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N U C L E A R I N S TR U M E N T S A ND M E T HO D S 9 5 ( t 9 7 I ) I 3 I - I 3 5 ; N O R T H - H O L L A N D P U B L I S H I N G CO .

D I F F R A C T I O N R A D I AT I O N D E F O C U S I N G O F A N E L E C T R O N R IN G *

E. KEIL

CERN, Geneva, Switzerland

C. PELLEGRINI

Laboratori Nazionali di Frascati, Frascati (Roma), Italy

an d

A. M. SESSLER

Lawrence Radiation Laboratory, University of California, Ber keley, California 9 4720 , ~].S.A.

Rec eived 23 February 1971

The influence upon axial stability in an electron ring of the criterion is obtained, and num erical examples show that thediffraction radiation reaction force, generated by a ring moving criterion is not an imp ortant constraint upon the choice ofin an acceleration column, is calculated theoretically. A stability parameters or the operation of a n electron ring acce lerator

1 . I n t r o d u c t i o n

I t is w e l l k n o w n t h a t t h e d i f f r a c t i o n r a d i a t i o n b y a ne l e c t r o n r i n g in t h e a c c e l e r a t i o n c o l u m n o f a n e l e c t r o nr i n g a c c e l e r a t o r ( E R A ) i s a n i m p o r t a n t e f fe c t i n s o f a r a si t c a n c a u s e s i g n i fi c a n t l o s s o f e n e rg y o f t h e r i n g 1 '2 ) .T h e e f fe c t o f t h e d i f fr a c t io n r a d i a t i o n u p o n t h e i n t e r n a ld y n a m i c s o f t h e r i n g h a s n o t s o f a r b e e n s t u d i e d ,

a l t h o u g h i t i s c l e a r t h a t t h e l a rg e e n e rg y r a d i a t i o nc o u l d e a s i ly h a v e a s i g n i f ic a n t e f f e c t u p o n r i n g s t a b i l i tyi n t h e a x i a l d i r e c t i o n , w h e r e t h e f o c u s i n g - c o m i n go n l y f r o m i o n s , i m a g e s a ) , a n d p o s s i b l y f r o m t h ea c c e l e r a t i n g f i el d - i s w e a k .

I n t h is n o t e w e s t u d y t h e c o n t r i b u t i o n o f d if f ra c t i o nr a d i a t i o n t o t h e a x i a l f o c u s i n g f o r c e s o f a r in g , l i m i t i n go u r a n a l y s i s , f o r c o n v e n i e n c e , t o t h e c a s e o f a r in gm o v i n g a t r e l a t i v i s t i c s p e e d s . We e v a l u a t e t h e d e f o -c u s i n g f o r c e f o r t w o d i f f e r e n t g e o m e t r i e s : I n s e c t i o n 2w e c o n s i d e r a c h a r g e d r o d a n d a c u r r e n t c a r r y i n g ro dm o v i n g p a s t a n i n fi n i te a r r a y o f s e m i - i n f in i t e p e r f e c t l yc o n d u c t i n g p l a t e s , w h i c h g e o m e t r y h a s t h e a d v a n t a g et h a t t h e p r o b l e m m a y b e a n a l y z e d a n a l y t i c a l l y. I ns e c t i o n 3 w e c o n s i d e r a c h a rg e d r i n g i n a n a c c e l e r a t i n gc o l u m n c o n s i s ti n g o f a n i n f in i te l y l o n g c o r r u g a t e dc y l i n d r ic a l w a v e g u i d e . T h e e f f e c t o f t h e r i n g c u r r e n ti s n o t i n c l u d e d i n t h i s m o d e l . I n s e c t i o n 4 w e e v a l u a t et h e a x i a l o s c i ll a t i o n f r e q u e n c y r e s u l t i n g f r o m d e f o c u s i n gf o r ce s , a n d i n s e c t i o n 5 w e p r e s e n t s o m e n u m e r i c a le x a m p l e s .

* Work supported in part by the U.S. Atomic Energy Com-mission.

