This system is called parallel-to-point, and this

location corresponds to the focal point in ordinary optics.

2. If (zlz’) = 0 then we get from the first row of the equation (2.4):

a: 2: (:vlz):co .

22

This means that final coordinate a: does not depend on initial inclined angle 2:6.
So all rays starting at the same point with different initial angles are focused
to the same point. This system is called point-to-point, and corresponds to the

paraxial image condition in ordinary optics.

3. If (z'lz) = 0 then we get from the second row of the equation (2.4):
:c’ = (x’lm’)z(, .
This means that final inclined angle 93’ does not depend on initial coordinate
2:0. So all rays starting with the same initial angles have the same final angles.

This system is called parallel—to-parallel.

4. If (:c'lz') = 0 then we get from the second row of equation (2.4):
:c’ = (:c’lz):co .
This means that final inclined angle does not depend on initial inclined angle.
So all rays starting at the same point with different initial angles have the same

final angles. This system is called point-to-parallel.

2.3 Phase Space Transformation

2.3.1 2-dimensional Phase Space Ellipse

A two dimensional beam phase ellipse is described by Twiss parameters [Wollnik 87a]

or Courant-Snyder parameters [Courant 58] by a standard equation of an ellipse:
72:2 + 2033' + (33'? = e . (2.9)

where

w — 02 = 1 . (2.10)

23

Alternatively, in matrix form:

(61:21:)-

The area of the ellipse is we, when 6 is the emittance of the beam.

Let the transfer map from point 1 to point 2 be M. If there are no couplings with
other coordinates of different phase plane like (y,y’) or (At, Aw), then the transfer

map is given by just a two by two matrix:

M11 M12
M 2 ( ] . (2.12)

”[21 M22
Also:
M22 —M12
M-1 = (detM)"‘ , (2.13)
—M21 M11

where M ‘1 and det M are the inverse matrix and determinant of the matrix M,
respectively. Note that det M is equal to one if there is no acceleration in a system
[Cotte 38],[Brown 67]. In general, det M is the ratio of the corresponding initial and
final momenta [Wollnik 87b], [Douglas 88].

At point 1 in terms of point 2:

(ml ) = M" ( $2) , (2.14)

(1:, 3’1 ) = (232 3'2 )(M_')tr, (2.15)

where superscript tr means the transpose of a matrix.

24

It follows that from equations (2.11), (2.14), and (2.15):

(32 $12 )(M—l)t1' ( 71 :1)M—1(3’2): 61. (2.16)

( ) ( ‘72 02 ) ( 132 ) ( )
2:2 :13; = 62 . 2.17
(12 £2 3'2

M22 —M21 ’71 an M22 —M12
T = . (2.18)
”"M12 M11 GI 31 "M21 M11

Then the determinant of T is:

At point 2:

Let T be:

detT = (detM)’2 . (2.19)

The Twiss parameters should also satisfy equation (2.10) at point 2, this will be dis-
cussed in section 2.3.2. Therefore we get the phase space transformation by comparing

equations (2.16) and (2.17):

( £92 ) ( H1 )
a2 = (detM)“‘A a1 , (2.20)
‘72 ‘71

62 = (detM)el , (2.21)
where
M121 —2MHM12 M122
A = —M11M21 M11M22 + M12M21 —-M(2M22 . (2.22)
M22l -—2M21M22 M32

If det M is equal to one, i.e. no change in energy, the equations for the phase space

transformation give the standard equations, where the area of phase space or beam

25

emittance is a constant of motion. For an acceleration system, the determinant of a

transfer map is less than one and greater than one for a deceleration system.

2.3.2 N-dimensional Phase Space Ellipsoid for an Axially
Symmetric Beam

We now extend the formalism to n-dimensional phase space. If a beam is axially
symmetric through a system the equation of an n-dimensional ellipsoid can be written

in matrix form as:

X(1)”T(1)X(1)= e, . (2.23)

where X(l) is the coordinate vector at the point 1, and T(1) is a n X 11 real symmetric
matrix corresponding to the a matrix in the Transport notation [Brown 79]. Also W6]

is the phase space area of the projection on one plane.

The volume of the ellipsoid Vn and area of the projection on one plane A are:

(«er/2
Vn ._.___.__________., 2.24
Fe“)- ( )
A = as. (2.25)

Now let’s consider matrix T:

1. Ti, = T], by symmetry.

2. In the two dimensional case (n22) Ti} corresponds to the matrix of Twiss
parameters:

T11 TI2 7 01
= . (2.26)
T12 T22 02 16

3. Its determinant is equal to one.

26

We consider an upright n-dimensional phase space ellipsoid:

3 2 3 2 mn— 2 3n 2
(.1) +01) 1 ...... +( I) +(_) :1, (2.27)
1310 1320 “310 320

where 2:,0 means the maximum of the 2:.- coordinate in the phase space ellipsoid. As

 

a result of the axial symmetry 2:10 and 2:20 are the same for any pair of phase space

coordinates.

Let s be beam emittance on one projected plane from n-dimensional phase space:
6 = 2:102:20 . (2.28)

We also define a matrix element of T to be a coefficient of an n-dimensional phase

space ellipsoid:

Ti, 2 coefficient of 2:?‘2?
i, 3'21, 2, 3, ...... n, (2.29)

n,+nj=n.

With these definitions equation (2.27) leads to:

2 2 2 2
T113131 + T2232 ‘1‘ """ + Tn—I,n—1$n_1 + Tnnxn = E I (2-30)
where

E

T.) = _2

”In
E
T22 : 2

:32”
e"

Tn-l.n—I : T

13m
6

Tun : _—

NM

8

27
To find the determinant of the T matrix:
detT : HleT,.- 2 1 . (2.31)

This was derived from the special case of an upright n-dimensional ellipsoid. But
since we can always find a rotated coordinate system in which the ellipsoid is upright,

this should be also true for tilted ellipsoids.

Equation (2.23) is rewritten using the transfer map M from point 1 to point 2:

X(2)tr(M“)trT(1)M“X(2) = e, . (2.32)

At point 2:
X(2)trT(2)X(2) = 52 . (2.33)

Let T; be:
r; = (M“)trT(1)M“ . (2.34)

Then it is required, with an appropriate constant C, that:
det (CTz') = 1 . (2.35)

Using the facts from matrix algebra:

det(AB) .—. det(A)odet(B), (2.36)
detA" = (detA)“, (2.37)
detAtr : detA, (2.38)

we get:
0 = (detM)2/" . (2.39)

Therefore, we get a transformation formula for the n-dimensional phase space
ellipsoid:
T(2) = (detM)2/"(M“)trT(1)M" , (2.40)

62 = (detM)2/"e]. (2.41)

28

Note if n=2 we get the same result as in section 2.3.1.
2.3.3 Envelope of Phase Space Ellipsoid

The envelope of the phase space ellipsoid in one projection plane is found by searching
for the maximum points of the projected two dimensional phase space ellipse at
various axial points through a system. The equation of the phase space ellipse in a

projection plane (2:, 2:’) is given by:
Tux? + 2Tur232:' + T113512” = a”: . (2.42)

Obviously they satisfy the relation:
TNT)”: — TL. 2 1 . (2.43)
We replace T5 with Twiss parameters which describe the shape of a 2-dimensional

phase space ellipse in a projection plane (2:, 23’):

T... = 7, T”: = a, T”, = )6. (2.44)

Then equation (2.23) is written as:
'72:2 + 2032/ + 32),) = s”: E e . (2.45)

Now we find the maxima of the phase space ellipse on the two coordinates. Define
f(2:,2:’) to be:

f(2:,2:') _=_ 72:2 + 201223’ + flmfl — e = 0 . (2.46)
Then take the gradient to get:

6f(2:,2:') = (272: + 2a2t',2a2: + 2323') . (2.47)

(I

—{—,2:. Inserting this

Since [6f(2:,2:’)]xr = 0 at the maximum value of 2:, we get 2:’ =

into equation (2.46) we get for maximum 2::

mm = ([875. (2.48)

29

But if we use equation (2.41) for a beam emittance transformation we get the beam

envelope of 2: in terms of initial beam emittance e]:

 

23m = \/(detM)2/"fl61 . (2.49)

Similarly, we get for maximum 23’:

 

2:' = \/(detM)2/"7€. . (2.50)

m

If the determinant of a transfer map is equal to one we get the standard equations for
beam envelopes of 2: and 23’, where [3 and 7 describe beam envelope characteristics of
a system. But in general the beam envelope characteristics of a system are described

by the transfer map and Twiss parameters of a phase space from equations (2.49)

and (2.50).
2.3.4 Example of n=4 Case

As an actual example we consider the phase space transformation of the ATLAS
Positive Ion Injector which consists of axially symmetric rf resonators and magnetic
solenoids. In this case the correlations exist between the phase space coordinates due
to rotation of an individual particle under axial magnetic field component of solenoids.
For example, in a magnetic solenoid coupling between transverse coordinates exist,
but longitudinal coordinates remain decoupled in first order. Therefore the transverse

phase space is 4-dimensional in rectangular coordinates.

The equation of the phase space ellipsoid is given by equation (2.23) by:

Tara: Trx' 0 0 13
I I T121.“ Tr’x’ O 0 I
( 13 23 y y ) 0 0 T11. Txx’ y

m :6. (2.51)
0 0 T”! Tr’x’ y

‘

Because of axial symmetry we can simply replace T5,- with the Twiss parameters which

describe the 2-dimensional phase space ellipse in the equations (2.44) and (2.45):

30

Therefore inserting Twiss parameters we get:

7 a 0 0 2:
, , a fi 0 0 2:’ _
(2: 2: y y) 0 0 7 a y —e. (2.52)
0 0 a B y'
The transfer map under axial symmetry is given by:
M11 MI2 M13 M14
M 2 M21 M22 M23 M24 (2.53)

—M.;; —M14 M11 M12
-1W23 —M24 M21 M22

From equation (2.40) for the n-dimensional case we get phase the space transformation

32 [31
( a2 ) = T.) ( a1 ) (detM)_l/2 , (2.54)
72 ’71

I A32 + A32 2(AI2A22 + Arm/112) Afg + A32 \

for n=4:

where

T = 14211422 '1' 14.111442 All/122 + A12A2|+ AHA” + AMA” (2 55)
A 2431.442 + 243214“ ’ .

 

 

K 2431+ A?” 2(AIIA2I + AMA“) Ajl + A?" I

24,-,- = (detM)1/2(M-‘),-,- (5,)“: 1, 2, 3, 4). (2.56)

For beam emittance and envelope transformations:

VdetMe, , (2.57)

232 = \/(dCtM)l/2fi2€1 , (2.58)

62

 

 

2:; \/(detM)‘/272q . (2.59)

31

The beam emittance does not change for magnetic solenoids, but in a system
including accelerating structures like rf resonators it changes by a factor of the square

root of the determinant of the transfer map of the system.

 

 

 

 

 

Figure 2.1: A two dimensional beam phase space ellipse. The equation of the ellipse
is given by 72:2 + Zazz’ + £3” = c. The area of the phase space ellipse is we. The e
is called beam emittance.

 

Chapter 3

The Equations of Motion of Linac
Components

3.1 Introduction

The optical elements of the P11 linac consist of radio frequency four-gap resonators,
superconducting magnetic solenoids, and drift spaces. Because of slow particle beam
velocities, less than 0.04c, the equations of motion are not solved by simple approx-
imations as in the high velocity case where rf cavities are treated as thin lenses.
Therefore they include nonlinear terms to be solved numerically in the equations of

motion.

In this chapter we derive the Taylor series expansion of radial and longitudinal
rf electromagnetic fields. We introduce a model solenoidal field with appropriate
parameters for the superconducting magnetic solenoids at the PH linac. These higher

order fields will be used to describe the nonlinear equations of motion.

Also we derive linear matrix optics both for the transverse and for the longitudinal

directions with the following assumptions [Vlasov 68]:

1. The longitudinal component of the transverse focusing field is negligible.

2. The radial velocity component of a particle is negligible relative to the longitu-

33

 

34

dinal component.

3. The longitudinal force of the rf magnetic field is sufficiently small to be replaced

by only an rf electric field.

4. The deviation of a particle is small enough to ignore the radial rf electric field

component.

Finally we describe the ways that the linear and nonlinear equations of motion

are implemented into computer programs.

3.2 Equations of Motion

Since the Positive Ion Injector (PII) consists of only axially symmetric elements the
cylindrical coordinate system would best describe related particle motions. However,
the equations of motion are written in rectangular coordinates for future possibilities

to add nonaxial symmetric elements like quadrupoles.

In rectangular coordinates a: and y are deviations in horizontal and vertical direc-
tions respectively from the reference trajectory. The distance 2: along the reference

trajectory is used as an independent variable.

The equation of motion of a particle of charge q with rest mass m under a Lorentz

force is written by:

_=q(E'+V'xB'), (3.1)

where E" and 3 represent electric and magnetic fields respectively. I7 is the velocity
of a particle.

In component form:

35

 

 

= qEz(0) , (3.2)

 

 

t 1—[3
d ,,
a; «3% ___ q(E,+cs,B.—cfi..B.), (3.3)

 

= q(Ez + cflszc — cfiyBx) ,

di( mCflzz) : q(EI+cfisz—Cflsz)v
( l
( l

where

Tb;
III

(3.4)

n|<1

The subscript .9 indicates the reference trajectory. Thus the first equation is for a
reference particle and the other ones are for a deviated particle from that reference

trajectory.

Now we change the independent variable from time to the position of the reference

trajectory by using:

 

 

d d
dt, - Cfis'cE ,
d d
a; Z CHI-J; , (3.5)
and
-+ d1?
Also, if we assume ,6 m H, the equations of motion (3.2) and (3.3) become:
dfis _ q (l —' fl3)3/2
dz _ (meg) fl, Ez(0) , (3.7)

36

 

 

21w. _ ( q,¢1-fif E $282328.) 1 1M
dz2 — mc2 ,8? x c dz c fiz(1-—,512) dz dz’

 

 

fl _ ( (1),/1—53 E +azs._a.Bzga_c_ ___1____£’£z.:’£ (38)
dzz “ mcz [33 ,, c 6 dz gz(1_fif)dzdz’ .

 

 

._ 23/
«if: ___ ( q )(1 fi.)2{Ez+éfi31_flfl}.

mc2 B2 c c

This set of equations of relativistic form completely describe both the transverse and

the longitudinal motions of a particle under acceleration along a reference trajectory.

3.2.1 RF Resonator

The electromagnetic fields for a cylindrically symmetric rf resonator were derived in
Appendix A. The equations of motion in an rf resonator are easily found by inserting

the fields into equations (3.7) and (3.8):

dfi.

_ q (1417.33)?”2
dz _( )

3..

(122: 1 dflzdz: : ( q)\/1—I322

3'27 + m1 — HD7522 m2 fl?

 

 

Ez(0,z)cos cf), , (3.9)

(Fer! + Fbr1+ Ferfl + Fbril)z ,

 

 

 

 

‘12:” 1 dflz dy ‘1 V 1 — fizz
2;; + 13;“ — (322) d2 2; — (mg?) 522 (Ferl + Fbrl + Fer3 + Fbr:;)y ,(3.10)
d . 1— 3* 3/2
d’: : (qu2)( 5:) (F620 + szl + F022 + szll) ,
where
18E,(0,z)
Fm " _ETC°S¢’ (3.11)

Fag : Ez(0,z)cos¢, (3.12)

37

 

 

F... : 522340,.)5, sinqS, (3.13)
(12: dy

szl — — (at + 5y) Flu] , (3.14)

l a:2 6Ez(0,z) 33Ez(0,z) 2 2
Fag-3 — 'i—6' {25—62—- + T (3 + y )COSQS, (3.15)
_ 1 a:2 62Ez(0,z) 2 2
F622 — —4 {§E2(0,z) + T} (a: + y )cosqS , (3.16)
1 w w2 6Ez(0,z) 62Ez(0,z) 2 2 ,

Fbrg — —igz{?—a—;— + ___-52'2— (23 +y )flz sm¢, (3.17)
dz: dy

szil — _ ($1: + $3,) Fbr3 , a (3.18)

where
()3 t + as dz + 4: (3 19)
z w : .
U Cfiz U a
as — t + ¢ —/ dz .5 3 20
.-.... ._ 623+... (.)

