I“
. L“ -‘

. ‘W'

-‘c—

3%

tr e _
. 33.35%
v‘ ' '

.“., - ,
—a-_-Q I

75.3:
pl“

\ul‘

. N
. -;"%‘-.~‘. é.” a: 5.:
.. kit-:1 a
i o. .
-.- .J.. JGV
a. t -
‘15 ‘K‘r- J32- ‘
a .

it
339

.

a

4w" 354” ' " E
.6, ’4: w ‘ A ‘ ‘ .
1:. ‘ .gfifr‘)‘: .

...~3;-:
3:: “:5 a; ’“

£~ ‘
.fi‘dw

. 'vv“, _ _ -~¢
y“ V‘I?:n135:93~g‘
A»: r \V'I 733;“
wEéTJLO‘a

\. \E—r‘u‘

p

' ~ 3::
Min

«3%

Q ;.L

..-;-'r'v;x ":Q'}

1‘.

_\

- was
‘ 1;:
a»:

 

-i \
V
3
4 (4.::.n‘|

, . mgug
‘

‘ ~ - ‘ v.‘
1”” on.“ “ll1-¢":‘, ‘L
‘ VI .'.

) '1) _'
~ g- «’J'- ‘ LL": .5" .
"“6931?” .~. " ‘ .-

“ml

‘F'E‘JJ’yy‘fi,
.. '- .1 l ‘ 9; ,
. - . \g-JI‘Z“, .\\_ 4391:2112 ‘- h ‘3‘ d.‘\-' ‘
I‘V)’ \ ,‘ h g 1".— , .j. ‘ , W ' , ._ . .1. at . .‘ ‘,' ‘i" is»;
7:..." r 11‘“). Ill” '5! .13 . -. 'I. ' . ‘ ’ , . . . '1'. ‘1 ~ §I. 't-i'
hm.- ' A ."‘ ‘ . ‘ ”3T: - .~‘ “ ,2“ . 7:. 3‘" '(Q' .‘L vl’a‘. L

L...\, r" g-‘ hm”.

 

 

JIWHTH mm

In aflf‘iim mm "L

3 1293 00799 7046

 

 

 

 

 

 

 

LIBRARY
Michigan State
University

 

 

 

This is to certify that the

dissertation entitled
Phase Selection in the K500 Cyclotron and the
Development of a Non-Linear Transfer Matrix Program

. presented by

Bruce Forrest Milton

has been accepted towards fulfillment
of the requirements for

PH.D. PHYSICS

degree in

 

 

 

 

Date ly/fl/gé
/ /

MS U i: an Affirmative Action/Equal Opportunity Institution 042771

 

 

MSU

 

RETURNING MATERIALS:
Place in book drop to
remove this checkout from

 

 

 

 

LIBRARIES
-_—. your record. FINES will
~ be charged if book is
returned after the date
stamped below.
Mari L
EJiE“
[c3

 

PHASE SELECTION IN THE K500 CYCLOTRON
AND THE DEVELOPMENT OF A NON-LINEAR
TRANSFER MATRIX PROGRAM

by

Bruce Forrest Milton

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astromony

1986

ABSTRACT
PHASE SELECTION IN THE K500 CYCLOTRON
AND THE DEVELOPMENT OF A NON-LINEAR
TRANSFER MATRIX PROGRAM
By

Bruce Forrest Milton

A method has been developed for the rapid calculation
of particle orbits in a cyclotron with spiral-shaped dees.
The method uses second order matrix transfer methods and has
been implemented in the FORTRAN program "SOMA". ( Second
Order MAtrix). SOMA has been checked against the slower
orbit integration program SPRGAPZ. A combination of SPRGAPZ
and SOMA has been used to investigate the phase selection
process in the Michigan State University K500 cyclotron.
'This study led to the design of hardware necessary for phase
selection and the ancillary beam diagnostic equipment.
Finally SOMA calculations and the phase selection

calculations are compared to experimental results.

ACKNOWL EDGMENTS

I would like to thank Dr. Henry Blosser for giving me
the Opportunity to work at this unique laboratory. I am also
deeply indebted to Dr. Morton Gordon for the help with the
transfer matrix program. and Dr. Felix Marti whose patient
guidance and assistance kept me going.

Thanks are also due to all those many cyclotron

employees with whom I have had the pleasure of working. I
have learnt much from their experience and will always

remember the comraderie.
Finally I would like to thank my parents. both of whom
showed me that this was possible. and without whose constant

love and support this wouldn't have been possible.

List

List

TABLE OF CONTENTS

of Tables
of Figures
Introduction

SOMA: A Cyclotron Orbit Code Using Second
Order Transfer Matrices

1 Introduction
2 Calculation of Transfer Matrix Elements
3 Crossing the Accelerating Gaps
4 Program Algorithms
.5 Starting Conditions
6 Comparision of SOMA with SPRGAPZ
7 Treatment of Harmonic Field Bumps
8 Vertical Motion

NNNNNNNN

>

Computational Examination of Phase
Selection in the K500 Cyclotron

Introduction

Coarse Selection

Phase Selection for Axially Injected Beams
Fine Selection

Further Considerations

0000030300
U’IhQN—e

Phase Selection Hardware and the VP Probe

Introduction

Phase Selection Hardware

Installation of the Phase Slit Hardware
Construction Of the Viewer Port Probe
The Gamma Probe

Link-Col:-
OIL-OJN-b

Experimental Results

5.1 Frequency Detuning
5.2 Phase Selection

Page

15
32
42
48
52
59
63

66

66
72
76
82
96

100

100
100
109
116
122

5.3 Radial Focusing Frequency 150

5.4 Axial Focusing Frequency ' 151

Page

6. Conclusions 154
7. Appendices 158
l SOMA Input 158

ll Orbit Code Parameters 174

References 176

Table

List of Tables

The elements of the arrays « and B. These
arrays contain the coefficients of the
displacements (x.px etc.) found in the

differential equations 2-5 ......................

The driving functions which appear in the
differential equations 2-9. The values of the

«'3 and the fl's are listed in Table 2-1 .........

The derivatives of the orbit coordinates

necessary for the calculation of the gap
correction. The symbol ”q" is any of the
particle parameters. The primes are

differentiation with respect to e ...............

The order of the various parts of the transfer

as a function of the input parameter N ..........

The phase width and the extracted beam current
at beam stop 0 for different combinations of

slits ...........................................

The input-output units used by SOMA .............

The lD codes for each of the parameters that

are saved .......................................

The source of the initial ellipse values as

determined by the parameter #42 .................

The input parameters 10 through 50, that are
entered on unit 5. The default values are in

brackets ........................................

Page

.. 26

.. 28

.. 36

.. 43

..148

..159

..161

..164

..172

List of Figures

Figure Page
1-1. Vertical section of the K500 cyclotron ............ 2
1-2. Median plane view of the K500 cyclotron. "A"

hill is located between the ”C” dee and the "A”

dee ............................................... 3
1-3. The K500 operating diagram ........................ 5
2-1. The geometry of the gap crossing correction.

The gap position is given at points Gapth(ir).
69 is the angle through which the orbit must be
moved ............................................. 33

2-2. The result of a fixed angle crossing a gap. The
transfer matrices of the orbits labeled 1.2,
and 3 are correct for a transfer from the
current gap to the fixed angle. while that for
orbit 4 is not. In this case the orbit being
calculated would use the values stored for
1.2.3. and 4 when the interpolation is being

done. and would give an incorrect result .......... 45
2-3. Starting conditions at the exit of the central
region. The solid lines are the values the

program would use for the centre of the
ellipse. The crosses are the values obtained
when orbits are numerically integrated from the

spiral inflector to turn 7. using the code

CYCLONE ........................................... 50
2-4. Differences between SOMA and SPRGAPZ as a

function of the initial displacements from the

E0. Each ray was run one turn without

acceleration. The initial conditions of each

ray are such that it falls on the boundry of an

eigen-ellipse of area (0.2)f2 mm-mrad. The

stability region ends around 1:150 ................ 53
2-5. The differences between SOMA and SPRGAPZ when

the spacing between stored values is changed.

The initial diSplacement was 30 O3”. and run

for one turn without acceleration ................ 55
v i

Figure

The difference in r (and pr) between SOMA and
SPRGAPZ for rays which have been run for 100

turns with acceleration .........................

The difference in energy for the same set of

rays as those shown in Figure 2-6 ...............

A sample precession cycle as calculated by
SPRGAPZ and SOMA for a field with a first

harmonic. The orbit began on the E0 .............

The vertical phase space after acceleration
forward for 300 turns. The dotted line is the
boundry of a 35.2 mm-mrad eigen-ellipse at
22.72 MeV/u. The Spread due to the x motion is

0.02” ...........................................

Plots of simple estimates of the turn
separation and turn widths associated with
different phase widths. The turn separation
(solid curve) is estimated using AR=AE'R/2'E
while the full width of a turn with a given Ad

is found using AR=(A¢)2'RI4 ......................

The radius of the central ray. shown as a
function of starting time. for four successive
turns. Note the typical horse-shoe shape

resulting from the cos(¢) dependence of the

energy gain ......................................

Electrode structure for the K500 firsr harmonic
central region using a PIG source. Four orbits
are shown corresponding to starting times (from
outer-most at 6:00 to inner-most).r0=230,240.
250. and 260 degrees. The peak electric field
between the source and puller is achieved at

10:270. A slit is located on the o0 hill
extension of the center plug allowing easy
installation and removal. This slit removes all

particles whose starting times do not fall

between 230 and 250 degrees .....................

Page

.. 57

.. 58

.. 62

.. 65

67

69

.. 71

Figure

The electrode structure for the first harmonic
central region with axial injection. The window
frame attached to the dummy dee following the
puller is used to neutralize the coupling
between the first and second dees. Five orbits
are shown corresponding to starting times of
230° (outer-most at window),24o°.2so°.2so°, and
2700 (inner-most). By narrowing the radial
width of this window it will be easy to remove

those starting times lying outside 2450 to 260°...

RF time differences for particles on the
boundary of a 100w mm-mrad phase space with

respect to the central ray. The dotted line
indicates the difference at the entrance of the
inflector. The solid and dashed lines show the

differences at the inflector exit. The abscissa
is just an arbitrary parameter around the

boundary of the phase space ......................

The phase space at the exit of the inflector
when the initial beam has an emittance of 25w
mm-mrad. The momenta have been divided by qB0

to express them in units of length ...............

The R and Pr plotted for a group of rays that
started on the perimeter of the ellipse shown
in figure 3-6. after 3 turns, as functions of

their average phase. In the middle frame the
results of interpolating to find each ray at an
average phase of -4° is shown. Note the ellipse

shows no distortion ..............................

Radius difference ri-r at 9:840 vs turn number

0
for a family of central rays. Ray 0 leaves the
source at 10:2350. the others at the times
labeled on the plot. At turn 33 a bar Of i.02
inches is shown to give an idea of the radius

variation expected from the r.pr distribution

around the central ray ..........................

A representative K500 phase curve. determined
using the standard procedures. The arrow is
located at the radius of the fine selection

system......, , ...............................

Page

73

75

. 77

. 80

.. 81

Figure

3-10.

A schematic of how a single post can act in a
manner similar to a slit. The inner edge of the
post scrapes off those particles with too large
a radius, while the outside scrapes off those
with too low a radius on the next turn. The
dotted region is the surviving beam while the

cross hatched region is the removed beam ........

Radius plotted as a function of the starting
time for turns 32.33 and 33 for the PIG
geometry in the upper sequence and for turns 29
through 32 in the lower plot for the ECR
geometry. Associated with each central ray is a
set of 8 rays that populate the circumference
of a .02 inch radius circle in R.Pr space. From

left to right: the first one shows the
situation at 9:84 before the blades are
inserted. the next one shows the situation

after a 60 mil blade has been inserted at 84°
and the third shows the effect of inserting a
second blade at 9:2040. The following three
frames give the the analogous situation at
e=204°. Note that the final phase width is
around 4 degrees and the full 0.02 inch phase
space around the central time survives. The
rays with different R.Pr values have a starting
phase that gives them the same energy gain per
turn as the central ray with which they are
associated. thus the horizontal label is

actually a measure of the energy gain per turn..

A simple demonstration of why two slits are
required to do a careful phase selection. Shown
are two bundles of rays with a one degree phase
difference. Note that at the first azimuth it
is impossible to remove all of one phase
without affecting the other phase. After the
particles have traveled 120° in azimuth they
have executed a third of a betatron oscillation
as highlighted by the cross marking the same
ray in both frames. As the shading demonstrates
it is now possible to remove almost all of the
unwanted phase. In this particular example the
two phases are sufficiently close together that
a small amount lthe unshaded portionl of the

unwanted phase passes the second slit ...........

ix

Page

.. 86

91

.. 92

The radius plotted at the azimuth of the lower
slit as a function of average phase (a good
measure of the energy gain per turn) for a
distribution of partiCles which pass through
the first turn slit. In this run the first and
second harmonics are present in the magnetic
field. so it should be compared to figure 3-10.
where the particles were run in a field that
had perfect three fold symmetry. Four different
turns are plotted. In the first frame on the
left no slits are present. In the middle frame
the upper slit has been inserted, and in the

final frame both slits are inserted .............

R plotted versus Pr for turns 507 and 508 at 9
= 336, corresponding to the entrance to the
electrostatic deflector. Pr has been divided by
mob to express it in inches. The shaded area
corresponds to a possible location of the
deflector septum. This plot shows that single
turn extraction of the resulting beam should be
possible. The energy spread of this group is

less than 6 parts in 104 .........................

The end of the shaft at the median plane of the
cyclotron. The sixty-two mil tungsten pin
intercépts the unwanted beam. The pin is
mounted in a copper cap which is easily removed
for rapid pin change. Note: the copper end plug
is water cooled so the tungsten will be

indirectly cooled ...............................

Cross section of the phase slit drive

mechanism. See text for a description ...........

The lower phase slit hole before the trim coil

leads were moved ................................

A schematic of the trim coil leads in their
positions before moving. It can been seen that
in both cases several leads seriously

encroached on the phase slit drive space ........

Page

.. 94

98

..102

..104

..106

..108

Figure

4-12.

The special pliers built to allow bending those
leads which were trapped under the dee stem
spinning. “Nth a little care they could be used
to move the leads without crimping the lead or

putting force on the feed-through ................

The fixture used for locating the hole in the
dee stem spinning. The other end of the
indicator shaft was at the median plane so it
could be tapped with a hammer to mark the
Spinning with the sharpened point; The drill
bushings provided alignment of the clearance
hole and the two threaded holes for mounting

the air cylinder on the hex flange ...............

The fixture used for setting the angle of the
drive. The cap on the end fits over the end of

the shaft when the normal cap with pin is
removed. The notch fits over the post on the
center locating fixture so the rotation of the

shaft is determined at either 0° or 180° .........

The lower drive mechanism installed. See text

for a description ................................

A schematic Of the new viewer port probe drive
shOWing the major components. The two separate
drive systems allow for a bellows to be used in
the regular range of travel and O-rings to be

used for insertion and removal ...................

A photo of the new drive installed on the

cyclotron. In this photo the drive is in the
'running' position and a probe has been
installed ........................................

The first probe used with the new drive. The
Kovar insulators were not sufficiently good
thermal conductors. resulting in their becoming
electrically conducting so this design was

discarded ........................................

The 2 jaw differential probe currently in
regular use. This design has proved to be very
robust .........................................

Page

.110

.112

.114

.115

.117

.119

.120

Figure

4-13.

4-15.

A schematic drawing of the phase probe. The PIN
diode is used to detect gamma rays produced
when the beam strikes the probe tip. The small
size of the diode and amplifier allows it to be
located near the probe tip so the count rates
are high and the source is distinct from the

background ......................................

A typical spectrum of intensity versus time.as
measured with the gamma probe. Notice that the
divide by two of the RF stop signal causes all

features to appear 360° apart ....................

A comparision between the PIN diode and a BaF
detector. Both detectors measured the same
beam. which was striking beam stop 1. The

measured beam widths are very similar ...........

The calculated and measured phase curves for
two different magnetic fields. The dashed curve

for the N5+ case is the calculated phase curve

when the mail coil currents are changed by 0.1

Amps .............................................

The phase width of the internal beam as
measured with the gamma probe at extraction
radius when the narrow first turn slit is

installed .......................................

A calculation of the percent beam which
survives the first turn in the cyclotron for
the wide and narrow first turn slits. The

inflector collimator is 4 mm in diameter ........

A calculation of the percent beam which
survives the first turn in the cyclotron for
the wide and narrow first turn slits. The

inflector collimator is 1 mm in diameter ........

A differential probe trace. taken with the
narrow first turn slit in place. The Cyclotron

was tuned for good extraction ...................

Page

..123

.124

..126

.130

..133

..134

..135

..137

Figure

5-6.

5-10.

A differential probe trace. calculated with the
program SOMA. Note .the similarities in
structure to the actual probe trace of the

previous figure .................................

The current intercepted by the upper slit as a
function of its radial position. The
calculation values were obtained with the code

SOMA ............................................

The current intercepted by the lower slit as a

function of its radial position. The
calculation values were obtained with the code
SOMA. The poor agreement at lower radii is due

to an encoder mal function ......................

The phase width as measured with the gamma
probe. for two different postions of the upper

Slit ............................................

A comparision of the phase width at extraction

- with and without the upper slit. In both cases

the narrow first turn slit is present ............

The phase widths at extraction with one slit
inserted. and the combination of two slits. The
extracted current is reduced by a factor of two

when the second slits is inserted ...............

The weigthed SOMA calculation is compared with
the measurements. Only the first turn slit is

in the machine ..................................

The weighted SOMA calculations are compared to
the mesasurements for two different locations
Of the upper slit. Note the relative
intensities Of the peaks. These calculations

assumed 100v mm-mrad initial emittance ..........

A differential probe trace taken during a 22N8+

35 MeV/A run. The wide first turn Slit is

installed. The radial focussing frequency can
be determined from the coherent oscillation
which is VlSIble as the large amplitude
oscillations.. ............................

Page

..137

..139

..140

..159

.143

..144

..146

..147

Figure Page

5-15. The radial focusing frequency as calculated
with the equilibrium orbit code. and the values
obtained from differential probe traces. The

horizontal bars indicate the region over which
the value of vr was averaged. The vertical bars

indicate the possible error in determining the
nomber of turns in a precession cycle ............. 149

5-16. The current hitting the center and lower jaw of
the main probe as a function of radius. A
coherent oscillation is induced in 2 by raising
the inflector 0.445" .............................. 153

5-17. The values Of the axial focusing frequency as
computed from probe traces such as the one
shown in Figure 5-14. The horizontal bars indicate the
region over which the value of v2 was averaged. The

vertical bars indicate the possible error in

determining the number of

turns in a precession cycle ....................... 153
7-1. A schematic drawing of a probe head, as defined

in SOMA. The two dimensions, AIDIFF and THICK,

are input in inches ............................... 163
7-2. An illustration of the meaning of the various

initial ellipse parameters. See text for an

explanation of how they are input to SOMA ......... 165

1. Introduction

At present the majority of accelerator studies at the
National Superconducting Cyclotron Laboratory are devoted to
two cyclotrons. the K500 and the K800. The K500 cyclotron
has been in operation since 1982. and is running a regular

schedule of experiments. The K800 is under construction and

is expected to begin testing with beam in 1987. In the case
of the K500 the effort is directed at improving beam quality
and intensity. while K800 work is devoted to more

fundamental design considerations. The two cyclotrons have
many similarities; size is their most obvious difference.
The material reported here will be dealing only with the

K500. but in most cases is equally applicable to the K800.
The K500 cyclotron at Michigan State University1‘2'3 is
a multi-particle. variable-energy machine. The bending limit

is K =520 MeV, and the focusing limit is K =160 MeV, where

b f
the energy limit in MeV/u is either Kb(O/A)2 or Kf(O/A).
whichever is smaller4. The compact magnet has a pill-box-

shaped yoke that completely encloses the cyclotron. The main
field is produced by two pairs of circular superconducting
coils located just beyond the extraction radius (see Figures

1-1 and 1-2). The flutter is created by three spiral-shaped

 

MSU —86- 322

........

        
   
 

Dee

      

r" .
artilll

f :-
l 1le

 

 

 

    

  

 

 

I .I
7‘. I. :1!!§.A NA a": ‘l_a @1ng Large coll
I'll!ll ills" .Illli\
i: ’J : _ Yoke

 

 

 

 

Figure 1-1. Vertical section of the K500 cyclotron.

 

 

 

 

  

/ c \
/ _'. .
“1&2 \ .
5 ”Wu . \\ '
HH/ ‘5‘ -. \\
I. .
/
’3
(3 1‘60“

0

\' I.\~

) A.
/.’ \

Figure 1-2. Median plane view of the K500 cyclotron. "A”
hill is located between the "C” dee and the ”A"
dee.

hills. The magnet gap on the hills is 6.54 cm. The RF

systems'6 consists of three dees located in the valleys
between the hills (where the magnet gap is 36”), and can be
operated over a frequency range of 9 MHz to 27 MHz. Figure
1-3 shows the range of energies and charge states that can
be accelerated. Originally the K500 ran with an internal
Penning Ion Source (PIG). but as of March 1986 it has been

coupled to an Electron Cyclotron Resonance (ECR) ion

7
source
The K500 cyclotron has several unique features that
make it an interesting case study. The high magnetic field

(BO=3T to 5T) leads to a very compact magnet design, which

in turn leads to a small separation between turns of the
internal beam. The small size of the cyclotron necessitates
that all the attached hardware must be compact in nature. as

space is at a premium. The small magnet gap and the tight
spiral result in a median plane field with large gradients.
In general these features place stringent requirements on
any approximations that are made. Because the dees are
spiral shaped. the azimuth of the gap crossing is a function
of radius. consequently dealing appropriately with the gaps

adds an additional complication to any orbit computation

routinesa.

