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This is to certify that the

thesis entitled

Evaluation of Analytical Formulas for
Calculating Cyclotron Orbit Properties

presented by

Dong-0 Jeon

has been accepted towards fulfillment
of the requirements for

M . S . Physics

degree in

 

 

447. 727‘ fi’WZ/A/l/

Major professor

Date May 17, 1990

 

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

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L University

 

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MSU Is An Affirmative Action/Equal Opportunity Institution
cMMpmG-pJ

 

 

 

EVALUATION OF ANALYTICAL
FORMULAS
FOR CALCULATING CYCLOTRON ORBIT
PROPERTIES

By

Dong-o J eon

A DISSERTATION

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

MASTER OF SCIENCE

Department of Physics and Astronomy

1990

(4.2; - 554:7

ABSTRACT

EVALUATION OF ANALYTICAL FORMULAS
FOR CALCULATING CYCLOTRON ORBIT PROPERTIES

Dong-o J eon

Improved analytical formulas for calculating cyclotron orbit properties have
recently been derived by M. M. Gordon. To test these formulas, a computer pro-
gram has been written that uses magnetic field data from any cyclotron to evaluate
these formulas and to compare the results with those obtained from a standard or-
bit integration code. This investigation has employed field data from three dierrent
superconducting cyclotrons all having magnet poles with three sectors. The results
show that the formulas give the best results in those cases where the average magnetic
field is relatively high (45 kilogauss). They also show that the new formulas are much

more accurate than the old simpler formulas.

To Father, my parents, brothers and Chris Park

ii

ACKNOWLEDGMENTS

I’d like to express my deep appreciation for the invaluable and painstaking advice

and kindness of Dr. Gordon. I’d like to thank Dr. Blosser and Dr. Mahanti and F.
Marti for their concern and time. I also like to thank S. G. Kim for his kind advice

on how to use a word processor.

iii

Contents

LIST OF FIGURES vii
1 Introduction 1
2 Theoretical Formulas 3
I Momentum and Energy .......................... 4

II Frequency Error .............................. 5
III Radial and Vertical Oscillations ..................... 5

A Radial Oscillations ........................ 6

B Vertical Oscillations ....................... 6

3 Computer Program Description 8
I Determining Fourier Coefficients ..................... 8

II Derivatives of Coefficients ........................ 9

III Interpolation Used in E. 0. Code .................... 10
IV Matching the Theory and E. 0. Code .................. 11

4 Data Analysis 12
I The Field of the Harper Cyclotron ................... 13

iv

II 130 MeV/A Field of the K1200 Cyclotron ...............

III The Field Data of the Orsay Cyclotron .................

5 Conclusions

A Appendix by M.M.Gordon

I Field Parameters .............................
II Equilibrium Orbit Calculation ......................
III Calculation of p and w ..........................
IV Radial Oscillations ............................
V Vertical Oscillations ............................
VI Born Approximation ...........................
VII Equilibrium Orbit Improvement .....................

LIST OF REFERENCES

20

42

44

44

47

48

00

52

55

57

List of Figures

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

Plots of fn and tn for n = 3,6,9 of the Harper cyclotron. ....... 24
Plots of kn for n 2 0,3,6,9 of the Harper cyclotron ........... 25
Plots of AE/E and —AT/T of the Harper cyclotron. ......... 26
Plots of Au, in units of 10"4 and AV, in units of 10‘3 of the Harper

cyclotron. ................................. 27
Plots of V, and u, of the Harper cyclotron. ............... 28
Two plots of Au, from two different theories ............... 29

Plots of fn and tn for n = 3,6,9 of the 130 MeV/A field data of the

MSU K1200 cyclotron. .......................... 30
Plots of kn for n = 0,3,6,9 of the K1200, 130 MeV/A field ....... 31
Plots of AE/E and —Ar/r of the K1200, 130 MeV/A field. ..... 32
Plots of Au, and Au, of the K1200, 130 MeV/ A field .......... 33

Plots of V, and V, from the E. O. Code for the K1200, 130 MeV/A field. 34

Plots of 11, obtained from the E. 0. Code and two dill'erent theories for

the K1200, 130 MeV/A field data ..................... 35
Plots of fn and tin for n = 3,6,9 of the Orsay field ............ 36
Plots of kn for n : 0,3,6,9 of the Orsay field. ............. 37

vi

4.1

C1

4.16

4.17

4.18

Plots of AE/E and —AT/T of the Orsay field ..............
Plots of AV, and A (V3) of the Orsay field. ...............

Plots of V, and V, from the E. 0. Code and from the theory by M. M.

Gordon for the Orsay field .........................

Plots of V, obtained from the E. 0. Code, from the theory by M. M.

Gordon and from the theory by Hagedoorn and Verster .........

vii

38

39

40

41

Chapter 1

Introduction

In determining whether a given magnetic field meets the performance requirements
for accelerating a particular ion to a given final energy, one must calculate various orbit
properties of the given field as a function of energy or radius. The most important of

these properties are the orbit period 1', and the radial and vertical tunes, Vr and V,.

Such calculations can be carried out either numerically, using the Equilibrium Or-
bit Code [1] or by analytical formulas. The accuracy of the the E. 0. Code is excellent,
but it does not provide the insight into the effects of various field characteristics which

is provided by the analytical formulas.

Cyclotron orbit properties have been analyzed by many authors [2, 3, 4] in the past
and although the resulting formulas have proved useful,their accuracy is quite limited
particularly when applied to machines with only three sectors. Dr. M. M. Gordon
has undertaken the task of deriving more accurate formulas by introducing certain
improvements. First of all, his analysis systematically maintains all terms, whether
large or small, that are of second order in the flutter field, that is, the part of the field
that varies with the azimuth 0. In doing so, an improved first-order calculation of
the equilibrium orbits themselves is made use of. Finally, the values of V, and V, are

calculated using a formula which is derived from a second-order Born approximation

and which should be valid all the way to the stop—band limit, V = N/2, where N is

the number of sectors.

The purpose of this thesis is to test these new analytical formulas by applying them
to field data from various cyclotrons and comparing the results with those obtained

from the Equilibrium Orbit Code.

All the theoretical formulas used in this paper are briefly summarized in Ch. 2

while a detailed discussion of their derivation is presented in the Appendix.

The theory uses a reference radius as an independent variable which is the radius
where various Fourier coefficients and their derivatives are calculated. In contrast, the
customary E. 0. Code uses energy as an independent variable. Thus it is necessary
to match these two independent variables to make a proper comparison between the

results. The solution of this problem is described in Ch. 3 on the computer program.

We use field data from three cyclotrons to test how well the results obtained from
the analytical formulas agree with those obtained from the E. 0. Code. The results
of these comparisons are discussed in Ch. 4. As will be seen, the theory works very
well for superconducting cyclotrons operating close to bending limit of their magnets.
In such cases, the magnitude of the flutter field is small compared with that of the

average magnetic field.

As a by-product of this thesis, a small improvement in the E. 0. Code has been
derived which removes certain fluctuations that occur in plots of V, and V, as a
function of radius (or energy). Specifically, this improvement provides a more accurate

method of calculating the radial derivatives of the field data.

Chapter 2

Theoretical Formulas

The analytical formulas were calculated by M.M.Gordon and are presented in more

detail in the Appendix.
We assume that the median-plane magnetic field is represented by:
B(r,0) : Bo(r) + Z[ 0,,(1') sin 11.0 + Hn(r) cos n0 ]

Bo(r) 4. Z Bn(r) cos [ n(0 — fln(r)) ] (2.1)

where n = N, 2N,. . ., and N is the number of sectors in the cyclotron magnet.