3.2.2 Magnetic Solenoid

A magnetic solenoid lens is used as a focusing element for both the horizontal and
vertical directions simultaneously. We introduce a solenoidal field which along the

optic axis has the form [Dragt 90], [Berz 93]:
Bu 2 z — l
82(0, 2) — —2— {tanh(2) — tanh(T)} , (3.21)

where BO is the maximum field strength. The length and half aperture of the solenoid

are l, d respectively.

38

Since energy is conserved 1n a magnetic field, the terms containingd 1’1;— in the

d—g—z was an approximation of

equations of motion (3. 8) vanish because the expression-—
exact form of $53, which is zero in a pure magnetic field. Also, the field components

of a magnetic solenoid are calculated from the model solenoidal field on axes and

equation (3.6) of Appendix B.

Thus from the equations of motion (3.7), (3.8) with calculated solenoid field com-

ponents, we obtain a set of differential equations of motion of the model solenoid:

 

 

dfis _
dz — 0 ’ (3.22)
Si; ___ (__)\/—fl{::’ (820 + Bbzz) - a:(B,.1+ 3'3)}’

 

if; = (i) fiz{::(Bz11--§£3 + Bbz2)_ y(B.1 + Br3)} 1 (3'23)

d z 12 d d
dflz : (_q_)(1_ fif)/ {iy—%E}(Br1+8r3)a

 

where
3,, = _éiflgfl, (3.24)
B.3 = £6£%£gfi(zz+y2), (3-25)
3.0 = B,(o,z), (3.26)
3.. = _gflgg—Sfl1.2+yz), (3.27)

Bz(0,z) 2 BO 0:{tanh( ——z)—tanh(

—2— )} , (3.23)

  
 

(9840,12) _ PE) 2 E 2 z—l .
62 —— 2d {sech (d) — sech( d )} , (3.29)

 

39

 

 

 

 

 

 

9.255%,” = %{—sech2(§)tanh(§)+sech2(z;l)tanh(z;l)} , (3.30)
W = %{sech2(§)[3tanh2(§)—l]}
_ %{sech2(z;l)[3tanh2(z;l)—-1]}. (3.31)
3.2.3 Drift Space
Drift space is considered as field free region:
‘3: z 0, (3.32)
3.}: = 0, (3.33)
if? — ..

3.3 Linear Transfer Map

Even though the derived equations of motion are nonlinear, it is useful to consider
linear solutions. The beam optics of a system is well described with linear solutions
in most cases if nonlinearity is not so strong. The lst order theory is mostly well

defined for phase space distribution and its transformation in accelerator physics.

The transfer map is represented in six phase space coordinates:

(x, 22’, y, 3’, At. Aw ). (3.34)

40

where
, _. if
:1: _ dz,
dy
I — _
y - dz’ (3.35)
At = t—t,,
Aw = 111—111,. (3.36)

i.e. At and Aw mean deviations of time of flight and kinetic energy from a reference

particle respectively.

3.3.1 Transfer Map of a Drift Space

The differential equations of motion in equations (3.32) and (3.33) are easily solved

for a drift space of length 3:

Transverse Motion

 

For transverse motion:

:1: = 330 + (1:6 ,
r 1
a: = 230 ,
y : yU + £316 ,
y, 2 311,1 1 (337)

where subscript 0 indicates initial values of phase space coordinates.

Longitudinal Motion

 

For longitudinal motion:

fls : 16.10?
H. = fizo. (3.38)

41

Obviously there is no deviation in energy for a drift space:

Aw = Awu . (3.39)

To find the deviation of time of flight we define:

A3 = fl. — [3. . (3.40)

Then from equation (3.36):

   

 

 

_ l _ i (A?)2
The relativistic relation:
11=\/1—(1+——)—2 — ~410— (3 42
— me? + w ’ ' )

where p is the momentum of a particle.

From equation (3.42) for an off axial particle we find:

mc2

{1
= {1-(11—1}
{

2(mc2)2 }‘/2

(me2 + w )3Aw

+w+Aw—21/2
1 * ___—___)}

 

flfmchw
———————3 .
Pa

22
F
_+_

(3.43)

42

Using Aw = Awo, from equations (3.40) and (3.43) we obtain:

2 2
g, m CAN)”

AB 2 2'3

(3.44)

Therefore from equations (3.41) and (3.44) we get for a drift space of length l:

2 4
m m c3,
1 A1011 + 6

8 8

 

 

At=At0+£{— Aw3+--- }. (3.45)
Note that the drift space is not linear for longitudinal motion unlike transverse motion.

If we combine equations (3.37), (3.39) and (3.45) in matrix form up to lst order

we get the transfer matrix for a drift space of length l:

 

 

 

 

 

 

1, (140000) 1,0

{3'3 010000 {mgl
001400

3] _ 310

yr — 0 0 010 0 116 ° (3'46)

At 011001—’«—;'§3 Ato

\Aw/ (00000 1)\Awo}

3.3.2 Transfer Map of a RF Resonator

Transverse Map

 

If just lst order terms of a: or y are taken in the differential equations of motion (3.9),
(3.10) with the approximation 32 It: H3, we find the linear differential equation of

motion:
112:1: d1:

32—2 + 04(2); + 5(2):: = 0, (3.47)

43

where

1 dfi.

0(2) = mg, (3.48)

 

V1 — 163 z ,Z w ,
"(2) : *1nqc2 ’62 {_%_6—El_a(-g_'_)—COS¢3+ ‘2—2E2(012)fl851n¢s} ' (3'49)

This equation is linear in form. But it is not solved in closed form because 01(2) and
11(2) are not constant due to acceleration and varying electric field profile respectively

along the resonator axes.

If the velocity of a beam is very high, enough to ignore change of velocity along
the accelerating gap structure, 0(2) is negligible. But in the low velocity case with

rapid energy gain such as the PH linac it can not be neglected any more.

Therefore we consider a thin part of an accelerating gap of a resonator where a
and 1: might be considered as constant. Thus the solutions of the equations of motion
in the thin part are easily solved in matrix form. In this way the whole transfer map
of a resonator is found by matrix multiplications of the transfer matrices of each thin

part.

The solutions of transverse motion for a thin part of a resonator are given by:

(i)ifa2—4n>0

1: : e"""'/2 { [cosh(p2/2) + gsinh(pz/2)] $0 + [g sinh(p2/2)] 1:6} 1

:11' 2: —e-‘”/2 { [g5 sinh(p2/2)] :13” + [cosh(p2/2) — gsinh(p2/2)] 2:6} ,(3.50)

(ii) if a2 —4n < 0

a: : e—m/2 { [cos(p2/2) + gsin(pz/2)] $611+ [3 sin(pz/2)] 3b} 1

44

2:! : _3‘02/2 { [if sin(p2/2)] :20 + [cos(p2/2) — 35in(p2/2)] :28} , (3.51)

(iii) if a2 — 4n = 0

:1: : cosh(az/2)$o + Z‘Sinh(aZ/2)x(' ’

:c' = ; sinh(a2/2):1:u + cosh(a2/2):1:{, , (3.52)

where

p E |a2—4n| . (3.53)

If we combine equations (3.50) - (3.52) in matrix form, we have transfer maps M for

a small SCCthIl Of a resonator:

(3)2M.(30), (3.54)

where

(i)if042—4n>0

' cosh(’;—z) + asinh(825)/P 25inh(p2_z)/P
M —uz/2 a (3'55)
-2n sinh(%)/p cosh("2—z) — acosh(l’2—z)/p

(ii) 110.2 —- 411 < 0

(cos(‘;—z)+asin(%)/p 25in(325)/p )
M _ ~Ol/2 , (3.56)
-—2nsin(E)/P cos(25) — asin(%3—)/p

2 2

45

(iii) if a2 — 4n = 0

M:

( cosh(%) 2sinh(9,f-)/a )
. (3.57)

asinh(‘—"§z—)/2 cosh(‘”)

7

Note that the determinant of the transfer map is not equal to one in general but

e"“"’/2 because of acceleration.

Since the resonator field is axially symmetric we get the same results for the

vertical direction (y, 2).

Longitudinal Map

 

For a longitudinal map it is more convenient to use the energy gain of a charged
particle under longitudinal electric field rather than to use the equations of motion

derived already.

The longitudinal electric field from equations (A21) and (A23) up to lst order

 

becomes:
E,(r, 2,t) : E,(0, 2) cos(wt + <15“) . (3.58)
Let 4) be:
2 d2
¢=wt+¢.=/O .3. +4.. (3-59)

Then the energy gain per unit length of a charged particle is given in differential form

 

by:
dw
a : qEz(0,z)cos¢, (3.60)
d 9
w. = qEz(0,2)cos qS, . (3.61)

112

46

From the definitions of At and Aw we obtain:

 

 

 

 

 

(13211) : 9Ez(0,2) (cos 45 — cos 43,) : —qE,(0,2) sin ¢,A¢ + . . .
dA¢ 1 1 wmz

Thus, up to lst order the above equations become:

 

 

dAw
dz :- —A](Z)A¢,
dAd)

where

31(2) = qEz(0,z)sin¢,,

 

wm
/\ 2 = —
Now we introduce a new variable:
X = / A2(2)d2 .

Then:

d d
— —_— A -— .
d2 2(z)dx

Thus, equations (3.64) and (3.65) become respectively:

 

dAw A1(2)
__ 2 _ A ,
dx A2(2) 35
fl = Aw.

(3.62)

(3.63)

(3.64)

(3.65)

(3.66)

(3.67)

(3.68)

(3.69)

(3.70)

(3.71)

47

By combining (3.70) and (3.71) we obtain a single second order differential equation:

 

 

 

d2A¢ A.(2) _
dxz + 32(2)A¢ — 0 . (3.72)
Let’s define:
2 __ 31(2)
— 32(2) . (3.73)

 

 

By the similar way as in the transverse case if we assume that the change of 162 is

negligible in a small section of a resonator, we find solutions for a small section:

(1)1fA1/A2 > 0

A115 2 cos(nx)A¢U + AwU sin(nx)/n ,

Aw = —nsin(nx)A¢u + Aw()cos(nx) . (3.74)
Using equations (3.68), (3.71), and A45 2 wAt we obtain:

At '2 cos(n/\22)Atu + Aw()_sin(m\22)/(nw) ,

Aw : ——wn sin(16/\22)At0 + Aw.)cos(m\22) . (3.75)
(ii) if Al/Ag < 0
At : cosh(m\22)AtU + Awusinh(n)122)/(nw) ,
Aw 2 am sinh(nA22)AtU + Aw(,cosh(nA22) . (3.76)
(iii) if Al/Ag = 0
At 2 At.) + AwUA22/w ,

Aw : Aw” . (3.77)

If we combine equations (3.75) - (3.77) in matrix form, we have transfer maps M for

a small section of a resonator:

48

At Ato
( ) = M o ( ) , (3.78)
Aw Awu

( cos(m\gz) sin(nA22)/(nw)
M _

where

) If Al/Az > 0 , (3.79)

~50) sin(nA22) cos(m\22)

( cosh(m\22) sinh(nz\22)/(nw) )
M = if AI/Ag < 0 , (3.80)
mu sinh(m\22) cosh(m\22)
1 Agz/w
M = ( ) if Al/Ag = 0 . (3.81)
0 1

Note that the transfer map is the same as for drift space if n = 0 .

3.3.3 Transfer Map of a Magnetic Solenoid

The differential equations of motion of a magnetic solenoid derived in section 3.2.2
are coupled between the a: and y coordinates due to rotation even in lst order. The

linear part of equations (3.22), (3.23) gives:
dfi,

 

 

dz = 0, (3.82)
3‘; — a{%Bz(0,z) + éggfgfly} = 0,
5:121 — a {ng(0,z) + éa—ngflz} = 0 , (3-83)
:22

where

_q\/1—fi3~q\/1—fl.3 (384)

_ moflz N mCH.

49

The differential equations are decoupled in the rotated coordinate system (X, Y,z),
which is rotated an angle 0 about 2 axis in the fixed coordinate system (2:,y, 2) (see

Appendix B):

 

 

(PX

421/ 2

d2—nYzO, (3.85)
2

where

K.

82 0, 1 — fl?
= ‘7 27:62) V a, . (3.86)

Since 5(2) is not constant, we also consider a small section of the solenoid to neglect

the change of n(z). Then the transfer map M, in the rotated coordinate system is

given by:
cos(nz) sin(nz)/n 0 0
_ —nsin(nz) cos(nz) 0 0
M, - 0 0 cos(nz) sin(nz)/n (3'87)
0 0 —nsin(nz) cos(nz)

The transfer map M in the fixed coordinate system is obtained by applying a rotation

matrix R(6) (see Appendix B) to the transfer map in the rotated coordinate system:

M = R(—0)M,

C-cosfl S-cosO/n C~sin0 S-sinO/n
_ —Sn-cos€ C-cos9 —Sn-sin0 C-sino (388)
_ —C-sin0 —S-sin0/n C-cos9 S-cosG/n ’ '
Sn-sinfl —C-sin0 —Sn-cos€ C-c030

where

C = COS(K.Z), (3.89)

50
S = sin(nz), (3.90)

9 = —/zn(z)dz. (3.91)

Therefore in the fixed coordinate system we get the phase space coordinate transfor-

mation by the transfer map M:

23 130
:c’ 2: M 0 36
y 310
y’ ya

(3.92)

Since the magnetic solenoid behaves like drift space for longitudinal motion from
equations (3.82) and (3.83) we find the same result as the longitudinal map of a drift

space.

3.4 Implementation by Computer Programs

The matrix optics and equations of motion under accelerating forces derived in Chap-
ter 2 and Chapter 3 are implemented to computer programs MINJI [Joh 93b] and
MINSOL [Joh 93c].

The first order matrix optics program MINJ I calculates linear transfer maps point-
by-point through a system both for the transverse and for the longitudinal directions .
This point-by-point search is necessary because of rapid changes of velocity. Therefore
it provides transfer matrices at any point of an element of the system to make it

feasible to investigate a beam inside an element.

This matrix optics program provides beam envelopes and Twiss parameters of
phase spaces for rotated or unrotated beams. Initial or final phase space matchings
are possible with appropriate fitting parameters by using the symplectic nonlinear

optimizer in it.

W.

 

51

On the other hand the program MINSOL just solves nonlinear differential equa-
tions of motion derived in this chapter analytically with off axial fields up to 3rd

order.

3.4.1 Numerical Integration

The spatial parts of the axial resonator field profiles including its lst and 2nd deriva-
tives were calculated using the method of moment [Shepard 85] which nicely fit ex-
perimentally measured data using perturbation method [Shepard 93]. To include 3rd
order effect of these fields, the 3rd order derivatives were calculated from the 2nd
order derivatives. As an integration routine the 4th order Runge-Kutta method with

constant step size is used [Press 86].

The more sophisticated integration routine would be used ideally but tests of an
integration routine using what we call the forward-backward method shows that using
4th order is enough to give consistent results. The forward-backward method takes
the initial conditions of a ray and calculates its final conditions. These final conditions
are then used as initial input conditions to try and trace back to the original set of

initial conditions.

