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ORAL-D-1148
CONF-650306-57
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STABILITY AND TOLERANCES OF THE
SEPARATED ORBIT CYCLOTRON*
Y
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R. E. Worsham
Oak Ridge National Laboratory
Oak Ridge, Tennessee
Summary
magnet period would cause the acceptance to be
increased by about 50%. Each sector is
adjusted so that the transit time for synchronous
particles is a constant in all sectors.
The physical arrangement of the magnets
and rf cavities in a separated Orbit Cyclotron
are described, and the equations of motion and
their solution are discussed. The stability
limits are determined for triplet and singlet lens
systems, and the choice of operating values to
give maximum acceptance are examined. The
treatment of random misalignment errors of the
magnets is described; the six types of errors
whose relative strengths are compared lead to
bench alignment errors of al mil. The align-
inent of adjacent lenses and alignment over a
distar.ce long compared with a betatron oscilla-
tion 'wavelength are shown, however, to have
relatively relaxed tolerances,
In the deflection sector there is a drift
space slightly longer than that of a normal
sector (i, e., WX > CQ), which upsets the
periodicity and requires special treatment.
The magnets QW and XS are treated as part of
a beam-matching system in that their config-
urations and gradients must be chosen to make
the beam area in phase space match that of the
sector at D, and of the following sector at T.
It has been shown that matching of about 99%
can be obtained. The problem is identical to
other matching problems for axial motion,
Introduction
Analysis
The first concept for the Separated Orbit
Cyclotron was originally described by F. M.
Russell. Some recent work at Oak Ridge is
described in other conference paper 82,3, 4,6
The principal change from Russell's work is
that construction in a single plane was found to
be more economical than the original "bee-hive"
configuration. In addition, the optimum
frequency is in the region of 50 Mc/s, where
large cavity gaps and, hence, large cavity
voltages are possible.
The properties of the normal sectors vary
smooth y so that the method of analysis devel.
oped for AG synchrotrons may be used. The
differential equation for radial or axial motion
is
døy + k(sly = 0,
Sector Geometry
where s is the length along the path and k(s) is
the focusing function:
.:
.
k(s)
-
for axial motion,
a driftsing or darity,
-, for radial motion,
, and
.
.
.
where
p = radius of curvature in a field, B.
:.
+
!
"
The physical arrangement of the rf cavities
and magnets in the Separated Orbit Cyclotron
is shown in Fig. 1. The synchronous ion crosses
a cavity perpendicular to the plane of the cavity,
receives an energy gain and a focusing or defocus
ing impulse, passes through a drift section AB,
enters the magnet BC, and so on. Because of
the requirement that gradients also alternate
radially" for rninimum spacing of turns, the
total number of magnets in a turn must be odd.
Two of the sectors such as AD constitute the
minimum magnet period. Also the gradients
must alternate along the beam path for this
shortest magnet period such that if QW goes
FDF, or focus, defocus, focus, XS must go
DFD. Therefore, SOC may consist of a series
of F, D, F, D, F... lenses (alternating singlets)
or FDF, DFD,... (alternating triplets). There
appears to be nothing gained by a more compli-
cated structure. There is a possibility that a
practical SOC could be built without this restric-
tion of two sectors per magnet period if the
flux concentrating plates* are used. The halved
n = hdp.
pdB
.
The matrix describing motion from one point
to another is
corp asinu β sin μ
M = 1
le sin u cosu - a sini
tw
Lita?
*Research sponsored by the U. S. Atomic
Energy Commission under contract with the
Union Carbide Corporation.
..
PATENT CLEARANCE OBTAINED: REHASE TO
The PURUS IS APPROVED. PROCEDURES
ARIMIDLERTES REDDYING FACTYON.
*** ...
I.
=
M
(%) - -(3)
well as by increasing the aperture, of course.
ß must necessarily increase with energy in a
given SoC since its value depends most strongly
upon the length of the magnet period for a given
focusing configuration.
where y' = dy/ds. The trace of M is 2 cos H.
For stability, the value of u must be real.
... 1 dB
B and a(= -2 , are periodic in a
periodic focusing system. The solution of the
cquation of motion may be written:
PIE 81/2,
cos (8)
sin y(s)
:
Y2
Table 1. Separated Orbit Cyclotron.
200-800 MeV Balanced alternating triplets
24 Sectors
1.5-in. aperture
20th Harmonic 7-kilogauss fields
Energy (MeV) 200 400 600 800
Gradient (kg/cm) 0.97 1.33 1.73 1.53
Fraction of pole 1.0 0.7 0.5 0.5
Pmax(M), axial 5.92 7.08 7.56 8. 30
radial 5.45 6.52 7.05 7.72
~3.2 3,2 -3,2 -3. 2
~3.6 -3.6 -3.6 -3.6
Acceptance, axial 27, 2
(mm-mrad), radial 29.6
Damping
1.0 0.94 0.89 0.88
where the phase o(s) =
.
To
The
ds
where
propagation constant Me is
L = the period length.
The value of B throughout a period may be
determined by the following differential equation:
Bill +
4 k(8) B' + 2 k'(s) B
=
0.
It can be shown that there is a constant of the
motion
Relative quantities for singlets and
triplets are shown in Table II.
W = 1/B [y? + (ay + By!)?).
Table II. Relative Values of , V, and Acceptance,
Sectors/period
ß
v
Acceptance
This quantity, which is an ellipse of area mW
in y - y' space, is an adiabatic invariant if the
