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APR 27 1955

LOWF-650316-52
CMCIAL
AICIAL
URT
YHTELST.
ACCELERATING CAVITIES FOR AN 800 MEV SOC*
N, F. Ziegler
Oak Ridge National Laboratory
Oak Ridge, Tennessee
Summary
equaduce a maxilength d and
The ininimum required energy gain per
turn for the 800-MeV Separated Orbit Cyclotron
increases by a factor of about five from the
injection radius to maximum orbit radius,
Ordinary rectangular cavities operating in the
TM 10 mode can be used to provide the accel-
erating voltage; however, the cavity length is
then about twice the distance between the inner
and outer orbits ( 18 ſt). This length, and the
rí power loss, can be reduced by shaping the
cavity to excite in addition the TM210 and
TM310 modes. Inclusion of these higher order
modes shifts the maximuin voltage from the
midpoint of the cavity out toward the end,
resulting in a shorter cavity and lower losses.
The performance of a 1/4-scale model of the
"shaped rectangular cavity" was found to agree
quite well with theory. "Wedge-shaped" cavi-
ties were also investigated. In this cavity,
which is a sector of a cylinder, the length of
the accelerating gap increases with machine
radius. The upper and lower boundaries of
either type of cavity can be shaped to excite
higher order modes.
operating wavelength , and n are selected to
produce a maximum orbit radius r, nearly
equal to the desired value at maximum energy.
In other words, ro = nbod /(21). The injection
radius is then r; = iBiN (201). The required
energy gain per turni may be determined from
AT (Ar)(T) (2 + (T/EO)(3 + T/E.)]/r, if
Ar car. Then the minimum cavity gap volta ge
is Vuin = AT/(mF cos Os), where m = number
of accelerating gaps per turn, F: transit time
factor and os = phase-stable angle. A curve of
V nin as a function of radius is shown in Fig. 2
for a typical case. Here the minimum voltage
increases by a factor of almost five between
injection and maximum energy, and the radial
distance (r. - r;) over which a specified voltage
must be maintained is 210 in, or 0.875 2. The
actual voltage developed by the cavities as a
function of radius may have any shape, provided
it is always greater than the minimum value.
Introduction
Several types of cavities appear to be useful
in SOC's. These may be divided into two gen-
eral groups--coaxial cavities and TM cavities
in which the rf magnetic field is transverse to
the direction of particle motion. Coaxial cavi-
ties operating strictly in the TEM mode would,
of course, provide an accelerating voltage
constant with radius. The distance between gap
centers must be at least BX12, however. At the
higher energies this distance, combined with the
large radial extent of the cavity, makes the
coaxial system unattractive. Since this paper
is concerned priniarily with an accelerating
system for a machine having a inaximum energy
of 800 MeV and an injected energy of 200 to
350 MeV, the coaxial system will not be con-
sidered further.
One of the distinguishing features of a
separated orbit cyclotron (SOC) is, as the name
implies, the relatively large distance between
adjacent orbits of the particles. The orbits
may be separated by placing the particles in a
three-dimensional spiral path'; however, with
sufficient acceleration per turn the orbits may
remain in a plane. For an 800 - Me V machine
the latter approach appears to be more eco-
nomical. A conceptual model of such a
machine is shown in Fig. 1.
Rectangular Cavities
The sirnplest cavity which could be used in
an 800-MeV SOC appears to be a rectangular
Cavity operating in the TM10 mode. The
"lectric field in such i cavity is given by
Esime = Eq sin (fx/l)(cos by) where
For a plane SOC the basic specilications
for the accelerating system can be delerinined
from the machine diameter, the injected and
maximum kinetic energies of the particles
(T; and Tol, and the minimum allowable
spacing (Ar) between adjacent orbits. The
operating frequency of the cavities rnay be
determined from wrf = n Wp, where n is an
integer and wp is the angular velocity of the
particles. Since wp = Vplr = ßc/r, then
wjf = nBc/r or 1 = Zar/lip). In practice the
b = (27/ 1 . (112212], and t = cavity length in
the radial direction. The gap voltage (where
the gap coincides with y = 0) is then
