The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

Spin Depolarization in Presence of Beam-Beam -Interaction
Yliri BATYGIN and Takeshi KATAYAMA*The Institute of Physical and Chemical Research (RIIffiN), Wako-shi, Saitama 351-01, JAPAN*center for Nuclear Study, School of Science, University of Tokyo, Tanashi 188, JAPAN
Abstract
Analytical and numerical treatment of spin
depolarization due to ion - ion collisions is given. It isshown, that in a collider with two Siberian Snakes in eachring spin depolarization due to beam-beam collision issuppressed. Degree of depolarization can be controlled by
an appropriate choice of working point of betatron particles
motion. In the absence of Siberian Snakes, beam-beamcollision results in monotonous spin depolarization.

and similar for Ap~. Parameter ~ is a beam-beam
parameter, which characterizes the strength of interaction:(3)

where N is a number of particles per bunch,r0 = q2/(41tE0 mc 2) is a classical particle radius, cr is atransverse standard deviation of _the opposite beam size andy is a particle energy.
--+1. Introduction
Particle colliders with polarized beams require careful
control of spin depolarization. During acceleration spin issubjected to intrinsic and imperfection resonances, resulting
in depolarization. Extra source of depolarization is beambeam collisions. In present paper effect of beam-beamcollision on spin depolarization in an ion - ion collider isstudied. Analytical treatme_nt of the problem provides
choice of the collider operation point, where depolarization
is minimized. It also indicates zone of relatively strong
depolarization.

Rotation of spin vector S =(Sx, Sy, Sz) is described bysubsequent spin matrix transformation [1, 2]. Matrix of spin
advance in an lattice arc is described as a matrix of spinrotation in dipole magnet with bending angle u:cos (coos)0Du=- sin (coOs)

0
10sin (coos)0cos (coos)(4)

where coos= (1+Gy)u and G = 1.7928 is the anomalous
magnetic moment of the proton. Matrixes of Siberian
Snakes are given by

2. Collider Model With Polarized Beams
Let us consider a collider ring with two installed
Siberian Snakes. We use a two-dimensional particle modelin phase space (x, Px), (y, py), where x andy are particle
positions, Px = ~; (dx/dz) and py = ~; (dy/dz)) are particle
momentum, ~; = R /Qx and ~; = R /Qy are average valuesof beta-functions of the ring, R is a ring radius, Qx and Qy
are betatron tunes. Particle motion between subsequentcollisions combines linear matrix with betatron angles
8x = 2nQx. 8y = 2nQy. perturbed by beam-beaminteraction:
Xn+lp~+l=Yn+lp~+lcos8x sin 8x00- sin8x cos8x0000cos00-sin 8y cos8y8y sin 8yr~~·J
Yn0Sl= 01[r~ /(2cr~)]

10'00s2 = 0-10-10-100. (5)Spin advance after crossing the interaction point is
described as follow :1- a(B 2+C2) ABa+Cb
ACa+BbSx
2
2
1- a(A +C ) BCa-Ab
Sy = ABa-Cb
ACa-Bb
BCa+Ab 1- a(A2+B 2)
Sz
A=Px,Po. (1)

B=Py'Po

C= Pz,Po

Po =Sx,oSy,o ' (6)Sz,oyPx2 + Py2 + Pz2

a= 1 -cos (Po oz) , b =sin (Po oz)(7)(8)where oz = l 12 is an interaction distance and l is a bunch

p~ + Apyn

length. Vector P = <Px. Py. Pz) is given as follow:

Beam-beam kicks Ap~. Ap~ are expressed as a resultof interaction of particles with opposite beam with
Gaussian distribution functionAn_~:
1- exp[- dt(2cr~)]
L\Px = 41t~Xn

0-10(2)-282-

1y ~EyPx = - [(1 +Gy)Bx + (Gy + - ) - ],Bp
1+yc(9)

Py = _1 [(1+Gy)By - (Gy + ___1__) ~Ex ], Pz = 0,
Bp
1+yc(10)

