I
I
71-31,189

DREISBACH, John, 1942PERFORMANCE ESTIMATES OF A MEDIAN PLANE
INJECTION SYSTEM FOR THE MICHIGAN STATE
UNIVERSITY CYCLOTRON.
Michigan State University, Ph.D., 1971
Physics, nuclear

University Microfilms, A XEROX Company, Ann Arbor, Michigan

PERFORMANCE ESTIMATES OF A MEDIAN
PLANE INJECTION SYSTEM F O R THE
MICHIGAN STATE UNIVERSITY CYCLOTRON

By
John Dreisbach

A THESIS

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

DOCTOR OF PHILOSOPHY

Physics Department

1971

PLEASE NOTE:
Some Pages have indistinct
print.
Filmed as received.
UNIVERSITY MICROFILMS

ABSTRACT
PERFORMANCE ESTIMATES OF A MEDIAN
PLANE INJECTION SYSTEM FOR THE
MICHIGAN STATE UNIVERSITY CYCLOTRON*
By
John Dreisbach

Ions from an external source can enter the central
region of a sector focused cyclotron by winding along a
sector edge in the median plane.

To use this method with

the MSU cyclotron, the injected beam first travels along
the inside of the dee stem to an "outer" electrostatic inflector located at a slightly greater radius than the last
accelerated orbit.

This outer inflector shifts the ions

onto a cycloid-like trajectory, the drift of the cycloid
carrying the particles inward, approximately following an
iso-gauss contour at one of the hi11-valley transitions of
the sectored magnet.

After approximately 35 cycles or

"turns" of the cycloidal motion, the ions end up in the

*Work supported by the National Science Foundation.

J o h n Dreisbach

central r e g i o n of the cyclotron w h e r e an "inner" el e c t r o ­
static i n f l e c t o r s y s t e m guides them onto a suitable normal
orbit.
In o r d e r to m a x i m i z e turn separation at the inner
inflectors,
be used.

the largest per m i s s i b l e injection energy should

This is h o w e v e r limited by the voltage ho l ding

capability of the inner inflectors.

Injection at 3/1000 of

the f inal cyclotron e n e r g y is a compromise bet w e e n these
factors and gives both adequate turn separation and r e a s o n ­
able i n f l e c t o r voltages.
The focus i n g proper t i e s of such an injection s ystem
were t h o r o u g h l y i n v e s t i g a t e d for the case of protons w i t h a
final c y c l o t r o n energy of 42 MeV.
zontal)

The median plane

(hori­

a c c e p t a n c e of the sys t e m is about 3 times the m e a ­

sured e m i t t a n c e of the M S U cyclotron using the p r e s e n t in­
ternal p r o t o n source,
in energy.

after adjusting for the difference

T h e v e r t i c a l acceptance of the injector, n e ­

glecting the effects of n o n l i n e a r verti c a l forces,

approx­

imately cor r e s p o n d s to the normal verti c a l emittance of
the cyclotron.
2

John Dreisbach

It was assumed that horizontal forces are indepen­
dent of vertical position.

Also,

for convenience,

the

azimuthal magnetic field variation was described by only
12 Fourier coefficients;

thus the field actually used cor­

responds to a cyclotron having a smoother flutter.

Test

cases estimate the effects of these various factors.
Since many external ion sources have low current
outputs, a buncher is often employed with external injec­
tors to increase the current out of the cyclotron.

De­

tailed studies herein show however that a lack of isochronism in the injection path of this type of injector
limits the improvement in transmission efficiency due to
a sine wave buncher to a factor of around 3.5, for a 2%
duty factor.

The buncher causes a momentum spread of

approximately 0.5% in the injected beam thus increasing
the horizontal width of the beam at the cyclotron central
region.

This effect, when added to a horizontal distor­

tion caused by the electric fields of the central elec­
trode system, effectively increases the horizontal emittance
by a factor of about 4 or 5.

John Dreisbach

One of the primary motivations for external injec­
tion systems is acceleration of polarized ions using some
form of polarized ion source.

So the influence of the

cycloid-like motion on the polarization of protons was
calculated:

there is essentially no depolarization.

If a polarized ion source having a luminosity
which seems feasible today is used w i t h this injector, the
estimated current out of the cyclotron should be around
50 nA.

The radial acceptance of the proposed system is

about a factor of 4 or 5 times smaller than that obtained
with some existing injectors;
somewhat smaller.

and the axial acceptance,

The energy spread of the beam from the

cyclotron should be slightly better than elsewhere because
the proposed duty factor is smaller.

Finally the esti­

mated increase in horizontal emittance, mentioned above,
is similar in magnitude to that found elsewhere.

4

TABLE OF CONTENTS

Page

LIST OF FIGURES........................................

iii

1.

I n t r o d u c t i o n ...................... . . ............

1

2.

General Description and Design Goals ............

5

3.

C o n s t r u c t i o n ......................................

12

3.1
3.2
3.3
3.4
4.

Ion source and buncher....................
Beam transport............................
......................
Outer inf lector
Inner inf lector system....................

Performance........................ ...............
4.1
4.2
4.3
4.4
4.5
4.6

Ion source and buncher....................
Beam transport............................
Outer inf l e c t o r ..........................
Drift along the sector e d g e .............
Inner inf lector system....................
Performance summary ......................

12
17
23
26
34
34
41
45
47
57
74

5.

C o m p a r i s o n ........................................

83

6.

C o n c l u s i o n ........................................

94

REFERENCES

96

LIST OF FIGURES

Figure

2.1

Page

Schematic diagram of the cycloidal type
injection .....................................

6

Cyclotron cross section w i t h injection
system ..........................................

8

3.1

Heliquad cross s e c t i o n ..........................

19

3.2

Outer inf l e c t o r .................................

24

3.3

Inner inf lector system— med i a n plane view . .

28

3.4

Cross sections of Figure 3.3.

.........

29

4.1

Median plane emittance at the exit of the
final i n f l e c t o r ..............................

66

Median plane emittance at the exit of the
outer i n f l e c t o r ..............................

67

Vertical acceptances at the final inflector
e x i t ............................................

71

Vertical acceptances at the outer inflector
exit.
(Note scale change from Figure 4.3).

72

2.2

4.2

4.3

4.4

. .

1.

Introduction

E x t e r n a l ion source facilities and a b e a m i n j e c ­
tion s y s t e m are frequently added to cyclotrons in order
to obtain beams w h i c h are incompatible w i t h the severe
r e s trictions of internal sources.

Most frequently the

d o m i n a n t interest is in obtaining p o l a r i z e d beams;
principle,

in

per f o r m a n c e should be improved for many other

beams as well.
F o u r main types of injection systems have been
utilized or proposed:

1) "Axial"

injection systems use

a hole b o r e d in the ma g n e t perp e n d i c u l a r to the med i a n
p l a n e — the b e a m of ions is guided through this axial hole
and then b e n t into the median plane by a 90° inflector or
m i r r or

(Po 65).

2) The "trochoidal"

injection s y s t e m uses

an e l e c t r i c field in the median plane to cancel the m a g ­
n e t i c force so that the injected b e a m follows a straight
line from the outside to near the center of the mac h i n e
(Be 6 7 b ) .

3) "Stripping foil"

injection systems d i r e c t a

2

beam of neutral or low charge ions from outside toward a
foil near the machine center— when the ions pass through
the foil charge exchange decreases their magnetic rigidity
and, if the foil is properly positioned,

the stripped ions

are left on a normal internal orbit ready to be acceler­
ated

(Ah 69, Be 70a) ..

4) The "cycloidal" injection system

uses the magnetic field gradient at a hill-valley boundary
in a sector focused cyclotron— in such a field ions drift
cycloidally along the sector edge from the outside of the
cyclotron to the center,

and then are shifted to a normal

orbit by some form of inflector.
In order to choose between the various injection
techniques, one would like to have detailed information on
the performance characteristics of each type of system and
also some idea of what equipment is required for each.
This thesis has the purpose of providing such information
for the case of a cycloidal injection system;

the other

types of systems listed above are already covered exten­
sively in the literature and can be evaluated from those
r e s u l ts.

3

M o s t of the injection systems for cyclotrons pres­
ently in operation are axial injectors.

Cyclotrons with

axial internal source holes can readily be adapted to this
type of injector.

The performance of axial injection

systems is fairly good and is well verified.

Probably

the main deficiency is the potentially large increase in
emittance as the beam comes out of the axial hole into the
central magnetic field

(Po 66a, Lu 68).

Injection of ions

in a low charge state and further stripping at the center
of the cyclotron is usable only for heavy ions, obviously.
If a neutral beam is injected,

it is not necessary to use

high voltage electrodes in the cyclotron but there are
problems of obtaining a small diameter beam in the central
region and of efficiently ionizing the b e a m at this loca­
tion

(Be 70a, Be 6 6 b ) .

The median plane method of inject­

ing a b e a m radially in a straight line to the central re­
gion requires a system of electrodes which extends all the
way from the outside of the machine to the central region
and a large number of high voltage supplies

(Be 6 7 b ) .

The

cycloidal type injector has the potential for alleviating
these deficiencies since there are no sudden magnetic

4

field variations that increase the beam emittance,

there

is no charge exchange after the ions leave the source, and
no electric fields are required over most of the injection
path.

2.

General Description and Design Goals

A sector focused cyclotron satisfies the dual re­
quirements of axial focusing and isochronism by means of
azimuthal variation in the magnetic field.
the simplified diagram

(Figure 2.1),

As shown in

this type of magnetic

field opens the possibility of using the cycloidal method
of median plane injection in which the ions spiral along
the interface between strong field regions called hills
and weak field regions called valleys.

This method was

first proposed by V. Gladyshev and others at the Lebedeff
Institute, Moscow,
cyclotron

(G1 65).

and was tested by this group in a model
The Lebedeff experiment demonstrated

that no electric fields were required over most of the in­
jection path.

Gladyshev's group also showed that the hori­

zontal and vertical focusing properties were good, at least
for the approximations of straight sector edges and sud­
denly changing magnetic field between the hill value and
the valley value

(assumptions which were reasonable for

their e x p e r i m e n t ) .
5

6

BEAK FROM
I0W SOURCE

H IL L

VALLEY

Fig. 2.1.— Schematic diagram of the
cycloidal type injection.

7

For the MSU case, it is proposed to inject positive
ions and, as will be indicated presently, an injection
energy of 0.3% of the final energy appears best.

With this

energy and with the large magnet gap of the MSU cyclotron,
the approximation of sharp hi11-valley edges is much less
valid and accurate knowledge of ion trajectories therefore
requires numerical integration of the equations of motion.
Major features of the proposed injection system are
shown in Figure 2.2.

The maximum injection energy, which

corresponds to acceleration across 155 kV, can readily be
obtained by floating the entire ion source at this voltage,
as is common in many types of accelerators.

The following

investigations hence assume that all the ion source equip­
ment is on a high voltage platform.
After preacceleration the beam must be transported
to the cyclotron magnet.

Since RF electric fields could

easily sweep the beam or increase its energy spread, it is
proposed to lead the beam down the inside of the dee stem
as shown in Figure 2.2.
Of several possible techniques for focusing the
beam and providing shielding from the magnetic fringe field

HELIQUAD

BEAM

SOUTH DEE

OUTER
INFLECTOR

VALLEY

DEE STEM
:r INFLECTOR SYSTEM

X

MACHINE

CENTER

SCALE

q

5

W INCHES

i-.i-i.fi I i i ^ i I

I 10

to C M
Fig. 2.2.— Cyclotron cross section with injection system.

9

inside the dee stem,
is a continuous,

one w h i c h looks par t i c u l a r l y attractive

twisted m a g n e t i c quadruple.

This consists

of a long iron pipe w i t h grooves in the inside containing
conductors.

The grooves are 90 degrees apart and rotate

along the length of the pipe
barrel.

like the rifling in a gun

The h e l i q u a d has been shown to have stronger fo­

c using than compar a b l e alternating gradient channels
(Pe 70).

The h e l i q u a d is d e s c r i b e d more fully in sections

3.2 and 4.2.
As is shown in Figure 2.2,

the exit of the heliquad

is about 48 inches from the center of the cyclotron since
if it were ne a r e r the iron in the pipe w o u l d be saturated
by the fringe field of the cyclotron.
and the outer inflector,

Between the h e l i q u a d

the b e a m m u s t be guided along a

s t r a i g h t p a t h and be focused.

Section 3.2 briefly describes

a few devices that could be used in this region.

The outer

inflector, w h o s e exit is located slightly b e y o n d the radius
of the last a c c e l e r a t e d turn, bends the b e a m into the cor­
rect initial conditions at the b e g i n n i n g of the cycloidal
p a t h so that it w i l l be correctly p o s i t i o n e d relative to
the inner i n f l e c t o r electrodes.
s c r i b ed in sections

3.3 and 4.3.

The outer inflector is de­

10

After about 35 turns of cycloidal motion inward
along a sector ridge# the beam comes to the inner inflector
system, a complex array of electrodes to focus the beam and
steer it into a path that becomes tangent to the centered,
accelerated orbit for its energy.

Until this point,

the

beam remains shielded from the HP field by staying within
the south dee.

Soon after the beam becomes tangent to the

centered orbit, it arrives at the

edge of the dee and is

then accelerated and extracted conventionally.

Sections

3.4 and 4.5 describe the inner inflector system in detail.
The following design goals were considered in ar­
riving at this injection systems

The acceptance of the

injector should be fairly large so that useful currents
from low luminosity sources, such as polarized proton and
deuteron sources, can be obtained.