All the theoretical formulas are expressed in terms of certain field parameters at

a given value of r, namely :

 

 

f" 2 E (2.2)
t.. I 4.4%,) (2.3)
k. : 13(132) ,kozé‘;(é£2), (2.4)
w = 2(a) #66263”), <25)
= ram)-

4

I Momentum and Energy

We take the charge and mass of the particle to be q and m, and define the

dimensionless parameter /\ as follows:

,\ = 9351(1) (2.7)

mop ’
where p is the momentum expressed in me units so that p : 73.
The value of A is found by iteration from the formula (see eq. A.22 in the Ap-
pendix):

I
l—A: szhzf. +2kn)y,.+%M”y,’,] (2.8)

where

fn

n2__fl2

 

yn : (2'9)

are Fourier coefficients measuring the deviation of the orbit from a circle (see eq. A.10

and eq. A.21 in the Appendix), and
M" = A(2k(', +7ko+3) .
Also, as described in sec. VII of the Appendix, the value of p. is obtained from:

1 sinno 3
' 3 — ' 3 3__-_° _ta___
sin (1100/2) .- sm (KOO/2) «l- 5 (00/2) n00 En la“ _ 453] (2.10)

where 00 = 21r/N,
52 = [W

with

il'I’=A(I+ko)+%(/\—I)

and

g3: i)? [(3f,+2k,,)’+4(nf,.t,)’]. (2.11)

The final value of A determines the momentum p and hence the energy E:

P2 2

= -. __x . 2.12
I+\/l+p2 me ( )

 

II Frequency Error

The Equilibrium Orbit is discussed in sec. 11 in the Appendix. Define r to be the

orbit period and w = 21r/r.

As shown in eq. A.26 of the Appendix, the frequency error is given by

wo rwo A2
z; “ =‘a‘ll‘ts¥(2"°“)”3l‘l (“3)

where fl = VIE-=7 and we is a fixed reference frequency, and y,, is given in eq. 2.9
+P

above.

III Radial and Vertical Oscillations

As shown in sec. VI of the appendix, the formula for the tune V is given as follows:

memo/2) = sin2(n00/2) + g (32;!)2 (Bin ”0° ) E": ——--’i 1 (2.14)

1:00 1:2 — 4n:2

 

 

where 00 = 21r/N is the period and both 9,, and n are given below for radial and
vertical oscillations. Note that this formula is the same as that used for calculating

u in eq. 2.10 above.

Caution is necessary when n —» N /2,because the sum has a singular term in
this case. The singularity is removed by the sin 15% in front of the sum. Thus, if

n = N(1 — e)/2,then
311:0; _
N2 — 4r;2 _ N3(2 — (3)6

which approaches w/ZN2 when 6 —+ 0 . One should therefore test whether 6 < 10“3,

say, and revise the sum accordingly.

Note further that if n” < 0, then
sin2(n00/2) —» —sinh2(|n| 00/2)
and

sin 1:00 sinh |n| 00

— —+
n00 '5' 00

 

in the above formula for V.

A Radial Oscillations

As shown in sec. IV of the Appendix, the values of It: and g: in this case are

n: = A(1+ko) +gA’Zy,(fn-1-3k,+k;—n’fnt:) +

1 1
§/\2[6/\(1+k0)-1lz n’yf. + 5430 + 111... +11% + 21:02:13.,

(2.15)

(92);”. = A’{(nfntn)’+ [1.1—k..+yn(-:—n’+A(1+3ko+k3))]2} ,
(2.16)

where (9,): is obtained from squaring the eq. A.36 of the Appendix and taking the

average over 0 and then multiplying by a factor of two.

B Vertical Oscillations

As shown in sec. V of the appendix, the values of It: and g}, are given by

N: Z —Ako + %A2 Z (ann i" nzfnt: '— 2k“ — k;) yn _

fl

é/VH +2Ako]z n2y2—%A3(6ko+9k(1+2kglz ll: 2

n

(2.17)

(92):. : A2{(nfntn)2+ [kn+(—21—n3+2/\kn+)1k6)yn]2} (2.18)

where (g, )3, is obtained from squaring the eq. A.39 of the Appendix and taking the

average over 0 and then multiplying by a factor of two.

Chapter 3

Computer Program Description

The analytical formulas make use of the Fourier coefficients and their derivatives.
Because the field data are given on a polar grid having a uniform Ar and a uniform
A0, the Fourier coefficients in eq. 2.1 above are computed as a function of radius
from the data given on each radial grid point, and their derivatives as a function of

radius are determined from these coefficients.

I Determining Fourier Coefficients

The Fourier coefficients of various harmonic numbers are obtained from numerical

integration. More precisely, the trapezoidal method is used.

Since the field data are given at 0 2 0°, 1°, 2°, . . . , the Fourier coefficients in eq. 2.1

are obtained from the following formula:

Nu. 1r
Hm) Z 2,; [B( ’( 180)?r ) CHM (i18fl) )l/N‘“
(3.1)
Na. 7',
cm) = 2; [m .(————— 180)" )s W —1——1‘,) )]/N..

(3.2)

where N", = 360/N is the number of 0 values within one sector, with N being the

8

number of sectors.

The amplitude of the am harmonic,B,,(r), is obtained from:

 

)=(/GZ(r) )1- 112(r (3.3)

We have harmonic numbers n that are multiples of N since the field has N - fold
azimuthal symmetry. Of course this is an idealization, since real field data include

imperfections.

II Derivatives of Coefficients

The data are given for 0 S r S rma, at values that are multiples of Ar. We
calculate the first derivative using a three-point formula. That is, for an arbitrary

function F(r), we use:

dF(r) __ F(r + Ar) — F(r — Ar)
‘11;— — 2A.. ° (3'4)

 

This formula is used to calculate the first derivative of B0, B,” G", Hu and used in

calculating kg and k7‘ defined in eq. 2.4 and in calculating tn from the following:

nfnt,1 = ii [11,611, — 0,121) (3.5)

where G; 2 ~35; (431‘) and H; = g; (13.1).

After the first derivatives have been calculated at each I value, the second deriva-
tives are obtained from the same formula by replacing F byd “7 .The second derivatives
are then used to evaluate kg and k; in eq. 2.5. Third derivatives of Bo required for

calculating kg in eq. 2.6 are obtained in the same manner.

We lose one point at 1‘ == rm“, in calculating first derivatives, a second one at
r = rm“, — Ar in calculating second derivatives, and so on. The extra points needed

beyond r :: 0, that is,at r = —Ar, ——2Ar, and -—3Ar, can be obtained by symmetry.

10

III Interpolation Used in E. 0. Code

Because the E. 0. Code uses numerical integration, it requires values of B, is?
and 95% at values of r that are not on the grid points where the field data are given.
Interpolation is therefore required and the method adopted for this code is somewhat

different from the usual four-point Lagrange form.

When a function F(r) is given at four evenly spaced points (r1, r3, r3, m), the
values of the function between 1'; and r3 is found as follows. First of all, using the
left three points (r1, r2, r3), we calculate F1(r) using a three-point Lagrange formula,
and then use the right three points (r3, r3, m) to calculate F3(r) again using the
three-point fromula. The function is then obtained by a linearly weighted average of

these two values:

13(1) = —-—r—-F2(r) + —--}F.(-:) (3.6)

where Ar = r3 — r;. The resultant function F (r) is not only continuous at r = 1‘, and
1'3, but also has a continuous first derivative at these points. However, the second
derivative is not continuous. The original E. 0. Code uses this formula to find not
only the field, but also its first derivative at each r value. As a result, the values of
V,, which are sensitive to the first derivatives, exhibit small cusps when plotted as a

function of energy or radius.

To remedy this problem, the method used to find first derivatives was changed
to the following. First, a table of first derivatives was constructed at the grid points
using the eq. 3.4 for 5%. We then use eq. 3.6 to interpolate in this table to find the
derivative at the given r values. This not only cures the difficulty, but is also consistent

with the method already used in the E. 0. Code for finding first derivatives of the

field with respect to 0 at a given r value.

The Lagrange interpolation formulas can be found in ref. [5].