focusing parameters vary slowly. For an
aperture of Y, the maximum value of Wis
given by
Singlets
Triplets
1.5
1.0
1.0
1.0
0.7
1.0
Da
Pmax
Random Error Analysis
The variation of ß through a sector of
alternating triplets is shown in Fig. 2. Since
the area in phase space is conserved, W varies
throughout the machine as the square root of
momentum. Therefore, the beam size will
The random error analyses for AG synch-
rotrons and linear accelerators 8, 10 are almost
directly applicable to the SOC. If there is an
error A(8) in an element of the focusing system,
the differential equation of motion becomes
vary as Vp/Bmax'
+ k(e)y = k(8) A(8).
ds
The stability diagram for an SOC sector
at 200 MeV, Fig. 3, shows that at high energies
the rf produces a negligible effect on focusing.
It is of particular interest that the betatron
oscillation phase shift per period can be changed
over a range of ~40° with only a very small
change in Romania Therefore, the betatron
frequency may be adjusted to prevent any
resonance from occurring due to magnet spacing,
and yet there will be but a small effect on the
acceptance.
The adiabatic invariant will have the approxi-
mate value
w1/2 = w 1/2
+ S as'A(8')k(8') [yz (8')c08 x - y, (8') sin x],
where W. and x give the amplitude and phase of
the betation oscillation without errors. Now
max
The results shown in Table I are typical
for alternating triplet lens in a 200-800 MeV
machine. To get any particular value of
gradient the pole may be made with partly a
flat field and partly a high gradient field.
Notice the acceptance for this 1.5-in. aperture,
This number could be increased by increasing
the number of sectors (thereby reducing B) as
Yg? = Pmax, FWF
AY má - -YP!?)
= Pmax, F7w1/2 - W.1/32,
References
F. M. Russell, Report ORNL-34 31 (1963).
m8
where is the final amplitude of oscillation
with errors, YF is the corresponding final
amplitude of oscillation if there are no errors,
Bmay, Fis the final value of Bmaxi and Ayrms
is the final rms amplitude of oscillation caused
by the errors. For example, if there is a
random error of mear square values in je
radial or axial position of a triplet lens, the
oscillation growth will be given by
N. F. Ziegler, "Accelerating cavities for
an 800-MeV SOC", to be published in the
proceedings of this conference.
- 3.
J. R. Richardson, "Meson Factories'', to
be pubiished in the proceedings of this
conference.
2
Arms 3 Pmax, fo Lamm P* i'm'
In=1
AY
4B
R. S. Lord and E. D. Hudson, "Magnet
for an 800-MeV SOC", to be published in
the proceedings of this conference,
where n = total niunber of elements, l = length
of mth element, BA and B. = values of B at the
focusing and defocusing lens respectively.
E. D. Hudson, R. S. Lord, and
R. E. Worsham, "A Low Energy
Separated Orbit Cyclotron, to be pub-
lished in the proceedings of this conference.
H. G. Hereward, Report CERN PS/INT.
TH 59-5 (1959).
S. Ohnuma, Yale University, private
communication.
8.
E. D. Courant and H. S. Snyder, Annals
of Physics 3, 1 (1958).
R. L. Gluckstera, Yale Internal Report
Y-7 (1964).
The values for the types of errors that
can occur in the magnet of the 200-800 MeV
machine and their relative sensitivity are listed
in Table III. Correlated errors refer to dis-
placement of the triplet sets of pole tips from
the synchronous path. Uncoräelated errors
refer to relative displacement of the individual
pole tips in a triplet, relative to the mean axis
of the individual lens. The worst error is
non-collinearity caused by displacement of the
center magnet of the triplet relative to the end
magnets. The available area in which the beam
may grow is determined from the difference
between aperture and the damped beam size for
no errors. Coupling between the radial and
longitudinal motion uses some of the space.
For this machine, the rms value of A must be
near 10-3 inches. Therefore, the bench
alignment of the poles must be completed to
about 1 mil rms. The alignment of the
assembled lens relative to each other can,
however, be ~5-10 mils rms. A smoothly
increasing radial error going from o at injection
to 1 inch at the final energy will rpoduce only
4 mils of radial oscillation. Therefore, it is
the bench alignment of the poles and the align-
ment from pole-to-pole that is critical
R. H. Helm, Report SLAC-14 (1963).
John Martin, ORNL. private communi.
cation.
се
Figure Captions
Fig. 1-soc, normal and deflection sectors.
Fig. 2--B and beam shape in typical sector,
Fig. 3 -Stability limits and Bmw in typical
sector, for alternating triplets.
These tolerances are based on no correc-
tion at all for the random errors. Since SOC
is very nearly a linear machine and since ea cli
random error amounts to a linear transforma-
tion, John Martin? has recently suggested that
a pair of small bending magnets could induce
an oscillation to compensate for the error-
induced oscillation. The magnets might appear
in each turn of the machine and, to be most
effective, should be spaced about a quarter-
wavelength of oscillation. It would appear that:
this scheme could increase the tolerance by
the square root of the number of turns. A
more detailed investigation is in progress.
Table W: Relative Ser.sitivity of Random Errors in a 200-800 MeV Soc.
Relative Sensitivity
Correlated Uncorrelated
Types of Errors
0.84
1. Non-collinearity
2. Displacement
3. Skew
4. Orientation in 2-X
plane
5. Length
6. Gradient
1.0
1.4 x 10-2
10-3
1.4 x 10-2
χωφ
0.84
alle
z/2AG/G
1.4 x 10-2
1.4 x 10-2
0. 84