A related goal is that

of minimizing all effects which would cause an increase in
the effective emittance of the injected beam.
such effects are discussed in section 4.

Several

Finally, if

bunching is to be used, it is desirable that rays from
different parts of the beam should have equal transit
times from the source to the cyclotron center if the

11

energies are equal; i.e. the injection should be iso­
chronous.

Section 3.1 deals with the buncher problem,

3.

3.1

Construction

Ion source and buncher

The design of the injection system depends to some
extent on the current available from the source.

The most

likely candidates for external sources usually deliver low
current beams, viz. present day polarized sources provide
nanoamps,

to a few microamps.

(G1 68, Cl 70, Pi 69, Mi 68),

and highly charged heavy ion sources also yield only sev­
eral microamps.

(Da 69, Kn 69).

With these currents it is

n ot necessary to consider the effects of space charge
forces.

And when a low current source is used on a cyclic

accelerator,
"buncher"

it is also clearly advantageous to include a

i.e. a device which accepts particles from the

source over a long time interval and groups them into a
small interval corresponding to the acceptance time of the
accelerator cycle.
The simplest buncher consists of an insulated elec­
trode through which the b e a m passes.
12

The voltage on this

13

electrode varies with time so that an ion may experience a
net change in energy as it sees an electric field first at
the entrance and then at the exit.

If the net energy gain

for an ion reaching the electrode earlier is less than for
a later one,

then eventually the later one will catch up

with the earlier one.

In the ideal case all the ions within

one RF period arrive simultaneously at the first accelerat­
ing gap of the cyclotron at the optimum instant.
simplest and cheapest

(although not ideal)

use a sinusoidal buncher voltage.

In the present case,

method is to

The frequency is that

of the bursts coming out of the cyclotron,
quency.

The

ie. the dee fre­

the energy change produced

by the buncher will be very small compared to the injection
energy,

because of the large path length.

age on the electrode is V cos wt,

dE — qV

[-cos (wt —

Tw

Then if the volt­

the energy change is

+ cos (wt +

lw

],

where
V = peak electrode voltage,
w = 2irf (dee frequency) ,
t = time at which the ion passes the center of
the electrode,
q = charge of the ion,
1 = length of the electrode, and
v = velocity of the ion before the buncher.

14

The v in this equation should actually be the velocity in
the electrode, but since dE is itself an incremental value
the approximation holds.

The first term is the energy

change due to the gap at the entrance of the electrode and
the second, that at the exit.

This becomes

lw
dE = -2qV sin — — sin wt.
2v
Assuming isochronous injection, ie, the total path length is
the same for all rays, the time T at which the ion arrives
at the center of the cyclotron is
L
.
L
j
dv
.
L
T = t + — —
= t + - (l-~) = t + v+dv
v
v
v

j dE
i ~ ,
2E

where
L = total path length from the buncher to the
cyclotron center and
E = energy of the ion before the buncher.

The combination of the two preceding equations is
L
LqV
T = — + t +
v
Ev

lw
sin — sin wt.
2v

Ideally T should be independent of t over an RF period;
but clearly this is not possible in the sinusoidal case,
In this case, all the ions at the buncher in a certain

1.1

15

time interval centered about t =

tt/w

will be bunched into a

considerably shorter time interval at the cyclotron central
region.

To decide what the length 1 of the buncher elec-

lw
trode should be note that sin —
should not be too close
2v
to zero or else a very large voltage V will be required.
This same problem was solved when the dee angle was chosen.
It is only necessary to make the length of the buncher
electrode equal to the path length in a dee at the injec­
tion energy.

That is.

where
A = angle subtended by the dee in radians, and
h = harmonic no. of the acceleration mode.

With a constant orbit geometry, 1 never changes.
In order to determine the only remaining unknown V,
use the fact that it is desirable to bunch as many ions
into a time interval AT as is possible with a single sine
wave buncher.

First differentiate equation 1.1 with re­

spect to t to find the maximum and minimum.

16

dT
dt

—— =

4

1 + K V cos wt,

where
_ Lqw
lw
K = —
sin — .
Ev
2v

, „
1.2

Setting the derivative equal to zero yields

w t = cos-

1

.

1.3

J\ V

This equation in general has two solutions for 0 < w t < 2 tt.
Call the smaller one t^;

then the other is

(2tt - wt^)/w.

Equation 1.1 shows that t^ corresponds to the maximu m T.
(Choose the sign of V to make K V positive.)

Next set the

difference between the maximum and m i n i m u m of T equal to
AT:

t-4 -

1

w

1 + —
w

(sin wt* - sin
1

(2tt - wt*))
1

= AT,

Simplifying this and solving for V yields
v _ 2 t .- .zwtj. + w a T

f

2K sin wt*

where K is given in equation 1.2, and t^ is the smallest
positive solution of equation 1.3

(which is to be solved

1>4

17

simultaneously with equation 1.4 to get V).

The time in­

terval at the buncher which corresponds to AT is obtained
from equation 1.1.
As examples,

these equations indicate that a two-

gap sine wave buncher w o u l d compress the ions from a time
interval of 99 RF degrees into 3.3°;
into 15.7°;

and 201° into 31.7°.

for K V are 1.1,

1.2,

137° into 9.0°;

164°

The corresponding values

1.3, and 1.5, respectively.

Note:

because of the lack of isochronism of the proposed i njec­
tion system,

the actual bunching efficiency will be lower

than the above numbers indicate.
conditions

Under typical operating

(section 4.1), a peak buncher voltage equal to

0.3% of the ion source potential would pack 7% of the cur­
rent from the source into a 7.2° cyclotron capture time.

3.2

Beam transport

The heliquad must act as a magnetic shield for the
cyclotron fringe field and have a sufficient acceptance to
transport the desired beam.

Preferably,

it should also fit

in the existing dee stem and have reasonable power

18

requirements.

A possible design, which would be usable

with the 1.75 inch dee stem aperture, is shown in Figure
3.1.

Typical parameters for this design include a 1.5 cm

diameter clear aperture and a maximum field gradient of
1500 gauss/cm.

The adequacy of these values will be dem­

onstrated in section 4.2.

Square wires 1/4 inch on a side

are located 1.1 cm from the heliquad center.

Replacing

the wires with filaments at their centers, and ignoring
all iron, the required field is produced with 2.67 kA per
wire.

If the average magnetic field line is actually

within iron for about 2/3 of its total path length, the
iron-free current should be divided by 3 as a first, rough
approximation.

Since about one half of the wire

(OFHC
—

3

copper)

is water hole, the resistance comes to .96 x 10

ohm/m.

A 2-meter heliquad would require a total power of

around 6 kW, maximum, for the numbers given.
Near the cyclotron, the fringe field becomes so
large that the iron in the heliquad would saturate.
sider a uniform magnetic field in the x-direction.
the magnetic scalar potential is

Con­
Then

19

X PIPE fFc)
POLE TIP(Fe)

W l RE

(Cu)

INSULATION

1 INCH
2 CM

Fig.

3.1.— Heliquad cross section.

20

d> = —H

o

r cos 0 ,

where x = r cos 0.

This field will then be distorted when an infinitely long
iron pipe of permeability y
z-axis.

is placed concentric with the

Let the outer radius of the pipe be a and the inner

radius b.

A straightforward solution of this magneto-

statics problem shows that if the permeability of the
material outside the pipe and in the bore of the pipe is
y^, then in the bore
2
a

= -H

[1 -

1+ ~ u

] r cos 0.

C-2
if w2 “ u l
C f . problem 13 of Chapter IV of
Meanwhile,

<->

(St 41) .

in the iron
=

(cr + d/r)

cos 0,

where

2a2 u H
(y + y )
1 O
2
1
c = — ---------2

2

2 " Ml>

2 c

(V*2 ”

d - b
V2 +

V1

_

2

2
2 * Vl’

euid

21

In Figure 3.1, b/a is about 0.65, not counting the pole
tips.

If it is desired to limit the field in the iron to

10 kG,

and taking

e<Jual to 1 0 0 0 t tJie allowed external

field is 1.45 kG; while in the bore,

a field of 10 G results.

At m a ximum excitation of the cyclotron magnet,

the fringe

field falls to 1.45 k G at approximately 48 inches from the
center of the cyclotron.

Note that since the calculation

assumed an infinitely long pipe,

the magnetic field in the

heliquad near the end w i l l be somewhat larger.

(It is as­

sumed that any perturbation of the internal cyclotron field
can be compensated by the "bump coils"

in the valleys which

are capable of producing a first harmonic of up to about
15 G at the outer turns at any azimuth.)
Between the exit of the heliquad and the entrance
of the outer inflector,

the beam is to travel in a straight

line for a distance of 23 inches.

Over this distance focus­

ing elements should match the emittance of the b e a m at the
heliquad exit to the acceptance of the outer inflector and
following portions of the injector,

and "bending" elements

should provide horizontal forces to balance the forces
arising from the cyclotron magnet fringe field.

The m a ximum

22

m a g n e tic field in this area occurs at the outer inflector
entrance and reaches about 12.5 kG at the highest cyclotron
m agnet excitation.

A system of electrostatic inflectors

w e a k e r near the heliquad and stronger near the outer ihflector could be used to balance the magnetic fringe
T o obtain focusing,

field.

an electric field gradient tranverse

to the beam w o u l d be desirable, but an electrode configur­
ation similar to the one at Saclay

(Be 6 7b) w o u l d give very

large fields at the electrodes leading to sparking.
injection energy is 5 keV at Saclay.)

(The

Weak focusing could

also be obtained by creating a magnetic field gradient.
By using air core coils to generate a field gradient but
n o t change the total number of flux lines cutting the
median plane,

the first harmonic contribution in the cir­

culating b e a m region could be held to a minimum.
necessary to use strong focusing,

If it is

it may be possible to

use a heliquad in this region with windings on the outside
of the pipe to shield it from the fringe field and to p r e ­
vent distortion of the field in the circulating b e a m re­
gion,

as is a standard technique

(Hu 65)

extraction channels for cyclotrons.

in the design of

If a design of this

23

type is used,

the dee stem w o u l d probably have to be e n ­

larged to contain the heliquad and windings.
Whatever device or combination of devices are used,
it will be assumed herein that the emittance at the h e l i ­
quad exit

(which can be adjusted somewhat as discussed in

section 4.2)

can be transformed linearly to match the ac­

ceptance at the outer inflector entrance.

The depolariza­

tion in a heliquad is treated in section 4.2;

the depolariza­

tion inside a weak focusing channel w o u l d probably be less
than in a quadrupole lens and thus negligible.

3.3

Outer Inflector

The construction of the outer inflector is illus­
trated schematically in Figure 3.2.

(Its position in the

cyclotron was shown in Figure 2.2.)

The trajectory calcu­

lations assumed that the electric field of the outer in­
flector is simply the same as the field of a cylindrical
condenser

(no electric field component out of the plane of

the d r a w i n g ) .

Also the field is assumed to end abruptly

at the ends of the deflector

(no fringe f i e l d ) .

Since

/

■
/ t / ) ) / / / /

7T7 / / /

f

\

J //

t //

t / / / / / / /

edge of dee stem

BEAM FROM HELIQUAD
ROUND PLANE

H I G H VOLTAGE
ELECTRODE

OUTER NORMAL
SCALE

I INCH

ORBIT

1

1 2 CM

Fig. 3.2.— Outer inflector.

J / / /

/

25

the height of the electrodes would be almost an inch and
their separation 0.235 inch, these assumptions are reason­
able .
Since all calculations were performed at the same
main magnet excitation,

it is not known whether the other

magnetic fields drop so fast with radius that it would be
necessary to make the inflector in two sections, with the
outer section having a lower strength.

In the case checked,

the radius of curvature of the beam d i d n ’t change too dras­
tically in the distance covered by the inflector.

An outer

inflector consisting of a polished stainless steel high
voltage electrode and a tungsten ground plane should reli­
ably hold the 60 kV
proton operation.

(ie. 98.8 kV/cm) necessary for 52 MeV
As a comparison, the electrostatic ex­

traction deflector of the MSU cyclotron which is fabricated
in this way achieves 137 kV/cm at a voltage of 96 kV.
After the beam exits from the outer inflector, it
begins its cycloid-like drift toward the central region
of the cyclotron.
of this motion.

Figure 3.2 shows the first few loops

26

3.4

Inner i n f l e c t o r s y s t e m

A f t e r completing their looping d r i f t in w a r d along
the h i l l - v a l l e y edge,
normal

the ions m u s t be s h i f t e d onto a

(ie. eventually centered)

orbit b e f o r e

leaving the

dee so that they w i l l thereafter a c c e l e r a t e conventionally.
In contrast to the outer inflector,

m o s t of the electrodes

in the inner system must not i n t e r c e p t the m e d i a n plane:
since the accelerating b e a m of the M S U c y c l o t r o n is 1 cm
high or less,

the inner inflector e l e c t r o d e s are p o s i t i o n e d

so as to leave a 1 cm clear a p e r t u r e .
In order to obtain an e l e c t r o d e s t r u c t u r e w h o s e
field could be conveniently computed,

a "point charge"

a l g o r ithm was employed for the design c a l c u l a t i o n s .
the first step in this calculation,

the field due to a

p o i n t charge inside the dee was c a l c u l a t e d
n u m b e r of image c h a r g e s ) .

Second,

As

(using a large

"electrodes" wer e s y nthe­

sized by rows of p o i n t charges p l a c e d 0.36 inches f r o m the
m e dian plane.

