PHASE SELECTION MECHANISMS
IN ISOCHRONOUS CYCLOTRONS

PRODUCING HIGH RESOLUTION BEAMS

BY

John Curtis Collins

A THESIS

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

DOCTOR OF PHILOSOPHY_
Department of Physics

1973

ABSTRACT
PHASE SELECTION MECHANISMS
IN ISOCHRONOUS CYCLOTRONS
PRODUCTING HIGH RESOLUTION BEAMS
BY

John Curtis Collins

Phase selection utilizing coupling between radial and
longitudinal motion has long been a useful tool at the
Michigan State University Isochronous Cyclotron for produc-
ing very high energy-resolution beams. A study was under-
taken to obtain a better understanding of this phenomenon
by identifying the relevant machine parameters and their
quantitative effects. While the M.S.U. machine provides a
convenient prototype cyclotron, our results are not confined
to this one example as various dee angles and magnetic field
structures are investigated. Other parameters of importance
are dee voltage, acceleration harmonic and phase history.
The role of orbit centering is given particular attention,
especially as it relates to the difference between actual
phase histories and those from the standard M.S.U. field
trimming program. Most orbit calculations were done with
a precise numerical integration code limited only by the
assumption of step function energy gain. Finally, the
data is used to discuss the general problem of placement

and effectiveness of phase selection slits.

,‘4 ACKNOWLEDGEMENTS

I would like to take this opportunity to thank Julie
Perkins for invaluable help and advice in the preparation
of this manuscript.

I would also like to thank Richard Au and Dave
Johnson for their time and aid in computer programming and
the use of orbit codes.

Thanks are due also to Larry Learn and Dr. Henry
Blosser for many helpful discussions and explanations of
cyclotron theory and operations.

My speCial thanks go to Dr. Morton Gordon whose
patient guidance gave this work direction and purpose and
to my wife Carolyn, whose impatient prodding kept the work
moving along. This thesis could not have been finished
without both of these peOple.

Finally, I wish to acknowledge the financial support
of the National Science Foundation throughout my graduate

career.

TABLE OF CONTENTS

Page
LIST OF FIGURES O O I O O O O O O O O O O i i
LIST OF TABLES O O O O O O O O C O O O O V
1 0 INTRODUCTION 0 O O O O O O O O O O O l
1.1 Historical Background. . . . . .. . . l
1.2 Approach to the Problem . . . . . . . u
2. ORBIT PROPERTIES AND CALCULATIONS . . . . . 5
2.1 Introductory Mechanics . . . . . . . 6
2.2 Computation Technique. . . . . . 1a
2.3 The Accelerated Equilibrium Orbit. . . . 19
2.4 The Central Ray. . . . . . . . . . 25
2.5 Basic Phase Selection. . . . . . . . 35
2.6 Effects of Radial Emittance. . . . . . H5
3. DATA AND ANALYSIS° - - - - - - - - ~ - 53
3.1 Dee Angle and Orientation . . . . . . 53
3.2 "Fielder" Phase Curve. . . . . . . . 52
3.3 Energy Gain Per Turn . . . . . . . . 71
3.4 Harmonic Number. . . . . . . . . . 7S
3 O 5 Four sector Field 0 O O O O O O O O 8 5
4. CONSEQUENCES FOR PHASE SELECTION. . . . . . 91
4.1 'Locating Phase Selection Slits. . . . . 91
4.2 Comparison with "Cyclone" . . . . . . 100
4 O 3 smry O O O O O O O O O O O O 0 lo 3
s O APPENDICES O O O O O O O O O O O O C l O 6
5.1 Formula Derivations . . . . . . 106
5.1.1 The Initial Condition for an

Accelerated Equilibrium Orbit. . . 105

5.1.2 The Energy Difference Between Phase
Displaced Rays. . . . . . . . 109
5.2 The Code "Devil" . . . . . . . . . 111
REFERENCES 0 o o o o o o o o o o o o 0 ll 8

Figure

1.

LIST OF FIGURES

Page

x(mils) vs. Px (mils) for a 14 MeV proton in

a three sector, 30 MeV field. Points are

plotted once per turn at 6=0° for no accel-
eration (a) and acceleration of 143 keV/turn

with two 138° dees located symmetrically

relative to e=o° (b) . . . . . . . . . 9

ABC properties in a four sector magnetic

field at 6=6 r=0° "Average" values are the

mean of px(0 ) and p (180°). Asymmetries are
values at 0° minus tfie average value. . . . 21

ABC properties as in Fig. 2 except with a
three sector magnetic field. Compare the
ordinate scales with those in Fig. 2. . . . 22

X-P history of the first 50 turns for a

welI centered proton with a=138°, Ef=30 MeV,

E .143 keV/turn, N=3, hal. Every turn is

given (dot) to 1:5 (circled) with every

fifth turn shown thereafter. Crosses mark

AEO locations at 10 turn intervals beginning 2
at 1.5. O O O O O I O I O O O O O 6

Phase history (deg vs. 1 at each gap) for the
cases.shown in Fig. 4. . . . . . . . . 29

x-Px history corresponding to Fig. 4 but with

N24. The first dot on each line is 121;

excellent centering makes recognition of

later turns difficult. All axes are in

mils . . . . . . . . . . . . . . 33

Phase history (deg vs. T at each gap) for the

cases shown in Fig. 6. Note that gap crossing
phase is unaffected by ear: unlike in Fig. 5 . 3“

ii

Figure Page

8. Schematic representation of the growth of
A between the CR (point C) and a phase
displaced ray (point D). (See text,
p. 37.) . . . . . . . . . . . . . 38

9. Difference in rf times of gap crossings for
zero (crosses) and non-zero (dots) phases
for 11-1 0 I O O I I I O O O O O O 3 8

10. 6x-6p history for the first 40 turns for
rays with A¢=12°. Properties of the reference
CR are shown in Figs. 4a and 5a. Points are
plotted once per turn at 6=6°r=0°. All axes
are in mils . . . . . . . . . .. . . 42

11. 6x-6px history for 16 rays initially located
on a circle of 20 mil radius. Properties of
the reference CR are shown in Figs. 4a and 5a.
All axes are in mils . . . . . . . . . 46

12. 6¢(deg) relative to the CR vs. T for rays with
the initial spatial displacements shown by the
correspondingly labeled points in Fig. 11a. . M7

13. (6X2+6px2)1/2 vs. 6x0 and 5Pxo at various dee
angles. Variation is close to linear over the
range of initial conditions shown for all dee
angles. . . . . . . . . . . . . . 49

14. Eigen-ellipse axis length ratios and flutter
vs. energy for an M.S.U. proton field with
N=3' Ef830 MeV o o o o e o o o o e 0 5‘4

15. Vr and u (see Eq. 11) vs. energy for the field
Of Fig. l4 O O O O O O O O O C I 0 51+

16. AR ,‘A and AR vs. T for N=3, Ef=30 MeV, TF=210,
a;I98° and h=l plotted at 6:601. for various
eor‘s. Ordinate values are in mils . . . . 60

17. ¢F(T) for M.S.U. proton fields for h=l and
Ef=30 MeV. See text, p. 62, for curve para-
meters. . . . . . . . . . . . . . 63

18. ¢F(T) for an M.S.U. deuteron field for h=2 and
Ef=1s MeV and an M.S.U. c3+ field for h=3 and
Ef=27 MeV. . g o o o o o o o o o o 76

iii

Figure

19.

20.

21.

22.

23.

24.

25.

Page

Orbit asymmetries for a=138o in the N=3

fields for h=1, 2, 3 used in Secs. 3.1

and 3.4. The px asymmetry is defined in

Sec. 2.3. 'The curve plotted with squares

is for TF=120. All others have TF=210.

The h=1 curve is the same as in Fig. 3 . . . 79

Zero phase gap crossing times with (1:1380
and h=2. Compare with cross locations
in Fig. 9. o o o o o o o o o o o o 81

¢F(T) for an N=4 30 MeV proton field. Com-
pare with curve C in Fig. 17 . . . . . . 81

Eigen-ellipse axis length ratios vs. energy
for a proton field with N=4, Ef=30 MeV. .
Note change of ordinate scale from Fig. 14. . 85

Vr and w (see Eq. 11) vs. energy in the field
used for Fig. 22. Note change of ordinate
scales from Fig. 15 . . . . . . . . . 85

AREO, A and AR vs. T plotted at 8:80 for
various Bor's for N=4, Ef=30 MeV, a=I38°,
TF=210 and h=1. Ordinate values are in mils . 88

AR. , A and AR vs. T for 60r=0° plotted at
8=88(a,b) and 6:180°(c,d) comparing Cyclone (a)
and Devil (b,c,d) results. See text, p. 100,

for comparison details. Ordinate values are

in mils . . . . . . o . . . . . . 101

iv

Table

II.

III.

IV.

VI.

VII.

VIII.

IX.

LIST OF TABLES

Comparison of 5¢CD from Eq. 3 and 5¢RL
from Fig. 5a between dee gaps i and j for
081380. 0 O O O O O O O O O O O 0

Rate of increase of 6¢ with 5x0 and 6Pxo
in (deg/10 mils) at T=3U. . . . . . . .
AB and AREO at e=80r and e=60r+n for various
dee angles. The magnetic field used corres-
ponds to curve C in Fig. 17. . . . . . .

A and Ax at 9=8 r=°° and 8=80r+n for the same
cases as shown in Table III. . . . . . .

AREO at e=eor=00 and 8=6°r=60° for the ¢F(r)
curves shown in Fig. 17. Included are the
areas under the curves and values normalized
to CASE E (parentheses) . . . . . . . .

A and Ax at 6=60r=0° and 6=60r=60° for the
same cases as shown in Table V. . . . . .

Comparison of AREO and A for a=180o and a=90°
at 6=80r=O° and 6=60r=6O° to show a connection
between a, eor and ¢F(T). See Fig. 17 for the
corresponding ¢F(r) curves . . . . . . .

Location and magnitude of the first maxima in
AREO and A at 8=6 r=0° and 8=8°r=60° for the
¢F(T) curves in Fig. 17 . . . . . . . .

Comparison of orbit properties in an M.S.U.
proton field for various values of E1 at
fixed Ef=30 MeV and e=eor=00 . . . . . .

AREOI A and Ax at e=eor for various dee angles
on h=2. The magnetic field used corresponds
to the ¢F (I) ShOWI‘l in Fig. 18 g o o o o o

Page

31

51

56

58

65

66

68

7O

73

77

Table

XI.

XII.

XIII.

XIV.

XV.

ARE A and Ax at 6=80r for various dee
angges on h=3 The magnetic field used
corresponds to the ¢F(I) shown in Fig. 18 .

ARE, A and Ax at 8= 6 or for various dee
ang es and N=4. The magnetic field used
corresponds to the ¢F(T) shown in Fig. 21 .

Turn number and value of the best Q at
6=6°r and 8=6°r+n for various dee angles
on h=1. This table complements Tables X
and XI. . . . . . . . . . . . .

Turn number and value of the best Q at
8=6°r and 6:8 r+" for various dee angles
on h=2 and h=3 This table complements
Tables X and XI. . . . . . . . . .

Turn number and value of the best Q at

r and 8= 6 r+w for various dee angles
Wltfi N=4 This table complements Table XII

vi

Paqu

83

89

93

94

95

1 . INTRODUCTION

1.1 Historical Background

Most cyclotrons operating normally are high current,
imprecise beam sources exhibiting large emittances
(40 mm-mr is not atypical) and beam pulse phase widths
(typically 200 to 300 phase widths corresponding to micro-
scopic duty factors of 6 to 8%). Such beams prove difficult
to extract cleanly and multi-turn extraction is commonplace
(with efficiencies of 50-70%), but with the consequence
that the external beam has an energy spread of approximately
the internal energy gain per turn. On the other hand,
modern nuclear physics experiments require small beam
spots and energy widths to obtain their ultimate resolution.
This conflict is usually resolved by designing the external
beam handling system with analyzing magnets and slits to
strip away unwanted particles. This technique involves
large aperture (expensive) magnets or high background
radiation and residual activity or, regretably, both.

A different solution is to obtain small emittance
and good energy resolution by extracting only a single
turn (at nearly 100% efficiency). The external beam would
then be an "image" of the ion source, which could be
constructed with a small slit to give small emittance. An

energy spread would still arise because of the sinusoidal

l

variation of the accelerating voltage with time but,
while possibly larger than desired, this would be much
smaller than the energy gain per turn. The fractional
energy spread generated in this way for a beam phase

width of 2A¢ in a perfectly isochronous field 18:1

AE/E = (A¢)2/2. _ (1)

From extensive studies,2 it was confirmed that single
turn extraction requires small A¢. Not independently one
notices from Eq. (1) that this also produces superior
energy resolution. The question then is how to obtain
such small phase widths.

Three methods have been used to accomplish this
selection of desired phase at M.S.U. The original idea
was to use the phase dependent axial focusing properties
of the electric field at the first few gaps with an axial

3 This system produced phase widths of about 7°,

slit.
good enough to observe single turn extraction. However,
Eq. (1) indicates that it is profitable to attempt to
reduce this phase width as much as possible. It was also
true that the axial slit passed particles of any phase if
they were emitted with small axial momentum near the
median plane. Analysis of certain empirical data showed
that radial slits at the half and tenth turns gave better

4

phase selection. Such a system worked through the coupling

between the radial and longitudinal (E-¢) motions of the

3

particles and produced approximately 3.50 phase widths,
a substantial improvement.
The third method, and the one presently in use,is

a refinement of the second retaining the half turn slit
and locating two more radial slits on turns 18 and 28.5
Beam pulse widths are now observed as low as 1.40 FWHM
with radial emittances of less than 1.0 mm-mr. These slits
thus make the M.S.U. machine a very precise nuclear physics
instrument. The position of these slits on early turns
also means that very little bothersome background radiation
is produced.

‘ The precise mechanism of the longitudinal-radial
coupling responsible for phase selection has not been
fully delineated to this point. It was therefore decided '
that a general study would be useful to determine which
cyclotron parameters affect this coupling, what the mechanism
of the effect is and how successful a system of phase
selective slits would be in other cyclotrons of various

design. This paper presents the results of that investiga-

tion.

1.2 The Approach to the Problem

The data presented herein is intended as only a very
rough guide to the importance of the parameters investigated
and the effectiveness of phase selection slits under specified
conditions. We have neglected details of source and early
turn electric fields which would have important consequences
in real cyclotrons but which would also be different in
every machine. What we wish to emphasize are the methods
we use to analyze the longitudinal-radial coupling since
these apply universally and serve as a very handy visualiza-
tion of the coupling process.

