WIHHHHI

WW

I”

133
307

 

RAPID TRAVERSAL OF 3/ 3 RADIAL
RESONANCE NEAR THE CENTER OF A
SECTOR FOCUSED CYCLOTRON

Thesis for ”IO Degree of M. S.
MiCHIGAN STATE UNIVERSITY

Joe Earl Stover
1960

was mumm;mavmmnmmrmmn

3 1293 01743 0137 J

 

LIBRAR Y *
Michigan Stan
University

oevsmfli'fl o
51 A“

mews"
99‘ ”a“

E WW

0 w

RAPID TRAVERSAL OF 3/ 3 RADIAL RESONANCE NEAR THE

CENTER OF A SECTOR FOCUSED CYCLOTRON

BY

JOE EARL STOVER

AN ABSTRACT

Submitted to the College of Science and Arts of
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

MASTER OF SCIENCE
Department of Physics

1960

WW

 

JOE EARL STOVER
ABSTRACT

A nonisochronous average magnetic field for a three sector
cyclotron is determined numerically that will give adequate axial
focusing to a proton accelerated by an energy gain per revolution
of 280 kev. without excessive phase slip.

The effects. on a beam of accelerated protons, of the traversal
of the 3/3 radial resonance which are introduced by using this
average magnetic field are investigated by use of the Michigan '

State University's digital computer, the MISTIC. It is found that
a beam of 70-kev protons a half inch wide can be accelerated through

this radial resonance without severe beam distortion.

RAPID TRAVERSAL OF 3/ 3 RADIAL RESONANCE NEAR THE

CENTER OF A SECTOR FOCUSED CYCLOTRON

By

JOE EARL STOVER

A THESIS

Submitted to the College of Science and Arts of
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

MASTER OF SCIENCE

Department of Physics

1960

Emit—b
H-7- {,9

ii

ACKNOWLEDGMENTS

I wish to express my sincere appreciation to Dr. H.
B. Blosser and Dr. M. M. Gordon for their advice and
suggestions during the progress of this work.

I am grateful to Dr. H. G. Blosser and T. R. Arnette
for furnishing the necessary computer programs and to the
United States Atomic Energy Commission for making this

work financially pos sible .

TABLE OF CONTENTS

Introduction ..............................................
Units and Coordinates ....................................
Axial Focusing and Nonisochronism ........................
Radial Stability and Resonance Traversal ............ . .......
Beam Distortion ..........................................

Conclusion ...............................................

Bibliography .............................................

iii

27

.42

59

60

63

Figure

10.

11.

12.

13.

14.
15.
16.

17.

18.

LIST OF FIGURES

The magnetic field in the medial plane ...............
The axial focusing of a three sector magnetic field .....
The flutter of the magnetic field .....................

The axial focusing frequency of the isochronous
magnetic field ......................................

The average magnetic field necessary for a constant Uz .

The nonisochronous average magnetic field with
acceptable phase slip ...............................

The axial focusing frequency of a proton in the
nonisochronous field ................................

The phase slip for a proton with an energy gain per
revolution of 280 kev. ...............................

The axial motion of a proton in the nonisochronous field.
The amplitude damping of the axial motion ............

The radial oscillation frequency of a proton in the
nonisochronous magnetic field .......................

The stable and unstable trajectories in the median plane

A phase diagram near resonance for a three sector
magnetic field . ....................... . ............

The stable regions at energies near resonance ........
A phase plot 220 Rev below resonance ................
The distortion of a 0. 27 inch central beam ............

The initial conditions for protons accelerated and
decelerated from resonance .........................

The distortion three revolutions above resonance ......

iv

Page
17
18

19

20

21

22

23

24
25

26

37

38

39
40
41

50

51

52

Figure
19.
20.
21.
22.
23.

24.

The

The

The

The

The

The

distortion five revolutions above resonance

distortion seven revolutions above resonance

distortion nine revolutions above resonance

distortion two revolutions below resonance

distortion four revolutions below resonance

distortion five revolutions below resonance

Page
53
54
55
56
57

58

INTRODUCTION

The performance and energy of conventional, fixed frequency
cyclotrons (i. e. those with azimuthally uniform magnetic fields) are
severely curtailed by the contradictory requirements placed on the
magnetic field shape by the separate considerations of focusing and
acceleration. These requirements are: (1) The magnetic field should
increase as a function of radius in order to maintain constant orbital
frequency for particles. (2) The magnetic field should decrease as
a function of radius in order to give axial focusing.

In 1938 L. H. Thomas (1) proposed a cyclotron (now commonly
called a sector focused cyclotron) which employs azimuthal variation
in the magnetic field in such a way as to satisfy both focusing and
acceleration requirements. The introduction of sectors into the field
introduces additional terms into the Kerst-Serber equations (2’) for
the focusing oscillation frequencies. These terms, which give axial
focusing, depend upon the mean square variation of the field about a
circle from its mean. Constant orbital frequency for particles (called
isochronism) is maintained by increasing the average magnetic field
as a function of radius in order to compensate for an accelerated
particle's relativistic mass increase.

Due to restrictions imposed on the possible shapes of magnetic
fields by Maxwell's equations, the isochronous average field gives

rise to axial defocusing forces. Sector focusing will overcompensate

for this situation at large radii. However, the mean square variation

of the field from its mean must tend to zero as the radius approaches
zero, hence axial defocusing is to be expected at small radii. Axial
focusing in this region has been successfully enhanced in the Delft (3)
and Illinois (4) four sector cyclotrons by the abandonment of isochronism
at small radii.

Cohen (5) suggested that the following procedures be used to
investigate the feasibility of overcoming axial defocusing at small
radii: (1) the numerical construction of an average magnetic field to
supplement the axial focusing in the central region, (2) an investigation
of the feasibility of the traversal of the radial oscillation resonance
which is necessitated by features of this average field. In this thesis,
these suggested procedures are carried out for the three sector
magnetic field shown in figure 1.

The numerical calculations to determine the appropriate average
magnetic field are based on formulas obtained from the smooth approx-
imation (6). These simple formulas are often rather inaccurate, but
they are useful for estimates. The orbital properties of ions in the
magnetic field obtained are determined from a sequence of computer
runs of the Michigan State University digital computer, the MISTIC.

The two principal computer programs which are used are the
”equilibrium orbit code" (7) and the ”general orbit code" (8). These
programs accurately determine the desired orbital properties by

integration of the particles‘ equations of motion. The equilibrium

orbit code takes a given median plane field and calculates the desired
properties of the equilibrium orbits and the small amplitude oscillation
frequencies as a function of energy. The general orbit code is used to
track ions initially displaced from their equilibrium orbits. This code
is also used to simulate acceleration by increasing the ions’ energy

at specified azimuthal angles, so that the effects of resonance traversal

can be investigated.

