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* Work s u pp or ted b y th e Office o f En ergy Resea rch , U.S . Dept o f E n e rgy, u n de r Con t r a c t No . DE - AC0 3 - 76 S F 0 0 09 8

Tra nsverse Bea m Brea k-Up in Linea rElec tron Ac c e le ra to rs

(u n d e r gr a d u a t e t h e s is )

b y

Gil A. Travish *

Coll ider Physics GroupAcce le ra t or an d Fu s ion Resea rch Div is ion

Lawrence Berkeley LabUn ivers ity of Californ ia

Berke ley, Ca lifo rn ia 94 72 0

J a n u a r y, 1 9 9 0

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ii

I a ls o w is h t o ack n o wled g e J a rv is Leu n g an d S i mi Tu rek fo r t h e ir h e lp

in p roofread in g th i s t ex t an d l is t en in g to my con vo lu ted exp lana t ion s .

To b e s u r e , t h e r e a r e m a n y o t h e r s w h o a s s i s t e d m e . I t h a n k t h e m

d e e p l y a n d a p o lo gi ze fo r n o t i n c l u d in g t h e i r n a m e s . F in a l ly, I w o u l d

l ik e t o d ed i ca t e t h i s t o my p a ren t s w h o ma d e a ll of t h is , i n c lu d i n g

myself, possible.

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1

C h a p t e r 1I n t r o d u c t i o n

Some h i s to r i ca l background on acce le ra to rs

a n d b e a m b r e a k - u p is give n fir s t . Th e n , b e a mbreak-up is descr ibed qual i ta t ively.Throughout , an overview of the thes is i s g iven.

1.1 Organ ization al Overview.

Th e organ iza t ion o f th is work is ou t l in ed in t h e t ab le of con ten t s . Th i s

ch ap t e r w ill co ve r t h e q u a l it a t i ve d e s c r i p t i o n s o f a cce l e r a t o r s a n d t h e

p ro b l em o f b e a m b r e a k -u p (BBU ). As s u mp t io n s t o b e u s ed t h ro u g h o u t

t h i s w o rk w i ll b e c l ea r l y o u t l in ed i n Sec t i on 1 .5 . A s yn o p s is o f t h e

an alyt ic an d n u m erical work is a lso g iven.

Th e s e c on d c h a p t e r w ill d e s c r i b e t h e c o m p o n e n t s of a n a c c ele r a t or

w h i ch a r e r e le va n t t o b ea m b r e a k - u p . Th e t h ir d c h a p t e r , a lo n g wit h

th e fir s t two App en d ices , g ives a de ta i led des cr ip t ion o f th e beam

b r e a k - u p p r o b le m . In c lu d e d is a d e r i va t i o n o f t h e BB U e q u a t io n a n d

the defin i t ion of a wakefield . The no ta t ion to be u sed fo r the

r em a i n d e r o f t h e p a p e r is a ls o in t r o d u ced . Th e fou r t h ch ap t e r

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2

pr esen t s a nu mber o f an a lyt ic so lu t ions to t h e equ a t ion der ived in

ch a p t e r t h r ee . Th e fift h ch ap t e r d is cu s e s m e t h o d s t o r ed u ce BBU.

Th e s ix t h ch a p t e r d ea ls wit h n u mer i ca l s t u d i es : t w o ca s e s a r e

cons idered . The fin a l cha p te r is a su mm ary.

Th e v a r i o u s ch ap t e r s an d s u b - s ec t i o n s a r e n u mb ered i n a s cen d i n g

order. All equa t ion s a r e nu m bered u s ing the fo llowin g conven t ion : c .n ,

w h e r e c is t h e ch a p t e r n u m b e r a n d n is t h e eq u a t io n n u m b e r wit h in

t h a t c h a p t e r . E q u a t io n s w h i ch a r e r e fe r e n c ed la t e r in t h e wo r k h a v e

b o ld n u m b e r s . Im p o r t a n t e q u a t i o n s a r e e m p h a s ize d w it h a * in t h e

l e ft ma rg in . Te rm in o lo gy i s g iv en i n b o ld w h en fi r s t u s ed . As u s u a l ,

ac ronyms are fi r s t given in pa ren t h es i s a f t e r th e fu ll ph ra ses . For

t y p ograp h ica l p u rp o s e s t h e s y mb o l (a G r eek i o t a ) i s u s ed t o d en o t e

th e ima gina ry i. Also , vectors are wri t ten in bold le t t ers , wh i le

scalars are in p lain type. Fin al ly, fu ll t i t les a re inclu ded for m ost

re fe rences in the hope tha t readers wi l l f ind th i s usefu l when

sea rch in g fo r lit e ra t u re .

1.2 Some Hist orical Background on Accelerators .

In 1 9 2 8 , Widere des cr ibed th e fi r s t su ccess fu l acce le ra to r. An d , ever

s i n ce t h en t h e r e h ave b een p ro b lems ! Wid e r e b a s ed h is d e s i gnloosely on the ideas put for th four years ear l ier by I s i n g .1 Th e m a c h i n e

w as a l in ea r i on a cce le r a t o r. Th e d e s i gn r e l ied o n a l t e rn a t in g vo lt a g e

a p p l ie d t o s e c t io n s o f t u b e o f e ve r i n c r e a s i n g le n g t h . 2 Th e p o t e n t i a l

d i ffe r e n c e s i n t h e g a p s (b e t w e e n a d j a c e n t t u b e s ) c o u l d a c c e le r a t e t h e

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3

i o n s . Th e t r i c k w a s t o h a v e t h e fr e q u e n c y o f t h e a p p l ie d volt a ge ,

a n d t h e le n g t h s o f t h e t u b e s , ju s t r i gh t s o t h a t t h e io n s w o u l d n o t b e

de-a ccelerated . A sch em at ic of Wider es des ign is g iven in Figur e 1 .1 .

Ions

Wid er e s d e s i gn m o t i va t ed Sloan a n d Law ren ce to inven t and bu i ld

t h e cyclot ron . Th e re d evi ce , a l o n g wit h a d v an ced ve r s i on s o f Va n d e

Graaff a n d Cockcroft-Walton acce le ra to rs , l aunched the f i e lds o f

a cce l e r a t o r an d ex p e r i men t a l h i g h en e rg y p h y s i c s . 3 Si n ce t h e l a t e

t w en t i e s , man y s ch emes h av e b een d evis ed t o acce le r a t e ju s t a b o u t an y

k i n d o f ch a rg ed p a r t ic le s . C ir cu la r m ach in es (t h e b e t a t ro n , d ev e lo p ed

b y K er s t i n t h e fo r t i e s , an d t h e cyclot ron ) qu ick ly p rov ided the

d es ir ed h i gh en e rg y p a r t i c le s o u rce s . Lin ea r a cce le r a t o r s t o ok a b ack

s ea t t o t h e mo re eco n o mi ca l c i r cu l a r mach i n es u n t i l t h e t e ch n i ca l

ad van ces o f t h e p e r i o d d u r i n g , an d i mm ed i a t e l y fo llo w in g , Wo r l d War

II. Th e s e a c h i eve m e n t s p r o vid e d t h e n e c e s s a r y t e c h n o lo gy t o m a k e

l in ear acce le ra to rs feas ib le an d economica l . Acce le ra to r sc ien t i s t s

F igure 1 .1 . Wider es acce le ra t o r was des ign ed toacce le ra te ions . A po ten t i a l d i ffe ren ce in th e gap s be tweent h e d r i ft t u b es w a s u s e d t o a c c e le r a t e t h e p a r t i c le s . Th een t ir e s t r u c t u r e was u n d e r a vacu u m.