We m a y o b t a i n a r o u g h e s t i m a t e o f t h e o r d e r - o f -m a g n i t u d e o f t h e d i f fr a c t io n d e f o c u s i n g f r o m a s i m p l ep h y s i c a l m o d e l . C o n s i d e r a c h a rg e , Q , m o v i n g a l o n gt h e a x is o f a n a c c e l e r a t i o n c o l u m n . T h e c o m p l e t es o l u t i o n t o M a x w e l l ' s e q u a t i o n i s , i n g e n e r a l , d i f f i c u l tt o o b t a i n , b u t r o u g h l y s p e a k i n g th e r e a r e i m a g ec h a r g e s m o v i n g i n c o n c e r t w i t h t h e c h a r g e Q . T h e s e

i m a g e s a r e s l i g h t l y d i s p l a c e d b e h i n d t h e c h a rg e ,l e a d i n g t o a n a x i a l f i e l d , E , , a t t h e c h a rg e a n d h e n c e an e t r e t a r d i n g f o rc e . T h e m a g n i t u d e o f t h e d i s p l a c e m e n ti s d i ffi cu l t t o e s t im a te . T he g rad ien t o f th i s f i e ld ,w h i c h i s w h a t d e t e r m i n e s t h e f o c u s i n g f o r c e , i s ,h o w e v e r, n o t s e n s i ti v e t o t h e i m a g e c h a rg e d i s p l a c e -m e n t . T h u s , i n a c o l u m n o f r a d i u s a , t h e f ie l d g r a d i e n td E J d z i n t h e f r a m e o f t h e m o v i n g c h a r g e i s a p p r o x i -m a t e l y g i v e n b y

(dEz)* (1)dz / a 3 "

T h u s i n t h e l a b o r a t o r y f r a m e ,d E z / d z i s p r o p o r t i o n a lt o t h e r e l a ti v i s ti c v - f a c t o r o f t h e c h a rg e .

Th e d e foc us ing fo rce o f (1 ) wi l l g ive a sh i f t in thes q u a r e o f th e a x i a l o s c i l l a t i o n f r e q u e n c y i n th e r i n gf rame , (o9~y)2 o f a m o u n t

a ( 9* v)2 - move* k d z / (2 )

w h e r e 09* i s t h e r e v o l u t i o n f r e q u e n c y i n t h e r i n g f r a m ea n d y * is t h e r e l a t iv i s ti c v - f a c t o r f o r t h e c i r c u l a t i n ge l e c t ro n s . T h i s f o r m u l a i s d e r i v e d i n s e c t i o n 4 , a l t h o u g h

13 1

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132 E. KE IL et a l .

m a n y r e a d e r s m a y c o n s i d e r i t o b v i o u s . T h u s , f r o m ( 1 )a n d ( 2) :

Av 2 ~ - U r e / ~ a , (3 )

w h e r e N i s t h e n u m b e r o f e l e c t r o n s i n t h e r i n g , a n d r ei s t h e c l a s s ic a l e l e c t r o n r a d iu s . T a k i n g N =1013,a = 1 0c m , a n d y ~ . = 4 0 - t y p i c a l p a r a m e t e rs o f a nE R A - w e o b t a i n Av 2 = - - 7 . 10 -3 , which i s sma l l inc o m p a r i s o n w i t h t h e e x p e c t e d s e l f - f o c u s i n g .

We h a v e , i n t h i s s i m p l e - m i n d e d d i s c u s s i o n , i g n o r e dm a g n e t i c i m a g e s w h i c h f o r a s m o o t h a c c e l e r a t i n gc o l u m n w o u l d g r e a t l y r e d u c e t h eAv 2 . H o w e v e r , t h es t r u c tu r e o f a n a c c e l e r a t in g c o l u m n d e s t r o y s t h en e a r l y p e r f e c t e le c t r ic a n d m a g n e t i c c a n c e l l a t i o n o f as m o o t h p i p e a n d t h u s o u r r e su l t - o b t a i n e d f r o m c o n -s i d e r in g o n l y e l e c t r ic i m a g e s - i s a f a i r e s t i m a t i o n o f th e

effect .