11

IV Matching the Theory and E. 0. Code

Energy is used as an independent variable in the case of the Equilibrium Orbit
Code while the radius called reference radius is used as an independent variable for
the theory. In order to avoid interpolating in the theoretical values, we therefore
modified the E. 0. Code to provide output at the same values of the reference radius

defined in eq. A.11 of the Appendix.

This is accomplished as follows:

1. Using a fixed energy step, the E. 0. Code is run to obtain a table giving the
reference radius and other orbit properties as a function of radius. The fixed
energy step is chosen small enough to insure the accuracy of the interpolation

required at step 3 below.

2. Using the analytical formulas given in Ch. 2, the program then calculates all of

the orbit properties, including the energy, for each value of the radius.

3. The program then interpolates in the table of E. 0. data to find the energy
corresponding to the given reference radius, and then reruns the E. 0. Code at
this energy to obtain the corresponding orbit properties including the reference
radius. The program then checks whether the difference in radius values falls

within an acceptible error, 1 x 10"6 inch.

4. For each value of the radius, the program then compares the values of E, T, V,

and V, obtained from the analytical formulas with those from the E. 0. Code.

Chapter 4

Data Analysis

Here we deal with three sets of field data with various characteristics. First there
is the field of the Harper Hospital cyclotron, second the 130 MeV/A, q/A = 0.33 field
of the K1200 cyclotron and third the 200 MeV proton field of the cyclotron being
designed at Orsay, France. All these are superconducting cyclotrons with magnet

poles having three sectors.

The difference for an arbitrary quantity, F, is defined as follows:
AF : FE. 0. Code " FTheor-y (41)

where F will be E, r, V,, V,. Also (AE)/E and Ar/r are defined as follows:

(AEl/E = (EE.0.Code—ETheory)/EE.O.Codea (4-2)
(As/1- : ( °° — .1. )/——“’—°——. (4.3)

WE. 0. Code wTheor-y ws. 0. Code

 

Because the field data are given on polar mesh, there are only a limited number of
harmonics which have significance, that is, we have at most 120 independent Fourier
coefficients because at each value of radius only 120 azimuthal points were measured.
The maximum number of harmonics added in the theory is chosen such that there

would be no change in the output when an extra harmonic is added.

12

13

I The Field of the Harper Cyclotron

The Harper cyclotron which is a superconducting cyclotron constructed at MSU

has the following characteristics.

It is designed to accelerate deutrons to a final energy of around 50 MeV. The field
data are given on a polar mesh with Ar = 0.1 inch and A0 = 1°, and the maximum

radius given is 12.3 inches. The magnitude of the central magnetic field is 45.9 kGauss.
The number of harmonics computed is ten, so that nma, = 30.

Figure 4.1 shows the magnitude of the three leading fn for n = 3,6,9 (refer to
eq. 2.2). As you can see, the fractional magnitude of f3 is the largest with maximum

6‘“ and 9th harmonics are smaller with

value of around 0.18. The magnitudes of the
maximum values of 0.05 and 0.02 respectively. These characteristics are favorable
relative to a theory which assumes the orbits to be nearly circular. Figure 4.1 also
shows plots of t,, for n = 3,6,9 (refer to eq. 2.3). As can be seen, these are quite
linear up to the radius of 11 in. for the third and sixth harmonics. At r = 6 in. ,

t9 shows the effect of a 180° phase shift associated with a zero crossing of f9. (The

theoretical formulas are functions of f: rather than f").

Figure 4.2 shows the plots of It:" for n = 0,3,6,9 (refer to eq. 2.4). The plot of k0
starts out from zero and increases up to a maximum value of 0.12 near 1‘ = 12 inches.
The curve of k3 starts out from 0.10 and at larger radii falls to a minimum of —0.70,
and this curve decreases sharply from the radius of around 10 inches. The maximum
value of he is 0.04 and the minimum is —0.12. The maximum of k9 is 0.08 and the

minimum is —0.02.

Figure 4.3 shows the plot of (AE)/E and (—AT/r) both in the units of 10—5.

Note a change of scale by the factor of 10 at the radius of 10 inches. Clearly these

14

fractional differences are very small up to the radius of 10 inches (corresponding to the
energy of 33 MeV ). In the last quarter inch ( 46 MeV to 48 MeV ), these quantities
change quite rapidly. Note the similar patterns of behaviour of the two curves with
the magnitude of the fractional difference of energy being approximately twice as
large as that of the fractional period difference. The polarity of the small fluctuations
of these two curves is also the same, that is, when one curve has a small upward peak,

the other has a small upward peak,too.

Note that since 1' a: 21rr/ v, the following equation holds:

£~£~ME

T ~ v ~272- (4-4)

This is why we have the factor of around 2 between (AE ) / E and —(Ar) / 1'.

Figure 4.4 shows the curve of AV, in the units of 10 ‘4 on top and that of AV,
in the units of 10‘3 at the bottom. Both AV, and AV, show the same patterns of
behaviour with the magnitude of AV, being about five to ten times as large as that
of AV,. This is not surprising since the theory involves V3, and one should compare
V,(AV,) with AV,. That is, the values of V, always lie between 0.99 and 1.04, while
the values of V, start out close to zero and then rise to a plauteau with V, = 0.2

between 9 to 11 in. ,and finally rise to 0.32 at 12 inches.

Note that the polarity of the fluctuations on the AV, curve and on the AV, curve
is opposite, that is, if one curve has a little sharp upward peak then the other one has
a. little sharp downward peak. This is due to the fact that most of the terms in the
formula for it: appear with opposite sign in the formula for a: ( refer to eq. 2.17 and
eq. 2.15). All four of the figures show erratic behaviour around the radius of 6 in.,
and between 10 to 12 inches. This can be attributed to the defects in the magnetic
field data which were derived from field measurements. Another trend worthy of note

is that the plots of AV, and AV, are positive over most range of the radii.

15

Figure 4.5 shows the plot of V, above that of V,. As can be seen, the radius of 12
in. corresponds to the extraction region. The small bump around the radius of 6 in.

is observed in both of the plots and is due to the defects in the field data.

Figure 4.6 shows the plot of AV, between the V, from the E. 0. Code and that
from the theory by M. M. Gordon, and the plot of AV, between the V, from the E. 0.
Code and that from the theory by Hagedoorn and Verster [4]. Both of them are in the
units of 10—3. As can be observed, the error of the analytical results by Hagedoorn
and Verster becomes larger as the energy of a particle increases, which means that
the particle becomes more and more relativistic, because the theory [4] assumes that
the particle is slow enough for the relativistic effects to be neglected, and the formulas

for the vertical tune do not have any term like kg.

According to the theory by Hagedoorn and Verster [4], the expression for V3 is

given as follows:

 

 

 

 

1x3 = —12’ +
"3431+ 1.3.132 _ £45391; _ AM: + 3nt . ("3)2 + (311)”
n 2(n2 — 1) 2(n2 — 1) 2(n2 — 1) 211.2
(4.5)
where

_, __ 1' (180(1')
”(r) — 1300‘.) dr ’

.11..) (1.43:2) WP”), (.7)

Age) = ,2 (11%,.(0) ,Bg(r) = ,2 (i559) , (4.8)

 

 

and the median plane magnetic field is given by:

B(r,0) : 130(1') [1 + Z ( 11,,(1') cos 120 + 3,,(1') sin 11.0)] . (4.9)

16

The expression for V3 is different from that given in the paper by Hagedoorn and
Verster in that the sign of the third term in the summation changed from plus to

minus when it was rederived. For more datails, refer to the paper by Hagedoorn and

Verster [4] .