Vy = Vinsin(tx/l), with Vm = -gEmTo provide
the required Ar at r; and ro
Vins in(tia/l) = Vi.
Vinsin[ola + ro - r;//l] = Vo,
1. Research sponsored by the U.S. Atomic
Energy Commission under contract with Union
Carbide Corporation.
PATENT CLEARANCE OSTAINED. RECEASE TO
THE PUBLIC IS APPROVED. PROGEDURES
ARE ON FILE MTS BECOME TO
v 2valo-Vicos (r, - r:llet?
ai Tsin r -ry
where Vi and Vo are the required gap voltages
at r; and ro. Since cavity power loss is a func-
lion of both V and ', there is an optimum
length for the cavity. In general the optimum
length is such that r, coincides with a point on
the gap somewhat greater than 1/2, in other
words V.mVO
The power loss in the walls (z = constant) of
either the simple or the "shaped" rectangular
cavity is a function of the electric field in the
cavity but the power loss in the perimeter (top,
bottom, and ends) is a function of electric field
and gap length. For a fixed azimutal cavity
space in iin SOC it can be shown that there is an
optimum :mber of cavities for minimum total
power loss in the machine. The ri power loss
in all cüvities may be expressed as
12
KO
PT-mäsintom! (mpw + Gpg)
If other TM0 modes are excited in a
cavity it is possible to produce a non-sinusvidal
gap voltage as a function of distance along the
gap. Consider the following field expansion.
P
E,(x,y) = !
C sin(nax/l) cost), y;
1
where K = (AT),/Cosos, m = total number of
cavities, G = total azimutal distance available
for cavities, o/m = TG/mßod transit time ár.gle
at l'o, Pw - power loss in cavity walls for unit
electric field it ro, and p = power loss per
unit gap length in perimeter of cavity for unit
electric field at ro. This equation is plotted in
Fig. 1, along with the power loss per cavity
PL and the transit time factors F; and F. for
Ko = 30 MeV, T; = 200 MeV, T. = 800 MeV,
G= 1200 in., pů = 1.39 x 1004 and
P = 2.04 x 10-6. The values of Pw and Pp are
chose calculated for the cavity of Fig. 3.
where on = (27/
1 1 - (nx / €114. Given an un-
limited number of terms in this expansion, the
gap voltage can be shaped to any desired func-
tion of x. In a cavity of finite length this result
cannot be achieved since the y boundary of the
cavity y, will exist only iſ the On's fall within a
liniited range. It is obvious from the definition
of b, that cos bny will become cosh boy for
terms with n>2!./. If y, is to be finite under
this condition then Cn must be very small com-
pared with Gli therefore, such tems can have
very little effect on the shape of the gap voltage.
Tapered-Gap Cavities
For an 800-Me V SOC the distance between
the inner and orbits will be about 0.7 to 1,01
if X = 240 in., and the cavity length may be
assumed to be about 1, 5. Under this assump-
tion about three terms could be included in the
field expansions. Where the cavity dimensions
are defined in terms of wavelength, X = x/l,
Y = 2y/. L = 21/1, and B = bod 12 the cavity
fields can be written as
Since the magnets in an SOC are approxi-
inately sectorial in shape, the space available
for cavities increases with machine radius,
Tlie gap length of rectangular cavities is fixed
by the space available at r; If the gap length
is increased with machine radius the cavity
volume can be increased, with a possible reduc-
tion in power loss. The electric field in tapered
gap cavities can be expressed as
P
Ego, y) = { 0,2,19,0 )cos b, V.
n
=
P
E, =) Csin naX cus B. Y,
where
2
n=
1
5,19,12)
2,19,0) = J,(4,0) - N, (a ) N, 19,),
H
= -(j/TT nl )
CB sinnX sinB Y,
111
11 = 1
b= (29/11NI - (,4/2012.
H
= -(j/La
nC cos naXcos B Y.
n=1
The y has been used in place of the conventional
2, and p is the radial coordinate whose origin
may or may not coincide with the machine
center. If the ends of the cavity are located at
py and o2 then the "a,, 's" may be determined
from
KU
In designing cavities to produce these
fields, the coufficients C, and the length are
selected to give a gap voliage which will meet
the minimum energy gain requirements. The
boundary YolX) and the power loss may then be
computed. A plot of Y. (X) and Eu(x) for a typi-
cal case (for a three-term expansion) is shown
in Fig. 3. A computer program was written to
perform the rather lengthy computation of
pertinent parameters,
J,(0,) - N, KU
N,CU,) = 0,