UNITS AND COORDINATES

For convenience, cyclotron units as defined by Gordon and
Welton (9) are used throughout this thesis. These are dimensionless,
relativistic units well suited to cyclotron work. The rest mass,

MO. and the velocity of light, c, are taken as unit mass and velocity.
Since the cyclotron considered here is a fixed-frequency accelerator,
the unit time is defined in terms of this frequency asT = 3. For a
particle with charge e, the magnetic field unit is
M can
B r— O 0
e
and the unit length is a = c/ .
(J
For a proton in the magnetic field considered in this thesis
a = 90.4 inches 5229. 4 cm
1/T = 12.18 x10”10 sec.-1

The axial and radial oscillation frequencies Uz and Ur are
given as the ratios of the actual frequencies of oscillation to the fixed
frequency of the cyclotron.

A right cylindrical coordinates system is used with the origin
located at the geometric center of the magnetic field. The z = 0 plane

is taken as the plane of symmetry of the field, parallel to the pole face

of the magnet.

SECTION I
AXIAL FOCUSING AND NONISOCHRONISM

Consider a magnetic field which has the property that there is a
plane such that the field at all points in the plane is perpendicular to
the plane. This plane of symmetry is called the median plane and is
taken to be the z = 0 plane in cylindrical coordinates. There exists
(10) a closed curve, called the equilibrium orbit, in this plane along
which a particle with a certain momentum can move. In order to be
usabl‘e‘afis a guide field for accelerators, the magneticifield must be
such that the motion of a particle near the equilibrium orbit is stable
in the following sense: if a particle, whose momentum is appropriate
to the given equilibrium orbit, is started with a small initial displace-
ment and small initial angle from the equilibrium orbit, it will remain
near the equilibrium orbit for all time.

Consider a charged particle moving between magnetic poles
which produce a field in the z = 0 plane shown in figure 1. If the
particle moves in this plane of symmetry it will experience no force
in the z direction. However, for a charged particle with a small 2
displacement from its equilibrium orbit a 2 component of the magnetic
fOrce is present. Unless this is a restoring force which will cause the
particle to oscillate about the median plane, its axial motion is unstable.

The axial force on the particle is given in cylindrical coordinates by

=' - 1
Fz eVrBe eVeBr . []

6
where e is the particle's charge, VI. and \gare the radial and azimuthal
components of its velocity, respectively; and B9 and Br are the azi-
muthal and radial components of the magnetic field at the location of
the particle.

Consider the second term on the right side of equation [1]. If
the expression for the magnetic field is expanded in a Taylor's series

about the median plane. equation [2] is obtained.

DB (r, 9, z)

e = e r . . . .
Br(r, ,z) Br(r. , O) + z 32 0+ [2]
z:

 

Due to symmetry of the field about the z = 0 plane the first term in
equation [2] is zero. Maxwell's equations give curl B = O in this region
or,

5B 5B

r z
= 3
az Br [I

 

 

Hence the second term in equation [1] is, to the first order, given by

BB

z
-eVez Br [4]
2:0

 

It is the axial force averaged around the particle's orbit that determines

the stability of the motion. If the average field given by
211
<13 >=-1— B (r,e)d8 [5]
z 211 o z

3
is isochronous, its partial derivative with respect to r is posnive,

then axial defocusing is obtained from this term.

7

The first term in equation [1] is the axial focusing force proposed
by Thomas (1). The basic concepts of this sector focusing can be
understood by consideration of a simple example. Consider the cyclo-
tron magnet to be composed of three wedge shaped magnets, each with
uniform magnetic field over the region of the wedge and a much weaker
uniform field interposed between the wedges, as shown in figure 2.
It can be seen from the geometry that as a charged particle enters
the strong field region the radial component of its velocity is positive.
The curvature of the magnetic lines at the edge of the magnetic wedge
is shown in figure 2. A particle with a trajectory not in the median
plane will encounter an azimuthal component of the magnetic field
whose direction depends on the side of the median plane in which the
particle is moving. For either positive or negative z, the cross
product of the radial component of the velocity and the azimuthal com-
ponent of the magnetic field is in a direction such as to give a restoring
axial force. In the region where the particle leaves the magnet, both
the radial component of the velocity and the azimuthal component of
the magnetic field have changed directions, thereby giving axial

focusing.

In principle, it appears that the focusing forces derived from the

azimuthal variations in the magnetic field could be made of sufficient
.strength to overcompensate for the defocusing forces arising from the
radial variation in the isochronous magnetic field. However, in actual

practice this cannot be done at small radii. Detailed analysis (6)

shows that this focusing strength is specified by the mean square
deviation of the field about a circle from its mean. This deviation
which is called the flutter can be made as large as necessary at large
radii by proper azimuthal variation in the magnetic field but for con-
tinuous fields must equal zero at r = 0; hence it is small for small
radii.

With the above principles in mind, the axial motion of an ion in
a sector focused cyclotron can be discussed. Figure 1 shows a
"contour map" of the magnetic field to be considered. This plot was
obtained from measured magnetic field values of a scale model of the
magnet for the proposed Michigan State University cyclotron described
elsewhere (17). It is a three sector, weak spiraled magnet designed

for a strictly isochronous average magnetic field which is given by

MIJ
0
21./2
2
C

 

<Bu)>= BA]

r

 

e1-

where MO and e are the rest mass and the charge. respectively, c is
the velocity of light and Uis the orbital frequency. The flutter, F,

as defined above is shown as a function of radius in figure 3. The
axial focusing frequency, Uz, for a proton in the isochronous magnetic

field is shown in figure 4. The axial focusing frequency was obtained

from. orbit integration by digital computer and is quite accurate.

It is noted from figure 3 that F is zero at r = 0 and quite small
at small radii. It then increases rapidly with radius. It is seen from
figure 4 that U2 has a relatively constant value over a large range of
r values, but falls off rapidly near the center of the field. There is
defocusing at small radii.

This behavior of the axial focusing frequency can be understood
in light of the approximate equation of motion for an ion with a small
axial displacement from its equilibrium orbit.which is given by several
authors as (6, 12)

2

d z
dGZ

 

+ U222 = o 1 [6]

2
where U2 is given by the smooth approximation as

2 r d<B>. 2
Uz "<B> dr +F(l+2tan a) [7]

 

where <B> is the average field, F is the flutter and a is the sector
spiral angle. The spiral is weak in the field considered here, hence
the last term in equation [7] can be neglected for approximate calculations

and equation [7] becomes

< >
U2: r dB +F [8]

z -<B> dr

 

Z .
As previously stated, 2] is negative where the flutter 18 not
z
sufficiently large to compensate for the defocusing due to the increase
with radius of the isochronous field and axial defocusing is to be

expected at small radii. Axial focusing can be successfully attained

10
in this central region from the first term in equation [8] by abandon-
ment of isochronism if the ions do not get out of phase with the
accelerating voltage.

The procedure used to determine the necessary average
magnetic field to give axial. focusing was as follows: (1) A <B(r) >
which would give a desirable axial focusing frequency was calculated
from the smooth approximation. (2) The phase slip for a proton with
a given energy gain per revolution was calculated; if it was greater
than 90 degrees, this field was abandoned and a <B(r) > with a less
desirable U2. was tried. It was assumed that the azimuthal variation
of the magnetic field would not be changed by the alteration of <B(r) >.