Finally, e q u i p o t e n t i a l surfaces of these

p o i n t charge distributions w e r e chosen to be the real e l e c ­
trodes and the strengths of the p o i n t charges w e r e adjusted
based on

(a) maximi z i n g the e l e c t r i c field in the me d i a n

27

plane,

(b) minimizing sparking by maintaining low field

strengths at the electrodes,

(c) keeping the 1 cm vertical

clear aperture, and (d) striving for electrodes with simple
geometrical shapes, eg. curved right circular cylinders
with hemispherical caps on the ends.

Because of interac­

tion among the electrodes and the several requirements
they must fulfill, they differ significantly from these
simple geometries; but the exact shapes are known and could
be duplicated, albeit with rather difficult fabrication
techniques.

Alternatively,

an electrolytic tank study

could determine the fields of the simple geometries,

the

focal properties then being checked by numerical ray trac­
ing.
The location of the 15 electrodes and the ground
planes is shown in Figure 3.3, while Figure 3.4 shows sev­
eral typical cross sections of the inflectors.

(The small

negative electrodes were omitted from the equipotentials
of Figure 3.4 A-A, clarifying the drawing.)

All the

electrodes marked with minus signs in Figure 3.3 are at
a potential of -48 kV in the case for which the calcula­
tions were made, viz. 42 MeV protons.

Since this is also

28

lS
GROUND
N

I

J INCH
r —

r

0 1 2 CM
Fig.

3.3.— Inner inflector system
— median plane view.

SECTION

A-A

-48

-12

SECTION

OkV

B-B

SCAUE

SECTION

Io

C-C
1.0 INCH

T

T"

z

cn

Fig. 3.4.--Cross sections of Figure 3.3.

30

the voltage of the outer inflector, only one power supply
and one lead-in are required for the negative high voltage.
Although the electrodes marked with plus signs are at a
potential of +2 kV, they could be enlarged and operated
at a lower voltage, or vice versa;

the thinner electrodes

closely approximate curved circular cylinders,

implying

easier fabrication.
As the ions drift toward the center of the machine
they are first influenced by the electrode pair D1

(Figure

3.3), one of three such separate or "dummy" electrode pairs.
The dummy electrodes are necessary to obtain good vertical
focusing properties:

without them,

the combination of the

1 cm vertical gap of the inner inflector system and the
rather strong vertical beam blowup gave an unusably small
acceptance.

Independently adjustable voltages for D2 and

D3, and the other positive electrodes as well, would pro­
vide for fine adjustment of the vertical focusing and of
steering.
After about 5 turns in the inner inflector system,
the ions arrive at the slit

(Figure 3.3).

If they hit one

of the jaws, a small change in the injection energy will

31

get them through the slit.

Monitoring the current on the

slit jaws wo u l d give an indication of the horizontal width
of the b e a m at this location and w o u l d signal possible
slow changes in many of the injector components.
Once through the slit,

the ions enter an inflector

that bends their trajectory into a tighter curve.

Ground

plane Gl, which constitutes part of this inflector and
shields the injected ions from RF fields, has two holes in
it to allow the accelerating b e a m to pass through.

Al­

though the median plane configuration of Gl is shown in
Figure 3.3,

this ground "plane" becomes wiggley near the

adjoining positive electrodes

(Figure 3.4 C-C),

again

causing fabrication difficulties.
The beam next experiences electric fields which
oppose the magnetic forces.

These electric fields are

provided first by the "open" inflector whose cross section
is shown in Figure 3.4 A-A,

and then by the "final" inflec­

tor consisting of ground plane G2 and a high voltage elec­
trode which intercepts the median plane.

In construction

the final inflector is very similar to the outer inflector
described in section 3.3; but w i t h a length of 1.1 inches.

32

a gap of 0.2 inch,

and a radius of curvature of 4.4 inches.

Fine voltage adjustments of the various p o s i t i v e e l e c t r o d e s
could be used to guide the b e a m through the final i n f l e c ­
tor,

the adjustments being p e r f o r m e d w h i l e o b s e r v i n g the

first few accelerated turns w i t h a b e a m probe.

The p r e s ­

ence of the final inflector w i l l ef f e c t the e l e c t r i c field
in the adjacent open inflector.

Other factors that w i l l

cause small discrepancies between the e l e c t r i c fields used
in the calculations and those of the actual p h y s i c a l d e ­
vice are the electr i c a l connections

to the e l e c t r o d e s and

the two holes in the gro u n d plane Gl.

Also,

the h i g h

vo ltage lead-ins w i l l have to be kept away f r o m the b e a m
or else e l e c t rostatically shielded so they w i l l n o t p e r ­
turb the ion trajectories.
The q u e s t i o n of w h e t h e r the p r o p o s e d inflec t o r s
w o u l d h o l d the required high voltage
p r o t o n operation)

(60 k V for 52 M e V

was investigated experimentally.

The

M S U c yclotron lab. mechan i c a l shop con s t r u c t e d a m o c k u p
of the upstream end of the second inner infle c t o r
w i t h a cross section like that in Figure 3.4 B - B ) .
m ockup h e l d -56 k V w i t h o u t excessive sparking.

(ie.
This

The high

33

voltage electrodes were stainless steel, and the ground
plane and spark plates

(above and below the pair of high

voltage electrodes) were tungsten.

After the test a 1 mm

long copper shaving was found stuck on an electrode at a
point of high electric field.

Probably the shaving fell

from the ground plane support, got attracted to the high
voltage and became welded to the electrode with the subse­
quent spark.
this shaving.

Most of the sparking occurred at or near
The maximum electric fields of the mockup

are very nearly equal to those of the actual proposed inflectors at the same voltage,

thus the inflectors will

likely hold the required voltage, assuming the absence
of copper shavings.

4.

4.1

Performance

Ion source and buncher

The ion source performance determines to a large
extent the performance desired from the rest of the injec­
tion system.

In section 3.1 several types of ion sources

were listed with maximum currents provided.

Other para­

meters of interest are the emittance, energy spread, and
if pulsed,

duty factor and repetition rate.

Almost all known sources for external injection
provide axially symmetric beams so the emittance is speci­
fied by one number.

The acceptance of the cyclotron as

well as the proposed injection system, on the other hand,
is larger in the vertical direction than in the horizontal
direction.

Also, since the orbit geometry is fixed, the

acceptances of the injector and the cyclotron are inde­
pendent of the type of particles accelerated and their
energy.

The transverse momentum of the particles from an

ion source tends to be fixed

(often determined by the
34

35

p l a s m a temperature)

I n d e p e n d e n t of the e x t r a c t i o n voltage?

thus the e m i t t a n c e

from the source w o u l d be less at h i g h e r

energies.

As a result,

if the source emittance w e r e

larger than the i n j e c t o r acceptance,
f i c i e n cy w o u l d d e p e n d on energy.

the t r a n s m i s s i o n e f ­

Since du r i n g a c c e l e r a ­

tion in the c y c l o t r o n there is no m i x i n g of l o n g i t u d i n a l
w i t h t r a n s v e r s e p h a s e space and very
tortion

(B1 69),

little n o n l i n e a r d i s ­

and since the energy coming out of the

c y c l o t r o n is 336 times the injection energy,

the e m i t t a n c e

at e x t r a c t i o n is 18.4 times smaller than at the e n d of the
i n j e c t i o n process.
protons

(Actually this ratio is 1% larger for

than for h e a v y ions due to the r e l a t i v i s t i c effect,

i g n o r e d here.)

A notable

lack of u n i f o r m i t y exists r e ­

g a r d i n g the units in w h i c h emitt a n c e is specified.

One

p o s s i b l e c o n v e n t i o n is to specify the emitt a n c e in units
of m m m r a d i a n s r e f e r r e d to 42 MeV.
be u s e d here.

This conven t i o n w i l l

If the e m i t t a n c e needs to be given at the

i n j e c t i o n energy,

for example,

it w i l l usually also be

c o n v e r t e d to the 42 M e V value.
F o r p o l a r i z e d ion sources,

a good criterion of the

q u a l i t y of the b e a m is the p r o d u c t of the luminosit y and

36

the square of the polarization.

{The information to compute

this product is not always complete in the literature on
ion sources.)

One of the best sets of figures yet quoted

is the design data of the polarized source used at the Berk­
eley 88 inch cyclotron

(Lu 69).

to lOya; the 42 MeV emittance,
zation,

The current expected is up
8 mm mr; the proton polari­

1; the deuteron vector polarization,

deuteron tensor polarization,

2/3; and the

1; with the following defini­

tions:

vector polarization, P

z

=

(N, - N )/EN,
+
-

tensor polarization, p zz = 1 ~ 3NQ/EN,

where ZN = N+ + N

+ N^; N q = 0 for protons, and N+ , N ,

and Ng are the occupation numbers of the corresponding
states.
At Erlangen, Germany, a 70% polarized proton beam
of less than 5.1 mm mr. emittance with a current of lya
was achieved (Cl 70).

The polarized source at Saclay

yields 3 to 5ya of protons.

The emittance is not specified

but assuming that the acceptance of the injection system
is the same as an earlier mockup

(not an entirely safe

37

a s s u m p t i o n ) , the 42 M e V emittance comes to around 2 m m mr;
the polarization at extraction is as much as 90%
Be 6 6 b ) .

(Be 6 7b,

A laser-pumped polarized 3He+ source has an emit­

tance of 1.5 m m mr at 42 MeV with a 4ya current and 5% p o ­
larization

(Pi 69).

A polarized lithium and cesium source

has achieved a 20% polarized Li++ b e a m at a current of 1.3 na
(no emittance data)

(Mi 68).

With one exception,

the articles on heavy ion

sources usually do not specify the emittances.

The "ordi­

nary" duoplasmatron or Phillips Ionization Gauge or Penning
discharge types would probably have emittance at least as
small as the polarized sources, based on the performance of
u ppolarized proton sources of these types

(Wr 6 8, Be 6 7a) .

Of course many factors influence the emittance for a given
general type of source, most especially the extraction
aperture size.

The types of ions and the currents available

were listed in section 3.1.

The only heavy ion source w i t h

an emittance estimate is the proposed "HIPAC" source
(Da 69).

It gives 32pa of Kr

420 m m mr at 42 MeV; however,

20 +

in an emittance area of

in this case the emittance

should be adjusted to the actual cyclotron extraction

38

energy:

the MSU cyclotron could accelerate such ions to

over 250 MeV,

at which energy the emittance is 170 m m mr.

The HIPAC is also unique in that it is the only type of
source with a nonnegligible energy spread,
fact,

as a matter of

it is around half the injection energy.

All the

other sources listed characteristically have very small
energy spreads,

eg.

the energy spread of polarized sources

comes mainly from the ionizer and is 20 to 50 eV,
(Gl 66).

typically

The energy homogeneity restriction for the injec­

tor system is examined in section 4.4;

the acceptance, in

section 4.5.
In order to be able to bunch the injected beam, it
is required that the energy spread before the buncher be
considerably less than the energy that can be gained in the
buncher, wh i c h is ±.6%

of the injection energy, roughly.

The e xact value depends on the usable capture interval of
the cyclotron acceleration process and whether the injection
is i s o c h r o n o u s .

Since the dee voltage of the MSU cyclotron

has sinusoidal time dependence,
acceleration is 1 - cos
where

uj

the energy spread during

(ooAt/2) , assuming isochronism,

is the angular dee voltage frequency and At is the

length in time of a beam pulse, if the starting phase is
adjusted for maximum energy gain per turn
energy spread).

(thus minimum

In order to achieve high extraction effi­

ciencies, it is necessary to limit the total energy spread
to less than the reciprocal of the number of turns.

For a

200-turn geometry this restricts the capture interval to
11 RF degrees.

Actually, because of finite electrostatic
\

extraction deflector thickness, RF voltage ripple and other
factors,

the phase width must be considerably less than

this for complete extraction.

The anisochronism of the

injection system amounts to much more than this, v i z . even
assuming perfect bunching, the time spread among different
rays at the center of the cyclotron would be some 40 RF
degrees, unless the emittance of the source were noticeably
less than 2 mm mr, referred to 42 MeV.

(Acceleration on

the first harmonic is assumed;

there is even less isochron-

ism at the higher harmonics.)

The optimum buncher voltage

actually depends on the distribution of current within this
time interval.

A reasonably fair estimate is that 3/4 of

the beam is spread over 20 RF degrees.

Take a buncher

voltage such that with isochronous injection the time

40

s p r e a d a t the c y c l o t r o n c e n t e r w o u l d be 9 RF degrees.
m e a n s t h a t V = 1.20/K,

a n d a time s p r e a d of 137 deg.

b u n c h e r is p a c k e d i n t o 9 deg.
(See s e c t i o n

3.1 for n o t a t i o n and equations.)

7% of the b e a m

N o w if the

for as m u c h as 7.2 RF

lies in t h a t time i n t e r v a l ,

for the case of n o b u n c h i n g .
for this case is 0.2%.

100%

A d d i n g the

that a c t u a l l y

at the b u n c h e r g o i n t o 29 RF deg.

e x t r a c t i o n e f f i c i e n c y is n e a r l y
degrees,

at the

at the c y c l o t r o n center.

time s p r e a d of the i n j e c t i o n p r o c e s s m e a n s
the 137 deg.

This

The

vs.

