SURVEY OF FULLY ANTISYMMETRIC ROTATORS FOR THE ELECTRON RINGS HERA AND LEP

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SURVEY OF FULLY ANTISYMMETRIC ROTATORS FOR THE ELECTRON RINGS HERA AND LEP

J. Buon

To cite this version:

J. Buon. SURVEY OF FULLY ANTISYMMETRIC ROTATORS FOR THE ELECTRON RINGS HERA AND LEP. Journal de Physique Colloques, 1985, 46 (C2), pp.C2-631-C2-635.

�10.1051/jphyscol:1985277�. �jpa-00224595�

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SURVEY OF F U L L Y A N T I S Y M M E T R I C ROTATORS FOR T H E E L E C T R O N R I N G S HERA AND L E P

J. Buon

Laboratoire de Z'Acce'le'rateur Line'aire, 91405 Orsay, France

Resume

-

Une e t u d e d e r o t a t e u r s a n t i s y & t r i q u e s est p r e s e n t e e pour LEP e t HERA. L ' a n t i s y m e t r i e e t l ' e s p a c e u t i l e c o n d u i s e n t a une p o l a r i - s a t i o n p l u s p e t i t e que c e l l e d e r o t a t e u r s s y d t r i q u e s h o r i z o n t a l e m e n t . Abstract

-

^As y s t e m a t i c s t u d y o f f u l l y antisymmetric r o t a t o r s made of

-- s i x magnets each is p r e s e n t e d f o r HERA and f o r LEP. The maximum p o l a r i - z a t i o n , t h a t could be o b t a i n e d w i t h them, is lower t h a n w i t h horizon- t a l l y symmetric m i n i r o t a t o r s a s it r e s u l t s from the a n t i s y m n e t r y and from t h e space a v a i l a b l e f o r i n s t a l l i n g t h e s e a n t i s y m m e t r i c r o t a t o r s .

A s e a r c h o f 900 m i n i r o t a t o r s f o r t h e HERA e l e c t r o n r i n g has b e e n achieved re- c e n t l y and i s d e s c r i b e d elsewherell-43. Each o f t h e f o u r i n t e r a c t i o n r e g i o n s is intended t o be equipped w i t h a p a i r o f such m i n i r o t a t o r s . The f i r s t one bends the e l e c t r o n s p i n d i r e c t i o n a l o n g t h e l o n g i t u d i n a l a x i s and t h e other one res- t o r e s t h e v e r t i c a l s p i n d i r e c t i o n i n t h e arc. These m i n i r o t a t o r s c o n s i s t o f 6 o r 7 magnets r o t a t i n g a l t e r n a t i v e l y t h e s p i n around the r a d i a l and t h e v e r t i c a l d i r e c t i o n s .

The two m i n i r o t a t o r s o f a p a i r produce two local v e r t i c a l bumps which are a n t i - symmetric w i t h respect t o t h e i n t e r a c t i o n p o i n t . However, i n t h e h o r i z o n t a l p l a n e , t h e corresponding h o r i z o n t a l l y bending magnets o f t h e two m i n i r o t a t o r s are symmetric. Consequently t h e v e r t i c a l s p i n d i r e c t i o n i n t h e arcs is o n l y r e s t o r e d at t h e d e s i g n energy. A c o r r e c t i o n is needed f o r r e s t o r i n g it when t h e r e is a s m a l l e n e r g y o f f - s e t . T h e r e f o r e t h e s e h o r i z o n t a l l y symmetric minirc- t a t o r s c a n o n l y o p e r a t e i n a s m a l l e n e r g y range. T h i s l i m i t a t i o n is n o t s e r i o u s f o r an e-p c o l l i d e r as HERA, b u t would be o f g r e a t concern f o r a n e e $%

c o l l i d e r a s m .

On the c o n t r a r y , r o t a t o r s , which are antisymmetric i n t h e h o r i z o n t a l and v e r t i - c a l p l a n e s , can o p e r a t e i n a l a r g e r energy range.

A s e a r c h f o r f u l l y antisymoretric r o t a t o r s is presented h e r e i n t h e c a s e o f LEP, b u t a l s o HERA f o r comparison. The same a n a l y t i c method h a s been used as f o r t h e h o r i z o n t a l l y symmetric m i n i r o t a t o r s E31.