Kitty
S
'
ORNL-DWG 65-4941
CAVITY E

-
MAGNET, O,
:
FOR DEFLECTION
MAGNET
MAGNET, O
+CAVITY E-
+CAVITY E
SOC Normal and Deflection Sectors.
Fig. 1. Soc, normal and deflection sectors.
ORNL-DWG 63-493
#
#

SLOPE
CHANGE
FREE
F, DF, FREE
FREE
D,F,,
FREE,
6
PATH( m )
RE
MAGNET
RF
MAGNET
RF
.
Beam Shape in 2-%' Space; T=200 MeV.
Fig. 2. B and beam shape in typical sector.
ORNL-DWG 65-492
T~200 MeV
GRAD
GRAD (kg/cm)


AT $=-30°
+
-
4.5
H=180° -
HR=180°
p=120° ---
=90° -
-
H* 600 -
--
---
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p=0° -
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- 60 -30 0
4-PHASE
OL
30
wala man
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Fig. 3. Stability limits and me in typical sector, for alternating triplets.
max
puu-
L.
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1
DATE FILMED
6 / 21 /65



121


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rpretation on below
. i en staan. Dit with any interno di inte
- LEGAL NOTICE
This report was prepared as an account of Goverament sponsored work. Neither the United
States, cor the Commission, nor any person acting on behalf of the Commission:
A, Makes any warranty or represcatation, expressed or implied, with respect to the accu-
racy, completeness, or usefulness of the information contained in this report, or that the use
of any information, apparatus, method, or process disclosed in this report may not infringe
privately owned rights; or
B. Assumes any liabilities with respect to the use of, or for damages resulting from the
use of any information, apparatus, method, or process disclosed in this report.
As used in the above, "person acting on behalf of the Commission" includes any em-
ployee or contractor of the Commission, or employee of such contractor, to the extent that
such employee or contractor of the Commission, or employee of such contractor prepares,
disseminates, or provides access to, any information pursuant to his employment or contract
with the Commission, or his employment with such contractor,

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