where Un = apy, and k = pole. Roote of this
equation are easily obtained by a computer
routine since Unlk - llt. As was the case
with rectangular cavities, the number of terms
which can be practically included in the field
expansion is about 210,-pla. The field
equations are simplifica somewhat is the fol-
lowing substitutions are made: R:2012,
A = 1/2, Y = 2yla, and B =D112. Then
rectangular cavities, then Vi/V.2 0.21. Where
Fi/F, = l for the tapered gap cavity, Vi/V, 2
0.1886. 1re coincides with X = 0.68 in the
shaped cavity of Fig. 3 then ri would correspond
to X = 0.68 . liro- ,)/(3, IX), or X = 0.116
and ViVo = 0.258. The cavity then provides
the required Ar when the maximum voltage is
properly adjusted. By the same reasoning it
can be shown that the tapered gap cavity of Fig.
5 also provides the required Ar.
1
C2,(AR)cos B. Y,
n=1
Table I provides a comparison of the three
cavily types for the assumed machine, The
length given for the simple rectangular cavity
HR = (/19m) CB, 2,(A R) sin B. Y,
n
1
1
1
1
n=)
TABLE I - Comparison of Cavities
р
Hv = (jloni " CA 2 (A Rlcos B Y,
11 11 O n
n=1
TypeNumber
Radial
Length
(in.)
Max.
Height
(in.)
Total
RF
Power Loss
(MW)
V = -(
RO
DC 2,(
AR),
1 T 11
n=1
408
372
372
J.(
AR)
Tn2
na 7. (AR) = J JARINIR"0"
SN (AR),
0 1 11 1 1 21
125.5
151
149.5
9
6.93
5.06
where 2 (AR)
*R-siniple rectangular, s-shaped rectangular,
T-tapered gap.
and . : angle between cavity walls. Again,
the coefficients C, and the cavity length p, -p
can be chosen to produce an acceptable gap
voltage. Yo(R) and power loss inay then be
calculated with a computer routine. Curves of
YA(R), E (R), and the normalized gap voltage,
V (R), are shown in Fig. 5 for a typical case.
is nearly optimum. For each case the number
of cavities is optimized.
Experimental
The total rf power loss in an SOC using
taperci gap cavities may be expressed as
A one-quarter scale model of the shaped
rectangular cavity, shown in Fig. 6, was used
to check theory, fabrication tolerances, and
tuning methods. Computed and measured char-
acteristics of the model are given in Table II.
ko
=/
TABLE II - Model Cavity Parameters
2
p __00
(inp. tp),
Tmj sin( mt|
where y = mo = total angle available for cavi-
ties, o/m = (
T u)/(mßon) = transit time
angle, and pe = power loss per radian in cavity
perimeter for unit electric field ai poThe
optimum number of cavities m may be deter-
mined from the equation
1 + Ped mbpw
tan(/in.) = (20/m,'T + 20 Pcm, P.
Comparison of Cavity Types
Effective
Resonant
Shunt
Frequency
Resistance
(Mc/s)
(MN
Calc. 196,83 22,000 0.922
Meas. 197.49 21,000 0.81
*R = (V x)2/(2PL)
'gmax''
The effective shunt resistance R and the rela-
live gitp field were measured by perturbation
techniques. The measured values are for the
Cavity as received from the fabricator, Fig. 7
provides a comparison between measured values
and theoretical values of the relative electric
field along the accelerating gap of this cavity.
The first "higher-order mode" observed in the
model occurred at a frequency of 220 Mc/s.
To compare the three types of cavities
which have been considered, the following SOC
parameters are issumed: :2:10 in, , T =
200 MeV, To = 800 MeV, n-alu = 20, Ar :
4.5 in., G = 990 in., and * = 2.287 radians.
Then B; = 0), 5662, Bo = 0,8418, 1; = (mB;1)/(211) -
433 in., 10 = 643 in., (AT); ~ 5.58 MeV,
(AT), 629.6 MeV, and V;!;/V.
(AT)/(AT), (),1886. if Fi/Fon1.9 for
Conclusion
Shaping vi SOC cavities to produce a non-
sinusoidal variation of voltage with machine
radius can signilicantly reduce rf power require.
ments for maciiines spanning a wide energy
range. The advantage of shaped cavities dis-
appears, however, as ile energy range is
decreased. For example, preliminary calcula-
tions indicate that cavity shaping would be
une coriomical in a 350 10 800 MeV machine.
Rolerence
1.
F. M. Russell, Nucl. Instr, and Meth, 23,
229 (1963).
Figure Captions
Fig. 1 - Conceptual Model of an SOC
Fig. ¿ - Minimum Accelerating Voltage as it
Function of Radius
Fig. 3 - Height and Gap Field for a Shaped Rec-
tangular Cavity
Fig. 4 - RF Power Loss vs Number of Cavilius
for a Typical SOC
Fig. 5 - Height, Gap Field, and Voltage for a
Tapered-Gap Cavity
Fig. 6 - Scale Model Cavity
Fig. 7. Measured Field in Gap of Model Cavity
PHOTO 68148 A
SECTOR MAGNET