S t a r t i n g w i t h a six-magnet m i n i r o t a t o r , an addi.tiona1 h o r i z o n t a l bend H a is added b e f o r e e n t e r i n g t h e r o t a t o r from t h e arc, i n o r d e r t o o b t a i n a n anti&- m e t r i c c o n f i g u r a t i o n o f the r o t a t o r p a i r ( f i g . 1). I t r e s u l t s a s m a l l a n g l e o f t h e beam l i n e i n t h e i n t e r a c t i o n r e g i o n with r e s p e c t t o i t s g e o m e t r i c a 1 , a x i s . The six-magnet m i n i r o t a t o r s depend on t h r e e parameters which have been chosen t o be _:t h e s p i n r o t a t i o n a n g l e s a

.,

a2 i n t h e two f i r s t h o r i z o n t a l l y bending magnets, and t h e t o t a l beam l i n e r o t a t i o n 8 = ( a + a + a )/v i n t h e r o t a t o r ( a is t h e s p i n r o t a t i o n i n t h e l a s t r o t a t o r mainet v3= E(GeV)/.44065 is t h d s p i n t u n e )

.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985277

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C2-632 JOURNAL D E PHYSIQUE

The optimum r o t a t o r scheme corresponds t o t h e maximum p o l a r i z a t i o n [ 5 3 P achie- vable i f a l l o t h e r d e p o l a r i z a t i o n e f f e c t s due t o quantum f l u c t u a t i o n s can be suppressed :

Pi -2 leil

l + &

-. -

c o s q l

-2 i

i p0 2 n

P = P -

0 -

Pi -2 lei l I t & - . -

-2

i Po 217

where pi' and pi1 are r

Ef

p e c t i v e l y t h e t r a j e c t o r y c u r v a t u r e s is t h e normal a r c magnets and i n t h e i r o t a t o r magnet, and 8 . is t h e b e a m l i n e r o t a t i o n pro- duced by t h i s r o t a t o r magnet. The a n g l e @.

is

t h e a n g l e between t h e magnetic f i e l d and t h e p o l a r i z a t i o n d i r e c t i o n i n x t h i s r o t a t o r magnet. A small term depending on t h e l o n g i t u d i n a l p o l a r i z a t i o n component i s neglected here. Po = 92.376 % is t h e maximum p o l a r i z a t i o n i n a p l a n a r r i n g .

I n t h e case of f u l l y antisynunetric p a i r s o f r o t a t o r s , t h e cosql. terms i n t h e numerator of t h e above formula cancel o u t . The depolarizatiAn AP/P can be w r i t t e n :

A t a given energy and f o r f i x e d normal c u r v a t u r e p -I, t h e d e p o l a r i z a t i o n AP/P s c a l e s l i k e t h e inverse square o f the r o t a t o r length L. Therefore t h e product L~.AP/P measures t h e f i g u r e o f m e r i t of any r o t a t o r . I n f a c t t h e s t r e n g t h of t h e a d d i t i o n a l h o r i z o n t a l bend depends s l i g h t l y on t h e length of t h e r o t a t o r , and one observes a few percent v a r i a t i o n of L~ AP/P when t h e length is varied.

This small v a r i a t i o n is neglected here.

Pig. 1 - Schematic lay-out ( t o p view) of a p a i r of f u l l y antisymmetric r o t a t o r s i n an i n t e r a c t i o n region of a s t o r a g e r i n g .

Ha : a d d i t i o n a l h o r i z o n t a l bend

nay, Ha , Ha3:horizontally bending magnets of a six-magnet r o t a t o r Va, V p , t y : v e r t i c a l l y bending magnets of a six-magnet r o t a t o r .

I - ^FULLY^ANTIS-IC ROTATORS FOR fiERA

me t h r e e parameters a , a and 0 f o r a six-magnet r o t a t o r have been varied f o r minimizing t h e f i g u r e &f krit ^'^L ^D/P.

According t o t h e r i n g geometry and t a k i n g i n t o account t h e space needed f o r matching t h e q u a d r u p l e t r i p l e t s on each s i d e of a r o t a t o r , t h e a v a i l a b l e space f o r a r o t a t o r is about 40 m, s t a r t i n g a t 173 m from t h e i n t e r a c t i o n p o i n t .