3.4.2 Model Solenoid

The superconducting magnetic solenoids in the PH linac have the following features
[Jaffey 74], [Jaffey 76]: The overall length including coil parts and iron shield is 25
cm and the aperture is 2.5 cm; The maximum central field of the present magnet is
7.6 T with fringe fields dropping rapidly to 10 G at a distance of 1 cm outside the
iron shield and 50 mG at a distance of 2.5 cm from the mu-metal shield; Therefore
the extension of the solenoidal fields into the resonators is negligible from the present

location of the solenoids and resonators in the P11 linac.

52

To describe the solenoidal fields analytically we use the model solenoid of equation

(3.21) with appropriate parameters of d, 1:

8(2) = %9 {tanh(§) — tanh(E—d:l} . (3.93)

The solenoid parameters, (1,1 are adjusted to best describe the real solenoid field.

In adjusting solenoid parameters d is found to be 1.5 cm for the half aperture
and l is found to be 10 cm for the body length of solenoid. Figure 3.1 shows the
magnetic field profile of the solenoid with the fitted parameters. The effect of fringe
fields is calculated both in the raytracing and in matrix optics by considering the

length of the fringe field tail and its relative strength to the maximum field strength
(see Appendix C).

 

53

 

 

 

 

 

.1.0. . . , , , .
s E"l"'l' I r[H’l'rg
0.87- _:
...: _§
0.45—— _3
0.2_:_—- __E
00:...) ..I....I....I.. |.,..:

'-0.15 —0.1 —0.05 0 0.05 0.1 0.15

z (m)

Figure 3.1: The normalized magnetic field profile on axes of the model solenoid with
half aperture 1.5 cm and body length 10 cm.

Chapter 4

Application to Resonator Study

4.1 Introduction

The four-gap superconducting resonators have been developed at Argonne for use
in the low-beta positive ion injector for the ATLAS linac [Shepard 87],[Shepard 89].
It was previously observed that at low velocities these structures can be focusing in
both longitudinal and transverse phase spaces due to an inherent alternating-phase-

focusing property [Pardo 87a], [Nolen 93].

Alternating-phase-focusing achieves both transverse and longitudinal focusing by
alternating appropriately the signs of the initial rf phases in each resonator. This
focusing has been reported in several papers [Good 53],[Chambers 76],[Okamoto 89],
[Sagalovsky 92]. But in the PII linac each resonator itself has this property even when
operated with a fixed initial rf phase. This might be understood from the relatively
fast velocity change within the multiple gap structures to cause effective sign changes
of rf phase offset. The strength of transverse focusing seems to be so strong that
it might not be necessary to use focusing solenoids at the beginning stage of PH

[Nolen 93] .

Since these resonators have the potential application to be used for extremely low-

beta low q/ m beams, it will be useful to understand the focusing properties of them

54

55

at low velocities. For this purpose a custom—made first order matrix optics computer

program is used.

In this chapter we investigate in detail focusing structures of the II and 12 type
resonators which are located at the beginning of PII. The limits of injection energy
and initial rf phase offset for transverse and longitudinal stabilities are calculated by

using the concept of focusing power to see possibility of resonator focusing only.

Throughout this and the following chapters, calculations will be made with 238U24+

which has a q/m=0.1 and 40Ar12+ which has a q/m=0.3 because of their relatively

low and high charge to mass ratios.

4.2 Focusing Power

The focusing power of a lens is defined as an inverse of the focal length. It can be
used to measure the focusing strength of a lens. The focusing power is calculated
from a transfer matrix of a lens. In the case of a thin lens which has only focusing
action without effecting the position of a ray, the transfer matrix is written in the
simple form of:

1 0
M. = . (4.1)
—1/f 1

In a system of a thin lens M. followed by a drift space of length f, the resulting transfer

matrix becomes:

1 f 1 0 0 f
M: = . (4.2)
0 1 —1/f 1 —1/f 1

Since the resulting first column, the first row transfer matrix element, M”, equals

zero this corresponds to parallel-to-point system discussed in section 2.2.2 of Chapter

 

56

2. All initial parallel rays are focused to a point of distance f. Similarly if the drift
space of length f precedes the thin lens Mg, the resulting transfer matrix element
of the second column, second row becomes zero. So all initial rays starting at a
point of distance f have the same divergences after passing a thin lens. Therefore f
is interpreted as the focal length and 1 / f as the focusing power of a thin lens and is

determined uniquely.

But the above statement is not true for a thick lens which has two principal planes

in general. This case is discussed in the following section.

4.2.1 Transverse Case

In a thick lens the focal lengths are defined with respect to two different principal

planes P1 and P2 in Figure 4.1. Let a transfer matrix of a lens be M,:

( (xix) (wlz’) )
M12: . (4.3)
(Z’Ix) (x’lx’)

The focal lengths of a lens can be calculated by two different ways from its trans-
fer matrix M1. Consider the two systems consisting of a drift length and a lens in

Figure 4.1.

The focal length fl is calculated from the point-to-parallel system in Figure 4.1.
The transfer matrix of a point—to-parallel system is found by applying a lens map to

a drift map:

((xlm) (mlz’) )(1 f.) ((mlw) (mlz)f.+(xlx'))
= . (4.4)
(m’lr) (m’lw’) o 1 (z’lx) (:v’lm)f1+(m’lw’)

Since the second column, second row element of the resulting transfer matrix should

 

57

be zero for point—to—parallel system the drift length fl is given by:

__ (z’lz’)

f‘ Z (w’lw) '

 

(4.5)

The focal length f2 is calculated from the parallel-to-point system in Figure 4.1.

The transfer matrix of the system is found by applying a drift map to a lens map:

( 1 f2 ) ( (film) (wlw') ) ( (wlz)+f2(=v’|m) (z’lz'm +(xlw’) )
= . (4.6)
0 1 (m’lm) (m’lz’) (x’lm) (m’lw’)

Since the first column, first row element of the resulting transfer matrix should be

zero for parallel-to-point system the drift length f2 is given by:

_ (xix)
(m’lx) '

It should be mentioned that the drift lengths f1 and f2 calculated from the two different

 

f2 = (4.7)

systems above are not the real focal lengths of a lens. Because they are calculated
from the entrance (f1) and the exit (f2) not from the principal planes [Wollnik 87a] of
a lens. So they are different by some constants. In the case of a thin lens f1 and f2 are

real focal lengths and have the same value as each other because (zlz) = (23’ [12’ ) z 1.

But in any cases the focusing power is proportional to a matrix element (:c’lz) of

a transfer matrix of a lens:

focusing power or —(a:'|:c). (4.8)

As a matter of fact, to investigate the sign of (z'lz) is a more convenient way to
study focusing property of a lens than to investigate the sign of focusing power. For
example if a lens is so strong that a focal point exists inside the lens then the matrix

elements (:clx) and (x’lz') of the lens become negative. Therefore it would look like

58

a defocusing lens if we observe the drift lengths that were derived in equations (4.5)
and (4.7). But (z’lx) provides correct information about the focusing property if the
number of intermediate images does not exceed two. In other word, the negative of
(z’lz) means that there is a net focusing action which is proportional to (:c’lx) and

vice versa, as shown in Figure 4.2.

4.2.2 Longitudinal Case

In the longitudinal phase space the transfer matrix of drift space is not the same as

in transverse case. The transfer matrix of drift length l was given in equation (3.46)

1 4012/1013
M. ... ( ] , (4.9)

of Chapter 3:

0 1

where p, is a momentum of a reference particle.

Let a longitudinal transfer matrix of a lens be M1:

M1:

(AtlAt) (AtlAw)
( ] (...)

(AwIAt) (AwIAw)

By the similar way as the transverse case, the drift length fl is calculated from the
point-to—parallel system in Figure 4.1. The transfer matrix of point—to-parallel system

is found by applying a lens map to a drift map in the longitudinal phase space:

( 1 —f1m2/P'31 ) ( (AtlAt) —flm2(AtlAt)/p:l +(AwlAw) )
M1. — . (4.11)

o 1 (AwlAt) -f1m2(Aw|At)/pz?. + (AwlAw)

The vanishing of the second column, second row element of the resulting transfer

 

59

matrix gives:
f 2 pg] (AwlAw)
‘ m2 (AwlAt) '

 

(4.12)

And similarly for parallel-to-point system, the vanishing of first column, first row

element of the resulting transfer matrix gives:

_ p3, (AtlAt)

f2 — m, (AwIAt) . (4.13)

Note that p,1 and p82 are momenta of the reference particle at the entrance and exit

of a lens respectively.

In both cases the focusing power is proportional to a matrix element (AwlAt) of

a longitudinal transfer matrix of a lens:

m

 

2(AwlAt). (4.14)

focusmg power or 3
By the same argument that was used in the transverse case, a convenient way to
examine the longitudinal focusing property would be to investigate the inclined angle
in the (At,z) plane of a ray after passing a lens as shown in Figure 4.2. From

equation (3.63) in section 3.3.2 of Chapter 3, with keeping in mind that A05 2 wAt,

the inclined angle in the (At, 2) plane is given by:

dAt _ p3

dz m2

 

Aw . (4.15)

For a parallel ray with Atu : 1, Awo : 0 at the beginning, the final value of Aw is
given by:
Aw = (AwIAt) . (4.16)

Therefore from the above two equations the inclined angle is proportional to a matrix

 

60

element (AwlAt):

 

3
d“ 0( -p“'2(Aw|At). (4.17)

717 777.

Since 45% is negative for focusing action, (AwlAt) is positive for focusing action

and negative for defocusing action.

61

4.3 Focusing Structures of II and 12 type Res-
onators

4.3.1 Transverse Focusing

The radial force on a particle of electric charge q in an axially symmetric accelerating
gap was derived in section 3.2.1 of Chapter 3. The radial force upward to first order
is given by:

16Ez(0, z)

w .
F1- -— Q{—§—-az——— COS 43, + '2—CE1(0, Z)fl, sm {fig} 23, (4.18)

where d), is local rf phase of a central ray.

The lst and 2nd terms are electric and magnetic forces coming from the rf fields.

Also, the equation of motion was given through equations (3.9), (3.10) of Chapter

3. For the transverse motion:

 

d2 d
$5 + a(z)2:—:— + 5(2):: 2 0 , (4-19)
where
_ 1 .13.
0(2) my“; , (4.20)
/1__ 2
n(z) = — 1 [3" F (4.21)

mcz [62 x ‘
The focusing action in the radial direction at a point 2 along a resonator axis is

determined by:

a2 — 4:: > 0 : Focusing ,

a2 — 45 < 0 : Defocusing . (4.22)

If the change of velocity through a gap is negligible, the focusing action is dominated

by the radial force only. So the action at the first half of a gap would be focusing and

62

defocusing at the second half to give net focusing or defocusing effect depending on
the rf phase change [Nolen 92], if the magnetic force term is small. But because of
the fast energy gain in 11 and I2 type resonators with low-beta structures the change
of velocity can not be negligible. Therefore the focusing action should be determined
both by the rate of the velocity change and the magnitude of the radial force. The

local focusing action may change even at the half of a gap.

4.3.2 Longitudinal Focusing

First order longitudinal focusing is completely determined by the local axial electric
field profile and the local phase of a central ray by equations (3.75)-(3.77) of Chapter

3:

Ez(0,z) sin (I), < 0 : Focusing ,

Ez(0,z) sin (b, > 0 : Defocusing . (4.23)

While a beam passes through a multiple gap structure resonator, the longitudinal
focusing action may switch to defocusing action due to a change of the local rf phase.

This effectively gives alternating phase focusing effect within a resonator.

4.3.3 Alternating Phase Focusing

Each type of resonator is subdivided into 200 thin resonators for matrix optics calcu-
lation. So the focusing power of each cell can be calculated by using transfer matrix
of each cell. Since each cell behaves almost like a thin lens the focusing powers give
nearly the same values for both point-to-parallel and for parallel-to-point conditions.
Also, we can calculate the focusing power of each accelerating gap from its transfer

matrix.

The focusing power plots of II and I2 type resonators in Figure 4.3 and Figure 4.4

63

show detailed focusing structures inside the resonators. The negative focusing power
means defocusing action at the corresponding cell and vice versa. Because of the
energy gain the strength of the transverse focusing power (solid line) decreases through
a resonator. In I2 type resonators the overall focusing structure is similar to the 11
type resonators. But the absolute value of the focusing power decreases considerably

because of the relatively high velocity after getting energy from the I1 resonator.

One thing to note is that at some parts of each gap both transverse and longitudi-
nal focusing can occur simultaneously. Because the change of velocity is not negligible
transverse focusing does not depend only on radial force but also depends on the rate
of velocity change through equation (4.22). Therefore this feature does not usually

appear in high-beta resonators where focusing is dominated by the radial force only.

Another thing to note is that the overall defocusing action in the longitudinal
direction is alternated to focusing action at the second half of each resonator. So
the initial rf phase offset is also effectively alternated in order to give an alternating-
phase-focusing effect. Table 4.1 shows the focusing powers of each gap, which is
calculated from the drift lengths f1 and f2 discussed at the beginning of this chapter.
Alternating-phase-focusing structures appear clearly between the first half and the
second half of each resonator. Also, the net transverse and the longitudinal focusing

actions of each gap are opposite to each other like in a high-beta structure resonator.

Table 4.1 shows that each accelerating gap does not behave as a thin lens in a low-
beta resonator because of its two different focal lengths. As a result the focusing effect
of a resonator might not be calculated from the same way as done in Table 4.1 because
of its very strong focusing property. Therefore we use the transfer matrix elements
(m’lz) for the transverse direction and (AwIAt) for the longitudinal direction. The
net focusing effects of 11 and I2 type resonators would be determined by observing

the matrix elements of their respective transfer maps.

64

4.4 The Limits of RF Phase Offset of Resonators

If alternating-phase-focusing properties of the P11 resonators are used without addi-
tional focusing elements each resonator itself should achieve both the transverse and

the longitudinal focusing for a given initial rf phase offset.

To find the stability region of the initial rf phase offset for a resonator at a given
initial velocity the transfer matrix of a resonator is calculated fora reference particle
with the custom-made matrix optics program. For a given initial rf phase offset
the transfer matrix element (z’lz) should be positive to achieve the net transverse
focusing. Also, the transfer matrix elements (AwIAt) should be negative to achieve

the net longitudinal focusing.

Figure 4.5 and Figure 4.6 show stability limits of the initial rf phase offset at
different initial velocities. The electric field gradients were 4.5 MV/m for type I1
resonator and 3.0 MV/m for type 12 resonator. The dotted points at an initial
velocity satisfy both conditions for the transverse and the longitudinal focusing. In
this calculation the initial rf phase offset was limited between —40 and 40 degrees

because above these limits the energy gains are not efficient.

The positive rf phase offset is mainly forbidden due to the condition of longitu-
dinal stability even though transverse focusing is stronger than in the negative case.
Some narrow bands of a negative rf phase offset at a given initial fl is mainly due to

transverse instability.

For example if we accelerate an “”Ar'“ beam of initial )6 = 0.0075 through an
11 type resonator, the range of the initial rf phase offset should be between —40 and
0 degrees. But it will be between —20 and 0 degrees in the 12 type resonator. In
fact it should be noted that the points near zero degrees might not be used as stable

points because the stability region is cut at zero degrees, where nonlinearity might

65

be considerable in the longitudinal direction.

According to the stability plots, the limits of stable rf phase offsets depends upon
the initial velocity especially for 12 type resonator. Thus the initial velocities should

be chosen appropriately to have a wide range of the incident rf phase angles.

l

4.5 Parameters on Focusing

In this section we investigate the effect of the injection energy and the initial phase

offset on the focusing power of 11 and 12 type resonators.

4.5.1 Injection Energy

The injection energy is usually determined by the transit time factors [Livingood 61],
[Hereward 70] of corresponding resonators to optimize energy gain if a transverse
focusing element is available. This is reasonable because longitudinal stability is ob-
tained rather independently of the injection energy if the initial rf phase offset is
chosen to be negative. But if resonator focusing is to be used for transverse focusing
the injection energy should be chosen appropriately. For example, the 11 type res-
onator does not give transverse focusing at an initial ,8 between 0.007 and 0.009 for

a uranium beam according to Figure 4.7.