The inclusion of phase selection slits allows us to
define a central ray from ion source to extractor. This
is done in Sec. 2.1. This central ray plays a crucial role
in this analysis, as it does in the design and operation
of any cyclotron built for high resolution. From one point
of view, a real beam is built up around the central ray
through small variations in position, momentum and phase,
so that beam properties in general and the properties of
individual particles in the beam should be intimately related
to the characteristics of the central ray.

We investigated various dee angles between 1800 and
90° since most existing cyclotrons use 180° dees with
some (Maryland being the prime example) using 90° dees
and a few using 150° or 120° dees. Both three and four
sector magnetic fields are used and acceleration harmonics

one, two and three were looked at. Other parameters

judged to be of general interest or possible heuristic
importance were energy gain per turn, average phase
history near machine center and relative orientation
between the dees and the field sector structure.

Ourdata on the longitudinal-radial coupling are
expressed in terms of the radial separation between the
central ray and a ray differing from it in initial phase.
In Chapter 2 we resolve this radius difference into two
components: the first depending on the total energy
difference between the rays and the second depending on
the details of how this energy difference was achieved.
We then discuss various properties of these components,
including their dependence on turn number, and detail the
role of the central ray. In Chapter 3 we analyze our
computer generated data in the framework provided by
Chapter 2 to determine how each component depends on the

various cyclotron parameters.

2. ORBIT PROPERTIES AND CALCULATIONS

2.1 Introductory Mechanics

To facilitate the study of particle orbits and their
differences, we may resolve the radial position of a
particle of energy E at azimuth 0, r(E,e), into two
components. The first is the static equilibrium orbit (EO)
radius, REO(E,6); the second is the deviation from that EO

value
x = r(E,6) - REO(E,6).

For any energy E we may define the E0 in a magnetic field

as the non-accelerated closed orbit having the same
periodicity as the field. This analysis proves useful
because the E0 is a function only of azimuth and total
energy (see Sec. 2.5). Thus we may separate out purely
energy effects in the radial differences between particles.
The canonically conjugate variable, pr, the radial component

of momentum, is likewise resolved such that

px = pr(E.e) - PREO(E,8).

Of course, x and px are also functions of energy and azimuth,
but we choose not to include that dependence in our notation.
Since x and px have equal footing in this formulation, it

is highly convenient to express them in the same units.

6

We choose to use mils (1 mil=0.001 in.=0.0254 mm) and
transform px from classical units into these units using
px(mils) = (A/moc) px, where m0 is the rest mass of the
particle under consideration and A is the cyclotron length
unit (= c/wo where mo is the ideal isochronous orbital
frequency) eXpressed in mils.

Completing the dynamical description of a particle
requires the conjugate variable pairs (t,E) and(z,pz).
These, with (r,pr) or equivalently, (x,px), define a six
dimensional phase space for the particle. We choose-to
express the time t in terms of the phase angle ¢ defined

here as

¢ = 8 - h8 =

rf wrft - he'

where wrf is the frequency of the accelerating voltage
and h is the acceleration harmonic, i.e., h = wrf/wo'

We neglect z motion entirely. This is justified as
long as we are considering motion near the median plane.
This condition implies that the maximum phase change per
turn expected from axial motion for a maximum vertical

amplitude d, given by
6 ¢<2nh (v d/2 )2
z —> 2 RBC ’

will always be small.
The static EO we have defined is a very useful tool

in analyzing single particle motion in sectored magnetic

fields, partly because one may visualize x and px values

as specifying the deviation of the actual particle orbit
center from the magnetic field center, and partly because

one can show that a particle coasting in the vicinity of

a reference E0 of the same energy executes harmonic (betatron)

oscillations about that E0.6

In the x-pX plane centered
on this E0, the phase point for the coasting particle
rotates about the origin at a distance equal to the pre-
cessional oscillation amplitude and at a rate such that,
at a given azimuth the phase point returns to its original
position after 1/(vr-1) orbits (turns) in the field. If
one observes the location of such a phase point over many
turns at a fixed azimuth relative to each field sector,
one finds that an eigen-ellipse is traced out. The ellipse
is a function of azimuth arising from the periodicity of
the magnetic field through Floquet's theorem. Detailed
accounts of its properties may be found elsewhere,7'°’9
and will not be reproduced here. As an example, one such
ellipse is traced out in Fig. la for a proton of 14 MeV
coasting in a field corresponding to a maximum energy of
30 MeV. (See Sec. 2.2 for the source of this field.)
Initial conditions used are x=20 mils and px=0 mils and
vr=l.05.

Since particles execute betatron oscillations about

any orbit which satisfies the equations of motion in the

given field, we are not limited to the static E0 as our

Figure l.--X(mils) vs. Px(mils) for a 14 MeV proton
in a three sector, 30 MeV field. Points
are plotted once per turn at 8=0° for no
acceleration (a) and acceleration of
143 keV/turn with two 1380 dees located
symmetrically relative to 6=0° (b).

10

 

soTx
Turn 20 H0“ ( )
A a
Turn O\\\30‘L
. ~20.
101
.L + + 31$
10 20 90 SO

 

 

 

 

90, X
F

(b)

2Q.
gag
so

-20__
Ami-

 

Figure 1.

ll

choice of phase space origin, while still maintaining the
characteristics of the phase point motion described above.
That other choices might be advantageous should be obvious
since we have not yet mentioned the acceleration process.
In a separated turn machine, it is highly useful to define

1 Just as the

an accelerated equilibrium orbit (AEO).
static E0 is closed in configuration space, the ABC is

taken to be that path of an accelerated particle which is
closed after one turn in the x-px plane, i.e., the repre-
sentative phase point is a "fixed point".

To illustrate the value of the ABC, Fig. lb shows the
motions for two phase points representing protons accel-
erated in the same field as used in Fig. 1a: particle A
started (x=-7.6 mils, px=42.5 mils, E=13 MeV) such that it
passes along an ABC on its seventh turn (=14 MeV) and
particle B started (x=-23.5 mils, px=68.3, E=13 MeV) such
that after seven turns it is displaced by 6x=20 mils and
6px=0 mils from the ABC of 14 MeV. Comparing the path
followed by B with that in Fig. 1a, we see that the AEO
acts as a Center of oscillation for accelerated particles
displaced from it just as the E0 (origin in Fig. 1a) did
before.

Were we to consider two beams filling identical ellipses
in the x-px plane, one centered on A, the other on B, that
centered on B would effectively fill a phase space area

larger than the beam ellipse itself and exhibit coherent

12

radial oscillations as B traverses the path shown for it. The

beam ellipse centered on A would effectively occupy an
area only slightly larger than itself since A does not move
far in the x-px plane. This is true regardless of incoherent
variations inside each ellipse. The beam corresponding to
the ellipse centered on A may be termed "well-centered".
In fact, at a given energy and phase, the beam centered on
the appropriate AEO will show the minimum possible coherent
oscillation amplitude. While this condition holds exactly
at only one energy, the coherent oscillation amplitude
increases only slowly with deviations from the energy of
the selected AEO, so that a beam which is well-centered near
the middle of its acceleration history will be so throughout
its entire history except on the first few turns after the
source-puller. Such centering is of great practical
importance because it reduces phase oscillations (see
Sec. 2.4), minimizes the effects of non-linearities and
makes extraction insensitive to dee voltage.

Like the E0, the ABC depends on the magnetic field.
But since the ABC characterizes the acceleration process,
it is also a function of the dee structure and the particle
phase, of which properties the E0 is independent. Details
of AEO properties of specific importance to this study

are given in Sec. 2.3.

13

Using the ABC of energy E and the corresponding
desired phase, we define the "central" ray (CR) as that
one which travels along the ABC on some turn near half
the final energy. The CR should then be a best centered
ray in the sense that any ray with other initial r,pr
values will exhibit larger coherent precessional oscilla-
tion amplitude. The precise energy used for the ABC
should not be at all critical. The CR might also be termed
the "design" ray since it would be used to locate the ion

source and any beam slits in a real machine.

14

2.2 Computation Technique

The information provided above may now be used to
guide our design of the calculations required to under-
stand the longitudinal-radial coupling. We are certainly
interested in E0 and ABC properties as well as in finding
a well centered beam. We are also interested in determining
how machine design affects these properties and, through
them, phase selection. The discussion below of the codes
and calculation methods finally used serves to list and
fix some limitations on the machine parameters at our
disposal, as well as showing how the concepts of the AEO
and well centered beam are actually put to use.

Most of the magnetic fields used were obtained using

10 Two Fielder

the field trimming program “Fielder".
features are particularly convenient for studying the
various effects of phase histories: (a) one may specify

a desired phase-as-a-function-of-energy curve to which the
final field should conform; and (b) an acceleration history
of phase and energy vs. turn number (T) is output along
with the field.' The phase history, denoted ¢F(T) or ¢F(E),
is quite valuable for estimating one contribution to the
longitudinal-radial coupling as shown in Sec. 2.5. It is
obtained by assuming that the particle gains energy
continuously and always remains on an BC. This is a good

approximation for a well centered particle when its energy

is very much greater than the maximum energy gain per turn.

15

But on early turns we shall note large and important differ-
ences between actual phases at gap crossings and ¢F(T).

Fielder utilizes measured field data from the M.S.U.
cyclotron, so, strictly speaking, our results are limited
to isochronous fields of low spiral with central cone and
three sectors. However, spiral is very nearly zero near
the center of any cyclotron, where phase selective slits
would be placed. The cone provides vertical focusing in
the central region. Its effect on ¢F(T) may be modified
by proper trim coil currents. Lastly, most low energy
cyclotrons have three sectors and Fielder does produce an
artificial four sector field, so we are not restricted in
choice of field periodicity.

As previously mentioned, the AEO at any energy depends
on the dee structure involved. Our investigations covered
the most widely used dee angles (180°, 138°, 90°). Since,
under assumptions given below, the only difference between
having one or two 180° dees lies in the achievable energy
gains per turn, we make no distinction between the two
cases. Situations with three dees were not considered.
Our standard dee set assumes straight edged dee gaps and
is described by two angles: the dee ang1e<1 subtended by
the dee edges and the orientation angle eor measured from
the radius at 8=0° in the magnetic field to the line of

reflection symmetry between the dees. Since we are using

16

M.S.U. field data, 6=0° is positioned near the center of
a valley. For convenience we number the accelerating gaps

counterclockwise beginning with gap 1 at

8 = 6 + (fl-a)/2

> 1 or

82 = eor + (n+c)/2
83 = 81 + n

84 = 82 + n.

The codes used to calculate particle orbits all use
the same exact median plane equations of motion and assume
step function energy gain. This assumption is a limitation
in applying our quantitative results directly to any real
machine if the first few acceleration gaps are included.
There, transit time effects are important as they modify
the CR x-px and phase (energy gain) histories. These
effects are so highly dependent on central region geometry,
it was thought best to omit them entirely from this work.

The particles are assumed to start from a virtual
source at 6=61 on the first turn (1:0). We consider 0.9
times the maximum energy gain per gap, E9, to be a reason—
able source-puller or axial injection energy and use this
value in step iii) below.

The outline of our calculational procedure is as
follows: '

i) We obtained a magnetic field from Fielder. The

input to Fielder consisted of the field periodicity N,'

17

the particle type (Eo and q), the final energy desired
Ef, the approximate number of turns IF, the acceleration
harmonic h and a phase curve for Fielder to attempt to
match. Aside from the desired magnetic field, Fielder

supplies EO data (REC, P vs. E) for the

REO’ vr
field, a proper energy gain per turn El, wrf and the actual

phase curve ¢F(T).

ll AEO's covering a

ii) Using the code "Disport-Z",
range of energies including Ef/Z were calculated for
specified ¢F(E), a and eor values. The ABC's are specified
by listing their r, pr, x, px, ¢ and E values at each gap,
as well as at e=eor and e=e°r+n.

iii) Selecting an energy near Ef/Z occuring at

6:6 , the corresponding r, pr and ¢ values were used to

or
initiate "acceleration" of a ray backwards by the code
"Goblin-4".11 An adjustment was made to the starting

energy so that the corrected Goblin results for the initial
-energy (130, 6=01), our last free parameter, would be

0.9 Eg. ,We then interpolated in the AEO table of step ii)
to find new starting r and pr for the adjusted energy and
the backward run was repeated. Three or four iterations
were usually sufficient to obtain CR starting conditions

at 6=01 in this way.

18

1v) Wlth the initial CR coordinates (ri, pri' ¢i’ Ei)

now determined, we used the code "Devil", written by the
author (see Sec. 5.2), to accelerate the CR and rays
differing from it slightly by Ari, 6pri and/or 6¢i. Devil
output the orbit properties (r, pr, ¢, E, x, px) at each
dee gap and any other azimuths desired (usually 0 and

or

60r+w) along with the differences in r, REO’ x and px

between the CR and the displaced rays. This output forms

the great majority of the data displayed in this thesis.

19

2.3 The Accelerated Equilibrium Orbit

To provide a framework in which to present our data,
it is convenient to discuss some of the properties of the
orbits which we shall calculate as outlined in the previous
section. In particular, the ABC, the CR and certain
properties of the beam surrounding the CR deserve special
attention, and in that order, because each item depends
intimately on the one preceding it. For the ABC, we are
interested in its variations with energy, phase, E1 and
starting azimuth in three and four sector fields.

We can derive a simple formula for the initial condi-
tions of the ABC by using a transfer matrix formalism in
which betatron oscillations may be represented as rotation

of the vector
X= X " 171px,

for n=Ro/(vrp)=l in our units. At each gap, the phase point
representing an accelerated particle experiences a shift

in the negative x direction equal in magnitude to the
corresponding GREG. If this shift is represented simply

as 6, we may determine the initial amplitude x0 for an ABC

to be

X = x - 31p = -i5cos(vra/2)/sin(vrfl/2), (2)

X0

(see Sec. 5.1.1). Including the px shift which also occurs

at each gap would add a small real term to X0, but little

20

to our understanding of the AEO, so we shall not treat it
specifically. The transfer matrix used here neglects all
field structure and is symmetric between half turns, that
is, after one half turn, x=xo and p¥=pxo.

Figures 2 and 3 show Disport-2 results for AEO's
in 30 Merproton fields with N=4 and N=3, respectively.
The ABC properties are plotted vs. energy at a=138° and
vs. dee angle at E=7.5 MeV, both at 6=6°r=0°. We plot
the average in mils of the px values at 6=0° and 180°
for comparison with Eq. (2). The average x values are
not included as they never exceed 1.5 mils. To measure
orbit asymmetry, i.e., the deviation from Eq. (2), we
plot the deviations from the average in x and p# values
at 0=0°. The x deviations have sign opposite that of

the other quantities plotted.