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magnet which is set a t ?he minimum v a l u e compatible with t h e length a v a i l a b l e f o r t h i s magnet i n t h e r o t a t o r ) C 3 I .

The minimum value LZ AP/P = 449 m2 h a s been found f o r t h e r o t a t o r parameters given i n table I. The t o t a l . l e n g t h of t h e r o t a t o r is 44.433 m and t h e maximum p o l a r i z a t i o n is 71.36 % f o r e i g h t r o t a t o r s . The beam l i n e is at a 5.695 mad angle i n t h e s t r a i g h t s e c t i o n .

TABLE f Ha

0

va ' a

v P

Ha V

Y Ha

The f i g u r e of m e r i t LZ AP/P is l a r g e r t h a n t h e optimum value (333 m2) obtained with t h e proposed h o r i z o n t a l l y symmetric m i n i r o t a t o r s A 2 f o r two reasons :

i ) The a d d i t i o n a l h o r i z o n t a l bend Ka i n c r e a s e s t h e r o t a t o r length and c o n t r i b u t e s t o t h e d e p o l a r i z a t i o n s i n c e one of t h e 0 two a d d i t i o n a l bends of a p a i r is bending the o r b i t i n t h e d i r e c t i o n o p p o s i t e t o t h e normal one. However,

t h i s e f f e c t can be suppressed i f t h i s opposite bend is obtained by shortening t h e l a s t arc magnet by an amount equivalent t o t h i s opposite bend. I n t h i s case the p o l a r i z a t i o n increases up t o 72.56 % and t h e f i g u r e o f m e r i t decreases down t o 395 m2.

HERA

SPIN ORBIT LENGTX

ROTATION ROTATION

0 deg -7.464mrad 3.064 m

31.67 8.843 3.630

-96.0 -26.81 11.004

-57.48 -16.05 6.588

124.0 34.63 14.999

25.81 7.207 2.958

19.11 5.337 2.190

i i ) I n h o r i z o n t a l l y symmetric m i n i r o t a t o r s , t h e h o r i z o n t a l l y bending magnets participate t o t h e nonnal bending of t h e beam-line. They g l o b a l l y c o n t r i - bute t o p o l a r i z e t h e beam by t h e Sokolov-Ternov e f f e c t , even i f one of them is bending i n t h e o p p o s i t e d i r e c t i o n . On t h e c o n t r a r y i n a f u l l y antisymrnetric pair of r o t a t o r s , i f one h o r i z o n t a l l y bending magnet i s b e n u n g i n t h e normal d i r e c t i o n , t h e n t h e corresponding magnet i n t h e o t h e r r o t a t o r of t h e p a i r is bending i n t h e opposite d i r e c t i o n . Their p o l a r i z a t i o n e f f e c t s cancel o u t , while t h e i r d e p o l a r i z a t i o n e f f e c t s add t o g e t h e r . This f e a t u r e is e x a c t l y t h e quoted c a n c e l l a t i o n of t h e cos@. terms i n t h e numerator of t h e formula giving t h e maximum p o l a r i z a t i o n P. The r e s u l t is a 1-r maximum p o l a r i z a t i o n than f o r

LEP

SPIN ORBIT tENGm

ROTATION ROTATION

0 deg -3.709mrad 5.742 m

33.64 5.174 8.811

-92.0 -14.151 21.907

-58.73 -9.033 15.382

130.0 19.996 34.620

25.09 3.859 6.571

11.36 1.747 2.704

h o r i z o n t a l l y synmetric m i n i r o t a t o r s .

F i n a l l y , one observes t h a t t h e space a v a i l a b l e f o r i n s t a l l i n g r o t a t o r s is l a r g e r f o r h o r i z o n t a l l y qmnetric m i n i r o t a t o r s , s i n c e t h e a r c can be made a l i t t l e s h b r t e r as t h e m i n i r o t a t o r s p a r t i c i p a t e a l s o t o t h e normal bending of t h e o r b i t . Therefore h o r i z o n t a l l y symmetric m i n i r o t a t o r s can be made longer and t h e i r maximum p o l a r i z a t i o n l a r g e r .