The first trial <B(r) > considered was the average field
necessary to maintain Uz at a constant value of 0. 24 over the range
of small radii where the sector focusing is weak. This field was
constructed by a numerical integration of equation [8] by Simpson's
1/3 rule. The flutter shown in figure 3 was used. <B(r)> was
calculated by this method from r = O. 17 to r = 0. 02 cyclotron units
at radius increments of 0. 01. The resultant average field and the
isochronous field are shown in figure 5. After the first and second
differences of this calculated field were smoothed, Uz and the phase
slip were obtained at successive energy values by use of the equilibrium
orbit code. The results of this computer run showed that Vz remained

relatively constant at the desired value of 0. 24 over the entire central

11
region of the field. However, the phase slip was excessive. Hand
calculations showed that a proton with an energy gain per revolution
of O. 28 mev would drop more than 90 degrees out of phase with the
radio frequency accelerating voltage before it traversed 1/4 the radius
of the magnetic field.

This average field was abandoned because of the unacceptable
phase slip but valuable information was obtained from the computer
run. As shown by Heyn and Khoe (3), if the non-isochronous and the
isochronous fields are plotted as functions of the radius squared, the
area between these curves is proportional to the cumulative phase
slip. The area above the isochronous plot will correspond to negative
phase slip and the area below to positive phase slip. This can be
approximately understood by noting that the phase slip acquired by
a particle during a given revolution is proportional to the difference
between the field experienced by the particle and the isochronous field;
and due to acceleration, the radius increase per revolution for the
particle varies approximately as the square of the radius of its orbit.
The phase slip of an ion is quite sensitive to the energy gain per
revolution, for it is this which determines the number of revolutions
that it will remain in a region where a given phase slip per turn is
acquired. The phase slip as a function of radius can easily be cal-
culated from the results of an equilibrium orbit code run if the energy

gain per revolution is specified. Hence, the area corresponding to a

12
phase slip of, say, 20 degrees can be determined once a computer run
is made with the equilibrium orbit code using a nonisochronous magnetic
field. This was done for a proton with an energy gain per revolution of
280 kev using the average field shown in figure 5.

A second trial average magnetic field was numerically calculated
with the general field shape suggested by Cohen and others (5, 3). At
the center, this second field is made larger than the isochronous one
and it falls off with increasing radius until it is well below the isochronous
field, then increases more rapidly than the isochronous field at radii
where the flutter is strong enough to give adequate focusing. The
phase is thus'. shifted negative and back toward and perhaps through
zero to a positive value.

With information obtained from the above mentioned orbit code
run, the phase slip can be approximated accurately and quickly by
graphical methods. The average field need only be calculated roughly
from equation [8] since it was found from the results of the previous
orbit code run that smoothing the first and second difference of
<B(r)> is sufficient to obtain a smooth Uz as a function of radius plot.

The procedure for this numerical calculation was as follows;

A field value, which was less than the corresponding isochronous
field value, was chosen at a radius where the square root of the
flutter equaled a desirable value for U . The field to be calculated

Z

was assumed to go throughla minimum at this point. The field at a

l3
radius 0. 01 cyclotron unit nearer the center was calculated from the

following approximation of equation [8]

 

 

 

 

 

 

 

 

 

+ B -B F
U2( )_ _ri ri-l 1.1 i ('ri)+F(ri-l)
z ri-l B,+B, r, -r 2,
1 1-1 1-1 1
01“ - 1)
2 ri' ri-l
1+I/ (r. )'F
p z '1-1 ri+r1-1
B. = B. [9]
1-1 2 _ r -ri_1 1
1- U (r, )-F 1
z 1-1 r +r,
_ 1-1 J
where
F + F
E‘: (ri) (ri_1)

2
The field value at each successive radius increment of r = i" . 01 was

2
calculated from the previously determined field value. ‘ U (r. 1) was
z

used as an adjustable panameter to maintain the proper area between
the calculated and the isochronous average magnetic fields. If 24 could
not be. maintained as a slowly varying function of radius, the
calculations were abandoned and another initial field value was chosen.
The average field which resulted from these calculations is shown in
figure 6.

The equilibrium orbit code was used to check the axial focusing
frequency and the phase slip at successive energy values of AB = 280
kev over the energy range where the average field deviates from

isochronism. Figure 7 shows the plot of the axial focusing frequency

which was obtained. The phase slip for a proton assumed to spend a

l4
half revolution on successive equilibrium orbits of AB = 140 kev is
shown in figure 8.

The axial focusing obtained from the radial dependence of the
magnetic field is very good in the central region of the field. Uz
falls off only slightly at small radii to a value of 0. 17, which occurs
at the radius which corresponds to the equilibrium orbit of a 70 kev
proton. The phase slip acquired is quite tolerable over this range of
radius values and seems to diminish at larger radii.

The axial focusing is demonstrated in figure 9 which shows the
z motion of a proton accelerated in this magnetic field with an energy
gain per half revolution of 9-2}?— cos q). The data for this plot were
obtained from a computer run with the general 'orbit code. The initial
conditions of the proton were: E0 = 70 kev, r = . 0062, Pr = . 00013,
zO = . 00300. PZ = . 00000, AE = 280 kev.

The maximum phase slip was found to be 1’ 18 degrees. This is
in good agreement with (p shown in figure 8. Table I shows the good
agreement between the NZ obtained from the equilibrium orbit code
and the Uz of an accelerated proton which was computed from 1/2
wavelengths of the z motion shown in figure 9. Uz is shown at energies
which correspond to the maxima and minima of this plot.

The damping of the axial motion shown in figure 9 can be under-

stood in light of an approximate solution to equation [6]. Assumethat the

z motion as a function of the azimuthal angle can be described by

15

 

de' Z
2 + 2/ zzo
as Z

2
where U is a slowly varying function of 9. If 2/ is positive as
z z

indicated by figure 7 the WKB approximation (10) gives

 

9
z: a COS[—[Ud9+b]
VUz 002

where a and b are arbitrary constants.
This is the equation of a pseudo-harmonic oscillation with

, a . .
varying amplitude of? and varying instantaneous wavelength of
z

A

ll
‘ N
it],

Hence the square of the amplitude as a function of the wavelength
should plot a straight line. Figure 10 shows the good agreement
between this approximation and the amplitude-wavelength relation-

ship obtained graphically from figure 9.

Table 1

U2 of an Accelerated and Nonaccelerated Protons

 

 

———=;=r—
E mev Uz (figure 9) U2 (orbit code)
0.616 0.168 0.176330
1.101 0.186 0.186017
1.817 0.187 0.187869
2.361 0 187 0.188186
3.194 0.188 0.188815
7.822 0.2154 0.212180

8.215 0.2156 0.218960

 

17

 

fixed point orbits

 

*'- unstable fixed point orbits

Fidd 26.|R

'— squilibriun orbit

—— stable

Figure l. The magnetic field in the medial plane.

Equilibrium orbit of an ion in a three sector cyclotron

/weak field region

strong field
I'. region

Magnetic field at edge of the wedge

 

 

 

 

 

 

 

 

 

 

Figure 2. The axial focusing of a three sector magnetic field.

 

 

,0.

5

5

no

0

0

C,

{1

O

1‘."

0

1’
11

$575 zomsoqowov .3

mm.

«um.

.-

ow. o

q

0

N
W’s-l
.

410-!