2 . S e m i - i n fi n i t e p l a t e s

I n t h i s s e c t i o n w e c o n s i d e r a s a m o d e l o f a n a c c e l e -r a t i o n c o l u m n , a n i n f i n it e s e t o f s e m i - i n f in i t e c o n d u c -t i n g p l a n e s ; i . e . a c o m b . T h e e l e c t r o n r i n g i s r e p l a c e db y a c h a r g e d r o d a n d a c u r r e n t c a r r y i n g r o d m o v i n gp a s t t h e c o m b . T h e a d v a n t a g e o f t h is m o d e l i s t h a tt h e d e f o c u s i n g f o r c e - j u s t l ik e t h e r a d i a t i o n l os s 4 ) -c a n b e c a l c u l a t e d a n a l y t i c a l l y.

W e e m p l o y e x a c t ly t h e n o t a t i o n o f re f . 4 , w h ic hr e f e r e n c e w i l l h a v e t o b e c o n s u l t e d t o m a k e t h e p r e s e n tc a l c u l a t io n u n d e r s t a n d a b l e . T h e p l a t e s a r e t a k e n inth e x - y p la ne an d ex tend f ro m - oo < y < oo, x > 0 .T h e y a r e s e p a r a t e d b y t h e d i s t a n c e 2 n L , w h i l e t he r o d ,l o c a t e d a t x = - x o , i s p a r a l l e l t o t h e y - a x i s a n d m o v e si n t h e z - d i r e c t i o n w i t h s p e e d v.

2 .1 . CHARGEDRO D

W e f i rs t c o n s i d e r a r o d h a v i n g c h a rg e q p e r u n i tl e n g t h . We w a n t t o c o m p u t e t h e e l e c t r i c f i el d i n th ez - d i r e c t i o n d u e t o t h e c h a r g e s a n d c u r r e n t s o n t h ep l a t e s , b u t w e o n l y n e e d t h i s f i e ld Es z e v a l u a t e d a t

x = - X o , z = v t + t r , a n d a v e r a g e d o v e r o n e p e r i o d o ft h e s t r u c t u r e . F r o m e q s . ( 8 ) a n d ( 2 3 ) a n d t h e a rg u m e n tl ead ing f rom eq . (23) to eq . (36) in r e f . 4 , i t i s easy tos e e t h a t

< E ( tr )> - < E , z ( - X 0 , z =v t + a , t ) > t =

{ ( ? ) ( q ) (1 - i/ /~ Y '~ x= I m ~ - L \ 1 + i / f l y /

x fo ~ d ; t P 2 ( 2 , y ) e x p [ - 2 x 2 + i ~ l ~ ( 4 )L r L _1) '

wh ere f l = v / c , a n d y = (1 _ f 1 2 )- , a n d P ( 2 , y ) i s g i v e nby eq . (34) o f r e f . 4.

Th e e va lu a t ion o f (4 ) , i n the l imi t o f y >> 1 , fo l low st h e p r o c e d u r e e m p l o y e d i n s e c t i o n 3 o f re f . 4 . I npa r t i cu la r, eq . (51) i s modi f i ed to

q (4 f ) ~ ~ x

{( ) fo }I m 1 - 2 i d 2 e x p ( - B 2 ) ( l + z ) , (5 )w i t h

B = o -,1 (6)Y

i 2 ~z = - + x / (2 ) ( I + i ) ( () - - , (7 )

and we h ave wr i t t en t r = ao /~ . The exp ress ion (5 ) isco r rec t , i n the l im i t o f l a rge Y, th r ou gh the f i r s t twot e r m s . E v a l u a t i o n o f t h e i n t e g r a l y ie l d s:

2 q~ ? 2 ~ \ Xo / \ Xo /J

f r o m w h i c h f o l l o w s :

q ~ ( ) , 2 n L ' ~ ,

= LY~ 2~7z \ Xo /( 9 )

d,~ /~=o 2xo ~ \ X o / J "

T h e f o r m u l a f o r < E ( 0 ) > s h o w s t h a t t h e a v e r a g e e n e r g y -l o ss de c r e a se s a s y - - w h i c h w a s t h e m a j o r r e s ul t o fr e f . 4 . O n t h e o t h e r h a n d , t h e l e a d i n g t e r m i n t h edefocus ing f i e ld va r i e s l inea r ly wi th y. I t i s easy to seet h a t t h is l e a d i n g t e r m c o r r e s p o n d s i n m a g n i t u d e 5)t o w h a t o n e w o u l d e x p e c t f r o m a n i m a g e r o d l o c a t e da t x = + X o .