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4

w e r e fo r t u n a t e t h a t t h e r e q u i r e m e n t s fo r r a d a r s o ve r la p p e d t h o s e fo r

a c c e l e r a t o r s . Th e p o s t - w a r p e r i o d a l s o s a w t h e a d va n c em e n t of t h e

k lys t ron (in p a r t b y t h e Varian b r o t h e r s ): a n id e a l p ow e r s o u r c e fo r

linear a cce le ra to rs .

Th e a cce le r a t o r fi eld q u ick l y s ep a ra t ed in t o t h r ee ca t ego r ie s : e l ec t ro n ,

io n a n d p r o t on m a c h in e s . Th e e le c t r on a n d io n m a c h i n e s w er e lin e a r

d es i gn s b y n eces s i t y 4 ,5 w h i le t h e p r o t o n m a c h in e s w e r e b o t h c i r c u l a r

an d l inear.

Hansens g r ou p a t S t a n f or d le a d t h e w a y in t h e d e ve lo p m e n t o f lin e a r

acce le ra to rs (l in acs ) fo r e lec t ron s . By 19 47 th ey ha d th e i r fir s t

m ach in e w o rk i n g , an d b y 1 9 5 5 S t an fo rd h ad a 4 0 MeV p ro t o t y p e fo r a

1 G eV ma ch i n e . O t h e r gro u p s a ro u n d t h e wo r ld fo llo wed w it h v a r i ou s

des igns .

1 .3 A Very Brief Hist ory of Beam Brea k-Up.

Beam brea k-u p (a l so ca l led bea m b low-up ) was p roba b ly fir s t obser ved

i n t h e m i d - fift i e s . H o w eve r , it w a s n o t u n t i l t h e m i d - s i xt i e s t h a t t h e

p h e n o m e n o n wa s u n d e r s t o o d . In fa c t , on e m i gh t s a y t h a t b e fo r e t h a t

t im e , BBU was n o t r e co gn i zed a s a s e r io u s p ro b l em, i n p a r t b ecau s e o f

t h e r e l a t i v e l y l o w b eam cu r r en t s u s ed . 6 Th e fi r s t o b s e r v a t i o n s w h i ch

wer e a c tua lly iden t i fi ed as BBU wer e o f beam cu r ren t redu c t ions .

D u r i n g t h e t e s t i n g a n d d e v elo p m e n t o f a c c e l er a t o r s , it w a s o b s e r ve d

t h a t i f t h e b e a m w a s i n j e ct e d w it h a h i gh e n o u g h c u r r e n t , t h e b e a m

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5

w ou ld b eg in t o lo s e cu r r en t a s i t t r av elled d o w n t h e acce le r a t o r. La t e r

it w as o b s e rved t h a t t h e b eam s t r an s v er s e d i men s i on s gr ew.

Th e t y p e of b eam b reak -u p o b s e rved i n t h e s e ea r l y ex p e r imen t s is n o w

cal led regenera t ive BBU . The beam i s d i s ru p ted by fi elds with in a

s in g le sec t ion (cav ity ) o f th e acce le ra to r (see F igu re 1 .2 ). As th e na m e

imp l ies , t h e fi e lds g row expon en t i a l ly th r ou gh feedb a ck (i .e . t he f ie lds

r eg en e ra t e ) .

Regenerative BBU

Feedback

Amplification

As b ea m cu r r en t s b ecam e h i gh e r a n d acce le r a t o r s l on g e r, an o t h e r t yp e

o f b e a m b r e a k - u p w a s o b s e r ve d . Th e m o r e s e c t io n s t h a t w e r e a d d e d

t o t h e acce le r a t o r, t h e w o r s e t h e p ro b l em b ecame . Th is n ew

p r o b le m w a s t e r m e d cum ula t ive BBU . I t began to harsh ly l im i t the

p e r fo rma n ce o f va r i o u s m ach in es . U n l ik e ot h e r m o re s u b t l e p ro b l ems

which l imi ted bea m qu a l ity, BBU pu t a defin it e l im it on h ow h igh beam

cu rren t cou ld be an d d ra s t i ca lly redu ced the beam qua lity.

Mul t isect ion BBU, as cumulat ive BBU is somet imes cal led , resul ts f rom

p ro g re s s ive ly la r g e r fie ld s b e i n g ex ci t ed in s u b s eq u en t s ec t i o n s o f t h e

Figure 1 .2 . Regen era t ive BBU is cau sed by th eam p l ifi ca t i on o f fi el d s w it h i n a s ec t i on w h e re a feed b ack

mech a n i s m ex is t s .

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6

acce le ra t o r (s ee F igu re 1 .3 ). In o th er words , cum u la t ive BBU is ,

lo gica l ly en o u g h , an e ffec t w h i ch b eco mes m o re p ro n o u n ced a s t h e

b e a m g o e s fu r t h e r d o w n t h e a c c e le r a t o r : t h e in s t a b ilit y a c c u m u la t e s

a long the a cce le ra t o r.

Beam

External RF

Fields

Th e g r o w in g fie ld s a c t o n t h e b e a m , c a u s i n g a s e c t io n o f t h e b e a m t o

b e d i s p l aced fu r t h e r fro m i t s in i t ia l p o s it i on . In t u rn , t h e fu r t h e r o ff-

ax i s a bea m s ec t ion i s , t h e h igher th e fi eld it w il l exc it e . Th e cou p l in g

b e t w e e n t h e b e a m a n d t h e fie ld s e x c it e d in t h e a c c e le r a t o r le a d s t o

th e BBU ins ta bi li ty (see Figur e 1 .4 ).

M a n y fix e s a n d c u r e s w e r e d e vis e d t o l im it t h e p r o b l e m s c a u s e d b y

B B U; h o w eve r , B BU r e m a i n e d a s e r i ou s p r o b le m , a n d r e m a in s o n e in

man y co n t em p o ra ry acce le r a t o r s . Th e p ro b l em h as n o t b een

e limina ted an d BBU has to be t aken in to a ccou n t in a l l fu tu re des ign s .7

Figure 1 .3 . Cu m u la t i ve BBU is c au s e s b y la rg e r an d l a rg e rfie lds being exci t ed in su bsequ ent cavit ies . Her e fie lds t r en g t h is r ep re s en t ed b y s h ad i n g .