17

II 130 MeV/ A Field of the K1200 Cyclotron

Presented here are the 130 MeV field data of the superconducting K1200 cy-
clotron [6] at MSU designed to accelerate an ion with q/A = 0.333, where A is the
number of nucleons, to the final energy of 130 MeV per nucleon. Ar = 0.5 in. and
A0 2 1° and the maximum value of radius given is 44 inches. The extraction radius is

around 40 inches. The magnitude of the central magnetic field is around 44.5 kGauss.
The number of harmonics calculated is ten, so that nma, : 30 again.

Figure 4.7 shows the plots of fn (refer to eq. 2.2) and tfl (refer to eq. 2.3) for
n = 3,6,9. The maximum magnitude of f3 is around 0.2 and those of f6 and f9 are
0.05 and 0.02 respectively. Their values form a plateau from 10 to 32 inches. The
plots of tn for n = 3,6,9 show similar patterns of behaviour. They decrease linearly
up to the radius of around 38 in. and then start to increase. Note that the magnitudes
of f" of this 130 MeV field data are almost equal to those of the Harper cyclotron

field data (refer to figure 4.1).

Figure 4.8 shows the plots of kg and kn for n = 3,6,9 (refer to eq. 2.4). The plot
of k0 starts out from zero and increases with a periodic ripple on it and reaches its
maximum of 0.30 at 38 in. and then decreases sharply afterwards. The plot of k3
forms a plateau with a periodic ripple on it and then decreases sharply from around
36 in. down to —0.5. The plot of kg is almost constant with a fluctuation on it and
decreases sharply from around 34 in. down to —0.18. The plot of k9 is almost constant

up to the radius of 36 in. and has a bump with maximum value of 0.20.

Figure 4.9 shows the plot of (AE)/E on top of that of —(Ar)/r with both in
the units of 10". The reason the values of (AE )/ E are very large when the radius
is small is that when the radius is small, E is almost equal to zero and hence the

error is amplified. Again they show similar patterns of behaviour besides that the

18

magnitude of (AE)/E is about 2 to 3 times as large as that of --(AT)/7', which is
quite natural (see eq. 4.4). The polarity of the fluctuation of these two curves is the
same, which is true for the Harper cyclotron field. Note the periodic ripples with a
period of about 3 in. which is well correlated with the gaps between the trim coils.
The general trend of these two curves is decreasing up to the radius of around 35
in. ( 95 MeV ), and then the curves start to increase rapidly and become positive
eventually. It should be noted that the values of these two quantities, averaging out
the fluctuation, are negative while those of the medical cyclotron field are positive.
Because the magnitudes of fn and kn of these two sets of field data are almost the

same, (AE)/E and —(Ar)/r of these sets of data are almost of the same magnitude.

Figure 4.10 shows the plot of AV, in the units of 10“ on top of that of AV, in
the units of 10-3. As is true in the case of fractional energy difference and fractional
period difference, rather remarkable behaviour of periodicity is observed which is well
correlated with the gaps between the trim coils. The values of these are positive,
averaging out the fluctuations over most of the range of radius. The polarity of the
fluctuation of these two curves is opposite, which is quite natural because most of the
terms appearing in n: (refer to eq. 2.17) are appearing with opposite sign in It: (refer
to eq. 2.15). Because the field parameters such as fn and k" are almost of the same
magnitude as those of the Harper cyclotron field data, the values of AV, and AV, of

these two sets of field data are of the same magnitude.

Figure 4.11 shows the plots of V, and V,. Observe that the amplitude of the ripple
on the V, curve is larger than that on the V, curve. This is probably due to the fact
that V, is sensitive to the first derivative of the field. This periodic ripple is well

correlated with the gaps between the trim coils.

Figure 4.12 shows the plot of V, of the E. O. Code and that of the theory by

M. M. Gordon above the plot of V, of the E. 0. Code and that of the theory by

19

Hagedoorn and Verster. As shown, there is some range of radius where the theory by
Hagedoorn and Verster predicts imaginary values of V, which are plotted as negative
values while both of the E. 0. Code and the theory by M. M. Gordon produce real
V, values. Clearly the new theory of M. M. Gordon is an improvement of the theory

by Hagedoorn and Verster. For the expression of V3, refer to eq. 4.5.

20

III The Field Data of the Orsay Cyclotron

These field data originate from the cyclotron being designed in Orsay, France [7]
to accelerate protons to a final energy of 200 MeV with the magnitude of the central
magnetic field 20.2 kGauss which is about twice as small as those of the two previously

discussed fields. The Ar is 2 cm, A0 is 2°, and the maximum radius of field data is

1.48 meter. The extraction radius corresponds to the radius of around 90 cm.
The number of harmonics computed is fifteen, so that nma, : 45.

Figure 4.13 shows the plots of f" and tfl for n = 3,6,9. What is unusal is that
the magnitude of f9 is larger than that of f3. The magnitudes of fn for n = 3,6,9
start to decrease sharply from around 70 cm and are about two times as large as
those of the two previous sets of field data. The values of tn for n = 3,6,9 are zero
up to the radius of 30 cm and show linear decrease up to the radius of around 75 cm
and then exibit quadratic decrease up to the radius of around 84 cm and afterwards
they increase. Clearly the magnitudes of fn for this Orsay field data are almost twice
larger than those of the two previous sets of field data and the magnitudes of t,, of

this Orsay field data are inbetween those of the two previous sets of data.

Figure 4.14 shows the plots of kg and Ian for n = 3,6, 9. The curve of k0 starts out
from zero and increases up to the radius of 84 cm and then decreases sharply. The
maximum value is 0.50 which is two times larger than that of the 130 MeV field and
three times larger than that of the Harper Cyclotron field. The maximum value of
k3 is 0.20 which is around two times larger than those of the 130 MeV field and the
Harper Cyclotron field but what is common is that 12;; decreases for these three sets
of field data and the minimum value is-1.0. The maximum value of k3 is 0.15 which
is almost the same as those of the two previous sets of field data and the minimum

value is -—0.30 which is almost the same as those of the two previous sets of field data.

21

The maximum value of k9 is 0.20 which is two times larger than that of the Harper
cyclotron field data and almost the same as that of the 130 MeV field data and the
minimum value is —0.50 which is almost the same as that of the 130 MeV field data.

Refer to the figures of 4.1, 4.2, 4.7, 4.8.

Figure 4.15 shows the plots of (AE)/E and (—Ar)/r both in the units of 10“.
Note that the values of these two quantities are in the units of 10‘'5 for the two previous
field data. The reason the value of (AE )/ E is very large for small radius is that the
energy is almost zero when the radius is small, so the difference is amplified. Both
of these show similar patterns of behaviour except that the magnitude of (AE)/E'
is around twice as large as that of -—(Ar)/T, which is quite natural (see eq. 4.4).
They show the same polarity of fluctuation except for the value of outermost radius
where the fractional period difference decreases sharply while the fractional energy
difference increases sharply. The absolute differences are 3 x 10"4 for (AB )/ E and
1.0 x 10“ for —(A‘r)/1' up to the radius of 60 cm and become erratic afterwards.
The maximum magnitudes of (AE)/E and —-(A1')/T are almost three to four times
larger than those of the two previous sets of data, and this is quite natural because
the magnitudes of fn and Inn of this Orsay field data are about 1.5 to 2 times larger

than those of the two previous sets of data (refer to figure 4.1, 4.2, 4.7, 4.8).