2%

cyclotron energy spread

If a s m a l l e r e x t r a c t e d e n e r g y

s p r e a d is r e q u i r e d the p h a s e w i d t h m u s t be r e d u c e d w i t h
a proportional

loss of c u r r e n t

(changing the b u n c h e r v o l t ­

age w o n ' t h e l p m u c h b e c a u s e of the lack of i s o c h r o n i s m in
the i n j e c t o r ) .
lower,

A l t h o u g h the e x t r a c t i o n e f f i c i e n c y b e c o m e s

a n e t gain in c u r r e n t can be a c h i e v e d by a large

i n c r e a s e in RF p h a s e w i d t h ,
spread.

The r e m a i n i n g

a g a i n w i t h a larger e n e r g y

7% of the b e a m

s p r e a d and a sine w a v e buncher)

(for a 0.2% e n e r g y

w i l l be fur t h e r d i m i n i s h e d

b y e f f e c t s to be d e s c r i b e d later.

41

4.2

Beam transport

There are no restrictions on the beam transport
line between the ion source and the cyclotron vacuum box.
The beam envelope may be as large as desired, many dif­
ferent types and sizes of focusing elements can be used
as well as auxiliary devices such as energy analyzers,
mass spectrometers, polarization rotators, and steering
elements.

Once inside the dee stem the previously de­

scribed helical magnetic quadrupole channel will be used.
With such a "heliquad," the axial component of the mag­
netic field can be ignored if the ratio of the period of
one complete rotation of the pole tips to the maximum
radius of the beam is greater than 20

(Pe 70).

In this

instance, G. Salardi has written the equations of motion
and their solutions

(Sa 6 8).

If the aperture of the

heliquad has a radius of 7.5 mm and the length of the ro­
tation period is 30 cm, this ratio will be 40 and the
resdlts of G. Salardi should apply very accurately.

By

choosing a magnetic field gradient of 1342 gauss/cm for
125 keV protons

(the value scales as momentum over charge),

the motion is periodic along the length of the heliquad

42

channel.

The periodicity of the ion beam is 47.5 cm in

this case.

If the total length is made an integer mul ­

tiple of this value,

the beam at the exit will have the

same emittance shape as at the entrance

(assuming a pe r ­

fect quadrupole field in the h e l i g u a d ) .

If all the fo­

cusing elements between the ion source and the heliquad
are axially symmetric, ec[. Einzel lenses, then the beam
at the entrance of the heliquad will be axially symmetric
also since the emittance from most sources is.

If these

axially symmetric focusing elements bring the beam to a
focus at the entrance of the heliquad,

a 42 MeV emittance

area of 6.8 m m m r will be completely transmitted to the
exit.

To attain this acceptance,

the proton beam should

have a radius of 1.19 m m and a divergence of 33.1 mr at
125 k e V at the heliquad entrance.

It should be noted that

the emittance of an axially unsymmetric beam can be larger.
For example,

if the heliquad is focusing in the Y direction

at the entrance;

the width of the beam in this direction

could be 15 m m total and the divergence same as before,
again assuming a focus at the entrance.

However, the area

of 6.8 m m mr is close to the maximum vertical acceptance

43

of the rest of the injection system and much larger than
the horizontal one.

Also, the acceptance of the heliquad

could be made even larger by using a tighter spiral of the
pole pieces.

Since the acceptances of the injection system

and the cyclotron, and indeed most nuclear experiments,
differ in the vertical and horizontal directions, it might
seem desirable to remove a little horizontal phase space
area and place it into the vertical one.

The heliquad

can exhibit strong coupling in the two transverse directions
(Oh 69) so it could be useful in this application.

Such a

scheme would of course be profitable only when the emittance
from the ion source (assumed axially symmetrical) was in
between the vertical and horizontal acceptances.
The focal properties of a heliquad can be approxi­
mated by an alternating gradient channel of quadrupoles
having the same field gradient as the heliquad, no drift
spaces and containing four quadrupoles per turn of the
heliquad pole piece.

The alternating gradient channel has

a slightly larger focal length.

D. Werren has examined

the problem of depolarization due to magnetic quadrupoles
(We 6 8).

Specifically, he considered a beam tube radius

44

of 2.5 cm, two 13 cm long quadrupoles separated by 13 cm
with a resultant focal length of around 15 cm, and found
a depolarization for the maximally divergent rays of a
few parts per thousand.

Of course the depolarization is

zero for paraxial rays.

The result is energy independent.

It might thus be expected that the heliquad channel would
also have an acceptably low depolarization.
As was shown in section 3.2, the heliquad must end
when the stray cyclotron field reaches 1.45 kG.

This is

2 3 inches from the entrance of the outer inflector.

Over

this length, the magnetic field increases from nearly noth­
ing at the exit of the heliquad to 80-90% of the field at
the cyclotron center.

Section 3.2 also describes possible

devices for use between the heliquad and the outer inflector.

If weak focusing is used, horizontal focusing can be

increased and vertical decreased or vice v e r s a .

It will

be assumed that the emittance from the heliquad

(which is

adjustable within limits, as previously indicated)

can

thus be matched to the acceptance of the rest of the system,
which will be described in the next sections.

45

4.3

O u t e r i n f l ector

The array of focusing elements b e t w e e n the h e l i q u a d
and the ou t e r i n f l e c t o r just balances the m a g n e t i c field so
that the central ray trajec t o r y is a straight line.
o uter i n f l e c t o r then b e n d s

The

the b e a m w i t h a radius of cur­

vature of 6 inches a ga i n s t the m a g n e t i c force.

This in­

flector ends about 1/2 inch b e y o n d the calculated p o s i t i o n
of the last turn of the n o r m a l internal b e a m of the cyclo­
tron,

In o r d e r to m a x i m i z e overall verti c a l a c c e p t a n c e ,

the h e i g h t of this outer i n f l e c t o r is the largest w h i c h
w i l l allow s u f f i c i e n t b r e a k d o w n voltage in the 1.75-inch
dee aperture.

Cal c u l a t i o n s herein assume an elec t r i c field

w h i c h is e f f e c t i v e l y u n i f o r m over a total height of 0.8
inch.

The h o r i z o n t a l gap of the outer infle c t o r was chosen

to a l l o w the h i g h vol t a g e electrode to operate at -48 kV,
the same as the inner inflectors,
g r o u n d e d to the dee;

and the other electrode

g i v i n g a gap of 0.235 inch.

The

m e d i a n p l a n e family of rays w h i c h determines the horizo n t a l
em i t t a n c e actu a l l y requires a gap of 0.273 inch and elec­
trode voltages of -53.5 k V and +2.26 kV.
rays w h i c h w o u l d fit in the

A n o t h e r family of

.235-inch gap and still give a

46

similar emittance probably exist, but finding such rays
is a lengthy task.

The next sections give the details of

the calculations.
There is an observable deviation from isochronism
between the ends of the outer inflector.

It is a small

effect, as would be expected from the short time the ions
spend in the inflector, and seems to partially compensate
the larger lack of isochronism arising from other sources.
The spin axis of polarized particles should be
alligned vertically along the field lines in the cyclotron,
if possible.

They should emerge from the heliquad alligned

this way since they immediately experience a magnetic field
which is nearly vertical for all rays.

The electric fields

before and within the outer inflector have no direct effect
on the spin, so any depolarization must be determined by
the small horizontal magnetic field components.

The prob­

lem is similar to evaluation of depolarization during ex­
traction, except that the velocities involved are some 18
times greater during extraction.

G. Budyansky has calcu­

lated a depolarization that is many times less than .03
for practical cases during extraction (Bu 60); also there

47

are experimental results which show no detectable depolar­
ization over the entire acceleration— extraction process
in cyclotrons

(Oh 70, Po 66b, Be 6 6 b ) .

Horizontal m a g ­

netic fields of the same order of magnitude are encountered
as the ions

drift down the sector edge, and no significant

depolarization occurs there

4.4

The

(see section 4.4).

Drift along the sector edge

magnetic field of the MSU cyclotron was mapped

at several main magnet currents using a grid consisting of
1 inch increments from R = 0 to R = 46 inches and 4 degree
increments from 0 = 4 to 6 = 360 degrees, inclusive.

The

azimuthal variation was Fourier analyzed into cosine

("H")

and sine

(“G " ) components in this way:
N

B ( R , 0) = B rt(R) +
0

E

_
n=l

[H

3n

(R) cos 3n0 + G_ (R) sin 3 n 0 ] ,
3n

where N will be called simply "the number of harmonics"
(Be 66).

Since the general orbit codes available pr e v i ­

ously were not w e l l suited for tracking ions as they loop

48

along the m a g n e t ridge line*

D. Johnson*

of this lab* wrote

a n e w comp u t e r code to integ r a t e the equations of m o t i o n in
C a r t esian coordinates.

He w r o t e versions of this code

using M i l n e - R e y n o l d s and R u n g a - K u t t a integration methods
and also a m o d i f i e d R u n g a —K u t t a utilizing selected double
p r e c i s i o n in t e g r a t i o n constants.

The m o d i f i e d Runga—K u t t a

version w a s used for m o s t of the calculations herein*

since

it e x h i b i t e d low a c c u m u l a t e d errors and p e r m i t t e d conven i e n t
step size changes.
p r o p o r t i o n a l to z

2

The o r i g i n a l versions included a term
in the e q u a t i o n for

p o r t i o n a l to z in the equat i o n s

and a ter m p r o ­

for $ and y

(the dots indi­

cating d i f f e r e n t i a t i o n w i t h res p e c t to t i m e ) .

A test p e r ­

formed w i t h and w i t h o u t these terms i n d i c a t e d that they
w e r e n o t n e c e s s a r y and thus the s u b s e q u e n t runs used only
linear z-raotion u n c o u p l e d f r o m the m e d i a n plane motion,
as far as the m a g n e t i c field w a s concerned.

A handy fea­

ture of the o r i g i n a l code made it easy to change the n u mber
of h armonics u s e d in c a l c u l a t i n g the azimuthal variation
of the m a g n e t i c field.

The orig i n a l code was m o d i f i e d many

times in the process of add i n g electric fields,

the capa­

bi l i ty of s i m u l t a n e o u s l y run n i n g two rays independ e n t of

49

each other in z (to reduce the required computer time),
and polarization.

As indicated previously, the electric

fields of the open inflectors were obtained from rows of
point charges.

For the case in which only median plane

motion is desired, the following procedure was used.
First a separate code calculates the potential and field
in the median plane due to a pair of charges each of unit
strength located at R =

0, Z = ±0.36 inch.

During inte­

gration of the trajectory, this data is then used to sum
the contribution of each point charge at each step.

(A

total of about 300 point charge pairs were used to generate
the fields due to all the open inflector and dummy elec­
trodes.)

For z-motion, the potentials are first calculated

in the median plane and also slightly above the median
plane.

Then the orbit code uses these two sets of poten­

tials to get the vertical electric field, assuming that
it is proportional to z.

The inner electrode fields are

important at distances out to r = 8 inches, approximately.
All of the programming is in FORTRAN IV and the time re­
quired for tracking one ray in the median plane

(two dif­

ferent z values) with depolarization determinations is

50

around 15 minutes on a Xerox Data Systems Sigma 7.

Most

of this time is spent in the electric field calculation
just described.

This method of calculation was used be­

cause it allows relatively easy modification of the elec­
trode configuration.

An analytic field is used in the

closed inflectors, viz.

the outer and final inflectors

(see section 4.5) and very little computer time is required
there.
As the ions drift along the hill-valley interface,
they spend a relatively long time
gions of high field gradients.
depolarization occurs,

(2 microseconds)

in re­

To verify that no important

the direction of spin was integrated

along with the other motions.

Classically the equation of

motion for a particle with angular momentum 3 and magnetic
moment

(Ts) is

~

at

= rs x S.

Although the spin of a particle is a quantum mechanical
effect,

the only quantum mechanics needed for the depolari­

zation calculation is a theorem which says that no quantum
mechanics is needed.

Usually called the correspondence

51

principle, it says that when averaged over many particle
states

{so as to become macroscopically measurable)

quan­

tum mechanical effects had better agree with classical
physics.

If the equation
P = < J > /jh,

where J is the angular momentum operator,
orientation vector?

defines the spin

then the equation of motion

dt

=

-r

h

x

p

holds for all values of j, where r is the gyromagnetic
ratio.

U. Fano gives this equation in

(Fa 57)

and also

gives references to complicated quantum mechanical deriva­
tions.
The depolarization test actually performed con­
sisted of running several median plane rays each with var­
ious initial conditions in the vertical direction.
initial polarization was P

z

= 0.9899, P

x

= 0.1, P

y

The
- -0.1.

(The beam would be polarized vertically initially ideally;
the nonzero median plane components simulate a partial de­
polarization before the exit of the outer inflector.)

The

depolarization of protons, investigated in all portions of

52

the injection system downstream of the outer inflector,
is roughly proportional to the maximum vertical displace­
ment attained.

The same magnetic field was used for the

spin and position coordinates, ie, the x and y components
taken to be proportional to z, and the z component, inde­
pendent of z.

Out of a total of 14 different rays whose

spin was tracked,

the maximum change in P

z

was 1%.

The

vertical initial conditions were chosen such that each
ray attained the maximum |z| permitted by structure in the
cyclotron.

The injected ions used were 125 keV protons

as in all computer runs; the number of harmonics used to
calculate the magnetic field was 9 in five of the cases
and 6 in the rest.

Since the total depolarization is the

average of all of the rays in the beam, including ones
that remain near the median plane, it is completely negli­
gible in this portion of the injection system.
The energy spread in the injected beam will broaden
the beam horizontally near the center of the cyclotron.
This is due to the spectrometer-type effect during the drift
along the magnet ridge.