.Emcom_u oc_.m.mao oomx och .m-F o.:m.m

3:: wuzmeamma Beam:

 

 

 

 

 

 

 

 

 

 

 

3 on em 3 e e m e n m a
TI _va
gum EA. 1 mum
(tr IV. on»:
v_ Tm
_t _ I_ mix
+u0m¢ 2&9 +nU§ +NU§ +023
43 a. new. on. new. n3. 3. S. no. .3.

 

 

\ \\ \ \ C

\
/\\.\\ -

 

om

 

 

 

 

 

 

 

 

 

 

 

 

on on on on 2 m on on no me
2x55 mm§\>ommzm

(9)!) man omnsvn mas

Transfer matrix programsg'37 provide the ability to

compute many orbits in a relatively short time. This greatly

facilitates the investigation of bulk properties of the
beam. In addition, these codes are quite powerful for
simulating the output that would be obtained from a

diagnostic device such as a beam probe under different
operating conditions. Unfortunately none of the transfer
matrix codes available were suitable for use with the K500
cyclotron because of the spiral shaped dees. and the large
field derivatives, so the program SOMA (Second Order Matrix
Approximation) was developed. The design of this program
will be the subject of chapter 2.

Most of the time it is desirable to run the cyclotron
in a manner that gives the most extracted current. Of course
there are exceptions. and in many such cases it is desirable
to reduce the phase spread of the beam. thus improving the

time resolution of the beam. and reducing the loss of

internal beam on cyclotron components1o. One such situation
is accelerator studies where the ability to Observe distinct
turns is a major advantage. Phase selection of the internal
beam using the coupling between the horizontal and
longitudinal motions has been used for many years. but

nevertheless every such system requires a detailed

7
investigation of its feasibility. Such an investigation will
be presented in chapter 3. In the case of the K500 the
highly non-linear nature of the central region and the low
number of turns involved makes orbit tracking with a
numerical integration program the preferred choice. After
the central region. the transfer matrix code provides a
rapid method of tracking the selected beam to extraction.
This makes comparison of experimental results to

computations much faster.

In order to perform phase selection in the K500
cyclotron a rather intricate set of hardware was
constructed“. and this will be presented in chapter 4. This

hardware had to provide the necessary control functions.
within the constraints of the limited space available and
the high magnetic field. As a result a large effort was put
into the construction and installation of the drive
mechanism. The construction of the viewer port (V P.) probe
allowed for improved beam diagnostics which helped in
understanding the beam behaviour with the slits in place.
Also the V.P. probe drive made use of the gamma probe
convenient and easy . The Z probe allowed the direct
measurement of the internal beam‘s phase width. so it was

invaluable for observing the results of the phase slits.

8

In chapter 5 the results from a set of measurements of
orbit properties in the K500 cyclotron are presented. Most
of the measurements seek to confirm the computed magnetic
properties rather than discover unknowns. As will be seen
the agreement between running conditions and computations is
quite good. The measurements also confirm that the phase
selection system operates in a manner consistent with the

calculations of chapter 3.

2. SOMA: A Cyclotron Orbit Code Using Second

Order Transfer Matrices

2.1 Introduction

For many years computer programs for the design of
charged particle transport systems have made use of a matrix
algebra formalism. The procedure is based on the fact that
to first order the final conditions may be expressed as
simple integrals of a few particular first order
trajectories (matrix elements) characterizing a system. In
these codes; beam-line elements are represented by idealized
components for which the trajectories were derived
analytically. The programs then compute a transfer matrix
for the whole system by multiplying together the transfer
matrices for each of the elements in the system. The results
provide rapid physical insight into the design of systems.
leaving ray tracing to final design confirmation. and the
computation Of higher order effects. In a procedure

described by K. Brown‘z. this technique was generalized to

include second order effects in the very successful program

”TRANSPORT". Some years ago a simple extension 0' the

10

transfer matrix ideas in beamline codes was made to allow
them to be used for the design of synchrotrons. Today
several matrix programs for synchrotron design exist that
correctly treat second and even higher order abberations.

The extension to cyclotrons is more difficult since the
beam path does not consist of a set of discrete single
function elements. but rather a single. very complex
magnetic field. which varies as a function of radius and
therefore as a function of energy. The well known solution
to this problem is to compute the first order trajectories
around a closed (equilibrium) orbit. (ED), for a set of
energies spanning the range of the cyclotron. Results of
this type of calculation are commonly expressed using the

variables 0%. vr and v2 (the orbital. radial and axial

focussing frequencies). Historically. as cyclotron running
time became more valuable. and computer time less expensive.
it became increasingly popular to track orbits in the
appropriate magnetic field, in order to have a better
understanding of the beam behaviour. \Nhen only a small
number of orbits need to be tracked in order to understand
the overall properties of the system this technique proved
to be very valuable. In cases where many orbits need tO be

followed. the large amount Of CPU time required to do the

11

numerical integration makes this procedure very demanding on
the available computer facilities. It was found that in
cases where bulk properties such as the radial-longitudinal
coupling are being investigated. a high degree of accuracy
in the individual orbits is not required. This meant that a
program that computed the transfer matrix elements and then
used them to determine the orbits of a large group of
particles. (each with different starting conditions) would
allow rapid investigation of these phenomena.

At TRIUMF (Vancouver, Canada) the first order transfer

matrix program "COMA"9 was developed based on these

principles. In this case the transfer matrix elements are

computed by the equilibrium orbit code "CYCLOPS"13 and

output at any number of azimuths. for a set of energies. The
minimum number of azimuths at which matrices are necessary
is determined by the number of accelerating gaps. These
matrix coefficients are then fed into COMA. and the program
selects a set of initial conditions for a set of particles
from a given distribution (see section 2.5). The initial
conditions are multiplied by the appropriate first order
- matrix thus determining the orbit parameters at the first
accelerating gap. At the gap a delta function model is used

to evaluate the energy gain of the particle. The conditions

12
at this gap can then be multiplied by the appropriate matrix
to determine the conditions at the next gap, and so on. The
actual matrix coefficients used are determined by
interpolating between the values that were computed by

"CYCLOPS". and stored at a set of discrete energies. Tests

of ”COMA” in a TRIUMF magnetic field9 show that for static
runs in regions away from the stop bands. and with initial
displacements as large as 2.0 in. from the E0 the errors
were similar to the changes that occur when the Hunge-Kutta
step size is changed. In the case of accelerated orbits a
particle with an initial 0.25 in. radial amplitude had an
error of 0.001 in. after 165 turns. Checks which involved
passing through stop bands showed larger errors. but the
results were still usable.

The sucess of "COMA" suggested that such a program would
be very useful for accelerator studies at MSU. However two
major differences between TRIUMF and the MSU superconducting
cyclotrons prevented the direct use of "COMA". The more

obvious difference is the accelerating gaps. which follow a

spiral in the K500 and K800 cyclotrons. rather than the more

conventional radial line. This difference implies a more

complicated gap crossing routine. similar to the one
8

implemented in the program ”SPRGAPZ" . It also complicates

13
the bookkeeping. as radial lines (used for output at a
constant angle) cross the accelerating gaps. The other
important difference is in the magnet structure. The K500
has a high field magnet (5T) with a 6.35 cm gap and a tight
spiral. while TRIUMF is a low field magnet ( 5T) with a 52.8
cm gap and modest spiral. The smaller magnet gap allows
larger azimuthal derivatives of the magnetic field. while
the tight spiral generates large radial derivatives. In a
transfer matrix program the first order matrix elements are
a function of the first derivatives of the magnetic field,
while the second order coefficients include terms involving

the second derivatives and so on. Thus the more rapidly

varying field allows the second order effects to be
significantly larger. in fact with initial displacements as
small as 0.010" there were significant differences between
the transfer matrix program and the orbit integration

routine (see 2.6). This was the motivation for developing a

transfer matrix code in which the second order effects were

included. It should be noted that a transfer matrix program
requires that the equilibrium orbit exists. if the magnetic
field contains large stop bands such as those that would

result if the first harmonic component of the field is large

in the region of vr=1.0 . then this would not be true. This

14

restricts the use of these programs to cases where resonance
crossings are fast and the field imperfections small. A
separate treatment for harmonic bump coils will be given in
section 2.7.

In the following sections the equations of motion of a
charged particle will be developed and then expanded about
the equilibrium orbit. After the method by which the

solutions to the first order differential equations are

found, has, been demonstrated. the second second order
contributions will be computed. Following this the effects
of the spiral gap shape will be discussed. Finally there
will be an outline of the routines used by SOMA. and some

comparison with an orbit integrating program.

The approach used to find the matrix elements is a
perturbation expansion. analogous to the Born aproximation
in quantum mechanics. First an exact solution is sought for
the case where the equations of motion (about the E0) are
linear. As is well known. the solutions to this can be found
by integrating the orbits of two rays. (displaced from the
E0). between the two points for which the transfer is
needed. Then the quadratic terms are added to the equations
of motion and treated as a perturbation. That is to say that
solutions are sought that are a combination of the exact

linear solutions (the eigenfunctions of the unperturbed

15
case). As in the Born approximation the solutions are formed
using a Green‘s function. It will be shown in section 2.2.4
that the Green’s function in this case is very simple. It
should be noted that because the equations of motion are
truncated we no longer have a Hamiltonian and so the

solutions are not symplectic.

2.2 Calculation of Transfer Matrix Elements
2.2.1 Equations of Motion
The Hamiltonian with 9 as the independent variable. for a

charged particle in a magnetic field with median plane

. . 14
symmetry IS given by.
H - - r - l r A (2-1)
‘ pe b e
As is done in all the orbit programs currently in use at

NSCL. a length unit ”a” and a field unit ”b" are defined as.

a=clob b=m0ublq (2-2)

where ub=2nvrflh if vrf is the nominal RF frequency. m is

0
the rest mass. q is the particle charge. 0 is the velocity
of light in vacuum. and h is the harmonic number.\Ne then

take the momentum unit to be whole so that momenta are

expressed in units of length.

16
Assuming that the magnetic field has median plane
symmetry. and is given in the median plane by B=B(r.6) then

to second order in 2, near the median plane:

- 1 2 828 1% 1 828

82- ' 8+ 2 Z (or2 + r r + r2 662 ) '
- a3

Br- - 2 ar’ (2-3)
- z 88

Be' ' r as

If the field. B. is divided by the field unit D such that
B(r.9) » B(r.9)/b.

then Hamilton’s equations yield.

 

 

 

.-gr- r.
- d9 _
Ji pz- f-pi)
dpe z 2 2
pr = 56— = J(p -p 'pz) ' r 8z + Z 89 '
r p -
dz 2
z = —— 2-4
d9 lez-f-pi) ( )
dp .
.__z_ §,I_§_B_
pz ‘ d9 ‘ Z ( ' ar r as ) and
. dr 1 r
‘r=——= _
de lip pr-pz)
(where z = 1 + E/mocz).

The zero order solution to these differential equations is

known as the Equilibrium Orbit (EO). which. in a magnetic

17
field having N sectors. and no imperfection. satisfies the
periodicity conditions.

r0(9 + 90) = r0(9). pro (.9 + 90) = pr0(9). 90 = 21r/N.

\Ne then wish to expand in terms of the diSplacements

(x,px.z, and p2 ) from the E0 where;

To simplify the results. we divide the equation for the
derivatives into terms of different order in the expansion
coefficients. As for notation. the digit in the subscript of
each term will refer to the order of that term. Also the
second order terms will be separated into those that depend

on x2 (as well as x px and p: ). and those that depend on

22. Thus the rxz contains the terms in the expansion of r
2

that depend on x . x px . and pi . The overall derivatives

using this notation are.

r' = r +r' + r’ + r’
b 1 x2 22
y = v + t + I + v
Dr pro Dri prxz Drzz
z' = z} + zé
t = 1 + 1
Oz 921 922

T' = Tb +7} + Tkg + 722

18
The results of expanding the equations 2-4 for the

derivatives in terms of x. px. z. and p2. and identifying

the orders of the various terms is given below. In each of

these equations. (which shall be labeled 2-5). where r or pr

appear on the right hand side of the equation. they refer to

the values of r or pr for the equilibrium orbit. Also the

equations will use p9 = (p2 - p30)“2 . the theta component

of the momentum for the E0. The zeroth order components are:

z_r

To = ;
pe

the first order terms are:

D 2
..1:_.Lx+r_Lp
De p3 X
e
p
. _L éfl
pr1- p px-(B+rar )X
9
2 L p
1:
092
1-2(r§§._p_r.§_§)
92“ a as

19

second order terms are:

 

 

 

2
2 r P D
_ E E f 2
”9 pa
r,_1rpl' 2
22‘ 2 pg pz '
. “11212-1 52 628 2
prx2 2 3 px 2 (2 or + r 5?? ) x
99
2
._1351,(628+1_a§+1 628)z2+1_9§zpz
przz‘ '2 p9 +2 or2 r or r2 892 pe 89
X rpr
22 = Dz ( 59 + p3 Ox )
e
as 828 EL 828 93 gg
p22 = z (or + r or2 pe arae ) x p3 as Z px
e
I p 1 2
T;(2= rXID+ Lr‘(13—)DZ
3 X 3 X
99 “9 p9
, 1 r
72:51793
De

The procedure for finding rO and pro is the same as that

.used in the equilibrium orbit codes "GENSPEO" and "CYCLOPS".

and is described in detail in reference 13. The method for

20
finding the first and second order matrices will be

discussed in the next two sections.

2.2.2 First Order Matrix
In order to find the first order transfer matrices X and

2 as defined by.

(x) =X<e.e.>(")
px 9f f ' px 9i
z z
() =Z(e.9.)()
pz 9f f ' pz 6|
we need two independent solutions. denoted (x1.px1) and
(x2.px2) to the equations for x .px and two solutions

denoted (21.pz1) and (22.p22) for the equations for z and
p . \Ne also require the correction. X. to the time

coordinate. (T). such that for a displaced orbit r a r + X.

For an orbit with initial displacements x(9i) and px(9i). X

will be given by.

X(9f) = X1(9f. 9i) x(9i) + X2(9f. 6i) px(9i)

were X1 and X are to be found. It proves most convenient to

2
choose the initial conditions.

x1(9i)

l
—L
"O

i
O

xziei)

21(9i) ‘ ‘ p21 (95) = 0
because then.
F

x (9 ) x (9 )
9.) = 1 f 2 f

X(ef' ' px1(ef) px2(9f)

1 (9,) 22 (9f)

 

 

 

2(9 ,9.) =
f ' 921(9r) p22(ef)
x1 r p
Xi = Z [ E; + 3 0x1]
99
x2 r p
x2=}[6—é+ 3pX2]
”e

The values of x1,px1 etc. are computed by integrating the

first order equations (2-5) along the equilibrium orbit
between 9i and 9f. As in all our orbit codes. the
integration routine uses the Runge - Kutta method of Gill15.

with a step size of two degrees.

2.2.3 Second Order Matrix Elements

In the case of a first order transfer. where the final
conditions are given as linear functions of the initial
conditions. the results are exact solutions of the

differential equations obtained when the equations of motion

22
are expanded to first order. \Nhen the final conditions are
given to second order in the initial conditions. the results
are an approximate solution to the differential equations
that are a result of a second order expansion. Inherent in
this difference is that the method of finding the second

order matrix elements must be different from that used to

find the first order elements. The approach outlined below

is similar to that used by K. Brown‘z. In this approach the
orbits are to be given as a second order Taylor expansion in
the initial displacements from the equilibrium orbit. It is
then required that. the expanded orbits satisfy a set of
differential equations that have been formed by expanding
the equations of motion to second order. For the first order
expansion coefficients this generates a set of first order
homogeneous. linear differential equations. For the second
order expansion coefficients the differential equations are
similar except that they are not homogeneous. The non-
homogeneous part of the equations has the form of a driving
function. Finally the second order coefficients are
evaluated via a Green 5 function integral containing the
driving function of the particular coefficient. and the

solutions of the homogeneous equations.

A convenient

the matrices

V(9) =

2(9

 

Inspection of

of the

I'OWS

A .D and E as defined by.

X(9f.

f 1

the differential

of

eliminate carrying

Vx( 9) =

 

 

9.)(

ei)( (Z) )9 +otef.ei) vtei)

 

statement

X

Z

A and

these

x
(3)9

23

of

- X1x(9i) +X2 px(9i) + E(9

D will

VZ( 9)

the problem

i S

+ A(9f.9i) V(9i)

that

.Gi) V(9i)

identically zero.

><><1

 

II) D

X
lN

rows we define.

 

equations 2-5 shows

we

that

require

some

SO 10

24

Thus the equations become.

X

X
( px)e = Xt9.ei) ( px)9i + Ate.ei) Vx(9i)
Z Z
( pz)9 = 2(e.ei) ( pz)9i + 0(e.ei) Vz(ei) . (2-6)

X(9) = X99) x(9i) + XEG) px(9i) + E(9 .Gi) Vx(ei)'

The elements of matrices A and D are simply the second order
coefficients of a Taylor's expansion of the coordinates. If
the differential equations (2-5) are written in matrix

notation. then they become:

g_ x _ x

d9( px)e - Kie) ( px)9 + «(9) the)

g_ z _ X -
ddpge—umtpge+vagm (2n
%(e) = m1 x(e) + m2 px(9) + HG) VX(G)

where the elements of «,B. and 3 are tabulated in table 2-1.

If the equations (2-6) are substituted into the
differential equations (2-5). the result will be a
differential equation containing terms to second order in

the expansion coefficients. Proceeding in this manner. and

retaining only terms of second order or less. Vx(e) and

VZ(9) are

25

 

 

Li1 x8 + 2 X11 x12 xo pxo + Xi2 pic 2

x11 X21 *3 +(X11 X22 + X12 X21)"o px0 + X12 X229:0
Vx = x21 “5 + 2 X21 X22 xo px0 + X22 9:0

zit 23 + 2 211 212 20 D20 + Ziz ”:0

211 Z21 2: +(Z11 Z22 + 212 Z21)20 pzo +z12 Z22 ”:0

221 25 + 2 221 222 20 p20 + 222 ”:0

_ .i
and.

x11211x020+x11212x0pzo+x12211px020+ x122129xopzt;
v2 = X11221x020+x11222xopzo+X12221px020+ X122229110sz

X21211x020+X21Z12Xo°zo+X22211pxozo+ X222129110sz

X21221xozo+X21222xopzo+X222219x020+ Xzzzzszopzo

 

 

I.

d

where the subscript zero implies that the initial values of

the coordinate are to be used.

Continuing the substitution and collecting the
coefficients of the initial values (x0.px0 etc.) a
differential equation for each first and second order
coefficient is obtained. The result shows a systematic
pattern.

X11 = k11 X11 + k12 X21 X21 = k21 X11 + k22 X21

X =k X +k X X =k X +k X

12 11 12 12 22 22 21 12 22 22

26

Table 2-1. The elements of the arrays a and B. These arrays
contain the coefficients of the displacements
(x.px etc.) found in the differential equations

 

2-5.
_ 2 3
0‘12 ‘ p I 99
_ 2 s
«13 _ 1 5 r p pr / pe
« = 5 r p / p3
16 r 6
as 1 aze
“21 ‘ ' 6r ' 2 ' 6r2
_ g 2 3
«23 - .5 p / De
_ 5,(_2628+1§§+1628,
o(24 - 6r r 6r r2 66
a _122
25 pe 66
0‘26 = ' 5 ’ pe
_ _ 3
as aze pr aze 93 as
B =—+r——.—- —— B — —-
21 6r 6r pe 6r66 23 pg 66
_ 3
ZZ—Zprlpe
D2
1 r r
)3 — — l—— ( 1 + 3 —* )
p3 p2
6 6
z 12r_r
6 - 3

27

+ f

a = k a + k a2n 1n

1n 11 1h 12

aZn = k21 ain + k22 a2n +f2n

where the f's are functions of the first order coefficients
and the elements of «. These driving functions are tabulated

in Table 2-2. A more compact statement of these results is.

g—éx -_- K(6) x g—e-Z=L(9) Z
(2-8)
9—A =K(6)A+F(6) 9—D=L(6)D+G(6)
d6 n n n d6 n n n
(2-9)
1 (9) 9 (9)
I:n‘e’ ‘ 1:19) G'n‘e) = 9319)
(2-‘10)
a (6) d (9)
An(6) — 3;:(9) Dnie) — d;:(6)
9— X = m a + m a + h
d6 n 1 1n 2 2n n
The equations in the first row are the differential

equations for the first order coefficients. the solutions of

which are already known from section 2.2.2 . The
differential equations for the second order terms An and On
are very similar in form to the first order equations.

except for the presence of a driving term (Fn or On). Since

28

 

Table 2-2. The driving functions which appear in the

differential equations 2-9. The values of the «'3
and the B’s are listed in table 2-2.

f11 = “12 X11 X21 + “‘3 X21

f12 = “12 ( X11 X22 + X12 X21 ) + 2 “13 X21 X22

f13 = “12 X12 X22 + “13 X22

f14 = “16 Z21

f15 = 2“16 Z21 Z22

f16 = “16 222

f21 = “21 X21 + “23 X21

f22 = 2 “21 X11 X12 + 2 “23 X21 X22

f23 = “21 X22 + “23 X22

f24 = “24 221 + “25 Z11 Z21 + “26 221

f25 = 2 “24 211 212 + “25 ( 211 222 + 212 221)
+ 2 “26 Z21 Z22

f26 = “24 222 + “25 212 Z22 + “26 222

911 = 212 X11 Z21 + 314 X21 221

912 = 212 X11 222 + 214 X21 Z22

913 = 212 X12 Z21 + “14 X22 Z21

Table 2-2 (cont'd).