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Cavity

Cut-away

Cavity

Cut-away

Cavity

Cut-away

Th u s , t h e p ro b l em i s t h i s : a b eam t r a ve r s i n g an acce le r a t o r is s u b j ec t

to an in s ta b i lity, ca l led beam brea k-u p , which m u s t b e con t ro lled if th e

per fo rm a n ce o f th e acce le ra to r is to be rea l ized . Th i s thes i s is

ded ica ted to an a lyt ica l ly defin ing an d u n ders t an d ing th i s p rob lem.

1.4 BBU Im posed Lim itat ions.

As w as men t io n ed i n Sec t io n 1 .3 , BBU p lace s a lim i t o n b ea m cu r r en t ,

p u ls e len g t h a n d in j ec t i on en e rg y. Th es e lim it a t i on s a r e p ro b lems for

t h e acce le r a t o r s c i en t i s t . Tro u b l e o ccu r s i f a p a r t o f a b ea m s c r a p es

t h e p ip e w a l l. N ame l y, cu r r en t w il l b e lo s t , t h e b eam w ill b r ea k u p o r

b e c o m e s h o r t e r - - p u l s e s h o r t e n i n g (s ee Figure 1 .5 ). Beam break -up

can be su ff i c i en t ly severe tha t beam propaga t ion i s imposs ib le because

th e en t i re beam i s los t (from h i t t in g the wal ls ).

F i gu re 1 .4 . Th e g ro w t h o f t h e t r an s ve r s e d imen s i on s o f t h e b eam d u e t o BBU is i llu s t r a t ed b y a s h o r t p u ls e. Th ebea m is in ject ed s l ight ly off axis (left). As i t t r avelsth rough success ive cav i t i es , t he d i sp lacement g rows .Even tua l ly, the pu l se becomes l a rge enough to h i t t hebea m p ipe an d i t w il l b rea k-u p.

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Time

Current

0

0

0z=0

z=0.5z max

z=z max

U lt i m a t e ly, t h o u g h , it is t h e e n d p r o d u c t , t h e a c c e le r a t e d p a r t i cle s ,

t h a t i s of co n ce rn . Wh erea s t h e p ro ced u re u s ed t o acce le r a t e p a r t ic le s

ma y be of l it t l e in t e res t to a n a cce le ra to r u ser, t h e b eam s fina l

ch a r ac t e r i s t i c s a r e o f g r ea t i mp o r t a n ce . It i s a t t h i s l eve l t h a t BBU

m u s t b e t a k e n in t o a c c ou n t . It is n o t s u ffic ie n t t o a c c e le r a t e a b e a m

fr o m o n e e n d o f a m a c h in e t o a n o t h e r. Th e qu a l it y o f t h e b e a m m u s t

co n fo rm t o t h e n eed s o f t h e u s e r s .

To un ders t an d th e e ffec t BBU ha s on b eam qu a l ity, we need to qu an t i fy

t h e co n cep t o f b ea m q u a l it y. Th e re a r e s ev e ra l fi gu re s o f m er i t fo r

beams dep en d in g on th e app lica t ion . For tun a te ly, two re la t ed

pa ram eters a re su ffic ien t fo r u n ders t an d ing th e e ffec t s o f BBU.1 0

Luminosity o f t w o co l li d i n g b ea m s is d e fi n ed a s t h e in t e r ac t i on r a t e

p e r u n i t c ros s s ec t io n . Alt e rn a t i ve ly, t h e lu mi n o s i t y can b e t h o u g h t o f

a s t h e d e n s i t y o f p h a s e s p a c e a t t h e t wo b e a m s in t e r a c t io n p o in t . In

Figure 1 .5 . Be a m c u r r e n t m e a s u r e m e n t s s h o w in g p u ls es h o r t en i n g d u e t o BBU a t t h r ee p o i n t s a l o n g t h eaccelera tor. This is a dr a win g of a typical osci l losc opesignal . 8 ,9

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t e rms o f t h e r ep e t i t i o n r a t e , , t h e n u mb er o f p a r t i c l e s , N t h e

lu minos i ty ma y be expressed as

L = N2

4 xy(1 .1 )

w h e r e x a n d y a r e t h e t r a n s v er s e b e a m s d i m e n s io n s . 1 1 Th e h i g h e r

th e lu m inos i ty, the more in te ra c t ion s . (Lu m inos i t ies a re o ft en g iven in

c m -2 s ec -1 wi th a typ ica l f igure a round 10 3 1 -1 0 3 2 fo r e l ec t ron

c o llid e r s . ) U n fo r t u n a t e l y, t h e l a rg e r t h e b e a m s c r o s s s e c t i on a l a r e a ,

t h e lo w er t h e lu mi n o s it y. Th u s , b eam b reak -u p lo wer s t h e lu m in o s it y

b y in c r ea s i n g t h e t r an s v er s e d i men s io n s o f t h e b eam .

O ft e n i t is t h e a r e a o f p h a s e s p a c e t h a t i s u s e d a s a m e a s u r e o f b e a m

qual i ty. The e m i t t a n c e o f a b e a m is r e la t e d t o t h e p h a s e s p a c e a r e a

(see F igure 1 .6 ). A g ross m eas u re of th e a rea i s ob ta in ed u s ing the rm s

n o rma l i zed emi t t an ce , xrm s = x 2 p x2 - xp x 2 ( w i t h a c o r r e s p o n d i n g

equation for y).

Th e h i gh e r t h e e m i t t a n c e , t h e lo w er t h e lu m in o s it y ( L ~ -1 / 2 ). BBU

act s to in creas e a beam s t ran sverse emi t t a n ce . For col liders , t h is

r e d u c e s t h e u s e fu l in fo r m a t i on t h a t c a n b e o b t a in e d fr o m t h e b e a m s .

An in c r e a s e d p h a s e s p a c e a r e a m a y r e n d e r o t h e r d e v ic e s in o p e r a b l e .

In t h e ca s e o f fr ee e lec t ro n l a s e r s , t h e g a i n is r ed u ced . It is fo r t h e s e

r ea s o n s t h a t BBU mu s t b e co n t ro lled t o a gr ea t e r ext en t t h an w ou ld b e

accep tab le fo r s im ply p ropaga t ing a beam th rough a n a cce le ra to r.

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p

x

x

rms is area"around"

(b)

p

x

x

is sum ofareas.

(a)

It i s t h e g row t h o f t h e s i ze o f t h e b eam fro m t h e s t a r t t o t h e en d o f t h e

acce le ra to r th a t is t h e c r i t i ca l measu re o f beam qu a l ity degrada t ion

du e to BBU. A qu an t i ty u sefu l fo r compa r ing th e growth u n der var ious

p a r a m e t e r s i s t h e d i s t a n c e ove r w h i ch t h e b e a m s ize gr o w s b y o n e e -

fo ld . Th i s leng th i s ca l led th e g ro wt h l en g t h .