The results in the four sector field are in excellent

1:

agreement with Eq. (2) using 6 varying as (AE) E- , where
AB is taken as the energy gain per turn divided by the
number of acceleration gaps (see Sec. 2.5). The asymmetries
are quite small, except at rather low energies where the
assumption of cOnstant 6 begins to fail markedly. Such
agreement with Eq. (2) is not too surprising since the field
and dee geometries are both symmetric between half turns.
AEO's in a three sector field (Fig. 3) are quite

asymmetric, however. Numerical agreement with Eq. (2) is

quite poor for both dee angle and energy dependence.

Obviously, Eq. (2) makes no allowance for the fact that the

VALUE (MILS)

3C

VALUE (MILS)

SC

‘33:

21

 

'3
F

 

- 0
“_" 138 x P, AVERAGE
5: - “3 KEV . + Px ASYMMETRY
o x ASYMMETRY

d

 

 

 

3 s s 12 15
ENERGY (MEVJ

(b)

 

g
I

9-.
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U

8
T

?.

 

F- = 7-5 "EV x P, AVERAGE -<
+ Px ASYMMETRY
o X ASYMMETRY

 

 

 

 

 

I .
90 120 150 . 180
DEE ANGLE IDES.)

. AEO properties in a four sector magnetic field at

6=60r=O°. "Average" values are the mean of px(O°)

and p (180°). Asymmetries are values at 0° minus
the aéerage value.

22

 

 

 

 

 

 

  
   
   

 

"5" °‘ = 138° x P, AVERAGE
mafi— E! = ”'3 KEV + Px ASYMHETRY .4
a. b o X ASYMMETRY _
£10m— ' -4
L51 80+- .1
.J
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HOA- -4
20L -
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ENERGY [MEV] -
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guar- ' '4
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J L 1 l

 

 

 

so 120 :so :80
DEE ANGLE [086.]

Figure 3. AEO properties as in Fig. 2 except with a three
sector magnetic field. Compare the ordinate scales
with those in Fig. 2.

23

transfer matrix is no longer symmetric between half turns
because the magnetic field no longer possesses two—fold
symmetry. Both asymmetry and average values do vary

directly as AE and roughly as AE/E. The curve labeled

"AE/E" in Fig. 3a was positioned by choosing a proportionality
constant C (=2864 mils) such that, at 7 MeV, the pK asymmetry
value is C(AE/E). (Such behavior is typical of the "gap-

crossing" resonance.)12

Both x and px asymmetry values
vary as cos a.

Since the four sector field is symmetric about any
diameter, changing eor has no effect on the curves in.
Fig. 2. However, the three sector field results are
functions of eor, the most striking of which is the reversal
of the x and px asymmetry values with a 30° change from
eor=o°. For both cases, sweeping eor through one sector
causes the phase point representing an ABC to "rotate”
about the origin counterclockwise by 360°. For the four
sector field, the point maintains a constant distance from
the origin, while for the three sector field, the distance
-varies according to the asymmetry values discussed above.
This effect will show up later in the x-px history of the
CR. (See Fig. 4, for example.)

In Eq. (2) above, it is assumed that the phase ¢ is
zero. When this is not the case we may consider 6 to be

an average radial shift which is proportional to sin(a/2)cosq>

and obtain the total shift at each gap by adding the

24

deviation term E, varying as cos(a/2)sin¢. Replacing
6 by 6-E at odd numbered gaps and by 6+8 at even gaps,

we find an additional term in Eq. (3) so

x0 = -E sin(vra/2)/sin(vrn/2), (2a)

indicating that initial x values only will be affected.
(See Sec. 5.1.) Numerical agreement between Eq. (2a)
and Disport-2 results is reasonable at high energies but
worsens at low energies as the centering dependent phase

shifts discussed below become important.

25

2.4 The Central Ray
Since our CR is chosen to conform with an ABC, we
expect to be able to trace its characteristics back to
the ABC properties just discussed. Let us begin with the
x-px history of the CR. U81ng 61,02,03,04,0

as observation points,an history of the first fifty turns

and 8 +w
or or

is shown in Fig. 4 for four separate cases using 138° dees
in a three sector, 30 MeV proton field (E1=l43 keV/turn).
The four values of eor are evenly spaced across one sector.
The dots represent each of turns 1 through 5 with only every
fifth turn accented for 10 through 50 for clarity. For
comparison purposes we include the crosses locating AEO
positions at e=eor and 80r+n for appropriate energy and
phase values every tenth turn_starting at turn five.
Qualitatively, this figure is the same as that for all
other dee angles, ¢F(T) curves and El values in a field
with N=3, with the one exception of 90° dees. In this
glast case, dee symmetry causes all histories to be found
in the fourth (x<0, px>0) quadrant of the x-px plane.
:Field symmetry gives this last result for all N=4 field
cases.

In. Fig. 4a, the e=o° history is close to the line
connecting the ABC positions and may be seen to begin
oscillation about that line near 1:40, in accordance with
the discussion of Sec. 2.1. Typical differences between

CR and ABC values (e.g., 6x=x(CR)-x(AEO)) are: 6x=30 mils,

26

Figure 4.--X-P history of the first 50 turns for a
welI centered proton with a=l38°, Ef=30 MeV,
E =l43 keV/turn, N=3, h=1. Every turn is
given (dot) to T=5 (circled) with every fifth
turn shown thereafter. Crosses mark AEO

locations at 10 turn intervals beginning at
T=5. All axes are in mils.

2"/

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28

6px=23 mils for 1:7, 6x=l7 mils, 6px=—8 mils for 1:28 and
6x=10 mils, 6px=3 mils for T=49. Starting from this
history, one can easily justify the locations Of the other
histories by assuming that a phase point rotates counter-
clockwise about the origin by an angle nearly equal to that
through which the actual particle coasts between gaps
and including the acceleration dependent shifts in the
negative x direction at each gap.

Increasing eor from 0° to 300 gives Fig. 4b which
may be related to Fig. 4a by reversing the roles of x and
px. This is a direct consequence of the exchange of x
and px asymmetry values for AEO's under the same dee
rotation. These asymmetry values also account for the
inequality between px:°° values at gaps l and 4 and those
at gaps 2 and 3 in Fig. 4a and c (b and d). Further incre-
ments of 30° in Get give Fig.'s 4c and d which are quite
similar to Fig.'s 4a and b with a reversal in gap labels.

The second CR characteristic of interest is its
phase history at each gap and the relation between these
histories and ¢F(1). A sample set of phase histories may
be found in Fig. 5a-d for the same four cases as Fig. 4a-d
while the corresponding ¢F(T) is labeled 'C' in Fig. 17 page .
There are two effects which cause deviations between ¢F(T)
and actual CR phase histories, ¢CR' Of lesser importance
is the form factor effect. In a sectored field the EO's

are scalloped, a consequence of which is that the E0 path

29

 

 

r Gap 2 i (a)
20m \ . 8 =00
lGap 3 or
10-
\bap u
“ Gap 1 _
4' A“IV'v-vvvwvvvvvv
J 10 0 u
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20- Gap 1 6 =”300
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10
p [\Gap 3
Gap 2
110 0 ‘-‘ vvvvv “r“ vvvvvv
20 " Gap 2 (d) o
\‘//,Gap 3 80r=90
10 -
0 /Gap 4
Gap 1
170- _, I " ' 0

Figure 5. Phase history (deg vs. T at each gap) for the
'cases shown in Fig. A. -

30

length is greater in the hills than in the valleys. Particle
phase tends to increase while crossing hills and decrease
while crossing valleys. For h=1, the phase oscillation
amplitude thus produced is of the order of only one degree.
This amplitude is directly proportional to h and becomes

more important with higher harmonic operation.

Of greater importance are those deviations arising
because the actual orbit center does not correspond to the
field center, i.e., because x and px differ from zero. In
general, one can show that the centering dependent phase'
shift involved with traversing an angle 8 may be well

approximated by

5¢CD=h(Px(l-cosB) + X(sin8))/Ro (3)

where RO should be taken as an average orbit radius and
x and px pertain to the beginning of the angle in question.
Note that 6¢CD is expected to be large only at small radii
and that px carries the greater weighting factor if B>n/2
and so will usually be of greater importance than x in
determining 66GB. Since °¢CD may easily be many degrees
on early turns even for h=1, we see that there is an
intimate and important connection between the centering of
a particle and its phase history. 3

In particular, we_may inspect an x-px history, such
as Fig. 4a, and predict the 66GB occurring between any

two dee gaps with Eq. (3). Table I contains such predictions

31

using reasonable Ro values for El=143 keV/turn and x and
px values from Fig. 4a for the first two turns. Agree-
ment between the quick calculation of Eq. (3) and real
shifts obtained from Fig. 5a is good and may be seen to
improve with energy. Because ¢CRI¢F at large T, the
larger 60CD's must straddle the ¢F(T) curve at small T
and because 6¢CD at 0=0l is positive for 00r=0° we can

now understand why 0 is small at 0=61 and large at

CR
0=02 in Fig. 5a.

TABLE I.--Comparison of 6¢CD from Eq. (3) and °¢RL from
Fig. 5a between dee gaps i and j for a=l38°.

 

 

T Gaps R0 3: px 6¢CD 6¢RL
i j (in.) (in.) (in.) (deg) , (deg)

0 l 2 1.37 -.020 .330 “ 23. . 20.
2 3 1.70 .010 -.190 -2. -3.4
3 4 1.93 -.400 -.100 -13. -14.S
4 1 2.17 -.O75 .340 1.3 1.2

1 1 2 2.41 -.030 .280 10.8 10.7
2 3 2.58 -.025 1.190 -1.7 -1.5
3 4 2.76 -.310 -.120 ~8.7 ~9.6
4 1 2.93 -.080 .310 .7 .4

32

The remaining three phase curve sets of Fig. 5 may like-
wise be explained using the corresponding x-px history of
Fig. 4. We take this opportunity to point out that, since
the histories of Fig. 4 are qualitatively representative
of all three sector field cases to be discussed below,

the phase histories of Fig. 5 may likewise be considered
to have all the qualitative features found in any case of
interest in a three sector field.

Similar data for a four sector field is found in
Fig.'s 6 and 7 respectively. Note how rapidly the crosses
representing AEO's approach the origin in Fig. 6a so that
only for T=5 and 15 are they clearly visible. The cross
locations are the same in Figs. 6b-d as in Fig. 6a and so
are not included. The highly symmetrical AEO's give rise
to very well centered beams at all dee angles, so the
6¢CD's are quite small. This implies that, in contrast
to the three sector case, the relative field-dee orientation

will have practically no effect on the phase histories, as

inspection of Fig. 7 immediately verifies.

33

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(a2 00

 

 

 

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Figure 7. Phase history (deg vs. T at each gap) for the cases

shown in Fig. 6. Note that gap crossing phase is
unaffected by 80F, unlike in Fig. 5.

2.5 The Components of Radial Separation
Between Phase Displaced Rays

Using the CR as a base, we may build up a typical beam
by surrounding the CR with rays initially displaced from it
in r, pr and t. Our basic problem in analyzing phase
selection is to determine, turn by turn, the radial separa-
tions of the particles constituting this beam so as to
specify the most effective slit locations for eliminating
those particles having phases differing from ¢CR by more
'than a predetermined amount. To do this we choose to resolve
the radial separation, AR, between any two phase displaced
particles, that between a particle and the CR being the
most generally useful, into two components: the radial
separation between the EO's having the particles' energies,
and the projection on the x axis of the distance between the

representative phase points on the x-px plane. 'That is,
AR = AREO + ARx. (4)

The first component depends only on the energy spread

between the particles and is easily understood and calculated.
The second component depends on a number of factors outlined
later in this section and resists simple calculation.

Using this decomposition, in this section we would like to
lay down some general principles on how the radial separa-
tion varies with turn number and the temporal structure

of the beam. We will discuss the effects of the spatial

structure in the next section.

36

Let us start with AREO. It is a straight-forward job

to determine that
REO(E.6) = RO(E)(1 + F(9)) (5)

describes the E0 of energy E in a periodic magnetic field

where
R (E) = A(2E/E )3. I (6)
O O

F(0) being the orbit form factor having the same periodicity
as the field.13 This description assumes that the field

is isochronous and that the energies are not relativistic.
Actually, F(0) is also a function of energy, but as long

as this dependence is weak (the case in all our fields)

we will suppress it, giving us.a simple formula for the

<:hange in the 30 with an energy change 6E:
i '8 '
AREO(E,6E,6) — A(6E)(2EEO) (l + F(e)). (7)

A good first order approximation for the energy

difference between phase displaced rays after any number

of complete turns is
AB 2 -E1(A¢)[:¢F(I’)dt’ (8)

(see Sec. 5.1.2) where A0 is the initial ray phase separa-
tion. The accuracy of Eq. (8), will be tested numerically
in Chapter 3. Its usefulness arises because, in conjunction
with Eq. (7), it may be used as a very simple method of

estimating the E0 Component of AR in Eq. (4).

37

The component AR.x in Eq. (4) does not yield to such
simple calculating techniques. Let us first define this
component more carefully. Consider the phase points
representing the CR and a ray displaced from it in time
by A6 but of equal energy. Let us label these rays C and
D, respectively. We will find it convenient to define the
vector A in the x-px plane between the points representing
such particles. The components of A are

Ax: XD - XC,

1p a pr - pxC'

while its length is

As defined above, Ax is equivalent to the ARx of Eq. (4)
and will be used in place of the latter from now on.

we now examine the processes affecting A and, in
particular, its length A. Let us start with C and D lying
on the same x, px point. We assume that their energy is
small compared with Bf and that lvr-ljso. Figure 8a shows
this initial situation. At the first acceleration gap, the
energy increase 6E produces a sudden jump in RE0 which appears
as a discontinuous shift by the phase points in the negative
x direction of magnitude 6RE0 given by Eq. (7). (See Fig.-
8a). (We shall neglect the Small concommittant changes in

P The points will then be separated along the x axis

REO') .
by a distance given by Eq. (7), where GE is taken as the

38

(a) (b) I (C) (d)

l'l a: ’D
l 3, 1.0 .3,

I I

D c

 

 

 

 

 

 

 

 

Figure 8. Schematic representation of the growth of A
between the CR (point C) and a phase
displaced ray (point D). (See text, p. 37.)

Gap 1,3

 

Gap 2,4

 

‘Figure‘9. Difference in rf times of gap crossings for zero
(crosses) and non-zero (dots) phases for h-l.

39

difference in energy gains between the particles, AEG.
(Figure 8b.) We term this process ”disjunction”. During
the drift between gaps, 0,‘A rotates counterclockwise
about its origin by the angle vre (Fig. 8c.). At each
succeedinggap, A undergoes a change in length, 6A, which
depends on AEG and the orientation of A, since only Ax
changes in the disjunction process (Fig. 8d).