W e h a v e n u m e r i c a l ly e v a l u a t e d < E ( tr )> f r o m ( 4) f o ra n u m b e r o f v a l u e s o f p = 2n L / x o a n d f o r y r a n g i n gf r o m 2 t o 5 0 . Ta k i n g p = 0 . 5 , f o r y = 5 t h e a s y m p t o t i cf o r m u l a ( 5 ) i s o n l y i n e r r o r b y 3 % , w h i l e f o r y 1> 1 0the e r ro r i s le s s tha n 1% . Fo r p = 3 .5 , the e r ro r i sa b o u t 7 % a t y = 5 , b u t l e s s t h a n 1 % f o r Y > / 2 0 .

T h e n u m e r i c a l c a l c u l a t i o n s a r e i m p o r t a n t f o re v a l u a t i n g h o w w e l l < E ( t r ) > i s a p p r o x i m a t e d b y i t sv a l u e a n d f i rs t d e r i v a ti v e a t ~ - - 0 . T h e c a l c u l a t i o n ss h o w e d t h a t t h e d i f f r a c t io n f i e ld s ( in c o n t r a s t t o t h es e lf -f ie l d o f a r o d ) w e r e w e l l - a p p r o x i m a t e d b y t h e f i r stt w o t e r m s o f a Ta y l o r s e ri e s o v e r d i s t a n c e s a < X o /y ;

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D I F F R A C T I O N R A D I A T I O N D E F O C U S I N G O F A N E L E C T R O N R I N G 133

i .e . a o < Xo. In a ppl ica t ions of th i s mo del to an ER Awe sha l l a lways sa t i s fy th i s con di t ion ; i . e . the r ingmin o r d imens ions ( i n t he r i ng f r ame) sh o u ld besmal le r than the d i s tance f rom the r ing to the acce le r-a t i ng co lum n wa l l. T hus , i n a r ing w i th non - ze ro mino rd imens ions , wh ich i n t he p r e sen t mode l w ou ld beapp rox ima ted by a compa c t bund l e o f th in r ods , thef ie ld due to charges a nd cur ren ts on the p la tes i sadequate ly descr ibed by (9) and (10) wi th q cor re -spon ding to the to ta l l ine charge of the ring . The se l f-f ie lds decrease as y-2 and we can safe ly neglec t themat large y.

2.2. C U R R E N T C A R R Y I N G RO D

A ro d hav ing cu r r en t i n t h e y -d i r ec t ion o f magn i t udeqf l ' c i s t rea ted in appe ndix A o f re f. 4 . Emp loying

M ax we l l ' s equa t i ons t o r e l a t e H~ to Ey one ob t a in s( n ( a ) ) =-- ( H ~ ( - X o , z = v t + a , t ) ) , =

- 2qfl 'Ylm{fo~d2p2(2,y)expV-2Xo2+i_~L2_l}"L L L ?

(11)

The lead ing te rms in the focus ing force a re eas i ly seento be

\ - - ~ a / ~ , = o 2 x 2 y 1 + 2 - - - ~ y \ Xo / d "

(12)

The energy loss , which i s eva lua ted in re f . 4 , varies as( f l ' y ) 2 y -L S ince t he t r an sv er se ve loc i t y o f e le c t rons ,i n t he r i ng f r am e , i s app rox ima te ly cons t an t a s t her ing i s acce le ra ted , the quant i tyf l 'y is essent ial lyy - independe n t ( a nd equa l t o un i t y, i f t h e e l e c tr ons havere la t iv i s t ic t ransverse ve loc i t ies before be in g acc e le ra tedaxia l ly ) . Thu s the ene rgy loss of a charged rod and acu r r e n t c a r ry ing ro d bo th va ry a s y -~ ( a t l a rg e y ) and i nfac t a re equa l in m agn i tude in th i s l imi t . In like ma nner,the foc usin g force con tr ib ut io ns [eqs. (10) and (12)]becom e equ a l in the l im i t o f l a rge y. W e be l ieve th i sequa l i ty to be a genera l (geomet ry- independent ) resu l t .