In this calculation the initial rf phase offsets are fixed to be —10 degrees for 11
type and —15 degrees for 12 type resonators. In the 11 type resonator the transverse
focusing power decreases as injection energy increases in the region of B,- > 0.01
according to Figure 4.7. But if we remember that the accelerating structure of
an 11 type resonator for the initial 5 was 0.0085 in Table 1.1 of Chapter 1, the
region of interest would be B.- < 0.01. In this region the transverse focusing has

a maximum or a minimum point of 3. The maximum transverse focusing power

66

appears at 3 x 0.0085 for 40Ar12+ and at H m 0.006 for 238U244“. Note that at these
points longitudinal focusing powers are approximately maxima. The structure of the

longitudinal case is similar to the transverse case.

In an 12 type resonator which has an accelerating structure of )6 = 0.015 the region
of interest would be 0.01 < 6 < 0.02. The characteristics of the overall focusing
power is very similar to the 11 type case. Unlike 11 type resonators the transverse
and longitudinal focusing power have maximum or minimum points of B with the

exception that the extreme points do not coincide approximately.

In conclusion the transverse focusing power in the low-beta range of 6 < 0.02
has a broad range from a strong focusing action to a rather weak defocusing action.
Therefore the injection energy should be taken appropriately to optimize transverse

focusing power. This will be discussed in the following section.

4.5.2 RF Phase Offset

The initial rf phase offset was already limited by transverse and longitudinal stability
conditions [Promé 70], [N ozomu 90]. It was restricted to be negative for longitudinal
stability. In this calculation for the relation between the focusing power and the
rf phase offset the initial beta of H = 0.0085 for 11 type resonator was used. The
final value of beta after passing the 11 resonator is used as an initial beta of 12 type

resonator.

Figure 4.9 and Figure 4.10 show how focusing powers depend upon the initial rf
phase offset. Note that the longitudinal stability is always satisfied for a negative rf
phase offset, but this is not true for the transverse case any more. If we consider
the efficiency of energy gain from the transit time factor and the nonlinearity of

longitudinal phase space near zero rf phase offset the region of interest would be

67

between —-30 and —10 degrees. In this range the transverse and longitudinal focusing
powers are inversely proportional to each other. i.e strong longitudinal focusing means

weak transverse focusing or defocusing etc.

Note that in a certain area of interest of initial rf phase offset the transverse stabil-
ity condition is not satisfied at the given injection energies for a 238U24+ beam. This
implies that the injection energy was not matched well to give transverse focusing.

This will be discussed in the following section.

In conclusion the transverse focusing powers increase as the initial rf phase offset
goes to zero while longitudinal focusing power decreases both for 11 and I2 type

resonators.

4.6 Initial Conditions for Focusing

In this section we investigate the possible initial conditions of injection energy and / or
rf phase offset for resonator focusing without a transverse focusing element. Since
transverse focusing power is very small for 6 > 0.02 in an 12 type resonator, it might

be enough to consider 11 and 12 type resonators only for transverse focusing.

First we consider the following rf settings for both 11 and 12 type resonators with

different initial betas at the entrance of the P11 linac:
11 type: E0 : 4.5 (MV/m) , (to : —10°
12 type: E” = 3.0 (MV/m) , 43” = —15°

Figure 4.11 and Figure 4.12 are plots of initial beta 6,- vs. final beta ,6, for 4UArl2+
and 238U24+ beams. The new initial beta at the 12 type resonator will be found from

the final beta of 11 type resonator.

First consider an ‘mAr12+ beam. An 11 type resonator has a wide range of initial

68

betas for focusing according to Figure 4.7. But the new initial beta at 12 resonator
between 0.01 and 0.015 can not be used due to transverse defocusing if the 12 resonator
is to be used as a focusing lens also. Therefore the range of the initial beta at
11 resonator which gives a final beta between 0.01 and 0.015 should be excluded.
Fortunately this condition is satisfied because the final beta for the 11 resonator is
always greater than 0.015 for an initial B > 0.005. Therefore an initial beta of 0.0085
would be a reasonable initial condition for an 4OAr12+ beam. But the transverse
focusing power at the 12 resonator decreases considerably compared with that at an

11 resonator without significant difference in the longitudinal focusing power.

Next, consider a 238U2‘1+ beam. In this case the 11 resonator has a relatively small
range of its initial beta of B < 0.007. The new initial beta at the 12 resonator should
be limited to below 0.012 for transverse focusing. Therefore the range of possible
initial betas at 11 resonator would be below 0.0075 from Figure 4.11. For example
we take 3 = 0.0063 for this calculation. Note that an initial beta of 0.0085 at the
11 resonator is not possible with an rf phase offset of -10 degrees. Because according
to Figure 4.9, the rf phase offset should be limited between -8 and 0 degrees but
in this range the focusing power is very low and also the 12 resonator does not give

transverse focusing-in Figure 4.8 and Figure 4.11.

In the case of 238U24+ we have another possibility to increase focusing power by
adding another 11 type resonator after the first 11 resonator. This possibility comes
from the relatively broad stability region of the 11 type resonator in Figure 4.6. For
example we take )6,- : 0.0063 at the 11 resonator then, after passing through the 11
resonator, the final beta would be )6, = 0.0137 at the second 11 type resonator. This

gives another transverse focusing to increase the net focusing power.

69

4.7 Conclusions

The 11 and 12 type resonators which have four-gap cylindrically symmetric accelerat-
ing structure behave as focusing lenses both for the transverse and the longitudinal
directions even without alternating the sign of the rf phase offset to give effective

alternating phase focusing.

The stability limit of the rf phase offset for transverse and longitudinal stability
might be well optimized in the range between —30 and —10 degrees if the incident
velocities are well chosen. Transverse and longitudinal focusing power are inversely

proportional to each other in this range.

The maximum limit of injection energy for transverse focusing would be 0.186
(MeV / A) or a beta of 0.02 because the transverse focusing power would be negligible
beyond this range. The range of injection energy for stability might be limited to a
low beta region less than 26 (keV/ A) or H < 0.0075 for a very heavy ion beam of
low charge to mass ratio like 238U2”. Also, if two 11 type resonators are used, the

injection energy for 238U2‘H beam would be lowered up to ,3 = 0.0065 or even less.

Since the tested beams were q/m=0.1 for 238sz+ and 0.3 for 40Ar12+ it is expected
that the wide range of heavy ions would be focused with 11 and 12 type resonators

only at the beginning stage of accelerations.

70

Table 4.1: Focusing powers of each accelerating gap of 11 and 12 type resonators
for “Arr“ with the rf settings and initial velocities shown in Figure 4.3 (11) and
Figure 4.4 (12).

 

 

 

 

 

 

 

 

 

Res. Gap Transverse Longitudinal
Type 1/f1(m") 132011") 131011-11 lfiztm")
1 28.902 16.926 -20.187 —19.276
2 7.402 6.602 -13.566 -8.044
11 3 -1.647 -1.510 5.330 3.299
4 -4.465 -4.392 11.401 11.144
1 2.662 2.380 -4.214 -3.527
2 , 0.775 0.733 -1.457 -1.101
12 3 -0.799 -0.765 2.122 1.681
4 -1.704 —1.685 4.063 3.712

 

 

 

71

 

 

 

 

 

 

 

 

P1: 2P2
‘1 j E ‘2
\1 ;/

 

 

 

 

 

 

Figure 4.1: The point-to-parallel system(left) and the parallel-to-point system(right).
P; and P2 are two different principal planes.

72

xor At

ray 1

 

 

 

 

 

 

 

Figure 4.2: The focusing and defocusing actions of a lens. (z’Iz) or -(Aw|At) is
positive for ray 1 to give net defocusing action and negative for ray 2 to give net
focusing action

73

n: E. = 4.5 (LN/m) ¢o=-10‘

“Ar 12. E/A=33.6 (keV/A)
0.0200 1 1 1 l l 1 l l l l l I 1 1 1 1 1 1
0.0175 I .

 

0.0150 ;—
50.0125 ;—
0.0100 5—
00075 :—
0.0050 =

Illlllllllllllllllllll Illlll

 

E.(0)

 

 

 

1111

1/f(m")
O u—t
g

V ......o". ... MAR...

 

 

 

Figure 4.3: The focusing power at each cell of an 11 type resonator for “Art“ beam
with initial value of ,6 = 0.0085 (bottom). The solid line is for transverse direction
and dotted line is for longitudinal direction. The top shows the [3 profile along the
resonator axis. The middle part is an axial electric field profile of the 11 type resonator

which has a four gap structure.

74

12: so = 3.0 (MV/m) ¢0=-15’

“Arm E/A=122.25 (keV/A)

 

 

 

 

 

 

 

 

 

 

0.024 : 1 1 1 1 1 1 1 1 I 1 1 1 1 I 1 1 1 [__i
0.022 _37— —§
50.020 ;— —§
0.018 E— 5
0.016 '
So:
all
0.4 57-. 1 I T l 1 1 I 1 1 1 1 I 1 1 1 [__E
78 0.2 E— ....fi... 4
b 0.0 : A 5 ”a: 3
—0.2 E- . -—2
—0.4 E— __E
- I 1 l l I J l 1 l l J l 1 J l I l 1 I
0 50 100 150 200

Figure 4.4: The focusing power at each cell of an 12 type resonator for ""Ar'2Jr beam
with initial value of [3 = 0.0164 (bottom). The solid line is for transverse direction
and dotted line is for longitudinal direction. The top shows the ['3 profile along the
resonator axis. The middle part is an axial electric field profile of the 12 type resonator

which has a four gap structure.

75

 

 

 

 

 

 

 

 

 

 

 

 

40
so ........
20 '''''
11: so = 4.5 (NV/m) “Arm
0
—20
-40
0.005 0.0075 0.01 0.0125 0.015 0.0175 0.02 18.
401111111111111111111111TT_
.. E- '::::..:::::..... l l I l :
20 _'._. “:22: __‘_
— 12: so = 3.0 (NV/m) “Arm 1
O '- _1
-20 _°"“.. .......... --............. -
E ........ .. ........... :
"11;""rl ;°“°;'“';'°"r"+". . . . l . . . . | . . . . l .11 5

 

 

 

-40
0.005 0.0075 0.01 0.0125 0.015 0.0175 0.02 18.

Figure 4.5: The stability region of the rf phase offset of 11 (up) and 12 type (down)
resonators as a function of initial 6 for 4”Arr“ beam.

 

76

40
*o

20

11: E, = 4.5 (LN/m) am’02“

0

-20

 

-40
0.005 0.0075 0.01 0.0125 0.015 0.0175 0.02 B:

40
1’0

20

12: so = 3.0 (“V/m) ””0”“

O

 

—20

 

 

 

-40
0.005 0.0075 0.01 0.0125 0.015 0.0175 0.02 31

Figure 4.6: The stability region of the rf phase offset of 11 (up) and 12 type (down)
resonators as a function of initial ,6 for “8U?” beam.

77

11: so = 4.5 (IN/m) so = -10°

”Arm" (solid). ”'0’“ (dotted)

 

 

 

 

 

 

 

 

 

5 I 1 I I I I I I I I I I I I I I I I I I
A I—
1 - 2
3 I I I I I :
0 _— ....................... -
E 5
: l E
_20 " 1 1 1 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1 1 ‘
0.005 0.01 0.015 0.02 0.025 0.03 131
0.0 ‘ .I I I I l I I I I I I I I ..L I I r T I I I I I .1
’3 1 ..--"'° I "' --------- I
E ..................... ...;
3 ‘2
I E
i
3
J
l 1 1 l 1 I 1 1 l 1 1 1 1 l 1 l 1 1 l 1 1 1 1

 

 

.0 '
0.005 0.01 0.015 0.02 0.025 0.03 131

Figure 4.7: The plots of (z’lz) and —(Aw|At) vs. initial beta [3.- for an Il'type
resonator. The solid line is for "0A9” and the dotted line is for 238U“"'.

78

12: s. ... 3.0 (IN/m) 41., = -15’

“Arm (solid). ”‘0” (dotted)

 

         

 

 

 

 

 

 

 

 

 

 

A 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r
.33. 3 I I I I j
x . “"°°-'---......_
V O _ .. -
-5 I —I
—10 Z— —I
-15 :— —I
_20 :'.I. l l I I 1 l l l I l I l l I l l l 1 I l l 1 l :
0.005 0.01 0.015 0.02 0.025 0.03 131
0.50 ‘ I I I l I I 1 I l I I I I 1 I
30.25
E
10.00
-0.25 .
-0.50 . ° ....................
—0.75 ' .............................
_1.00 l I I 1 l I l I l l I l I l 1 J 1 1 l l 7

 

0.005 0.01 0.015 0.02 0.025 0.03 18:

Figure 4.8: The plots of (z’lz) and —(Aw|At) vs. initial beta £1,- for an 12 type
resonator. The solid line is for "UAR“ and the dotted line is for 238U2”.

79

u: so = 4.5 (NV/m) p.=0.0005

”Arm (solid). “”0"“ (dashed)

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

A 10 1(1 1 1 111 1 1 1 1 1 1 11 1 1 11 1 1 11..
x \ ._
z _:_ L \ \ I I I I _5
_5 E. _:
40:—
‘152— '1
_20:I I I II I I I I I I I I I I I I I I I I I I I I I I I I
-40 —3O —20 —10 O 10 20 950
2 _1 1 11 I 1 1 1 1 l 1 I 1 1 I 1 11 1 [11 1 I 1 11 _
3 1 —— / J
I E / I
.. / -
0 ’ , ,1
: / I
_1 —— —— -— '— " ...:
_2 J J I I I I J I I I I l I I I I I I I I I I I I I I I Id
—40 —30 —20 —10 0 10 20 960

Figure 4.9: The plots of (z’lz) and —(Aw|At) vs. rf phase ofl‘set'oiu for an 11 type
resonator. The solid line is for 4UAr12+ and the dashed line is for 2”Uz’”.

80

12: so = 3.0 (IN/m) p,=0.0129

“Arm (solid). ”'0‘“ (dashed)

 

  
   

 

 
 
 
 

 

 

 

 

   

 

 

 

 

 

A 10 ‘ 1 111 1111 I11 1 1111 1 111 111
x ..
E I I I I 3
5 —Z
0 : 1 ‘ “ ~ -— ~ _ j
—5 :— —:
__10:I I 14I I I I I I I I I I I I I I I I I I LI I I I I I-
—40 —30 —20 -10 0 10 20 ¢o
2 I I I I I I I I I I I I I I I II I I I I I I II_
a; : I I I I -
E IE- .2
I _ I
0 Z 3
_2: II I I J I I I I4 I I I I I I I I I I I I I I I I I Id
-40 -30 —20 —10 0 10 20 9%

Figure 4.10: The plots of (z’lz) and —(Aw|At) vs. rf phase offset qbu for an 12 type
resonator. The solid line is for 40Ar12+ with ,8.- = 0.0164 and the dashed line is for

238v?” with 3.- = 0.0164.

fit

81

11: so = 4.5(IIV/m) ¢o=-10°

”Arm’ (solid). “U2“ (dashed)

 

0.0200 11l1111l111111111l1111l1

 

 

 

 

0.0175 _— //_:
0.0150 _— , , ’ _2
0.0125 -_—— / , ’ ’ _‘
- / I
I / I
0.0100 _— / ...
I / I
- / I
0.0075 :7 __
It 2
0.0050 '- I I I l I I I I I I I I I I I I I I I I I I I I-
0.006 0.008 0.01 0.012 0.014 I3i

Figure 4.11: The plot for initial beta E,- vs. final beta [3, of an 11 type resonator. The
solid line is for “Arm’ and the dashed line for 2"WU“+ beam.