Figure 4.16 shows the plots of AV, in the units of 10‘3 above that of A (V3) in the
units of 10’3. The reason we have the plot of A (V3) instead is that there is a small
region of values of radius where the theory shows imaginary values of V, , while the E.
0. Code shows that, even though being small, the values of V, are real. Both of these
show rapid change from 70 to 90 cm. Around this range of radius, we observe that
the plots of ta, to, and t9 change more sharply than in the range of radius, 0 S r S 70
(cm). As is true for all the field data, the values of AV, and All: are positive, if

averaging out the fluctuations. The polarity of the small fluctuations on these curves

22

is opposite. What is different from the case of the two previous sets of field data is
that we can hardly see periodic fluctuations and these two plots of the differences
are rather smooth. This is due to the fact that the field data of the Orsay cyclotron
were obtained not from the actual measurements of the field but from the numerical
calculations. In the case of the two previous sets of data, the plot of AV, is in the
units of 10 '4 and that of AV, is in the units of 10"“. The magnitude of AV, is about
five to six times as large as those of the two previous sets of data. This is due to the
fact that the magnitudes of f" and kn of the Orsay field data are about 1.5 to 2 times
larger than those of the two previous sets of data and hence the effect of these terms
is squared because the terms in It: (refer to eq. 2.15) and (5),): (refer to eq. 2.16) are
functions of f: and fnkn, etc. The magnitude of A(VZ) is about three to four times
larger than those of the two previous sets of data. Because you can’t compare A(1/3)
with AV,, you should compare A(Vf) of the Orsay field data with 2V, X A(V,). This
is also due to the fact that the magnitudes of f" and is, of the Orsay field data are
about 1.5 to 2 times larger than those of the two previous sets of data and hence the
effect is squared because the terms in It: (refer to eq. 2.17) and (9,): are functions
of f: and fnkfl. Hence the final effect is the amplification of these differences by a

factor of around four.

Figure 4.17 shows the plot of V, above that of V,. The result obtained from the
E. 0. Code lies above that from the theory. Note that some of the values obtained
from the analytical formulas are negative which, for convenience, indicates that the

values are actually imaginary.

Figure 4.18 shows the plot of V, obtained from the E. 0. Code, that from the theory
by M. M. Gordon and that from the theory by Hagedoorn and Verster. As can be
seen, the deviation of the values obtained from the theory by Hagedoorn and Verster

from those obtained from the E. 0. Code is much larger than those obtained from the

23

theory by M. M. Gordon. The theory by Hagedoorn and Verster predicts imaginary
values of V, over broad range of the radii where imaginary values are represented by

negative values.

24

 

omTTIYTTUITTTUIVTIUITUITIVUTr

fa

    
  
    

0.15

0.10

0.5

 

 

I‘IUVUUITI'1l'1r1

 

/ i

l j l 1 1 l l l l l l l L I L l l l L l l I

 

0.00

 

UflUTrU'rUUUIUUleIrTV

1.5

1.0

0.5

 

I'UV'

-0.5 .,
H

 

UUVU

 

_,.oa.1.n..1....1...li....1...
8 4 O 8 10 18
William
Figure 4.1: Plots of f, and tfl for n = 3,6,9 of the Harper cyclotron.
t; is depicted with a solid line, to with a dotted line, t; with a dotdashed line.

 

25

 

 

 

 

 

- '\‘-.
[- ~ ,_. \
1 l
4.2.)—
P
P
-o.4[-
1.
~03]-
TA 1 [4 l L l l l l_ l L l 1_L l l l J l L L l L l l l
2 4 6 8 10 18

Win)”

Figure 4.2: Plots of In, for n = 0,3,6,9 of the Harper cyclotron.
Ito is depicted with a solid line, k; with a dotted line, It. with a dashed line, ’69 with
a dotdashed line.

26

 

  

 

 

  
     

 

 

 

16 -I I I I I I I I I I I I I I I I I l I I I I I I I I I d 160
10 - -—‘ 100
s L — so
I 1
o - o
—s '— - -so
; AE/E x 10" 1
-1o :— — -100
I 1
-16 PI L I I I I L I I I I I l L I I I I I I L I I I I I I --150
P I l I I I T T I I I r I I I I T f I I I I I I I I
0.0 P- _. 5°
C .
as :- — as
l- '1
: 1
0.0 -: 0
F m
b
.. -A‘r/r x 10 - .
- 1
~ 1
-0.0 P -: -50
f I
rI I l I L I L l I I I I I I I I I I I I I I I L I L
s 4 s a 10 12
Radius(in).Medcyc

Figure 4.3: Plots of AE/ E and -Ar/r of the Harper cyclotron: in units of 10“.

Note there is a change of scale at 10 in. The left scale is valid to the radius of 10 in.
and the right scale valid from 10 in. to 12 in.

27

 

 

 

 

 

 

 

 

 

4— .—(_m
p- d
1- -1
1 LI 1 1 1 1 L1 1 1 1 I L1 1 1'1 1 LI l 14 1 1
F_l I I I I I I s I t s I IT I s Iii r fifi I
_ I l l I 1
4r- — so
I- 1
.1- d
.- -1
p I
2r - 20
1- cl
h
P
[-
0‘ 0
t-
b
1- .(
3 ..
P AV x10
-3F' ' “'m
)- d
I d
h .
b -
-4— —-40
D ‘
l'1 1 I LLL 111 LILI 1 1 1 1111 1 1111 1 1'I
3 4 0 10 13

0
Win): "
Figure 4.4: Plots of AV, in units of 10“ and AV, in units of 10‘3 for the Harper
cyclotron: Note there is a change of scale at 10 in.

28

 

I010 I I I I I T r] I I I I I I rI I I I I I TI I I I T

)-

)-

 

o.“ILLLLIIILIIIJIILIIJIIIIIILI

 

 

OA-fI I I I I T] I I I IrI I I III r I I I I I I I

 

 

 

0.0 L L I l 1 m1 I 1 L L L I L L L L I L l I 1 I 1 l I
s 4 s a 10 is

Radius(in);Medcyc
Figure 4.5: Plots of V, and V, of the Harper cyclotron;
These data are obtained from the E. 0. Code.

29

 

rj I I I I I I I I I I I I I I I I I I I I I I

O»
I I I I I I I I I I I I I I l I I4
‘11‘111111111

 

 

 

 

a an
o o
-2 l l J_ l l l l l l l L l_ L l l 1 LL I l l l l l l A l l . -m
2 4 s s 10 12
Winkledeyc

Figure 4.6: Two plots of Au, from two different theories.
The plot of the difference between the u, obtained from the E. 0. Code and that from
the theory by Hagedoorn and Verster is depicted with a dotted line, and that of Au,
between the u. from the E. 0. Code and that from the new theory by M. M. Gordon
with a solid line. Note there is a change of scale at 10 in.

 

I

 

.- r.

0.15 '—
;

0.10 —

0.05 L to __
: f9 \
. J

 

 

 

 

p
L.
r
p
p
r
p
p
b
:-

 

 

 

.r..l..4.lfl.g..l..L.
.5 10 80 so 40
MuflnJfliao
Figure 4.7: Plots of f... and t" for n = 3,6,9 of the 130 MeV/A field data of the MSU
K1200 cyclotron: t; is depicted with a solid line, t. with a dotted line, t. with a
dotdashed line.

.W‘l E1“

31

 

 

IjIIIIWIIITII

 

 

 

Radius(in).fl130

Figure 4.8: Plots of kn for n = 0,3,6,9 of the K1200, 130 MeV/ A field.
k0 is depicted with a solid line, Is; with a dotted line, ls. with a dashed line, kg with
a dotdashed line.

if“! "f ‘ H

 

q
d
1
d

—
i
I
q
q
—
q
.1
1

 

a
I‘IITIITrIIIrIWT‘IlIIIIIIII

 

 

/\/

 

 

‘-25
AE/E x 10"
-so
. 4 I L I l I J I I I I I J I l 4 I
m: I I F I r r I f I l I I I r T I I
30-F—
10:;-
° C-
-10 L.
_'_ -A1'/1' x 10°
.... g.

 

 

 

 

 

O
p
O
8

Radius(in).fl 130

Figure 4.9: Plots of AE/E and -Ar/r of the K1200, 130 MeV/ A field: in units of

10“.

40

-ss

—00

Figure 4.10: Plots of Au, and Au. of the K1200, 130 MeV/ A field: AV,. is in units of

 

IIIIIWIIIIIIIIIII

 

 

 

IIIIIII

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I I I I I I I I I I I I l L L I I I
. I I I I I if I I I I f I I r I I I I I
’ s
; All, X 10
: W
E [
C
P
i

I I— I I I L I I I I I I L I I I IL I I
o 10 so so 40

Radius(in.):M130

10“ and Av, in units of 10”.