2.3. F O C U S I N G F O R C E

The foc us ing force on an e lec t ron , in the ax ia ldirect ion, is given by

ea [O E~ (a , t ) f l' OH r(c r 't ) ] (13)F ( t ) = L ~-~ ~-~ .=o'

w hich we wr i t e i n t he fo rm

F( t ) = K ( t ) a .

Fo r the semi- infin i te p la te model , then , t ak in gq = N e / 2 n R ,with R the r ing rad ius

K - ( g ( t ) ) , =

- 4NeZYrcRx~1 + 3(()~ \(2nL~x o . 3[1 +(fl 'Y )2] " (15)

3. Corrugated cylindrical waveguideIn th i s sec t ion we represen t the acce le ra t ing co lumn

by an inf in i te ly long , per iod ica l ly cor rug a ted cy l indr ica lwavegu ide w i th geome t r i c a l pa r ame te r s a s sh o w n i nf ig . 1 . W e em ploy the n o ta t io n o f re f . 1 which i snece s sa ry fo r t he un de r s t and ing o f wha t f o ll ows .

The com ple te vec tor po ten t ia l A ( r, t ) i s g iven as asum ove r t he e igen func t i ons o f t he emp ty w a ve g u ide

A~(r) :A ( r , t ) = ~ q z ( t ) A z ( r ) , (16)

A

where t he func t i onsq z ( t ) obey t he equa t i on

#z+~o~qa = N~ 1 ~ j .A *d V = f~ .(17)dr"

N is the nu m ber of ce ll s and VN is the i r vo lume.I f the az imu tha l m ot io n o f the e lec t rons is neg lec ted ,

an e l ec t ron r i ng w i th cha rge Q an d geom e t r ic a l pa r a -meters as shown in f ig. 1 , t ravel l ing with speed v, has

the cur ren t dens i ty

0 9Jz ~ X

~ h ( R ~ - R ~ )

x H ( h - l z - v t [ ) H ( p - R O H ( R 2 - p ) ,(18)

[ ]

L

!

Fig. 1. G eom etry of a currugated cylindrical waveg uide with a

(14) charged ring.

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134 E. KEIL et a l .

w h e r e H ( x ) i s t h e H e a v i s i d e s t e p f u n c t i o n . P e r f o r m i n gthe in te gra t io n in (17) wi th A~ f ro m re f . 1 y ie lds :

fz = _ Qv____~i~ A m S ( flm h ) J(Z m ) eX p( iflm Vt ).( 1 9 )N e Ogz m

T h e f a c t o r s S ( x ) = x - ~s in x a n d J t a k e i n t o a c c o u n tthe f in i te d imen s ions o f the e le c t ron r ing ; J i s g iven by

J (Z m) = 2 [ R 2 J I ( Z m R 2 ) - R 'J ' ( z ' R ' ) ] (2 0)

Zm(R22 " R~ )Jo (Zm a )

T h e p r o p a g a t i o n c o n s t a n t s fl~ a n d Xm a r e d e f i n e d i nref. 1 by flo =o g a / v - 2 M / d w i t h l c h o s e n s u c h t h a t[ flo l 1 T = N d/v ,a n d h e n c ef o r t >~ T, q~(t)is given by

i 7a( t ) = o9~-1 f a ( t ' ) s i n o g a ( t - t ' ) d t ' , (21)w h i c h b e c o m e s

Q d iq x ( t ) = - 2 co Z ~ A m S ( f l m h ) J ( Z , . )

x {[S(49 +) + S( q~ -)] sin o9~ t +

+ [S(49 +) - S( q6 -)] i cos o9~ t}, (22)

w h e r e 49 + = N~ d(o ga/v+ flm),and S(~b)= 49- ' s in 49a s a b o v e .