1.5 The Assum pt ions .

In ch ap te r 3 an equ a t ion (t h e BBU equ a t ion) is d er ived wh ich

d es c r i b e s a b eam s t r a n s v er s e g row t h d u e t o BBU. It is d o n e s o u n d e r

a nu m ber of as su m pt ions which a llow fo r a t ra c tab le der iva t ion . Th ese

a s s u m p t i on s w ill b e ou t l in e d h e r e , w ill a p p l y t o t h e r e m a in d e r o f t h e

pa per, a n d wi ll s e rve to defin e th e scope o f th is work .

Figure 1 .6 . Th e e m it t a n c e c a n b e t h o u g h t o f a s t h e s u m o f t h e p h a s e s p a c e a r e a s a s s h o wn i n (a ). Alt e r n a t ive ly, a n dm o r e u s e fu lly, it c a n b e d e fin e d a s t h e a r e a e n c l os e d b y aboun dar y as in (b).

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Th i s t h e s i s w i l l o n l y co n ce rn i t s e l f w i t h cum ulat ive , t ran s ve rs e b e a m

break-up in l inear , e l e c t ro n acce le ra to rs . Regen era t ive e ffec t s a re

ignored ; long i tud ina l mot ion due to such e ffec t s as synchro t ron

m o t io n an d lo n g it u d i n a l BBU a r e a l s o i gn o red . So me of t h e an a l y s isd o e s c a r r y ove r t o ot h e r d e vi ce s , s u c h a s p r o t o n li n a c s , b u t t h i s wi ll

no t be con s idered . In a dd i t ion , man y o f th e ca lcu la t ions as su me a

rela tivis tic b eam , an d d o n o t ap p l y a t l ow en e rgie s . Th e on l y s o u rce of

BBU wh ich wil l be d i scus sed is w ak efields . In ad d it io n , t h e en t i r e

th eory is l inear in t h e b e a m s t r a n s v e r s e d i s p l a c e m e n t : h ig h e r o r d e r

e ffec t s a r e i gn o red (o r a s s u med s m a l l). In fac t , t h e r ad iu s o f t h e b eam

i s n eg lec t ed : o n l y t h e t r an s v er s e m o t io n o f t h e b eam s ce n t ro id is

an a lyzed . Severa l o f th e ca lcu la t ions as s u m e th a t the beam pa ram eters

(i .e. en ergy) vary s lowly over a bet at ron wavelen gth (see Ch ap ter 2 for a

d e fin i t io n ). In t u r n , it is o ft e n a s s u m e d t h a t t h e b e t a t r o n w a v ele n g t h

i s l es s t h a n a g ro wt h l en g t h (i. e .

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Ev en u n d e r t h e ap p ro x ima t io n s u s ed t o ob t a in t h e BBU eq u a t i on , it is

s t i ll n o t poss ib le to so lve th e equ a t ion in fu l l genera l ity. Desp i t e th i s ,

severa l spec ia l cases an d a ppr oxim at ions l ead to s imple , us efu l m odels .

In Cha p ter 4 , th e fo llowin g cas es a re cons idered :

1 ) No wak e2) Cons tan t wake3) Linear wak e4) Genera l wak e fo r a coas t ing beam5 ) Th e t w o -p a r t i c le m o d e l6 ) Gener a l wak e in c lu d ing acce le ra t ion

Th e s e s o lu t io n s p e r m i t u s t o b e t t e r u n d e r s t a n d h o w B BU a ffe c t s t h e

bea m . Th is kn owledge al lows us to find ways to coun tera ct BBU.

1.7 Fixes and Cures .

O n l y o n e m e t h o d t o co u n t e r B BU is e x p lo r e d in t h i s w o r k : t h e BNS

effect .1 2 Th e BNS effec t is cau sed by a sp rea d in en ergy a long th e

en t ire beam. It is in ves t iga ted in Ch ap ter 5 .

To b e s u re , t h e r e a r e m a n y o t h e r r em ed i e s fo r BBU: cavit y d e s ig n

changes , RF phase ad jus tmen t s , s t ronger focus ing e lements , e t c . .

H o w eve r, t h e s e w i ll n o t b e co n s id e red h e re . 1 3 In o t h e r w o r d s , w e wi ll

o n l y exam in e h o w ce r t a in ch a n g es t o t h e b eam can r ed u ce BBU, r a t h e r

t h a n ch an ges t o t h e acce le r a t o r.

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Referen ces an d Notes(C h a p t e r 1 )

1P. M. Lap ostolle an d A. L. Septier , ed., Linear Accelerators (Amsterdam: N orth H ollandPu blishing Com pa ny, 1970), pp . 1-18.2The length of the tubes ha d to increase along the accelerator to comp ensate for the increasedspeed o f the ions because all the tubes wer e connected to the same AC source (i.e. a fixedfrequency).3The Van de Graa ff accelerator is just an electrostatic device wh ere p art icles are movedmech anically against a po tential difference. The Cockcroft-Walton accelerator u ses a series of capacitors, charged in par allel and discharged in series to pr odu ce a large potential difference(up to 1MV) w hich accelerates part icles.4Circular d esigns could on ly be used to accelerate electron to tens or h un dred s of MeV. Beyond

that energy range, rad iative losses begin to d ominate for electrons (radiation p ower 4) andlinear machines become m uch m ore efficient. Modern circular electron m achines can achieve~100 GeV.5Historically, the magn etic fields need ed to bend and focus th e heavy-ions in a circular orbit

mad e high energy heavy-ion circular machines impractical. Adv ancements in magnet d esignshave raised som e new possibilities for circular electron machines.6See J. Haimson, High Current Traveling Wave Electron Linear Accelerators, IEEETransactions on N uclear Science NS- 12 3 (1965) for a d iscussion of early BBU ob servation s.7Even a cursory review of the literatur e w ill rev eal sever al pap ers w hich discuss th elimitation impo sed by BBU. For a consideration of BBU in a n ext generation linac design seeF. Selph and A. Sessler, Transverse Wak efield Effects in th e Two-Beam Accelerator ,Law ren ce Berkeley Lab Prep rint LBL-20083 (1985).8For actual photo grap hs see Fig. 1 of R. H. Helm a nd G. A. Loew, Beam Breaku p in LinearAccelerators , ed. by M. Lapostolle and A. L. Septier (Amsterd am: No rth H olland Pu blishingCom pa ny , 1970), p. 175.9Cu rren t red u ction an d pu lse shor tenin g are shown in Fig. 4 of R. J. Briggs, et al. , BeamDynam ics in the ETA and ATA 10 kA Linear Ind uction Accelerators: Observations and Issues,IEEE Transactions on Nuclear Science 28 , 3 (1981).10P. B. Wilson, Linear Accelerators for TeV Collid ers, in Laser Acceleration of Par ticles , A IPCon ference Proceed ings N o. 130 (1985).11This expression for lum inosity assum es identical beams with g au ssian profiles (i.e. ~exp[-x2 / (2x2)] ).12BNS is an acronym for V. Balakin, A. Nov okhatsky, an d V. Smirnov three Soviet scientistw ho pr oposed a dam ping scheme for VLEPP. See V.E. Balakin, e t a l . , VLEPP: Transver seBeam Dynamics, Proceedings of the 12th International Conference on High-EnergyAccelerators , ed. by F.T. Cole and R. Donald son (Batavia, Illinois: Fermi N ationa l Accelerato rLabora tory , 1984), pp . 119-120.13For a t reatment of other BBU rem edies in mu lt ibun ch l inacs see K. A. Thom pson and R. D.

