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I- UNITED STATES A TO M I C E N E R G Y CO M M i S S I ON
AECU-1111
FORCED BETATRON OSCILLATION IN A
SYNCHROTRON WITH STRAIGHT SECTIONS
By
Nelson M. Blachman
FNGR LIBRARY
JUN 1 O Too f
UNIV. OF WASH
April 5, 1951
ITIS Issuance Datel
Brookhaven National Laboratory
ls
– Technic a l l n form a ti on Service, Oak Ridge, Te n n e s see
NIVERSITY OF
IIII
8647

y
PHYSICS
AEC, oak Ridge, Tenn., 4-5-51—350-wis's
FORCED BETATRON OSCILLATION IN A SYNCHROTRON WITH STRAIGHT SECTIONS
By Nelson M. Blachman
ABSTRACT
The effect of azimuthal variation in the height of the magnetic median plane on the vertical oscillations
of a particle in a synchrotron with four straight sections is studied, particularly in the neighborhood of
resonance between the frequency of these oscillations and the frequency of revolution. It is found that
Fourier analysis, which is needed in the case of a circular magnet, is replaced by analysis using modi-
fied sine and cosine functions. This investigation also applies to the effect on the radial betatron Oscil-
lations of azimuthal variation in the equilibrium radius.
INTRODUCTION
In a circular synchrotron equality between the frequencies of revolution and free vertical (betatron)
oscillation occurs where the index of the fall-off of the magnetic field with radius, n = -(r/ H,)(6 Hz/ 6 r),
becomes unity, at the limit of the region of radial focusing. When straight sections are inserted into
the orbit, this equality occurs at a lower value of n, where there is still radial focusing. The beam
may then be close to this resonance during acceleration and may pass slowly through it during ejection.
It is therefore important to determine the effective loss of aperture due to the resulting vertical oscil-
lations, which are excited by the azimuthal variation in the height Z(9) of the magnetic median plane
(locus of H. = 0); 6 denotes the azimuth in the associated circular magnet.
This analysis relates to a synchrotron or betatron having four straight sections, each of length L,
separating four magnet quadrants of radius R. The resonance occurs where” 4p1 = 21, i.e., where
n = n.1, With
ctn n;/*/2 - n:/*L/2R (1)
We shall see that at resonance the increase per revolution in the rms amplitude of vertical oscillation
is given by the square root of the sum of the squares of the two integrals
1/2 3/2T
2T n1 II/2 1/2 ſ 1/2 -
== Z(6) cos ni’ “6 d6– Z(9) cos n " " (II – 9) dº
O Z(9) C(9) d6 Sine nº 21/2 |ſ 1 T/2 1
2
+ |...” COS n}/*2n - 9) * (2)
> and
tº .2 n1/2 T 2T 1/2
; : J Z(9) S(0) d6 = ivº ſ Z(9) cos *(;- 9) d6 -) Z(9) cos n / (#1 *- 9) dol (3)
"o sine ni T/2 |Jo T. 2 T 1 \2
; which we may denote by TZc and T2s, respectively (see Fig. 1). In the case of a circular magnet
* (L = 0) this increase per revolution becomes simply T times the amplitude of the first-harmonic com-
Ç
... ponent of Z(9).
:
~
*N. M. Blachman and E. D. Courant, Rev. Sci. Instruments, 20: 596 (1949).
3.
AECU-1111
2 AECU-1111
Near this resonance, as we shall see, the principal part of the steady-state response of the beam
has amplitude
n . Sine n!/ 2/ 2 X 1(z. + zº)/ 2
1 1 C (4)
... .1/2 -
n}/ 21 + sine n1 –1 |n n1
We shall see also that the mean position of the beam is displaced by the mean value of Z(6), namely,
ſº Z(9) d6/2T. -
O
RMS AMPLITUDE
The straight sections are at 9 = k1/2, with k any integer. This azimuth will denote the center of the
straight section, ki/2 – 0 will denote the end toward lower 9, and kT/2 + 0 will denote the end toward
larger 6. Letting y = n-1 2 dz/ d6, with z the vertical displacement of any particle, we can write the
following relations governing the free vertical oscillations (Z = 0)*
y(kT/2 + 0) = y(kT/2 – 0)
z(k1/2 + 0) = z(kT/2 – 0) + ay(kT/2)
(5)
yſ(k+ 1)1/2] = -sz(k1/2) + (e - #as) y(kT/2)
zſ(k+1) 1/2] = (c - #as) z(kT/2) + (s + 2C -#aºs);(k/2)
2
with a = n!/ *L/R, C = COS n!/ 2n/ 2, and s = Sine n!/ 21/ 2. The general solution to the difference equa-
tions (Eqs. 6), obtained by assuming that y and z depend exponentially on k, is
(6)
z(kt/2) = A cos (kpı 4-4), y(ki/2) = -A(l + ac/s -pº)” Sine (kpı + ºp) (7)
with cos pi = c – 1/2as and A and T arbitrary.