CO
to
to

ll
be
m
d
X
A
d

h1 = “2 X11 x21 +
h2 = “2 ( X11 X22
h3 = “2 x12 X22
h4 ‘ “6 221

h5 = 2 “6 221 222

3'
II
N
0')
N
m

30
the driving terms are only functions of the first order
expansion coefficients (which are known). the solutions to

equations 2-9 can be found using a Green's function.

9
An(6) = I Fn(6') A(6.6') d6' . and (2-11)
0 .

Bn(9) = Gn(6') A(9.9') d6' . ' (2-12)

(ah—,6:

where A is the Green's function.

The solution for the second order expansion coefficients
of the time. T. are much simpler. since the right hand side
of equation 2-10 involves functions of the first order

expansion coefficients only. Thus.

de' . (2-13)

6
3(6) =[(m1a1n(6‘) +m2 a2n(6') +hn)
0 .

If the Green's function is known then the second order
matrix elements can be obtained by numerically integrating
equations 2-11 .2-12 and 2-13 . along the E0 at the same

time as the first order equations are being computed.

2.2.4 The Green‘s Function16

The problem requires the solution of.

31

IO.

[1 -K(9)]An(9) =Fn(9)

0.
(D

where l is the unit matrix and K is either K or L as defined

in equation 2-7. The solution. X of the homogenous equation.

'0.

[l -K(9)]Xn(9)= 0

Q.
d)

is known. The Green's function must be a solution of;

'0.

[Id

0

- K(6)] An(6.6') = I 6(6‘6')

subject to the conditions.
A = O 6 < 6'

and A(G = 9'+6 ) = I.

Since A is a solution of the homogenous equation for e > 6‘
(or 6 < 6') it must be a linear combination of X. If Y(e )
is a matrix to be determined. then.

A(6 > 6')

X(6) Y(6')
The boundary conditions at 6 = 6' give,
A(6 = 6 +6) = l = X(e') Y(6‘)

f. Y(6') = x"(e’)

:. A(6.6’) = X(6) x'1(e')

A(6) = x1119) X22(6) - X12(6) x2119) = 1 so.
X22‘9') ‘ x12(9')
x‘1(e') =
—Lx21<e') X1119 a

 

 

32

2.3 Crossing the Accelerating Gaps17

\Nhen the equations for the matrix elements are integrated
along the equilibrium orbit they start and end on radial
lines that pass through the point where the E0 crosses the
gap. As a result when the displaced rays are transferred up

to the gap. the values for the displacements (x.px. etc.)

are the values along a radial line. In order to compute the

effects of the acceleration correctly. the values of r. pr

and r are needed at the point the displaced orbit crosses
the gap. Since the values of the orbit at this point are not
known. they must be estimated. Moving the displaced orbit
onto the gap first requires the calculation of 66 (see
Figure 2-1). which in turn requires the value of 6r. The

angle of the gap 69(r) is a given function of the form.
99(T) =90' @(r),

which has been input to the program. In our case the gaps
have been entered as a table. (GAPTH). of theta values at

equal ARg intervals in radius. A double three point

Lagrangian interpolation18 is used to find the first and

’second derivatives;

0. O.

fiLDa)
I
$2

0.

1\)

.L.
I
Q
N

33

 

”no 0
mspucsp 0mm ‘ 60 - ( )
6mm 1121-1
GAPTHaR /”“‘G*P
) ’ \ R=(IR+1)I:AR.
Rn .
so
R-IRtt-AR.

 

 

Figure 2.1. The geometry of the gap crossing correction. The
gap position is given at points Gapth(ir). 66 is
the angle through which the orbit must be moved.

34

The required 669 corresponding to a 6r (unknown) can be

divided into first and second order components. as can 6r.

802

Using the Taylor expansion for 669. and 6r.

d6 d26
66 :d—rgar+1—96r2
g dr2
d6 1c126 .
3 ——9 (6r + 6r ) + — ___9 5r2 (2-13)
di’ 1 2 2 2 1
dr
6r-x+r’ 66+%r"662
: . . . 1 u 2
x f (r0 + x ) 661 + rO 662 + 2 r0 <561

where the zero subscript indicates the E0 value. Collecting
terms of the same order.

6r1 = x + r0 1

5r = f 69 + X' 59 +

n 2
2 r <561

1
0 2 1 2 0
If these values for 6r are then substituted into equation

(2-13) and the first order terms collected.

35
so,

1 1 - r0 (d69/dr)

 

66

Now that the values of 661 and 6r1 have been determined.

it remains to find 6r2.

 

- 1 . l .. 2
6r2 _ r0 662 + x 661 + 2 r0 661
d6 1d26 2
592 = ——gdr 6r2 + 5 —9‘2 6T1
d r
d6 d6 1 2 1d26
_ , g _ n _ 9 2
_ dr r0 662 + dr (x 661 +2 rO 661) + 2 dzr 6r1
_ . 1 n 2 1 2 2 2
662 — (x 661 +2 rO 661)(d69/dr) + 2 6r1 (d eg/dr )
1- r6 (d6 / dr)

9

So, using the appropriate values for the derivatives gives.

 

XCX1
591 = 1 r' a
01
(x 66 +lr”662)ot+-1-6r (x
1201 1212
592 = 1 - r‘ a
01

The procedure for calculating the changes in each of the

orbit coordinates introduced by 66 follows the same method
used to find 6r. If q is any coordinate then.
q * q + 50

where.

36

 

 

 

Table 2-3. The derivatives of the orbit coordinates
necessary for the calculation of the gap
correction. The symbol ”q” is any of the particle
parameters. The primes are differentiation with
respect to 6.

Q 00 Q1 “0
2
p 2 D L_E_
r r r
r r p / p —— x + ——%— p —— r‘ + 3 p’
r 6 p9 pe x pa 0 p9 r0
0 D
r 68 r . 66 .
pr pe- r B - p px (B+rar)x -p pr0 (B+rar) r0
6 6
z 0 r pZ / pe O
p
68 r 68
pz 0 z (r 6r ' p 66 ) 0
6
l r l . r .
T 2 r./ p6 p x + 3 pr px p ro + 3 pr pro
6 p9 6 p9

37

6q 1 q' 66 + q" 662

Nl-b

1 1 _1_ ” 2
(Q0 + q1 ) 66 + 2 Q0 66

The values of the necessary derivatives are tabulated in

Table 2-3. In order to compute all these derivatives the
p
values of p - r B B + r fig and r 66 - —L §§ are
6 6r r 9 6

stored along with the other EO values when the transfer

matrices are being computed.

Now that the values of r.pr and T are known on the gap

the effect of the RF voltage can be computed. The
computation of the acceleration process is identical to that
used in the numerical integration code "SPRGAPZ". and is
described in detail in reference 8. It suffices here to say

\

that both the energy and pr are modified by this routine.

Before proceeding to perform the transfer up to the next
gap. the values of the displacements on a radial line are
again required. Since the energy has changed. so has the
equilibrium orbit. and thus this is not simply a reversal of
the previous process This time both the azimuth of the E0
and of the orbit are known. ie.

66 = 669 - 66EO

38
This time the difficulty arises because the values of the
displacement from the E0. along the radial line. are not

known \Nhat is known is.

x9 = r9 - rO

px = prg ' pr0

zg and ng
The subscript "g" refers to the values on the gap. In the
following. coordinates without a subscript will be
understood to be evaluated on the radial line that passes

through the point at which the E0 crosses the gap. The

required corrections 6r and 6pr are.

As before.

6r = r‘ 66 + x' 66 +

1
0 2

n 2
r0 66

59 = 592 (2-14)

. ~ 1
r prO 66 + px 66 +

2 pr0
The displacements on the gap (which are known) are given by.

x = r -r = x +6r
g g 0

p = p - 9,0 = p +<Spr

39

Putting these values for x and px into equations (2-14)

gives.
2
6r = v 66 +-1 M662 + (-l-x ..1—2- p ) 66 -J-66 6r
0 2 pa 9 3 xg 9
p
6
r 2
—§— 66 6p
D6
6 - ' 66 + 1 " 662 [ EL - t B r 99 ] 66
Dr - pr0 2 pr0 + ' p9 pxg + 6r ) xg
p
+ [ -L 69 + ( B + r 92 ) 6r ] 66 .
pe r r

Rearranging these equations and identifying the derivatives

of x and leads to.
g pxg
Pr r 2
6r(1+-6—66)+(——;L-66)6pr=Ar
9 De (2-15)
6r (8 r 22) 66 (1 EL 66 ) 6 - A
‘ + 6r + ' pr ‘ or

where Ar and Apr are the values of 6r and 6pr respectively

if x9 and pxg are used in place of x and px. Equations

(2-15) are a set of linear equations in 6r and 6pr. the

solution to which is given by.

Dr

2
6r = [Ar - i —— Ar + L—B— Apr} 66] / DET
pe 9%

40

6p = [Ap + { -L Ap + ( e + r as ) Ar: 66] / DET
r r pe r r
p2 68 r 2
DET = 1 - r 662 + ( B + r —— ) ——E— 662
32 r p“
6 6
It is necessary to retain all the terms so that if the

accelerating voltage is zero. moving onto and back off the
gap will not result in a change in the values of the

coordinates. Calculation of the changes in z and p2 follow

the same pattern. so:

62 = - p 66
De 2
5 _Z(.2§_E_r_22)59
pz _ 6r p9 66
62 — L 66 6 )
- p ( 929 oz
6
6 - ( r fig - EL fig ) 66 ( z 62 )
p2 ' 6r pe 66 g
r f
62 + — 66 6p = — 66 p = A2
p z p 29
6 6
(.22,p_r_a_§)5952.5 _(,22-p_r§_§)592 _Ap
6r pe 66 p2 _ 6r pe 6 g - z
62 = [ Az - 1 66 Apz ] I OETZ

p
68 r 68
6pz = [ Apz - ( r 6r - pg 66 ) 69 62 ] / DETZ
DETZ _ 1 - L ( 1' LB. . _p._r is. ) 592
— pe 6r pe 66

The calculation of 67 is considerably simpler. because
there are no coupled equations. The correction to r is.

6T=T'66+r’66+16662

O 1
in which the only unknown value. 7} . depends on x and px.
¢‘ = —l— x + l—L p p = -l— (x - 6r) + 1—L p (P ' 50 )
1 De 3 r x De 9 3 r X9 r
D6 ‘36
_ l x + Li D p UL _ u p 6p
99 9 D3 1' X9 De D3 1' 1'
6 6
= ML, 14 p 5.,
19 D 3 r
6 p 9

So.

Once the corrections have been computed the substitution.
q * Q +5Q
is made. and everything is set to make the transfer to the

next gap.

42

2.4 Program Algorithms

SOMA is designed to operate as a self-contained unit with
the exception of the magnetic field grid which must be
produced by a separate program (the grid is the same as that
used by SPRGAPZ and CYCLONE). At the beginning of each run
the program either computes the transfer matrix elements or
reads them in from a binary file produced'during a previous
run. If the matrices are to be computed then the magnetic

field and gap table are read in. Then the program searches

for equilibrium orbits using the procedure of Gordon and
Welton19 . for a set of specified energies. After each E0 is
found a search is made for the points at which the E0

crosses the gaps. and when found the values of r.pr and 6 at

these points are stored. A separate routine is then used to
integrate the equations of section 2.2 along the E0 from one
gap location to the next. The integration technique is again

a standard Runge-Kutta15. in the input stream it can be

specified whether first order elements or both first order
and second order elements are to be collected. The input can
also specify up to 10 fixed angles at which the transfer
matrix coefficients will be stored. Currently the fixed
angles must fall on a standard Runge-Kutta step. After this

procedure is repeated for all the selected energies. and if

43
the main probe option is selected. a probe transfer matrix
is then computed for each energy. This is done by
integrating from gap 6 up to the track location. which has
been input as a table of r.6 values. Finally all the
transfer matrices are stored on binary files.

Once SOMA has an appropriate set of transfer matrices. it
then reads in the parameters common to all particles. This
includes the dee voltage. the locations of slits and probes.
and the set of transfer equations required. In fact several
options exist for the transfer as shown in Table 2-4.Care
should be taken when choosing options other than 1 and 6.
since these two are the only cases in which a complete

expansion to a given order is done. That is to say that case

Table 2-4. --The order of the various parts of the transfer
as a function of the input parameter N.

 

 

 

N x z

1 first first
nd . st . .

2 2 In x0,1 in 20 first

3 1st in x and 2 second

0 0

nd . st .

4 2 In x0. 1 In 20 second
nd . .

f

5 2 in xoand 20 irst
nd .

6 2 in x and 2 second

 

 

44

one is the first order solution. case 6 is the second order
solution. and all the others are a mixture of the two. For
example. if N=4 then the program uses exact median plane

equations of motion. allows x to couple into 2. but does not
couple 2 into x. Cases 2 through 5 are useful for
determining what terms are responsible for a given effect.
The next step is computing the starting conditions for
all the particles (up to 5.000 may be run) and storing them.
The various possible sets of starting conditions are
discussed in section 2.5 . Particles are then run one at a
time. each being run until it reaches a turn limit. an

energy limit. or a radius limit. whichever comes first.

At each gap a test is performed to determine if any
requested fixed angles fall between the current gap and the
next gap. If a fixed angle is found then the particle

parameters (x.px.z.p .r.r.p ) are computed for that angle

and stored. The fashion in which the parameters are stored
depends on what the fixed angle has been designated to
represent. After all the fixed angles found have been
computed. the program proceeds to compute the transfer to

the next gap. At the gap. the parameters (E.x.px.z.pz.r) are

updated as described in section 2.3. and the process repeats

itself as often as necessary, It should be noted that the

     
 

Figure 2-2.

45

 

\ — — — EO'S
\ 11 STORED VALUES
0 mmapmnmn

CURRENT GAP

The result of a fixed angle crossing a gap. The
transfer matrices of the orbits labeled 1.2. and
3 are correct for a transfer from the current
gap to the fixed angle. while that for orbit 4
is not. In this case the orbit being calculated
would use the values stored for 1.2.3. and 4
when the interpolation is being done. and would
give an incorrect result.

46
computations for the fixed angles in no way affect the
values at the gap.
There is a difficulty that occurs in the region where the
fixed angle crosses the accelerating gap. The problem arises
because the values of the matrix elements are found by

interpolating between values that are stored for fixed

energies. In Figure 2-2 one possible scenario is
illustrated. In this case the transfer matrices stored for
orbits 1,2. and 3 correspond to transfers from the current

gap to the desired fixed angle. but the one stored for orbit
4 is a transfer from the next gap to the fixed angle (see
arrows). If, as in this case. the interpolation for the
matrix elements on the orbit use the values for orbit 4 and
orbit 3 then the results will be incorrect. It should be
noted that' there are currently no structures in the K500
cyclotron that cross the accelerating gaps. Nevertheless the
program prints an error message when a transfer of this type
happens.

The fixed angles can be designated as one of two things.

either a flag or a probe. The flags themselves are divided

into two groups. intercepting and non-intercepting. At a
flag if the particle lies between the minimum and maximum
values for that flag. the orbit parameters are stored. If

the flag is intercepting then the particle IS considered

47

removed from the beam. and the next particle is begun.
otherwise the run continues unchanged. After “all the
particles have been run the program will produce scatter
plots of any pairs of the orbit coordinates, at any of the
possible 20 flag locations. A slit can be described as 2
intercepting flags located at the same azimuth. For detailed
ray tracing the particle parameters at all gaps and azimuths
(or some combination thereof) can be printed out.

A probe consists of a differential and a main jaw. which
can have up to 3 axial divisions or 60 phase divisions. The
probe is considered to move outward in radial steps. Upon
finding the orbit parameters at the probe azimuth the
program determines in which steps the particle would give a
current reading. The requirements for this determination are

that the probe location does not intercept an earlier turn

of the same particle. but does intercept the the current
turn. For each bin that these requirements are met the bin
count is augmented by 1. There are 20,000 bins available to

be divided between the z (or 6) bins and the radial bins. At
the end of the run the probe bin values are written in a
binary file which can then be used as input to a plotting

routine.

48
2.5 Starting Conditions

A transfer matrix program is used to run large groups of

particles, so the generation of starting conditions is of
great importance. As a result the program offers a variety
of methods. each with its own particular use. The input
routine has the ability to calculate the horizontal and
vertical eigen-ellipses. the accelerated equilibrium orbit

(AEO). and the average central phase. all at a given energy
known as the central energy. Any combinations of these
values can be used by the various routines used to generate
the starting conditions. or the values of these parameters
can be set in the input stream.

The simplest routine reads in the values E (energy).x.px

.z. and p2 on gap 1 for each particle. A similar but more

complicated routine reads in the values of E.r.pr.z. and p2

on gap 1. These two crude techniques are oriented towards
cases in which either specific orbits are being tracked. or
the initial conditions are being determined by another

program. The first technique can also be used to re-start a
previous run from the stored final conditions.

The next group of routines are those designed to generate
the starting conditions for a group of particles which fill

ellipses. The ellipses can be either eigen-ellipses or be

49
input as a major axis. minor axis and a tilt. The center of

the x.p ellipse can be displaced from the equilibrium

x
orbit by either an amount determined to be the offset of the
AEO or by an amount given in the input stream. In all these
cases a specified interval of phase is divided into equal

steps, and each starting phase is given an ellipse to be

filled using one of the technrques discussed below.

In chapter 5 a special input routine is used. In this
case it was desired to run a set of particles which matched
the conditions at the exit of the central region. In Figure

2-3 the energy and the displacements x and px for a group of

particles which were run outwards from the spiral inflector
for seven turns with the program CYCLONE are shown. Each of
the rays run would in effect be a central ray for a given
phase. In each frame the solid line is a function of the
form noted in that frame. It can be seen that with
appropriate choice of the slope. the linear approximation

for x and px is quite good.\Nhen using this special routine.
instead of keeping the displacements in x and px for the
center of the x-px ellipse. and the starting energy. the

same for all starting phases. the coefficients of these

functions can be input to SOMA and it will use them to

50

.szJO>O once 6:. mc_m: n can. 0, Lo.om__c_
.m._am ms. Eco. um.m.mm_c_ >__mo_LmE:c mam
m__n.o :62; nwc_m~no mm:_m> 6:2 mum mommoao och
.mwo_._m on, .o m..:mo on“ .0. mm: 6.30; Em.mo.a
on. mm:_m> mg. 6.6 mm:.. u_.0m och .co_om.

 

    

    

 

 

 

 

 

.m..cmo ms. do ~_xm mg“ .m mco_._ucoo mc_.cm_w .m-m m.:m.u
93: kahuna Amado—.5 Hm Amadeus N
95%.: 03.6 mmod o mwcdl mod mmod
ONI. q q q «a a q. . . . q . . . a a - ... ONI
. a _ H _ fl _ H _ _
31 T 1H! [HI 3.1368. [”21
1 1.. ..i l m+o.flu i .d
n H H n m
SI nl Iii inl lat)
- “m w- m 6
at ”I [#1 neiscmoofhl L61 0
m 313666.616 H + 8911.6 mm m
D r. . _ . . _ . _ _ .. iJ _ . _ . _ WI _ _ _ p _ FI— p p i O

51
calculate the ellipse center and the Initial energy for each
starting phase.

In the other ellipse-type routines the starting energy
can be randomly distributed about the central energy if
desired. In all of these cases the ellipse can either be
uniformly populated. or randomly populated. The most

convenient choice is to populate the ellipse being studied

uniformly. and to populate randomly the other ellipse. (eq.
uniformly populate the x-px ellipse and randomly populate
the z-pz ellipse). to get an idea of the spread caused by

the coupling. Uniformly populating both ellipses implies a
large number of particles. The random population is produced

using a standard random number generator to select x and px

values between 0 and the ellipse maximum. Then the program
checks to see if the coordinates fall within the ellipse

proper. and if not it selects new values for x and px. until

they do. The uniform population is done by assigning a
square of fixed area to a point which is located at the
center of the square. Points are placed in phase space until

no more squares will fit into the ellipse.

52

2.6 Comparison of SOMA with SPRGAPZ

In this section the results of median plane calculations
with the program SOMA will be compared with the results
obtained with the orbit integration code SPRGAPZ. The tests
discussed are only a sample of the many checks preformed.
The program SPRGAPZ integrates the exact median plane
equations of motion. and the linearized z motion equations.
This allows the coupling of the x motion into the z motion.
but not the z motion into the x motion. In the following
section comparison of the z motion will be done with the
program SPRGAPZ4 which uses equations for the vertical
motion that are valid to fourth order in 2. There are three
areas from which one expects to generate differences between
SOMA and SPRGAPZ. The most obvious source is the transfer
matrices themselves. As the transfer matrix technique is an
approximation of a given order there will be contributions
from the higher order terms. In this case it is expected
that the error would be proportional to the next term in the
Taylor expansion. In Figure 2-4 the differences after one
turn (without acceleration) are shown. As in all the figures
involving comparisions of orbits between SPRGAPZ and SOMA
the differences are plotted against a measure of the initial
displacment from the E0. In each a set of rays lying on the

boundry of an eigen-ellipse was run. and the maleum

53

1 TURN STATIC

 

 

 

100 fl I I l T TUTTI I I r rrrfin
10-1
rs -'-' FHEH?ORDER ,
' /
:3 — SECOND ORDER / /
3 10"2
A
.l
% 10-3
'5
V
‘> -
10 ‘
10-5
10-6 ,
1 5 10 50 100
Initial Condition T
Figure 2-4. Differences between SOMA and SPRGAPZ as a
function of the initial displacements from the
E0. Each ray was run one turn without
acceleration. The initial conditions of each ray
are such that it falls on the boundry of an

eigen-ellipse of area (0.2)f2 mm-mrad. The
stability region ends around i=150.