Ruth, Controlling Transverse Multibunch Instabilities in Linacs of High Energy LinearColliders, Stanford Linear Accelerator Center Preprint SLAC-PUB-4801 (1989).

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C h a p t e r 2Li n e a r Ac c e l e r a t o r s

Th e va r iou s com p o n en t s w h i ch m ak e u p a

lin ea r a ccele r a t o r a r e d e s c r i b ed . Th e u s e s an deffec t s o f the dev ices a re d i scussed , and am o d e l fo r a n a c c e le r a t o r is p r o p o s e d . F ig u r e san d p h o t o g rap h s o f co mp o n en t s a r e g i v en a tt h e e n d o f t h e c h a p t e r.

2.1 The Bas ic Com ponen ts .

Th e p u r p o s e o f a n a c c e le r a t o r is , a s i t s n a m e im p lie s , t o b r in g m a t t e r

to h igh ve loc ity. Faced with th e p rob lem of acce le ra t ing cha rged

p a r t i c le s t o v e ry h i gh ve lo c it i es , o n e m ig h t b e t em p t ed t o u s e a d ev ice

ana logous to a s l ing-sho t . Ind eed one cou ld se t u p a po ten t i a l

d i f f e r en ce b e t w een t w o p o i n t s an d acce l e r a t e a p a r t i c l e i n t h i s

man n e r. P ro b lems a r i s e an d t h e co mp l ex it i e s b egin w h en o n e is

r e q u ir e d t o d o t h i s fo r m a n y p a r t i c le s , a n d ve r y h ig h e n e r g ie s , w it h

g rea t con s i s t en cy an d p rec is e co n t ro l o ve r a l l p a r am e t e r s . In fac t , t h e

very fir s t p r ob lem i s ob ta in in g the p ar t i c les t o be acce le ra ted .

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2.1.a Injectors.

Fo r an e l ec t ro n l i n ea r a cce l e r a t o r, t h e r e i s l i t t l e p ro b l em i n o b t a i n i n g

elect rons . Various e l e c t r o n g u n s h av e b een d e s i g n ed t h a t p ro d u ce

s t r e a m s o f e le c t r on s . Mo s t gu n s w or k o n t h e s a m e p r in c i p le a s t h e

e lec t ron gun in t e l ev i s ion p ic tu re tubes . 1 A ca thode i s hea ted to

indu ce e lec t ron s to l eave i t s su r face . Th e e lec t ron s acce lera te towar ds

t h e an o d e . O n ce t h e e lec t ro n s ov e r s h o o t t h e an o d e, t h ey a r e

focused , acce le ra ted to some i n j e c t i o n e n e r g y , a n d s e n t t o t h e

acce le ra to r. Th is en t i re as s emb ly i s re fe r red to as th e i n j e c t o r .

In j e ct o r s c a n b e m u c h m o r e s o p h i s t ic a t e d t h a n d e s c r ib e d a b o ve (s e e

Figu re 2 .11 a t th e en d of th i s ch ap te r ). Never th e less , t wo bas ic

r e q u i r e m e n t s a r e c o m m o n t o a l l in j e ct o r s : 1 ) p a r t ic le p r o d u c t i on a n d

2 ) p a r t i c l e p r e - a c c e l e r a t i o n . If t h e p a r t i c le s t o b e a c c e l e r a t e d a r e

n o t r e a d i ly a v a i la b le , t h e n a s c h e m e m u s t b e d e v is e d t o ge n e r a t e t h e

pa r t i c les . Assu ming th i s is done , th e par t i cl es mu s t s t il l be p r e -

acce le ra ted to th e in jec t ion energy. The pu rpose fo r th i s is qu it e

s imp le : acce le ra t ion t ech n iqu es work mos t e ffi c ien t ly in cer t a in

sp ec ifi c en ergy ra n ges . Th e in ject ion ener gy i s p icked to be a low one .

Met h o d s , d i ffe r en t f rom t h o s e u s ed i n t h e ma in a cce le r a t o r, w h i ch a r e

m o r e e ffi c ie n t fo r l ow e r e n e rg ie s c a n b e u s e d t o p r e - a c c ele r a t e t h e

pa r t i c les (s ee Tab le 2 .1 ). Also , d i ffe ren t cu r ren t reg ime s r equ i red i ffe r e n t gu n s a n d p r e - a c c ele r a t i on t e c h n iq u e s : t h e r e q u i r e m e n t s o n

a g u n p r o d u c in g ~1 Am p (p e a k c u r r e n t ) a r e ve r y d iffe r e n t fr o m o n e

t h a t p ro d u ces ~1 0 k A.

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In lin ear acce le ra to rs , i n j ecto rs a l so p lay a c r it i ca l ro le in b eam qu a l ity.

Unl ike c i rcu la r mach ines , where beam qua l i ty can be improved in

s t o r a g e r i n g s , li n a c s a r e l im i t ed b y t h e o ve ra l l q u a l i t y o f t h e in j ect ed

b eam. Th u s , an i mp r o vemen t in t h e in j ec t o r w i ll mo s t p rob ab lyimpr ove th e final beam qu al ity.

2.1.b Cavities.

During Wor ld War I I ma ny m eth ods we re dev ised to p rodu ce h ighp o w er a t m i c ro wav e fr eq u en c ie s (~ 3 -3 0 G H z). 2 Th e s e m e t h o d s w e r e

soon pu t to u se in l inea r accelerat ors . At fi rs t , t h e avai labi lity of

m i c ro w av e p o w er s o u rce s f ro m s u rp l u s mi l i t a ry r ad a r s w as ex p l o i t ed .