The simplest possible approxima­

tion to the situation is a straight boundary line

53

separating a constant weak magnetic field and a constant
strong one.

Now consider ions starting at the boundary

with an initial direction which is perpendicular to the
boundary.

After the first half-turn the distance from the

starting point is proportional to the momentum of the ion.
And on the second half-turn the trajectory is a different
size semi-circle because of the sudden change in the field.
The nonstartling but important point is that the distance
to the starting point is still proportional to the momentum.
In fact, even after 70 half-turns the distance to the start­
ing point is still proportional to the momentum.

The dis­

tance from the outer inflector to the inner inflector system
is around 30 inches.

The proposed buncher causes a peak

to peak momentum spread of about 0.5% if the buncher is
4 m from the vacuum tank.

(This is for first harmonic

acceleration;

the momentum spread is 0.25% for second har­

monic, etc.)

This means that the beam would sweep back

and forth 0,15 inch at the cyclotron center due to the
0.5% momentum spread.

This result is approximately con­

firmed by the actual integration of the equations of mo­
tion of the injected ions.

The maximum energy spread

54

which can be -tolerated is determined by this broadening
of the beam.

For instance,

if the motion of the bea m were

perpendicular to the axis of the final

(closed)

inflector,

the permissible sweeping would be equal to the inflector
aperture minus the zero-energy-spread b e a m width;
difference is about 0.14 inch.

this

Thus it might be desirable

to increase the inflector path length so that the buncher
w ould not have to produce such a large momentum spread
transport system is assumed to be i s o c h r o n o u s ) .

(the

Locating

the ion source system outside the shielding w a l l w o u l d also
enable operators or technicians to manipulate it while the
cyclotron is running.

Even if all the b e a m fits into the

final inflector, momentum spread causes an effective in­
crease in the median-plane emittance.

The inherent m o m e n ­

tum spread of the ion source itself and of the preac c e l e r ­
ator limits how far the buncher voltage can be lowered by
increasing the path length.
One problem which arose was how many harmonics
must be used in calculating the azimuthal dependence of
the magnetic field.

(The conventional general orbit codes

for the accelerating beam require 3 harmonics,

ie. up to

55

and including sine and cosine of 96.)

The amount of de­

p o l a r ization for a given maximum vertical displacement
varies only insignificantly between the 6 and 9 harmonic
cases.

The median plane motion differs by up to approx­

imately .005 inch per loop

(or t u r n ) .

This is due to the

fact that if the number of harmonics is 9, the maximum
azimuthal gradients available become greater and the radius
gain per loop increases with respect to the 6 harmonic
case.

The effect is evident only at the outer radii.

The

m edian plane error between the 6 harmonic calculation and
the true motion in the exact magnetic field could probably
be compensated for by a small change in the injection
energy;

thus it was decided to do the feasibility studies

u sing 6 harmonics.

But after completing most of the

lengthy computer r u n s , it was discovered that the number
of harmonics has a somewhat more important effect on the
axial motion of the beam.

To evaluate the dependence of

the z motion on the number of harmonics used,

several rays

were run backward from the final closed ihflector out to a
radius of about 31 inches.
tions,

For the same starting condi­

the ratios of the maximum axial displacement attained

56

in the 9 harmonic case to that in the 6 harmonic case were
5.2, 4.1, 3.7, 4.6, 1.7, 0.36, 3.8, and 1.5.

Since the

central geometry and especially the dummy electrodes were
adjusted to give the best z focusing properties for 6 har­
monics, it is not surprising that the beam tends to grow
larger axially when the number of harmonics is 9.

To get

a very accurate picture of the z motion in the actual
cyclotron magnetic field, it may be necessary to use even
more than 9 harmonics.

After finding out how many har­

monics are required, the inner electrode structure would
have to be changed to optimize the axial acceptance again.
Since this would take a lot of computer time, only the re­
sults for 6 harmonics will be considered, corresponding to
a field with a smoother azimuthal variation, eg. one pro­
duced by a magnet with a larger gap.

By optimizing the

inner electrodes for more harmonics it may well be that
the resulting axial acceptance would be as large as that
found for the case investigated, since relatively small
changes in the size of the dummy electrodes,
cause a very marked change in the acceptance.

for instance,

57

As with axial motion, it was observed that the
median plane motion in the electric field free region was
essentially linear for the maximum usable emittance area.
The maximum nonlinearity is somewhere around 10% which,
although noticeable, is not very important.

The nonlin­

earities in the central inflector system, on the other
hand, cause a large distortion of the median plane emit­
tance.

Both horizontal and vertical motions in this region

are described in section 4,5.

The overall performance of

the injection system from the outer inflector to the final
inflector is summarized in section 4.6.

4.5

Inner inflector system

Drawings of the inner inflector system were given
in section 3.4.

The method of calculating the electric

field of the open inflector and dummy electrodes was
briefly described in sections 3.4 and 4.4.

The require­

ments of maintaining a degree of simplicity in the calcu­
lation, minimizing computer time, and ending up with a
comprehendible set of results suggest that z motion

58

computations be limited to vertical forces which are pro­
portional to z and horizontal forces which are independent
of z.

This means that only two rays with independent ver­

tical initial conditions are needed to specify completely
the vertical motion;
tions are separated

and the vertical and horizontal equa­
(the vertical motion depends on the

horizontal motion but not v i c e - v e r s a ) .

These are the same

approximations used in the magnetic fields,
they are reasonable

for which case

(due mainly to the fact that the magnet

gap is large compared to the vertical size of the b e a m ) .
In order to try to obtain a very rough idea of the errors
involved in applying these approximations to the electric
field of the inner inflector system, special computer runs
were made.

(It might be expected that nonlinear effects

would be noticeable since the vertical clear aperture be­
tween electrodes is only 0.4 inch.)

Most of the calcula­

tions used vertical electric fields obtained from poten­
tials on the median plane and potentials on the plane at
z — 0.05 inch, assuming that E
this were true,

z

is proportional to z.

If

the same field would result using poten­

tials from some other pair of z values.

Computer runs

59

w e r e made using vertical electric fields calculated from
z equal to 0 and .05 and also 0 and .15 inch using rays
with the same starting conditions.

(Again the code tracked

the ions backward from the final to outer inflectors.)
The ratios of the maximum vertical displacement attained
by the ions in the 0.15 inch case to that in the 0.05 inch
case are 0.59,

1.3,

1.3,

2.2,

and 1.5.

The change in the

magnitude of the axial acceptance was less than the ratios
of individual rays for the cases tested.

Of course,

the

shape of the acceptance areas will change.
The voltage on the electrodes of the open inflector
was chosen to provide the correct radius of curvature for
ions in the median plane.

In the approximation used for

the bulk of the computer runs,

the same horizontal electric

field components are used regardless of how far from the
median plane the ions might be.
actly valid since an ion above

Obviously this is not e x ­
(or below)

the median plane

w i l l be closer to the electrodes and will be in the p r e s ­
ence of a considerably stronger horizontal electric field.
It is quite easy to get an upper limit on the result this
e ffect wo u l d have by comparing the bending strength of the

60

open inflector configuration at the median plane vs. that
away from the median plane.

Since the field at 0.2 inch

is as much as 30 to 35% higher than at the median plane,
it can be seen that particles near z = ±.2 inch in the
open inflectors will all be lost, as an upper limit.

Ac­

tually the runs with the horizontal force components inde­
pendent of z and linear vertical forces indicate that the
beam envelope tends to be small vertically in the open
inflectors and, since the vertical focusing is fairly
strong here, any one ray is at large values of z only for
a short time.

About the simplest calculation that would

give some indication of the motion of the particles in the
more complex field uses horizontal force which can assume
two values depending on how far the particle is from the
median plane.

The calculation consisted of a series of

computer runs all with the same horizontal initial condi­
tions but with different vertical initial conditions.
There were two sets of horizontal components of electric
field:

the standard ones ie. median plane values, and the

non-median plane ones

(z = 0.12 inch).

The standard hori­

zontal components were used whenever the particle was less

61

than 0.0 7 i n c h from the m e d i a n p l a n e a n d n o n - m e d i a n p l a n e
ones w e r e used otherwise.
d e pends on the z motion.

Thus
Also,

the r e s u l t i n g x-y m o t i o n
a l t h o u g h the v e r t i c a l e l e c ­

tric fields are just the s t a n d a r d

linear ones o b t a i n e d f r o m

the p o t e n t i a l s at z — 0 and z = 0.05 inch,

the a c t u a l r e ­

s u l t a n t z m o t i o n is n o t linear s i n c e d i f f e r e n t rays fo l l o w
d i f f e r e n t x-y paths y i e l d i n g d i f f e r e n t v e r t i c a l forces.
T h e c o m p uter code tracked the p a r t i c l e s b a c k w a r d s

from the

e x i t of the final i n f l e c t o r u s i n g v a r i o u s ini t i a l conditions
in the z d i r e c t i o n w i t h the r e s t r i c t i o n that e a c h at t a i n
n e a r l y the m a x i m u m d i s p l a c e m e n t f r o m the m e d i a n p l a n e p e r ­
m i t t e d b y the structure.

The n o n l i n e a r i t y is e x e m p l i f i e d

by two rays starting on the m e d i a n p l a n e w i t h a 16% d i f ­
ference in v e r t i c a l divergence,

o t h e r w i s e identical;

almost

a f o u r f o ld change in the m a x i m u m v e r t i c a l d i s p l a c e m e n t
occurred.

The p r o j e c t i o n s o f the v a r i o u s t r a j e c t o r i e s o n t o

the m e d i a n plane create an e n v e l o p e that is a b o u t
w i d e for m u c h of the p a t h length.
exit,

.2 inch

A t the o u t e r i n f l e c t o r

the h o r i z o n t a l dista n c e s b e t w e e n the ray lying in the

m e d i a n p l a n e and the other rays

(measuring p e r p e n d i c u l a r to

the m e d i a n p l a n e ray)

-.14,

are +.17,

-.05, -.04,

a n d -.25

62

inch.

T h e t u r n s e p a r a t i o n a t this p o i n t is a b o u t

.25 inch.

If the e f f e c t c o n s i d e r e d h e r e causes a h o r i z o n t a l s p r e a d of
a r o u n d a q u a r t e r i n c h or m o r e

in a d d i t i o n to the i n h e r e n t

s p r e a d d u e to the i n c o m i n g h o r i z o n t a l e m i t t a n c e ,
the b e a m w i l l be

lost.

vertical displacements
beam not

The

lost rays w i l l b e those of large

in the i n n e r i n f l e c t o r system.

lost w i l l e f f e c t i v e l y r e d u c e

fields w i t h
torted,

z, the h o r i z o n t a l e m i t t a n c e gets f a i r l y d i s ­

as y o u w i l l see q u i t e soon.)

In the case t e s t e d

a b o u t 6 5% of the v a l u e o b t a i n e d

for the s i m p l e r case of h o r i z o n t a l forces
This number w i l l shrink

Actua l l y ,

of

a l l t h e r e s u l t s of this very a r t i f i c i a l t e s t run,

using horizontal

forces w h i c h

c o r r e c t a t z = 0 to the v a l u e
e v e r the p a r t i c l e ' s

ness.

i n d e p e n d e n t of z.

f u r t h e r b e c a u s e some of the rays

s t r a y t o o far h o r i z o n t a l l y and are lost.

inch,

(It is little

b u t e v e n a s s u m i n g n o v a r i a t i o n of the h o r i z o n t a l

the v e r t i c a l a c c e p t a n c e w a s

course,

The

the h o r i z o n t a l a c ­

c e p t a n c e of the i n j e c t o r b y w i d e n i n g the beam.
consolation,

some of

change s u d d e n l y f r o m the v a l u e
correct at z = 0 . 1 2

vertical displacement exceeds

inch w h e n ­
0.0 7

a r e n o t m e a n t to b e i n t e r p r e t e d w i t h n u m e r i c a l e x a c t ­
T h e y o n l y g i v e a r o u g h e s t i m a t e of the k i n d s of

63

errors involved in the linearized approximations which are
necessary to carry out quantitative calculations.

It

should be emphasized that all of these nonlinear effects
result from the electric fields.

As said previously the

nonlinear and emittance mixing capacities of the magnetic
fields are negligibly small.
The closed inflectors are assumed to be large
enough vertically so that

is negligible inside in the

region occupied by the beam which is .4 inch high in the
final inflector and .8 inch in the outer inflector.

At

the exit of both closed inflectors the potential changes
abruptly from the cylindrical capacitor distribution in
the inflectors to zero outside.

At the entrance to the

final inflector the potential changes suddenly from that
determined by the open inflector electrodes to the cylin­
drical capacitor form.

The sudden potential change means

a sudden change in the speed of the ion.

A delta function

electric field effectively exists at these points directed
parallel to the inflector axis.
locity does not change,
particle direction.

Since the transverse ve­

there is a sudden change in the

And since the potential in the closed

64

inflectors is more negative than outside#

usually,

the

positive ions go faster inside and are more nearly parallel
than outside the closed inflectors.

This slightly unphys­

ical delta function electric field is simply an easy way
of approximating the fringe field of a real inflector
w h i c h w o u l d extend over a measurable distance.

This dis­

tance is so short in the case of an inflector with a h o r i ­
zontal gap of around 0.2 inch that the sudden approximation
used is prob a b l y accurate.
T o determine the acceptance area in the median
plane and its shape,

a series of rays were tracked from

the final inflector to the outer inflector.