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C2-634 JOURNAL D E PHYSIQUE

I1 - FULLY ANTISYP(MCrm1C m A ' I O R S FOR LEP

The same o p t i m i z a t i o n procedure h a s been performed f o r f i n d i n g f u l l y antsym- m e t r i c r o t a t o r s f o r LEP.

A reasonable s p a c e o f la0 m has been considered, s t a r t i n g at about 300 m from t h e i n t e r a c t i o n p o i n t ( f i g

.

²⁾

.

^The c u r v a t u r e o f the hor_izontally bending magnets has been fixed t o 2 . 0 t i m e s the normal c u r v a t u r e pol (apart f o r one magnet as for &). The c u r v a t u r e o f t h e v e r t i c a l l y bending magnets h a s been f i x e d 10 % lower khan t h e c u r v a t u r e o f t h e h o r i z o n t a l l y bending magnets. So one o b t a i n s a better balance between t h e c o n t r i b u t i o n s o f each magnet t o t h e depo- l a r i z a t i o n and a s l i g h t l y improved p o l a r i z a t i o n degree.

Fig. 2 - General lay-out o f the optimum f u l l y antisymmetric r o t a t o r f o r LEP at 50 GeV : a ) s i d e view, b ) t o p view.

Fig. 3 shows a bidimensional p l o t of L' AP/P as a f u n c t i o n o f t h e s p i n r o t a t i o n a n g l e s a and a , at a fixed v a l u e o f 8 corresponding t o the minimum v a l u e o f L

' AP/P, h f o r drily one p a i r o f r o t a t o r s .

The minimum v a l u e L' AP/P = 554 mz has been obtained f o r the r o t a t o r parameters given i n t a b l e I, and f i g u r e 2 shows t h e g e n e r a l lay-out o f t h i s optimum rota- tor. l t t e t o t a l l e n g t h o f t h e r o t a t o r i s 95.737 m and the maximum p o l a r i z a t i o n

is 73.47 % f o r e i g h t r o t a t o r s (L'AP/P = 1876 mZ ).

A comparison between LEP and HERA i s o b t a i n e d by remarking that the c u r v a t u r e and t h e o r b i t r o t a t i o n i n the r o t a t o r magnets s c a l e like t h e i n v e r s e o f t h e energy. Therefore one must compare the v a l u e s o f A L ~ ~ - ' E ~ f o r e i g h t r o t a t o r s .

0

HERA Lt?P

hor. -tric f u l l y antisym. f u l l y antisym. r o t a t o r s

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Bidimensional p l o t o f t h e f i g u r e of merit 'L dP/P v e r s u s t h e s p i n r o t a t i o n a n g l e s a , a o f t h e twu first h o r i z o n t a l l $ bending magnets Hax, H a for

IEP a t 50 GeV ; (e=3.592

mrad, o n l y one p a i r of rota- t o r s ).

Hatched area : no s o l u t i o n .

I n c o n c l u s i o n t h e optimum f u l l y antisymmetric rotatzrs-fo: LEP _andHEM have v e r y c l o s e v a l u e s o f t h e i n v a r i a n t parameter : AL po -E

.

^{I t}^{i s an}^obvious

consequence of t h e v e r y close s p i n r o t a t i o n a n g l e s I n t h e i r magnets ( s e e t a b l e I ) . Both o f them have a worse f i g u r e o f merit t h a n t h e proposed h o r i z o n t a l l y symnetxic m i n i r o t a t o r s A2 for BERa.

The a u t h o r is i n d e b t e d t o B. Montague and K. S t e f f e n f o r many f r u i t f u l d i s c u s s i o n s .

IV - REFERENCES

I11 S'L'EFFEN K . , DESY HERA r e p o r t 83/06 (Kay 8 3 ) [ 2 1 STEPPEN K., DESY HERA r e p o r t 83/09 (May 83 )

t 3 3 BUlN J., K U r e p o r t LAGRT/83-10 ( J u n e 8 3 ) t 4 1 B W N J., DESY HERA r e p o r t 84/03 (Feb. 8 4 )

t 5 3 DEWO'NEV Y a . S., KONDRATENKO A.M. and SKRINSKII A.N.

Sov. Phys. Dokl. 14 ( 1970) 583.