Further insight into this process may be obtained by
considering the net disjunction from a pahrof opposite gaps.
For 08180° and vr=1, AA over one turn is parallel to the
x-axis with magnitude given by the difference between the
6A's at each gap because vre should increase by nearly 180°
between such gaps. (This assumes that AEG = 6E(A¢)-6E(CR)
has the same sign at_both gaps, the usual case.) This
difference is zero under only the most extraéordinary
circumstances since 6A depends on energy and phase by
Eq.'s (7) and (8) while the same half turn asymmetry discussed
in connection with AEO's modifies the 6A of the first gap
when seen at the opposite gap in a three sector field.

With four accelerating gaps, the net disjunction is
quite different for the second gap pair, AA24, than for the
first, AA13. Mest apparent is the difference in the actual
rf times of gap crossing involved for a constant nonézero
phase. Figure 9 shows such times for some (arbitrary) values
of a and °CR (dots) on a part of an rf wave, along with the

corresponding zero phase times (crosses). Vertical distance

40

between the dot and cross for a given gap translates directly
into a difference in disjunctive effects between gap pairs.
We shall term this the rf asymmetry effect.

Furthermore, in any four gap situation, the net change

in A over one turn as measured at 0=00r, for example, is

AA,= (AA13)K1 + (AA24)K2. (9)

The K1 and K2 are vectors depending on the dee angle since,
to first order, they would parallel the respective gap pairs.
But K1 and K2 are not simple to calculate accurately since
they also depend on the past acceleration history of A.

The disjunction at an intervening gap effectively changes the
angle of rotation of A from the expected value of nearly

180° between opposing gaps, a change which must be reflected
in K1 and K2. We shall refer to Eq. (9), linking the net
disjunctions at each pair of opposing gaps to the change in

A per turn, as defining the gap-pair coupling effect.

We now can see that it would be quite difficult to
calculate this second component of AR. To obtain an expression
like Eq. (5) for A or Eq. (7) for AA, one would need an
analytical form for the phase as a function of gap and turn
numbers, which is not readily available (cht) is absolutely

inadequate for this), and a similar form for the vectors

K1 and K2 inEq. (9) above.

41

In lieu of the optimal situation, we present Fig. 10
showing the x-px history over the first 40 turns at 0=0°
for two rays separated by A¢=i2° from the CR, used as the]
origin of the figure. This CR is the ray, the x-px and
phase histOries of which were used in Fig.'s 4a and 5a.
Note that precession becomes apparent near turn 6 or 7
where vr=l.01, that A becomes constant at about 11 mils
after 10 or so turns, where vr=1.02, and that only precession
is important thereafter.

From a broader viewpoint, the process whereby A
increases in length may be seen as a resonance phenomenon
driven by the asymmetries present in the acceleration
process. In both three and four sector fields, if ¢F(T)
alone were accurate at each gap of turn T, a driving
asymmetry would be present since the first derivatives of
the rf waveform yielding AEG(=(A¢/wrf)(dvrf/dt)) would be
different at each gap crossing (see Fig. 9, for example)
so the 6A's from different gaps would rarely cancel (the rf
symmetry effect). The centering dependent phase shift which
actually always occurs serves to accentuate this asymmetry
in almost all cases since ¢F(T) is large where 6¢CD is
large. ( Exceptions to be noted later occur when the
'phase at gaps 2 and/or 4 exceeds (n-a)/2.) In three sector
fields, there is an additional driving asymmetry represented
by the gap-crossing resonance discussed in connection with

the ABC. As will be shown clearly by data below, A shows

Figure 10.

U2

 

 

-201

o: = 138°
E! = 1‘33 KEV

. PA
18 20

Ac = +20

6x-6px history for the first 40 turns for rays
with A0212O. Preperties of the reference CR are

shown in Figs. 4a and 5a.
once per turn at e-eor

Points are plotted
. All axes are in mils.

43

much larger variations with changing acceleration parameters
in three sector fields than in four sector fields because
of this second driving term. A decrease in the rate of
resonant growth of A occurs as ¢CR approaches ¢F(T) which
in turn approaches zero, as the energy in the denominator
of Eq. (7) becomes large and as vr-l begins to deviate
significantly from zero. It is this last condition which
really destroys the resonance regardless of the influence
of the first two.

We have now seen that both ARE0 and A depend on the
ray phase separation A0, ARE0 through Eq.'s (7) and (8)
and A through a more circuitous route involving the
disjunction process. Discussion of phase selection and the
evaluation of parameter effects may be facilitated by

introducing a quality factor Q as

Q = 0E0 + 0x = (AR)/(A¢). (10)

where
QEO = (AREo)/(A¢)I

0x = Ax/(A0).

Note that, although we do not show it explicitly, Q is a
function of T and 6. Obviously, large Q implies good phase
selection. It is the behavior of Q with 1,6) and various
machine parameters which we shall actually be investigating

in the next chapter.

44

We shall regard Q as constant with respect to A0 in
any given situation. Orbit calculations show that, for
most cases below, this is true for A0 as large as 32°.
Even for the exceptional cases, Q may easily be assumed to

be constant over the small A0 we would expect a practical

slit to pass.

45

2.6 Effects of Radial Emittance

We now turn to the consequences of having sources of
finite width emitting particles in many directions. The
most important mechanism to be considered is the centering
dependent phase shift of Eq. (3) which imparts a time
displacement relative to the CR, 60, depending on the
initial particle position in the 6x-6px plane (6xo,6px°)
having the CR as origin. On early turns, Ro increases
rather rapidly, even on a single turn, so that an average
phase shift 60av develops between a spatially displayed ray
and the CR. Consider the five rays labeled A through E
in Fig. lla at 1:0. Their subsequent phase behavior
relative to the CR appears in Fig. 12. Again, the CR is
that used in Fig.'s 4a, 5a and 10. Note that the rays of
larger 69x0 tend to develop the larger 60av's. 'Note also
that there is a small amplitude (~0.l° compared with ~1.l°
average shift) phase oscillation appearing at larger turn
numbers where Ro varies only slowly. The phase oscilla-
tions are related to the slow oscillation of the:displaced
ray about the CR, particularly the changing 6px values.

Figure 11 may be taken as showing the evolution of
a section of phase space assumed occupied by beam at 5 turn

intervals from its initial circular condition at T=0 to T=25

U6

 

 

Figure 11. 6x-6px history for 16 rays initially located on
a circle of 20 mil radius. Properties of the

reference CR are shown in Figs. 4a and 5a. All
axes are in mils.

u 7

Fig. 11. Cont.

 

 

 

 

 

34,,

 

A
l 1 I / 1T

10 2'0 3'0 4'0 30

Figure 12. 60(deg) relatice to the CR vs. T for rays with the
initial spatial displacements shown by the
correspondingly labeled points in Fig. 11a.

48

(Fig. If). The distortion of the original (arbitrary)
circular shape may be seen to be essentially complete by
T=15 and the resultant beam ellipse mainly rotates there-

after. A nice explanation of this phenomenon has been

14

given by Bolduc and Mackenzie. The occupied area on the

6x-6px plane varies from 1256 mils2 at T=0 to nearly 1200
mils2 at T=15, a pleasing result since the accompanying
6E-60 co-ordinate changes are small.

There is a fixed relationship between the orientation
of the ellipses in Fig. 11 and the phase oscillations of
Fig. 12 such that, for a given ray, the turn where its
extreme 6px~value occurs corresponds to the turn with an
extremum in its 60. Finally, note that, just as the rays
of larger 6xo experience larger distortions in the 6x-6px
plane, so do they also show larger oscillations about their
60av (”0.3° with ~0.39average shift).

While we will emphasize the effects of temporal over
spatial displacements in this paper, we will present some
data pertinent to the behavior of rays with the initial
conditions used above for Fig. 11. We should, therefore,
establish whether the effects we see are linear with 6x0
and 6pxo. Figure 13 gives values of (6x2 + 6px?)° between
the CR and the ray marked 'A' in Fig. 11 as functions of
initial 6x0 and 6px0 for three different dee angles. For
values of 150 to 200 mils one observes deviations from

linearity, especially for 90° dees, but for smaller values

49

Figure l3.---(6x2+6px2)1/2 vs. 6x0 and 6pxo at various dee
angles. Variation is close to linear over the
range of initial conditions shown for all dee
angles.

 

amnezixi AEHEHZH

 

 

8N one 2: on
A A A A
m
e on
_D
o ,
e as
> .
0
Jam.
D. d
nu IJOON
e :58
0
48m
com .03 o
a emmaux 6 18m
OomHHVO Q
L8:

,ee

tn
1%

(Sim) NOIlvanas 33Vd8 BSVHd‘

.ma mszwfim

6.5: x .ZHEZH

 

ooN ow" on: on.
A A a A m
m
.0 Room
G
D
G D
b , 18m
0
D com
0 A
00m “8 D AOONa
OWmQHR..D
U Coma"! Q

 

.1 com“

[SWIM] NOIlVSVdBS 33Vd8 BSVHd

51

corresponding more closely with the Ax-Apx areas to be
expected in a high resolution cyclotron, any deviations
are unnoticable.

While it does not affect our discussion, one should
note that 6x and 6px related phenomena are not generally
linear as emphasized in Fig. 4 of the work of Blosser
previously cited.5

As the data in Fig. 13 implies, the 60av are, however,
found to vary quite linearly with both 6x0 and 6pxo in the
range used. Table II gives approximate rates of increase
of 60av with 6x0 and 6px0 observed on turn 20 with a=138°
(a=90° for h=3) on acceleration harmonics h=1, 2 and 3.
The 60av for the 6px0 displaced rays grow at better than
twice the rate for the 6x0 displaced rays on all harmonics.
That 60av shows unexpected behavior for h=2 has to do with
precisely how the 60av is generated and is net germane to

our purpose.

TABLE II.--Rate of increase of 60

and 6x0 and 6px0 in
(deg/10 mils) at 1:20.

3V

 

 

h=1 h=2 h=3
6x0 .22 .10 .45
6pxo .56 .50 1.40

 

Sinceh60av as a function of T increases from zero
at T=0 while ¢F(T) decreases from its maximum at that point,

from Eqs. (7) and (8) it is not surprising that the rays

52

under discussion show very small ARBO values. In fact, the
rate of increase of ARE0 with 6x0 and 6px0 is about an order
of magnitude less than that shown in Fig. 13 for the increase
in phase space displacement.

We make these comments by way of eXplaining that
the radial separation between the CR and spatially dis-
placed rays is insensitive to the parameters in Chapter 3
except for h, the effect of which is summarized in Table II.
Therefore, we present no further data on such rays in
Chapter 3. we will, however, return to this subject when
.the question of locating phase selective slits arises in

Chapter 4.

3. DATA AND ANALYSIS

Without exception, the data quoted for AREO, A and
Ax in tables in this chapter are values measured between
two rays initially displaced by A0=iZ° from the CR. There-
fore, using the appropriately labeled columns in any table,
one can obtain the Q values as defined in Eq. (10) from
QEO=AREO/4 and Qx=Ax/4. To minimize the number of minus
signs used, the table entry ARBO is taken as positive if
0 o . .
AREO (—2 )>AREO(+2.)' the usual case for a pOSitlve phase

curve. Likewise, the table entry AX is positive if

0 O
Ax(-2 )>Ax(+2 ).

3.1 Dee Angle and Orientation

In this section we wish to discover how the dee angle
(a) and orientation (00x) affect AR corresponding to an
initial A0. All results in this section were obtained from
Devil using a three sector proton field with Ef=30 MeV,
E1=l43 keV/turn and the 0F(1) curve labeled "C" in Fig. 17.

Figure 14 gives, as functions of energy, the eigen-
ellipse axis length ratios and the flutter, here calculated
as H3(r)/<B(r)> where H3(r) is the co-efficient of cos(36)
and <B(r)> is the a independent term in the Fourier repre—
sentation of the field (6=0o is a valley center). Figure 15

contains Vr and the peak energy gain precession angle

53

54

 

 

 

 

35’ 1’ r ‘r l' r* I 743
1.5» - 6.0
p
O
:m— -J 5.0 S
< —l
x 3'3
(D
H . a 9.0 "‘
><<1 3*- . . - FLUTTEB 2%
1.2- A 3.0
L L L l L L 2.0

 

1 2 3 ‘1 5 S 7
ENERGY (HEV)

Figure 14. Eigen-ellipse axis length ratios and flutter vs.
energy for an M.S.U. proton field with N=3, Ef=30

 

MeV. . .
1°06 1 T t I r I 530
1.05 950
1.0'1 360
1.03 270
(I 1 02
a O

1.01

 

 

23
O
('930) NOIiVIOU ‘WNOISSBDBUd

 

 

0
he.
88°29

L_ L l L L L
l 2 3 '1 S S
ENERGY (HEV)

\I

Figure 15. vr and 0(see Eq. 11) vs. energy for the field
of.Fig. lu.

55
E
w = (Zn/El)!o (vr-l) 00’, (11)

also plotted against energy. Both figures are presented for
future reference. Four sector field results will be dis-
cussed separately in Sec. 3.5.

In the tables appearing below, the eor values used are
evenly spaced throughout one sector while the labels "H"
and "V" are used to indicate a hill or valley center where
appropriate.