.., .
.4
' ' All-L

 

1.2—

10*-

0.0 h-

I

I I I I r I r r 1
I'
I 4 1 1 I I 1 L L

 

 

 

 

0.5 u

 

04

0.3

0.8

0.1

IIIIIIIIIIIIIIIIIII'IIII

 

0.0 4

 

 

 

"z
I I I L I I I I I
no as
wmmao .

40

Figure 4.11: Plots of u, and u, from the E. 0. Code for the K1200, 130 MeV/ A field.

35

 

I

 

 

 

 

 

 

 

 

 

I...L.L..l....l..-.
.030 10 30 N 40
Radius(in);u130

Figure 4.12: Plots of u, obtained from the E. 0. Code and two different theories for

the K1200, 130 MeV/ A field data.

Upper scale: The plot of u, obtained from the E. 0. Code is in a solid line and that
from the theory by M. M. Gordon is in a dotted line. Lower scale: The plot of u,
obtained from the E. 0. Code is in a solid line and that from the theory by Hagedoorn

and Verster in a dotted line.

 

 

P I I I r I I I I I 1 I l I I I I I I I I
l.
0.3 r-
h fa
0.2 P-
. f.
p
p
0.1 -

0.0‘...

 

 

 

p

 

akLLILLIkILIILL.ILIII
-

 

 

 

 

L
0 00 40 00 00
WWW
Figure 4.13: Plots of f" and t" for n = 3,6,9 of the Orsay field.

t3 is depicted with a solid line, t. with’a dotted line, t. with a dotdashed line.

F1 Aux-2‘21

37

 

 

 

 

 

 

_1.o llll I114 IJII n 1'1 II II 1::
0 00 40 00 so

Radius(cm).0rsay

 

Figure 4.14: Plots of It, for n- — 0, 3, 6, 9 of the Orsay field.
k0 is depicted with a solid line, It, with a dotted line, It. with a dashed line, In. with
a dotdashed line.

 

16 fit I I I I r u y I r.

I

10

I I I‘I

a
Ifi'fiiu

 

O
I I I I I I
E

H

X

H

O

b

 

 

b
b
p
I-
h—
b
p
b
-

 

 

 

-A‘r/'r x 104

IIITIIIIIITITIITIIIIIIIIIIIT‘

 

ILILIIIILJJIILILIILIIJ
.2

 

 

0 80 40 00 00
Radius(cm).0rsay
Figure 4.15: Plot of AE/E and —A‘r/‘r of the Orsay field: in units of 10“.

 

Av, x 103

IIIIjITIIjjIIIITIIITIJI

1 l l l L l L L L l l l L l l
-

 

I
llllllllllllll]lllllllll

 

 

l
r! I I l Ti I III—r I I I

7.5

Au,“ x 103

g .
IITIIIIIIIIITIIIIIIIII1II

 

l L L l— l l l l L l I L L l l

l 1 1

WP

 

 

 

ljlllIllljlllllllllljjljl

 

 

Redius(cm): Orsay
Figure 4.16: Plots of Au, and A (v’) of the Orsay field.
Note there is a change of scale at 50 cm.

L.
no

10

 

IIIT‘IIIIIITIIIII1II1IIII

 

 

 

 

 

 

30 60
Figure 4.17: Plots of u, and u, from the E. 0. Code and the theory by Gordon for
the Orsay field: The tunes obtained from the E. 0. Code are depicted with a solid
line, and those from the theory by M. M. Gordon depicted with a dotdashed line.
The imaginary values of u, are represented by negative values.

41

 

 

 

 

0.60 I I I I I I I IvI l I T I I l I I I I l l
oas—
y-
: ....... .. ..... \. I "
b.. "\ I
0.m'— | '
. I
. \.,°
425%-
r
y.
_ ”2
F'
-0.50r-
.4 J l l 1 LL L l l l l #21 l 1 1 LL 1 l
0 20 40 60 80

Figure 4.18: Plots of u, obtained from the E. 0. Code, from the theory by M. M.
Gordon and from the theory by Hagedoorn and Verster: The plot of u, obtained from
the E. 0. Code is depicted with a solid line, that from the theory by M. M. Gordon
depicted with a dotdashed line and that from the theory by Hagedoorn and Verster
depicted with a dotted line.

Chapter 5

Conclusions

For the Harper Cyclotron field, the absolute value of (AE)/ E and (AT)/T are
less than 50 X 10" and 20 x 10“ respectively which are fairly small (refer to figure
4.3). The absolute values of AV,. and AV, are less than 60 X 10" and 20 x 10”3
respectively (refer to figure 4.4). From this, we can say that the matching of the E.

0. Code and the theory is very good.

For the 130 MeV field, the absolute value of (AE)/E and (Ar)/T are less than
50 x 10’5 and 25 x 10"l5 respectively (refer to figure 4.9), which are quite similar to
those above. Also the absolute values of AV,. and Au, are less than 80 x 10" and
60 x 10"3 respectively (refer to figure 4.10). As can be seen, the absolute values of
the differences of the tunes are somewhat larger than those for the Harper Cyclotron

field. Nevertheless, the theoretical formulas are still quite good.

In the case of these two sets of field data, the flutter field is relatively small
compared with the average field. Hence the assumptions of theory are fairy well

satisfied.

For the Orsay field, the absolute values of (AE)/E and (AT)/T are less than
15 x 10"4 and 8 x 10'“ (refer to figure 4.15), which indicates that the matching of

the E. 0. Code and the theory is good, but not as good as for the other two cases.

42

“.l

L:-_:‘ n
I
-

43

Also the absolute values of AV,. and A(Vf) are less than 40 x 10‘3 and 80 x 10’3
(refer to figure 4.16) which are around five times as large as those of the 130 MeV

field.

The reason why the results for the Orsay field are worse than those of the Harper
Cyclotron field and the 130 MeV field is that the average value of the Orsay field (20
RC) is less than half that of the other two (45 RC). Since the flutter field is about
the same for all three, it is relatively more important for the Orsay field so that the

assumptions of the theory are less well satisfied.

One result worthy of note is that the theory seems to predict smaller values for
both 1!, and 1!, than those obtained form the Equilibrium Orbit Code. This conclusion
holds for all three fields investigated here, provided one averages out the fluctuations

produced by irregularities in the measured field data.

Appendix A

Appendix by M.M.Gordon

I Field Parameters

We assume that the median plane field B(r,0) is given as a function of the polar
coordinates 1' and 0. This function is usually specified in a polar mesh with fixed

intervals Ar and A0. Actually the field is represented by a Fourier series:

B(r,0) : 80(1') + Z 3,,(7') cos [n( 0 — fln(r))] , (A.1)

where Bo(r) is the average field, 8,,(7') is the amplitude of the nth harmonic, and
the function 0 = fin(r) gives the spiral curve for this harmonic. For a cyclotron with
N sectors and no imperfections, n takes on the values n = N,2N, . . . , so that the

angular period is 00 = 27r/N.

Assuming Bo(r), 13,,(1‘), and ,Bn(r) are known, we calculate the following dimen-

sionless parameters:

 

Bn
ft: = 23—0 , (A2)
_ dfln
tn _ r( d1. ) , (A.3)
r (18.,
kn _ E; (7;?) a (A 4)

__3|

A 75......1 ’

, r2 (1an
kn 2 E (71‘?) , (A.5)

where these kn and k; apply for n = 0 as well as the other n values. In addition, we

need:

II 7‘3 (1380
ha —- E; (“a—73‘) . (A.6)

All of the above parameters are required at each r value in order to carry out the

calculations.