The magnetic

field was generated from 6 harmonics# creating an error of
a few mills per loop which can be compensated by a small
change of injection energy,

as described previously.

In

the m edian plane the electric fields are quite accurate,
limited only by any fabricational difficulties in the con­
struction of the various electrodes and ground planes.
The m edian plane motion is, however, nonlinear, neces s i t a t ­
ing a large number of rays in order to determine the ac­
ceptance accurately.

The important objectives are

65

ma x i m i z i n g the acceptance area and m a i n t a i n i n g a shape
that is rather compact during the a c c e l e r a t i o n in the
c y c l o t r o n to insure passage through various slits and
apertures and provide both low diverg e n c e and small b e a m
size on target.

The median plane e m i t t a n c e of the b e a m

as it emerges from the final inflector p r i o r to crossing
the first accelerating gap is shown in Fi g u r e 4.1.

It

meets the criterion of compactness w h i l e the lumpy appear­
ance arises from an attempt to include as m u c h area as
is feasible.

The area enclo s e d by the figure corresponds

to 2.4 m m m r referred to 42 M e V

(or 43.7 m m m r at 125 k e V ) .

The area possibly could be increased by b u l g i n g out the
b o u n d a r y at some places, b u t additional calculations w o u l d
have to verify acceptable v e r t i c a l focusing and horizo n t a l
p o s i t i o n and direction at the outer inflector.

Because of

the substantial nonlinearities in the m e d i a n p l a n e caused
by the electrodes of the inner inflector system,

the ac­

ceptance area at the exit of the outer i n f l e c t o r is in bad
shape,

as shown in Figure 4.2

arbitrary).

(the choice of the origin is

Assu m i n g that the emittance f r o m the ion

source is elliptical and the b e a m optics b e t w e e n it and

66

ft

Fig.

4.1.- - M e d i a n p l a n e e m i t t a n c e at the
e x i t of the final inflector.

a
Mmr (IZfktV)

0>
>1

Fig. 4.2.— Median plane emittance at the exit of the outer inflector.

68

the o u ter infle c t o r e x i t are linear

(both likely true,

a p p r o x i m a t e l y ) , then these optics m u s t be a d j u s t e d to p r o ­
duce the e l l i p s e w h i c h coincides b e s t w i t h Figure 4.2.
T h e r e are these three d i f f e r e n t casess

1) The ion source

e m i t f a n c e is a lot sma l l e r them 2.4 m m mr.
a r o u n d 1 m m mr or less

(again at 42 MeV)

An ellipse of

fits entirely

w i t h i n the shape of Figure 4.2 and thus w o u l d e n d up as
a s u b s e t of Fi g u r e

4.1 w i t h

tance area and no loss.

little dilu t i o n of the emit-

M o s t sources co n s i d e r e d for ex­

ternal injec t i o n do n o t h a v e such small emittances.

2) The

ion s o urce emitt a n c e is around 2.4 m m mr referred to 42
MeV.

In this case the e m i t t a n c e elli p s e w i l l overlap w i t h

the a c cept a n c e a r e a of Figure
be

4.2;

some of the b e a m w i l l

lost and n o t all of the area of Figure

filled.

4.1 w i l l be

F o r an ion source emittance of 2*4

1/3 of the m e d i a n plane area w i l l be lost;
emittance,

perhaps 2/3 w o u l d be

lost.

m m mr, perhaps
for twice that

3) W h e n the source

e m i t t a n c e is large compa r e d to 2.4 m m mr at 42 MeV,

the

e l l i p s e can be m a d e to enc l o s e the area shown in Figure 4.2
and 2.4 m m mr w o u l d be a c c e p t e d by the injector regardless

69

of w h a t

the e m i t t a n c e of the s o u r c e m i g h t be*

would be

lost.
The vertical acceptance,

p l a n e r ays of F i g u r e
approximations

4.1 w a s

of h a r m o n i c s

u s e d to d e t e r m i n e

are

linear and

z.

The number

the m a g n e t i c f i e l d s w a s

The total

l i m i t e d to 0.8 i n c h in the o u t e r in-

0.4 i n c h in the i n n e r i n f l e c t o r sys t e m ,

in b e t w e e n .

In a d d i t i o n ,

ances

As in the m e d i a n p l a n e

case,

fields

of the i n n e r i n f l e c t o r system.

ously,

the d u m m y e l e c t r o d e s

was

the e l e c t r i c

for improving

Much experimentation

the v e r t i c a l a c c e p t a n c e

a t 42 M e V for the b e s t m e d i a n p l a n e
dummy electrode configurations

l i m i t e d to

As mentioned previ­

are n e e d e d o n l y

f o c u s i n g in this region.

r e q u i r e d to o b t a i n

1.6

the v e r t i c a l a c c e p t ­

are a l s o i n f l u e n c e d v e r y s t r o n g l y b y

the v e r t i c a l

and

the h a l f - a n g l e of d i v e r ­

g e n c e i m m e d i a t e l y a f t e r the f i n a l i n f l e c t o r is
25 mr.

6.

( r e q u i r e d b y s t r u c t u r e i n the

are p l a c e d o n the v e r t i c a l m o t i o n :

h e i g h t of the b e a m is

inches

forces

are i n d e p e n d e n t of

The following restrictions

f lector,

f o r s e v e r a l of the m e d i a n

d e t e r m i n e d m a k i n g u s e of the

that all vertical

that hori z o n t a l forces

machine)

the r e s t

ray tested.

of

7.76 m m m r
Most trial

g a v e v e r y low v e r t i c a l

70

acceptances for m o s t of the ions having horizontal initial
conditions different from the central ray.
however,

In this case,

the vertical acceptance areas for rays chosen at

random around the periphery of Figure 4.1
are 6.73,

4.76,

to 42 MeV.

7.55,

7.23,

(or Figure 4.2)

7.43, and 5.9 4 m m mr referred

Since the vertical acceptances tend to be low

for rays away from the center portion of the area shown in
Figure

4.1,

the average of all the horizontal rays wo u l d

probably give a number fairly close to 7 m m mr;

the approx­

imations used in these calculations were stated at the b e ­
ginning of this paragraph.

Figures 4.3 and 4.4 show the

variety of vertical acceptances for different horizontal
rays.

The figures show only the positive z regions since

the boundaries are symmetric about the origin.

The areas

shown are one half the max i m u m possible acceptances limited
by structure in the cyclotron;

areas read off the graphs

m u s t be divided by 18.4 to give the 42 M e V acceptances.
(Since the shapes are not ellipses it is n o t immediately
obvious by looking at the drawings that the connecting
transformation is linear;
shown

it is though.)

The largest area

(bounded by the solid lines in the drawings)

is from

A
2.0 m

^

r

UZ? kmV)

\
*

k

\

•

Fig.

4.3.- - V e r t i c a l a c c e p t a n c e s
final i n f l e c t o r exit.

a t the

72

%

?m rw

\

Pig. 4.4.— Vertical acceptances at the
outer inflector exit.
(Note scale
change from Figure 4.3)

73

a ray in the central port i o n of Figure 4,ly
all b o u n d a r y points.

the rest are

F r o m F i g u r e 4.4 it is e v i d e n t that

if the e m i t t a n c e of the source is somewhere ar o u n d 6 m m m r
(42 M e V ) , some of the b e a m w i l l be lost and some of the
v e r t i c a l a c c e p t a n c e area w i l l n o t be filled.

A g a i n be c a u s e

the rays on the p e r i m e t e r of the h o r i z o n t a l acceptance area
tend to have sma l l e r and m o r e e c c e n t r i c vert i c a l accept a n c e
areas a s s o c i a t e d w i t h them. Fi g u r e
m a x i m u m vari e t y of such areas,

4.4 which shows the

fails to indicate that ac­

tually the v e r t i c a l acceptance areas coincide w i t h each
other fairly w e l l for the m a j o r i t y of the h o r i z o n t a l rays.
Thus it w o u l d be reasonable to assume that w e l l over half
of the v ert i c a l emittance area of a source w i t h a 42 M e V
e m i t t a n c e of 6 m m mr w o u l d be t r a n s mitted by the injection
system,

for example, w i t h the approximations of these cal­

culations .
Thus it has b e e n d e m o n s t r a t e d that the inner in­
flector s y s t e m plays a m o s t i m p o r t a n t role in de t e r m i n i n g
the o p tical p r o p e r t i e s of the i n j e c t e d beam.

The moti o n

in the m e d i a n plane becomes m a r k e d l y n o n l i n e a r under the
i n f l u e n c e of the hi g h l y localized elec t r i c field from the

74

several electrodes.

The verti c a l acceptance depends

crit­

ically on the inner inflector sy s t e m b e c a u s e of the s t r o n g
v e r t i c a l focusing and defocusing forces of the electr i c
fields and also because the prese n c e of the electrodes re­
stricts the vertical aperture to about a centimeter.

The

usable vertical acceptance w o u l d be uselessly small w i t h o u t
careful tailoring of the inner electrode configuratio n to
y i e l d reasonable vertical focusing properties.

Section 4.6

gives operating characteristics of the overall injection
s y s t e m w h e n used w i t h typical exter n a l ion s o u r c e s , and
section 5 compares this p e r f o r m a n c e w i t h different types
of injection systems and w i t h internal ion sources.

4.6

P e r f o r m a n c e summary

Because of the low current output of m a n y ext e r n a l
ion sources

(higher current sources usually can be modified

for internal o p e r a t i o n ) , an o f t e n used criterion of the
q u a l i t y of an injection sys t e m is the transmission effi­
ciency.
First,

This is somewhat inadequate for two reasons.
the transmission of a given injector depends

75

strongly on the proper t i e s of the ion source used.

For

i n s t a n c e , the b e a m quality of an u n p o l a r i z e d p r o t o n source
is usually very high,

resulting in a higher transmission

e f f i c i e n cy than for p o l a r i z e d sources at the same currents.
S e c ondly transmission alone provides no indication of the
b e a m q u a l i t y after e x t r action from the cyclotron;

eg.

it

may be p o s s i b l e to increase transmission by increasing the
cyclotron RF phas e width, b u t this m i g h t make the energy
s p r e a d unacceptably large.

Rather arbitrarily assuming

that the energy spread should be 0.2%,
w i d t h is limited to 7.2 RF degrees,
of energy spread.
traction.

total,

the RF phase

ignoring other causes

This phase width permits complete ex­

Section 4.1 d e m o n strated that 7% of the beam

c ould be b u n c h e d into this time interval w i t h a single
sine w a v e b u n c h e r for the case of first h a r m o n i c cyclotron
operation.

This is the m o s t important source of loss of

t ransmission effici e n c y for m o s t ion sources;
at h i g h e r h a r m o n i c acceleration;
e x actly isochronous,
times b e t t e r

it is worse

if the injection path were

the efficiency w o u l d be about four

(and n o additional

loss at hi g h e r h a r m o n i c s ) .

The 7% e fficiency applies to the case in w h i c h the 2.4 m m m r

76

horizo n tal emittance

(42 MeV)

is filled and assumes that

rays w i t h different vertical motion but the same median
plane p rojection are isochronous w i t h each other.

A typ­

ical m odern p o l a r i z e d ion source m i g h t have a current out­
p u t of 5ya and an emittance of 5 m m m r at 42 MeV.

Such

an e m i t tance w o u l d fill a large portion of the distorted
acceptance area of Figure 4.2 with pr o p e r horizontal fo­
cusing and about 2/3 of the median plane b e a m w o u l d be
lost.

In this case the lack of isochronism w o u l d lead to

the 7% time structure efficiency quoted earlier.

The v e r ­

tical acceptance is nearly adequate for the entire emit­
tance,

under the approximations of section 4.5.

Assuming

that the path length from the buncher to the central re­
gion is such that the lack of isochronism causes a hori­
zontal emittance increase to around 7 m m mr at the cyclo­
tron central region,

about 1/2 of the b e a m within the 7.2

RF deg. phase width should come out of the cyclotron.

In

this case the overall transmission efficiency w o u l d be
about 1%.

If the source delivers about 5ya, an approxi­

m ately 50 n A b e a m w i t h a total energy spread of about 0.2%,
and an effective vertical emittance of 6 to 7 m m mr

77

and a h o r i z o n t a l one of h a l f

that should be extracted from

the c y c l o t r o n .
The increase

in r a d i a l e m i t t a n c e

sweep during injection caused by
I t is p o s s i b l e

to b e a m

the b u n c h e r e n e r g y spread.

tha t the e x t r a c t e d e n e r g y

increased a little

is d u e

spread w ill be

d u r i n g the a c ce l e r a t i o n p r ocess

or that

the h o r i z o n t a l e m i t t a n c e w i l l e f f e c t i v e l y i n c r e a s e b e c a u s e
o f b e a m s w e e p i n g in
magnets
fects

due

the c y c l o t r o n

to the e n e r g y

are n o t d i r e c t l y

be seen
10 ram m r

that

As

related

(see B 1

transmission

the s q u a r e

of

(Ba 68 a n d F i

69)

4 . 1 as h a v i n g a n e m i t t a n c e
and a current of

4 yA.

of

greater than

a higher
3

He

about

to the

luminosity,

source

1.5 m m m r r e f e r r e d

to 42 M e V

M o s t of this e m i t t a n c e a r e a w o u l d

the h o r i z o n t a l a c c e p t a n c e o f

addition

the d e v i a t i o n

the "extremities"

It can

d e s c r i b e d b r i e f l y in s e c t i o n

fit i n t o

since

systems.

the e mittance.

consider the p o l a r i z e d
that was

69) , b u t s u c h e f ­

is p r o p o r t i o n a l

an e x a m p l e o f a s o u r c e w i t h

on the o t h e r hand,

field or bending

to i n j e c t i o n

for sources w i t h e m i t ta n c e s

(42 M e V ) , t h e

reciprocal of

spread

fringe

the injector

a n d in

from i sochronism w o u l d be much
of F i g u r e

4.2 w o u l d n o t b e

less

fi l l e d .