54
difference for each ellipse was plotted. The area of the

eigen-ellipse was 0.2f2 mm-mrad. If the expansion is done to

first order the error function goes as (5.6E-6)f2. At this
radius. (16"). an emittance of 5 mm-mrad corresponds to a
maximum orbit center displacement of o 03". If the expansion
is taken to second order the the error is proportional to
(1.6E-8)f3. in other words the errors are third order in x.
In the first order case the results are exactly the same as
that found if the first order equations of motion are
integrated numerically. Note that this is not true for the
second order case where the solution is to second order in
the exact first order solution. not an exact solution of the
non-linear differential equation.

There is also a difference generated by the interpolation‘
of the matrix elements when the orbit's energy lies between
the stored values. In Figure 2-5 the differences between
SPRGAPZ and SOMA are plotted for different interpolation
step sizes. For each step size a ray was run whose energy

was exactly halfway between two stored values. (the worst

possible case). The initial condition of the ray was a
displacement of 0.003". and the results are plotted after
one turn without acceleration. At this energy (11 MeV) a

step size of .2 Mev results in differences of less than a

55

1 TURN STATIC

 

0.0006

 

0.0005
’1?
C
‘3
5 0.0004
57.
3 0.0003
’3
2’ 0.0002
0.0001
Figure 2-5.

 

 

I I T r I l I l I r I I T I I

IllllllllllLlllLllllLllll11

 

 

The
the
The
one

0 02 Q4 &0
Distance Between 80': (MeV)

differences between SOMA and SPRGAPZ when
spacing between stored values is changed.
initial dISplacement was 30.03". and run for
turn without acceleration.

56
tenth of a mil. Larger step sizes lead to much larger
errors.

The third source of differences is the gap crossing
routine. These are the hardest to meaere as they only occur
when the accelerating voltage is on. so both of the other
two effects will be present at the same time. The situation

is also confused by an uncertainty in the location at which

the orbit crosses the gap. Since the final orbit is not

sensitive to small variations in the gap position this
uncertainty is only a problem when looking at specific
values on a gap. In Figure 2-6 the differences after 100

turns with acceleration are shown as a function of the
initial displacement from the E0. These differences are the
sum of all three sources of error. As can been seen in the
figure the differences for first and second order do not
have different slopes as they did in Figure 2-4. This is
mostly caused by the fact that accelerated orbits are always
displaced from the E0. so even the F=0 ray has a displacment
of at least 0.02” from the E0 at many points. There are also
errors caused by the gap crossing routine. which are
seperate from those caused by the expansion technique. A
comparison of a first order transfer with first order gap
crossings. and a first order transfer with second order gap

crossings is also shown in Figure 2-6 . This illustrates

101

10°
’3‘
O
'8
:13,

A 10-1
.I
.4-

3, to-2
5

10—3

10—4

F i gu r e 2- 6.

57

100 TURNS WITH ACCELERATION

 

r I I l l I

q

II I I I IITIII

FIRST ORDER. 2” ORDER GAP CORRECTION
FIRST ORDER

III
I l llllll

     
 

r

5 — SECOND ORDER /’"§
: ’5' 2
/ .' ‘
t- / _' .1
E / Z‘.
/ .-
_ , 2

- / .......
,. -
F —:
= .x' a
I / -
./ ‘
"' -i

Hun—a -..-- b'“ -.r

I I IlllIl
l Illlllll

 

 

lllll l I II

I l I 1

[III

 

1 5 10 50 100
Initial Condition 'f'

The difference in r (and pr) between SOMA and

SPRGAPZ for rays which have been run for 100

turns with acceleration.

58

100 TURNS WITH ACCELERATION

10*1 I] I I I ITITII I I IfIIIT

 

- - FIRST ORDER. 2"“ ORDER GAP CORRECTION
----- raunroanna

  
  
 
 

 

 

 

10"2
— SECOND ORDER .
-3
" 10 _§
ii 3
a , g
/
5! 10-4 .3
10-5 ,
10—6 _J] 1 1 1 1 ILIJL 1 1 +1 1111'
5 10 50 100
Initial Condition '2'
Figure 2-7. The difference in energy for rays for the same

rays shown in Figure 2-6.

59
that the orbits are relatively insensitive to the gap
position. Those differences that do occur, arise because of
differences in the energy gain as shown in Figure 2-7. In
this figure the the difference in energy is plotted for the
same rays as shown in Figure 2-6. For the rays of small
initial emittances the error is 5 parts in 107, which is the

same magnitude as the round-off error. The difference in x

for a ray which is initially 30 mils from the E0 (e=5mm-
mrad) after 100 turns. is only 1 mil in x and pX combined.
certainly adequately small unless ray tracing is being done.

2.7 Treatment of Harmonic Field Bumps20

As shown by M.M. Gordon35

the perterbations of the radial
oscillations due to asymetric accelerating kicks. can be
duplicated using an equivalent field bump. It is therefore

reasonable to assume that the effects of the field bump can

be represented by making appropriate changes in pr (and p2)

at each of the 6 accelerating gap locations. That is to say
the field bump is to be represented by a series of delta
functions. such that.

b(r.9) = E uk(r) ale-ek(r)). k=1.2 ..... 6.

60

The values of the uk(r) can be chosen to give the

appropriate first and second harmonic bumps, while
suppressing all 3N components.
Using equations 2-4 we find that the appropriate momentum

kicks must be such that.

2 6pr(k) = -r b
p
Q 4.219.
2592(k) =2[ 7 r' p 59]
9
b = g1sin(9) + h1cos(9) + gzsin(29) + h2cos(29).

where g1.h1.g2 and h2 are the measured bump components at

this r. VVe therefore define q(,w<. and w( such that at gap k

the impulses are;

p
r
éprlk) = -r uk . and 602(k) = z r vk + zfpe wk
so:
E uk6(9-9k) = g1srn(9)+h1cos(9)+g25in(29)+h2cos(29)

99 - Qfl 99 - EH
2 vkate-ek) dr‘Sln(9)+dr‘COS(G)+dr28ln(29)+ 2cos(29)

k dr
E wk6(9-Gk) = h1sin(G)-g1cos(9)+2h25in(29)-292cos(28)

Using orthoganality these three equations give us four
linear equations for each of u.v. and w. The 3N harmonics

can be surpressed by requiring that the Sum of the even k

61
terms and the sum of the odd k terms are zero. and thus
there are six equations for six unknowns In matrix notation

the linear system to be solved is,

 

 

 

 

 

 

'sin(61) sin(92) ...... sin(66)1 fu1 v1 W11 '9‘ dg1/dr h1 '1
008(61) 005(92) ...... cos(66) u2 v2 w2 h1 dh1ldr '91
sin(291) sin(292) .... sin(296) U3 v3 w3 = v 92 dg2/dr 2h2
cos(29,) 003(292) .... cos(296) U4 v4 w4 h2 dh2/dr -2g2
1 0 1 0 1 0 U5 v5 W5 0 0 0
Lo 1 o 1 o 1 j _u6 v6 wsj Lo 0 o J
SOMA uses the IMSL. (International Mathematical and

Statistical Libararies INC). subroutine LEQIF to solve this
system of equations at each radius value at which a bump
profile has been given. The values of uk.vk and WR are
stored in a table at the beginning of the run. Each time an

orbit crosses a gap, (k). the program interpolates in the

table to find the values for uk.vk,wk at the orbit radius.

and then computes the impulses.

6p = -r u

r k’
pr
ODZ=Z(IVk+b—Wk)
9
Figure 2-8 shows a typical example of an orbit calculated

with SPRGAPZ and SOMA In th1s case the orbit begins on the

62

31 =0.018 ¢1 = 180°

 

 

I I I I I I I I I I I I r r’I I I I I _

0.02 —" SPRGAPZ“

I 3011A :

- ..l

0.00 — .—

3? ~ ‘
3 . 1
0 b q
:9, -o.oz +— —
H F -

31 g -
-0.04 — '—

I— -l

"1 1 I 1 1.1 1 I 1 1 1.1 I 1 1,1 1 I 1 1d

 

 

 

-0.04 -0.02 0 0.02
X (inches)

Figure 2-8. A sample precession cycle as calculated by
SPRGAPZ and SOMA for a field with a first
harmonic. The orbit began on the E0.

63
ED for E=4 MeV and then the bump causes it to precess. VVhen
run with a wide varity of initial conditions. and different
bump magnitudes the results were always qualitatively the

same as those shown in Figure 2-8.

2.8 Vertical Motion

In order to observe the non-linear z-motion offered by
SOMA a simple comparision case was run. A set of 64 rays was
formed from all the possible combinations of eight rays
located on the perimeter of the vertical eigen-ellipse with
emmitance of 75 mm-mrad, and eight rays located on the
perimeter of a 35 mm-mrad eigen-ellipse in the horizontal

plane. The magnetic field was the same 12C3+ 3O MeV/u field

used before. so the emittances correspond to a final

emittance at extraction (30 MeV/u) 0f 62:30 mm-mrad and 5x Z

14 mm-mrad. The same rays were also run with the code

SPRGAPZ436 which correctly treats the magnetic field to

fourth order in 2 (ie. the equations of motion have terms of
fourth order in 2). All particles were run for 300 turns
(field geometry was for a 500 turn total). so most of the
acceleration region is covered.

In Figure 2-9 are shown the results from both the SOMA

and SPRGAPZ4 runs In both cases the points all lie very

64

near the eigen-ellipse, and the spread due to the different
x-p values is approximately j"0.02" on an ellipse with a

half major axis of 0.2". The fact that this spreading is

very similar in both cases indicates that the contribution

of the 23 term in SPRGAPZ4 is very small. since it is not
included in SOMA. There is however a small difference in the
amount of rotation around the ellipse boundry in the two

cases which leads to a possible combined z.pz error of

0.026". This is probably again caused by the poor EO closure
on the spiral gaps. lf SOMA is run without the x motion
coupling into the z. (N=5 in Table 2-4), then the 8 points

in each group become one as would be expected.

3.0 A Computational Examination of Phase

Selection in the K500 Cyclotron

3.1 Introduction

Phase Selection is generally used in cyclotrons when it

is desired to lachieve single turn extraction and its
associated benefit521. Although we may wish to take
advantage of single turn extraction eventually . our initial
goal is to achieve. separated turns over most of the

acceleration region so that detailed accelerator studies can
be carried out. As shown in Figure 3-1, the high magnetic
field in the K500 cyclotron leads to a turn separation which
is rather small compared to the turn width associated with
the phase spread. If one also includes the spatial extent of

the beam (the x.px size) then it is apparent that with the

:150 phase width transmitted by the central region. distinct
turns would be observable only for the first few inches. If
on the other hand the phase width were reduced to i 20.
separated turns would be observable for all of the
acceleration process and beam centering COuld be
determined. Centering is of great practical importance

66

67

 

0.150

 

 

 

 

0.125 '— J
r- -i
0.100 E— -3
‘3 E I. :
O i- -1
'5 0.075 -— _
S - n
—' ' 1
g : :
0.050 :- *1
0.025:— , , , ’ ._‘
: .—""’ 32° _____ :
0.000 L L'P'l‘l'fi' +'f"L—1'-1"I'1-I1—I.1.-I. 1.1 :d

0 5 10 15 20 25

R (inches)
Figure 3-1. Plots of simple estimates of the turn separation

and turn widths associated with different phase

widths. The turn separation (solid curve) is
estimated using ARzAE'R/Z'E while the full width
of a turn with a given Ad is found using

AR=(A6)2'RI4.

67

 

 

 

 

 

0.150 I I T‘r
0.125 L -3
C I
0.100 -— —‘
'3 F r :
O - ..i
f. 0.075 —- —
S I i
I; ' 4
<‘ 0.050 .__ J
0.025; ’ , , ’ _‘
: ’ ’ ’ ’ 3 2! ....... - :
0.000 1 l 1 I}.r-a-rfl'i'fi'1—i-1.-I.F.1‘I 1 1 1 1 1 1 I L‘
0 5 10 15 20 25
R (inches)
Figure 3-1. Plots of simple estimates of the turn separation

and turn widths associated with different phase

widths. The turn separation (solid curve) is
estimated using AR=AE'R/2'E while the full width
of a turn with a given A0 is found using

AR=(A¢)2'R/4.

68
because it reduces phase oscillations, minimizes the
effects of non-linearities and makes extraction much less

22

sensitive to the dee voltage . Separated turns also allow

the measurement of the radial focusing frequency vr . and.

with an induced coherent oscillation. the axial frequency

Phase selection in cyclotrons is performed by taking

advantage of the coupling between the radial (r.pr) and

longitudinal (E-¢) motions of the particlesza. Figure 3-2
gives a typical plot of radius versus starting time. Note
the horseshoe shape. with the peak occurring at the starting
time corresponding to the largest average energy gain per
turn up to that point. The shape of this curve is a direct

consequence of the cos(¢) dependence.(where d is the average

phase). of the energy gain. In the case of an axially
injected beam it is possible to populate all the starting
phases that will clear the posts in the central region. VVhen
running with an internal ion source this is not necessarily
true as the source to puller voltage is used to pull ions
from the source and thus the density of ions will be
dependant on the starting tHne.\Nith the flexibility offered

by an axial injection system it is pOSSlble to deSign the

69

 

 

 

 

 

7.3- I I I I I I r I Ia‘T—I I I I I fi] I ‘
'lz - "
.- 33 -1
A " -4
I
o - .
5. ° - q
a - 32 .
a _ a
a . i
10 F- —‘
,_ -i
,_ -l
69 I I
ShuiuulTuno(TJ
Figure 3-2 The radius of the central ray. shown as a
function of starting time. for four successive
turns. Note the typical horse-shoe shape
resulting from the c03(¢) dependence of the

energy gain.

7O
central region to allow only the desired portions of the
curves in Figure 3-2 to survive the first turn.

At the center of the horseshoes in Figure 3-2 there is
no radial dispersion with phase. On the other hand. on
either the leading or trailing edges, the radius is strongly
dependent on the phase. As we wish to separate particles
with different phases on the basis of the radius differences
this feature will be very much needed. (In section 3-4 there
will be a discussion of how the slope of the curves can be
modified). Given that we require a one-to-one correspondence
between radius and phase. it will be necessary to insure
that only one side of the horseshoe is p0pulated.lNhen the
beam is axially injected into the K500 the particles with a
starting time of 'ro=250O have the largest energy gain per
turn in the middle of the cyclotron. ie. by the time they
reach 15” they are at the center of the horseshoe. Also the
particles with 70:260O have the least centering error at
15", so it would be advantageous to populate the starting
phases between 2500 and 260°. The process of selecting which
starting times are populated will be referred to as "coarse
selection”, since lit will limit the phase width in the
machine to i100. while the more careful selection at 7" will

be referred to as "fine selection".

71

 

 

Figure 3-3.

 

 

Electrode structure for the K500 first harmonic
central region using a PIG source. Four orbits
are shown corresponding to starting times (from
outermost at 6:00 to innermost). 70:230.240.250.
and 260 degrees. The peak electric field between
the source and puller is achieved at 10:270. A

slit is located on the 00 hill extension of the
center plug allowing easy installation and
removal. This slit removes all particles whose

starting times do not fall between 230 and 250
degrees.

72
3.2 Coarse Selection
The first stage of the phase selection process is a
coarse selection made near the center of the machine where
the large turn separation allows the installation of a slit
with a large enough frame to avoid the possibility of
undesired phases passing outside the frame. If such a system

were designed to transmit only 200 of phase the situation

illustrated in Figure 3-2 would be single valued. In Figure
3-3 such an aperture is shown for the first harmonic central
region using a Penning Ion source. In this figure we have
superimposed 4 orbits on a median plane section of the
central region electrode structure. The tour rays have

0 O 0

starting times .70 .of 230 .240 .250 , and 260° and an

initial x=px=0. As shown. only the 2400 and 2500 rays pass
through the slot formed by a U shaped block mounted on the

hill portion of the center plug. (2300 almost does). (By
locating the slit on the center plug it can be removed and

inserted by pulling the center plug. a considerably easier

task than raising the magnet cap.) Since a To of 2700

corresponds to the peak electric field in the source-to-

73

 

 

 

 

 

 

 

 

 

 

 

MSU-86498
3 b I I I T I I I l I Ifl I I I I ‘l I I I I ‘I—I I I F
2 __ —l
I -
r .
1 L— .2
- -1
P 4
o f\
b J
..1— .—
l l l l l 1 L L I 1 L L 1
0 1 2 3
Figure 3-4. The electrode structure for the first harmonic
central region with axial injection. The window
frame attached to the dummy dee following the
puller is used to neutralize the coupling
between the first and second dees. Five orbits
are shown corresponding to starting times of
230° (outer-most at window).240°.250°.260°. and
270° (inner-most). By narrowing the radial width
of the window it will be easy to remove those

starting times lying outside 245° to 260°.

74

puller gap it is unlikely that many ions outside the 230°-

260° range shown can enter the first turn. In the case of

later starting times (towards 270°) there is insufficient
time to cross the source-to-puller gap, and for the earlier
times there is insufficient electric field to pull the ions
from the source. The solution in the case of an axially
injected beam is quite different as shown in Figure 3-4 for
a first harmonic mode. In this central region the RF
coupling between dees is neutralized by a window frame
structure mounted on the dummy dee which screens one dee
from another. By enlarging the radial extent of the vertical

sections of this frame it can also be used to select a group

of starting times. in this case between 245° and 260°. “Nth

\

this particular central region the r =250° ray has the least

0
centering error at 15". whereas. in the PIG case best
centering occurred for the ray with 10:2400. In both cases
the phase spread transmitted by the narrow slit is

approximately 1 10° around the "centered" orbit.

AT (deg)

Figure 3-5.

 

 

75

-------------- Inflector entrance
inflector exit (from x-spnce)
------ inflector exit (from y—spnce)

 

Q/u=0.5 80:36.2 kG

 

I I I T

 

 

 

 

 

 

 

 

g l 1 1 J

2 4 6 8
RF time differences for particles on the
boundary of a 1000 mm-mrad phase space with
respect to the central ray. The dotted line
indicates the difference at the entrance of the
inflector. The solid and dashed lines show the
differences at the inflector exit. The abscissa
is just an arbitrary parameter around the

boundary of the phase space.

76

3.3 Phase Selection for Axially Injected Beams

Conventional wisdom would say that a phase selection
system is not necessary when the beam is being axially
injected. since the beam can be pre-bunched before entering
the cyclotron. In fact we do use a buncher located just
before the entry into the cyclotron yoke. but there is a
fair amount of de-bunching of the beam as it traverses the

yoke2°and inflector. This debunching is illustrated in

Figure 3-5. where the difference in starting times is
plotted as a function of particle number. (There are 8
particles distributed around the perimeter of an ellipse.)
As the bunched beam will have a phase spread in the

neighborhood of ten degrees the beam entering the cyclotron

will again have a phase spread of thirty degrees: the only
difference now is that the buncher phase is another
adjustable parameter. In the case of the axially injected
beam there is a further concern that the non-linearities in
the spiral inflector will produce a distorted phase space

which could make phase selection difficult. To reduce this
effect we can work with a small beam spot. It was found that
if the analysis System in the beam transport system was used
to select a beam with an emittance of 250 mm-mrad

(unnormalized). that after the strong focusing that takes

Pr (inches)

Figure 3-6.

77

 

0.11:31r'"|""l”"l""l_";
0.10;— .5:
0.09;- ..i
0.00;— _
'0.07 :Hlnul”Ulnnlunli

 

 

 

0.40 0.49 0.5 0.51 0.52
R (inches)

The phase space at the exit of the inflector
when the initial beam has an emittance of 250
mm-mrad. The momenta have been divided by qBO to
express them in units of length.

78
place during the yoke traversal. the beam spot size at the
entrance to the inflector would be 1 mm in diameter. The gap
in the inflector is 4 mm. so a 1 mm beam should pass through
sufficiently far away from the electrodes to avoid serious

non-linearities. (See Figure 3-6, a plot of x-px at the exit

of the inflector ) To insure that the spot size at the
entrance of the inflector is indeed 1 mm in diameter the
collimator at the inflector entrance could be replaced with
one that has a 1 mm hole instead of the usual 4 mm. In cases

where the beam intensity from the ECR ion source is high.

this will still leave sufficiently large beam currents to
run experiments. So far as distortions are concerned the
only remaining question is whether or not the electric

fields on the first few turns would distort {he phase space.
In Figure 3-7 we show the results when a group of eight rays

populating the perimeter of the ellipse shown in Figure 3-6

are accelerated forward 3 turns using the Program CYCLONE25.
This program integrates the equations of motion in the
measured magnetic field and in an electric field which has
been computed with a relaxation code. For each ray several

different starting times were run. so the values of R and Pr

at the final position are plotted as a function of the

average phase on the last turn, By interpolating to get the

Figure 3-7.

79

The B and Pr plotted for a group of rays that
started on the perimeter of the ellipse shown in
figure 3-6. after 3 turns. as functions of their
average phase. In the middle frame the results
of interpolating to find each ray at an average
phase of -4° is shown. Note the ellipse shows no
distortion.

80

 

 

 

 

 

PHI AVERAGE

 

 

 

 

Anon—03V ..L

0.12 —

2.58

2.55

2.54

3.52

R(inches)

 

 

*1‘44dd—4111 dfifld

44—4‘111

-‘

PHI AVERAGE

 

 

 

A3553.

Figure 3-7,

I r I

O.|P .
4; I- a
O
.8
gear \ 1
GI ' q
10 230

-OJI- .

8 I6 24 32 40

Figure 3-8.