«V

‘C

‘c

C

\x

C

.Eofl ofiocmme 05 Ho

0

C

S.

C

C

Amv

 

AmV1A mV

noun—5G 03H.

11
111

.m madma—

1H0.

1No.

Inc.

.Two.

umo.

Lb .

rwo .

 

 

moo.

mm.s
_

m

.vm.

8.235

.363 030:me mdoconsnvog 65 mo >oaodw9~w 9:353 338 2:.

5.375 zomsoqowov H
mm .
_

N. o
n

66.4“.

ow. mm. 9.2.3. 2. 3. 8. so. so.
_

.6 6.23mi

 

 

N

1

_

a»

at

c

a

O

‘3

0

0
‘0

fl

4.

L
q

‘1

is .

LIQH o

1:.

ram .

.uww .

A
\O
In

1
I
O
\O

 

 

3».

cm.

d).-

N
.\~ unnumcoo .m no“ taummmooo: 30¢ oflocwme owmno>m och. .m ondmfim

$9st 292.040on 6
mm. em. cm. on.
L L b L _

we. c

 

 

 

.I III! III ‘1. t

_ _ a . _ _

3on mdoconsnvofl 1

paofl mdoconcooflaoc

III II I I

I

I

cfio.

fi-

11:0.

l

IMHO .

-mHo .

{vac .

L

logo .

LAO .

 

 

wdo.

madam cm: x Amv

.mfiw 6693 63.056666 £53 633 oflu6sm68 6wmn6>m mdoconnoogcoc 6£H .0 6ndwfim

. . 3.375 755.046on 6
6m mm aw 6w. cm. 62. N2. mo. so. o
_
_ r w n n + mmm .

.4.—

 

-]+—

 

. _ _ 3

36¢ mSOGOHAUOmEOG \J DLU lumwm.
. C C

 

@H6G mdocoufiuomfl l fl
L QVN.

\V
(D
0)
AI
0

 

‘1
‘3

110mm .

o
0
o
o
0
1.1
0

L

Ivmm.

l

rme.

JINoN .

 

 

 

 

.263 mdodon£606aos 65 a“ Gouonm .6 mo >6n6dw6nm wcfimdog H363 6FH. .N. 6ndmwm

6.32: 293.040on .6
3. J. 2.. cm. mo. 8. s
_ _

NO. 0
L L

_

4CO

 

(j
0
I) .
O /

0
0
u
0
(D

0
o
o

 

_
q _ . A _

llwo .

Q
0
(I

G

.v

Q

(I

 

 

 

 

 

 

56% owm mo dofldfio>6n “6% “Saw >mn6c6 s6 5:3 cohoum m .HOH Q36 66663 6FH. .w 6.3%er
3.575 zomsoqowov a
ma. 0H. 6H. NH. OH. mo. mo. 60. No. C
n + i n n 3. h __
1100M-
1T ON:
0
.1 OH:
0
o
o
11 OH
0
/ II
oom

 

 

.@ midwfm

 

 

 

 

 

 

 

 

.E6fl mdocousoogcon 6S» a“ cohoum .6 mo Gofloe H.603 63.;
moo .1
1- moo ...
1. Moo .1
G
. _ _ o
] . .
25% pom 6N... 6
AW
.t :5 .
..
L
1..
1: N00 .
C
0 nos .
6.52.: 205338 N

 

.GofloE H.363 65 mo wsagmp 6632Q86 65H. .oH 6ndmwm

Amzfigv <

.5. to pm :6 :m :N
p

L

 

.-

 

h
1 a

-—1.—
d

_
a J

 

 

m

56.64me @375 zomeoqowo 6-3.3.6....

27

SECTION II
RADIAL STABILITY AND RESONANCE TRAVERSAL

The feature which was so carefully built into the average mag-
netic field in Section I to give axial focusing introduces another problem;
namely, that the ions must pass through a radial oscillation resonance
near the center of the cyclotron (5). The radial oscillation frequency

is given by the smooth approximation (6) as

2 r d<B>
U :1+<B> dr. [10]

 

It is seen from equation [10] that Ur is necessarily unity at r
equals zero. However, if the ion source is situated between the dee

gap in such a way that the ions are immediately accelerated to an energy

 

d<
of, say, 70 kev dr

is negative; then, in effect, Ur increases from

a value less than unity to become equal to unity when the average mag-

netic field changes from a decreasing to an increasing function of

radius then continues to increase with radius. The radial oscillation

frequency for a proton in the magnetic field under consideration was

calculated by using the equilibrium orbit code and it is shown in figure 11.
Ur = 1 corresponds to a quadratic radial resonance in a three

sector cyclotron. An approximate mathematical treatment of the

traversal of 3/ 3 radial resonance by Sturrock (13) shows that a large

fraction of the ions are lost if the beam of ions spends many revolutions

28
near this resonance. However, its effects may not be catastrophic,
since accelerated ions spend only a small amount of time near the
resonance. The effect of this resonance on a beam of ions must be
carefully investigated before the method of achieving axial focusing
described in Section I can be used.

The reason that a radial resonance is to be expected at Ur equal
unity can approximately be understood from a qualitative description
of the trajectories of ions in the magnetic field plot shown in figure 1.
First consider the effects of the radial variation in the magnetic field
on the motion of an ion which initially has a small radial displacement
from its equilibrium orbit. The radius of curvature of such an ion is
given by

P

r=-—-— [11]

e<B>

where e and P are the ions charge and momentum, respectively and
<B > is the average magnetic field along its trajectory. If the radial
dependence of the magnetic field is such that U2, as given by equation
[10.],remains positive the ion will undergo radial oscillations about its
equilibrium orbit. Differentiation of equation [ll] shows that if the
average magnetic field as a function of radius decreases less rapidly
than l/r radial instability is not to be expected from the radial
variation in the magnetic field. The average magnetic field described
in Section I satisfied this requirement for radial stability.

Since the radial oscillations of interest are those which have an

oscillation frequency near unity, it is convenient to consider the

29
possible trajectories of ions in terms of this frequency. The resonance
occurs at a radius approximately 1/ 7 the radius of the magnetic field
plot shown in figure 1; hence the equilibrium orbit of an ion near
resonance is almost circular and quite small in comparison to the
dimension of the field. Consider the trajectory of an ion, near, but
not at resonance, which initially has a small radial displacement from
its equilibrium orbit. The trajectory after one revolution will also be
almost circular with its center displaced. However, since Ur is not
unity the ion will reach its maximum radial displacement at a different
azimuthal angle so that the orbit will not close. The azimuthal angle
traversed by the ion before this maximum displacement occurs will
be greater or less than 211, depending on whether Uris greater or less
than unity. After many revolutions the trajectory will form a rosette-
shaped trace which is illustrated in figure 12. It follows that whether
the trajectory for an ion which is revolving in a counterclockwise
direction advances around the rosette in a counterclockwise or clock-
wise direction depends upon whether )4 is greater or less than unity.