78

The more

central portions of

equivalent transit times
pears

t h a t i n the

better by

at

the area have m o r e nearly

to the machine

1.5 m m m r case,

least a factor of

better bunching efficiencies.
small

320 n A

(same e n e r g y

the i o n s o u r c e

to be

is e x p e c t e d

ion

2 , l e a d i n g to c o n s i d e r a b l y
This

combined with very

Only

2.4 m m m r
energy)

has been assumed

caused by the buncher,

(Da 69),

20+

b e a m of

in s h a r p

contrast,

32 y A in a n e m i t ­

corresponding injection energy

system.

spread will be

T h e b u n c h e r is u s e l e s s

ions p e r

7 and

to the p r e s e n t h i g h e r e x t r a c t i o n

means t hat the extracted current wou l d be

two million

trans­

and h o r i z o n t a l e m i t t a n c e s b e i n g

(now r e f e r r e d

which

r e f e r r e d to 250 M e V w i t h a n e n e r g y

a b o u t a 1% e n e r g y

The vertical

The energy spread

for m o s t ion sources.

of t h e

mi tted by the injection
here.

as b e f o r e ) .

that

to d e l i v e r a K r

s p r e a d o f a b o u t 50%
743 keV.

8% o r an e x t r a c t e d c u r r e n t o f

source

t a n c e a r e a of 170 m m m r

or

the i s o c h r o n i s m is

(and p r e a c c e l e r a t o r )

s m a l l c o m p a r e d to

The HIPAC

of

spread

is a g o o d a p p r o x i m a t i o n

of

It a p ­

losses due to e m i t t a n c e m a t c h i n g s h o u l d give a total

transmission efficiency

of

center.

seco n d ;

7.4 p A

i t is a s s u m e d t h a t the

79

v a c u u m is g o o d enough, to p r e v e n t c h a r g e — c h a n g i n g c o l l i s i o n s
before or during acceleration.
R.

B e u r t e y is p l a n n i n g a p o l a r i z e d i o n s o u r c e h a v ­

i n g a n i o n i z e r u s i n g the H+ + H°
very high magnetic

f i e l d t h a t is e x p e c t e d to d e l i v e r 0.4 m A

of p o l a r i z e d p r o t o n s
d i s p e r s i o n of

+ H + r e a c t i o n in a

in 3.1 m m m r

(Be 6 8 ) w h i c h w i l l

±230 e V

of b u n c h e r efficiency.

w i t h an e n e r g y

cause some

loss

This high current raises a question

t h a t h a s b e e n i g n o r e d u n t i l now,
of the s p a c e c h a r g e

<42 MeV)

forces?

viz. w h a t is t h e e f f e c t

Especially with a buncher,

the

peak

currents w o u l d be milliamps.

This

is the s a m e m a g n i t u d e of c u r r e n t e n c o u n t e r e d f r o m the

internal nonpolarized proton
case,

reliable

calculations

w i t h s u c h a source.

ion s o u r c e s .

In the p r e s e n t

of the s p a c e c h a r g e

l i m i t of

the i n j e c t i o n s y s t e m w o u l d b e v e r y t i m e c o n s u m i n g a n d d i f ­
ficult,
near

b u t it c a n b e s u p p o s e d t h a t it w o u l d b e s o m e w h e r e

the r a n g e of s p a c e c h a r g e

jection systems

i.e.

limits

0.3 to 0.8 mA,

f o u n d in a x i a l i n ­

peak

(Co 62,

C l 69).

T h e i n j e c t i o n e n e r g y is h i g h e r in the p r e s e n t c a s e b u t
there

are m o r e s e v e r e r e s t r i c t i o n s

A t a n y rate,

the p o l a r i z e d s o u r c e s

o n the size of the b eam.
in o p e r a t i o n now,

and

80

also m a n y of the h e a v y ion sources unsuitable for location
inside the cyclotron, y i e l d b e a m currents far b e l o w the
space charge l i m i t .
The p o l a r i z a t i o n of the e x t r a c t e d b e a m should be
es s e n t i a l l y the same as that of the source,

the only real

p o s s i b i l i t y of d e p o l a r i z a t i o n is in the heliquad,
there it is unlikely.

and

The rest of the injec t i o n sy s t e m

has b e e n shown to be free of serious d e p o l a r i z a t i o n e f ­
fects

(sections 4.2 through 4.5).

The q u e s t i o n of d e p o l a r ­

ization during the acc e l e r a t i o n pro c e s s in isochronous
cyclotrons of less than a b o u t 100 M e V has been exami n e d
t h e o r e tically

(Po 66b, Bu 60)

Be 66b, Oh 70),

and e x p e r i m e n t a l l y

(Be 70b,

and the a m o u n t of depol a r i z a t i o n was

found

to be small in all cases.
E x t e r n a l sources of highly cha r g e d heavy i o n s , p a r ­
ticularly the one at Saclay
Li

3+

, C

5+

, O

6+

, and Ne

*7+

(Kn 69)

giving 0.5 to 1 yA of

, could be u s e d w i t h this injection

s y s t e m if the e m i t t a n c e w e r e acceptable.

U n f o r tunate l y

there is a lack of d a t a on the emitt a n c e of such hea v y ion
sources,

both inte r n a l a n d external.

The fact they operate

fairly successfully w i t h conv e n t i o n a l cyclotrons indicates

81

that the b e a m quality

is n o t t o o bad.

that of p o l a r i z e d ion sources#

If it is s i m i l a r to

for example#

e s t i m a t e s of the t r a n s mission efficiencies

the previous
and e x t r a c t e d

b e a m q u a l i t y apply.
The

transmission efficiency

creased somewhat by
cyclotron#

could always be in­

l e n g t h e n i n g the d u t y f a c t o r in the

a t the e x p e n s e of r e d u c i n g the e x t r a c t i o n e f ­

f i c i e n c y a n d i n c r e a s i n g t h e e n e r g y spread.
mission efficiency

then depends

The exact trans­

on the t h i c k n e s s

of the

e l e c t r o s t a t i c d e f l e c t o r septum,

h o w w e l l its s h a p e is m a t c h e d

to the shape

a n d the c e n t e r i n g as w e l l

of t h e

a s t h e d u t y factory
estimate.

l a s t orbit#

making it difficult

to g i v e a n u m e r i c a l

T h e M S U cyclotron n o r m a l l y operates w i t h re­

s t r i c t e d RF p h a s e w i d t h s
the p e r f o r m a n c e

and single turn extraction#

so

estimates given have assumed these condi­

tions .
T h e e f f e c t of the i n n e r e l e c t r o d e s y s t e m on the
first

f e w a c c e l e r a t e d turns

could probably be compensated

by the centering

coils.

t u r n is o v e r

t i m e s the i n j e c t i o n e n e r g y #

2.5

T h e e n e r g y of t h e b e a m a f t e r o n e

i n f l u e n c e of the s t r a y e l e c t r i c

fields.

r e ducing the

The injection

82

system at Saclay uses electric fields to counteract the
magnetic field for the entire median plane injection path.
The electrodes in this case have voltages of +20 kv and
-23 kV

(and other lower voltages)

and the accelerated

orbits are disturbed by less than 2 mm radially and negli­
gibly vertically

(Be 67b).

It would be expected that the

results in the case of the proposed MSU system would be
similar, thus the beam could be recentered.

Section 5 will

compare the estimated performance of this proposed injection
system with that of existing injectors and with internal
sources.

5.

Comparison

A t p r e s e n t axial i n j e c t i o n systems are the m o s t
p o p u l a r w a y of get t i n g ions to the center of a cyclotron
from an e x t e r n a l source.

The s y s t e m designed by Powell's

g roup at B i r m i n g h a m was the first r e p o r t e d o p e r a t i o n of
an i n j e c tion sy s t e m in an a c t u a l cyclotron

(Cl 6 9 b ) .

In

this s y s t e m the ion sour ce is located about two meters
above the cyclotron m e d i a n p l a n e and 11 k e V deuterons pass
t h rough a 2 inch diamete r hole in the m a g n e t pole focused
by 6 e i n z e l lenses o n t o an e l e c t r o s t a t i c m i r r o r at the
m e d i a n plane.
eter;

The aperture of the lenses is 2.2 cm di a m ­

a n d the hole is r e d u c e d to 1 inch in the pole tip.

V a rious versions of the s y s t e m are d e s c r i b e d in the liter­
ature

(Co 62, Po 65, P o 66 a, P o 66b, and Cl 6 9 b ) .

A spe­

cial feature of this s y s t e m is the p r e s e n c e of grids of
ve r t i c a l wires in the dees to improve v e r t i c a l focusing.
The grids also i n t e r c e p t 30% of the b e a m and d i s t o r t the
radial emittance.

The gauze of the e l e c t r o s t a t i c mirro r
83

84

also con'tributes to the distor t i o n of the emittance and
i nterc epts another 35% of the beam.

A l l axial systems have

the p o s s i b i l i t y of e n l a r g e m e n t of the transverse emi t tance
as the b e a m comes out of the pole tip hole into the c e ntral
m a g n e t i c field.

Rays w i t h a transverse m o m e n t u m com p onent

w i l l e x p e r i e n c e a transverse force due to the longitudinal
field and the total transverse m o m e n t u m of these rays w i l l
i n c rease w h i l e the longitudinal m o m e n t u m decreases.

The

change in p h a s e space area due to the varying electr o m a g ­
netic m o m e n t u m term is discu s s e d by Powell

(Po 66 a ) .

The

e f f e c t is m i n i m i z e d by adjusting the focusing properties
of the s y s t e m so that the b e a m is at a n a r r o w w a i s t b e t w e e n
the hole e x i t and the mirror.

A l s o there is a disto r t i o n

of the e l e c t r i c field of the einzel lenses due to el e ctron
trapp i ng w h i c h ap proximately doubles the emittance.

The

n e t r e sult of all these effects is that the 42 M e V e m i t ­
tance is 10 m m mr at the source and about 44 m m mr in the
c i r c u l a t i n g beam.

The expansion factor is thus similar to

that in the M S U cycloidal sy s t e m
d i f f e r ent reasons.
jector is larger,

(horizontal ) , b u t for q u i t e

The acceptance of the B i r m i n g h a m in­
around 20 m m mr

(42 M e V ) .

A sine wave

85

b u n c h e r is u s e d a n d -the t o t a l t r a n s m i s s i o n e f f i c i e n c y is
2%.

A

l a r g e R F p h a s e w i d t h is u s e d d u r i n g a c c e l e r a t i o n ;

in fact,

the

finite injection energy allows

a r o u n d 90 d e g r e e s .

l i m i t e d to a b o u t

h igh:

impractically

large

the B i r m i n g h a m g r o u p

low

fairly
injected

4.5 m e t e r s

88 - i n c h c y c l o ­

a polarized proton and deuteron

above

the m e d i a n p l a n e a n d a b e a m t r a n s ­

p o r t s y s t e m c o m p o s e d of t h r e e sets

axis

limit c a n b e

a x i a l i n j e c t i o n s y s t e m for the

tron at B e r keley uses

triplets.

these

y A t o the m i r r o r .

The

source

for i n j e c t i o n e n e r g i e s

In s p i t e of

the s p a c e c h a r g e c u r r e n t

u s i n g an RF source,

a b o u t 250

in o r d e r to

A l s o the v o l t a g e r e q u i r e d o n the

m u c h g r e a t e r t h a n a b o u t 15 keV.
energies,

the i n j e c t i o n e n e r g y

1/5 o f the d e e v o l t a g e

o b t a i n c e n t e r e d orb i t s .
mirror becomes

to b e up

If t h e e l e c t r o s t a t i c m i r r o r is p l a c e d

a t t h e e x a c t c e n t e r o f the c y c l o t r o n ,
is

this

of e l e c t r i c q u a d r u p o l e

A g r i d d e d e l e c t r o s t a t i c m i r r o r p l a c e d o n the

in the m e d i a n p l a n e

is s i m i l a r to the o n e a t B i r m i n g h a m

e x c e p t a f i n e r g r i d m e s h is u s e d

(Cl 6 9 a ) .

The

s o u r c e is

e x p e c t e d to g i v e up t o 10 yA e v e n t u a l l y a n d the m e a s u r e d
e m i t t a n c e is

8 m m m r r e f e r r e d to 42 M e V

(Lu 69).

The

86

7.3 c m I .D. o f the e l e c t r i c q u a d r u p o l e s a l l o w an a c c e p t a n c e
o f the i n j e c t i o n line of

12.4 m m m r

tion e n e r g y is 10-15 keV.

(42 MeV).

The injec­

The Berkeley group calculates

t h a t 0 , 6 - 0 .8 m A can be t r a n s m i t t e d if the 42 M e V e m i t t a n c e
a r e a of the s o u r c e is n o m o r e
correctly

(Re 69).

than 3.1 m m m r a n d is s h a p e d

In s i d e the a x i a l h o l e t h r o u g h the

m a g n e t y o k e a n d pole,

it is u s u a l l y n e c e s s a r y to take the

m a g n e t i c f i e l d into a c c o u n t since it is usu a l l y a r o u n d 1 k G
a t m a x i m u m m a g n e t excitation.