82

12: E. = 3.0 (MV/rn) ¢o=-15‘

“Arm (solid). amug“ (dashed)

 

0.05 1 1 1 1 1 1 1 1 1 1 1 1 1
fl. 1 1 [ T 1 I l 1 1

0.04

0.02

—
_
h-
p
b
003 ~—
. _—
—
_
-
h
—
h
-
_
_

0.01 / '—

1
\
I

 

 

 

0.01 0.02 0.03 0.04 0.05 3.

Figure 4.12: The plot for initial beta [3.- vs. final beta [3, of 12 type resonator.‘ The
solid line is for "(’Ar12+ and dashed line for 238U24+ beam.

83

11: so = 4.5 (IN/m). so = -10
12: so = 3.0 (IN/m). oo = —15

”Arm. po=0.0085 (up)

“‘11”. p,=0.0063 (down)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10
’3‘ 11 12
N ’— __d/
0
¥ fi\
-5 \—————\
—10
0 0 2 0 4 0.6 0 8 1 2(m)
10
’g‘ 11 12
V 5
N /
0 5%?”
-1O
0 0.2 0.4 0.6 0.8 l 20!!)

Figure 4.13: The transverse raytracing plots for "”Ar'“ (up) and 2""‘Uw‘i (down)
beams with parallel rays at the beginning.

84

11: E0 = 4.5 (IV/m). Ibo = -10
12: E0 = 3.0 (“V/m). ¢° = -15

“AI-“3*. po=o.0085 (up)

“u“. p,=0.0063 (down)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o 0.4
5 " /\
b’ 02
0.0
\_,
-02 \M $7
—0.4
0 0.2 0.4 0.6 0.8 1 2(m)
’g‘ 0.4 u
‘2’ 0.2 f\
~’/\‘\
0.0 ><
-0.2 \x—M
_0.4 /

 

 

 

 

 

 

 

0 0.2 0.4 0.6 0.8 1 Z(m)

Figure 4.14: The longitudinal raytracing plots for "(’Ar12+ (up) and 238U24+ (down)
beams with parallel rays (AW = 0) at the beginning.

Chapter 5

Applications to Acceptance Study

5.1 Introduction

The maximum emittance of a beam that a system can accept is called the accep-
tance or the admittance of the system; the particles within this acceptance will
be transmitted without striking the walls. In accelerators which have a periodic
property in motion, the magnitudes of acceptances would be calculated analytically
[Chambers 76],[Edwards 87], [Okamoto 89], [Sagalovsky 92]. Even in a linear accel-
erator we might approximate periodic structure of motion [Bollinger 78] if the length
of a system is long enough to assume periodic motions. But this is no longer the
case in the Positive Ion Injector which has neither periodic structure nor long system

length.

Raytracing calculations by the brute force method have been used as a general way
of calculating transverse and longitudinal acceptances [Game 70]. These calculations
start from a square lattice large enough to incorporate the expected acceptance.
However, this method is too time consuming to apply to a large system. Also, it does

not give quantitative information about the nonlinearity of the system.

In this chapter we introduce a very straight forward way to find the transverse

and longitudinal acceptances of a system by using the first order beam envelope

85

86

functions calculated from the transfer map. We subdifferentiate the terminology
of acceptance into geometrical acceptance and linear acceptance. The geometrical
acceptance [Lejeume 80b] is understood as the acceptance of a. system calculated
from first order phase space transformation theory. This would deviate from the
real acceptance if nonlinearity appears. The linear acceptance is understood as the
acceptance limit which behaves linearly through a system within given deviations

from the higher order distortions of a phase space.

One of the interesting things in beam dynamics might be to quantify the nonlin-
earity of a beam. It has been reported in several papers [Lysenko 88], [Dragt 91] that
some moment invariants for symplectic systems exist. These invariants are formed
with some linear combinations of the average of the monomials of the phase space
coordinates. The physical meaning of them are not clear at this time [Pusch 93]
except for a very simple kind of invariant called generalized rms emittance. But
these invariants might be used to quantify the degree of nonlinear motion as Alex J.
Dragt pointed out [Dragt 91]. In this chapter we also present a quantitative measure
of nonlinearities using the concept of rms emittances and deviations of phase space

coordinates between linear and nonlinear transformations.

5.2 Geometrical Acceptance

The geometrical acceptance of a system is defined as the acceptance calculated from
linear phase space transformation. This is larger than the useful acceptance if non-

linearities are present.

87
5.2.1 Transverse Acceptance
The beam envelope at a position in a system with acceleration was derived for an

axially symmetric beam in equation (2.49) of Chapter 2. In n-dimensional phase

space:

 

2:2 2 \/(det M)2/"figel . (5.1)

where det M is the determinant of the transfer matrix and ,82, 61 are the beta-function

at the position 2 and the initial beam emittance in one projected plane, respectively.

The beta-function at point 2 is found if the initial Twiss parameters of the phase
space and the transfer map are known. Equation (5.1) is still valid for a nonaxially

symmetric beam if n22.

We define:
0* : (det M)2/"fl, (5.2)

so that ,6’ is the effective beta-function.

Then the geometrical transverse acceptance of a system is limited by the apertures

of optical elements and effective beta-function:

GTacc : (Signmin . (5.3)

where R is the aperture of the optical element. The subscript min indicates the

 

minimum value.

In actual calculations the effective beta-functions are searched through optical el-
ements except drift spaces, to find the transverse acceptance at each element which
have different apertures. Then the maximum transverse acceptance will be deter-
mined by the smallest one. The transverse acceptance when calculated this way is

what we call the geometrical transverse acceptance of a system.

88

5.2.2 Longitudinal Acceptance

For longitudinal phase space there is no physical aperture which limits acceptances,
but to maintain the phase focusing aspect of the resonators phase excursions or time-
of-flight deviations must be limited. For example if the rf phase offset is 030 for the
reference particle, then the deviation of rf phase angle should not exceed 450 for any
particle because at zero rf phase nonlinearity would be very important. Hence an

estimate of the longitudinal acceptance is obtained by requiring:

Atgflz
w

Rlong . (5.4)

where At is the deviation of the time-of-flight and w is the angular rf frequency of a

resonator.

Now Rlong is analogous to the radial aperture for the transverse case. With
this concept of aperture for the longitudinal case we can also find the longitudinal
acceptance of a system. Since there is no coupling between transverse and longi-
tudinal phase spaces in the first order calculation the longitudinal phase space is
two-dimensional. Also, longitudinal emittance is constant through the system. One
difference in longitudinal acceptance calculations from the transverse case is that the
apertures of transverse focusing elements, like solenoids, do not affect the longitudinal

acceptance. So we have for an estimate of the longitudinal acceptance:

R: (2)
G" I 13(2) I... (55’

where Rlong was defined above.

The longitudinal acceptance calculated this way is what we call the geometrical
longitudinal acceptance of a system. In practice, we find that the longitudinal accep-
tance defined in this way is a good starting point for determining the linear acceptance

which will be discussed soon.

89

This concept of the aperture for the longitudinal case does not seem to be self-
consistent because there is no limitation on the longitudinal acceptance if a system is
purely linear. But, if we consider a nonlinear system, for longitudinal acceptances we
might find or estimate the stability limits of longitudinal motions without much diffi-
culty. So these limits would be used as the limits of linear deviations of longitudinal

motions .

5.2.3 Fitting of Geometrical Acceptance

The geometrical acceptances can be found numerically by fitting the Twiss parameters
of the phase spaces with an appropriate optimizer. The transverse or longitudinal
acceptance is calculated for an assumed initial set of Twiss parameters. Then the
optimizer varies the Twiss parameters until the maximum values of the geometrical

acceptances are determined.

5.3 Linear Acceptance

5.3.1 Scaling of Nonlinearity by Linear Transformation

The nonlinearity of a system was not considered in the geometrical acceptance cal-
culations described above. In practice only a fraction of the geometrical acceptance
transforms linearly. We define the linear acceptance of a system as an acceptance
which is transformed linearly within given deviation limits from the result of linear

and nonlinear transformations.

To establish a quantitative measure of the degree of nonlinearity, a new quantity,

called the deviation of the particle i, is defined:

.1... IzI—sz

(5.6)

31111

90

where subscript 1,17. indicate linear and nonlinear transformations of the particle 1',
respectively, and an," means the maximum value of the coordinate a: in the linear

transformations of the geometrical acceptance.

We introduce a method to find linear acceptances by comparing linear and non-
linear transformations. As an example, let us find the transverse linear acceptance

with dev. g 1%:

1. Find the geometrical acceptance using an optimizer.

2. Find maximum values of position 23m and divergence 23],, from the linear trans-

formation of the initial geometrical acceptance ellipse.

3. Find deviations of the phase space coordinates of individual rays with coordi-
nates chosen on a grid bounded by the. geometrical phase space ellipse from

linear and nonlinear transformations.

4. Discard all initial rays for which one of the deviations is greater than 1% to get

the linear acceptance from the remaining rays.

Note that this deviation method gives the area and the orientation of a phase space
ellipse. The degree of nonlinearity is controlled through the choice of the magnitude

of the dev. parameter.

5.3.2 Scaling of Nonlinearity by RMS Emittance

It is well known that the normalized emittance [Billen 75], [Lejeume 80a] is conserved
even in an accelerating. system. But because of distortions of the phase spaces sub-
jected to nonlinear forces the effective area of phase space tends to increase. So
to avoid this weakness we consider the transformations of their effective area called

root-mean-square (rms) emittance. The concept of rms emittance has been used to

91

measure the effective area of a distorted phase space ellipse subjected to nonlinear

forces [Lapostolle 71], [Lawson 73], [Lejeume 80a].

The rms emittance is defined for a pair of phase space coordinates a: and 13’ by

[Lapostolle 71], [Sacherer 71]:

 

em, : 4\/< m2 >< 3’2 > — < 23’ >2. (5.7)

This rms emittance reverts to the emittance of the phase space ellipse if the phase

space ellipse is not distorted.

If a system is linear the coordinates will transform by the transfer map M of the

system:

I
332 = M11$1+M12311

23,2 2 M21131 ‘1" M2213; . (5.8)
Using equations (5.7) and (5.8) we get the rms emittance transformation [Dragt 91]:
6rms2 : (det M) ' 6rmsl. - (59)

The rms emittance defined above transforms in the same way as the two-dimensional
phase space transformation derived in equation (2.21) of Chapter 2. Thus (det M)“l cm,
is a constant of the motion. We use it to modify the usual definition of normalized

rms emittance:

6mm, 2 (det M)—l - com, . (5.10)

Equation (5.10) would yield the common definition of the normalized rms emittance
for the transverse case if cm, were multiplied by the initial value of the relativistic
factor ['17. Because of the invariance of the normalized rms emittance under linear
transformations it might be used as a measure of the nonlinearity of a system by

comparing its values between linear and nonlinear transformations.

92

The invariant result of the normalized rms emittance is true only when no first
order couplings exist with the other pairs of phase space coordinates. The general
invariants theorems for first order couplings have been developed by several authors
only for symplectic systems as pointed out in the introduction of this chapter. But
in our case the first order couplings do not appear or can be decoupled by rotating

the coordinate system.

5.4 Application to an Alternating-Phase-Focusing
System

It was shown in Chapter 3 that transverse focusings by multi-gap resonators were
possible without additional focusing elements because of their inherent alternating-
phase-focusing properties. As a first application we investigate geometrical and linear
acceptances of such a system consisting of rf resonators and drift spaces. Since the
focusing power of the P11 resonators for the transverse direction may be negligible
after considerable energy gains we consider just three successiVe resonators at the
beginning of the P11 linac which was shown in Figure 1.2 of Chapter 1. This system

configuration is also shown schematically in Figure 5.2.

5.4.1 Geometrical Acceptances on Incident Velocities

Figure 5.1 shows geometrical transverse and longitudinal acceptances on incoming

velocities. These acceptances were calculated from the first order matrix optics code

MINJI [Joh 93b].

For these acceptance calculations "”Ar'2+ (solid) and ”RUN+ (solid+dots) were
used. The rf settings for the electric field gradients E0 and the initial rf phase offset

¢0 for the resonators are:

93
11: E0 = 4.5 (MV/m), d)” = —10°
12-1: EU 2 3.0 (MV/m), 050 : —15°
12-2: E0 = 3.0 (MV/m), (120 = —15°

We assumed that all three resonators had the same half apertures of 8 mm of
a type 11 resonator which has the smallest diameter. Also, we assumed that the
magnitude of the initial rf phase offset, 10 degrees for a type 11 resonator, would

correspond to a radius of longitudinal case.

Figure 5.2, as an example, shows geometrical transverse (upper) and longitudinal
(lower) beam envelopes starting with corresponding acceptances calculated from the
matrix optic code MINJI. For a 23'i‘Uz'” beam with an incident velocity 0.007c, the
calculated geometrical acceptances were 140.49 mm-mr for the transverse direction
and 51.07 keV-nsec for the longitudinal direction. The deviations of time of flight
were used instead of rf phase angles to include drift spaces for longitudinal phase
space. The radius of 10 degrees for longitudinal wall corresponds to 0.57 nsec in the

deviation of time of flight for each resonator which has an rf frequency 48.5 MHz.

According to Figure 5.1 the incident velocities of ion beams for this alternating-
phase-focusing system should be compromised between the transverse and the longitu-
dinal acceptance; the low velocity region would be limited by longitudinal acceptances
and the high velocity region would be limited by transverse acceptances depending

on the beam qualities from the ECR ion source.

5.4.2 Estimation of the Linear Acceptances

It would be useful to measure the nonlinearities of a system by comparing normalized
rms emittances between linear and nonlinear transformations. In these calculations

the linear transformations of individual rays were calculated from the first order trans-

94

fer map and nonlinear transformations were calculated from raytracing calculations.
But basically if a higher order nonlinear map is available it would be used for nonlinear

transformations to avoid time-consuming raytracing calculations.

Figure 5.3 shows the linear acceptance of 238U24+ beam at dev. g 1% with
incident velocity 0.007c. The large ellipse in the top left plot is the geometrical
transverse acceptance which has a phase space area 140.4911 mm-mr. This large
ellipse transforms under acceleration to have a maximum beam radius of 8 mm and a
beam divergence of 10.9 mr with pure linear transformations at the end of the system.
To select a linear region from the geometrical acceptance, as an example here with
dev. _<_ 1% from the nonlinear transformations, only the initial rays are kept which

satisfy certain deviation conditions:

]1‘f2 - 1‘“ ] < 0.01 X 8 = 0.08 (mm),

I r32 -— r}, I < 0.01 x 10.9 = 0.109 (mr) .

where subscript fl and f2 means linear and nonlinear transformations of a ray at the

end of the system.

The rays inside a small phase space ellipse are the linear region of the geometrical
acceptance with deviation 1%. This gives the transverse linear acceptance of 4.01
mm-mr (area 4.0111 mm-mr). Since it is not necessary that the linear region be an

exact ellipse we will use rms emittance to define the linear phase space.

The two plots in the lower part of Figure 5.3 show the transformations of the
linear phase space under linear (left) and nonlinear (right) transformations. The
deviations of individual rays with dev.: 1% does not appear clearly in the nonlinear

transformations. But some deviations appear with dev.: 5% as shown in Figure 5.5.

We assumed in the nonlinear transformations that longitudinal phase space was

zero at the beginning. It is noticeable that there is relatively large growth of the

95

longitudinal phase space due to coupling between transverse and longitudinal phase

spaces as shown in the top right plot.

In the same manner as the transverse case, Figure 5.5 shows longitudinal phase
space plots on axis. We assumed in the nonlinear transformations that the transverse
phase space was zero at the beginning. In this case the growth of a new transverse

phase space does not appear.

