SUPPRESSION OF BEAM-BEAM DEPOLARIZATION

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SUPPRESSION OF BEAM-BEAM

DEPOLARIZATION

J. Buon

To cite this version:

J. Buon. SUPPRESSION OF BEAM-BEAM DEPOLARIZATION. Journal de Physique Colloques,

1985, 46 (C2), pp.C2-637-C2-640. �10.1051/jphyscol:1985278�. �jpa-00224596�

SUPPRESSION O F BEAM-BEAM DEPOLARIZATION

J . Buon

Laboratoire de 2'Aece'ZQrateu.r Line'aire, 91405 Orsay, France

R e s W I1 est d i s d e des l i m i t a t i o n s e t d e l a compatibilite des

- -

conditions d e "spin matching" q u i o n t

ete

proposees pre&emnent pour reduire la depolarisation due a l ' i n t e r a c t i o n faisceau-faisceau dans un collisionneur.

Abstract - "Spin matching" conditions have been previously derived f o r suppressing beam-beam depolarization i n e l e c t r o n s t o r a g e r i n g s indepen- d e n t l y o f t h e magnitude and o f t h e non-linear f e a t u r e o f the beam-beam i n t e r a c t i o n . Limitations and compatibility with o t h e r "spin matching'' conditions are discussed.

Spin matching conditions have been previously derived C11 f o r suppressing t h e depolarization by t h e beam-beam i n t e r a c t i o n i n a storage r i n g .

The p r i n c i p l e is t h e o v e r a l l c a n c e l l a t i o n of s p i n r o t a t i o n s produced by a s i n g l e beam-beam k i c k a t one i n t e r a c t i o n point and by the a d d i t i o n a l betatron o s c i l l a t i o n i n i t i a t e d by t h i s kick.

I n an e l e c t r o n storage ring, b e t a t r o n o s c i l l a t i o n s a r e damped. After damping one is l e f t with a n e t s p i n r o t a t i o n which only vanishes i f these s p i n matching conditions a r e s a t i s f i e d . Then t h e normal s p i n d i r e c t i o n is restored. Before complete damping t h e r e is

a

trembling of t h e s p i n d i r e c t i o n , with i n f a c t t h e same amplitude as the s p i n r o t a t i o n given by the beam-beam k i c k i t s e l f a t t h e i n t e r a c t i o n point Cll. However, t h i s trembling disappears w i t h a relaxation time equal t o t h e betatron damping time T Now t h e depolarization r e s u l t s from

P'

t h e a c t i o n of successive beam-beam k i c k s on spin. Without s p i n matching one has t o consider t h e n e t s p i n r o t a t i o n s o f a l l t h e kicks during one p o l a r i z a t i o n time T

.

On t h e contrary, when s p i n matching is achieved, only t h e tremblings,

produ& by t h e k i c k s i n one damping time T

,

add together. Consequently it is expected t h a t beam-beam depolarization

i g

reduced by about t h e r a t i o T

which is usually l a r g e .

d 7 ~ '

I n a proton s t o r a g e ring, t h e r e is no b e t a t r o n damping. The trembling8 of spin are not damped. Only their t i m e average is suppressed by the proposed beam-beam s p i n matching Ell. The reduction of the beam-beam depolarization depends on t h e degree of coherence of these tremblings f o r successive kicks, compared t o t h e coherence of the averaged s p i n r o t a t i o n , without s p i n matching.

Let r e c a l l t h a t t h e proposed s p i n matching conditions a r e independent o f t h e b e e b e a m i n t e r a c t i o n strength. They apply whatever are the b e a m i n t e n s i t y and Cross-section and t h e non-linearity of t h e beam-beam i n t e r a c t i o n does not matter.

C2-638 JOURNAL DE PHYSIQUE

However, t h e s e s p i n matching conditions are v a l i d only out of the betatron depolarization resonances. Linearity of the r i n g o p t i c s is a l s o assumed. The e f f e c t of non-linear lenses, as sextupoles, would have t o be investigated. These l i m i t a t i o n s are general and apply t o any type of s p i n matching condition. Finally, t h e assumed o p t i c a l l i n e a r i t y assures t h e decoupling between t h e spin matching conditions f o r photon emissions i n magnets and those f o r beam-beam i n t e r a c t i o n s . Rawever, they could not be r e a l i z e d simultaneously. This compati- b i l i t y is studied i n the next s e c t i o n f o r a r i n g equipped with 90- s p i n rota- t o r s .