81

 

 

 

 

 

 

TURN NUMBER

Radius difference ri-r at 6:840 vs turn number

0
for a family of central rays. Ray 0 leaves the
source at 70:2350. the others at the times
labeled on the plot. At turn 33 a bar of :.02
inches is shown to give an idea of the radius

variation expected from the r.pr distribution
around the central ray.

82

Ft.Pr values for the particles with the same average phase we

get the ellipse shown in Figure 3-7, which is almost
distortion free. as desired. The use of particles with the
same average phase is particularly important since these are
the particles which will have the same energy gain per turn
and therefore they will all arrive at the deflector with
almost the same energy. Also by using this grouping of the
particles one avoids an apparent distortion which is

actually due to the energy dependence of R and Pr'

3.4 Fine Selection

Upon leaving the central region the beam is well

behaved and has a phase width of approximately 20°. lt

§

. . . o
remains to .reduce this 20° to something of the order of 4

0. As is apparent in Figure 3-8. at this level of phase

or 5
selection the radius spread due to the phase is comparable
to the beam spot size. so the interaction between these two

must be taken into account. To achieve the best possible

selection one would like to place the next set of

83

 

SIN“)

 

 

 

 

5 10 15 20 25 30
R... (inches)

Figure 3-9. A reprsentative K500 phase curve determined
using the standard procedures The arrow is
located at the radius of the fine selection

system.

84

obstructions where 0. defined by2°z

Q=IAR)/(A¢) , (3-1)

AB = R(¢2) - R(¢1) A0 = a - a

is a maximum. At the same time it is advantageous to do the
selection as near as possible to the center of the machine.
where the beam energy is low. so as to reduce the possible
activation of the cyclotron components. At any given radius
the AR term in equation 3-1 is a result of two separate
effects. First there is the change in radius associated with
the energy difference between two particles with different

phases. It can be shown2°. that a good first order

approximation for the energy difference is.
r
AE 3 - A0 J sind dE

and the resulting radial difference is given by.

AR =

85
From these equations it can be seen that the maximum value
of 0 would be obtained when Jsind is a maximum. Figure 3-9
shows a representative phase curve and the JsinadE for the
K500 cyclotron. At 7” the integral of sine is large but is
not at a maximum. Increasing the value of the integral at 7”
would entail sharpening the initial drop in the phase curve.
which would lead to a quite different phase history for the
inner part of the machine. The phase curve shown in the
figure has been chosen to meet several important criteria.

First the large initial positive phase is chosen to gain

electric focusing in the first few turns. The negative
excursion and subsequent rise back to zero, which is
centered about 6". is a result of tailoring the field so
that vz at 6.5" does not become too small. At this radius
the energy is too large for there to be much electric
focusing. but the flutter is not yet at a maximum. so a
small gradient is added to the field to raise the vertical

focusing. At the same time the initial fall-off of the phase
curve (inside 2") is determined primarily by the iron
geometry of the cyclotron and thus is not easily changed
with the trim coils. It will be shown below that with a
phase curve determined by these criteria (such as that shown

in Figure 3-9) the Q will be sufficiently large at the phase

86

  
 
 
 
 

MSU-86497

To=250°
I’ll/IIIIIIII[IIIIII/Illlllllllllllllllll[I’llIII I I
.,_. . . . turn n+l

z[I’ll/III/llllllllIII/[III] | I
} turn 11
To = 230°

Figure 3-10. A schematic of how a single post can act
manner similar to a slit. The inner edge of

post scrapes off those particles with too large
a radius. while the outside scrapes off those

with too low a radius on the next turn
dotted region is the surviving beam while
cross hatched region is the removed beam

87
slit location. Since this process also produces a magnetic

field with good values of yr and v2. trim coil power. and

the integral of sind at extraction (near zero). it will be
used in the subsequent calculations.

The second source of radius variation with phase is
centering. \Nhen particles cross the initial gap between the
source and the puller the energy they gain is significantly
larger than their initial energy. thus their total energy at
the exit from the first dee is very sensitive to the voltage
present on the dee at the time of crossing. Since the dee
voltage is a function of phase the energy will be strongly
dependent on the phase of the particle. In the K500
cyclotron the details of the central region require that the
phase at the first gap crossing be different from zero.
otherwise the radius spread could be made small by running
near 10:2700. As seen in Figure 3-4 the variation in energy
leads to a large difference in radius at the exit of the

first dee. From this gap onward the percentage change in

energy at a gap crossing decreases. Soon all the particles
have similar rigidities. but the differences in radii
remain. so each starting phase will have a different orbit
center. All the particles begin their trajectory at the same

point (the source exit). so as the orbits precess about the

88

equilibrium orbit. there will be approximately two locations
per turn at which the orbits are all at the same radius. In
the K500 the fine selection will be done at two locations,
120° apart (but on the same turn). If the phase dependent

centering were arranged such that the radii were either
dispersed or focussed at one slit location. they would not
be at the other location. Phase dependent centering is
extremely sensitive to the central region geometry. so the

best method of studying it is direct orbit integration. the

results of which will be discussed below.
In Figure 3-8 we plot the radius differences ri-r0 at a
fixed azimuth, where r is the ray which leaves a Penning

0

Ion Source at T0=235° ( the results for an axially injected

beam are qualitatively the same). At the radius of turn 33

(approximately 7") there is a space between trim coil number
2 and trim coil number 3, so at this radius in the center of
the hill a 112” diameter access hole passes from the liner

through the pole and exits on the magnet cap. To preserve
the magnetic Symmetry there are six such holes but because
of the space requirements of the system only two of them are
usable. one on hill A and the other on hill B. As can be
seen in Figure 3-8 the Q at this turn is quite good ( for

particles near To :235° ). while the turn separation is

89
still about 100 mils (see Figure 3-1) and the energy is only
606 of the extraction energy. Included in this figure is a
bar of i .02 inches. which is intended to give an indication

of the radius variation expected from the x.px distribution

around the "central ray”.

The 100 mil (.100”) turn separation at this radius is
insufficient to allow the insertion of a slit. but will
allow the insertion of a post between turns. Although posts

are less common than slits they have been used at other

laboratorie52° with good results. In principle, after two
turns the post has had the same effect on the beam as a
slit. so long as the size of the post is such that it
scrapes beam from both the turn before it and the turn after
it. as modelled in Figure 3-10. In this mode of operation
the high radius particles are scraped from turn ”n” on the
inner edge of the blade and the low radius particles from

the turn "n+1” on the outer edge of the blade.\Nith vr 31.01

the composition of the beam at turn "n+1" is little changed
from that at "n" and the post has removed the high and low
radii from the successive turns just as a slit would from a
single turn.

A more detailed analysis of the selection

process involving the x.px spread of the beam. is presented

Figure 3-11.

Radius
time

90

plotted as a function of the starting

for turns 32.33 and 33 for the PIG

geometry in the upper sequence and for turns 29

through

32 in the lower plot for the ECR

geometry. Associated with each central ray is a

set of

8 rays that populate the circumference

of a .02 inch radius circle in R.Pr space. From

left

to right: the first one shows the

situation at 8:84 before the blades are
inserted. the next one shows the situation

after

a 60 mil blade has been inserted at 84°

and the third shows the effect of inserting a

second
frames

9:2040.

around
space

blade at 0:2040. The following three
give the the analogous situation at
Note that the final phase width is
4 degrees and the full 0.02 inch phase
around the central time survives. The

rays with different R.Pr values have a starting

phase that gives them the same energy gain per
turn as the central ray with which they are
associated. thus the horizontal label is

actually a measure of the energy gain per turn.

 

 

 

 

91

com on”

.F—-m o.:o_u

Anooumovv m2; ozfimdhm

Qua

can can

cow

can

can

 

v-0
1.
[IIIITIIITrT—Irrd

\

 

 

i—IIII

_\

I—F

III I

IIII

-
1111L111I1111I1IJ

i—IIII

l

__\r

L11|1111

i—TIIIIIIIIIFIIIIIIII-

L111LL111

 

_1 4

_

1
l
l
l

     

1111
IIIIIIIII'IIII

1111I1111

 

i—IIII

)—

fl
——1

_
11L1I1111I11111111

\
_ _

 

IIIITIIIIIIIIrl'IIII

_

L.\

-—t
d
—-1
11111111

IIIIIIIIIIIIIIIIIIIq

 

L ‘1 ‘1
11'1 1111

—
d
i
1111 L1L1 1111 1111

 

 

.fih

Mg.

as.

.43 u o

 

€85.03 may ozzmfim
can ova can gum ova can can oem on" new new can

.emuo

cum ch can can ofim can

 

 

 

 

 

_ . _ _ _ . _ fl .
r Al II.
I ~11! 5 1'
r [I III
r I." [I
T .11 _ Hi
I [I- [T
.I [I l!-
r § III Ill
I 1T 1!-
10 [Fe 1...!
r 1:. II
rl 1.5 1r

— u _ ~ — l— p — - — I

 

      

I r

l

llllllllllllll
IIIIIIIII

l
l

1111
IIII

 

_
-
_

_
-
_

1111

I

I

— —

 

1111

IIII

 

1111

IIII

 

1

l
IIII

 

_

u

s
Illll

IIIIIIIII'IIIIIIII

 

—
-
—
d

   

IIIIIL111

111I1111

 

_
_

 

:2 :3
(canon!) $1110sz

‘1
p

..wb

2 :2
(sauna!) 51110sz

as.

043

92

 

042

Pt (in)
c
y...
H

040

 

 

 

043

042

II'II
\

0.11

1

(111) .1d

IIII

040

 

IIII]

 

 

11111I11111111111

 

0 1111111111 111L111

'007 0.00 0.09 0.9 0.91 0.01 0.02 0.03 0.01

Figure 3-12.

 

 

 

0.09
R (in) R (in)

A simple demonstration of why two slits are
required to do a careful phase selection. Shown
are two bundles of rays with a one degree phase
difference. Note that at the first azimuth it
is impossible to remove all of one phase
without affecting the other phase. After the
particles have traveled 120° in azimuth they
have executed a thifd of a betatron oscillation
as highlighted by the cross marking the same
ray in both frames. As the shading demonstrates
it is now possible to remove almost all of the
unwanted phase. In this particular example the
two phases are sufficiently close together that
a small amount (the unshaded portion) of the
unwanted phase passes the second slit.

93
in Figure 3-11. In this figure we have plotted the radius
against a pseudo starting time for three successive turns at
the location of the phase slit holes. The horizontal label
is referred to as a pseudo starting time because the actual

starting times for rays with different values of r.pr have

been adjusted27 so that all rays with the same horizontal
label will have the same energy gain per turn. In the first
and fourth (numbering from left to right) the situation is
shown with both slit mechanisms retracted. In the second and

fifth frames the results are shown after a 60 mil blade has

been inserted at an azimuth of 840 (upper slit mechanism).
It can be seen that this blade provides most of the phase
selection desired, but it is still necessary to eliminate
those unwanted particles whose betatron oscillations have
placed them at the same radius as particles with a desired
phase. To do this final cleaning up operation a second slit
is required. At this radius in the K500 the radial focusing

frequency vr is close to 1.0. After the azimuth has changed

by 120° those particles whose large x components moved them

towards the radius of the desired phases will now have moved
away from the desired phases. as illustrated in Figure 3-12.

In the figure two bundles of rays that differ in phase by

94

 

 

 

 

 

 

0=204°
i 72
”.1 '15. l-—l l-:'- 13—1 '1 ,1:

I IE ._ -- .
7.1 L —C- -.j:' -~— —-_«7.1
i I . .E 1
: ji '.' .3” :: -<
7.0 l""‘ _._ . 3.3; ~F- —: 7.0
t j: ~-- -1
C 3C .,:.=, i ._.°-. -
59 —- -*— 2 gg' -jf- ~ I? ":69
: :; [1.3: 1: -. 1

' .41.. -. _l .0. .i
6.8 I 1 I _L i J 1"I 1 I I 1 'I 1 I 6.8
~30 ~20 ~10 ~30 ~20 ~10 ~30 ~20 ~10

Figure 3-13.

PHI AVERAGE (degrees)

The radius plotted at the azimuth of the lower
slit as a function of average phase (a good
measure of the energy gain per turn) for a
distribution of particles which pass through
the first turn slit. In this run the first and
second harmonics are present in the magnetic
field. so it should be compared to figure 3-10

where the particles were run in a field that
had perfect three fold Symmetry. Four different
turns are plotted. In the first frame on the
left no slits are present. In the middle frame
the upper slit has been inserted. and in the
final frame both slits are inserted.

95
one degree are plotted. As the shading demonstrates. with
two slits it is possible to remove almost all of one phase
(to the right of the line) white hat removing any of the
particles whose phase is one degree different. In this
example the two phases are sufficiently close together that
a small amount (the unshaded portion) of the unwanted phase
passes the second slit. The closer the phase of a bundle is
to the desired phase. the larger the unshaded region
becomes. (In other words. more of the horizontal phase space

associated with that phase passes the slit.) It is the

particles in the unshaded area which eventually produce the
sloping sides of the gaussian-like peak seen in a beam
current versus phase plot. In the third and sixth frames of

Figure 3-11 the result of placing such an obstruction at an

O

azimuth of '204 (lower slit mechanism) is shown. The group

of rays present in the final turn has a phase spread of 12°
around the chosen central phase. By correctly tailoring the
phase curve and the RF frequency, the particles with the
central phase will have the maximum energy (within a given
turn) at the outer radii of the machine. In chapter 5 these

computations will be compared to experimental results.

96

3.5 Further Considerations

The discussion up to this point has made use of orbits
which were computed in a magnetic field having perfect three
fold symmetry. In actuality there are small, but nonetheless
important, first and second harmonic components in the K500
magnetic field. \Nhen the imperfections are present. care
must be taken to center the beam if a truly well defined
beam (a primary goal of phase selection) is to reach the
extraction radius. At the radius of the phase slits the
strongest controls on the beam centering are the relative
dee voltages and the relative dee phases. At the same time
the dee voltage can be used to fine tune the position of the
turns relative to the posts. The technique would then be to
provide a reasonably centered beam at the posts by using the
relative dee voltages and phases. and the right positioning
(fine adjustment only) by using the average energy gain per
turn (sum of the dee voltages). The centering at the posts
cannot be perfect. as it is desirable to obtain the best
centering at a higher radius where the turn number is
larger. This centering is achieved by using the center bump
coil. In the case of the K500 there is not complete freedom

to set the bump coil since it should be set to values which

97

reduce the first harmonic in the region of the vr = 1.

resonance at 5.5".
In the case of a main field for a particle with a
charge-to-mass ratio of 0.25 and a final energy of 25 MeV/n.

a center bump setting of 81:12.1G ¢,=-70.2° meets these

criteria. \Nhen orbits are run in this field the selection
process is not significantly changed as can been seen in
Figure 3-13 where data similar to that in Figure 3-11 is
shown. In fact those differences that do occur between the
case plotted in Figure 3-13 and that in Figure 3-11 can be
attributed to the different phase curve used in the two
cases. This highlights the need to optimize the phase curve
and the starting phase (by positioning the first turn slit)
and the size of the posts. The next Figure (3-14) continues
the calculation in a field of three fold symmetry the rest
of the way out to the deflector entrance. This shows that at
least in principle. single turn extraction can be achieved
in the K500 cyclotron when running in first harmonic mode.
To achieve single turn extraction. however. much work is
needed to verify the stability of the various systems. and
to determine the correct centering conditions at the

resonances near extraction. The phase curve and frequency

98

 

I.22_ I. . I R I l r ‘
LZOL .' f-.- $S .
o l.|8- '
s x .-.
i LISP §$ .-. .
IJ4~ §§ 'é.2 .
|.|2l' L E . -

 

 

 

 

 

Figure 3-14.

26. 48 26.5 2 26.56

R (inches)

R plotted versus Pr for turns 507 and 508 at 9

= 336. corresponding to the entrance to the
electrostatic deflector Pr has been divided by
mob to express it in inches The shaded area
corresponds to a possible location of the

deflector septum. This plot shows-that single
turn extraction of the resulting beam should be
possible. The energy Spread of this group IS
less than 6 parts in 104.

99
need to be tailored carefully as well by using the procedure

28
described by Gordon .

4. Phase Selection Hardware and the VP Probe

4.1 Introduction

During 1985 hardware was constructed to perform the
experiments discussed in chapter 5. All of this equipment
was constructed for general operational use on the K500
cyclotron. These components fall into two groups: those
concerned with phase selection only, and those oriented
towards general beam diagnostics. In the second group is the
viewer port probe that has proven to be a valuable every day
diagnostic device. In the first group is the hardware
required to accomplish phase selection as described in

chapter 3.

4.2 Phase Selection Hardware

As described in chapter 2. phase selection in the K500
cyclotron requires two small tungsten blades to be inserted
between turns 32 and 33 on successive hills. Easy insertion
and removal of the blade from the beam chamber and
adjustability of the slit position (to accomodate different
orbit patterns) were seen to be desirable features of a
phase selection System. Also. since the lifetime of the

100

101
blades was uncertain, a large effort was expended on
developing the ability to change blades with minimum
disruption of cyclotron operation. This last feature would
also allow for adjustment of blade size (by changing the
blades), which is analogous to changing the slit size in a
conventional system. Access to the median plane on the hills
consists of two. half inch diameter holes located near the

center of hills "A" and ”B" (see Figure 1-2) at a radius of
7.038". The hole on hill A (9:83.50) emerges on the upper

pole cap. while the other (9=203.50) emerges on the lower
cap. This requirement is imposed by the relative locations
of the center plug gate valves.

The features required are realized by mounting a one
inch long tungsten blade (pin) off-center in a small copper
cap that is bolted onto the end of a forty inch long
stainless steel shaft traversing the magnet poles. Since the
pin is located off-axis, rotating the shaft in the access
hole results in the desired adjustability in the pin 5

radius (and an unimportant change in its angle). Moving the

shaft up and down by one inch will move the blade in and out
of the beam chamber, while pulling the shaft all the way out
will allow changing the blades. In actuality the shaft is

two concentric stainless steel tubes set up in a spray tube

RF LINER

102

‘l'_Ul1l_G_$Jl'-.'N PlN

 

 

\QAJ.’

 

TRIM Cbll.\

POLE TIP

Figure 4-1.

 

 

The end of the shaft

cyclotron. The

in a copper cap
rapid pin change.

 

\ -

 

\.

 

-M/P
‘x/END PLUG

 

 

 

\

SHAFT TUBE

WATER SUPPLY

\

thER TU§_E_

 

at the median plane of the

sixty-two mil tungsten pin
intercepts the unwanted beam. The pin is mounted

water cooled so the

cooled.

which is easily removed for
Note: the copper end plug is

tungsten will be indirectly

103
configuration (as shown in Figure 4-1) so that the beam end
of the shaft is water cooled. This feature indirectly cools
the tungsten blade which will intercept the unwanted beam.

Control of these functions is provided by a mechanism
located between the center plug and the dee stem on the pole
cap at the exit of the access hole. The access hole lies
beneath the flared portion of the dee stem spinning; the
limited space available requires the compact and intricate
device shown in Figure 4-2. It can be seen in this figure
that the mechanism consists of two major groups of parts.
those below the bellows which are rigidly fixed to the pole
cap. and those above which are moved up and down on a pair
of rails by the pneumatic cylinder. It is this one inch of
bellows motion that allows the shaft to move so that the
blade can be inserted into. or retracted from. the beam
chamber.

Located at the top of the moving section is the
rotating water manifold. Inlet water surrounds the shaft in
the small chamber formed by the first two coaxial O-rings.
passing through the four radial holes in the shaft to reach
the center tube. The water returns from the tip between the
inner and outer tubes. flowing out through another set of 4
radial holes into the the small chamber formed by the second

and third O-rings Just below the third O-ring the shaft

 

 

 

 

Figure 4-2.

 

 

 

104

MSU-u-ZIZ

 

 

 

 

 

 

l
LL13—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Cross section of the phase slit

See

text

for

a description

 

 

 

drive mechanism.

105

diameter is reduced to form a shoulder which rests on the
worm gear. \Nhen the thrust plug screw is tightened this
shoulder and the ball bearing in the thrust plug trap the
shaft axially so the shaft must travel in and out with the
action of the air cylinder. A key on the shaft just below
the shoulder mates with a keyway in the worm gear; turning
the worm gear Causes the shaft to rotate. On the opposite
side of the worm. a second worm gear is mounted on a servo
potentiometer, providing information about the rotational
position of the shaft. The driving worm is driven by a motor
mounted three feet away on the dee stem support beam. Torque
is transmitted from motor to worm gear by flexible drive
cable. This system was chosen so that there would be
sufficient room to shield the synchronous motor from the
high magnetic field.

The ability to change the blades with the minimum
disruption of cyclotron operation requires removing the
shaft without raising the cap or otherwise breaking high
vacuum. The blade-changing operation begins by removing the
thrust plug. exposing a set of threads in the top of the
shaft. Removing the two copper patches in the dee stem
spinning allows a rod to be passed through the spinning and
threaded into the the top of the shaft Pulling on this rod

draws the shaft out of the cyclotron. while the sliding seal

106

 

Figure 4-3. The lower phase slit hole before the trim coil
leads were moved.

107

O-ring maintains the vacuunL VVhen the piston is in the
retracted position the bellows forms a lock chamber
sufficient to accomodate the pin and cap assembly. The ball
valve can then be closed using a 1/4" drive rachet wrench to
turn the the brass ball valve coupling one quarter turn.

Once the valve is closed the shaft can be retracted the rest

of the way. Insertion is accomplished by reversing these
operations.