5.5 Nonlinearity of a System

We found linear regions of phase space with certain deviation limits between linear
and nonlinear transformations. So we might use the relative deviations of the two

transformations as a measure of nonlinearity of a system.

In this section we investigate the relationship between the scaling of nonlinearity
by linear transformation and by rms emittance discussed in section 5.3.2 of this chap-
ter. We calculate the normalized rms emittance enrms, which is the rms emittance

divided by the determinant of the transfer map, by:
15mm, : (det M)"l 'Erm, . (5.11)

Since this is invariant under acceleration we define two kinds of nonlinearity by nor-
malized emittances:

I I

 

 

N1 E Enrrtns — EnrmsU , (5.12)
EnrmsO ' p0
61.! __ ELI
_ nrms nrmsn

nrmst)

where the superscripts l,t indicate longitudinal and transverse phase spaces respec-
tively, and efiimw means normalized transverse (t) or longitudinal (l) emittance at the

beginning of a system.

96

Since the area of longitudinal phase space has dimensions of angular momentum
we multiply the initial momentum of a beam 19., by the transverse phase space in
equation (5.12) of which area has dimension of mm-mr. Therefore the nonlinearities

defined in the above equations are dimensionless.

The first kind of nonlinearity N1 might be interpreted as the degree of coupling
between longitudinal and transverse phase spaces. i.e. the longitudinal emittance
growth per unit transverse phase space area in angular momentum units. N2 might
be understood as the relative growth rate of the normalized rms emittance due to

nonlinear forces.

The upper part of Figure 5.6 shows linear transverse acceptances on different
linear scales (solid). For this calculation we assumed the geometrical transverse ac-
ceptance to be 140.49 mm-mr at the beginning without longitudinal phase space.
The linear acceptances increase linearly about the linear scalings or deviations, but
this will converge to the geometrical acceptance at a certain limit. The growth of

longitudinal phase space (dotted) increase linearly and fast.

The lower part of Figure 5.6 was calculated from linear and nonlinear transfor-
mations of two dimensional transverse phase space without any deviations of energy
and time of flight at the beginning. It shows the relationships between two kinds of
nonlinearities N1,N2 with respect to the linear scaling. The coupling between trans-
verse and longitudinal phase spaces increases gradually (dotted) while the relative
growth of normalized rms emittance tends to decreases (solid). In this plot negative
growth rate means compression or the decrease of normalized rms emittance relative

to the initial values.

The new creation of the longitudinal phase space does not violate Liouville’s the-

orem [Lichtenberg 69], because in the coupled 2n-dimensional phase space only 2n-

97

dimensional phase space volume is conserved.

Figure 5.7 shows the same kind of plots as in the transverse case for the longi-
tudinal phase space. The initial geometrical acceptance was 51.07 keV-nsec without
transverse acceptance. The relative growth rate of longitudinal phase space increase

faster than in the transverse case.

5.6 Application to the PII Linac

In this section we investigate geometrical and linear acceptances of the beginning
part of the P11 linac with superconducting solenoids as focusing elements. For this

calculation we use the model solenoid discussed in Chapter 3.

We also consider just three successive resonators with solenoids between them at
the beginning of the P11 linac which was shown in Figure 1.2 of Chapter 1. This

system configuration is also shown schematically in Figure 5.8.

5.6.1 Incident Velocity on Geometrical Acceptances

If transverse focusing elements like magnetic solenoids are available, the incident
velocities need not be limited, at least for transverse focusings. Since longitudinal
acceptance increases for higher velocities the optimized velocity for maximum energy
gain might be used as an incident velocity. For example 0.0085c will be used instead
of 0.0070c as an initial velocity of 238U2”. This increased initial velocity increases

geometrical longitudinal acceptance more than two times relative to lower velocity.

5.6.2 Fitting of Solenoid Strength

The basic concept of the beam optics of the P11 linac was to make beam waists at

the center of each rf resonator to minimize nonlinearities due to the off axial fields

98

of the resonators. But since beam waist points depend upon the initial shape of the
phase space ellipse it is not easy to find the initial shape of a phase space for the
geometrical transverse acceptances and the magnetic field strength of solenoids. But
as an approximation for the beam waist condition we use point-to-point conditions
between the centers of resonators. This corresponds to the paraxial image conditions
in ordinary optics and gives a good approximation for the beam waist if the beam size
is not so big. This condition is useful because it does not depend on the initial shape
of a phase space but on the transfer map of two corresponding points. The deviations
from the waist points might be tuned for beam waists without changing the results of
transverse acceptance calculations. The result of fitting the solenoids field strengths
using the matrix optics code is shown in Figure 5.8. The beam waist points exist

approximately at the centers of each resonator with fitted maximum magnetic field

strengths of 4.24 for the lst, 7.08 for the 2nd , and 7.17 Tesla for the 3rd solenoid.

5.6.3 The Selection of Nonlinear Optical Elements

Figure 5.9 shows the transformations of the linear phase space enclosed by the small
ellipse within the large ellipse with dev. S 1%. Here the difference between an
alternating phase focusing system i.e. transverse focusing by resonators only and
a solenoid focusing system is the orientation of linear phase space ellipse. In the
solenoid focusing system the orientation of linear phase ellipse is quite different from

that of geometrical phase ellipse.

To understand this the effective beta-functions for the two different orientations
were plotted in Figure 5.13. Figure 5.13 is the effective beta-function plot through
the each element calculated with Twiss parameters from the geometrical acceptance
(dotted) and linear acceptances (solid) with dev. S 1%. Overall, for the transverse

direction, the beam envelope with Twiss parameters from the geometrical acceptance

99

looks much better than in the linear acceptance case. But the geometrical case is

much worse than the linear case for nonlinearities.

Qualitatively, these curves indicate that nonlinearities are minimized by keeping
the effective beta-function small in 11 while allowing it to be larger in 81. This might
be a very useful concept in identifying the nonlinear elements and to minimize their

effects.

5.7 Analysis of the Two Focusing Systems

The one advantage of the solenoid focusing system over the resonator focusing system
is that the large transverse acceptance is independent of the incident velocity. So the
incident velocity would be adjusted for longitudinal acceptance. On the other hand
in the resonator focusing system the incident velocity is limited by both transverse
and longitudinal acceptances: the low velocity gives large transverse acceptance but

small longitudinal acceptance in general from Figure 5.1.

Here we analyze the two focusing systems with two different incoming velocities:
0.0070c for the resonator focusing system of which the initial velocity was chosen by
compromising between transverse and longitudinal acceptances; 0.0085c is used for
the solenoid focusing system of which initial velocity is well matched to the P11 linac

with the uranium beam.

The big difference in the two systems comes from the linear transverse acceptance:
The geometrical transverse acceptances of the solenoid focusing system were about
two times that of the resonator focusing system both for transverse and longitudinal
acceptances. But the linear transverse acceptances of the solenoid focusing system

are much more than two times at the same deviations as seen in the upper plots of

Figure 5.6 and Figure 5.11.

100

The explanation might be that most of the nonlinearities against the transverse
phase spaces come from the resonators rather than the focusing solenoids. The lower
plots of Figure 5.6 and Figure 5.115how quantitatively that the solenoid focusing

system is more linear than the resonator focusing system.

But even the growth rate of the longitudinal phase space per linear transverse
acceptance (N1) in the solenoid focusing system is better than in the resonator fo-
cusing system, the absolute values of the growth of the longitudinal phase space is
much larger than the resonator focusing system because of larger linear transverse
acceptances. Therefore the actual linear transverse acceptance will be limited by the

absolute values of the growth of the longitudinal phase space.

Figure 5.3 and Figure 5.4 show transformations of the linear transverse accep-
tance in the resonator focusing system with deviations of 1% and 5% respectively.
Figure 5.9 shows the transformation of the linear transverse acceptance in the solenoid
focusing system with deviations of 1%. In the 5% deviation of the resonator focus-
ing system the nonlinearity clearly appears in the substructure of the plot with the

creation of longitudinal phase space.

On the other hand for the longitudinal linear acceptances on axes in the two sys-
tems, there is no significant differences in the nonlinearities from Figure 5.7 and
Figure 5.12. This is mainly due to the fact that the growth of the transverse phase
space does not appear from the longitudinal acceptances. The geometrical longitu-
dinal acceptance of the solenoid focusing system was about two times that of the
resonator focusing system. This ratio does not change so much in the linear accep-

tances from the upper plots of Figure 5.7 and Figure 5.12.

Figure 5.5 and Figure 5.10 show transformations of the linear longitudinal accep-

tances on axes in the resonator focusing and the solenoid focusing systems respectively

101

with deviations of 5%. The new creations of transverse phase space do not appear.

5.8 Conclusions

The scaling of nonlinearity was defined from deviations of linear and nonlinear trans-
formations (dev. ). It has complementary relations with the scaling of nonlinearity
by N1,N2 defined from rms emittances. For example if we require a relative nor-
malized rms deviation of 10 % (N2 = 0.1) at the end of the system for longitudinal
phase space we could correlate this scaling by rrns emittances with the scaling by the
dev. . The N2 = 0.1 corresponds approximately to dev. 20.1 in the lower part of
Figure 5.7. So by using dev. 20.1 we can find the corresponding linear acceptance
from the geometrical longitudinal acceptance. Therefore this will quantitatively give
detailed information about the matching conditions of the initial phase space ellipses

that will be linearly transformed.

The calculated longitudinal and transverse acceptances are going to be enough
to include acceptances of beams from the ECR ion sources at ATLAS [Bollinger 92]
even for the heaviest beam like uranium with very low velocities less than 0.008c. But
nonlinearities in the resonator focusing system were relatively large compared with

solenoid focusing system due to the strong nonlinearities of the resonators.

One thing to note is the relatively fast increasing of longitudinal phase space
coming from the phase space coupling, so longitudinal beam dynamics would be
more important than transverse ones to consider nonlinearities of this kind of phase-

alternating-focusing system with very low ion velocities.

In the calculations we did not consider the couplings of transverse and longitudinal
phase coordinates at the beginning. If higher order transfer maps are known for a

given system it might be done easily and fast to study the nonlinearity of a system

102

starting from real coupled phase spaces between the transverse and the longitudinal

directions without time-consuming raytracing type calculations.

103

Table 5.1: Focusing structures of the resonator focusing system for “0 Ar12+ beam with
incident velocity 0.0085c.

 

 

 

 

 

 

 

 

 

 

Res. Gap Transverse Longitudinal
Type ”Mm—I) ”Mm-II l/fllm-T) l/fzfm—II
1 28.902 16.926 -20.187 -19.276
2 7.402 6.602 -13.566 —8.044
11
3 -1.647 -1.510 5.330 3.299
4 -4.465 -4.392 11.401 11.144
1 2.662 2.380 -4.214 -3.527
2 0.775 0.733 -1.457 -1.101
12-1
3 -0.799 -0.765 2.122 1.681
4 -1.704 -1.685 4.063 3.712
1 1.959 1.881 -3.171 -3.286
2 0.824 0.800 -1.681 -1.403
12-2
3 -0.659 -0.642 1.617 1.375
4 -1.202 -1.210 2.603 2.655

 

 

 

104

Table 5.2: Focusing structures of the resonator focusing system for 238U24+ beam with
incident velocity 0.0070c.

 

 

 

 

 

 

 

Res. Gap Transverse ‘ Longitudinal
Type 1/f11m-I) 1/f2(m_1) ”Mm-II ”Hm-r)
1 -5.994 -5.562 23.073 13.219
2 1.373 1.218 -2.471 -1.831
11
3 1.506 1.389 ~2.841 -2.172
4 -0.044 -0.041 0.449 0.353
1 -2.911 -2.988 8.847 7.112
2 -0.361 -0.340 0.912 0.782
12-1
3 1.428 1.369 -2.776 -2.463
4 1.999 1.948 -3.780 -3.588
1 -0.722 -0.704 1.703 1.452
2 -0.261 -0.252 0.620 0.546
12-2
3 -0.185 —0.l79 0.455 0.404
4 -0.309 -0.301 0.753 0.672

 

 

 

 

 

 

 

105

11: 4.6W/m-10 m-1:3.0.-15 12-2:3.0.-15
“Ara" (solid). NU.“ (solid+dots)

 

300
250
200
150
100

50

-
—
q
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—-1
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q
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—-1
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IIIIIIIIIIIIII

  
 
  

I...

GTacc(rmnernr)

 

 

 

I l I I I I I I I I I I I J I I I I I

0
0.006 0.007 0.008 0.009 0.

 

o IJII IIII

151

 

:1:5() 1 1 1' 1 1 1 1 1 1 1 17 1
I I l

125
100

Gucc(keV.nsec)
P}.
l1111'1111]1111]1111l1

I IIIIIIIIIIIIIIIIIIIIIII

 

 

 

 

() I I I I I I I I I l I I I I I I I
0.006 0.007 0.008 0.009 0.

h

C III
p.
3

Figure 5.1: The geometrical transverse (upper) and longitudinal (lower) acceptances
as 3. functions of incident velocities.

106

“U2“. p,=0.0070 (22.82 keV/A)

II: 4.5KV/mo-10 12- l:3.0.- 15 12-2:3.0.- 15
G'I'acc.= 140.49 (mmomr)
Glace. =5 1.0? (keVonsec)

 

15.0 111r1111ll111l1111|1111111

12.5
A 10.0
7.5
5.0
2.5
0.0

I—
u-
-

r(mm

IIIIIIIIIIIIIIIIIIIIIIIIII

 

 

 

1111

_

IIIIIIIIIIIIIILJIIIIIIIIIIII

0 0.25 0.5 0.75 1 1.25 1.5 2(m)

 

 

1.01111'1111III11'1111l111ll111
0.8

11 ‘2" 12-2
0.0 I I I I I I I I J I I I I I I I I I I I I I I I I I I I
O 0.25 0.5 0.75 1 1.25 1.5 z(m)

0.6

At(nsec)

0.4
0.2

IIII

 

 

1111]1111l1111l1111l111

 

Figure 5.2: The beam envelopes of beam radius (upper) and deviation of time of
flight (lower) starting from maximum emittance.

 

107

”u“. p.=0.0070

11: 4.5MV/m.-IO 12-lz3.0.-15 12-2:3.0.-15
Mace-4.01 (mmomr). dev. < l (X)
p. 24.488. a, =5.sss

 

 

  

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20 L:IIIII 1111l1r11|1111[_4 50 IIII IIII IIII III:
: r( )-r‘(mr) E 25 E_ At(nsec)-Aw(keV) _i
10 — -— : .. :
' 3 0 E- "ii’ear -E
o I— —Z -25 E— 21" -;
10 : “5° :— “3
2 a -75 :- —;
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I IIII IIII IIII IIII I IIII 1111 IIII IIII I
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0 E- -3 0 E- -€
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—4flIIIIIIIIIIIIIIIIIIII—L: _4ITI-IIIIIIIII1IIIIIIIIIII—I:
-4 -2 0 2 4 -4 -2 O 2 4

Figure 5.3: The phase space transformations of the transverse linear acceptance for
a 238U21+ beam with an incident velocity of 0.007c and with a deviation of 1%.