-BEAM S P I N MATXiING I N A RING WITff ROTA.IDFS

In a r i n g with 90° s p i n r o t a t o r s , assuming an o p t i c s , symmetric with respect t o

straight section arc

R2

the i n t e r a c t i o n p o i n t s IP1, I P 2 ,

...

and t o the arc c e n t r e A , t h e general spin matching conditions f o r beam-beam i n t e r a c t i o n i n t h e v e r t i c a l plane*):

where =

m

+

i a

(see r e f . 1 f o r t h e n o t a t i o n s ) , can be rewritten, a f t e r some algebra : 1 s i n p

-

+

Sz

+

C - E AZ = 0 P i p z dos p

-

cos p s i n

-

2 defining the o p t i c s i n t e g r a l s :

sz

=

j'

Gz

6

cos

n

d s

I

5

=

j..

4

s i n

+

dS

The similar s p i n matching conditions i n t h e h o r i z o n t a l plane can be w r i t t e n : 1 s i n p X - Q + s x + C X = O c o s p - c o s p X X

H e r e the i n t e z r a l on the a r c vanishes as t h e s p i n closed s o l u t i o n is t h e r e v e r t i c a l and k.; = 0

Now s e v e r a l s p i n matching c o n d i t i o n s are required f o r avoiding l a r g e depolari- z a t i o n induced by the r o t a t o r s C21. Among them are t h e c o n d i t i o n s :

I n t h i s c a s e t h e previous s p i n matching c o n d i t i o n s f o r beam-beam i n t e r a c t i o n become simply :

The p h y s i c a l meaning of t h e s e c o n d i t i o n s is very simple. E f f e c t i v e l y t h e y are

equivalent t o t h e following o p t i c a l c o n d i t i o n s a t t h e e n t r a n c e R1 o f the rota- t o r s :

t a n *

-

X,Z

-

l/ax,z (a = -1/2 dp/ds)

expressing t h a t a s i n e - l i k e t r a j e c t o r y w i t h z e r o amplitude a t t h e i n t e r a c t i o n point IP1 h a s a z e r o slope a t t h e point Rl.

s t r d i g h t s e c t i o n

~1

r o t a t o r a r c

Therefore t h e s p i n r o t a t i o n produced by t h e beam-beam k i c k a t t h e i n t e r a c t i o n point IP1 is e x a c t l y compensated by t h e s p i n r o t a t i o n produced by t h e sub- sequent a d d i t i o n a l o s c i l l a t i o n between IP1 and R1. The i n i t i a l s p i n d i r e c t i o n (normally l o n g i t u d i n a l ) is r e s t o r e d a t t h e e n t r a n c e point Rl. After t h i s p o i n t

R1, t h e s p i n transparency given by t h e s p i n matching c o n d i t i o n s f o r the rota- t o r s a s s u r e s t h a t there is no e f f e c t on t h e s p i n d i r e c t i o n f o r , e v e r .

C2-640 JOURNAL

DE

PHYSIQUE

Unfortunately, t h e s e beam-beam s p i n matching conditions a r e uncompatible with

the s p i n matching conditions Cx = 0 f o r t h e r o t a t o r s . E f f e c t i v e l y t h e l a t t e r

are

equivalent t o the o p t i c a l coh8itions :

t a n 4, = - a

x , Z X . Z

a t the point R1, expressing t h a t a cosine-like t r a j e c t o r y with zero slope a t IP1 has a zero s l o p e at t h e point Rl too.

me

v a r i a t i o n from the ewct beambeam s p i n matching is measured by t slope di,z/ds =t the point Rl of a sine-like t r a j e c t o r y w i t h an i n i t i a l

l/q,z

Slope a t CP1 :

a,

z 1

I n the absence o f s p i n r o t a t o r s , ewct s p i n matching conditions can be f u l f i l - led, w h i c h must g r e a t l y reduce the beam-beam depolarization. No condition is needed i n t h e h o r i z o n t a l plane, and only two conditions i n the v e r t i c a l plane, which are energy-dependent

.

I n a storage r i n g equipped w i t h s p i n r o t a t o r s , s p i n matching conditions f o r t h e r o t a t o r s and f o r the bean-beam i n t e r a c t i o n cannot be f u l f i l l e d simultaneously. I f one maintains those conditions f o r t h e r o t a t o r s , one can only reconmend l a r g e

ex

.values and small a values at the entrance R1 of the rotators i n ,

order to' mmmuze the v i o l a k g n of the beam-beam s p i n matching conditions. Hence one reduces t h e slope o f a sine-Pike t r a j e c t o r y a t point R1, i.e. t h e r o t a t i o n o f the s p i n d i r e c t i o n following a beam-beam kick.

The author is indebted t o D r . K. S t e f f e n f o r questioning f i r s t the compatibility of s p i n matching conditions C23.

E l 3 BUON J., Proc. of t h e 12th I n t . Conf. on High Energy Accelerators, Batavia (1984) and LAL R e p o r t LAL/RT/83-12