F o r A rc ~ ~ , t h e c o n t r i b u t i o n s t o q a c o m e f r o m t w oreson ance s w i th oga_+fltv = 0 in the no ta t io n o f re f . 1 .In tha t l imi t we f ind :

l im qa ( t ) = - Qdiog"~2 At S ( ogz h/ v) J (oga/yv)sin o9~ t .N c --* or)

(23)

T h i s r e s u lt i s m u l t ip l i e d b y a f a c t o r o f t w o b e c a u s et w o r e s o n a n c e c o n d i t i o n s a r e f u l fi ll e d a t t h e s a m ef r e q u e n c y b y w a v e s t r a v e l l i n g i n o p p o s i t e d i r e c t i o n s

w h i c h a r e c o u n t e d a s o n e m o d e i n r e f . 1 .T h e e l e c t r ic fi e ld g r a d i e n t f o r t h e 2 t h m o d e i s

OE~a c~AzaOz z = , , = - ~a( t ) ~---~1~=~,"

(24)

T h e z - d e r i v a t iv e o f th e v e c t o r p o t e n t i a l , a v e r a g e d o v e rt h e m i n o r r i n g d i m e n s i o n s , f o l l o w s f r o m r e f . 1:

OoAzZ = vt : - og-~ Z Am flm S ( flm h ) J (zm )

x e x p ( - i f l mv t ) . (25)

Us ing (23) , (24) and (25) we f ind the e lec t r i c f i e ldgrad ien t in the l imi t N~ ~ oo :

lim t3Eza = Q d i o g f 2 A t S ( o g a h / v ) j ( o 9 a / y v ) N c ~ o o ~ Z z = v t

~ A m f l m S ( f lm h ) J ( Z m ) C O S o g z t e x p ( - i f l m V t ) .(26)m

W h e n t h is e x p r e s s i o n is a v e r a g e d o v e r t h e t i m en e c e s s a r y to t r a v e r s e o n e p e r i o d o f t h e s t r u c t u r e t h esum reduces to a s ing le t e rm and y ie lds :

lim O E za '~ = Q d i [ A t S ( o g a h / v ) j ( o g ~ / y v ) ] 2 "Nc-~ o~ O z z =v t/ 2 oga v

(27)

S ince At i s no t ava i lab le in c losed fo rm, i t i s advan-t a g e o u s t o c o m p a r e ( 2 7 ) to t h e e n e rg y U x r a d i a t e d i n

t h e 2 t h m o d e i n o n e p e r i o d o f th e s t r u c t u r e , c a l c u l a t e din ref . 1 . We f ind

l im OEza ~ = ogzUa (28)

The to ta l e lec t r i c f i e ld g rad ien t

- ~ - ~ I z =o,/

i s o b t a i n e d b y s u m m i n g ( 2 8 ) o v e r a l l m o d e s . B e c a u s e

of the fac to r oga i t converges l ess rap id ly as a fu nc t io nof ogz tha n the energ y loss Ua.F i n a l l y, w e w i s h t o r e m i n d t h e r e a d e r t h a t i n t h i s

sec t ion we have ne g lec ted r ing cur ren t e ffec ts .

4. Evaluat ion of the axial f requency

I n o r d e r t o e v a l u a t e i n t e r n a l r i n g d y n a m i c s i t i sc o n v e n i e n t t o w o r k i n th e f r a m e o f r e f e r e n c e i n w h i c hthe r ing i s a t r es t . In th i s f rame, ax ia l mot ion o fe lec t rons i s desc r ibed by

d 2 z * / d t * 2 + o g * 2 V 2 o Z * =F * ( t * ) / m o Y * , (29)

where o9* i s the revo lu t ion f requency,Vo descr ibes thef o c u s i n g d u e t o i o n s , i m a g e s , a n d t h e a c c e l e r a t i n gwave , 7" i s the re la t iv i s t ic y - fac to r fo r the c i rcu la t inge l e c t r o n s , a n d F * is t h e a x i al f o r c e o n a n e l e c t r o n d u e t ot h e d i f f r a c ti o n r a d i a t i o n . T h e a b s e n c e o f a s t a r o n Vof o l l o w s f r o m i t s i n v a r i a n c e u n d e r L o r e n t z t r a n s f o r -m a t i o n . W e h a v e n e g l e c t e d th e c h a n g e i n e n e rg y o f a ne lec t ron in the in te rva l o f an ax ia l osc i l la t ion .