Let us first examine the radial separation component
given by Eqs. (7) and (8) to both assure the accuracy of
these equations and determine the effects of a and eor'
Table III gives AE and AREO for four values of eor' Turn
14 entries are included because-AREO(T=14) is the maximum
value for the chosen ¢F(T) curve while turn 21 entries are
present because ¢F(T=21)ZO°. If one compares the AE
values in the table with the values found using ¢F(T)
and Eq. (8) of AE(T=14)=28 keV and AE(T=21)=30 keV one

finds a variation of AE with 60 mirroring the form factor

r
phase shift. This shift corresponds to crossing half a

valley for eor=3o° or half a hill for eor=9o° and is present
because ¢CR was forced to be 0° at eor at the same large turn
number in all cases. The AE's from Eq. (8) are slightly higher

than the values at eor=o° because the 6¢CD values tend to

56

.mm .mm

mm .:m
.H: .mm
.m: .mm
.m: .mm
.m: .mm
.m: .@N
.nm .mm
.H: .mm
.H: .mm
.:: .mm
.om .mm
.Hm .nm
.om .NN
.m: .:m
.0: .mm

.mm .mm .mm .mm .mm

.m:

.mm .:m .mm .oem
.0: .mm .0: .mm .0: .mm .mm .mm .om
.H: .mm .0: .mm .0: .mm .H: .mm A>v.o:m
Jo: .mm .0: .mm .m: .mm .s: .mm Amv.omw
.om .mm .m: .mm .m: .mm .m: .mm . .oam
.m: .mm .m: .mm .m: .mm .m: .mm .om
.mm .5: .mm .5: .mm .:: . .mm Amv.oma
.H: .mm .H: .mm .m: .mm . .mm .mm A>v.o Hm
.m: .:m .:: .:m .:: .:m .m: .:m .osm
.:: .mm .:: .mm .:: .mm .m: .mm .om
.:: .mm .:: .mm .:: .mm .m:. .mm A>v.o:m
.m: .:m .m: .:m .m: .:m .:m .mm Amv.om
.mm .mm .mm .mm .mm .mm .mm .mm .OHN
.m: .em .Hm .sm .Hm .sm .mm .mm .om
.om .mm .:m .mm .:m .mm .m: .mm Amv.oma
.:: .mm .m: .mm .m: .mm .m: .mm A>v.o ::

 

Amaaev A>mxv

0mm: m:

cans

AmHHEV A>oxv AmHHEV A>oxv AmHHEV A>oxv Amafiev A>mxv

om om om no

m: m: m: m: m: m: m: m: a p
omaua wmaua omaua owaua
omnue: cpsp\>mx m:Hu:m an: mnz

 

.NH .me Ca 0 m>a30 on monogamppoo pom:

LO ho
cameo ospmcwms wee .mmfiwcw mme msofisw> poo =+ one one one as 0mm: was m<nn.HHH mom:e

57

push the average of ¢CR(T) below ¢F(T) on early turns.
Note that the AREO's vary as the corresponding AE's, but
modified by F(6). The 13% variation between hill and
valley values of AREO may be calculated using the field
flutter; comparing T=l4 and T=21 data Shows this variation
to be independent of energy, at least in this small region
of interest. In summary, our data show that dee angle is
of rather little consequence in determining AREO, but that
eor is of definite importance.

Table IV lists A and Ax values. At this point we will
explain only the qualitative details of the A data by using
the appropriate ¢CR’ exemplified in Fig. 5. We shall return
to the Ax values in Chapter 4.

The simplest case to discuss is, of course, a=lBO° where
there is no gap pair interaction (cf. Eq. (9)), where
positive phase always means that GA(=AREO(+A¢)-AREo(-A¢))<0
at each gap and Where, on early turns, Ax=l. The net AA-
per turn will depend on the difference in AE's between the
gaps and should, therefore, vary roughly as the area between
the phase histories at the two gaps. However, when E is
small, the (E)45 dependence of Eq. (7) must be considered,
so that the question of whether the larger phase (and, hence,
03) occurs at 01 or 62 becomes important. With regard to
achieving large final A's, the area dependence favors
6 r=0°,60o while the energy favors eor=30°,60°, adding to a

0
double blessing on 6013600 and a double curse on eor=900.

8
5

 

 

.m: .0» .m: .mm .mu .s .m: .es .mu .ma .osm
.ma .mu .m .om .2 .w .H .2 .HI .HH .om
.mal .mm .HHI .mw .mu .ww .wl .om .0 .mm .02m
.ma .m2 .2H .om .mH _ .mm .HH .hm .H .02 .00
.HH .wa .mn .Hm .mu .2m .51 .mm .m .2m .oam
.mfiu .:m .m: .:m .m: .:m .0 .mm .m: .mm .om
.om .m2 .oa .oa .2 .NH .m .mm .m .mm .omH
.mml .mm .OHI .oa .5: .mm .nl .mm .m: .5m .0 .Hm
.mml .Hw .mml .2m .ml .o .m .m .m .HH .owm
.wm .Hw .Hm .mm .m .2 .ml .m .NHI .oH .0m
.5:: .mm .mm: .05 .mm: .ms .mmu .ms .:m- .mm .o:m
.mm .om .05 .ee .em .:e .mo .ms .m: ..mm .00
.mml .2m .22: .w2 .mml .m2 .mml .m2 .mml .mm .oam
.mH .om .mm .mm .mm .Hm .om .mm .om .mm .om
.2ml .M2 .m .2 .ON .mm .om .mm .mm .2m .owa
.2H .m2 .ml .m .mHI .2m .mml .mm .mml .2m .0 2H
AmHHEv AmHHEv AmHHEv AmHHEV AmHflEv AmHHEV AmHHEv AmHHEV AmHHEV AmHHEV
x x x x x no
A 2 x A 2 K A 2 2 4 a p
omud omaua mmaua omHnd omfiud
omnue: caspx>mx m:Hu:m muz
so no N
.HHH magma ca czonm mm momwo mean mgu pom :+ one 6cm ouo no new <|I.>H mqmde

 

59

If one studies data at fixed eor for decreasing a,
one observes systematic changes as gap pair interactions
take precedence. Among these is a rather subtle inter-
‘action, not previously mentioned, between the magnetic
field sector structure represented by the eigen-ellipse
and the gap pair interaction which produces the early turn
orientation of A on the x-px plane. First of all, the
eigen-ellipse major axis rotates as one continuously changes
his observation angle 6 so that it forms an angle with the
positive px axis of 6e(6) which, for N=3, at 9=0°,30°,60°,
90° is given by ee=o°,45°,9o°,135°. Since for N=3,
6e(6) is periodic over 120°, upon traversing 1800 of azimuth,
one finds the eigen-ellipse "rotated" by 90°. (Recalling
our comments on the ABC's in the same type of field, it
becomes apparent that the eigen—ellipse gives us an easily
visualized method of determining orbit asymmetry effects.)

Now, when A is parallel to the eigen-ellipse major axis,

A will be maximum (note the eor 30°, 6=210o and 60r=90°,

6==90o entries in Table IV), and when A parallels the minor
. . . . o o = o
aXis, A W111 be minimum (6or 30 , 9=30 and eor 90 ,
e=270°).
Certainly a much more graphic demonstration of this
phenomenon is found in the oscillations of the curves of

A as a function of T in Fig. 16. The data in Table IV for
0L=l38o represents two points on these curves for each eor
value. The oscillation amplitude is determined by the

requirement that the extreme values be in the ratio of the

60

IL 003 = 00 L 903 = 300
80» i 80% f

r " I \
60y ’\ 30' \

_ / . \ A- ....J. \
40le ---- -‘,.. /\ 40- .' / " -----

'l \..°} .....

 

 

  

20W 20 .

. . 1’.
2'03'0H‘05'0

an . ---
AREO a .......
A 3 —

(d)

 

 

_ _ 00.. = 90°
80%
' L
60+
‘IO - 'Q'Q.
- In Or. T. \
20 T .K”.

   

 

 

 

10 50 30 i0 ST)

F {01250 30 #0510
\ / - ' l
i I

Figure 16. AR, AREO and A vs. T for N=3, Ef=30 MeV, TF=210,

‘ 6'1380 and h=1 plotted at 6:6 for various 6 's.
or or

Ordinate values are in mils.

61

major to minor axis lengths, which ratio is plotted in
Fig. 14.

Getting back to the interaction mentioned above, let
us now examine the case of 900 dees. With these the
asymmetry in rf times of gap crossing causes A, after a
few turns, to point at about 1350 from the positive px
axis until precession becomes important. Comparison with
6e at various eor's shows that for Bor=90o the orbit
asymmetry term driving AA is in phase with the rf driving
term, while for eor=300 the two are 1800 out of phase.
This is why the pattern of large and small numbers shifts

among the 60 columns with decreasing a.

r
In summary, Table IV shows that 60r=60 gives the
greatest A's and, therefore, the best Qx's for all dee
angles except d=90o where 60r=90o is superior, all other
things being equal. Since the A's are comparable with the
AREO's, it follows that the dee orientation must be givenl

some thought whenever phase selection is desired and more

important considerations do not dictate a specific eor‘

62

3.2 Fielder Phase Curves

In this section we shall investigate the effects of
varying the magnetic field contour with a fixed dee
geometry, our study here concerning the effects of the
different ¢F(T) we may produce. We use the initial phase,
¢OE¢F(T=0) and the turn number To such that ¢F(T=To)=0,
to characterize these curves. In any real situation, one
would locate the ion source so that the CR corresponds
to the centroid of the maximum intensity ion group from
the source. Thus a change in 00 corresponds to a change
in the source-puller location relative to 61. Changing
To at a fixed 00 is a change in the magnetic field.

The five curves used here are presented in Fig. 17.
The curve parameters (¢O,To) are: A=(10°,20), B=(20°,10),
c=(20°,20),.n=(20°,30), E=(30°,20). Curves going strongly
negative on early turns are not considered because of the
possible associated axial defocusing difficulties. An
attempt was made in all cases to obtain the straightest
possible line from 00 to zero for DEFETO with Fielder.
Behavior of ¢F(T) on the first few turns is dominated by,
the central cone which extends some three or four inches
in radius and causes the initial rapid fall-off in phase
observed for T<3. Control over all other sections of ¢F(T)
is about as good as is possible since the average trim coil

spacing at M.S.U. is nearly half the magnet gap (6.75 in.).

63

.mLmuoEmLmo m>pso pom .mo .c
.pxmp mom .>mz omuwm was an: pom wcamfim couOha..D.m.z pow Aevme .NH mhswfim

mwmzaz zmah
ow . or on ON an
. .41 J

 

 

 

 

 

1 w m>mao
a m>¢zo
... ‘ o wise
Am m>¢zu
< m>§o

cm

C9039

 

 

['930] Did

64

To obtain the data on AREO in Table V and on A and
Ax in Table VI, we performed the procedure outlined in

Sec. 2.2 for protons in three sector fields with Ef=30 MeV,

a=130°, eor=0° and 60° and E =143 keV/turn. The columns

1
labeled "Area" in Table V contain the results of using

Eq. (8) and the different ¢F(r)'s with trapezoidal rule
integration. The values in parentheses in each column are
"normalized" with the corresponding values for case B

(note 1's in those columns). These normalized values of
AREO(eor=0°), AREO(60r=60°) and "Area" should be the same

for a given ¢F(T) if Eq. (8) is accurate. Agreement averages
about 5% except for case A where 20% to 30% differences are
average. Since all beams are similarly well centered, it
should not be unexpected that the 5¢CD's are of similar
magnitude in all cases. In case A, this means that on the
very first turns ¢CR at 91 and 93 will be negative, decreas-
ing AREO and accounting for the apparent discrepancies.

Such evidence leads us to say that Eq. (8) is valid and

useful.

The A's of Table VI show that ¢F(T) for T<10 or so
is a most important characteristic and that the final A is
rather insensitive to the fine details of the ¢F(T) curves.
Roughly one may group cases A with B, C and D, leaving E
by itself, using the effect on A as the distinguishing
feature. From Fig. 17 it is seen that this grouping also

is a rather natural one for the average of the ¢F(T)'S,

65

 

A.Hv Am:m.v Am:o.v Ammm.v AHmH.v
.25 .m0 .m2. .2m .2H .om
A.Hv A.Hv Ammm.v Ammw.v A2mm.v Ammo.v Ammm.v A2om.v “Ham.v AN©H.V
mm.: .mo Hw.m .om mo.m .m: mH.H .Hm om. .HH .0 om
A.Hv Amww.v Awmm.v Ammm.v AhmH.v
.se .H@ .m: .em .ma .oo
A.Hv A.Hv Amw>.v Awm>.v Ammo.v Aomm.v Ammm.v n2mm.v Amam.v AHNH.V
:m.: .ms o:.m .mm mm.m .m: mm.a .:m mm. .NH .o ma
A.av Aomm.v fi2mm.v AHH2.V Ame.v
.oe .Hm .m: .mm .mH .oo
A.HV A.Hv Ammw.v Amou.v Am2w.v Amom.v Ammm.v Ammm.v AHNN.V ANmH.V
mm.m .so mo.m .m: Hm.m .:: om.H .om as. .HH .o oa
A.Hv Am2m.v Ammm.v Aom2.v Amma.v
.om .mm .mm .2m .5 .0m
A.Hv A.HV Ammm.v Amao.v Ammm.v Aawm.v Amm2.v A2m2.v A2om.v Amaa.v
mm.m .mm m:.H .mm o:.H .mm :o.H .mm o:. .o .o m
Aces» ACLSp Amps» Acne» Acne»
Iompv AmHHEV nomnv AmHaEv nomuv AmHHEV loopv AmHHEv Iompv AmHHEV
won: one: one: omm: son: our: song ommo song 0mm: goo e
m mmgo o mmgo o mm:o m mmgo < mm<o
omnuea onso\>ox m:HuHm owmauo Hus muz

 

.Ammmmnuconmav m mm<o ou pmNHHMEpoc wmzam> one mm>p50 ocu pots: moons on» one
no no
ooosHooH .sa .wam on ozone moenso Aevme one son ooou one one oou one on omm:un.> mamas

.mv .om .mH .ww .«N .Hm .Hm .mm .NH .hv .om

 

.oau .H: .m. .:m .oH- .om .oHu .o~ .m: .o: . .o om
.mm .ma .oo . .Hs .mo .:e .Ho .oo .Hm .oo .oo

.mm: .mm .eau .:m .m:: .:m, .mau .m: .:H. .m: .o ma
.mm .sm .os .ms .mo .oo .oo .oo .oo. .mo .oo

.mm. .:m .omu .:m .Hmn .:m .m:: .oH .mHu .oa .o on
.mm .mm .om .Hm .m: .m: .o: ..m: .:: .m: .oo

.:mu .:m .oau .oa .::: .:H .mau .ma .oau .o: .o m

%
Indeed finesse finesse finesse Andes“ finesse finesse Anaes. finesse Andes.
an MA x um um HO
: r : a : : a a : : o e
m ammo o mmeo o mmeo m mmso g mmgo
omens: susux>0x m:HuHm owmaus an mnz

 

 

.H

o o
.> manna CH czonm no women mean on» MOM com" 000 can Ocuu one as x« can «II.H> mamma

67

particularly before turn 10. As ¢F(T) increases at a fixed
I from case A to case B, the asymmetry in the rf times of
gap crossings between the two pairs of Opposing gaps
increases, thereby increasing AA13 and decreasing AA24
at 6°r=0. This rf asymmetry driving term is not linear
with phase since a factor of four change in ¢F(T) between
cases A and E gives only a factor of two change in A.

We present Table VII with AREO and A vs. I data for
a=180o andt90o to demonstrate that we cannot treat a,
eor and ¢F(T) disjunctive effects as being totally inde-
pendent. One difficulty is brought out in the a=180°,
eor=0° data for case A showing larger A's than for case C
or even case B for T>10, apparently contradicting the last
paragraph. But we mentioned above the negative phases occurring
with curve A at 61 on early turns. On these turns, AA=6A1+6A2
instead of AA=6Al-6A2 as is true when all phases are positive
as was assumed above. A second difficulty is the decrease
in A as one moves from case A to case B for a=90° and
eor=o°, again, an apparent contradiction. However, the
explanation is that 6A1 and 6A3 are becoming closer in value.
As ¢F(T) increases, dVrf(63)/dt-dvrf(61)/dt increases since
dVrf/dt changes non-linearly with phase with the result

that 6A3-6A decreases because the energy denominators of

1

Eq. (7) make 6A3<5A1.
The last phase-curve-related changes in AR we wish to

mention are the locations and magnitudes of the first

maxima in both ARBO and A. For AREO there is only one

68

TABLE VII.--Comparison of AREO and A for o=180O and a=90O
at 6=80r=0O and 6=60r=60O to show a connection
between a, eor and ¢F(T). See Fig. 17 for the

corresponding ¢F(T) curves.