Now y(kit/2 + 0) = y(kt/2) and z(k1/2 + 0) = z(kT/2) + 1/2ay(kT/2). The amplitude of vertical
betatron oscillation in the k-th quadrant is the square root of the sum of the squares of these two
quantities. In general, it is different for each value of k. Near the resonance at n = n1, the rms value
of these various amplitudes, averaged Cver all values of T, is given by
1/2
A H*k/ 2) + yºkº/ 2) cscº n: 2 T/ 2 (8)
(Exactly at the resonance the amplitude varies periodically with k, and it is not correct to average
over b, but the beam will not be exactly at resonance for long.)
RESONANCE
The vertical betatron oscillations are simple harmonic oscillations about the magnetic median
plane. Thus, in the curved sections
dºz/do” + nz = n2(6) (9)
while between curved sections the connection formulas (Eqs. 5) apply. The Sturm–Liouville theoryf is
applicable to this differential equation with such boundary conditions. Given any two independent So-
lutions, which we may call C(6) and S(6), of the associated homogeneous equation -
*Consideration of the values at the centers of the straight sections leads to a simpler result (Eq.
7) than consideration of the values at 0 = k1/2 + 0, as in Ref. 1.
fSee, for example, Margenau and Murphy, “The Mathematics of Physics and Chemistry,” pp. 253,
516, D. Van Nostrand Company, Inc., New York, 1943. l
AECU-1111 3
d°z/d6** nz = 0 (10)
subject to Eqs. 5, we can write the Solution to Eq. 9 as
z(0) = ſ* S(0) cºſ) = c(o) S(0) na(0) do"/(Cds/d0 – Sac/d0) (11)
in which the denominator, the Wronskian, is a constant.
The functions C(9) and S(0) defined through Eqs. 2 and 3 are seen to be independent solutions of
Eq. 10 for n = n.1, satisfying the connection relations (Eqs. 5). Their Wronskian is n; %31, the subscript
denoting evaluation at n = n.1. Since both they and Z(9) have period 27, each revolution at resonance will
augment the value of z(6) at azimuth 0 = 0 by the value of Eq. 11 with 6 set equal to 27, namely, -TZs.
Now y(9) is ni!/ * times the derivative of z with respect to 6 in the absence of further perturbations.
Thus it is obtained from Eq. 11 by differentiating under the integral, keeping the upper limit fixed. The
increase per revolution in y/ st at azimuth 0 = 0 is consequently TZ C” It follows from Eq. 8, then, that
the increase in rms amplitude per revolution at the n = n1 resonance is the square root of the sum of
the Squares of 2 and 3.
During passage through resonance, the amplitude will first increase slowly, then at resonance
attain its maximum rate of increase, the foregoing value, and finally oscillate about its new asymptotic
value, as described by the Fresnel integrals and the Cornu spiral. The effective duration of the res-
onance, corresponding to the maximum rate of increase of amplitude, is given by the - 1/2 power of
the rate of change of the difference between the frequencies of revolution and of vertical betatron
oscillation, namely, fo sº-me fy -1/ 2. This rate of change is best computed by assuming that the ratio of
the latter frequency to the former is increased by the straight sections by half the ratio of their total
length to the total length of curved sections.” Thus, if the variation of n with radius is rapid at res-
Onance, iye nf w/ 2n, - nf_/ 2nt and i.e. 0, so that the amplitude of vertical oscillation set up by passage
through resonance becomes approximately
(2nliſi)/*-(z. zº)” (12)
STEADY STATE
We can find the steady-state response of the beam to the variations in the height of the magnetic
median plane by expanding both Z(9) and z(9) in terms of the eigenfunctions of the operator (d”/ d6% + n),
subject to the connection equations, Eqs. 5. Each eigenvalue Ai except the lowest, A O? is doubly de-
generate; we may denote the two corresponding eigenfunctions, assumed orthogonal, by ºi and l'i. Thus
2T 2T
CO / \ ... [ {}^ r. ſº f Z(97) p.(97) d6'
z() --> *|% º'º" •o." g-—º,0) (13)
A: P / i T
iTE O 1 q (9') d6 2 (a’) aa’
!, l ſ w;(0 ) d6
The eigenvalues are the values of X for which the equation
dº/d6°4 (n + x)4= 0 (14)
has, subject to Eqs. 5, a solution ºf (0) = b(0 + 21). The lowest is Ao = -n, which has the single eigen-
function do = 1. It corresponds to the vanishing of the frequency of vertical oscillation where n = 0.