In the above discussion only the effect of the radial variation
of the magnetic field was considered. The radial instability of the
3/3 resonance arises from the threefold symmetry of the azimuthal
variation of the magnetic field. Hence, the effect of the azimuthal
.variation of the field on the rosette-shaped trajectory shown in figure

12 must be considered. Due to the threefold symmetry of the field

30
shown in figure 1, the equilibrium orbit is slightly triangular. This
can be seen by consideration of equation [11]. The amplitude of the
radial oscillations off this orbit will be enhanced or diminished during
a given revolution by the azimuthal gradient of the field, depending on
the azimuthal position at which the ion crosses the equilibrium orbit.
If the radial oscillation frequency is not equal to unity. the rosette
trajectory is maintained and the amplitude of oscillation is alternately
enhanced and diminished. This gives radial stability for small amplitudes
of oscillation near resonance. What is meant by small amplitudes will
be shown shortly.

As )4 approaches unity, many revolutions are required for a
particle's trajectory to complete the rosette pattern. As a consequence,
an ion with a 24 nearly equal to unity will traverse the same azimuthal
sector of the magnetic field for many consecutive revolutions. In doing
so, it will experience the effect of a similar magnetic gradient during
each revolution. Consider such an ion whose radial amplitude is
increasing as it sweeps through an increasing magnetic gradient near
one of the three magnetic "hills" shown in figure 1. If its radial
amplitude is larger than a certain critical value, the amplitude will
increase with each successive revolution. The trajectory of the ion
will deviate from the rosette pattern and form a trochoid—shaped
pattern, shown in figure 12, which will move radially outward in the
field gradient of the hill, slowly at first then more rapidly. This is

what is meant by radial instability. If, initially, the amplitude had

31
been smaller than the above mentioned critical value, it would require
many revolutions before disastrously large amplitudes could build up
and instead, the rosette shaped trajectory would carry the ion azi-
muthally out of this region of the field. As this ion progressed around
the rosette, it would sweep through the decreasing field gradient on the
other side of the hill. and its amplitude of oscillation would be decreased
by the reversal of the above described process and radial stability
would result. The limiting case between the stable and unstable
trajectories occurs when the increase or decrease in radial amplitude
introduced by the field gradient during one revolution is just large
enough to compensate for the azimuthal shift of the trajectory so that
the orbit is closed. This discussion is a simplification in that both the
increase and decrease in amplitudes are enhanced by the gradient of
the other two hills which the trajectory traverses each revolution. In
summary; near resonance, the radial oscillations about the equilibtium
orbit are stable if the initial amplitudes are smaller than a critical
value, and unstable if they are initially larger than this critical value.
If the initial amplitude is equal to this critical value closed orbits are
formed. It follows from the above discussion that the magnitude of
this critical amplitude depends upon the amount by which the particle
is off resonance.

An analysis of the trajectories of ions in the central region of

the magnetic field shown in figure 1 which have Ur equal to unity shows

32
that any amplitude of oscillation about the equilibrium orbit will be
unstable. All these trajectories will form trochoid shaped traces in
the magnetic field gradient about the magnetic hills.

A useful procedure for investigating radial stability is the use
of phase plots. If x, the radial displacement from the equilibrium orbit,

dx
-d—e, are known, the center of curva-

and the conjugate momentum, pX =
ture of the ion is defined. From geometrical considerations it can be
seen that if pX is plotted as a function of x, once per revolution, the
resultant phase plot corresponds to the movement of the center of
curvature of the trajectory at that azimuth angle. These phase plots
can be used directly to gauge the range of radial oscillation amplitude
for which the orbits are stable. It is easily seen that curves in phase
space which close about the equilibrium orbit point correspond to the
stable rosette shaped trajectories described earlier. The open curves
correspond to the trochoid shaped trajectories of the unstable orbits.
A typical phase plane diagram (13) for an ion near resonance
in a three sector magnetic field is shown in figure 10. The feature
of interest is the triangular shape of the stable region which is defined
by the three "unstable fixed point" labeled U. These points not only
mark the boundary separating the stable and unstable region in the
phase diagram, but behave in a manner analogous to that of the central
equilibrium orbit, in that a particle starting at such a point on each

successive revolution comes back to the initial point (14). Hence,

they correspond to the limiting case between the rosette and trochoid

33
trajectories discussed earlier in this section.

As an ion is accelerated through resonance, the size of the stable
region in phase space decreases to zero, and then steadily increases.
Approximate behavior of particles accelerated through resonance can
be inferred if the location of the unstable fixed points are known for
successive energy values above and below resonance. For example,
the number of revolutions required for ions with a given energy gain
per turn to traverse the region of resonance effects can be determined.
The effective size of the stability region can be found. The coordinates,
in phase space, of the unstable fixed points were found by use of the
orbit codes. The stable‘triangles are shown at energy increments
of 0. 25 mev in figure 14. The energy corresponding tonresonance is
1. 47 mev. The azimuthal angle in the field was chosen to correspond
to the first dee gap crossing, for convenience. To view the effects
of resonance traversal, consider a beam of protons in this magnetic
field with an energy gain per revolution of 280 kev which were emitted
from an ion source situated at the center of the cyclotron between the
dee gaps. After two complete revolutions, the beam of ion will reach
an energy of . 50 mev. At this time the beam can be considered to be
a spot in phase of, say, 0. 003 cyclotron units in diameter which is
located near the center of the large stable triangle. Three revolutions
later, the stable region has decreased in size, leaving part of the
beam in the unstable region where the amplitudes of the radial oscillations

increase. The stable region continues to decrease in area, ceases

34
to exist at resonance, and then begins to expand. It is hoped that
this expansion of the stable triangle will overtake the beam before
excessive increases in the amplitudes occur.

The rate of expansion of the radial amplitudes while the beam
is in the unstable region is determined from phase plots obtained by use
ofthe general orbit code. Figure 15 is a phase plot at an energy of
1. 25 mev, which is approximately one revolution below the resonance
energy. The full radial scale of this graph is 4. 52 inches. The stable
region at this energy has decreased from 1. 5 inches at 0. 50 mev to
0. 32 inches. Six orbits are shown; three within the stable region
and three in the unstable region where the radial amplitude expansion
occurs.

At Oak Ridge (5) and MSU (17), sufficient phase plots have been
made for azimuthally variation magnetic fields of the type considered
here, so that it is assumed that plots in the stable region will close
about the equilibrium orbit point. These orbits were abandoned to
conserve computer time since the number of revolutions required for

the plot to close is approximately , which is 342 revolutions at

 

 

__1___
24. -1
this energy.

In the unstable region, amplitudes of the radial oscillation
about the equilibrium orbit increase each revolution as indicated by

the movement of phase points along the "outflowing asymptotes. "

This corresponds to the formation of the trochoid-like trajectories.

35

It can be noted from figure 15 that a particle near one of the unstable
points in this region will move out radially, slowly at first, then
more rapidly with each successive revolution. The rate of amplitude
growth then depends upon the location of the orbits in phase space
with respect to the unstable fixed points and upon the number of
revolutions during which the particles have been in the unstable region.