F r o m ( 1 4) , a n d t h e i n v a r i a n c e o f lo n g i t u d i n a l f o r c eu n d e r L o r e n t z tr a n s f o r m a t i o n , w e h a v e

F* ( t* ) = K ( t ) a , (30)

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DIFFRACTION RADIATION DEFOCUS1NG OF AN ELECTRON RING 135

w h i c h , s i n ce z * = aT N , b e c o m e s

F * ( t * ) = K ( t ) z * /7 1 l. (31)

In the case of re la t iv is t ic axia l r in g ve loc i ty (711 >> 1)a n d f o r c l o s e l y s p a c e d a c c e l e r a t i n g c a v i t i e s , t h e t i m ev a r i a t i o n o f K ( t ) i s r a p i d c o m p a r e d w i t h a n a x i a lo s c il l at i on p e r i o d , a n d w e m a y a v e r a g eK ( t ) o v e r t i m e .T h u s l e t t i n g

K = - ( K ( t ) ) , (32)

a n d c o m b i n i n g ( 2 9 ) , ( 3 1 ) , a n d ( 3 2 ) , w e h a v e

d 2 z * / d t * 2+ o9" 2 v 2 z* = O, (33)

w h e r e t h e t o t a l a x ia l b e t a t r o n o s c i l l a t i o n f r e q u e n c y vi s g i v e n b y

~2 2 * *2= v o - K / ( m o T ~ o 711)"

W e i n t r o d u c e t h e q u a n t i t y B , b y w r it i n g

K = (N e E 7 11 ~ 2 f iR ) B ,

(34)

(35)

w h e r e N i s t h e n u m b e r o f e l e c t r o n s i n t h e e l e c t r o n r i n ga n d R i s t h e r i n g m a j o r r a d i u s . C l e a r l y B h a s t h ed i m e n s i o n s o f in v e r se l e n g th s q u a r e d . T h e f a c t o r711h a s b e e n i n s e r t e d m e r e l y f o r c o n v e n i e n c e . F r o m ( 34 )a n d ( 3 5 )

v 2 = v 2 - N r e R B / ( 2 IrT*r ) . (36)

I n ( 3 6) , r e i s t h e c l a s s ic a l e l e c t r o n r a d i u s a n d w e h a v ee m p l o y e d ~o~ ~c / R i n d e r i v i n g t h e e q u a t i o n .

T h e q u a n t i t y B , u p o n w h i c h v 2 d e p e n d s , i s a f u n c t i o no f th e g e o m e t r y o f th e a c c e l e r a t in g s t r u c t u r e a n d o fr i n g s p e e d . F o r B > 0 , t h e d i f f r a c t i o n r a d i a t i o nr e a c t i o n i s a d e f o c u s i n g e f f e c t . A x i a l s t a b i l i t y f o l l o w si f v2 i s p o s i t i v e , a n d h e n c e t o o b t a i n s t a b i l it y w h e nB > 0 r e q u ir e s t h a t a n o n - z e r o a m o u n t o f fo c u s i n gb e s u p p l i e d b y i o n s , i m a g e s , o r t h e a c c e l e r a t i n g w a v e .

We h a v e n o t c o n c e r n e d o u r s e lv e s w i t h r a d i a l m o t i o ni n t hi s n o t e a s th e f o c u s i n g - f r o m i o n s , im a g e s , a n d t h ee x t e r n a l f ie l d - i s s t r o n g i n t h is d i r e c t i o n , a n d t h e r e i sn o n e a r d a n g e r o f l o ss o f ra d i a l s t a b i l i ty. C r o s s i n go f a r e s o n a n c e b y a r e l a t i v i s ti c r i n g w o u l d - p r e s u m -a b l y - n o t b e s e r i o u s .

5. Numerical examples

I n t h i s s e c t i o n w e e v a l u a t e ( 3 6 ) f o r t h e s t r u c t u r e sd i scussed in sec t ions 2 and 3 .