The parts below the ball valve form the mount for the
rest of the assembly. The base flange is threaded so that it
screws onto the tube which separates the liner vacuum from
the main vacuunL VVhen fully threaded on. it also compresses
the O-ring making the liner vacuum seal. To prevent this
flange from backing off. screws pass through 4 of the 12
clearance holes in it and thread into the pole cap. The
redundant clearance holes were necessary since the point at
which the threads on the liner tube would bottom out was
unknown. Above the base flange is the ball valve body. which
is also the support for the guide rails. Since the guide

rails must have the correct orientation relative to the dee

stem, all orientations were made possible by having the
bolts holding the ball valve body to the base plate pass
through thirty-degree-wide clearance slots to pick-up four

of the twelve available threaded holes in the base plate

108

   

.momam o>ucu udflm
manna 9: co nonomococm 23367:; momma Hmcm>mm mommo noon 5 ans... comm comp
:mo .3 .mc:os mcouon mcoduamoa :9: a“ momma Hfioo ET: on» ho odumeonom < .zi: mcsm:

m<u mm304

 

 

m<u mmmm:

 

109

4.3 Installation of the Phase Slit Hardware

The photograph in Figure 4-3 shows the lower phase slit
hole before installation of the hardware. As can be seen in
this figure and in Figure 4-4, some trim coil leads passed
through the space required by the phase slit drive. This
would not have been true had the leads conformed to the
designed configuration. Therefore the first part of the
installation process was devoted to re-routing the errant
leads. To facilitate this operation both the upper and lower
center plugs were removed. Next the center plug extensions
and the gate valves were removed. After removal of the the
hill extensions. the center plug liners were unbolted and

pulled out as far as the dee spinnings would allow. about

six inches. This disassembly exposed the flare fittings on
trim coil #1. and improved the access to the fittings on
some of the other trim coils. In all cases the leads were

either cut or unsoldered at the transitions between the 1/4"
and 3/8” copper tube. In those cases where the feed-throughs

were exposed. the flare fittings were undone and the segment

of 1/4” line was re-manufactured. For trim coils #3 and #4
on the upper cap and trim coil #4 on the lower cap. the
feed-through lies beneath the dee stem spinning. Here. with
the cut end free. the leads were bent using the modified

pliers shown in Figure 4-5. During this operation care was

 

Figure 4-5.

The special pliers built to allow bending those
leads which were trapped under the dee stem
spinning. Mch a little care they could be used
to move the leads without crimping the lead or
putting force on the feed-through

111
taken to apply as little force to the feed-throughs as
possible, because they are known to be fragile.)Nith a
little practice this bending technique proved to be quite

efficient and simple. After the bending was complete the

leads were re-insulated. the transitions were soldered
together. and the leads water tested. At this point we also
cleaned all the feed-throughs and repaired any damaged

insulation on any of the other coils. (This work seems to
have removed a short that existed in lower trim coil #1.)
\Nith the trim coil leads in the proper locations. the
phase slit base plate could be screwed on and the center
plug liner re-installed. thus restoring the liner vacuum.
Next came the task of making the access holes in the dee
stem spinning and spinning flanges. Location of these holes
was accomplished with the special tooling fixture shown in
Figure 4-6.\Nhen the rod with the sharpened point was passed
through the pole to the median plane. striking this rod with
a hammer created a locating mark on the spinning. Using a
right angle drill and a 19/32” bit. a hole was drilled
perpendicular to the surface of the spinning and centered on
the punch mark. Using a hand file. the hole was enlarged in
the vertical direction so that an oval shaped hole was
formed which would allow a 9/16” shaft to pass lhrOugh

freely. The holes in the upper flange were drilled with a

Figure 4-6.

 

 

 

       

   

 

 

IIIMIIII I a

   

\ ‘ 'llllll. '1’;
\ ‘ \ ‘

\\Ԥ1

  

   

"I/A:

   

/

..

 
 

W
4,

   

\

      

 

 

 

 

 

 

 

 

 

 

i .
It I
H
ll ‘
N (i l
1" main '
u
ram \
NW \ ‘
F“
+ I] I
‘ '/ 1 l
‘ r‘ll l :
. I!
.1 .
/ -
// a ' ‘—-_\
- --., . 4; , —-'—* ,
\\ , £ ”fin: §\\$\T
K ’ ’i‘” x.“ s\.r~1?x\\
The fixture used for locating the hole in the
dee stem spinning. The other end of the
indicator shaft was at the median plane so it
could be tapped with a hammer to mark the
spinning with the sharpened point. The drill
bushings provided alignment of the clearance
hole and the two threaded holes for mounting the

air cylinder on the hex flange.

113

portable drill press. while the lower ones were made with a
hand drill motor. In both cases the drill was guided by the
drill bushings in the locating fixture.

in order that the half-inch shaft might pass freely
through to the median plane, the bead formed where the liner
tube is welded to the liner was cleaned up with a
combination of oversized hand reamers and a file \Nith this
work complete. the drive mechanisms could be mounted and the
shafts installed.

After the motors were installed it was found that
there was excessive wind-up in the flexible drive cables.
This difficulty was resolved by changing to a larger
diameter drive cable and improving the alignment of the worm
gear and the worm. “Nth these modifications the drive
performed quite well.

To calibrate the position of the pin. the fixture shown
in Figure 4-7 is used. By placing the U shaped slot over the
dowel pin in the dee locating fixture and rotating the phase
slit shaft until the cap fits onto the the end of the shaft.
the location of zero degrees can be found.

The complete installation of the phase slits excluding
the drive cable modification and alignment took close to two
weeks to complete Over half of that time was devoted to re-

routing the trim coil leads

 

 

 

 

Figure 4-7.

1"

The f
drive.

ixture used for setting

The cap on the end

the shaft when the
removed. The notch

center
shaft

 

the angle

normal cap
the post
rotation

fits over

locating fixture so the

iS determined at

either 00

O 1’

fits over the

with

1800.

of the
end of
pin is
on the
of the

115

 

Figure 4-8. The lower drive mechanism installed. See text
for a description.

116

4.4 Construction of the Viewer Port Probe

During commissioning of the K500 cyclotron. a
relatively simple two jaw radial probe was installed in the
extraction region. The success of this device led to a
decision to construct a more sophisticated model which could
be removed under vacuum and would increase the travel from
4" to 12". After extensive design and assembly work in 1985
such a device was installed on the cyclotron in November of

that year.

Figure 4-9 is a schematic view of the probe as
constructed. During. normal operation the lower table is
locked in the innermost position and the upper table (a
precision slide table) is moved in or out with a ball screw
driven by a stepping motor. In this 'running‘ condition the
moving vacuum seal is provided by the stainless steel

bellows mounted between the front plate and the upper table.
To remove the probe the lower table is driven to its outer
limit so that the end of the guide tube is in the lock
chamber. and then the gate valve is closed. During this

operation a pair of viton O-rings separated by a floating

spacer ring (all labeled sliding O-ring in Figure 4-9)
provides the vacuum seal. This division of seals means that
the bulk of the time (during regular probe travel) the

bellows are used. but for changing the probe. when the

117

_m>oEw_ 0cm co_..mmc_

cod com:

on o. mmc_..0 ucm _m>mc_ .o poem. .m_:mm. on“
c. com: on o. mso__mc a .o. 30..“ mEm_m>m m>_cu

m_m.mamm oz. och .m.cmcanoo .o_mE cc“

oc_305m

m>_.u mnoca ~coa cm;m_> so: on, .o o.~mecom < m.w m_:m_u

 

 

 

>

 

 

L

 

WJodh CNBOJ

 

 

 

 

Eul

 

 

|II|||||

 

uqm<h mutt:

 

 

 

u>i_<> uhdo

uncan—
wh<4a w3044m¢\

mBOJme

wh(4n hzoxu
.umDh mo.3w

02.10 02.0.Jm
xwmz<xu 1004 h<hmo>mu

.rDo QZDQ ZDDU<>

mxo >

118
travel is 40". the O-rings are used. The lower table is
mounted on Thomson pillow blocks. and driven by a ball screw
attached to a synchronous motor. A complete probe change can
be accomplished in half an hour. making repairs and
modifications relatively simple.

Another important feature of this design is the ability
to accept any probe that meets the following simple
requirements: that it match the bolt pattern on the plate.
and that the maximum diameter from the plate inwards be
5/8". The length of the probe from the mounting flange to
the tip may be anything from 45.5" to 59.325”. Presently
four different probes have been constructed. The first of
the tested designs is shown in Figure 4-11. The object here
was to provide isolated water and electrical circuits within
the limited space It also provided fairly good positioning
of the tungsten jaws to ensure the correct differential

lengths. Unfortunately the Kovar feed-throughs were poor

thermal conductors. consequently after beam hit the probe
for any length of time the temperature of the jaws rose
dramatically and eventually the feed-throughs became
electrical conductors. The design currently in regular use
is shown in Figure 4-12. This one is the same as that
orginally used with the 4" drive but with length changed

Also the jaws are now a single piece of molybdenum whiCh

119

.0r
.1 i
101

. . O
.OMOQHNO

‘.
9".

 

the

installed on

e new drive
this photo

h

photo of t

A

Figure 4-10.

he
been

I

i n

S
a

h

the drive

In
position

cyclotron.

S

a probe

and

r

'running

installed

120

.omucmomfiu mm: cmfimmo ads» on mcfiuozccoo xdamOMLUomHm
mcdeoomn Lawn» :H mcfiuasncc .mLOuosccoo Hmscmnu coom zaucmdoquusm
go: one: accumasmca Lm>ox one .m>fico so: on» no“: com: mnoca uncuu one ._T;~mcsmfim

34... zwkmozahJ
Iozomrhiowmu m<>ox

...? :0: .LT;

gym?

I . / . \.\
IP34) n‘v fig“ fig» ark l i. if
253:... 35:... . . .

OJiZJQ any
thing; {3)-};

 

\Tx

2394.6 (.3
/

  

     

\

 

 

 

 

 

 

 

 

 

 

   

G. .-
zmmwum_o

31... 44....

o .‘ e

z’v
/“

 

 

 

 

 

 

121

.uosnoc >Lo> on on
no>oLa no: :mfimoo mace .oo: cmflsmoc cw >Hucocczo onocq Hofiucocouufio 3m“ m one .m_i= ocsmfim

 

media mama

 

 

 

 

 

 

 

3 <.< 79kg
afrwum. P12 .035.)
A1313 09:.
90.9.4530.“ fllvriv . .
/ 310an 3. 3.
t
. h.-
.34
.nw .

 

 

 

 

 

 

 

 

 

 

 

 

. V \
fit _ i'v ,q//
> It“. > a . e Tc.
u: *Jw-._.-.~.i.s_...... H H. Mini . @ e e -
Air . l . , e e
.3, \ _ 2 m_i'.v .e 9/ \
24mm 4.0. U_ii' /<\

122
should improve the cooling while at the same time reducing
the possibility of producing alpha emitters. One of the
other probes developed is being used as the target for the

radioactive beam experiments of M. Mallory et 3.39

4.5 The Gamma probe

The fourth probe. which has been constructed for use in the
VP probe drive. is designed to measure the phase of the
beam. This is to be done by detecting the gamma rays
generated from the beam striking the probe tip with a PIN
diode located just behind the probe tip. A schematic of the
phase probe is shown in Figure 4-13. At the exit of the
probe the signal is again amplified, and then sent to the
control room. In the control room the signal is fed into a
constant fraction discriminator (CFD). The output of the CFD
is then used as the start signal for a time-to-amplitude
conveter (TAC). The stop signal for the TAC is a pulse
generated at every second positive zero crossing of the RF
signal. Dividing the stop signal by two means that all the
features in the time spectrum will appear twice. 360O apart.
This then gives an immediate calibration between channels
and degrees of phase. without any worry about cable length.

The TAC output is digitized using a multichannel analyser

123

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PIN DIODE
PREHAMP
[CABLES B M
I 1 I
I f
/ if 1’
\
4X
\ \ X
CIRCUIT TUNGSTEN TIP
BOARD PROBE TUBE

Figure 4-13.

A schematic drawing of the phase probe. The PIN
diode is used to detect gamma rays produced
when the beam strikes the probe tip. The small
size of the diode and amplifier allows it to be
located near the probe tip so the count rates
are high and the source is distinct from the
background.

124

 

15000 i

12500

10000

7500

COUNTS

5000

2500

[IIIIUIIIUIIIIIIIIIUTIIIIUU

 

...11....l....l...Julia...

IIIIIIIIITIIIIIrj'IIUrrIIYI

 

 

l<e 380' —4>

 

 

 

 

 

 

 

D

Figure 4-14.

100 200 300 400 500 600
(fiflUflflfl.NUflBER

A typical spectrum of intensity versus time.as
measured with the gamma probe. Notice that the
divide by two of the RF stop signal causes all
features to appear 360O apart

125

for a preset length of time. A sample output is shown in
Figure 4-14.

If the peak width in a spectrum. such as that in Figure
4-14, is to be attributed to the phase width of the beam. it
is necessary to measure the time resolution of the PIN
diode. This proved to be more difficult than expected.
because the diode's small size resulted in too low a count
rate for a coincidence measurement.lNhat was done. was to
measure the rise time of the diode when pulsed with a fast
laser. These measurements concluded that the rise time was
better than 500 ps. In Figure 4-15 the time spectrum of the
external beam. as measured with the PIN diode. is compared
to a measurement of the same beam made with a BaF detector.
The time resolution of the BaF was known to be better than
300 ps. and the two detectors produced very similar results.
Given that the gamma probe can make a reliable determination
of the phase. it can be used to measure the phase width of
the beam between 20" and extraction. As well it can be used
to measure changes in the phase as a function of radius. but
the determination of the absolute value of the phase will
need to be done with a different technique. such as

frequency detuning.

126

BMI 1‘14” 14.6187MHZ 8/6/86
1500 I I I I I I r I I T I I I I I. f I I T T I I I

 

-I

 

1250 - BaF

— — "PIN DIODE
1000

750

COUNTS

llllLlLLllllllllllll

 
  

500

250

 

 

 

IIIITfifirIlIIIIlll—tfirtiriI11!

 

40 50 00 70 00 90
CHANNEL NUMBER

Figure 4-15. A comparision between the PIN diode and a BaF
detector. Both detectors measured the same
beam. which was striking beam stop 1. The
measured beam widths are very Similar.

5. Experimental Results

5.1 Frequency Detuning

The phase history of the beam is a very sensitive
function of the magnetic field, thus an independent measure
of the phase would verify the procedures used to compute the
magnetic field. Several methods are available to measure the

phase. but the frequency detuning method first proposed by

Garren and Smith31 is the most straightforward. The

separated longitudinal equation is.

'i

. _ . 2vh Eb - _
Sin ¢ _ Sin ¢0 + —EE J ( u 1 )dE (5 1)
O
. 2wh
=Sln ¢0+6—E—'F(E)
Er
= 90 '

where. F(E) J ( c) 1 )dE

0

and 6E is the energy gain per turn.

Since F(E) depends only on the energy and not the initial

value of the phase (do). the width of the beam in sin dIE)
is constant when equation 5-1 is valid. If the phase of any
part of the beam reaches : 900. it decelerates inward to

127

128
the machine center and is lost. If the central ray of the
beam reaches 900 at a given radius this can be observed on
an integrating probe as a reduction of the beam current to
half amplitude at that radius.
Rewriting the longitudinal equation for a new RF

frequency w’ gives.

(w'- 9b)
2 h
+ a; “b { F(E) + E L (5-2)

 

Sin¢'(E) = Sin¢(E)

where ¢'(E) is the phase obtained with the new RF frequency.
and ¢(E) is the phase at the operating frequency “b-

Defining the frequencies w; and w. as the frequencies which

drive sind to +1 and -1 respectively. equation 5-2 becomes.

 

 

 

2”(“h - uh) I '
+1 =sin¢(E) + l F(E) +51
6E db
2n(ci - cfi) ‘ ’
-1 = sin¢(E)-+ 65 w i F(E) + E r
0
Let 6+ = (c4 - c%) I “b and we find the expression for sine
is.
sin¢(E) = (:1 2gE + 6+ sin¢0)/(1 + 6:).

. . . . 32
USing this formulation instead of the more common

129

2 “b - cE(r) - c{(r)
cu(r) - c{(r)

 

sin¢(r) =

means that the phase can be found at radii for which either

“i or “i cannot be found. The trade-off is that the energy

gain per turn 6E must be known.

Ideally. determination of the phase using this method
requires only that the machine parameters (RF voltage.
magnetic field etc.) are stable in time. In practice however
there are several limitations. First of all the centering of
the beam affects the results by shifting the radius at which
the half intensity is observed. As is well known it is not

. .. 33
pOSSlble to center the beam at all radii at once . so

usually the best centering is reserved for the large radii

where the turn density is greatest. In the K500 even this is
not possible. because. in order to minimize beam losses
during the resonance crossings at the outer radii. it is

necessary to run with the beam off center for most of the

machine34. Another difficulty arises because the main probe

in the K500 is a complicated set of train cars that travel
on a spiral shaped track located on the center of one of the
hills (see Figure 1-1). Because of its complex nature.
determining the exact location of the probe is difficult. so

the position read out can be wrong in some places by as much

 

1.0 -— I I l I l g
~ 53 MeV/A ‘3.“ 4
L + MEASURED -
- - COMPUTED -
, .
0.5 l— _.
r- -i

 

 

 

 

 

A P
3 l—-
E :
E .
L- _.
- -i
-l.o : 1 4 J l l I L l l 1 l l J 1 Pl 1 L I l l I l L 1d
0 5 10 15 20 25
3! I... (inches)
1.0 L—T—r r I I I I I r I I I 1 I l I I T I l T T I fir r—
- 35 MeV/A “N“ -
- + MEASURED -
- —- COMPUTED -
- COMPUTED-0.1 AMP .-
o.5 — e.-
P
S l"-
a b
m -

 

 

 

 

R prob- (inches)

Figure 5-1. The calculated and measured phase curves for two
different magnetic fields. The dashed curve for

the N5+ case is the calculated phase curve when
the mail coil currents are changed by 0.1 Amps.

131
as 0.5”. If the phase curve changes rapidly, as it does at
large radii, the frequency required to drive the phase to

900 becomes a constant. For example in Figure 5-1b the c)

for radii larger than 26.0" will be a constant because the
current will always be lost at 25.0"

In Figure 5-1 the results of measurements in two
separate .magnetic fields is shown. The agreement in both

cases is quite good. except for a sharp bump in the N5+

case. Either this discrepancy is caused by a probe
calibration error or it is a result of a change in the main
field. During this particular run the main coils currents
drifted by as much as 0.1 amperes. as the lead temperatures
changed. Unfortunately the problem was not detected until
late in the run so there is no way of determining when it
happened. Since the data was taken in sequence it is
possible that the bulk of the excursion took place as data
between 14" and 12” were being collected. The dotted Curve
in the figure is the phase curve If the coil currents are

reduced by 0.1 Amps.

5.2 Phase Selection
The previous chapters have discussed the phase

selection hardware and how it is intended to achieve its

132
goals. In this section experimental results of a test run of
the phase slits will be presented. As described In 3.2 the
coarse selection is done with a slit located on the dummy
dee 'between the puller and the second dee ("C" dee). The
only way the aperture of this slit can be changed is to
raise the cap, which is a one day job. As a result it is
difficult to compare directly beam conditions with and
without the small slit. In Figure 5-2 the output from the
gamma probe. collected when the coarse selection slit was in
place. is shown. This can be compared with the computed
results in Figure 5-3. The computation was done by tracking
particles through the axial injection system and around the
first turn in the cyclotron. The initial beam filled the
available aperture and any particle which hit either the
inflector (including the collimator) or a dee post was
removed from the beam. This calculation predicts that the
phase width after the first turn would go from 430 to 140
(lfiNHM) when the narrow slit is installed. This is in
excellent agreement with the )-probe measurement done at

extraction radius (Figure 5-2). that gave a result of 14.80.

Figure 5-4 is shown for reference. It gives the results of a
calculation similar to that shown in Figure 5-3. but with
the collimator aperture reduced to Inmm VVith a 1 mm

collimator the Starting emittance is reduced to 25v mm-mrad

133

 

10000IIIITIIIIIIIIIIITIIIIIIIIIIITI

0000

 

A¢=14.e°

COUNTS

L

r
8000 r-

r-

4000

 

.
i-
-
2000 —
P
h
b
_

 

 

o lLlllll llllllllllllLJ lllj

460 470 480 490 500 510
CHANNEL NUMBER <- 1’

 

Figure 5-2. The phase width of the internal beam as measured
with the gamma probe at extraction radius when
the narrow first turn slit is installed

134

FULL EMMITTAN CE

 

100

S

8
IIIIIIIIIIIIIITIIIIITIIT

PERCENT SURVIVING
AFTER FIRST TURN

 

Figure 5-3.

IIIIIWITIIIIlrIII[IITIIIITIII

O
‘r
b
h
h
h
p
_
h
b
h
—
h
"/

‘--S|IALLSLIT
————LARGESLIT

 

 

 

 

220 230 240 250 260 270

"’0 (098)

A calculation of the percent beam which survives
the first turn in the cyclotron for the wide and
narrow first turn slits. The inflector
collimator is 4 mm in diameter.

135

REDUCED EMMITTAN CE

IIIIIrIIIIIjIIIIII'WIIIIIIIIIII

 

 

100

40

 

PERCENT SURVIVING
AFTER FIRST TURN

IFII1TIIFF7IIIITIII—FITIIII

 

 

 

To (DC!)

Figure 5-4. A calculation of the percent beam which survives
the first turn in the cyclotron for the wide and
narrow first turn slits. The inflector
collimator is 1 mm in diameter.

136
as was discussed in section 3.3. At the time of the test run
(when the first turn slit was installed) there was an
alignment difficulty with the spiral inflector and so it was
not possible to run the reduced emittance case. but in the
future it may be possible to do so.