Consider the motion of a proton, with an energy of 1. 25 mev,
which is displaced 0. 7 inches from the equilibrium orbit along the
outflowing asymptote in figure 15. After ten revolutions, its radial
amplitude has increased to 2 inches; but. if its energy gain per revolution
is 0. 28 mev, in ten revolutions the proton will obtain an energy of 4. 0
mev. Its amplitude stability limit would then be greater than 2 inches.
Therefore, 1. 25 mev protons with radial amplitudes less than 0. 7
inches will probably survive the resonance traversal. The "effective"
stable amplitude limit will actually be larger, since only a minority of
the ions with this amplitude will have such unfavorable coordinates
with respect to the unstable fixed points. Phase points of protons
which have initial conditions that exceed the effective stable amplitude
limit will not be overtaken by the expanding stable region. such
particles are said to fail to survive the resonance traversal. The
conclusions obtained from this consideration are very encouraging,
since the radial amplitudes are not expected to exceed 0. 35 inches at

this energy.

36

The investigation of radial stability thus far has been inferred
by means of static phase plots and knowledge of the size of the stable
region at energy values of interest. However, the problem of radial
stability can only be completely settled by tracking the phase points
of groups of ions as they are accelerated through resonance to an
energy where the amplitude stability limit is quite large. The procedure
for doing this is discussed in detail in the next section. It will be
shown that an acceptably large central beam will survive the traversal

of the 3/3 radial resonance in this magnetic field.

463$ 636:me mdoconaooficoc 62.3 a“ Con—0.2m 6 mo >6c6dv6um Goflmdfiomo H6362 69H.

NN. 0N. ma.

\0
‘H-l-i

8.575 292.046on 6
v.7 . mm. eta.

ma.

.2 668E

 

 

.-

0
G

0

(I

(I
C!
0

G
(\

t!

0
0

. n

‘x

G
‘\

«v

6

O
(I
0)

I
M

No.
0 0

(\

loo .H

L

112C;

LING;

T33

J

:34

umoé

 

 

be;

38

 

wrc'sv‘a’v‘ '

////// v

{{[f

v v.‘

 

 

"\NX\\\

9'

 

A

 

\

l

ry of unstable radial oscillation

\\\\

l

 

 

AA

0

[ff

 

 

\O$&\\

 

plane .

the median

stable and unstable trajectories in

choid-shaped trajecto

The tro
Figure 12; The

 

 

 

 

 

 

 

 

 

Figure 13. A phase diagram near resonance for a three sector magnetic
field.

Ar (CYCEOTRON UNITS)

 

. 010“”

Resonance energy = 1. 47 mev

.008-

.006-

.0044

.002 ‘

 

. 002 “

.004 -

. 006

.008-

.010‘

 

l l I l

 

41 I. i

 

 

f T T l
-. 008 -. 006 —. 004 -. 002

0 . 002 . 004 . 006

APr (CYCLOTRON UNITS)

Figure 14. The stable regions at energies near resonance.

. 008

 

r (CYCLOTRON UNITS)

\

 

F1
II

1.25 mev

. 080“ X

unstable fixed points

. 076+ 5

. 072"

.068”

. 064—

l

. 060“

 

. 056-
<——- 20 revolutions

. 052+

20 revolutions

 

.048“

 

. 044-

 

 

 

 

.040 L, 1 1 1 %
-.01-2 -. 008 -. 004 0 .004 .008 .012 .016

Pr (CYCLOT RON UNITS)

Figure 15. A phase plot 220 kev below resonance.

42

SECTION III
BEAM DISTORTION

In Section I it was stated that the general orbit codewas used to
simulate the acceleration of a proton from an initial energy of 70
kev to 9 mev. The radial orbital properties obtained from this com-
puter run indicated that this particle survived the traversal of the
3/3 radial resonance. However, the survival of one ion means very
little. It is the survival of a beam of ions with a minimal amount of
distortion that is necessary for the efficient operation of a cyclotron.
To investigate the feasibility of resonance traversal, a group of
particles must be accelerated through resonance. It follows from
Liouville's theorem (15) that the area enclosed in phase space must
remain invariant during the acceleration process. Therefore, if the
area in phase space enclosed by this group of particles is the size of
the desired particle beam, the effect of resonance traversal on the
entire beam can be predicted.

If four particles whose initial r and Pr values form a square in
phase space are accelerated through resonance, the final phase area
enclosed, although equal to the initial area, may be very distorted in
shape. A beam of particles initially inscribed within this phase area
would likewise be distorted.

The reason that beam distortion is to be expected can be under-

stood by a consideration of the phase plot. shown in figure ‘15, and the

43

relative sizes of the stable regions at various energy values, shown

in figure 14. Assume that initially the phase area, at an energy below
resonance, is bounded by particles whose initial conditions do not
exceed the effective stable amplitude limit discussed in Section II. The
shape of the phase area will be unchanged as long as all the particles
remain in the stable area. As these particles are accelerated, the
stable region decreases in size, leaving the particles in the unstable
region where their amplitudes of oscillation either increase or decrease.
This change in radial amplitude corresponds to movements in various
directions of the phase points along the flow lines shown in figure 15.
The direction and rate of this movement will depend on the coordinates
of the phase points with respect to the unstable fixed points and the
number of revolutions that the particles remain in the unstable region.
This will obviously change the shape of the phase area since its size

is invariant. As the particles are accelerated, the resonance is
traversed and the expanding stable region overtakes the points so that
the distortion of the area ceases.

The distortion of the area can be very severe, since phase
points with less favorable location with respect to the unstable fixed
points move more rapidly, and hence remain outside the stable region
for a greater number of revolutions. Obviously, the larger or less
compact the initial phase area, the more severe the distortion.

For another view of the effects of beam distortion, consider

a group of phase points enclosing a circle in the stable region at an

44
energy well below resonance. If, after resonance traversal, the area
is distorted into a long, narrow shape, a much larger stable region at
energies above resonance will be required to retain the phase points.

With these general principles in mind, the technique for inves-
tigating beam distortion can be understood in light of the relative sizes
of the stable region shown in figure 14. If an ion is accelerated from
the center of the cyclotron and reaches an energy of, say, 4 mev with
r and Pr coordinates which lie within the stable region at that energy,
the orbit thereafter will be stable (5). Also if a 4 mev ion initially on
the equilibrium orbit at that energy is decelerated and traced to near
zero energy its phase coordinates will locate the position of the ion
source. In particular, if a group of ions with initial conditions that
enclose an area in phase near the resonance energy equilibrium orbit
phase point are accelerated to 4 mev and decelerated to near zero
energy the proper location of the ion source can be found and the
distortion of the centralrays of the ion beam can be investigated.

The general orbit code can be adapted to investigate beam
distortion and'fto determine the necessary initial conditions for the
central ray of the ion beam. A slight modification of this code will
simulate acceleration by increasing an ion's energy by a constant
amount each half revolution. Hence, groups of particles with different
initial r and Pr values will have a constant energy gain per revolution
regardless of their respective phase slip. Also, deceleration of parti-

cles can be simulated by a similar procedure.

45

The distortion of the central rays of the particle beam was
investigated by using the general orbit code to track a group of five
protons with an energy gain per revolution of 280 kev from the resonance
energy of 1.47 mev to a final energy of 3. 99 mev and backward from
resonance to 70 kev. The initial phase coordinates of the protons
were chosen to form a 0. 0028 cyclotron unit square, with one proton
at the center, near the equilibrium orbit corresponding to resonance
energy. The phase space coordinates were obtained for each particle
at the end of successive revolutions. This computer run was then
repeated for a proton initially at the center of this square with an
axial displacement of 0. 005 cyclotron units, to investigate possible
coupling of the radial and axial oscillation by the "sum" or "difference"
resonance.(16).