T h e s e m i - i n f i n i t e p l a t e m o d e l , a f t e r r e p l a c i n g t h ec h a rg e p e r u n i t l e n g t h q b yN e / 2 7 r R a n d s e t t i n gf l ' 7 = 1 , y i e lds , f ro m (15) ,

B = (1/x 2) [1 -0 .7 7 8 (27rL/711Xo)~] . (37)

T h e c o r r u g a t e d c y l i n d r i c a l w a v e g u i d e m o d e l f o r ac h a rg e d r i n g ( r in g c u r r e n t s i g n o r e d ) y i e l d s

B = 2 7 r R ~ l / a 3 , (38)

w h e r e t h e c o e f f i c i e n t r l ( b /a , d / a , g / a , 7 ) i s a w e a kf u n c t i o n o f a ll o f i ts a r g u m e n t s . C o m p u t a t i o n s f o r al a rg e n u m b e r o f c a s e s i n d i c a t e t h a t r/ < 0 . 5. A sr e m a r k e d a t t h e e n d o f se c t i o n 2 .2 , w e e x p e c t t h a t i n t h er e l a ti v i s ti c l i m i t r i n g c u r r e n t e f f ec t s s h o u l d i n t r o d u c ea n a d d i t i o n a l f a c t o r o f 2 in ~/.

Ta k i n g a s t y p i c a l v a l u e s , N =1013, R = 3 . 0 c m ,7" = 40 on e f inds , f rom (36)

v 2 = v 2 - 3 . 2 x 1 0 - 2 B . ( 39 )

T h u s ( 37 ) w i t h x 0 ~ 1 0 . 0 c m a n d 711 >> I y i e l d s

v2 = v 2 - 1 . 6 x 1 0 - 4 ; w h ile (3 8) w i th R = 3 . 0 c m ,a = 1 0 . 0 c m a n d r / = 1 .0 ( t o i n c l u d e m a g n e t i c e f fe c t s)y i e ld s v 2 = v g - 6 . 0 x 1 0 - 4 ) .

T h e s e d e f o c u s i n g e ff e c ts a r e s m a l l , a n d p r e s u m a b l yc a n b e e a si ly o v e r c o m e i n p r a c t i c e b y m e a n s o f io nf o c u s i n g o r i m a g e f o c u s i n g .

We a r e g r a t e f u l t o o u r c o l l e a g u e s i n t h e B e r k e l e yE l e c t ro n R i n g A c c e l e r a t o r G r o u p f o r s t i m u l a t in gd i s c u s s i o n s a n d h e l p f u l c o m m e n t s . We w i s h t o t h a n kR i c h a r d H a z e l t i n e f o r h a v i n g c a r e f u l ly c h e c k e d t h e

c a l c u l a t i o n o f s e c t io n 2 a n d , i n p a r t i c u l a r, f o r c o r -r e c t i n g a p r e v i o u s e r r o r i n s e c t i o n 2 .2 . O n e o f u s( C . P e l l e g r in i ) i s t h a n k f u l f o r h a v i n g b e e n a b l e t os p e n d a y e a r i n B e r k e l e y, d u r i n g w h i c h t e r m t h i sp r e s e n t w o r k w a s i n i t i a t e d .

References1) E. Keil, On the energy loss of a charged ring passing a

corrugated cylindrical waveguide, CERN Internal ReportISR/TH/69-49(1969); and in Proc. 7th Intern. Conf. Acceler-ators (Yerevan, USSR , Sept. 1969) to be published.

2) j. D. Lawson, Rapporteur's paper, ibid.z) L.J. Laslett, On the focussing effects arising from the self-

fields of a toroidal beam, Lawrence Radiation LaboratoryInternal Report ERAN-30 (1969).

4) R. D. Hazeltine, M. N. Rosenbluth and A.M . Sessler,J. Math. Phys. 12 (1971) 502.

5) The sign is just wrong, however. The reason for this is thatin the plate structure, since the plates are perpendicular to thedirection of motion of the rod, the boundary conditions aresatisfied, to fair approximation, by an image rod of the s am esign as the rod (thus minimizing hrx along the plates); hence

the reversed sign in (dE/dtr).