 

N=3 h=1 E1=1A3 keV/turn A¢=i2°
CASE A CASE 0 CASE E
a eor r AREO A AREO A AREO A

(mils) (mils) (mils) (mils) (mils) (mils)

 

180. 0. 5 8. 19. 28. 16. 50. 16.
10 1A. 29. A0. 27. 66. 30.

15 16. 37. A3. 3A. 70. 38.

20 15. A1. A1. 38. 66. A3.

60. 5 12. 31. 3A. 37. 60. 52.

10 18. A6. A7. 53. 73. 80.

15 20. A7. 51. 53. 78. 87.

20 18. 37. A9. A3. _ 7A. 73.

90. 0. 5 1 25. 23. 21. A8. 16.
10 7. 38. 36. 3A. 67. 25.

15 9. 5A. A0. A9. 73. A0.

20 9. 61. 39. 58. 70. 50.

60. 5 3. 35. 25. 39. A9. AA.

10 11. A9. A2. 57. 7A. 67.

15 15. 50. 50. 60. 83. 7A.

20 15. 38. A9. 48. 81. 6h.

69

maximum and its properties depend on how quickly ¢F(1)
approaches zero. However, the (E)-8 factor in Eq. (7)

moves the maximum away from T the point of greatest AB.

0'
Table VIII shows I for the maxima in AREO and A and the

reSpective maximum values. The strong correlation between
To and the location of maximum AREO is easily seen. In
contrast, the location of the first maximum in the resonance

driven A's is independent of To and depends on 60 and 6e.

r
This section may be summarized by saying that Q

increases with increasing ¢F(T) at small I, less than 15,

perhaps. Q unconditionally increases under this condition

so
as does Qx for 60r=60°. However, the behavior-of Qx for

80r=0° is a dependent with a generally weaker tendency to

increase than Qx(eor=600)' In any case, having ¢o as

large as possible for a fixed I will improve Q.

0

70

TABLE VlII.--Location and magnitude of the first maxima in

- =0 = = O
AREO and A at e-eor 0 and 6 eor 60 for the

¢F(T) curves in Fig. 17.

 

 

N=3 h=1 a=138O El=l43 keV/turn A¢=i2o
CASE eor AREO A
Turn Value Turn Value
(mils) (mils)
A 0. 14 12. 19 19.
60. 15 15. 13 63.
B O. 9 26. 21 20.
60. 10 29. 15 66.
C O. 15 45. 19 26.
60. 16 49. 13 75.
D O. 19 57. 18 24.
60. 19 62. 13 74.
E 0. 15 72. 20 Al.

60. 15 77. 14 93.

71

3.3 Energy Gain per Turn

We have now seen the effects of varying the dee and
magnet geometries and the magnetic field contour for a
fixed acceleration harmonic. The last geometry we have
to vary is that of the orbits themselves, that is, the

total turn number, inside the magnetic field. For our

TF,
purposes, TF=Ef/El. A fixed TF implies equally good center-
ing for any type of particle which may be used with a given
h and, along with identical, or nearly so, ¢F(T) curves,
ensures that all AR's, AREO's and 1's will be the same

in all such cases. (Orbit calculations with TF=210 for

15 and 36 MeV protons and 60 MeV helions show this very
nicely for d=138o and h=1.

One may, of course, design different rF's into one's
machine, usually with a tendency toward lower numbers to
enhance turn separation and extraction, although this
leaning is limited by voltage holding capabilities and the
a chosen. Unlike M.S.U., most cyclotrons operate near their

maximum dee voltage so that T changes with particle type

F
and Ef. In either case, as demonstrated in the previous
section, both ARE0 and A will depend on the ¢F(T) involved,
which, therefore, must be specified before we continue.

In making a comparison between cases of different rF's, we
consider two viable alternatives. First, we may take

¢F(E) as invariant (or as nearly so as Fielder will allow),

in which case we already know enough to state that, at least

72

after it reaches its maximum, AREO should vary roughly as
(E1)-k at a fixed I by Eqs. (7) and (8). Notice also that,
while decreasing El should decrease the resonant driving
"force" acting to increase A, it will also maintain ¢CR

at a higher value over more turns, increasing the resonant
drive. Therefore, Q is not a sensitive function of E1 and

increasing El cannot be eXpected to improve Q greatly.

The second alternative is to maintain ¢F(I) as constant.
Doing so introduces a rather subtle effect so we therefore
present Table IX comparing data on 0F(I), x, px (for the CR)
AREO and A for protons with Ef=30 MeV and E1 values such
that TF=120, 210, and 300. Since the ¢F(T) curves are the
same (at’ least near the machine center), the integral in
Eq. (8) is constant and AREO should vary as (E1)8, which can
be verified as true from the table entries. Values of A
should likewise change by this factor since all resonance
driving terms are combinations of terms similar to the left
hand side of Eq. (7). However, AE depends on the docn's
of Eq. (3). The x-px data included in Table Ix shows that
centering improves with increasing IF, as should seem
intuitively correct, but more slowly than as (E1)%. Then
60GB, at a given gap and T, actually tends to decrease

with decreasing T since the denominator of Eq. (3)

F
increases faster than the numerator. This decreases the
rf asymmetry driving term, at a rate other than proportional

to (E1)%. It is this difference in rates, between the

73

TABLE IX.--Comparison of orbit properties in an M.S.U.

proton field for various values of El at

= = 80
fixed Ef 30 MeV and 8 eor O .

 

 

N=3 h=1 o=138O A¢=i2°
T TF ¢CR x px AREo A
(deg) (mils) (mils) (mils) (mils)

5 300 6.02 26. 214. 18. 9.
210 7.7A 33. 262. 2A. 11.

120 7.28 59. 292. 33. 8.

10 300 5.71 7. 179. 21. 13.
210 A.52 23. 206. 28. 16.

120 2.93 11. 216. 36. 10.

15 300 3.18 16. 150. 23. 16.
210 1.27 16. 16A. 28. 19.

120 1.57 -16. 183. 3A. 12.

20 300 .56 9. 125. 22. 17.
210 -.07 8. 13A. 25. 21.

120 .A0 -15. 171. 31. 12.

74

variations with'El in the l.h.s. of Eq. (7) at a constant

¢F(T) and indocD that gives rise to the larger A values

for TF=210 compared with either of the other cases.
Inspection of the data also shows that the 210 turn

geometry is possibly close to producing the largest possible

A's under the condition of fixed ¢F(T). Since Q is insen-

sitive to T (the table shows only a 20% variation with a

F
factor of 2.5 change in IF). no computer time was spent
attempting to find this optimal TF. It is interesting to
note, however, that in this case, as for constant ¢F(E),

Q does net increase monotonically with El as might be

expected at first glance.

75

3.4 Acceleration Harmonic Number

The final acceleration parameter to be investigated for
a three sector field is the acceleration harmonic h. Three
effects should arise with increasing h. First is simply the
fact that for a fixed a the rf times for zero phase gap
crossings will change so that the rf term driving disjunction
will be different for each h value. Second is the increased
importance of the orbit form factor phase shifts by the
factor h. Third is the multiplication by h in Eq. (3) for
6¢CD’

The data presented in Table X for h=2 is for a 15 MeV
deuteron field with the same TF(=210) as in Sec. 3.1 and
3.2. We treat only 1380 and 900 dees for h=2, but the
particle type and final energy are quite arbitrary: 7.5 MeV
protons and 30 MeV alphas give equivalent results. The
¢F(T) curve may be found in Fig. 18.

Most easily explained in detail is the ae90° case
since zero phase gap crossings occur at the rf wave peaks.
AREO's should be somewhat below those listed in Table III
for h=1 since this ¢F(T) is about 20% below curve C of
Fig. 17. Values of A are between 4 and 10 mils for all
values of eor and T, since rf asymmetry is almost non-
existant. The excellent centering which we have previously
associated with d=90° leads to very small docn's which might

affect the rf asymmetry, even with the extra factor of two.

+

76

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mwmtaz ZED».

on 0... cm oN on

A J 4 4 J a 4

L .. ..l...‘ v .

..IIIII'I; .(‘rlln-Il.) -.. . Arc
fil I.
I S
T
T. ON
I m u I D 8
N A. I a
P . F b L s b

 

 

 

o...

('030) IHd

 

77

gulf..- . :3 WM”?

.Hm .em- .Hm

 

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78'

The 6A's at opposing gaps are almost equal, differing mainly
only by the energy denominator of Eq. (7), and, therefore,
nearly cancel. As a measure of the orbit symmetry involved,
Fig. 19 gives px asymmetry values for AEO's with a=138° vs.
percentage ofEf on h=1, 2 and 3. By Sec. 2.3 we expect
values for a=90° to be lower than the h=2 curve actually
plotted, which is already lower than the h=1 case presented
for comparison.

Of greater interest is a comparison of AREO and A
data for 1380 dees and h=2 in Table X and previously given
data for h=1 (Tables III, V and VI). The h=2 AREO data
shows larger variations (approximately twice as large) with
eat as follows from the more important form factor phase
shifts. As to the data on A, note first that h=2 does not
demonstrate variations with ear as large as does h=1, nor
does it have the very strong Preference (in terms of larger
Qx) for 9°r=60o and the poor showing at 90r=90o discussed
before for h=1. In fact, Gor=90o gives the best Qx on h=2.

To understand these results one must realize that the
zero phase gap crossings occur as in Fig. 20. (Cf. Fig. 9.)
Large phases now tend to produce small 6A's at 81 and large
6A's at 82, in direct contrast to the h=1 situation. Small
phases are now to be preferred at 81 for larger resonant
increase in A. The better Qx for eor=9oo may now be seen to
be connected with the very low phase occurring at gap 1
and the very large phase at gap 3. The reversal in relation-

ship between eor and A is the reversal in the sign of

79

Figure l9.--Orbit asymmetries for a=138° in the N=3

fields for h=1, 2, 3 used in Secs. 3.1
and 3.4. The px asymmetry is defined in
Sec. 2.3. The curve plotted with squares
is for TF=120. All others have T =210.
The h=1 curve is the same as in Flg. 3.

8O

 

 

 

I
7.3qu

2 1 1

>¢hmzz>w<

xi

1

HO

 

20

30

20
PERCENT OF FINAL ENERGY

10

 

90

Figure 19.

81

rf

Gap 1

t+

Gap 2

 

Figure 20. Zero phase gap crossing times with d=l38O and
h=2. Compare with cross locations in Fig. 9.

 

 

 

 

 

 

 

I I I I t
4 80 NEV
.4
4
.7
'i
.f :__;==='I-————I
» A
L L 4 A _L_ L l 1 L J
10 20 30 HO 50
TURN NUMBER

Figure 21. ¢F(r) for an N-4, 30 MeV proton field.

82

dvrf/dt at each gap when comparing h=1 and h=2.
Again referring to Fig. 19, we see that ABC asymmetry
greatly increases with third harmonic operation. This,
along with the fact that 60GB now includes a factor of
three, makes acceleration very difficult in some cases and
impossible in others: one finds that one cannot deceler-
ate particles backward past 1 or 2 MeV to find source
positions using a=138o and 1F=210. As demonstrated by
Learn, gt_gl.,ls we may expect 90° dees to perform adequately
.if we switch to a 120 turn geometry although 138° dee per-
formance is still unsatisfactory as supported by the large
asymmetries in Fig. 19. (We should mention that our
difficulties on h=3 stem from our centering requirement
for the CR.) We shall, therefore, consider only a=180o
and 90° for h=3 and will, for convenience, use the same
field as Learn. For further details on this problem and the
performance of 600 dees, we refer the reader to Learn, at_al.
The relevant data is contained in Table XI for 27 MeV

C+3

ions. Again we see the dependence of AREO on ear
becoming stronger from the form factor effect for both
0181800 and 90°. The extremely large A values observed

(as much as an order of magnitude greater than most of our
previous cases) are due partially to the increased sensi-
tivity to the gap-crossing resonance shown in Fig. 19 and
partially to values of 6¢CD of 200 to 30° between gaps on

early turns. This is also, of course, the reason for the

order of magnitude change in A between 9or=3°° and 90°.

83

 

eniw In I. ,AUr

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Anaeev Anaesv Anessv finesse Anaeev 2nHHsV
x2 2 omma x2 2 omma goo e
oomuo ommano
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.ma .mfim ca CZch Apvme on» on mocoomoaeoo some
m<II.HN mqmde

84

In summary we can say that Q is highly dependent on
h because of the different gap crossing times involved
at fixed a. A phase selection system which works well on
one harmonic should not be expected to perform satisfactorily
on other harmonics. For example, with 90° dees, citing
the best values at any turn in each case one finds

_ _ ~ _0
Q(h—l)/Q(h—2)-4 at Ger-0 .

85

3.5 Four Sector Fields
Lastly we wish to consider AR in a four sector field.
We shall again consider a 30 MeV proton field. The ¢F(T)

for this field is in Fig. 21 while the v and w curves are

r
in Fig. 22 and eigen-ellipse properties are in Fig. 23.
Both 0 and Vr as functions of energy are smaller in four
than in three sector fields, asaue the eigen-ellipse
eccentricities.

Unaffected by the change in N are Eqs. (7) and (8),
as supported by comparing the ARE0 columns in Tables XII
and V. The same form factor variations with eor are
present in AREO and the magnitudes are consistent since
the area under ¢F(T) of Fig. 21 is 70% that under curve
c in Fig. 17.

What we must give some attention to is Qx' Of great
importance to Qx is the fact that the field symmetry now
eliminates the gap-crossing resonance so that AEO's are
quite symmetric at all energies and dee angles (cf. Fig. 2 and6).
Accordingly, the particle orbits are symmetric,with CR
x-px histories all appearing in one quadrant (x<o,px>o).
This is quite a contrast to the N=3 case of Fig. 3 and 4.