The results of these computer runs are shown in figure 16. It
is seen that the shape of the square is only slightly distorted by the
effects of resonance traversal. It was found that the maximum effects
introduced into the phase coordinates of the central proton by including
axial motion was less than 0. 0006 cyclotron units. These results are
very encouraging, in that the radial width of a beam enclosed by this
phase area is 0. 27 inches and the distortion is certainly tolerable.
The feasibility of 3/3 radial resonance traversal for a sizable beam
of protons is thus established.

It is desirable to know the dimensions of the region in phase

space from which 70 kev protons can be accelerated without severe

46
beam distortion. This is accomplished by simulating the acceleration
and deceleration of protons, with the array of initial r and Pr values
shown in figure 17, in the manner described above. These phase points
can be considered to form annuluses of various sizes and with different
orientations with respect to the equilibrium orbit phase point. By
tracking this array of phase points, the distortion of various parts of
a large proton beam can be determined. As can be seen from figure 17,
these phase points are symmetrically located with respect to the initial
conditions of the five protons with radial displacement of . 004, . 006 and
. 010 cyclotron units. The array is slightly off center with respect to
the phase coordinates of the equilibrium orbit corresponding to this
energy, due to a computer error.

The distortion of this array of phase points during the acceleration
process is shown at the end of the third, fifth, seventh and ninth
revolutions in figures 18 through 22, respectively. The stable region
and equilibrium orbit coordinates are also shown. The full scale of
the radial axis is 5. 42 inches on these plots.

From these graphs, it is seen that the distortion of the phase
areas in the third and fourth quadrants is relatively slight, whereas it
is very severe in the first and second quadrants. The reason for this
can be seen by considering figure 18 in light of the static phase plots
discussed in Section II. All but two of the particles in the third and

fourth quadrants are in the stable region after two revolutions. These

two particles have favorable phase space coordinates with respect to

47
the unstable fixed points, so that in a static phase plot their phase
points would move parallel to the side of the stable triangle. These
points are overtaken by the expanding stable region within two more
revolutions and the distortion ceases.

The phase points in the first and second quadrants are more
unfavorably located with respect to the unstable fixed points, and hence
are moving more rapidly. It is seen that the distortion is very severe
and that those particles with the largest radial values will not survive
the resonance traversal.

The results of the computer runs simulating deceleration of these
particles are shown in figures 22, 23, and 24. The distortion is less
severe in this case since only five revolutions are required. The com-
puter runs for the phase points in the third and fourth quadrants were
not made since it can be safely assumed that distortion would be less
in that region. The justification of this assumption can be seen in
figures 22 and 23, since the phase points in question would have been
nearer the equilibrium orbit phase point, hence less distorted than the
corresponding first and second quadrant points. All the third and fourth
quadrant phase points would have been in or very near the stable region
in figure 23.

At 70 kev, the distortion is again severe in the second quadrant
but is less severe in the first quadrant. The stable region is not
shown in figure 22, but as shown in the appendix it is approximately

the full scale of the graph.

48

The results shown in figures 17 through 24 indicate that there
exists an area in phase space at 70 kev of perhaps . 006 x . 006 cyclo-
tron units from which a beam of protons can be accelerated with an
energy gain per revolution of 280 kev without severe beam distortion.
This area is centered near but above the equilibrium orbit phase point
in figure 24. The distortion of the beam is diminished as the central
region of this area is approached. In summary, all particles in a
0. 70 inch beam of protons with an energy gain of 280 kev per revolution
would very likely survive the 3/3 radial resonance traversal. A beam
composed of these protons, whose radial displacements are a half
inch or less, would not be severely distorted.

The data given in figures 16 through 24 were obtained by use of
a modification of the general orbit code which assumes a square wave
acceleration voltage. In an actual case both the phase slip and the
spread in phase of particles must be taken into account. The phase
slip as a function of radius is shown in figure 8. It can be seen that
a sinusoidal voltage would never be farther than 15 degrees from its
peak when particles cross the dee gap. The maximum phase spread
at successive dee gap crossings for the deceleration run shown in
figure 16 is given in table II. From these data it is seen that the square
wave assumption is good and that the use of a sinusoidal voltage would

not significantly change the results.

Table II

Phase Slip Spread for the Deceleration Run Shown in Figure 14

 

 

 

E mev A0) (max.)
1.33 5. 3O
1 19 4 1O
1 05 7 40

O
0.91 8.2
0.77 8. 7°
0.63 2. 00
0.49 11.50
0.33 7. 7°

0
0 21 14 4

 

r (CYCLOTRON UNITS)

 

 

      

  

 

 

 

 

-. 02 -.

. 10 “"
( 3. 99 mev
0 ’v 3'
09 .. o ') F— 43 mev
0N
(9.§>=3 9—2. 87 mev
. 03 .11.. 33;!
$ é——-‘ 2 31 mev
. 07.3-
<—— 1. 75 mev
06-- r
%% {—— initial conditions
(— 1. 19 e
. 05‘s m V
. 04"-
&% (—— . 63 mev
. 03-. a?!
02“ .
<€% (— . 07 mev
.01 l '

 

: . A. .
01 0 . 01 . 02 . 03
p1. (CYCLOTRON UNITS)

Figure 16. The distortion of a 0. 27 inch central beam.

 

r (CYCLOTRON UNITS)

 

 

 

 

 

. 078
.074a—
.070jr
1"."
. 066 -L c
v 0';
Q :5 ,
h‘
-1. '4
. 062 ‘ - 9
'5‘ ‘ 0; c
12‘ ’3'
0:.
. 058 -- A?)
‘; :; r 04"
‘< f;
\‘I
0}
.054-~ \- .;.
c.
'5
. 050 ~-
. 046 “
. 042“ 81 equilibrium orbit phase point
E = 1. 47 mev ‘
. 038--
. 034 1 1 l 7' 1 1 I
-.012 -.008 -.004 0 .004 .008 . 012 .016
Pr (CYCLOTRON UNITS)
Figure 17. The initial conditions for protons accelerated and

decelerated from resonance.

 

r (CYCLOTRON UNITS)

 

. 0964-

.0921~

. 088-1—

. 084‘”

. 080-1-

 

. 076——
. 07%.
.0684
X unstable fixed point
. 064-— ® equilibrium orbit phase point 1‘ //
/
E = 2. 31 mev )1<
.060-
. 056"“

 

l 1

 

l l I

 

-. 0'12 -. 008

Figure 18.

-. 004 0

. 0'04 .009
Pr (CYCLOTRON UNITS)

The distortion three revolutions above resonance.

 

r (CYCLOTRON UNITS)

 

 

 

 

 

 

 

 

The distortion five revolutions above resonance.