\Nith the coarse selection in place the differential
traces (Figure 5-5) showed a much more pronounced turn
pattern than usual. The large periodic structure is a result
of a coherent oscillation. In Figure 5-6 the computer
program SOMA has been used to estimate what a probe plot
would be like. given the calculated orbits. For this case it
was assumed that the phase width was 140 and the starting
conditions were an eigen-ellipse. The area of the ellipse
was chosen to be 100v at the inflector entrance. and the
center of the ellipse was located using the prescription of
Figure 2-3. The computed and measured turn patterns are
similar in all aspects: 1) the turn spacing is the same.
indicating that the energy gain per turn is close to the
calculated value, 2) the space between turns is reduced in
the same regions. 3) the region of bunched turns due to the
coherent oscillations occur at similar radii. The last point
indicates that most of the centering error is induced by the
central region (in the calculation the centering error is

built into the initial conditions). since the field used in

137

PRO“ MIN 00?

PLOT 227. comm SM-OO V9,)! 3” 120.26! K “.279 mu: 2..-2..

 

 

I it
. [I ' II. ‘ ..i.!)\--"r“'-.: v'
1. iii 1 9 9‘. I It 37 I {I :‘i :I ii. i. {I it 1 {i 'l 1”.“ II A H Ilif": i‘ig‘-l}’v'Q--".~fu‘v‘ - --~~~
; "I I- my") ,3 3;. f‘- i {it} St- 1L} 5 I ‘i :I’hji .115 .11!!! lint} 1' r Iii'l-Higi’ iii-9‘9"“ i" .
jg; :9; ‘(uil iv! It} '1}, {If U It} “‘3 lg 1 *‘3 ‘d ‘u‘ ‘-.I ’5} " ‘4' ’f I Kr; v L fig ‘1 (I
V 'L L 1 - L 1 L L g 1 L 4 1 1 A g 4% L L L n "
U c T I v I.
PIOIC WING OUT PLOT 22?. “306820 3-906-06 9 I In 120.261 "It “.279 W! 200,-200
. . 9" I
if . I " .k
-4 ii fitttr; “11g . ..- r“
W-EMEA’“IEV‘I'RRS..2: f ’. 9 9. 9. 11:9,}; (9)319) I‘Ih“1W"A"-"NJ'LUAK\ H939. 19'. it's .
f ‘91.»?§'1:.1"Er‘!§-.'33’i!i'£i'3'“! I . i, . '1 . ,
1 \ ‘ I I J
‘ l L 1 I? l L l 1% 1 L L I: l l L ‘1‘ A L {as
Figure 5-5. A differential probe trace. taken with the
narrow first turn slit in place. The cyclotron
was tuned for good extraction.

lS-SEP—BB 04:56:51

PROBE -- 2

 

40

 

 

 

 

 

; f r I r I r r I r R‘ r f i T I T r T T T V T l {I
[_- : _4
m ..
05 Z
23' 20 :-
U E : 5 ‘l‘
g 10 ‘ u I '
a ' .
N O J 9 ' ‘ ' 7 , . i 1 1 l I l 1 1 J
5 ' 6 7 3 9 10
RADIUS (inches)
5 <1
2:1 -i
n:
m .3
:3
U
:3 ‘9
D
b I
V- L-.L.__L-_L__1.WI_LL._L J L 4 J g i I 1 L i i I L i_ L L _

 

10 12

RADIUS (inches)

Figure 5-6. A differential probe trace. calculated with the
program SOMA. Note the similarities in structure
to the actual probe trace of the previous

figure.

138
the computations does not contain the imperfection

harmonics. Attempts to center the beam with the center bump

usually result in large beam losses. probably from the
reduced stability region which results when the bump coil is
not cancelling the first harmonic at v = 1.

r
As described in section 3.4 the fine selection is

accomplished with a pair of tungsten posts whose positions

are adjustable. If proper selection is to take place. one
must find the locations at which the posts lie between
turns. Figure 5-7 shows the measured and calculated current

distributions at the upper post azimuth. The arrows in the
figure show optimal post locations. The computations were
done with the program SOMA. using the differential probe
option with the probe width equal to the post thickness. The
measured points were obtained by placing the main probe at
310." and recording the current as a function of slit
position. Since the only possible loss mechanism is the
post. the current lost must be the current hitting the post.
The values for the lower slit are shown in Figure 5-8. The
reduced amplitude variation at this location is caused by
the increased turn width at this azimuth. The plots of
Figure 3-10 show that at the lower slit position (6:2040)

the radius changes much more rapidly With phase. than at

139

 

 

 

 

UPPER SLIT
100 ' ' ' l I I r l l r T r
l l I .
‘ + MEASURED d
- - cucunmzo j
80-- ' __
- l + I .
- + + q
60*—- _a
. + .
é - + Cl
0 + 4
4.
lg 40 +4, + +++
R ' A
20-—- _s
" -i
o J L l J 1 1 I l l 1 l l l l 1
69 7 71
RADIUS 0F POST
Figure 5-7. The current intercepted by the upper slit as a

function of its radial position. The calculation
values were obtained with the code SOMA.

ZFUILCURRENT

100

40

Figure 5-

140

 

 

 

 

I l’ I T f r I 1 I I 1 I I T— I l I T T
c + MEASURED ‘
: - CALCULATED I
l- + ‘1 -I
... + + + + + ' -"
: ++ + + + + 1
.. + ++ + + -
i- d
r- l l L l l l l l l l J l I l l l l l l .‘
6.9 7 7.1 7.2
RADIUS OF POST
8. The current intercepted by the lower slit as a
function of its radial position The calculation
values were obtained with the code SOMA. The
poor agreement at lower radii is due to an

encoder mal function.

141
9:840. and so the turns are more smeared. The poor agreement
between measured and calculated results for the lower
mechanism at smaller radii is a result of an encoder error
(later measurements indicated the device is non-linear).

Another method of determining the optimum location for
the posts is to look at the gamma probe data. The Z-probe
output for two different positions of the upper slit is
shown in Figure 5-9. “Nth the upper slit at 71 (arbitrary
display unit) the beam is being split by the post. As the
post is moved to 125. more of the earlier phases survive the
post and some of the later ones are removed.lNhen the upper
post is in position 125 the beam is being scraped from the
outside of one turn and the inside of the next turn as it
should.

The phase width with the upper slit at 125 is compared
to no slits in Figure 5-10. Mch the post at this location
the lfiNHM was 5.20. The same procedure was then followed for
the lower slit. The reduced turn separation at 9:204 (lower
slit) made placing this slit a more difficult task. but it
could be used to reduce the shoulders on the peak in Figure
5-10. as can be seen in Figure 5-11. The phase width with
both slits inserted was 4.20 (FWHmM. Table 5-1 compares the
beam current on beam stop 0 (after extraction) with and

3

without the slits Re-examing the calculations of chapter a

142

 

 

 

 

 

8000 I ' r ' I ' j' l V I r I l I I r T r r
_ -l
. ‘ .
aooo — I \ — — — PSU-125 —
” \ ' pan-71 ‘
- I \ .
' I \ .
u I \ g
g 4000 —' I \ —l
o. F I \ ‘
I)
i— I \ 4
I- I d
i- , \ .4
2000 "" I \ \ _-
T / \ n
- \ 4
/
. , ‘ / \\ a
i- \ / \
o ‘ ’1 l l L l_ l l l L I l l I 1‘ }
470 400 400 500 510

CHANNEL NUMBER <— ¢

Figure 5-9 The phase width as measured with the gamma
probe. for two different postions of the upper
slit

143

 

 

 

 

 

10000 _I I I I I I f I I I T I I I I I T I ' I I I I I ' I I 4

~ +

0000 — —comss -+

- — — PSU=125 -

l- I -4

0000 i- ,\ —

8 [ A¢=14.0' - I’ 1 — 2

4000 -- I ( .4

_- ‘ d

t I - A¢=5.z' .

2000 -- l '\ __

- I \ -

.. \ .

- “ I, \ 4

a P1 L l l l l /l,l J ‘1’ I l l I 1 l 1 L T T , q,
400 470 400 400 500 510

CHANNELNUIIBER <-¢

Figure 5-10. A comparision of the phase width at extraction
with and without the upper slit. In both cases
the narrow first turn slit is present.

144

 

0000 I

0000 _-

4000

COUDHHI

2000

 

Figure 5-1L

 

 

‘ ui
I \ — - - PSU=123 —
I’ ‘ PSU-123 ‘
, x + PSL-IZI :

 

 

 

400 490 500 5t0
CHANNEL NUMBER <—¢

The phase widths at extraction with one slit
inserted. and the combination of two slits. The
extracted current is reduced by a factor of two
when the second slits is inserted.

145
it would appear that if the lower post had a smaller
diameter. the phase width could remain the same but the
amount of beam surviving both posts would increase.

In Figure 5-9 the main peak is accompanied by tails on
either side and this substantially increases the full width
of the peak. As may be noticed these tails were not
predicted by the computations shown in chapter 3. During the
experiment the initial emittance was 1000 mm-mrad. while the
calculations assumed a starting emittance of 25w mm-mrad. To
see if this increase in emittance could account for the
tails a SOMA calculation was preformed. The calculation
began at turn 8 with a phase width of 200 and an emittance
of 26.5 mm-mrad. As seen in Figure 5-3. when 1007 mm-mrad is
run with the small first turn slit. the slit also acts to
eliminate some of the horizontal emittance. To simulate this
effect the dashed curve in Figure 5-3 was used as a
weighting factor for the different starting times. Figure
5-12 compares the result of weighting the SOMA output to the
measured phase width. As can be seen the weighting results
in a very close agreement with the experimental data. Again
using this weighting procedure the calculation was preformed
with the upper slit in two different locations. These
results are presented in Figure 5-13 In both cases the

calculations predict the locations and relative heights

146

 

 

 

 

30 - I r I r 1 r I I t I I I I r r r 1
g -
25t- tnemnuwd —~
I calculated 1
20 —- __‘
E :- ‘
E 15 F— _‘
10 -- _j
C :
5 _ —
O I. l l l l l l l l l
0 —4o —50

 

PHI (deg)

Figure 5-12. The weighted SOMA calculation is compared with
the measurements. Only the first turn slit is
in the machine.

 

25 I

20

15

 

T I I If I I r I I T I I I I

--CAUflHJIEDIu=6£B
MEASURED PSU=71

 

llllljlllllllilllLllllLi

 

 

 

25

20

15

flflENflTY

 

—— CALCULATED RI=7.035
""" MEASURED PSU=125

LAJAljllillllllljlllllli

 

 

Figure 5-13.

 

PHI (dog)

The weighted SOMA calculations are compared to
the mesasurements for two different locations
of the upper slit. Note the relative
intensities of the peaks. These calculations
assumed 1000 mm-mrad initial emittance

of the peaks.

calculations is

between peaks.

this. One

located at

7.3” there has

possibility
the initial
that unbalanced

the the turn

Nevertheless the

emittance Will

that
There
possiblity
extraction

been

emittance are not

dee
separation at
similarity

remove

148

the calculations
are several
because

is that

radius. while

some smearing in
correct.
voltages changed
the slit

suggests

the tails.

tend

It

that

the

to go
possible explanations

detector

between.

The one difference between measurements and

to zero

for

was

the calculations end at

Another

is that the various starting conditions such as

is also possible

reducing

the centering and so

has changed slightly.

the

 

 

 

Table 5-1 The phase width and the extracted beam current at
beam stop 0 for different combinations of slits.
SLITS PHASElNIDTH CURRENT AT 880

Coarse only 14.80 46 nA
Upper at 125 5.20 20 nA
Lower at 121 20 nA
PSU=125. PSL=125 4.20 10 nA

The calculations shown in Figure 3-10 suggest that the

phase widths should be 7.50 (FWD for the upper slit alone

and 4.50 when both the posts are inserted. which is in good

agreement with

the 5 2”

and 4.2 obtained experimentally.

149'

 

l 1 1 1 1 L 1 1 1 1 1 11 1 L 1 1 IL . 1
3r 1‘ ' I. I9 20
PRO“ m! N? PLOT 4.? 11012. 12-80-06 0! In .313 in 143,170 I 217,-!”
I '. i, I
’ ~/

My

1 1 1 1 1 L 1 1 1 1 L L L L 1

IS 1‘ 17 I. I I! 20

 

 

Figure 5-14.

A differential probe trace taken during a 22N8+

35 MeV/A run. The wide first turn slit is
installed. The radial focussing frequency can

be determined from the coherent oscillation
which is visible as the large amplitude
oscillations.

 

 

 

 

 

 

1.10 I I I I T r I I I I T I I I I I I I T I I I I l I I I ‘
- ”Ne" '35 MeV/A ~
1.05 (_- + _
l- .i
‘ LOO—— -
a ,. ..
r- -t
. l
0.95 -— —-
" l l l J l ‘
0.90 1 J 1 1 1 l 1 4 LL 1 1 1 1 1 1 1 1 1 1 1 1 1
O 5 10 15 20 25
R"... (tnchos)

Figure 5-15. The radial focusing frequency as calculated
with the equilibrium orbit code. and the values
obtained from differential probe traces. The
horizontal bars indicate the region over which

the value of vr was averaged. The vertical bars

indicate the possible error in determining the
number of turns in a precession cycle.

150
5.3 Radial Focusing Frequency
The differential probe plots of Figure 5-14 show a
large coherent oscillation. This coherent oscillation is
produced by the initial centering error mentioned in section
5.2. The radial focusing frequency is related to the number

of turns in a cycle. N . by:

As can be seen in Figure 5-14 the number of turns in a cycle
cannot be counted directly, because beyond 10." the spacing
between turns is comparable to the thickness of the
differential wire on the probe. Since we do know the
relationship between energy and radius. we can estimate the

number of turns in a cycle using.

 

E2 - E1
Nr = 3(q/A)Vdee Cos(¢) ' (5'3)
Given in Figure 5-15 is the computed vr versus radius for
the 22Ne8+ 35 MeV/A field. along with the experimental
values. The horizontal bars indicate the radial interval
over which the number of turns is calculated. and the
vertical bars indicate the error in the average vr value

arising from the possible error in determining the radii.

Each data point is determined by measuring the radius of the

151
beginning of a precession cycle and the radius of the end of
the cycle. Then an equilbrium orbit program is used to
determine the energies for the two radii. and the average

value of cos(¢) in this range.

5.4 Axial Focusing Frequency

The head of the main probe is divided into three equal
vertical sections. each 0.25" high. For these measurements a
large coherent oscillation is induced by placing a .455"
shim under the spiral inflector. thus moving the position at
which the beam exits the inflector off the median plane.
Plotting the current on each of the sections as the probe is

drawn outwards gives a profile of the vertical position of

the beam spot. In Figure 5-16 the current on the center and
lower jaw are plotted over a 10" radial range. As can be
seen in the figure the current moves back and forth between

the two jaws. The negative values of the current result when
electrons are knocked from the jaw. since the electron

trapping system is only effective on the sum of the three

jaws. From plots such as Figure 5-16 the number of vertical
oscillations over a radial range can be determined. The
number of turns in the same range can be determined using
equation 5-3. The number of turns in a complete cycle of the

axial motion is related to the focusing frequency;

152

N
M|-a

N=1. ifv<
V

The result of these measurements in the 14N4+ 20 MeV/A field

is shown in Figure 5-17 along with the computed values. The
horizontal bars give the radial range over which the average

value of v2 is found. while the vertical bar gives the error

which arises from incorrectly estimating the number of

cycles in the radial range.

153

 

PRO“ NOV!“ 00" PLOI ”2‘, C. 123 8‘2 34106-06} 21,?3: MI! I ?.25| M: 36.27, “St 2”.'2..
14' H E; 9;? 5‘2 {‘1‘}. ii i I. l :1: Ii}-
i I. I' I 1 -; ' 1‘3. 6 :I I 1‘ I_ a | -! I. I i‘ It {I} \ P!
C lsx" j i | f}. :2 p i f '9: ’3‘: 1': I I '3 .5 :1 i 1“: z . :1 i 3. (J I
- a fit It if . tifiii, g: tuiw.tig (#49 gags
: I q! I I? I A£ I ‘- . . : 1 3 1i 1 ‘,f ‘,t ' ! r3, 1 -1 i I L
3' 1 A'té '1‘1 ! I I, u £51112; H. '“1 5‘; I (J
- . ‘i.qvuv
I [It '
r .-zu

"inn"... out I f IAEI.I.II&:)%I-V ‘737'1’1’

 

Figure 5-16. The current hitting the center and
the main probe as a function
coherent oscillation is induced in
the inflector 0.445”.

lower jaw of
of radius. A
2 by raising

 

I I I I l I I I I I I I I I l I I I I l I I I I I I
1.0 :— “N“ 20 how/i
0.8 3-
_ 0.6 :-
3 I
0.4 '—
0.2 _—
0.0 - l l I l l l l I J. l I l l l l l l J l l l l I I l

 

"llllllllllllllllllllllllllll

 

l l

 

0 5 10 15 2O 25
R"... (Jnchos)

Figure 5-17. The values of the axial focusing
computed by the equilibrium orbit

values obtained from probe traces such as

one shown in Figure 5-14.The ho
indicate the region over which th

was averaged. The vertical bars
possible error in determining
turns in a precession cycle.

frequency as
code. and the
the
rizontal bars
e value of vz

indicate the
the number of

6. Conclusions

If the sucess of a computer program lay solely in its
speed, then SOMA will be a huge sucess. In a run such as
those done to produce the plots shown in chapter 5. 300
particles were accelerated for approximately 90 turns. using
a total CPU time of 15 min. By comparison, a rule of thumb
for the orbit integration code SPRGAPZ is, 1 minute to run
one particle 100 turns. That is to say a similar run with
SPRGAPZ would have used approximately 270 minutes of CPU
time. a factor of 18 increase. Of course speed is not the
only determining factor. For example flexibility is of great
importance. Currently SOMA has several 30phisticated input-
output (IO) routines. that make the handling of large
numbers of particles much easier. The bookkeeping alone
would make an orbit integration code run with 300 particles
a tedious task. A case in point is the probe option; it
reduces the problem of determining the patterns that would
be exhibited if there were a given set of orbit conditions.
to a trivial problem. mfltrt the orbit codes previously

available. this was a difficult undertaking. and it seemed

154

155
as if one could never run enough particles to get a truly
smooth curve.

As with all new programs SOMA will need constant
upgrading for some time, as users request more specific
features. As mentioned in section 2.6 the equilibrium orbits
as calculated between gaps do not close perfectly. and this
could be improved by adding an extra iteration to the E0
search. Last. but not least. the IO routines could use some
cosmetic improvements. In particular. software needs to be
developed for the plotting of the calculated probe traces.
as the present system is cumbersome and slow.

A transfer matrix program such as SOMA lends itself to

the study of several sets of phenomena . two that come to
mind immediately are an investigation of beam conditions
that lead to better extraction. and an investigation of
centering. Obviously for the centering study SOMA will not
be useful for the first few turns. and care must be taken

when displacements are large. Nevertheless centering often

takes over 50 turns. and is phase dependant. so the
increased speed of SOMA would be useful. It may also be
possible to use SOMA. along with other programs. to

determine the centering error from a probe trace. The

situation for extraction studies is very similar. Again a

156
large number of particles needs to be run since the radial-
longitudinal coupling is very important. Also the region of
interest (the two resonance crossings), requires close to
100 turns, so the turn number times the number of particles
is very high.

The results presented in section 5.2 show basic
agreement with the computations. which Suggests that the
calculations presented in chapter 3 could be expanded upon
using SOMA. The programs also show that with the correct
operating conditions it is possible to achieve a narrow
phase width. Combined. the computations and the experiments
show that phase selection in the K500 is feasible. Naturally
this leads one to wonder about the possibility of single

turn extraction. which Figure 3-13 suggests is theoretically

possible. In practice it will require considerable effort.
For example the small, slow oscillation of the main coil
currents in time must be reduced if sufficient stability is

to be maintained. At the same time a large effort is needed
to reduce the centering error which appears to arise from
the central region. On the mechanical side. these studies
could be made considerably easier if the main probe were
improved. in particular. correctly calibrating the probe.
reducing the vertical bounce as the probe moves. and

improving the electron traps on the Jaws. are all things

157
that would make probe data more reliable. From a practical
point of view. if the phase selection system is to be useful
for beams other than those produced in copious amounts by
the 'ECH. the transmission into the cyclotron has to be
improved. Separated turns (and single turn extraction) would
be helped significantly by reducing the initial emittance.
Perhaps this can be done by reducing the inflector

collimator to 2 mm in diameter. but again this would result

in unacceptably low beam currents unless the transmission
improves.
It is probably fair to say. that in most cases the

computed values are more easily interpreted than the
measured ones. but often ignore important machine
conditions.\Nhen it comes to making measurements in the K500

it can be very difficult to achieve repeatable results. so

there will always be a need for both computational and
experimental studies of the K500. Now that the basics of
phase selection are understood and confirmed. it is hoped

that much further progress towards single turn extraction

can be made.

APPENDICES

APPENDIX l

SOMA INPUT

The input and output to SOMA has been distributed among
many FORTRAN IO units in order to simplify modifications.
The primary input file (unit 5) contains most of the program
switches and the cyclotron parameters. Other information
such as the magnetic field. harmonic bumps, and the spiral
locations are on separate files. Table 7-1 gives a complete
listing of all the IO units used by the program and a short
description of the information kept on each one.