The lith Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

where Bp is a rigidity of particles, E =(Ex, Ey, 0) is an

Finally, matrix of spin advance of a particle ininteraction point at the n-th turn is:

electrical field and B =(Bx, By, 0) is a magnetic field of the
opposite bunch. Taking into account, that particles arerelativistic ~::::1, y >>1 and electromagnetic field in theinteraction point is created by an opposite beam with round[1- ~ cos 2(n8+'¥)] [- ~ sin2(n8+'¥)] [- <p cos(n8+'¥)]->

Gaussian space charge distribution, vector P is simplified:
Px = 4G 1. y_ [1 - exp (-Ic r 2~ 2 )],(11)22242242[-~sin 2(n8+'¥ )] [1- ~ sin\n8+'¥)] [- <p sin(n8+'¥)] . (18)
1-<p sin(n8+'¥ )<p cos(n8+'¥)<p2-220"2

Py =- 4G 1. .X. [1- exp ( ~ _r_)],2cr2
Ic r2(12)where I is a beam current and Ic = 47t£o m 0 c 3/q =(A/Z)·3.13·107 Amp is a characteristic value of the beamcurrent.
3. Analytical Treatment of Spin Depolarization

It is easy to verify, that determinant of the matrix (18)equals unit, as it is required by conservation of absolutevalue of spin~~ = 1.
Subsequent multiplication of matrixes (13) and (18)gives us possibility to predict effect of beam-beaminteraction on spin depolarization after large number ofturns. SuppQse, initial spin vector has only one transverse
component Sy = 1 and other components are equal to zeroSx = Sz = 0. Spin advance after n turns is as follow

To make an analytical treatment of spin depolarization,
let us consider a collider with two installed Siberian Snakes
and one interaction point. Matrix of spin advance after one
revolution in the ring between beam-beam interaction is.
2 n-1
Sx = ~ (L (-l)t+n- 1 sin 2(i8+'¥) + ... ] ,42.
2 n-1
HY sin (iS+'¥)] ,Sy = 1- ~ [i=O2L-1 0 0
Mring = D1tt2 ·S2 · D7t · St ·D1t/2 =0 1 00 0 -1

(13)

Sz=(-l)n<p

L (-1) sin(i8+'¥),

i=OAverage values of spin components are achieved byintegration of eqs. (19-21) over all initial phases andaveraging over turn number:

pB =....L =- cos(n8 +'¥).
Poq;

where andall particles:2<p1_ ),
Sy=1--(8 cos (8/2)

Sz=O,

Sx=O,

A= Px = sin(n8 +'¥),Po-e =21t (Q - -)s .

Averaged root-mean-square values of spin components aregiven by.

q;2 - ]2, <Sy>. =__]___ [?.

4] ,<p512 cos (8/2)

8 (cos8)Hence, the matrix parameters a and b in eq. (8) can beapproximated as follow:
2b = sin <p :::: <p .

(23)

2

<SX>= [a = 1 - cos <p "" ~ ,2(22)e are average values of parameters <p, e among(15)

Then, let us take into account, that parameter <p = P 0 ·0z
in eq. (8) is small:

(21)

1

X= r COS(n8 + 'f'), y = r sin(n8 + 'f'), 8 = 8 + ~{} , (14)where '¥ is an initial phase of betatron particle oscillations
and ~e « 27t is tune perturbation due to beam-beamcollisions. Parameters A and B in eq. (6) can be expressed
as follow:

(20)

.

n-1Suppose, the ~tat~n t~nes in x and y directions areequal each other Sx = Sy =e. We will consider particle
motion far enough from low order resonances, therefore,
particle trajectory can be expressed as a linear oscillator
with perturbed betatron tune e:

(19)

i=O

(24)

2

<Si> =[<p _ ]2 cos(S/2)

(25)

(17)
Attained formulas indicate, that spin depolarization due
to beam-beam collisions is suppressed and depends on-283-

The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

betatron tune in the ring. The most dangerous working
point is close to half-integer value of betatron tune, because

1600
1400
1200

in that case the. val~e of cos (9/2) is close to zero and spin
depolarization becomes large. Due to small value of <p, alleffects, proportional to cp4 are negligible as compare with
that, proportional to cp2. Therefore, among possible
depolarization effects, the most pronounced are change ofaverage value of Sy component and rms value of <Si>.