Before looking at any data we may conclude from our
previous work that AA will be significantly smaller than
for N=3 because of smaller docn's accompanying the more
symmetric orbits producing more equal disjunctions at

opposite gaps. The oscillatory structure of A vs.T will

86

 

 

 

L3H" I I r r I I
h r o = 67.50 d
o = 22.50
IDIJZL- .—
H v
0- /
E I. O = 45° .7
a:
33
< 1014- . = 00 +
t .J
LQL: L, l L i l. l

 

 

l
l 2 3 ‘0 5 6 7
ENERGY 0150
Figure 22. Eigen-ellipse axis length ratios vs. energy for

a proton field with N-4, E =30 MeV. Note change
of ordinate scales from F1 . l4.

 

I I r I I I 19"

 

 

 

L
9
0930) NOIlVlOU ‘WNOISSSOBUd

 

 

0
1 1 1 1 1 1 J. .50
l 2 3 Q 5 . 6’ 7
ENERGW’IHEV)

Figure 23. v and w(see Eq. 11) vs. energy in the field
u§ed for Fig. 22. Note change of ordinate scales

from Fig. 15.

87

also be decreased since the precession is slower and the
relative orientation of A and eigen-ellipse at a given
azimuth is of lesser import. From the resonance viewpoint,
although the resonant condition will exist over a signifi-
cantly larger number of turns than in the three sector field
due to slower change in vr, only the rf driving term will

be large with a negligible field asymmetry driving term.

The conclusion to be reached then is that a four sector

cyclotron as at Maryland will have a small Qx value and

 

one which is insensitive to eor when compared with a similar
VN=3 machine. I
We present AREO and A compiled in Table XII and
Fig. 24 (analogous to Table III and Iv and Fig. 16) to bear out
these assertions. Dependence on ear is qualitatively the
same as for N=3 for both ARE0 and A, with the extra feature
of decreasing A with increasing I for 80r=0°, 22.50 and
d=180°. This is mainly an energy effect depending on (E).-15
in Eq. (7) which we can now see because it is not masked
by any field asymmetry. The quantitative dependence of A on

eor is much weaker at each a value (for 0L=90o compare the

worst caSe A change of 20% here to the N=3 worst case change

of 400%); Since only the rf driving term is appreciable it

is hardly surprising that we find a very strong a dependence.

Finally, notice that there exists the same preference as

with N=3 in terms of larger Qx's for ear to be a hill center.
In summary, for h=1 we see that a four sector field

offers one much less sharply defined choices toward

 

 

‘

a; ’Er.
a.

88

 

 

 

 

 

 

. (a)
80. 90R = 00
80-

L+0» .

..9' """" ..J’./ ‘—

.- /\\_, """""
20_/

1'0 20 30 4:0 50
AR

L (C)
80-

. .. 903 = 800
80-
40” . 00000 ’-

b,‘ ...... /

_. /\ 70......
20 x’ - a:

r 10 2'0 330 4‘0 50

 

Figure 24. AREO,

OI'

‘ 2:2?

 

 

 

 

 

' (b)
80F 00,. = 300
80-

40- ..... ,
b.. ..‘o... /
20.- r"-~. _.-’"’“”"
./' 11,2—-- EJ
10 20 .30 4'0. 50
80- (d)

: eon = 900
60-

90- ...

_.,- ’-..f. ,r

_. a’ -““':%/3 ......
20”,

_, 10 20 250 4:0 50

A and AR vs. I plotted at 8=80P for various
0 's for N=4, Ef=30 MeV, o=138°, TF=210 and h=1.

Ordinate values are in mils.

89

 

 

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90

optimizing Q. Only for 900 dees is there a strong
dependence of AR on anything other than energy difference
and this is independent of eor. Even after correcting
for the difference in ¢F(T) curves, the best possible Q
for N=4 will be seen to be worse than the best for N=3.
These same general statements should also apply to
higher harmonic operation. One further effect we would
find is lessening of the variations in ARE0 with 60

because the form factor F(8) varies as N-z.

r

4. CONSEQUENCES FOR PHASE SELECTION

4.1 Locating Phase Selective Slits

To this point we have presented data which can certainly
be used as at least a semi-quantitative guide showing how
AR varies with various accelerator parameters. The question
to discuss now is how to use this information in planning
for a phase selection system in either an existing or a
proposed machine. We shall assume that a CR as defined in
Sec. 2.1 exists in either case and that the aim is to
locate a slit or slits, centered on this ray, for their
most effective use at the lowest possible energy. That is
to say, we wish to know where Q of Eq. (10) is maximum in a
region where thermal and radiation problems will be least
severe and where turns in the unfiltered beam are still
separated.

In general, slits could be placed anywhere along the
CR and perform phase selection. With few exceptions, phase
displaced particles will always have some radial separation
due to their energy difference, and, as a first order approxi-
mation, one would, if one could, place a slit where ARE0
was maximum. This approximation amounts to simply neglecting
A and is a good approximation for cases like d=1800, N=4,
h=1 or a=90°, N=3, h=2. One can combine Eqs. (7), (8), and

(6) to write
91

 

92
0E0 = AREo/A¢ = -(Ro/2)<¢F>I(l + F(8)) (12)

an expression which the data in Sec. 3.2 showed to be fairly
accurate. To apply Eq. (12) to all azimuths, one must
require that GREG from Eq. (7) at any gap is much less

than ARE0 used above. We shall assume that I is large
enough for such to be the case so that QE0 is independent

of 6 in all our discussion in this section.

0

In the other cases, e.g., d=90°, 80r=90 , N=3, h=1 or

d=90°, 6 =30°, N=3, h=3, where A and ARE0 are of comparable

or
magnitude, they must be added, with due consideration given
to the orientation of A, to determine the optimal slit
location. In the examples previously given in Figs. 16 and
24, we have included the end results of these considerations:
AR vs. I. Tabulated results in the form of Q values fill
Tables XIII, XIV and XV. Maxima in these curves
obviously are optimal slit locations, but the curves are
accurate only near the azimuths at which they are drawn.
Since we are formally presenting data only at 9=60r (Sec. 5.3
contains data at other angles), we must discuss the principles
governing these curves so we may have a basis for deciding
what AR will be at arbitrary angles.

At this point we return to the data on Ax which has
been supplied in most of the tables of Chapter 3 but has
not been formally discussed. These Ax depend not only on
A, which we have talked about at length, but also on the

orientation of A in x-px space. For a=180°, before the

93

 

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a e a p a e o
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94

 

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96

precession angle 0 becomes large, A parallels the x axis.
However, for any other a, the interactions represented by

the vectors K1 and K2 in Eq. (9) cause A to take on some

other orientation. The sign of Ax results from the dependence

of ¢CR on gap number. For example, in Fig. 16 AR may be

seen to start as the sum of the other two Curves for

o
and

6=0° and 900 (AXEO) and their difference for 60r=30
60° (0x30).

If we define u as the angle between A and the x axis
so that we could calculate it as u=Tan-1(Ap/Ax), we find
that u=uo, a constant, over the early turns until
precession takes over. Given such a no at e=o, for

instance, on a turn where w is small, one may rewrite

Eq. (4) a3
AR(9,T) = AREO(T) + A(T) cos(vr9-uo+w(1)). (13)

This equation is particularly useful for T large enough
(usually T>lO) that AREO is independent of 6 and A varies
only according to eigen-ellipse effects. With Eq. (13)
and the data in Chapter 3, one can now determine AR at any
location in his machine. (Strictly speaking, given only
the data in Chapter 3, one must resolve the sign ambiguity
involved with po=Cos—1(Ax/l) by obtaining the difference
loetween “o and u on some later turn and comparing that

ciifference to the A0 expected for the corresponding AT.)

 

‘0

00.00030} 4.

97

Suppose we have now positioned one slit so that
Eq. (13) predicts a satisfactory Q. To determine the
necessityfbr,and possible placement of any second slit,
we refer to Sec. 2.5 to remind ourselves that particles
displaced in px from the CR will display significant
phase displacements from the CR during their acceleration
history. We must then consider eliminating these particles.
Since the incoherent oscillation amplitude of such particles
is usually greater than their AREO from the CR, the proper
location of a slit follows from considering precessional
motion.

There is one important difference between the spatial
and temporal displacement cases which makes a two slit
phase selection system necessary. At eor and before 0
becomes important, “0 is usually small while a vector similar to
A on the x-px plane between rays of 05px is 900 "out of phase"
with A, approximately parallel to the p; axis. This means
that to achieve total phase selection requires two slits
either 900 apart on the same turn or k precession cycle'
apart on the same azimuth. Having both slits the same
size yields a final beam which occupies a compact area in
phase space: its area and all dimensions are small. Such
a beam will retain its high quality (small phase width and
x—px area) through the extraction region and present a small

lenergy width and emittance to the beam transport system.

98

One assumption implicit in the discussion of this
section is that we are dealing with separated turns. In
fact, such would not be the case for T=20-30, where we
would place our slits, because of the large beam phase
width to that point. To cure this problem, the M.S.U.
cyclotron contains a slit (10 mils wide for h=1 proton a

operation) about 1800 away from the source-puller. This 0

 

"half turn” slit performs a rough phase selection, passing i
at most two particle groups at different centroid phases, ;0
each with a relatively narrow phase width; One group is
the maximum intensity group, the Centroid of which corres-
ponds to our CR. It is this group which the cyclotron is
tuned to accelerate and on which our slits perform the
final phase selection. The other group is lost on the jaws
of the first slit, the difference in centroid phases not)
being enough to cause overlapping of turns.
One final comment to be made concerns the minimum
practical slit width which should be used in any system.
Certainly, one can obtain arbitrarily good phase resolution
in one's beam by decreasing his slit width, but at the cost
of decreased beam current. In fact, if one assumes that AR
varies linearly with A0, it is easy to show that the beam
phase width decreases (improves) at nearly the rate at
which beam current decreases as slit width is decreased,
until the slit width is about the same as the zero-energy-

spread beam width. This latter is just the x width of the

 

99

ellipse in Fig. lle or f, for example, and is directly
related to the source slit width. Thus, the use of
phase selective slits more narrow than the source slit
would probably prove to be a poor choice, giving too
little beam current for the beam phase width achieved.
We now turn to one last tOpic, the effects of real
central region electric fields, before summarizing the

‘results of this paper.

100

4.2 Comparison with "Cyclone"

As mentioned in Sec. 2.2, Devil suffers from one
major simplification: 'the neglect of the exact electric
fields on the first few turns. The justification for this
neglect is that such fields are different for each central
region design, even for equal dee angles. It is also, of
course, important to separate other influences on Q from
the effects of these electric fields. However, this study
could in no way be called complete without some comparison,
albeit brief, of results from Devil with those from

16 an orbit code utilizing measured electric

"Cyclone,"
potential data for the source-puller and early turn regions.
Figure 25a gives Cyclone results for AREO, A and AR vs. T

at 6=O° for a value of 0 to=-22o for the CR at the source.

RF
The Ef of 42 MeV and the central region geometry were
chosen to correspond to data previously published by
Blosser.5 Devil results for this case are in Fig. 25b.
ARE0 may be seen to be the same in both cases as expected
since ¢CR from Cyclone agrees with ¢CR from Devil at 6=0°
and 1800 to within about 1°. The Cyclone A is smaller than
its Devil counterpart by a factor of about 0.5. Note,
however, that the extrema in the AR curves occur at similar
T values in both cases, the Cyclone values peaking near
1:18 and Devil values near T=l4.

The differences in A may be traced to the more gradual

nature of the energy gain on early turns in Cyclone. Our

101

80
60
HO
20

lilTrfT
0
a

 

 

 

f0 éo i0 E+0 sro _ {b 50 :30 Lia so

 

'-T—-1

AR=---

(d)

T

/“\ ‘x
0

I ’ ..looooooo‘ooooopl...
V i ,4 2‘ \‘r/ ‘ ... I: k
f \10/ 20 30 00 30 F 10' 2'0 30,00 so

Figure 25. AREO, A and AR vs. 1 for eor=o° plotted at e=o°
(a,b) and 6-1800(c,d) comparing Cyclone (a) and
Devil (b,c,d) results. See text, p. 100, for
comparison details. Ordinate values are in mils.

 

 

 

 

 

 

III .I
. 0". III I I.lllll...|.ul‘| ' I
III' 4. nl II I

102

picture of discrete 61's is no longer valid and must be
replaced by one in which the change in A is spread over
some rotation angle at each gap crossing. But since
disjunction acts only parallel to the x axis, it should be
expected that the net effective GA will be less than the
discrete 61 we have discussed before. The discrepancy in
extreme locations is also a function of the details of the
electric fields. The no from Cyclone has A about midway
between the axes (uo=-45°) while that from Devil hasA
‘closer to the positive x axis (00:00). This orientation
difference must be made up by precession, hence the extra
turns required to bring A parallel to the x axis, the
extremal condition.

Figure 25c shows the Devil results for AR and its
components in this 42 MeV case at 6=180°. Comparison with
Fig. 3 in the work of Blosser cited above shows the same
agreement as between Figs. 25a and b. Note also the
exchange of node and antinode locations in AR between Figs.
25b and c and between Fig. 25a and Blosser's Fig. 3. This
follows directly from Eq. (12) with a change in 6 of 180°.
We also show Fig. 25d with AR and its components for a
ray displaced by 6px from the CR. This should be compared
with Blosser's Fig. 5 to show that Devil and Cyclone agree
on the location for a slit to eliminate spatially displaced

particles.

103

4.3 Final Summary

We may now draw some general conclusions about the
parameters affecting the longitudinal-radial coupling and
the process of locating slits to perform phase selection.
QE0 depends mainly on ¢F(T) and only secondarily (10%)
on form factor effects. Its maximum value and the
corresponding turn number may be determined from Eq. (11)
quite easily. It is quite independent of all other
influences. (See Sec. 3.1.)

There are two basic influences on QX: the rf times
at the gap crossings and the orbit asymmetry. The first
determines the 6A's at each gap, which latter then combine
to form the AA over each turn (Eq. 9). The second deter—
mines the CR centering properties (x-pX history) which in
turn affect the rf times in question through Eq. (3).
Orbit asymmetry also has a role in determining the K1 and
K2 of Eq. (9). The parameters we have studied are important
to phase selection to the extent that they modify one of the
above influences.

Probablythe parameter with the single greatest effect
on phase selection is the field periodicity, N. For the
two-dee cases we have restricted ourselves to, N=4 removes
the orbit asymmetry term with the result that AR=ARE0
and QzQEO’ N=3 involves a large orbit asymmetry term
related to the gap-crossing resonance which tends to make
Qx at least comparable to QEO’ thereby introducing effects

which make the situation more complicated but also make

 

'0 I'll: iiiiiliilc {III

104

Q larger. This also means that three sector cyclotron
have a definite advantage over their four sector counter-
parts when it comes to performing phase selection. (See
Sec. 3.5.)