The B e r k e l e y g r o u p m a d e d e ­

t a i l e d s tudies of this region,

e s p e c i a l l y the e x i t of the

hole

(Lu 68 ).

A t this p o i n t the d e c r e a s e in l o n g i t u d i n a l

v e l o c i t y w a s "less them 5 p a r t s p e r thousand."
e n e r g y does n o t change,
to i n c r e a s e

Since

the t r a n s v e r s e v e l o c i t y w o u l d h a v e

to one tenth of the l o n g i t u d i n a l velocity,

if

there w e r e a 5 p a r t p e r t h o u s a n d d e c r e a s e in the l o n g i t u d ­
inal velocity!

thus i n d i c a t i n g a g a i n the i m p o r t a n c e of

t a k i n g care to m i n i m i z e the i n c r e a s e in e m i t t a n c e a t the
e x i t o f the hole.
d a t a is listed:

In

(Cl 69a),

the f o l l o w i n g e x p e r i m e n t a l

w i t h a 12 k e V p o l a r i z e d p r o t o n i n j e c t i o n

b e a m the total t r a n s m i s s i o n e f f i c i e n c y w a s
w a v e buncher.

3% u s i n g a sine

(The c u r r e n t o u t of the s o u r c e w a s

2 yA.)

87

A t r a n s m i s s i o n e f f i c i e n c y of 1.5% was a c h i e v e d w i t h o u t the
bu n cher.

In a test w i t h a n o n p o l a r i z e d source,

ci e n c y w a s

the e f f i ­

4.5% w i t h o u t the b u n c h e r , i n d i c a t i n g t h a t the

t r a n s m i s s i o n m a y d e p e n d on the e m i t t a n c e o f the source.
At

the 60 M e V c y c l o t r o n of G r e n o b l e an a x i a l i n ­

j e c t i o n s y s t e m is in o p e r a t i o n w h i c h uses an e l e c t r o s t a t i c
d e f l e c t o r ins t e a d of a mirror,
b e a m losses on the grid.

The d e s i g n p r o b l e m s

m o r e c o m p l i c a t e d b u t a sligh t l y
feasible

are s o m e w h a t

larger i n j e c t i o n e n e r g y is

(19 k e V for the h i g h e s t c y c l o t r o n e n e r g y ) .

a n o n p o l a r i z e d source
a t 12 keV.
grees

in or d e r to e l i m i n a t e the

With

the t r a n s p o r t opt i c s a l l o w 150 y A

The RF p h a s e w i d t h is c a l c u l a t e d to be 50 d e ­

(Pa 69).

The total t r a n s m i s s i o n w i t h a p o l a r i z e d

p r o t o n s o urce is 0.4%

(Cl 69b),

or s o m e w h a t less than the

o t h e r a x i a l systems m e n t i o n e d or the e s t i m a t e s of the M S U
c y c l o i d a l method;

this v a l u e w a s h o w e v e r o b t a i n e d w i t h o u t

a buncher.
T he a x i a l i n j e c t o r of the C y c l o t r o n C o r p o r a t i o n ' s
15 M e V H

m a c h i n e uses e l e c t r o s t a t i c g u a d r u p o l e

e l e m e n t s and a g r i d d e d e l e c t r i c mirror.

focusing

The m aximum injec­

tion c u r r e n t a n d en e r g y are 2.5 m A a n d 15 keV,

demonstrating

88

the a b i l i t y of this m e t h o d to c o n q u e r s p a c e c h a r g e pr o b l e m s .
The t r a n s m i s s i o n e f f i c i e n c y w i t h a b u n c h e r is

1.5%

(Cl 6 9 b ) .

T h e p e r f o r m a n c e d a t a of t h e s e e x a m p l e s

are p r o b a b l y r e p r e ­

s e n t a t i v e of the a x i a l s y s t e m s

than a d o z e n i n s t i ­

tutes

at m o r e

(Kh 67).
The s e c o n d i n j e c t i o n s c h e m e

to be used on a full

scale c y c l o t r o n a n d the f i r s t to f u n c t i o n w i t h a p o l a r i z e d
beam,

w as m e d i a n p l a n e i n j e c t i o n of an a t o m i c b e a m a t

thermal velocities

(Cl 6 9 b ) .

In e s s e n c e ,

s o u r c e is m o d i f i e d by r e m o v i n g

the i o n i z e r a n d p l a c i n g it

a t the c e n t e r of the c y c l o t r o n .
used,

a p o l a r i z e d ion

S i n c e a n e u t r a l b e a m is

there is n o p r o b l e m of the c y c l o t r o n fields d i s ­

t u r b i n g the trajectory,

b u t i t is d i f f i c u l t to focus.

the fixed e n e r g y c y c l o t r o n at Saclay,

this m e t h o d w a s u s e d

to o b t a i n a p o l a r i z e d d e u t e r o n b e a m of
w i t h an en e r g y s p r e a d of

1.5%

In

.03 n A on ta r g e t

(Be 66 b, p.

80).

This

m e t h o d w a s also u s e d to i n j e c t p o l a r i z e d p r o t o n s int o the
S a c l a y vari a b l e f r e q u e n c y c y c l o t r o n

(in the case of p r o t o n s

it is n e c e s s a r y to use a l i q u i d n i t r o g e n c o o l e d i o n i z e r to
re d u c e the u n p o l a r i z e d b a c k g r o u n d ) .

Up to 0.1 n A in the

e x t e r n a l b e a m was o b t a i n e d w i t h a p o l a r i z a t i o n of 50 to 70%

89

resulting

f rom contamination b y residual gas
(Be 66 b,

of the c y c l o t r o n
efficiency of
ever

attain
A

the

are

82).

a value

comparable

70a,

Be

next electrons

atomic beam enters
str i p p ed at the
double

charge

the

center.

exchange

neutralized with

about

are

added

cyclotron
The

at

o f t h e n e u t r a l b e a m is

sides

of

the

the n e u t r a l i z e r

or d eu t er o ns

ions

radially,

and

and
the

fast

the atoms b e i n g
the Rez

45 k e V a n d t h e b e a m is
in a 20-cm

.01 torr.

but

Minimum diver­

small

are desired
Only

22%

long

implies

a large

apertures

on

to r e d u c e

of

the

atomic

1 cm d i a m e t e r s t r i p p i n g foil n e a r the center

of the cyclotron,
as

to the

required which

flow and p u m p i n g i n s t r u m e n t s .
b e a m hits

Czechos­

then accelerated

40% e f f i c i e n c y

beam diameter at the neutralizer,
both

is t o i n j e c t

injection energy of

s y s t e m is

chamber containing hydrogen
gence

approach

Polarized protons

first produced conventionally

focused;

atomic beam method will

as a t D u b n a a n d a t Rez,

70b).

tha t the

to oth e r i n j ection schemes.

somewhat more promising

(Be

I t is u n l i k e l y

thermal velocity

fast neutral p a r t i c l e s
lovakia

p.

in the center

a l m o s t half of

4.5 y A of p o l a r i z e d p r o t o n s

this

is

ionized.

are avai l a b l e b e f o r e

As m uch
the

90

neutralizer and the total transmission efficiency is 0.14%
with a buncher.

The injection system at Rez is still rela­

tively new and it is thought that modifications can be made
to increase the efficiency
C.

(Be 70a).

A. Tobias first proposed a method of injecting

heavy ions at a low charge state into the median plane in
an arc with a large radius of curvature and then stripping
them to a higher charge state near the cyclotron center
(To 51).

Following his suggestion the group at Orsay is

building an injector system using a heavy ion linac for
preacceleration to 1.-1.6 MeV per nucleon.
was not yet operational as of 1969

The system

(Ah 69).

The method presently in use at the Saclay variable
frequency cyclotron consists of balancing the magnetic
field with an electric field for the entire length of the
radial median plane injection path.

The system of electrodes

uses 13 separately regulated voltages from ±1 kV to +20 and
-2 3 kV to compensate both the fringe and central magnetic
fields and to shape the transverse electric gradient to
provide vertical and transverse focusing (Be 6 7b).

The in­

jection energy for protons is 5.2 keV with an 18.4 kG central

91

field.

Up to 5 yA of current is provided by the source.

The polarization after extraction from the cyclo t r o n is
90%.

A total transmission of 1.4% has been attained

70 n A external)

(Cl 6 9 b ) .

(ie.

The acceptance of the system

now in use is not known, but an experimental mockup having

8 instead of 13 separately regulated voltages had a 42 M e V
(Be 66b, p.

acceptance of around 0.33 m m mr

84).

Cyclotrons w i t h internal ion sources produce unsur­
passed nonpolarized beams of light positive ions, at least
up to lithium.

Here there are problems since there are no

room temperature gaseous compounds of lithium,

the second

and third ionization potentials are high, and lithium vapor
is chemically active

(Va 69).

A satisfactory internal

lithium source was developed at the Kurchatov Atomic Energy
Institute in Moscow, even though the limited space avail­
able is a definite disadvantage,
is 40 hours.

and the source lifetime

Since lithium contamination of the dees lowers

the work function and causes sparking,
washed every two weeks.

the dees have to be

The size of the extractor slit

in the source is 1 m m x 2 m m and the external b e a m is 100 nA
of 60 M e V Li

3+

ions, or 8 yA of 36 M e V Li

2+

(Va 69).

92

Lithium sources

suitable

for e x t e r n a l i n j e c t i o n s y s t e m s

s e e m to b e a b s e n t f r o m t h e
the s o u r c e

used at Saclay

o n l y 500 n A o f L i ^ + , the
be v e r y m u c h

less

(Kn 69), b u t s i n c e i t p r o v i d e s
c u r r e n t f r o m the c y c l o t r o n w o u l d

group has

that operates

a t a n a r c p o w e r of 2.4 kW.

obtained.
ful.

(4 y A ) , N

5+

(1

yA) , a n d O

charge states

viz. N e

O

(Kn 69).

, and

The b e a m

t h e s e ions.

o u t o f the

N o w a n d in t h e

of c o u r s e .

o u t p u t t e n d to b e

the c y c l o t r o n

5+

5+

Extracted

(0.4 yA)

have been

beam was

unsuccess­

large

of the o t h e r

elements,

, 500 n A of e a c h o u t of the s o u r c e
cyclotron would

foreseeable

of p o l a r i z e d source w o u l d hav e
cyclotron,

inside

s o u r c e a t S a c l a y has p r o v i d e d b e a m s o f

Ne and a lso higher
, C

(tungsten),

life to 20 h r s .)

A n a t t e m p t to g e t a g o o d N e

The external

Yoshitoshi

(The c a t h o d e s e r o d e a t 0.3

g r a m s p e r h o u r a n d l i m i t the

4+

(Mi 70).

developed a dual cathode

water cooled anode source

beams of C

case.

of h e a v i e r e l e m e n t s e a s i l y o b t a i n e d

are a c h i e v e d in J a p a n

Miyazawa's

A n e x c e p t i o n is

than in the internal source

Fairly good currents
in a gas

literature,

Negative

to b e

be weak

future,

any type

located outside

ion s o u r c e s

for

the

of h i g h c u r r e n t

and have high power r e q u i r e m e n t s .

93

The Cyclotron Corporation,

for instance, has very success­

fully used an axial injector in this application
as d e s c ribed previously.

(Cl 6 9 b ) ,

The system d e s c r i b e d in the p r e s ­

e n t feasibility study is not suitable for negative ion
injection because of the difficulty of h o l d i n g large p o s i ­
tive potentials on the electrodes.

A l s o the space charge

limit has not been evaluated.
It thus appears that as a general principle,

if a

given type of ion source can be constructed to operate in
the center of the cyclotron,

superior results w o u l d be o b ­

tained by doing so; external injection is n e e d e d for the
types of sources w h i c h cannot be squeezed into the cyc l o ­
tron.

6.

Conclusion

The method of injecting ions from am external
source into a sectored cyclotron by cycloid-like motion
along the sector edge, which was first used by Gladyshev's
group in Moscow on a model cyclotron using an injection
energy 8 .6 % of the final energy
in a full scale application.

(G1 65), can also be used

The system studied in this

paper w o u l d provide performance estimated to be roughly
comparable to that of the other types of injectors now
in use at cyclotrons of around the size as the one at MSU.
The injection energy of 0,3% of the final energy is the
maximum that will allow the injected beam to be bent into
a suitable orbit near the cyclotron center with attainable
electric field strengths.

A buncher can be used to in­

crease the transmission efficiency although a lack of isochronism in the injection path puts limits on its useful­
ness.

Ions of different energies end up at different points

near the machine center thus causing an effective increase
94

95

in horizontal emittance area.

The optical properties

during drift along the sector edge are almost linearr but
not while in the central inflector system.

This necessi­

tates making some questionable approximations.

A reason­

ably large vertical acceptance of the system can be ob­
tained by a carefully optimized set of electrodes in this
region.

The effective horizontal emittance is consider­

ably less than some other injection systems but for high
quality present day polarized ion sources the total trans­
mission efficiency is estimated to be comparable.

There

is essentially no depolarization in the cyclotron magnetic
field and fringe field.

External sources of highly charged

heavier ions could be used but the current out of the
cyclotron would be very low.

The injection system investi­

gated here is not appropriate for negative ions.

With a

2% duty factor the energy spread in the beam is about 0 .2%
which should give high extraction efficiency.

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