1000

z 800
600

I

400.IIJ d.ll ll.l

20000.9999 0.99992 0.99994 0.99996 0.999981Sy

4. Numerical Simulation of Beam-Beam Effect on Spin
Depolarization
Computer simulations utiliting numerical model ofSection 2 were performed for the beam parameters,presented in Table 1.

z

Table 1. Parameters of the interacted beams
Particle energy, 'Y
Rms beam size at interaction point (IP), aBeam-beam tune shift per collision ~Beta function of a ring, ~*,Number of modeling particles, N

IlllllILLI I.I.IHJI

II.
IIIII.
Ill I.IJJ 111

0-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5

Sx260
0.08mm-0.01250.65m
5000

cp = 4nG 'Y ~JL = 9 -10- 3 , e= 2n (Q - 0.006) .

300250z 200
15010050,II,In IIt... LII IIlii

Jill lltu,

0-0.02-0.015-0.01-0.005 0 0.005 0.01 0.015 0.02

(26)

Results of numerical simulations indicate that inpresence of Siberian Snakes spin depolarization issuppressed. Depolarization is expressed as a manifestationof spin components distribution (see Fig. 1). Distribution ofSx component is much narrow than that of Sz component. It
also follows from eqs. (24), (25), where <Si> isproportional to cp4, while <S~> is proportional to cp2.
Average value of Sy oscillates around stable point close tounit (see Fig. 2). Average values of Sx and Sz oscillate
around zero (not presented in Fig). The most significant
depolarization is observed if betatron tune is close to half
integer value. In other cases depolarization is substantiallysmaller. Numerical results are in good agreement withanalytical predictions of the preceding section.
Without Siberian Snakes beam-beam interaction
results in slow spin depolarization. In Fig. 3 average valueof spin component Sy exhibit monotonous decreasing. Itindicates, that presence of Snakes is important to prevent
depolarization due to beam-beam collision.

xl0-3350

For that combination of collider parameters, the valueof matrix parameters are as follow:
~*500
450
400
350
300
250
200
150
100
50sz
Fig. 1. Spin distribution after 500 000 turns in a ring with
one IP and two Siberian Snakes, Qx =Qy =14.249.
J.--,--,--,--,--,- -,--.--.--,---,

0.999995 1--1--1--1--+--+ --+-+-+-+-4
0.99999 1-t--t--t--t--t-- t--t--t--1--l
0.999985 ~-iiii$iii$ii;;liiiiii*-~iiiiiifiiiil~ 0. 99998 l--l--l--l--l--l--l --1--l--1--1It'll 0.999975 1-t--t--t--t--t-- t--t--t--1--l
0. 99997 1-t--t--t--t--t-- t--t--t--1--l
0.999965 1-t--t--t--t--t-- t--t--t--1--l
0. 99996
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5TURN X 100000
U.W.u.u..Lu.u..Lu.u..LLU.U.LU.U .LU.U.LJ.J..U.LU.U.W..W

Fig. 2. Average value of spin component Sy as a functionof tum number in a ring with one IP and two Siberian
Snakes, Qx = Qy =14.249.
10.9995
0.999
0.99850.998
0.99750.997
0.99650.996
0.9955

References[1]. A.Chao, NIM 180 (1981) p. 29.
[2]. T.Katayama, "Basic theory of spin dynamics in RHIC",RIKEN Internal Note.

.... ,.

....
Ill..

"' '1.....

!".

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
TURN X 100000

Fig. 3. Spin depolarization due to beam-beam collisions ina ring without Siberian Snakes, Qx = 14.095, Qy = 14.59.

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