II Equilibrium Orbit Calculation

For a point charge q moving in the median plane with a constant momentum p, the

differential equation for the trajectory can be derived from the following Lagrangian:
L(r,1",0) = p(r’ + 19 )’2‘ — q/ rB(r,0)dr (A.7)
where dots are used to denote 0 derivatives so that 7" = dr/dfi, for example.

We assume that the orbit is nearly circular so that (ii/r)2 is small. Expansion of
L then leads to :

1r°=

Loni-.0) = 5:»;- — / ( qrmno) — p) dr (M)

where terms of fourth and higher order terms have been dropped. The resultant

differential equation is :

%(;)+;(;)221—%r}3(r,fi). (A.9)

The equation for the equilibrium orbit is written as follows:
r..=ro[1+y(0)]’, (AJO)

where y ( 0 ) has the same periodicity as the field. For simplicity, we take ( y(0) ) x 0,
and our reference radius r0 therefore has an unconventional definition:

1
2

)3. (A.1l)

ro=((r.o)

- ‘Jr :37 h

 

46

It is this n, which is used as the independent variable in the formulas given below,
and it should be clearly understood that for every circle of radius r in the given field,
there is an equilibrium orbit having r = r0 . The field parameters associated with this
value of r0 then determine the momentum p of the particle and hence the remaining

orbit properties.

The differential equation for 31(0) is then given by:

§=%(l+y)[l—%r8(r,0)] , (A.12)

which is correct at least to second order. Expanding the right side in powers of y, the
result is:

g 2 — M — 17(0) — M'y — F'(0)y — -;-M"y’ — , (A.13)

where we have dropped all terms of third and higher order in the flutter field, that

is, the oscillatory part of B(r,0). The expansion coefficients are given by:

M = $(A—1), (A.14)

W) = gzgf.cos(n(0-a)). (A45)

M’ = A(l-lko)+-12~(/\~—l), (A.16)
F'(0) ——- gxgusf.”moses-a»+2nf.t.sin(n(o-—a))1,

(A.17)

1)" = A(3+7k.,+2k;,) (A.18)

and where the parameter A is defined by:

A = imam) . (A.19)

If the flutter field vanishes so that the field is axially symmetric, then we have

only the zero order solution yo = 0, so that rm 2 1'0, and A E 1, or p : qroBo(ro)

47

which is obviously true. The first order value of y(0) is obtained by solving:
:7 -+ My = — F(0) . (A.20)
with the solution given by:
M) = 5A2“: y.cos(n( o -— A. )) , (A.21)

Will] 3],, : £51513?“
A technique for obtaining an improved 801M501! for 11(0) will be discussed in a later

section. We should note that a more detailed development of the material in this and

the following section can be found in a previous paper [8].

III Calculation of p and w

Since ( y ) = 0, the average value of eq. A.13, above yields:

1
0 = M + ( F'(0)y ) +51» "( y2 ) . (A.22)
Hence we obtain the following formula for A:
1 2 1 n 2
1—»\=Z,\ Z[(3fn+2kn)yn+§M y" . (A.23)

Since the right hand side is of second order, an iteration scheme is used to obtain the
value of A and hence p. That is, we start by setting A = 1 on the right and use the
resultant value of ( l -— A) for the next approximation and so on. Once A is found,

we obtain the value of p and hence the energy E.

If C is the circumference of the equilibrium orbit, and if r is the orbit period, then:
C 2 v1” = 21rR. (A.24)

Thus we have:

sz/w=((r’+1=’)%), (A.25)

m...-

48

where w = 27r/1'. If we is a fixed reference frequency, we find:

we Towo

1 2 2 2
I: - 2 .
v 1+8A 2“} n +1)y,, , (A26)

 

7.7
correct to second order. This formula is then used to determine the deviation from

the isochronism condition, that is, the condition w = wo independent of energy.

In addition to R defined above, it is useful to obtain the average radius given by:

F=(r,o)=ro(l+%x\22y2) (A.27)

u
again correct to second order. This f is the average radius occurring in the standard

output of the equilibrium orbit code.

IV Radial Oscillations

We start from the exact Lagrangian given in eq. A.7 and set:
7' : reo 'l' 3
and
1" = 1"“, + :i: .

An expansion of the Lagrangian, L, to second order in a: and :i: then gives:

1 '2 1
L = 5:; + bani: + -2—c:c2 , (A.28)

where a, b, and c are periodic functions of 9 given below. If we now set x = afi, then

L can be transformed into the standard form:
1 ., 1 , ,

where K(0) 2 (12(1) — c) —- h2 + h , with h : ci/a

q...
h.

49
In this case, the equation for the linear oscillations becomes:
30+ K(0)€ = 0 (A.30)
and since K(0) is periodic, this is a Matthieu - Hill equation.

For the radial oscillations, the parameters a, b, and e are given by:

a.2 = (1'2+1i2)%/r2 , (A.31)

b = —r1"(r2+1"2)—%, (A.32)
_ .2 2 -2 -% __ 9 I? r

c — r(r +r) p[3r( 8)] (A.33)

where all quantities are. evaluated at r : r,o(0) and 1‘ = 1i,o(0). Hence we find:

Kz(0)=gl-:[-;;(1'B)](1+:;)g -l-h.1—hi, (A.34)

where here, h, :- z — f .

This expression for K,(0) is exact, but it must be evaluated along the equilibrium

orbit and this can be done only approximately.

We divide K(0) into two parts:
K(0) : n2 — 9(0) (A.35)

where ( K (0) ) = n2 so that ( 9(0) ) = 0. Thus 9(0) is the oscillatory part and
contains the alternating gradient focusing. If n2 is calculated to second order, then
9(0) should be obtained only to first order since its contribution to the focusing is

then of second order.

For the radial motion specifically, a detailed calculation gives the following values:

gz(9) = 4}: (nfntn)si"(n(0-fin))-

1
A: [f,.+k..1—Ay..(1+3ko-1—k3)+-2~n2yn c08(n(0-fln))

(A.36)

‘L

”.1

l'

50

and

1
2 _ _2 I_ 2 2
n, — A(1+ko)+2/\;yn(f.+3k.+kn nfnt,)+

%A2(6A(l+ko)—l)2n2y:+

Tl

—:§,\3(1+11kO +11k3+ 2kg); yfi. (A.37)

7%

It is these quantities that are used to calculate V,., the radial tune.

In the limit of large N values, we obtain the simple smooth (or high frequency )

approximation value:

g,(0) ——+ —Z(nfntn)sin(n(0—fln)), (A.38)

1
n: ——. ”kc—5211313,. (A.39)

Thus g. reduces to the alternating gradient focusing produced by the spiral , while
n: reduces to the constant gradient focusing term and a negative term corresponding

to Laslett focusing ( or defocusing here).

V Vertical Oscillations

Considering only the linear oscillations, the equation of motion for z is:

d pi 313 1‘68
55(W)*ql"3¥“755 2’ (“0)

which corresponds to the following Lagrangian:

12°2 1
L = 5;; + Ez-cz2 , (AAl)

. l ' . . .
where a2 = (1"2 + 1'2 )’ and c = g [ r955:- — 5%? ] . The motion here is directly analo-

gous to that of the previous section.

A.-.‘ A...

.ll

51

Changing variables from z to C such that z : aC, the Lagrangian can then be

tranformed into standard form:
1 ‘2 1 2

where K,(0) : —a2c + h -— h”, where h = ti/a. . Using the above values for a and c,

we then find the following exact form of K,(0):
1
3

41-91225 :‘3 --_2
K,(0)—p[rao r 61‘ 1+7.2 {~h h. (A.43)

Here again all quantities must be evaluated along the equilibrium orbit which can

only be. done approximately.