 

108

”'02“, s,=0.0070

11: 4.5HV/m.-10 12-1:3.0.-15 12-2:3.0.-I5
[trace-10.66 (mmomr). dev. < 5 (X)
3. =2.023. do 28.125

 

 

       
 

 

 

 

 

 

 

20 1 111 1111|1111|1111I'_ 50 1111 1111 1111 111Tfl
..-. _ c :

_. =;:. r( )-r'(mr) _ 25 ~__ At(nsee)-Aw(keV) __-:

- - : - :

10 :- _. Z . '."'.-., z
- 0 :- .213}? —:I

r " : j

0 :— —_ —25 :— "E
: .. —50 :— —.‘.
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- - -75 ..— —:
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-4 -2 0 2 4 -0.2 -0.1 0 0.1 0.2

 

 

IO 111111111111111111 10111111111111111111
l I I I I- I I | I I

b .
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hIIIIIIIIIIIIIIIIIIIIIII bLIIIIIIIIIIIIIIILIJIIII
_10 -10

 

 

 

 

 

 

-4-2024 -4-2024

Figure 5.4: The phase space transformations of the transverse linear acceptance for
a 23302.14- beam with an incident velocity 0.007c with a deviation of 5%.

z“11““, s.=0.0070

11: 4.5HV/m.-10 12-1:3.0.-15 12-2:3.0.-15
[Luce-5.21 (keVonsec). dev. < 5 (X)

p, =s.254:-09. a, =0.ssv

 

 

   
 

 

  

 

 

 

 

 

100 111T IIIIIIIII
b d
0 — -—
F d
7 -1
- -I
3 I
_IOO’IIIIILIII
-1-05 0 05 1
100 _1111]1111]11T1|1111_
: Istordg' 1
50 t— 5 ‘1
i
I- ? -I
b g _
— é _
0 :— 5 —:
" 7'. -I
5 d
3 Z
/
-50:— z {I
_ é -
_IOO'IIII IIIIIIIIIIIILI'I
-1-0.5 O 0.5 1

Figure 5.5: The phase space transformations of the longitudinal linear acceptance for

 

10 111l1111l1111l111_
C r(mm)-r’(mr) I
5 ___. ___:
I I

0 L _-
I—- and

II- c-
5: I
I Z

p— ..

1 I I I -
J1 UII IIII ILJI

 

 

 

-0.1-0.05 0 0.05 0.1

 

 

 

 

IOO IIII 111111111 ITIr
_ ] ] ..

E raytraiflng E

50 _— § ___
o 5— g -—:
-50;— E —:
: § :
_IOOUIIIIIIIIIIIIIIIIIW
-1-0.5 O 0.5 l

a 2"‘8U2H beam with an incident velocity 0.007c.

110

mun“. Bro-007° E/A=22.82(keV/A)

11: 4.5MV/1n.-10 12-1z3.0.- 15 12-2:3.0.- 15
G‘I'aec.- 140.49 (mmomr)

 

r111l1111|11r1I1111l111

  
     

 

 

 

25 dotted: longitudinal (keVensec) .5
E 20 solid: transverse (mmomr) ..... .3
w 15 —:
10 -—'j
5 ....... —f

O I I I I I I I I I I I I I I I L

0 0.02 0.04 0.06 0.08 0.1
dev

0 3 I I I I I I I I I I I I I I I I T I I I I I I I _

Bond: N‘ dotted: N3 .-

 

 

 

 

0 0.02 0.04 0.06 0.08 0.1

Figure 5.6: The relationship between scalings of nonlinearity by linear transformation
and by rms emittance for a 238U24+ beam with an incident velocity 0.007c. The initial
geometrical transverse acceptance was 140.49 mm - mr without longitudinal phase

Space.

lll

amat“, 3,-0.0070 E/A-22.82(keV/A)

ll: 4.5IN/m.-10 [2-1:3.0.-15 lZ-2:3.0.-15
Glace-51.07 (mmomr)

 

100IllrlrllllllIII—TIIIIITTIIIIII

   
 

dotud: nonlinear
solid: linear

em (kthnsec)
O)
O

o...
.e
.e
0'.
..
00.
.e
0'.
...
..
.....
.-

   

 

lllllll lllllll»

 

 

 

 

 

0

0 0.05 0.1 0.15 0.2 0.25 0.3

dev.
0.5 l l l l I l [1 l I l l l l l l I l l I l l l l I l ll ...
0.4 :— —5
N; 0.3 "E— —;
0.2 :— f
0.1 :— -—.‘.
0.0 - l I 1 111] l l l l I ll 11] ll 1 I 1 l1 1 [1‘
0 0.05 0.1 0.15 0.2 0.25 0.3

dev.

Figure 5.7: The relationship between scalings of nonlinearity by linear transformation
and by rms emittance for a 238U24+ beam with an incident velocity 0.007c. The initial
geometrical longitudinal acceptance was 51.07 keV - nsec without transverse phase

space.

 

15.0

12.5
“10.0
7.5
5.0
2.5
0.0

r(mm

1.0
0.8
0.6

At(nsec)

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.p.

o IIIIIIIIIIIIIIIIIIIIIII

0.2
0.0

112

mufifl 350.0085 (33.65 keV/A)

ll: 4.5W/m.—10 12-113.0.-15 [2-2:3.0.-15
81: 4.238841 ('1') 82: 7.075668 83: 7.173162
GTaec.=263.0 (mmemr)

Glace-117.4 (keVonsec)

 

 

 

 

 

 

 

 

I I I I I I I I I I I I I I I I ..

3 s: so so __-'_-

3 II Iz-I Iz-z :

I I

:— _:
: I l I I I I I I I I I I I l I

0 0 5 1 1 5 2
I I I I I I I I I I I I I I I _
3: II 32 Iz-I ss Iz-z :

 

 

2(m)

Figure 5.8: The beam envelopes of beam radius (upper) and deviation of time of
flight (lower) starting with a maximum emittance in the system using solenoids for

the transverse focusing. The incident velocity of the 2”U2” beam was 0.0085c.

 

“u”, p.-o.ooas

113

11:4.5HV/m.-10 12-1:3.0.-15 [2-2:3.0.-15

81: 4.238841 (1') 82: 7.075688 83: 7.173182

LTacc=51.40 (mmomr). dev. < 1 (Z)

a, -0.173. a, - 0.974

 

  

 

    

 

 

 

 

 

 

 

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100 [III [III [TIT T'qu
0 =__ At(nsec)-Av(lreV) 3
‘_IOOE}_ ; —:
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-10 -5 0 5 10

Figure 5.9: The phase space transformations of the transverse linear acceptance in
the system using solenoids for the transverse focusing. The incident velocity of the

238U24+ beam was 0.0085c.

”U”, p.=o.ooss

I1: 4.5W/m.-10 12-1:3.0.-15 12-2:3.0.
[Lace-19.04 (keVonsec). dev. < 5 (x)

5, =2.231:-09. a. =-o.594

 

 

 

 

 

 

 

 

 

 

-15

 

 

 

 

 

 

 

 

200 IIIIIII I III II d 10 IIII IIII llllllllq
I A nsec)-Aw( V) I I «Impala» Z
100:— ‘ —E 5 E— ‘3
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200 _llTlIllllllerIll: 200 :TlTlIllllIllllIlll:
_ lst order _ - raytracing ..

L. \‘ ...: I... _Z
100: \\\ : 100: § :
: \ : : Kx :

0 :- \ "'2 0 :— § —_
5 \\\\ 3 5 It 3
—100E— \\\\ —_ —-IOOE- I ~—-:-
_Zoobljllllllllllllllll _ZOO'IIIIIIIIIIIIIIIIII
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 l

 

Figure 5.10: The phase space transformations of the longitudinal linear acceptance
in the system using solenoids for the transverse focusing. The incident velocrty of the

“MU?” beam was 0.0085c.

115

”"8“; 31'0'0085 E/AB33.65(kOV/A)

11: 4.5IN/m.-10 12-1:3.0.-15 12-2:3.0.-15
sl: 4.238841 (1') .2: 7.075688 83: 7.173162
GTace.=263 (mmsmr)

 

 

 

 

 

 

 

 

3OOIITI rIIIIIrIIIIIIIIIIII:
250 :5:— dotted: longitudinal (keVonsec) _;
200 E—— solid: transverse (mmomr -—':"
~ 150 ;— .............................. ...... ._.._.
100 E— .......................... __=
50 E .......... —::
0’.i..IIII1IIIIIIIIIIIIIIIILIm
0 0 02 0.04 0.06 0.08 0 l
dev
0.20IIIIIIIIIIIIIIIIIIIIIIII
0.15 solid: N! dotted: N.
2“
30.10
2"
0.05
0.00 -
0 0.02 0.04 0.06 0.08 0.1
dev.

Figure 5.11: the relationship between scalings of nonlinearity by linear transformation
and by rms emittance for a 238U24+ beam with an incident velocity 0.0085c in the
system using solenoids for transverse focusing. The initial geometrical transverse
acceptance was 263 mm-mr without longitudinal phase space.

116

”‘0’“, s.=o.ooss E/A=33.65(ireV/A)

11: 4.5IV/m.-10 12-1:3.0.-15 12-2:3.0.-15
s1: 4.238841 (‘1') 12: 7.075668 s3: 7.173162
GLaoc.=ll7.41 (mmernr)

 

150 IIII IIII IIII IIII [III II I

l l l l l '
125
100

1111

ll

dotted: nonlinear
solid: linear

1111111

50
25

111I1111I1111I

 

 

111L11I1111I1111I1111I111

0.05 0.1 0.15 0.2 0.25

 

o IIIIIIIIIIIIIIIIIIIIIIIIIII

.9
co

dev.

 

0.20 IIIIIIIIIIIIIIIIIIIIIIIIIIII

IIII

0.15

z
0.10

l I | ‘
11111111111

0.05

IIITIIIIIIIIIII

111

 

 

0.00 111l111‘1I1111I1111I1111I1111
o 0.05 0.1 0.15 0.2 0.25 0.3

dev.

 

Figure 5.12: The relationship between scalings of nonlinearity by linear transforma-
tion and by rms emittance for a ”SUI” beam with an incident velocity 0.0085cin the
system using solenoids for transverse focusing. The initial geometrical longitudinal
acceptance was 117.4 keV-nsec without transverse phase space.

117

”'0'“. “41,0035 (33.65 keV/A)

ii: 4.5llV/m.-10 12-1:3.0.-15 12-2:3.0.-15
sl: 4.238841 (1') s2: 7.075668 s3: 7.173162
solid: Geometrical Twiss parameter

dotted: Linear hiss parameter

 

 

 

 

 

 

 

 

 

 

 

 

3.0 I I I I I I I I I I I I I I I I I I f -I
p,’2,55—— _8L Lil—£3; I2—2 3
2.0 E— -‘ —-§ F“.
1.5 E— _:
1.0 E— ..E
0.5 : ...... .5
0.0 L“? I . . - i ----- 4- . . 7...-.. i I
0 0.5 1 1 5 2 z(m)
5 I I I I I l l
a, 4 I— _3‘.
? I
6:: 3 _-
nn. 2
2 _—
1 E—
0 : I I I 1 I 1 1
Figure 5.13: The effective beta-functions of the transverse (upper) and the longitu-
dinal (lower) phase spaces for 238U24+ beam with an incident velocity 0.0085c in the
system using solenoids for the transverse focusing. The dotted lines are for Twiss pa-
rameters from the geometrical acceptances and the solid line is for Twiss parameters

from the linear acceptances.

Chapter 6

Summary and Discussions

First order matrix optics has been applied to the system which includes very-low-beta
heavy ion radio frequency resonators. In spite of its rather time consuming point-
by-point calculations through each resonator to find its transfer matrix, it has been
shown that the transfer map found can be used to investigate the focusing structures
of multiple-gap resonators quantitatively. The calculations of focusing powers by
using the transfer maps clearly showed the alternating-phase focusing properties of
II and I2 type resonators of the PH linac. We expect that lst order matrix optics
will be used to design multiple-gap resonators for very low-beta rf resonators in the
various sides. For an example, the distances between accelerating gaps might be
optimized for required transverse and longitudinal focusing powers with appropriate
optimization routines. Also the electric field gradients or incident velocities might be

optimized for needed optical properties of each resonator.

The lst order matrix theory has been extended independently to the phase space
transformations under acceleration. By searching beam envelope functions it was
possible to find the so called “geometrical acceptances”. This, when used with non-
linear calculations such as ideally higher order matrix optics programs or raytracing

calculations, was used in finding “linear acceptances” with matched phase spaces.

118

119

The model solenoid [Berz 93] has been introduced through the calculations instead
of existing superconducting solenoids with appropriate physical parameters to fit the
real field profiles. The transverse optics of the model solenoids do not seem to be
sources of significant nonlinearities even for relatively large size beam envelopes. We
expect that this model solenoid will well approximate real solenoidal fields in computer
calculations for the whole system of the PH linac with a little more investigation about
transverse optics of the model and real solenoids. By using these model solenoids it
will be possible to investigate in detail the nonlinear effect of the solenoids for the
transverse direction. The nonlinear effect of the solenoids was not included in the
code LINRAY [Pardo 89], which has been used for the ATLAS linac, because it has

only linear radial focusing terms for resonators and solenoids.

The possibilities to use resonator focusings at the beginning part of the PII linac
have been investigated. The stability limits of rf phase offsets and ranges of incident
velocities for 11 and 12 type resonators to achieve transverse and longitudinal focusings
simultaneously during the accelerations were calculated for two different beams of
40A?“ and 238U2”. It has been found that there is an intermediate velocity range

in which transverse resonator focusing does not occur for 238U2”: for this beam both

lower and higher velocities do display transverse focusing.

The possibilities of using resonator focusing partially at the P11 linac for very
low velocity heavy ions might be contingent upon transverse beam emittances from
the ECR source. There are no differences in longitudinal beam qualities between
resonator focusing systems and focusing systems using transverse focusing elements
like solenoids. The absolute values of transverse (geometrical) acceptances calculated
in Chapter 5 might be enough to accept beam emittances from the ECR source. But,
as expected, it turned out qualitatively that the resonators were major sources of

nonlinearities which reduced linear acceptances significantly. As a result the resonator

120

focusing scheme is inferior to the existing system using superconducting solenoids as

transverse focusing elements in terms of overall linear acceptance.

The other aspect to note is that lowering incident velocity to increase the trans-
verse acceptances in the resonator focusing system causes decreasing of the longitu-
dinal acceptances. Thus the initial velocity should be determined by a compromise

between the transverse and the longitudinal emittances from the ion source.

We have also proposed a new method to match phase space ellipses by comparing
linear and nonlinear transformations of individual particles. The consistency of this
method was checked with the growth of rms emittances which measure the order or
disorder of phase space distributions. We expect that this kind of analysis would be

useful also in other beam transport systems.

Finally we would like to point out the effect of the higher order coupling of phase
space coordinates in the rf resonators. The longitudinal phase space is newly created
even if its initial area was zero because of coupling of the transverse coordinates due to
nonlinear radial forces. This coupling effect also arises from the interactions between
transverse and longitudinal coordinates. As a consequences of phase space couplings
the new created or distorted longitudinal phase space distributions are asymmetric
both in time and energy. Therefore this asymmetric distribution of longitudinal phase
space should be considered in beam dynamic calculations to redefine the central ray
or synchronous particle of a bunch at the end of each resonator. But this would not
be a simple procedure because the asymmetric distribution of phase spaces is not be
known until initial phase spaces are given. This effect is corrected automatically in
actual tuning of the accelerating system by measuring the center of the energy and
the rf phase angle for each resonator. This is one advantage of an accelerator which

consists of a series of independently phased rf resonators.

Appendix A

Electromagnetic Fields in an RF
Resonator

We derive the time dependent electromagnetic fields of a rf resonator which has axial

symmetry, when the axial electric field on the resonator axis is known.

We assume that the If resonator electric field components are written in cylindrical

coordinates, (r, 0, 2) by:

E,(r,z,t) = E.(r,z)cos(wt+¢u),
Eg(r,z,t) : 0, (A.1)

Ez(r, z, t) = Ez(r, z) cos(wt + Q3”) .

where w is angular frequency of 'rf field and 4’0 is initial rf phase angle.