La t e r, cu s t o m p o wer s o u rce s w ere b u ilt . It w as w e ll k n o wn a t t h e t ime

tha t a condu ct ing cavity cou ld be exc i t e d by ex te rna lly su pp l ied

p o w er. 3 In fa c t , t h e id e a o f u s i n g c a v it i es a s r e s o n a t o r s t o a c c e le r a t e

p a r t i c l e s h ad it s o r i gi n s b e fo re Wor l d War II. Th e id ea w as s i mp l e :

t h e ab i li t y t o ex c i t e l a rg e fie ld s w it h m o d es t amo u n t s o f RF 4 p o w e r

cou ld b e exp lo it ed t o acce lera te pa r t i c les .

Ta ble 2 .1 . A comp ar ison of four d i ffere n t accelera tors ;t h e ir in jec t io n en e rg ie s , fin a l en e rg i es an d p eak cu r r en t s .No t e t h e w id e v a r i a t i on , a n d h e n c e t h e r a d i c a lly d iffe r e n tin jec to r des igns .

N a m e SLC (SLAC) SSC ATA (LLNL) E TAII (LLNL)

Part ic les Used e+ / e- p e - e -

In jec t ion Energy 1.21 GeV 1 TeV 2.5 MeV 2.5 MeV

Fi n a l En e rg y 50 GeV 2 0 TeV 5 0 MeV 6 MeV

Peak Cu r ren t 1 .2 k A 7 0 m A 1 0 k A 1 .5 k A

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Cavi t i es a re o f t en made o f very good conducto rs , copper be ing the

m o s t c o m m o n ly u s e d . 5 Th e acce le ra t ion cav i ty is c o n s t r u c t e d wit h a

mi n i mu m o f t h r ee co n s i d e ra t i on s :

1 ) Th e b ea m mu s t b e a b l e t o en t e r an d ex it .2 ) Th ere mu s t b e an in pu t fo r exte rn a l ly su pp l ied RF.3 ) App ropr ia t e fi elds ha ve to exc it ed so as to a cce le ra te th e

b eam.

A cav i ty i s o ft en rep res en ted b y a pil lbox . Cav it i es a r e o ft en des ign ed

to be cy l indr i ca l s ince th i s geomet ry fac i l i t a t es eng ineer ing ,

co n s t ru c t io n an d an a l yt i c w ork . F igu re 2 .1 is a r ep re s en t a t io n o f a

cav it y wi t h t h e fir s t t w o r eq u i r em en t s i d en t ifi ed . Al s o , F ig u r e 2 .1 2 a t

th e end o f th e cha p te r i s an a r t is t s con cep t ion o f a spec ia l cavity

des ign .

Beam Line

CavityRFPort

RF Feed

BeamPort

Th e t h ir d c o n s i d e r a t io n r e q u ir e s t h a t t h e r e b e a n a x i a l e le c t r i c fie ld .

Th er e a re m an y (even in fin i t e ly man y) modes which sa t i s fy th i s .

H o w eve r , it is u s u a l ly d e s i r a b le n o t t o e x ci t e c e r t a i n m o d e s , s u c h a s

BBU m odes with t ra n sverse e l ec t r i c fie lds . In ad d i t ion , th e lower

Figure 2 .1 . A p i ll b o x m o d e l o f a c a v it y s h o w in g t h e b eama n d RF po r t s as wel l as a sec t ion o f b e a m p i p e to bea t t a ch ed a t o n e b e a m p o r t .

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modes o f t en a re eas ie r to exc i t e and p rov ide g rea te r acce le ra t ion .

Th i s t o p i c i s a n en t i r e fi e l d i n it s e lf. For t h e p u rp o s e s o f s t u d yin g

BBU , it i s s u ffi c ien t t o b e aw a re o f t h e s i m p l e mo d e l fo r a c a vi t y an d

t h a t fie ld s can b e an d a r e ex cit ed in t h em. Cu m u la t i ve b eam b reak -u pcon cern s i t se l f no t wi th th e fi e lds exc i t ed by th e ex te rn a l RF, bu t

r a t h e r w it h t h e fie ld s i n d u c e d b y t h e b e a m . Th is w ill b e a n a l yz ed in

t h e fo llo win g ch a p t e r s . H o weve r, t h e r e a r e ad d it i on a l s t r u c t u r e s t o b e

co n s i d e red b e fo re w e con t in u e .

Th e s ing le cav ity is u su a l ly no t su ffic ien t to acce le ra te th e beam to th e

d es i r ed en e r g y. Th e re is a f in i t e l im i t o n t h e s t r en g t h o f t h e fi el d s i n

a cavity. 6 An d , t h e r e is a n e n e r g y r a n g e w h e r e a p a r t i c u l a r c a v it y is

m ost effic ien t . Th u s , severa l cavit ies are often u sed (see Figu re 2 .2).

Beam Line

Cavity RF FeedPort Cavity

Cut-away

RF FeedPort Cavity

RF FeedPort Cavity

RF FeedPort

Powersupply

Powersupply

Powersupply

Powersupply

F r o m a m a n u fa c t u r i n g s t a n d p o in t , if o n e c a vit y ca n b e m a d e , s e ve r a l

cav it i es can be m ad e fo r a mu ch lower cos t per u n i t . However,

s t r i n g i n g cav i t i e s t o g e t h e r r eq u i r e s s o p h i s t i c a t ed p o w er s u p p l i e s

an d s w i t ch in g c ir cu i t s . Th e RF p o w er mu s t b e s u p p l ied a t t h e

Figure 2.2. A s eries of cavit ies is of ten re qu ired toa c c e le r a t e a b e a m t o t h e d e s i r e d e n e rg y. E a c h c a v it y willh ave it s own RF feed.

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a p p r o p r ia t e m o m e n t a n d a t t h e co r r e ct p h a s e (s e e S ec t io n 2 . 2 ). F r o m

a b e a m b r e a k - u p s t a n d p o in t , a s t r i n g of c a v it ie s c o m p l ic a t e s m a t t e r s

becau se fie lds from a cavity can exci te a djacen t cavit ies : th is

p h e n o m e n o n i s t e r m e d c ross - ta lk . However, as s t a t ed in th ein t rod u c t ion , we wi ll as su me tha t c ross - t a lk is n eg lig ib le . In o t h er

words , th e fie lds a re cu t o ff in th e beam p ipe which a d joins ca vit i es .

To al leviate some of the d i ff icul t ies wi th mul t ip le cavi ty s t ructures ,

an o t h e r s t r u c t u r e w as d evis ed : t h e d i sk - lo a de d s t ru c tu re . It is

p r e s en t ed h e re s i mp ly fo r co mp l e t en es s (s ee F i gu re 2 .3 ).

In s t ead o f a s e r i es o f c avit ie s , t h e d i s k - lo ad ed s t ru c t u r e h a s a s e r i es o f

d i sks wi th i r ises in th em. Adjacen t d isk s fo rm a cav ity - like enc losu re .