RF CAVITY
INJECTOR MAGNET --
*
:
0
5
10
FEET
15
20
EXTRACTOR MAGNETY
lily
1. Cine e pothud medisit an SOG,
ORNL-OWG 65-1435
2.0

1 = 240 in., fə 50 mc/sec
NUMBER OF CAVATIES, m = 22
GAP LENGTH, g = 45 in.
0s= 30°
Ar= 4.5 in.
Ti = 200 MeV
7. = 800 MeV
ri = 433 in. --
16 = 643 in.
V min (MV)
400
440
480
520
600
640
680
560
(in.)
lib. 2. Mummin actuluraling village is a function of raclius.
ORNL - DWG 65-1436

TO
Eg (RELATIVE )
L = 3.1
6,= 4.0
Cy=-0.25
C3= 2.1
Oo
0.2
0.4
0.6
0.8
4.0
toy. 1. ledgedal and gup bold lur . shafered reclullgular civity.
ORNL-DWG 64-9928R

TO
11!
TOTAL POWER (MW)
r
.
P = {
uinter/m] (mpur + GPP!
5
..
10
....
15 20 25 30
NUMBER OF CAVITIES
35
40°
Hox. 1. Kto power 119N VA number of civities for it typical Soc.
ORNL-DWG 64-9927R

0.7
0.6
0.4
C = -2.29
C2=-0.735
Cz=-0.353
-
+
~
.
3.77
4.39
5
.01
5.63
6.25
R
Fig. 5. Inglit, fap field, and voltage dur a tapered-yap cavily.

>
.
ii
.
.
.
:
fin. 6. Sudlo model cavity.
ORNL-DWG 65-1437

Eg (RELATIVE)
THEORETICAL CURVE
• MEASURED VALUE
-
o
0.2
0.4
0.6
0.8
1.0
Fig. 7. Measured field in gap of model cavity.

I
.
AIM
DATE FILMED
3/ 2 /65
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