.1047”
.100“
.096-~
.092-—
.0883-
. 084 g /
/
/
,’ , ’/
.080» ‘3. ® .
l ‘o ' v. /
‘ ~ 1’
‘ s /
“ 9‘ /
° 076+ x unstable fixed points ‘ //
1
@ equilibrium orbit phase at //
\
E- 2 87 mev ‘ /
.072~- " ' ‘ //
1‘ /
1 /
1 /
. 068" 1‘ /
‘ /
1/
.1
.064 { 4 1 1 .1, t? .1
-.012 -.008 -.004 0 .004 .008 .012 .016
Pr (CYCLOTRON UNITS)
Figure 19.

~108

.104

.100

. 096

~ 092

. 088

- 084

. 080

. 072

 

.020

r (CYCLOTRON UNITS)

 

.136 0

. 132*

.128"-

.124-

‘T

.12031

.116“

.112"

 

T

.108“

.104-1'

.1000”

   

1" unstable fixed points
® equilibrium orbit phase 1 .

o O96"I \
E = 3.99 mev

h \ V
. 092 \l

l

 

q)-
d
qu—

 

 

-.012 -.008 -. 054 0 . 004 .008 .012
Pr (CYCLOTRON UNITS)

Figure 21. The distortion nine revolutions above resonance.

 

r (CYCLOTRON UNITS)

 

. 066.-

. 062-—

. 058--

. 05+

. 050--

. 046‘“

 

. 042-1—

. 038--

. 034-” X unstable fixed points
8 equilibrium orbit phase point

. 030-- .E = .91 mev

. 026-

I

 

 

l l 1 I l

 

 

1)-

t l I I l

-. 012 -. 008 -. 004 0 . 004 . 008 . 012
Pr (CYCLOTRON UNITS)

Figure 22. The distortion two revolutions below resonance.

r (CYCLOTRON UNITS)

 

I

. 048‘

. 044—-

T

.040‘

.036-

 

.0324

.028‘

. 024‘

. 020*

 

 

X unstable fixed points

.A.— .-3

 

 

 

. 016'" ,
® equilibrium orbit phase point 1
E = . 35 mev '
. 012—1‘ 1
. 008 11 1L : 1'1 1 1
-.016 -.012 —.008 -.004 0 .004 .008 .012

Pr (CYCLOT RON UNITS)

Figure 23. The distortion four revolutions below resonance.

r (CYCLOTRON UNITS)

 

. 046

.038“—

. 034--

. 030"

. 026*

. 022“

 

. 018“”

. 0144—-

 

.010“

I

8’ equilibrium orbit phase pt.

I

E = . 07 mev 1

. 006~- !
i

 

1)—

. 00.. ,L - 1 1 I .

 

1 r r r -- ~ -
-. 012 -. 008 -. 004 0 . 004 . 008 . 012

Pr (CYCLOTRON UNITS)

Figure 24. The distortion five revolutions below resonance.

59

CONCLUSION

Numerical calculations show that good axial focusing can be
obtained near the center of a sector focused cyclotron from a non-
isochronous average magnetic field without excessive phase slip.

The average field given in Section I is especially designed for a proton
with an energy gain per revolution of 280 kev in the azimuthally varying ‘
field shown in figure 1. The procedure described can be applied to
other ions in simularly varying magnetic fields if the magnitude of

the accelerating voltage is chosen properly.

Ions accelerated in the field described above must traverse the
3/3 radial resonance near the center of the cyclotron. The effects
of this resonance must be carefully investigated for three sector
cyclotrons. The feasibility of accelerating a practical-sized beam of
protons through the 3/3 resonance without loss of particles or severe
beam distortion is established in Sections II and III. It is shown by
orbital integration with a digital computer that a 1/2 inch wide beam
of protons with an energy gain per revolution of 280 kev can be
accelerated through this resonance without intolerable beam distortion.
This beam width is sufficiently large for the practical operation of a

cyclotron.

60
APPENDIX

Radial Stability in the Central Region Due
to Radial Variation in the Magnetic Field

A qualitative consideration of the trajectories of 70 kev protons
initially displaced from their equilibrium orbit indicates that large
amplitudes of radial oscillations are stable in the field considered
here. The radius of the equilibrium orbit of 70 kev protons is less
than 1/ 30 the radius of the magnetic field. AS can be seen from figures
3 and 6, the effect of the azimuthal variation in the field can be neglected
in comparison with the effect of the radial variation of the field.

Since Ur = . 9855 at this energy, the trajectories of ions with small
initial displacements from the equilibrium orbit will form the rosette
shaped patterns discussed in Section II. For larger radial amplitudes
the trace of the trajectories, after a given revolution, deviate from
circular as the ions sweep through the radial gradient of the magnetic
hill centered at r = 0. However, the rosette shaped pattern is
maintained for all trajectories which circumscribe the center of the
field each revolution. If the initial radial displacement of an ion is
such that its trajectory does not circumscribe the center of the field
each revolution, then its trajectory forms a trochoid-like trace in the
magnetic field gradient about the central hill. Since these trochoid
shaped trajectories move along the equilibrium orbit the criterion

for stability given by Courant and Snyder (10) is satisfied. That is,

61
an ion moving along such a trajectory will remain near the equilibrium
orbit for all time.

As can be seen by a consideration of figures 1 and 6, only if a
70 kev proton's initial radial displacement from the equilibrium orbit
is large in comparison to the radius of its equilibrium orbit will its
trajectory form a trochoid-like trace which will move out radially in

the manner described in Section II.

(1)
(2)
(3)
(4)
(5)

(6)
(7)
(8)
(9)
(10)

(11)
(12)

(13)

(14)
(15)
(16)

(17)

62
BIBLIOGRAPHY

L. H. Thomas, PhyS. Rev. 2, p. 580 (1938).

D. W. Kerst and P. Serber, Phys. Rev. _6_0_, 53 (1941).
T. A. Heyn and Khoe Kong Tat, R5122, p. 622 (1958).
S. Chatterjee, Private Communication.

B. L. Cohen et al. , Oak Ridge Relativistic Cyclotron, ORNL-
2648.

K. R. Symon e_t;a_1_., Phys. Rev. 1&3, p. 1837 (1956).

T. Arnette, Michigan State University, unpublished (1960).

H. G. Blosser, Michigan State University, unpublished (1960).
M. M. Gordon and T. A. Welton, ORNL-2765.

E. D. Courant and H. S. Snyder, Annals of Phys. 2, p. 1 (1958).

H. G. Blosser, Sector Focused Cyclotrons, Submitted for
publication in Proceedings of AIEE (1959).

B. L. Cohen, Cyclotrons and Synchrocyclotrons, Handbuch der
Physik, Vol. XLIV (1959).

P. A. Sturroch, Annals of Phys. _3_, p. 113 (1958).

H. G. Blosser and M. M. Gordon, A Proposal to U.S. Atomic
Energy Commission, Michigan State University publication (1959).

H. Goldstein, Classical Mechanics, p. 266, Addison-Wesley
Press, Cambridge (1951).

P. H. Sturrock, Static and Dynamic Electron Optics, Cambridge
Press (1955).

H. G. Blosser et a1. , Proposal for a Nuclear Research Facility,
Michigan State University publication (1958).

HICHIGQN STQTE UNIV. LIBRQRIES

31293017430137