We assume that the spatial part of the longitudinal rf field is expanded by Taylor

series in terms of r:
°° 1 6kE1(0,z) k
Ez(r,z) = Z III—61'" 1‘

k=U

(11.2)

We first apply Gauss’ law to the infinitely thin cylindrical pill box of radius 1‘ shown
in Figure A.1. If there is no source of electric charge Gauss’ law of divE = 0 leads
to:

f”. {Ez(r’, z + A2) — 1341-32)} 27rr'dr' + 21rrE,(r,z)Az = 0 , (11.3)

121

122

Of
_ BE (1", z) , ,
Er(1‘,2) — —;‘/:)r ___—621‘ Md

(A.4)

From equations (A2) and (A.4) we find a radial component of rf electric field:

2 1 6k+lEz(012)rk+1
UM(1: + 2) 6281‘" ’

 

E,(r, z)

E,(r,z,t) = E,(r, z)cos(wt + 430) .

Since the electric field is axially symmetric Faraday’s law of curlE’ =

8E,(r,z,t) _ 6Er(r,z,t) + BBQ
3r _ (92 ET ’

in other words magnetic field has only one component in 9 direction.

Also Ampere’ 5 law of curlB— — ’2 ‘7’): gives:

6.80 w .
_ E— _ C—2E,(r,z)sm(wt + (1)0) ,

where c is the speed of light in vacuum.

Thus from equations (A.5) and (A.7) we obtain rf magnetic field:

' °° 1 0"]5'),(0,z)rk)H

B
”(’3 2) 2:10 k!(k + 2) 61‘" ’

 

w
7

Bg(r,z,t) = —( )Bg(r,z)sin(wt+¢u).

('3

(A.5)

— %’E’— becomes:

(A.6)

(A.8)

123

From equations (A.5), (A.6), and (A.8) we get the rf longitudinal field:

 

 

1 6k+2E2(0,2) w2 6kEz(0,2)
Ez(r,2) Z .Ez,(0 Z)“— Z k!(k+ 2)2{ 62261“: ‘I" "g ""6 r—k Tk+2 1

1:20

(A.9)

Ez(r,2,t) = Ez(r,2)cos(wt + (15“) .

The partial derivative 93%.9—‘1 in the above equation can be found iteratively in terms

of the longitudinal component of rf electric field and its partial derivatives on axis:

 

 

6E,(0,2) __ 0
6r — ’
62Ez(0,2) _ 62 E (0, 2)
a2 ‘ ‘22[ 622 32-19(0 2" ’
63Ez(0,2) _ 0
67:3 _ ’
6"Ez(0,2) _ 4! i 62Ez(0,2) +I_:)_:62E'z(0,2)
67°4 _ 22 42 _622 67'2 c2 67‘2
_ 4! ’6"Ez(0,2) wz 62Ez(0,2) wz
- 22.42_ 3.4 +2?(2——a.2 +§Ez(°’z) ’
65Ez(0,2) _ 0
61:5 ~— ’
66Ez(0,2) _ 6! 29:6 5(0, 2) +£64E2(0,2)
61'6 _ 22-42- 622 c2 6r“

 

_ 6! 66Ez(0 6’E;(0, 2)
" 22 .42 626 c2 624

 

 

 

124

6! w2 w262Ez(0,2) w"
+ 22-42-62[27(3c_2 622 +ETE‘(0’Z))]

In the general expression:

6(2"+’)Ez(0,2) __
3,.(2n+1) - ’

62("+’)E2(0,2) _ C" 62("+’)Ez(0,2) + C, (‘13)nHEm z)
n 2 Z 7

6,201“) "' m 3220:“)

 

 

 

 

n—l>0 wz "- -k 2k+1
+ Z [0;(n+1)(‘~:— ) 6( )Em’z) , (A.10)

2 k+l
kzll 62 ( )
where

[2(n + 1)]!

C 22-42-62---[2(n+1)]2

Zfl

 

=(—-1)"+1 (A.11)

(n = 0, 1, 2, 3, ...).

Thus from equations (A.9) and (A.10) we obtain:

E.(r,z) = i0..[8—:———ng;§3’ ————z)(—C-—:) Ez(0,z)]r2"+

11:0

 

°° "-2 w? " ’° 1(92<k+')13(0, 2)
2:22;, [CM ("+1’ (:2) 622ml) " ’

(A.12)

Ez(r, 2,t) = Ez(r, 2) cos(wt + (150) .

125

where

1 1
C... :(_1)n[2-4-6----(2n)]2 = (—1)" W' (A.13)

 

From equations (A.5) and (A.13):

 

_ 0° 62"+’Ez(0,2) wz n6Ez(0,2) 2"“
E7032) — garnI 622m!" (—) T 1‘ +

02

n 2

 

M8

w2 n—k—l 62(k+1)+lEz(0,Z) 2n+l
Cm(n+1) 25- 622(k+l)+l 7

 

3
II
to
*
II
C

E,(r,2,t) = Er(r,2)cos(wt + 45”) , (A.14)

where

I
22"“(71 + 1)(n!)2 .

 

C... = (—1)"+l (A.15)

Also from equations (A.8) and (A.13) :

00 2n 2 n
Bo(1‘,2) : 20m [W (1) 6Ez(0,2) 7,271+

 

":0 622" c2 62
oo n—2 W2 .n-k-l 62(k+’)Ez(0,z) 2"
"2:20": k=U [(71 ‘I' 1) (22-) az2(k+l) 1' +’ 1
(A.16)
3.11321) 2 -—(:—:—)Bg(r,z)sin(wt + (1.).

Note that the terms related to w are the result of a time dependent rf electric field.

126

Summary

 

The time dependent electric and magnetic fields have been derived for a cylindrically

symmetric system in general form when the rf electric field on axis is known. If we

take just terms up to 3rd order in equations (A.13) - (A.15) we get:

E,(r,z,t)
E9(r,z,t)
Ez(r,z,t)
B..(r,z,t)
Bo(1‘,2,t)

32(1‘, 2, t)
where

EAT, z) 2 2 (92

2 2
Ez(1‘,2) I Ez(0az)_Z{%E2(O,Z)+W}1-2

1
Bg(r, z) = §EZ(0, 2)

In rectangular coordinates:

Ex(r, 2,t)
Ey(r,z,t)
Ez(r,z,t)

Bx(r,z,t)

= E,(r, z) cos(wt + (130) ,
= 0 ,
:: E,(r, z) cos(wt + (#0) ,
: 0 ,
w .
: —?Bo(r, z) sm(wt + (to) ,

:0,

IE
1
622

Ez(0,z) +

r_.__.

16c—2

623

: E,(r, z) cos(wt + 45”) ,
: Ego, z) cos(wt + <15”) ,
: Ez(r, z) cos(wt + (bu) ,

w o
: _C—zBo(r’ z) sm(wt + <15”) a

1 {w2 63Ez(0,z)

_13E2(0,z)r 1 {£6E2(0,z) + 63Ez(0,z)
c2 62 023

9

},.3.

(A17)

(A.18)

(A19)

(A.20)

(A.21)

127

By(r,z,t) = —§;B,,(r,z)sin(wt + 45”),
Bz(r,z,t) = 0, (A22)
where
Ez(r,z) = E,(o,z) — i{§;Ez(0,z) + W} (22 + 312) , (A.23)
Bx(r,z) -_— —;E2(0,z):c —11—6{5§E2(0,z)+ 23%9‘3} (22 + y2):c,

1 1 w2 63E; 0,2
Buy-,2) :- 5153101)!) — — {—Ez(0,z) + —a—:3—)

16 cg }(=c2 + my. (A24)

128

 

Figure A.1: Cylindrical pill box.

Appendix B

The System in a Cylindrically
Symmetric Magnetic Field

B.1 Magnetic Field in Magnetic Solenoid

We expand the off axial field of a static magnetic solenoid which has axial symmetry,

when the axial magnetic field on axis is known.

The relation between the magnetic field Iii and magnetic vector potential A is

given by:

at

:2 curl If . (B.1)

In component form:

 

B,- : {6A2 _ 6(1‘Ag)} l ,
1‘

 

 

 

 

3? 82
(9A, 6A2
Be — (92 _ 6r ’ (B2)
B _ (6(1‘Ag) — 6A,. 1
z — 81' 80 r .

Since the vector potential is dependent on only the angle 0 in an axially symmetric

129

 

 

 

130

system equation (B.2) becomes:

 

 

_ 6(7‘A0)1
B,- _ — (92: r,
Bg = 0, (B.3)
624,1
3‘ ““ 69 i

It is well known that the rotationally symmetric magnetic vector potential can be

expanded by the following power series [Forest 84], [Jiye 86]:

1 (; >21“ 62*Bz(0, z)

=E:(—1)k—!————(k+l)! 622* , (3.4)

Thus from equations (B.3) and (BA) we find the magnetic field expansion in terms

of the magnetic field on axes and its derivatives. Up to the 3rd order of r:

_1aBz(O,z)r+_ 1 63B (0, z) ,

3'0”) z 2 62 16 623 ’

Bg(r,z) = 0, (3.5)
2

32032) = 32(0,Z)_1Wr2

:1. 622

In rectangular coordinates:

_188(0, ———————~—3——y) 163B(O,z)

31032) 2 62 +16

___—3+“ fly,

 

 

 

131

_1aBz(0,z) 133Bz(0,z)

 

By(r,Z) : 2 Ta: + 16—6—2?($2 + 3’2)“: ’ (B'6)
16232 0,2
Bz(r,z) = B;(0,Z) — Z BET-20132 + y2) .

B.2 Angular momentum conservation

The Lagrangian function of a particle of electric charge q and rest mass m in static

magnetic field [Jackson 75] is given by in cylindrical coordinates:

L — me 1 2+qV-A
c

= —mC2¢1 — (7‘2 + 7.20.2 + éz)—l 'i' qréAfl , (B7)

 

where l7 is velocity of a particle.

From the Lagrangian function the equations of motion are given by:
d 6L 0L
"‘* _"" = a . : :6: ' B8
dt (64.) aqj (’ " z) ( )

Since 9 is cyclic in the Lagrangian function, from equation (B.8) the canonical mo-

mentum P9 is a constant of the motion [Goldstein 59]:

_BL_ mrzd
_69_ 1__'_’;

(.'

 

P9 + qer . (B.9)

i.e. angular momentum is conserved in static magnetic field.

 

 

132

B.3 Rotation of a Particle in Magnetic Field

The rotation angle of a particle under a magnetic field can be found from equation

(B.9):

. do A
02—22—1 1—[32—9. (B10)

dz m r

Thus integrating the above equation with respect to the axial distance we obtain:

0(2) = —(%),/1 — (92]—”:3 {(C5)2 — 52 — (1-(.'v2)}“/2 dz . (B.11)

If we neglect 1‘2 and (rd)2 terms in the bracket and take the first term of A0 in equation

(BA), then we get the rotation angle along the optic axis:

 

21(2) W

m 20,8 ABZ(O,z)dz . (B.12)

B.4 Decoupling of Equations of Motion

The linear differential equations of motion for a magnetic solenoid in section 3.3.3 of

Chapter 3 are coupled in rectangular coordinates. For the x-direction:

J21: dy 1632(0,Z)
2;; — a {8232(0) Z) + 5 82 3]} — 0 1 (8'13)

where

 

 

133

a _ (ml—fl?

: mcfi, °

(8.14)

To decouple equation(B.13) we introduce a rotated coordinate system (X, Y, 2), which

is rotated by an angle 0(2) with respect to the fixed coordinate system (23,31, 2) by:

X : zcosg+ysin9,

= —xsin9+ycos€ .

Thus from equations (B.13) and (B.15) we find:

 

(122 C05 (122 am dz sm dz COS dz

dX, d6 dY d9,
{ESln9+Xa';COSQ-+ ECOSO—YESIIIO} 0

d0 1 6340,25) c129 _
{aB,(0,z) + E} — {Ea—5;— + 32‘}y -— 0 .

Also from equation (B.12):

d6

0
“d: = —'2—BZ(O,Z) ,
fl _ _SaBz(0,z)
dz:2 — 2 62 .

Therefore from equations (B.16) and (B.17) we obtain:

(PX aB,(0,z) " .121» aBz(0,z) " . _
{E+(—T—) X}C089—{2—z?+(——2—_ Y 31116—0-

(B.15)

(B.16)

(B.17)

(B.18)

 

"-11

I'll.

 

_J'

 

134

Since c030 and sin 9 are independent each other it follows that:

dz2

d2Y aBz(0 z) 2

—— —————-’—— Y = . B.19
1122 + ( 2 ) 0 ( )

In other words equation (B.13) is decoupled in rotated coordinates system. The

equation of motion in the y— direction is also decoupled by the same way.

B.5 Rotation of Transfer Map

Since the motions under magnetic field are coupled in a fixed coordinate system the
first order transfer map of such a magnetic element is found in the rotated coordinates
system to decouple the motions. To find the transfer map at the fixed system the

transfer map is rotated inversely relative to the rotated coordinates system.

In this section we find the rotation matrix to get a transfer map in the rotated
coordinate system by an angle 0 counterclockwise about the z-axis. (x, y, z), (X, Y,

z) are the fixed and the rotated coordinate systems respectively.

The position vector in the rotated system is related to the fixed system by:

X = c0501: ——sin0y ,

Y 2 cos 0m + sin 03/ . (3.20)

The corresponding momentum vector (:c’,y’) transforms by the same way as the

position vector (2:, y):

X’ = cos 923' — sin Oy' ,

 

 

135

Y' = cos 923' + sin 9y' . (B.21)

Therefore from the above two equations we get a rotation map 12(9):

cos 9 0 — sin 9 0

0 cos 9 0 — sin 9
sin 9 0 cos 9 0

0 sin 9 0 cos 9

3(9) = (B.22)

If we apply the rotation matrix to a transfer map in a fixed coordinate system, we

find the transfer map in the rotated coordinate system.

7!

 

(Ti ..'

 

Appendix C

Calculation of Fringe Field

C.l Length of Fringe Field Tail

The limit of the solenoidal field can be calculated directly from the model axial field

equation rewritten by:

Bo

3(2) 2 —2— {tanh(

z+l/2

 

 

In terms of an exponential function equation ((3.1) becomes:

2B(z) __ 22 —l 22 -— a2

BU 22+1 22+a2’

 

 

where

z :2 82”, a : el/d .

136

)—tanh(z“d’/2)} .

(C.2)

(0.3)

(0.4)

137

Let R be the ratio of the maximum field strength Bm to the field strength at the

distance of z 2 2*:

 

82*
R — Bm (C.5)
Then from equations (C.2) and (0.3) we obtain:
2*“ + {02(1— 3T)+1}z*2 + (12(1— 3:) .—-. 0, (0.0)
RB RB
where
f? 2 tanh(-2-la- . (C.7)

Equation (0.6) is easily solved to get the half width of the full range of the solenoidal

fields at a specific ratio R:

2* 2 9ln {an-E2173 —— 1)} . (0.8)

C.2 Transfer Map of a Solenoid

Since a solenoidal field has a long tail the total transfer map of a solenoid includes a
pretty long drift space besides its physical length. So it would be convenient to filter
the drift space out to have a transfer map independent of the length of the fringe field

tails.

Let M be the total transfer map of a solenoid. The map is then decomposed into

a map of a right tail field M,, a map of a body field M5, and a map of a left tail of a
solenoidal field M):

M 2 MrMle . (C.9)

 

 

138

We also decompose the transfer map of a tail field into a map of the drift space
corresponding to the length of the tail and a map of the fringe field including only a

field effect of a tail:

M, = MflM(d) ,

M, = M(d)M,, , (0.10)

where M (d) is a drift map of length d.

Thus from equation (C.10) we find the fringe field maps:

M1,: M,M(—-d),

M), = M(-—d)M,. , (C11)

in other words, fringe fields map are obtained by multiplying the negative drift map

to the map of the fringe field tails.

Also from equations (C.9) and (C.11) we obtain the transfer of a solenoid independent
of the drift spaces of tails:
M, 2 Merbe[. ((3.12)

 

 

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