In t h is w ay, RF in p u t c an b e fed in f rom o n e en d a n d ext r ac t ed o u t t h e

o t h e r. D is k - l oad ed s t r u c t u r e s can b e d e s i gn ed i n s t an d i n g o r t r ave lin g

wave configu ra t ion s . Ca lcu la t ing beam break-u p in su ch a s t ru c tu r e

r eq u i r e s e i t h e r co n s i d e r i n g t h e en t i r e s t r u c t u r e t o b e o n e u n i t , o r

analyzing the cross- ta lk . 7

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RF Input

Iris

Cavit i es can b e us ed fo r a n u mb er of o ther pu rposes . Cav it i es a re u sed

to b u n c h o r d e -b u n ch a b eam 8: par t s of th e beam are m ad e to

aggrega t e . An o ther p r ima ry u se is in k ly s t r ons . Klys t ron s a r e dev ices

w h i ch a r e co mmo n l y u s ed t o p ro d u ce RF i n a p ro ces s w h i ch i s

opp os i t e to tha t in an acce le ra t o r : en ergy is ex t rac t ed from a n

e le c t r o n b e a m a n d i s u s e d t o e xc it e fie ld s . Klys t r o n s a r e b u t o n e o f

s e v e r a l d e vi ce s u s e d t o p r o d u c e t h e R F n e e d e d t o d r i ve a c c e le r a t o r s .

Th e s e w il l n o t b e c o n s id e r e d i n t h is t h e s i s s i n c e t h e y d o n o t d i r e c t ly

per t a in to beam b reak -up . Never th e less , a pho tograph o f su ch a device

i s in c lu ded a t th e end o f th is s ec t ion (see F igu re 2 .1 3).

Nex t , w e exam in e a n o t h e r m e t h o d o f p a r t i cle a cce le r a t i o n .

Figure 2.3.A cut-away sect ion of a d isk-loaded s t ructure .

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2.1.c In duc t ion Cells.

Th e i n d u c t i o n ce l l an d t h e l i n e a r i nduc t io n a c c e l e r a to r (LIA) were

d evelop ed a t th e La wr en ce Liverm ore Nat ion al Lab (LLNL) bas ed on

co n cep t s p u t f o r t h b y Christofilos . The idea is s t r a igh t fo rward ;

h owever, th e en g inee r in g is no t . The LIA ce l ls work by sen d ing a

s h o r t p u l s e in t o an a r ea w h e r e a p o r t i on of t h e en e rg y p ro p aga t e s

t h ro u g h a f e r r it e co re wh il e m o s t o f t h e en e rg y t r a ve ls d o w n a n a r ro w

c h a n n e l. Th e c h a n n e l e n d s i n a t a p e r ed ga p . Th e w a ve p r o d u c e s a

p u l s e d fie ld a c r o s s t h e g a p (a n d h e n c e a p o t e n t i a l d i ffe r e n c e ). T h efi e ld ex is t s fo r th e l eng th o f t im e i t t ak es t h e wave to p ropa ga te

th rough th e fe r r i t e . Th i s fi e ld acce le ra tes th e pa r t i c les pass ing

b e t w een t h e gap . In s i mp l e r t e rm s , t h e fe r r i t e in s u l a t e s (b y p ro v id i n g

a t r a n s i en t i n d u c t a n ce ) t h e t w o s i d e s o f t h e cavit y. F igu re 2 .4 is a

diagram of th e ba s ic LIA m odu le.

Accelerating gap

RF reflector

Beam

Pulseinput

Vacuum

Focusing coilFerrite

A n i n d u c t i o n mo d u l e can a l s o b e u n d e r s t o o d i n t e rms o f a s i mp l e

equivalent c i rcui t (see Figure 2 .5) . 9

Figure 2.4. A LIA module or induction cell .

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Zt

ZL IB

Al t h o u g h t h e fie ld s i n a LIA m o d u l e ma y b e q u it e d i ffe r en t fro m t h o s e

in a r e s o n a n t c a vit y, t h e B BU e q u a t i on s a r e m u c h t h e s a m e . F u r t h e r ,

t h e r e a r e m o d e ls w h i ch a l lo w LIA m o d u le s t o b e t r e a t e d in t h e s a m e

manner as a pil lbox cavity. 1 0 Th e n u m er ica l ex am p l e s gi ven i n Ch ap t e r

6 are of LIAs.

2.1.d Magnets .

Th e n e e d fo r t r a n s v e r s e fo c u s i n g in a n a c c e le r a t o r i s a p p a r e n t i f o n e

cons iders a l l t h e des ta b i liz ing fo rces a c t in g on th e beam: space

ch ar ge , wakef ie lds , RF in ca vi t ies , e t c . Th i s need i s p r ima r i ly sa t i s f iedb y mag n e t s , 1 1 a l though in th eory e l ec t r i c fi e lds cou ld be u sed to

m odi fy th e beam mot ion . In p ra c t i ce , m agn et i c fi e lds e xer t mu ch

higher forces on a h igh veloci ty beam. 1 2

Figu re 2 .5 . An equ ivalen t c i rcui t for an in du ct ion cell . Th eg rey s ec t i o n o n t h e l e f t r ep re s en t s t h e p o w ert ransmiss ion : Z t i s the equ iva len t impedance o f thet ran sm iss ion cab les . The compen sa t ion load , Z L, isd e s i g n ed t o ma t ch t h e b eam lo ad : t h is r ep r e s en t s t h efe r r i t e c or e . Th u s , t h e d r i ve c u r r e n t is d i vid e d , e qu a l ly,b e t w een t h e b eam a n d t h e lo ad .

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Perhaps the s imples t magnet i c f i e ld which i s o f any use i s the

solenoidal fie ld . I f we cont em pla te a pa r t ic le t ravel ling ins ide a

s o l en o i d a l fi e ld , w e s ee t h a t t h e Lo ren t z fo r ce w il l t en d t o co u p l e t h e

t r a n s v e r s e d i r ec t io n s o f t h e p a r t i c le m o t io n . To u n d e r s t an d t h i s mo requ a n t i t a t ive ly, we wi ll s o lve fo r th e equa t ion o f mot ion o f a p ar t i c le

m oving thr ou gh a solenoidal field .

V B

I

e1

e 2

e3

Th e b a s i c ge o m e t r y is g ive n in F ig u r e 2 . 6 ; t h e u n it v e ct o r s a r e e 1 , e 2

a n d e 3 (c o r r e s p o n d i n g t o g en e r a l iz ed d i r e c t io n s 1 , 2 a n d 3 ). We t a k e

t h e p a r t i c l es ch a r g e t o b e q , i t s v e lo c it y t o b e v w it h v z>>v a n d t h emagnet ic f ie ld i s g iven by B=B e 3 . Work ing in cgs-Gauss ian u n its

(w h e r e c h a rg e i s i n s t a t c o u l o m b ), t h e Lo r e n t z fo r c e o n t h e p a r t i c le i s

given b y

F = qc v B = qBc v e 3

= qBc ( )v2e 3 - v1e 2 . (2 . 1 )

For a relat iv is t ic par t ic le ,

d vd t

= qB m c ( )v2e 1 - v1e 2 (2 .2 )

F igu r e 2 .6 . S olen oid a l focus ing. A pa rt icle of velocity vt raverses a reg ion where a so leno ida l magnet i c f i e ld i su sed to focu s th e pa r t i c le .