The first line of the unit 5 file must contain the
values of the two logical variables LMATRICES and LPROBE. in
that order. If LMATRICES is true then the run begins by
calculating the transfer matrices requested on unit 11.
otherwise the program attempts to read the transfer matrices

from units 51 and 52. VVhen LPROBE is true then the program

will either calculate or read in (again depending on the
value of LMATRICES) the transfer matrices for the main
probe. Fo|IOWlng this first line are any number of lines
containing the values of various input parameters. Each line

158

159

 

Table 7-1. The input-output units used by SOMA.
Unit Description
INPUT
5 input parameters
11 initialization of transfer matrix computation
12 initial conditions
13 spiral locations
14 harmonic field bump data
44 magnetic field
OUTPUT
30 orbit parameters of particles which hit flags
31 printout at constant theta
33 probe output (binary)
34 printout at gap locations
35 printout at end - for restart
BOTH
51 gap to gap matrices (binary)
52 constant theta matrices (binary)
53 main probe matrices (binary)

160

begins with the number (between 1 and 50) of the input
parameter being set. All input parameters have default
values so only those differing from the default need be
entered. The parameters may appear in any order and the list
is terminated by a "-1”. The input parameters divide into
three groups. Those with ID numbers 1 through 9 give the
descriptions of the probes and flags to be used. The
parameters 10 through 34 are all single entry real numbers
while 35 through 50 are single integer values. so this last
group contains most of the switches.

Input parameter number 2 is used to indicate the number
of flags to be described. Immediately following the line
which began with a ”2” there must be one line containing the

values of; 9 ITYPE RMIN.RMAX. for each flag. The flag is

f !

assumed to be located at angle 9f and is only in effect when

the orbit radius lies between RMIN and RMAX. The flag angle
must have a transfer matrix. but many flags can share the
same matrix if they have the same azimuth. Up to 20 flags

may be requested. There are three types of flags. The first

type (ITYPE :1) is called a transparent flag. In this case
when the orbit radius lies inside the flag. then the orbit
parameters are saved but the orbit is unaffected. If lTYPE

is 2 then the flag is opaque. in which case when RMIN <

161

R(ef) < RMAX then the orbit parameters are saved. the

particle is considered lost. and the program proceeds to the
next particle in the input distribution. VVhen ITYPE=3 then

the orbit parameters at 9f are recorded on unit 31 whenever

the particle radius lies inside the flags range.

A line beginning with a 3 contains information on the
snapshots to be produced at the end of the run. A snapshot
is a scatter plot of any two of the possible 9 orbit
parameters. A snapshot will be produced for the initial
conditions. the final conditions. and all flags of types 1
and 2. For each snapshot desired a pair of integers are
given. with the first integer specifying the orbit parameter

to be put on the horizontal axis. and the second doing the

Table 7-2. -- The ID codes for each of the parameters that
are saved.

 

 

 

lD Parameter Description

1 r orbit radius

2 pr radial component of the momentum

3 2 vertical position

4 p2 vertical component of the momentum
5 T time

6 E energy

7 x displacement in radius from the E0
8 p displacement in pr from the E0

9 b = PGAV average of the phase at the last 6 gaps

 

162
same for the vertical axis. The labeling of the various
parameters is given in Table 7-2. For example the line

"3.1.2.9.1." would produce scatter plots of pr vs r and r vs

B at the run’s conclusion.

The ”4" line provides the description of the probes to
be considered during the run. A maximum of two probes. one
radial and the other the main probe are allowed. The
parameters on the ‘4 card’ are;

4.NP.Ri.R .AR.IBIN.ABIN.ADIFF.9.THICK.

f
Probe information will be accumulated between Riand Rf. with
a bin size of AR. IBIN determines if a second parameter is
to be binned as well. If IBIN=1 no other binning is done.
\Nhen IBIN is 2. z is binned and when lBlN=3 the phase is
binned. ABIN is the bin size for the z or phi bins (up to 60
phi bins are available).\Nhen NP is 1 the probe is radial
and it is located at an azimuth a. On the other hand if NP=2
the main probe matrices are used and 9 is ignored. The
parameters ADIFF and THICK give probe head dimensions as
shown in Figure 7-1.

The various possible initial conditions discussed in

section 2.5 are selected using parameter 42. A value of 2

for this parameter runs the restart option. In this case

163

 

 

 

 

 

BEAM
—«> <——-ADIFF
MAIN JAW
DIFF. JAv I T:
THICK

Figure 7-1. A schematic drawing of a probe head. as defined
in SOMA. The two dimensions. AIDIFF and THICK.
are input in inches.

164
unit 12 contains a line for each particle to be started. The
format of these lines is;

d.r.pr,E.T.d.d.Z.pz(10X.3f12.5.f12.6.f12.3.4f8.4)

where d is a dummy variable. A file which meets these
requirements is produced on unit 35 at the end of each run.
\Nhen lP(42)=1 the program expects the initial conditions to
have been generated by a separate program and stored on unit
12. The unit 12 file should contain one line for each
particle to be run. with the following format:

Energy,x.px.z.pz.¢ (6f12.5).

Table 7-3 shows the remaining possiblities for parameter 42.
each of which gives a different method for calculating the

initial ellipses.

Table 7-3. -- The source of the initial ellipse values as
determined by parameter #42.

 

 

 

 

lP(42) X-PX ellipse ZoPZ ellipse

3 calc. eigen-ellipse calc. eigen-ellipse

4 calc. eigen-ellipse input

5 input calc. eigen-ellipse

6 input input
The eigen-ellipses are computed for an E0 energy equal to
the central energy. P(28). The definitions of the various

ellipse parameters are given in Figure 7-2. The X-Px ellipse

165

P:

 

 

 

Figure 7~2. An illustration of the meaning of the various
initial ellipse parameters. See text for an

explanation of how they are input to SOMA.

166

center (x0 and pxo) can be set in three different manners.
If IP(48)=IP(45)=0. then the values of X0 and pxo will be

P(30) and P(31) respectively. If lP(48)=O and lP(45)=1 then
the program will calculate the AEO for a particle starting
with the central energy, P(28). The AEO is found by running
3 particles for one turn with acceleration. A linear fit is
then used to determine the starting condition which will

result in the x.px values remaining the same after one turn.

These initial conditions are then run and the process

. . -8
repeats untll the closure ls better than 10 or 10

iterations have been made. which ever comes first. The final

method of setting x0 and pxo is to make them a function of

the initial phase. If lP(48)=1 then unit 12 contains one

line which i&

a1. b1. a2. b2 (4f12.5).
Then
x0: 31 + (fl - ¢r) 32.
px0 - D1 + (6 - ¢ ) b2.
E = Ecen cos(¢ - fir).
where. Ecenz P(28) central energy.
d = P(29) reference phase.

167
A full ellipse is started for each value of the phase (a).
where the phase. d. is:

¢ = ¢ + i r Afl. i=0.n¢-1.

where.

do = P(27). ao = P(26). and n¢ = IP(41).

\Nhen IP(44)=1 then %) is calculated. The calculation is

performed using;

 

Sin(¢1)= sin(¢0) + 1) dE

so d1 is the phase at E = E . when do is the phase at the

first energy EO at which transfer matrices are stored. The

new value for do (to be used as described above to determine

the starting phases) is computed to be:

$0 = ¢1 - P(26)*|P(41)/2.

so that the phase group is centered about the phase given by
an E0 code.

Table 7-4 contains a listing of the input parameters 10
through 50 and their default values. It should be noted that
the RF frequency is.

wrf=h(1+e) Loo. lf, cooquo/m.

168
where e is the frequency error and h is the harmonic number.

Parameters 16.17.18 are a,b.+ respectively for the X-Px
ellipse. while 20.21.22 are the same for the Z-PZ ellipse.

The area per point (#24) is the area assigned to each
particle in the uniform distribution. so there are
approximately nab/P(24) particles populating the ellipse.

The number of transfer equations (#39) is explained in Table

2-4. VVhen IP(40) is zero the Z-PZ ellipse is uniformly
populated. If it is one then the X-Px ellipse is uniformly
populated. and a value of 2 causes both ellipses to be
uniformly populated. Those ellipses not uniformly populated
will be randomly populated. IP(46) is used to stop the

printing of the initial particle parameters in the log file
(unit 6). IP(47) determines whether the gap correction is

done to first or second orden

Unit 11 is read whenever transfer matrices are to be
calculated. The five lines in this file contain;

Ei.AE.NE

NVANGLE

TH ANGLE(i). i=1.N_ANGLE

N_EOU

q.BO.NSEC.NR.NSWI

169
Matrices will be computed for the energies.

E = Ei + AE ' i. i=0.NE-1.

N_ANGLE is the number of fixed angles for which transfer

matrices will be calculated. Up to 10 fixed angles may be
requested. The next line contains the azimuths of the fixed
angles. and these must fall on a regular 20 Hunge-Kutta

step.\Nhen N EOU is 1 only the first order transfer matrices
are computed and stored. but when N-EOU=2 the full second
order transfer matrices are calculated. The final line gives
the particle charge in units of e, the central field. the
number of sectors. and the number of radius values at which
the field is stored. If MSW/is non-zero the magnetic field
is assumed to have a header which is concluded by an end of
file mark.

The magnetic field is input on unit 44. This file
begins with a header of up to 10 lines. concluded by an end-
of-file mark. Following this is the magnetic field values in
a regular r.e grid. The data is stored so that theta varies
most rapidly. The theta step size is one degree and there
120 theta values if NSEC=3 and 360 theta values if NSEC=1.
The input format is 8F9.5 and there are NR radius values

beginning at R=0.0" and increasing in 0.5" steps.

170
The gap locations are given on unit 13. The first
record is.
NRGP.RO,AR (l5.2F10.5).
The gaps themselves are given in the subsequent records as a
table of angles.

91(80). 92(R0) (2F12.5)

91(RO+AR).92(RO+AR)

etc.

(NRGP entries in the table).

61 is the entrance to dee1 and 92 is the exit (gap2). It is

assumed that there are 3 dees evenly spaced around the

machine. so 6 gaps are defined. The e vs R function for a
gap is assumed to be linear between data points (a spiral in
real space) for the purpose of interpolation and
differentiation. There must be at least two r values. and
any orbit to be computed must be between the r limits of the
gap table. It is advisable to keep the table interval small
enough that large discontinuities in the derivative are

avoided. There is no restriction on the value of 6. but the
following 8 values must satisfy these conditions:

62(r) >e(r) +10O

171
I 9(r+Ar) - 6(r) I < 90°.

These limits ensure that no two spiral lines fall in the
same integration step. and there is no +/- 3600 ambiguity in
the 'table. Note that in most cases. a table with only two r
values would require more than a 900 difference in
successive theta values. Gap print-outs are done at the
gaps;

IG = IGO +|DG x i. i=0.ng-1
where.

IGO= IP(37), IDG=IP(36). and ng=lP(38).
These print-outs appear on unit 34. and if a large number of
particles are run this file can become extremely large.

\Nhen IP(49) is different from zero then the program
expects information about the harmonic field bumps to be
located on unit 14. The file should contain:

(4F9.5)

RO.AR.NR (2F.|5)
BUMP1(i) i=1.NR (8F9.5)
BUMP2(i) i=1.NR (8F9 5)

where b1 and d1 are the amplitude and angle of the first

harmonic bump. and b2 and d2 are the same but for the second

harmonic bump. The bump field will be speCIfied at the

radial positions:

172

 

Table 7-4. The input parameters 10 through 50. that are
entered on unit 5. The default values are in
brackets.

PARAMflO) DEE VOLTAGE (KV) 0.00000 ( 50.00000)

PARAM(11) HARMONIC NUMBER 1.00000 I 1.00000)

PARAMI12) FREQUENCY ERROR 0.00000 ( 0.00000)

PARAM(13) PHASE ERROR OF DEE1 IN DEGREES 0.00000 ( 0.00000)

PARAMI14) PHASE ERROR OF DEE2 IN DEGREES 0.00000 ( 0.00000)

PARAM(15) PHASE ERROR 0F DEE3 IN DEGREES 0.00000 ( 0.00000)

PARAM(16) x RADIUS 0F ELLIPSE (INCHES) 0.01071 ( 0.01000)

PARAM(17) Px RADIUS OF ELLIPSE (INCHES) 0.01071 ( 0.01000)

PARAMIla) TILT OF x-Px ELLIPSE (DEG) 0.00000 ( 0.00000)

PARAM(19) AREA OF X-PX ELLIPSE (MM-MRAD) 9.14804 ( 5.00000)

PARAM(20) z RADIUS 0F ELLIPSE (INCHES) 0.01071 ( 0.01000)

PARAM(21) Pz RADIUS OF ELLIPSE (INCHES) 0.01071 ( 0.01000)

PARAMI22) TILT OF z-Pz ELLIPSE (DEG) 0.00000 ( 0.00000)

PARAMI23) AREA OF z-Pz ELLIPSE (MM-MRAD) 9.14804 I 15.00000)

PARAMI24) AREA PER POINT (IN**2) 0.00500 ( 0.00500)

PARAM(25) +/- % DELTA E 0.00000 ( 0.00000)

PARAM(26) INCREMENT IN PHASE (DEG) 1.00000 ( 1.00000)

PARAMI27) PHASE OF FIRST GROUP (DEG) 0.00000 ( 0.00000)

PARAM(28) CENTRAL ENERGY (MEv) 0.00000 ( 0.00000)

PARAM(29) REFERENCE PHASE (DEG) 0.00000 ( 0.00000)

PARAMI30) DELTA x (IF 45=0) 0.00000 ( 0.00000)

PARAMI31) DELTA PX (IF 45=0) 0.00000 ( 0.00000)

PARAMI35) NUMBER OF TURNS 50 ( 100)

PARAMI36) SPACING BETWEEN GAP PRINTS 1 ( 1)

PARAM(37) INITIAL GAP PRINT ( 1)

PARAM(38) NUMBER OF GAP PRINTS 1 ( 0)

PARAMI39) NUMBER OF TRANSFER EQUATIONS 6 ( 6)

PARAM(40) IF 0 THEN z-Pz UNIFORM 1 ( l)

PARAM(41) NUMBER OF PHI GROUPS 1 ( 1)

PARAM(42) INPUT DATA TYPE 1 ( 3)

PARAMI43) RANDOM NUMBER SEED 249279641 ( 249279641)

PARAM(44) IF NE 0, CALC PHI INITIAL 0 ( 0)

PARAMI45) IF 1 THEN USE ACCEL E0 0 ( 0)

PARAM(46) IF 1 SUPRESS INITIAL PRINT 0 ( 0)

PARAM(47) ORDER OF GAP CORRECTION 2 ( 2)

PARAM(48) IF 1 READ ELLIPSE CENTER 0 ( 0)

PARAMI49) IF 1 USE FIRST HARM. BUMP 1 ( 0)

FINAL VALUE OF RANDOM NUMBER SEED WAS 249279641

173
R = R0 + i I AH i=0.NR-1.

The product b1: BUMP1(i) should give the first harmonic

component of the field at the radius step i in kilogauss.

 

APPENDIX ll

ORBIT CODE PARAMETERS

Chapter 2
All tests were run in a K500 magnetic field trimmed for
12 4+ .
30 MeV/u C . The central field was 80:34.50535 kG . and

the RF voltage was Vdee=60.4 kV. The field had perfect 3

fold symmetry and the RF frequency was equal to B \Nhen not

0'
specified test runs began at E=11.0 MeV/u. All tests with

acceleration began with ¢=0.0 . The 2 motion study began at

E=5.0 MeV/u and ran for 300 turns.

Chapter 3

The PIG cases were run in the K500 30 MeV/u 1204+ field

with 80:34.50535 kG and Vdee=60.4 kV. The ECR case studies

16 4+

were done in a K500 field for 25 MeV/u O with

80:42.20057 kG. and Vdee=73.06 kV. Except where noted. the

magnetic fields had perfect three fold symmetry.

174

175
Chapter 5
Comparision to the experimental results in section 5.2

was done in a K500 field for 20 MeV/u 12N4+. which has

80:33.3192 kG. Vdee= 51.5 kV. and 3 fold symmetry. The

initial phase width was ~180 to 2° at E=0.38717 MeV/u. The
centering conditions using the technique of Figure 2-3 are.

x =-0.034 - 0.0033(¢-¢r)

0

px0= +0.030 - 0.0061(¢-¢r)
E = Ecencos(d-¢r)
¢r=-13.0°

Ecen= .38717 MeV/u.

The relative densities of the different starting times was
determined using Figure 5-3 for the case of 100W initial

emittance and a small first turn sliL

REFERENCES

REFERENCES

H.G. Blosser. ”The Michigan State University
Superconducting Cyclotron Program ”.IEEE Trans. NS-
gngL.2040(1979).

H.G. Blosser and F. Resmini, ”Progress Report on the 500
Mev Superconducting Cyclotron". IEEE Trans. NS-

26(3).3653(1979).

M.L. Mallory. "Initial Operation of the MSU
Superconducting Cyclotron". IEEE Trans. NS-30(4).

2061(1983).

H.G. Blosser. 9th Int. Conf. on Cyclotrons and their

Appl.. G. Gendreau. ed.. Les Editions de Physique 147.
(1981)

J. Riedel. ”RF Systems".IEEE NS-26g2). 2133(1979).

J. Riedel. "Three Phase RF Systems for Superconducting
Cyclotrons". IEEE Trans. N§;§Q. 3452(1983).

F; Marti. to be published in the procedings of the 11th
Int. Conf. on Cyclotrons and their Appl.. Tokyo (1986).
M.M. Gordon. ”Effects of Spiral Electric Gaps in
Superconducting Cyclotrons." NIM. 169.327(1980)_

—

176

10.

1L

12.

13.

14.

15.

177

C.J. Kost and G.H. Mackenzie. "COMA a Linear Motion
Code for Cyclotrons.” IEEE N§;gg. 1922(1975).

J.C. Collins. "Phase Selection Mechanisms in lsochronous
Cyclotrons Producing High Resolution Beams". Ph.D.
thesis. Michigan State University. 1973. pg. 1.

B.F Milton et al.. "Design of Beam Phase Measurement and
Selection System for the M.S.U. K500 Cyclotron". Proc.
10th Int. Conf. on Cyclotrons and their Appl.. F. Marti
ed.. 55.(1984).
K.L. Brown. "A First and Second Order Matrix Theory for

the Design of Beam Transport Systems and Charged

Particle Spectrometers.” Adv. in Particle Phy. 1.
67.(1967).
M.M. Gordon, ”Computation of Closed Orbits and Basic

Focusing Properties for Sector Focused Cyclotrons and
the Design of 'CYCLOPS‘”. Particle Accelerators. 16.
39(1984).

P. Kramer, H.L. Hagedoorn and N.F. Verster. "The Central
Region of the Phillips AVF Cyclotron". Proc. of the CERN
Cyclotron Conference 193.(April 1963).

S. Gill. "A Process for Step-by-Step Integration of
Differential Equations in an Automatic Digital Computing

Machine". Proc. Cambridge Philos. Soc.. 7.96(1951)

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

178
M.M. Gordon. "Green's Function for the Mathieu-Hill

Equation”. internal NSCL memo. June 1985.

M.M. Gordon. "Notes for 2 Order Transfer Matrix Code".

nd
internal NSCL memo. 1986.

R.E. Berg, ”Precise Methods for Pre-Calculation of
Cyclotron Control Settings”, Ph.D. Thesis. Michigan

State University. 43 (1966).

M.M Gordon and T A.\Nelton, ORNL-2765 (1959).

M.M. Gordon. ”Possible Treatment of Field Bump Effects
\Nithin a Transfer Matrix Code". internal NSCL memo. Feb
2 1986.

M.M. Gordon. ”Single Turn Extraction” IEEE Trans. NS;

13(4).48 (1966).

Loc. Cit. 10. pg. 12.

Ibid. pg. 3

F. Marti. "Design of the Axial Injection System for the
NSCL Cyclotrons" IEEE Trans. NS-32(5).2450(1985).

T.l. Arnette. "Program CYCLONE”. Michigan State
University internal report (1966).

Loc. Cit. 10. pg. 43.

\N.B. MHlson et al.."Beam Diagnostics and Improvements at

TAMVEC". IEEE Trans. NS-18I2).299(1971).

28.

29.

30.

31.

32.

33.

34.

35.

179
M.M. Gordon. "Canonical Treatment of Accelerated Orbits
in Sector-Focused Cyclotrons” Part. Accel. 12.13(1982).
Loc. Cit. 20. pg. 51.
M.M. Mallory. to be published. in the procedings of the

11th International Conf. on Cyclotrons and Their Appl.

Tokyo (1986).

Garren and Smith. "Diagnosis and Correction of Beam
Behavior in an lsochronous Cyclotron". Proc. of the Int.
Conf. on Sector Focused Cyclotrons and Meson Factories.

Cern. April 1963.

R.E. Berg. "Precise Methods for the Pre-Calculation of
Cyclotron Control Settings". Ph.D. Thesis. Michigan
State University. 1966. pg. 86.

Ibid. pg. 87.

F.Marti et al. ”Effect of Orbit Centering and Magnet
Imperfections on Beam Properties in a Superconducting
Cyclotron",Proc. 10th Int. Conf. on Cyclotrons and their
Appl.. F. Marti ed.. 46,(1984).

M.M. Gordon. "Perturbation of Radial Oscillations in
Superconducting Cyclotrons Due to Asymetric Dee Voltages

and Phases". IEEE Trans. NS-30.2439(1983).

36.

37.

180

M.M. Gordon and v. Taivassalo. ”The z4 Orbit Code and
the Focusing Bar Fields Used in Beam Extraction
Calculations for Superconducting Cyclotrons”. NIM 5247.
423(1986).

D.A. Johnson and H.G. Blosser, "Computor Program for
Tracking of Linear Cyclotron Orbits". Michigan State

University Internal Report MSUCP-4(1960).

 

EUN

nICHIan smt IV. LIBRARIES
I I III III III IIIIIIIII IIIIIIIIII IIIIIIIIIIIIIIIIII
31293007997046