The Matthieu - Hill equation for C then becomes
5+ K,(0)C :- 0 . (A.44)
We again separate K, into two terms:
K40) = .3 — 940) , (A45)

where ( K,(0) ) 2 n3. If we then evaluate gz(0) to first order and n: to second order,

the following values are obtained:

g,(0) = Aannt,,sin(n(0—)6,,))+
A2:(kn+Ay,,(2koel-k6)-l gnzyn)cos(n(0—fln)) ,
(A.46)

and

n: : —Ak0+ éA’Z yn(n2fn+n2fnt:—2kn—k;) —

31,—)?(14 MHZ n’y: — §~A3(6ko + 91.3 + MHZ y}.-

n

(A.47)

52

If we again consider the limit of large N values, these become

g,(0) —2 Z nfntnsin(n(0—fln)) , (A.48)
1 .-.. 1
n: _+ —k0 + 5 2. f3 + ~2— 2 fjtf, . (A.49)

We again have the alternating gradient term in g,(0) and the combined effects of
constant gradient, Thomas, and Laslett focusing in 5:. The more complete formulas
above ( and in the sec. IV) show additional focusing effects of gradients in both the
flutter field and the average field that become important in the edge region of the

magnet.

VI Born Approximation

The Matthieu — Hill equation encountered in the preceding two sections can be

written:

55 + 1:22: 2 g(0):c , (A.50)

where g(0) is periodic so that g(0 -l 00) = g(0), where 00 : 2x/N . We choose a
Green’s function G(0, 0') which is a solution when the right side is replaced by a delta

function 6(0 — 0’) so that:

G(0,0’) = 0 ,if0<0’

G(0,0') = isin n(0 — 0') ,if0 > 0' . (A.5])
In this case, the general solution becomes:
0
22(0) = 20(0) + —— / sin ( A(o — 0' ) ) g(0'):1:(0') (111' , (A.52)
0

and

:13::i:°(0)+/00 cos(n(0—0'))g(0'):c(0')(10', (A.53)

2:1

53

where 130(0) and its derivative 2:."(0’) are the zero order values.

We use two independent solutions 31(0) and 233(0) to construct the transfer matrix:

__ 221(0) 32(9)
111(9)- ( 131(0) 42(0) ) , (A54)

with initial conditions chosen so that AI( 0 = 0) = I, the unit matrix. Thus we take:

312(0) 2 cos n0 , (A..55)
1
223(0) 2 it sin n0 . (A.56)

When these values are inserted into eq. A.54 above, we then obtain 2:1 and 2:; at all

0 values.

The tune, u, is obtained from the trace of M at 0 = 00 :

2cos(u0o) = Tr[M(0o)l

: 131(00)‘l'2f32(00) . (A.57)
Using the solutions given above, we then find:

0
2cos(u00) = 2cosn00+ifo g(0')sin(n(00—0'))x,(0')+
0

cos(n(00—0')):cg(0') d0'. (A.58)

Thus, if 9(0) 2 0, we find 11 = n, as expected.
If 2:, and 2:; are calculated to first order in g(0) using the eq. A.58 above, we

obtain cos ( V00) correct to second order:

sin n00 ‘0

2cos(u00) = 2cos(n0o)+ —-——- g(0')d0'—|-
I: o

g. 9(4)]: g(0")sin(~(o'—0"1)

x sin ( 14(0. — 0' + o")) d0’d0" . (A.59)

54

We should note that this expression gives the exact result when g(0) corresponds to

the focusing produced by two thin lenses located within the sector.

For the case of interest here, we can write:

9(0) 2 2: 9,, cos (n0 + 7n) (A.60)

with yo = ( g(0) ) = 0 so that the first order term in cos(u00) vanishes. A straight-

forward but tedious calculation of the integrals yields for this case:

 

 

92
n2 3E . (A.61)

cos(u00) : cos(n00) — 211-00 (51111900) 2

K’ n

Using the half-angle formula, the alternative form is obtained:

 

- 2
sin2 ( 1100/2) = sin2 ( n00 /2) + $0381“ K00 Z 2 9" (A.62)

1:00 11. —4n’ '

n

We should note that the above formula can be used not only below the stop-band
(u < N/2), but also within the stop-band where |cos u0ol> 1. Also the singularity in
the term with n = N where n = N / 2 is cancelled by the factor sin 1:00 in front. That

is, in the limit n —-2 N/Z and 1:00 ——2 1r, we have:

 

grit—es (A...)
which is finite.
The above formula can also be used when K.2 < 0 simply by setting
n = i In]. That is, by making the following changes:
cos 1:00 —-2 cosh [ml 00 , (A.64)
sinz (1900/2) ——» —sinh2( |n| 00/2) , (A.65)
sinn02 sinh [ml 00 (A.66)

 

 

.___+ __...__.._.
I400 '5' 00

55

It often happnes that the vertical oscillations are characterized by quite small

values of It: as well as V3. In such cases, the above formulas simplify to:

1/2"’Is:2Jr-1-Z:g'21 (A 67)
z z 2 n2 ' '

n
This formula shows why the most important terms in g: are those proportional to
11’. Also, this formula leads directly to the smooth approximation result when use is

made of the values given in eq. A.67, namely,
2 1 2 2
uzz—kM 2: fn(1+2t,,) . (A.68)

Of course, this result could be improved by using the more complete formulas for n:

and g,(0) given in eq. A.62.

VII Equilibrium Orbit Improvement

To improve the value of g(0) given in eq. A.13, we consider the linear terms:
2+ [ M' + F’(0) ]y = — 17(0). (A.69)
The homogeneous part of this equation has a particular ( Floquet ) solution given by:

y; = w(9)61(1)(i14<15(49)) (A70)

where 111(0) and [(M0) —- 0] are real and periodic with the same periodicity as the field.
The “tune” here is )1 which can be calculated using the formulas in the preceding
section by setting n” : Ilf’ and 9(0) : —F'(0), where M' and F'(0) are given in

eq. A.16 and eq. A.17.

The function w and 43 can be obtained from :

1112925 2 1 (A.71)

171+ [ 111' + F'(0) ]w = nil/w3 . (A.72)

56

Note that if F'(0) = 0, then w : 1, Q5 : 0, and p2 = M'. These then are the zero-order

values.

To obtain the required solution of the inhomogeneous equation above, we change

variables from g(0) to ”((12) using a Floquet transformation:

11(9) = w(9)n(¢) (A-73)

where 17 is the solution of:

12.71 + 2 _ —w3(0)F(0) (A 74)
44’ u n — - .

These equations could, in principle, be used to determine the equilibrium orbit at

least to second order.

For simplicity, we treat here only the first order solution. That is, we set w = 1,
(b = 0, and obtain as a result the exact same 31(0) as given in eq. A.21, but with yn

now given by:

y=—f"
" 122—112.

 

(A.75)

That is, Al’ is now replaced by p2 which can be calculated as described above. Since
[1.2 differs from M ' by the inclusion of second order terms, it should improve the
results. We should also note that all the formulas given in the preceding sections are

in terms of 31,. rather than its explicit form so that either expression can be used.

N

57

References

M.M. Cordon, Particle Accelerators 16 (1984) 39.

L. Smith and A.A. Carren, Lawrence Berkeley Lab.,UCRL Report #8598, January,

1959.
G. Parzen, Ann. of Phys. 15 (196]) 22.
H.L. Hagedoorn and NF. Verster, Nucl. Instrum. Methods 18,19 (1962) 201.

Samuel D. Conte and Carl de Boor, Elementary Numerical Analys — an algorithmic

approach, McCraw-Hill,New York, 1980.

II.Blosser et.al.,Advances in Superconducting Cyclotrons at MS U, 11th int. Conf.

on Cyclotrons and their applications (1986) 157.

S.Cal‘es for Agor project group, Agor A Superconducting Cyclotron For Light And

Heavy Ions, 11th int. Conf. on Cyclotrons and their Applications ( 1986) 184.

M.M. Gordon, Calculation of Isochronous Fields for Sector-focused Cyclotrons,

Particle Accelerators 13 (1983) 67.