The next is A1 = n.1 - n, corresponding to the resonance at n = n.1, where Eq. 14 becomes identical
with the equation of free vertical oscillation (Eq. 10) for n = n1. The latter has the periodic orthogonal
Solutions C(6) and S(0), which we may take for our p, and iſ 1, respectively. The next eigenvalue is
A2 = n2 − n, where ctn n; *1/4 = n}/*L/2R. These eigenvalues Ai = ni – n are given by the values ni
of n for which L = iT/2. However, so long as L & 2R, no > 1, and there will be no chance of any A1
with i = 2 vanishing where there is orbital stability (0 < n < 1).
*N. M. Blachman and E. D. Courant, Rev. Sci. Instruments, 20: 596 (1949).
4 . * AECU–1111
The important terms of Eq. 13 will be the first two, in which Ai can become small. These two
terms give us
2T ... 2 1/2
1 P / 1 sin" n' T 1/2
z0 =#|ſ Z(0) de' - -1/2 sin n1/2,
2T 2T
coſ Z(9') C(9') d6' + S(9) ſ Z(9')S(9') d6'
O O
SII] Il (1 5)
1 T + n
The effect of the lowest eigenvalue, given by the first term, is independent of the value of n; it merely
displaces the orbit by the mean value of Z. The second term represents an oscillation which in the
curved Sections is sinusoidal and of amplitude four.
GENERALIZATION
The entire present analysis applies equally well to the radial betatron oscillations excited by
azimuthal variations in the equilibrium radius R(0) if only (1 – n) is substituted everywhere for n,
r for Z, and R(0) for Z(9). Resonance between the frequencies of radial betatron oscillation and of rev-
olution will occur where n = 1 - n1, i.e., where n = 0.183 in the Brookhaven proton synchrotron. This
value, however, is below the range of n in the machine. The present analysis can also be modified to
apply to a number of straight sections different from four.
The insertion of straight sections into a synchrotron magnet thus requires the replacement of
first-harmonic Fourier analysis of the inhomogeneities with cos 0 and sine 6 by analysis using the
functions C(6) and S(0). The latter analysis turns into the former, as it should, when the length L
of the straight sections is taken to be zero. The effect of the straight sections is to introduce Small
amounts of odd harmonics into the component of Z(6) that is resonant at n = n1. Fourier analysis could
lead to an erroneously optimistic view if the first-harmonic component is small while the third is
large, since slow passage through this resonance might excite such large vertical oscillations as to
lose the entire beam. As pointed out by Ernest Courant; if ai and bi denote the usual Fourier coef-
ficients, then
2T CO 1/2
ſ zc d6 = 4 ctn nº/*/2 X (–1)"a / 2m 1 "1
Iſl = O 1/2 2m + 1
2 n1 1/2 (16)
T CO
f zs do - 4 cm n"/2 X. "2m 1/|*#– n1
O II] = O 1/2 2m + 1
n1 -
which, with short straight sections, are, to first order in L/R, respectively
1 3 5
TZe = Tait (L/R) ( 2*1 ~ *3 * 12*5 ) (17)
1 3 5
"Zs - Tbi + (L/R) ( 391 + *3 + 12°5 + - - )
Thus, a third harmonic is 3L/4TR times as bad as a first harmonic, this figure being 8 per cent for
the Brookhaven proton synchrotron.
*Private communication.
AECU-1111
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