In particular, N=3 cases are sensitive to eor. As
the dees are rotated with respect to the magnetic field,
CR centering is modified, altering ¢CR and, ultimately, A.
Thus we observe as much as an order Of magnitude change in
A(a=l38°, h=1 and a=90°, h=3) as eor changes to sweep the

dee symmetry line through one sector. eor=60° tends to

give the best h=1 results for most dee angles (6015900 is
best for a=90°), while results on other harmonics are
mixed, being highly a dependent. (See Sec. 3.1.)
Improvement in Q may be brought about in most cases
by having ¢F(T) stay as large positive as possible through-
out the region herein Vr is close to one. This increases
AREO and usually also A, but exceptions there do arise
(0:900, 60r=0°, h=1) because disjunction is not linear with
¢CR‘
for phase selection. It should be borne in mind that having

Thus, a central field cone may be seen to be beneficial

¢ (T)=0 for all T is not really the necessary condition for

CR
obtaining minimum final energy spread, although it is suffi-

cient. The necessary condition is that the integral,1

E
'h f(sindJ/cosBMdE = 0.

Thus, large regions of large positive phase are not
detrimental to final energy spread if eventually balanced'

by regions of negative phase. This condition is always

105

fulfilled by the Fielder fitting procedure. (See Sec-
3.2.)

For either N=3 or N=4, E1 has surprisingly little
effect on Q and so, given a centered beam, changing El
will not affect the performance of phase selection slits.
(See Sec. 3.3.)

Changing acceleration harmonics mainly changes the
rf gap crossing times (although this, in turn, has some
effect on orbit symmetry). One should not expect, there-
fore, "fixed" slits to perform equally well on all harmonics
at fixed a. In particular, we find that Q with 0:138o
should be only slightly less on h=2 than on h=1 but with
a large change in 00. Operation on h=3 for the dee angles
examined looks to be substantially better than on h=1
because of large 60Gb values. (See Sec. 3.4.)-

Lastly, from our observations on comparing Cyclone
and Devil results, we may infer that the true electric
fields present in the central region act to decrease Q from
‘the step-function-energy-gain value by decreasing A.

They also modify “0 which controls the I of maximum Q for
fixed 9. This result serves to point out that one should
include these fields in the design calculations for a slit

system for an actual cyclotron. (See Sec. 4.2.)

APPENDICES

106

5.1 Formula Derivations

5.1.1 Calculation of the Starting Condition
for an Accelerated Equilibrium Orbit

In x-px phase space, the point representing a freely
coasting particle rotates about the origin. At an accel—
erating gap the point jumps in the negative x direction a
distance equal to the E0 radial shift corresponding to the
energy gain AE. This shift, AREO, is given in Sec. 2.4.
Let us neglect F(6) and assume that (E)—% varies negligibly
over any one turn so that ARE0 depends only on AE. Further
assuming a two dee system with dee angle a less than 1800
(one or two dees of 1800 is just a simplification of the
argument below), a constant phase ¢ within a turn, harmonic

h=1 and setting Eg=qu we may write the energy gains at the

gaps as
AE1 = A33 = Eg cos(0/2-a/2+¢) = Eg Sin (a/Z-¢)
= Eg (sin(a/2)'cos¢ - cos(a/2) sin0)
AE2 = AE4 = -Eg cos(0/2+a/2+¢) = Eg sin(a/2+¢)

Eg (sin(a/2) cos¢ + cos(a/2) sin0).

If we define

09
ll

‘8 .
A(2EOE) Eg Sin(a/2)cos¢

0')
ll

-5 .
A(2EOE) Eg cos(a/2)sin¢ ,

107

then the shifts ARE0 at gaps 1,2,3 and 4 are

ARl = AR3 = 6-6

AR2 = AR4 = 6+5.

Notice that, in general, 6>>e because sin a/ZZcosa/Z for
1800303900 and because cos¢>>sin¢ for the small phases
(<20°) we are interested in.

We may represent betatron oscillations with a transfer
{matrix formalism such that X(6+A6) = M(A6)X(6) where X(6) =.
x - inpx with n==Ro/(vrp) and approximate M(Ae) =
exp(ivrA6). If ARE is as derived above and all drifts

0
between gaps have the same v =vr we have, for one turn,

x(20) = X(0)exp(iv20)-(6-e)exp(iv(30/2+a/2))
-(6+e)exp(iv(3n/2-d/2))-(6-e)exp(iv(0/2+a/2)
-(6+6)exp(iv(0/2-a/2)).

For an ABC, we require that X(20) = x(0) so we regroup
the terms above and apply the relations between trigonometric

and exponential functions to get
X(0) = -i6(cos(va/2)/sin(v0/2))-e(sin(va/2)/sin(v0/2)).

These two terms appear as Eqs. (2) and (2a) in the text in

Sec. 2.3.

108

One should observe that, under our assumptions, the
geometry repeats after 8 turn so X(0) = x(0) by symmetry.

This condition also yields the same X(O) as above.

109

5.1.2 Differential Energy-Gain per Turn
We assume that the particle energy gain at the ith

gap is

+ N.0)
i

6Ei = Eg cos(6RF

where Eg=qu is the maximum possible energy gain per gap
and Ni=0, l, l, 0 for i=1, 2, 3, 4 with h=1. The difference
in energy gains at the ith gap between particles separated

initially in phase by A0 is then

6Ei = -(A¢) Eg sin(0 + N10).

RF

Over one turn, with gaps at 61, the total energy difference

will be

2 = _ 2 °
6E i6Ei (A¢)Eg i Sln (Bi + N10 + 01)

and over the acceleration history of the particles

. _ Z 2 -
AE — (A0) Eg T i (s1n(6i + N10) cos¢i(r)

+cos(6i + N10) sin¢i(T)).

For any dee angle a, if we assume that 01(0) is small
and nearly constant over any turn, we find that the first
term above sums to zero. Then we may write the remainder

as

110
AB = -(A0) Eg sin(a/2)% E ¢i(T). (14)

If we replace Eg sin(a/2) by E1/4, then this result holds
for any h.

We may use the Euler—Maelaurin formula to replace

the sum over T by an integral as follows:17
Tf
If
1 ¢i(T)= { ¢i(T) d1 + k(¢i(Tj)+¢i(Tf)).
T=Tj j

In the cases we are interested in, Tj=0 and If is usually
greater than 15 so that f 01(1) dr>>¢i(1j) and ¢i(rf)=0.

If we recognize that ¢F(T) is nearly the average value
of the 01(0) over one turn and replace Eg sin(a/2) by E1/4,

Eq. (7) in Sec. 2.5 follows immediately from Eq. (14).

111

5.2 The code "Devil"

"Devil" is a code for calculating median plane orbits
and deviations between them using a fourth order Runga-
Kutta-Gill integration technique with roundoff error

18 It is similar to the code "Goblin-4' written

correction.
for this laboratory by D. Johnson, while incorporating
improvements in technique over Goblin and performing direct
comparisons of the orbits of up to nine rays. Devil can

also plot its results. It does give up Goblin's ability

to "accelerate" a ray backward and does not include provision
for calculating axial motion. Like Goblin, Devil assumes

a step-function energy gain at each gap. Following are the
equations of motion solved by Devil, the program flow and

the input parameters with definitions, units and FORTRAN
formats.

The input to the code consists of the “standard"
magnetic field deck (suitable for use in all M.S.U. orbit
codes now in general use) followed by a string of running
parameters defining the condition of the machine and the

particle source location. The field deck defines the

following quantities:

particle rest energy (=mOc2) (MeV)

till
ll

particle charge (units of e)

00
II

number of field sectors

n = number of field harmonics in deck

112

B = B(r=0) (kG)

A = c/wo

qBo/ (moo) (MHz)

8
ll

Note that A, B0 and 00 are not independent. If A is
specified the code assumes that Bo and we are also. Other—
wise Devil calculates its own self-consistent set of
constants. The field is specified by the coefficients
in its Fourier decomposition:

m

B(r,6) = B(r) + .Z

(H.(r)cos(jN6) + G.(r) sin(jN6)).
3‘1 3 3

The running parameters are listed in full with proper units

and formats on the last three pages of this section. The rf

frequency is set by defining an Srf such that

wRF = 00 (1+erf)h.

Devil performs its calculations in terms of modified
cyclotron units so that r, pr and B(r,e) are in units of
inches,moe4\ and 30' respectively. In these units the

equations of motion are:13

y E 1+E/Eo
_ 2_ 2 8
P - (P Pr )
dr/de = rpr/P
dPr/de = P - rB(r.e)
derf/de = h(l+e)Yr/P

113

th

The energy gain at the i gap is

AEi = q Vd COS (erf(ei) + Nifl)l'

where Ni = 0,1,h,h+l for i=1,2,3,4. The gap locations, ei,
are given by
61 = eor + a1,

62 = eor + 0 - oz,

03 = eor + 0 + oz,

to
II

4 6or ’ “1'
where the a of Sec. 2.2 is a = 0-01-02. We have used only
al=az in this paper.
The program begins with the computation of EO properties
RE0 and PREO vs. energy at each output azimuth using the

11 as a subroutine. The angular steps to

code "Sigma-E0"
be used in the integration are determined and the total
field at the end and mid-points of each step are calculated
and stored. The rays (the CR is always included) are
"accelerated" a specified number of turns one at a time,
results at the angles of interest being stored until all
rays are finished. Then x and px values are obtained using
the tabulated E0 data, deviations from the CR in 0, r,

REO’ x and px are found and stored and a printed record

of 0, E, r, pr' x, px and the above deviations is output.
Finally, these deviations are plotted if desired. This

modular program form is adopted to trade core storage for

computing speed, a worthwhile endeavor on the M.S.U.

114

Cyclotron Lab's XEROX SIGMA-7 computer, which has 64K
words of real core. Average running time accelerating
five rays for 50 turns with six output azimuths and creat-

ing three plots is about 7.5 minutes.

115

Devil Input

Standard field deck
Card Format
A. (F7.2, Fll.3, I3, 3F11.6)

Field identification number
Particle rest energy (MeV)

Particle charge (e)
Cyclotron field unit (kG)
Cyclotron length unit (in.)

mU'lubUNI-J
0

Ideal isochronous frequency (MHz)

B. (213, 2F11.6, I4)

Number of sectors

Number of field harmonics to follow

First radius value in field tables (in.)
Radius increment between table entries (in.)
Number of table entries

U'lnwaH
O

C. (7F11.7)

Tables of B(r) , Hj (r) , Gj (r)

D. (213)

1. Number of imperfection harmonics to follow.
=0, none read, skip to next input
2. Imperfection harmonic number

E. (7F11.7)

Imperfection harmonics (if any)

Running parameters ~ each parameter on a separate card

F.

00000000
mxlmUlwal-J

010
011
012

013

014

116

(212, F12.5)

Frequency scale factor, default=l

a
rf
Acceleration harmonic

Dee-To-Ground Voltage (kV)

0
or

l

0‘2

Phase constant for h#l, default=90°(h=2),
~180°(h=3) .

00

Ray ID (see below) at A vector origin
Ray at A vector head
Number (maximum=6) of plotting angles to follow
-0, no plotting done, skip to next input
a.) (4A4) Plot Title
b.) (7Fll.5) Plotting angles (degrees)
Number (maximum=105) of values in following
F0 table ‘ '
=0, no table follows, skip to next input
a.) (Fll.5) Angle for which EO data is presented
b.) (3F12.5) EO table (E, REC, FREQ)
Number (maximum=16) of extra printing angles
to follow.
Dee gaps are automatically included.
a.) (7Fll.5) Output angles

Initial (source) conditions

GO

ID,

(212, F12.5)

IN, Z

where

ID = Ray identification number
1 implies CR (value of Z copied into
variable IN for all other rays when

this ID occurs)

Variable index

IN

117

l Azimuth (deg.)
2 Radius (in.)

3 Radial momentum (in.)

4 Starting phase (deg.)

5 Energy (MeV)

Z Variable value
(Note: Z replaces the CR value if ID¢1 and IN=0.
However, 2 adds to the CR value if ID¢1 and IN<0.)

H. (12)
Run command = -1 begins calculations

LIST OF REFERENCES

9.

118

REFERENCES

M. M. Gordon, "Single Turn Extraction," IEEE Trans.
Nucl. Sci. N§;1§ (4), 52(1966).

H. G. Blosser, M. M. Gordon and T. I. Arnette,
"Resonant Extraction from Three-Sector Low—Spiral
Cyclotrons," Nucl. Instr. and Meth. lgllg, 488(1962).
H. G. Blosser, "Problems and Performance in the
Cyclotron Central Region," IEEE Trans. Nucl. Sci.
N§113_(4),1(1966)

H. G. Blosser, "Performance of a Modern Medium Energy
Cyclotron," Bull. Am. Phys. Soc., (Nov. 1967).

H. G. Blosser, "Optimization of the Cyclotron Central
Region for the Nuclear Physics User," Fifth Inter-

national Cyclotron Conference (Butterworths, London,

 

1971), 257.

D. W. Kerst and R. Serber, "Electronic Orbits in the
Induction Accelerator," Phys. Rev. 62, 53(1941).

E. D. Courant and H. S. Snyder, "Theory of the
Alternating-Gradient Synchrotron," Ann. Phys. (N.Y.)
3, 1(1958).

M. M. Gordon, "Orbit Properties of the Isochronous
Cyclotron Ring with Radial Sectors," Ann. Phys.
(N.Y.), 29, 571(1968).

A. J. Lichtenberg, Phase Space Dynamics of Particles.

(John Wiley & Sons, Inc., New York, 1969), p. 119.

10.

ll.

12.

13.

14.

15.

16.

17.

18.

119

M. M. Gordon and D. A. Johnson, "Application of a New
Field Trimming Program to the M.S.U. Cyclotron,"

AIP Conf. Proc. 2, 298(1972).

M. M. Gordon, T. I. Arnette and D. A. Johnson, Bull.
Am. Phys. Soc., (April, 1964).

M. M. Gordon, "The Electric Gap-Crossing Resonance

in a Three Sector Cyclotron," Nucl. Inst. and Meth.
gig, 268(1962).

H. S. Hagedoorn and N. F. Verster, "Orbits in an

AVF Cyclotron," Nucl. Instr. and Meth. l§:12, 201(1962).
J. S. Balduc and G. H. Mackenzie, "Some Orbit Calcula-
tions for TRIUMF," IEEE Trans. Nuc. Sci. ggzig (3),
287(1971).

L. L. Learn, H. G. Blosser and M. M. Gordon, "An
Optimized Multi—Particle Central Region for the
Michigan State University Isochronous Cyclotron,"

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L. L. Learn, D. A. Johnson, private communication.

P. J. Davis and P. Rabinowitz, Numercial Integration,
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R. W. Hamming, Numerical Methods for Scientists and

Engineers, (McGraw-Hill Book Co., N.Y., 1962), p. 212.

 

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