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w h ere m is t h e p a r t i c le s m as s , i s the Loren tz fac to r, and the

p a r t i c le s e n e rg y is a s s u m e d t o b e co n s t a n t . We s e e t h a t t h i s c a n b e

s ep a ra t ed i n t o t w o cou p l ed fir s t o rd e r eq u a t i o n s (o n e i n e 1 a n d o n e in

e 2 ). Th u s , t h e p a r t i c le wi ll t en d t o m o ve in a h e li x. Th e fr eq u en cy o f

oscil lat ions is given by

* c =qB

m c . ( 2 . 3 )

Th i s i s k n o w n a s t h e cyc lo t ron (o r gyro ) frequ ency. The so leno id

p rov ides a gu id in g fi e ld : th e p a r t i c l e wi ll fo l low t h e f ie ld l ines . Of

course , i f t he par t i c l e ' s t ransverse ve loc i ty becomes too g rea t , t hemagnet ic f ie ld wi l l not be suff ic ient ly s t rong to def lect the par t ic le .

A m ore care fu l an a lys i s m u s t be don e to ob ta in th e focus in g e ffec t fo r a

p a r t i cle en t e r in g an d t h en ex it in g a s o len o id . In t h is c a s e , t h e p a r t ic le

w il l b e i n f lu en ced b y t h e r a d i a l ma g n e t i c fi el d s a t t h e s o l en o id ed g es .

In a d d i t io n , t h e a x i a l fie ld w i ll r a m p u p (d o w n ) a t t h e e n t r a n c e (e x it )

o f th e so leno id . The ra d ia l fi e ld c ros sed in t o th e par t i c le s ax ia l

v e lo c i t y p r o d u ces a k i ck i n t h e azimu t h a l d i r ec t io n . In t u rn , t h e

az imu tha l ve loc ity c ross ed in t o th e ax ia l m agn et i c f ie ld p rodu ces a

k i ck t o wa r d s t h e a x is a n d in t h e r a d i a l d ir e c t io n . Th e e q u a t i o n o f

m ot ion fo r th e pa r t i c le is

m d vd t

= qc (vB) - m ( r ) - 2m ( vr ) (2 .4 )

w h e re ag a i n t h e Lo ren t z f ac t o r i s a s s u med co n s t an t an d cy l i n d r i ca l

co ord i n a t e s h av e b een u s ed . H e re = ( v / z) in th e z -d i rec t ion .

Separa t ing th i s in to coord ina tes , we have th ree equa t ions :

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r : m d v rd t

= qc vBr + m v

2

r

: m d vd t

= qc vzBr - 2 m vr v

r

z : m d v zd t

= - qc vvr . (2 . 5 )

We c a n a s s u m e t h a t t h e t ot a l a n g u la r m o t io n is s m a ll th r o u g h o u t t h e

s o l en o id an d t h a t r is n ea r l y co n s t an t . Th en , t h e Co r io lis fo r ce in t h e

eq u a t i o n can b e n eg l ec t ed an d v zcon s t an t . N ex t , t h e paraxial

app roxima t ion is us ed : B r (- r / 2 )( Bz / z)| r=0 . Thus ,

v +q B z

2 m c r = 0 . (2 . 6 )

The p a r t i c l e s az imu tha l ve loc ity wi ll be th e s am e u pon ex i t in g th e

s o l en o id a s it w as u p o n en t ry. Th i s s m a l l a z im u t h a l ve lo c it y p ro vid es

th e focu s in g e ffec t via th e Loren tz fo rce . Th e pa r t i c le wil l ro ta t e ab ou t

th e ax is a t a frequen cy

* = 1

2 qB

m c , (2 . 7 )

ca l l ed t h e be ta t r on f r e que nc y , 1 3 which i s one ha l f the cyc lo t ron

fr eq u en cy. Th i s fr eq u en cy is i n d ep en d en t o f t h e p a r t ic le s d i s t an ce

from th e ax is . Thu s , a par t i c le t ra vers ing severa l c lose ly spa ced

solenoids wi l l undergo focusing whi le osci l la t ing at the betat ron

frequ ency. Not i ce tha t to ma in ta in a cons ta n t focu s ing fo rce , th e

m ag n e t i c fie ld m u s t i n c r ea s e i n p ro p o r t i o n t o . Fo r lo n g e r s o l en o id s ,th e cyc lo t ron os c il la t ion s wi ll dom in a te over the b e ta t ron m ot ion .

An o t h e r t y p e of m a g n e t , c o m m o n l y fo u n d i n c i r c u l a r a c c e le r a t o r s , i s

t h e d ip o le mag n e t . Th is m ag n e t p ro vid es a n a n g u l a r d e fl ec t i on o f t h e

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b eam in o n e d imen s i on . Th e d i p o l e ma g n e t is u s ed fo r b en d i n g

(s t ee r i n g ), n o t fo cu s i n g . Th u s , d i p o le s a r e n o t u s e fu l in co n v en t io n a l

lin a c s . H o weve r, h i gh e r o rd e r mag n e t s , n a me l y q u ad r u p o le s , a r e u s ed

ex tens ive ly (see F igu re 2 .14 a t t h e end o f th e cha p te r ) .

A q u a d r u p o le m a g n e t c a n fo c u s o n ly o n e t r a n s ve r s e d i r e c t io n , a n d i t

d o es s o a t t h e ex p en s e o f t h e o t h e r d i r ec t io n : o n e d i r ec t io n is

focused , the o ther i s defocused . Thus , a pa i r o f quadrupo les i s

requ i red to p rov ide symmet r i c focus ing .

Th e t r an s v e r s e mo t i o n o f p a r t i c le s t h ro u g h a l a t t i c e o f q u ad ru p o le s i s

o ft en ca lcu la ted u s ing t r a n s f e r m a t r ic e s . A focus in g sec t ion com pr i sed

o f a fo cu s i n g (F ) an d a d e fo cu s i n g (D ) w i t h d r i ft s p a ces (O ) b e t w een

t h em i s o ft en ca l led a FODO l en s (s ee F i gu re 2 .7 fo r t h e g eo me t ry ).

Th e o rb i t of a p ar t i cle th rou gh a se r i es o f FODO lens es i s ca lcu la ted by

mul t ip ly ing toge ther the mat r i ces o f each l a t t i ce e l ement . 1 4 Th e

r e s u lt , w h e n t h e m a g n e t s a r e p e r io d ic a lly s p a c e d , 1 5 a n d t h e t h i n le n s

ap prox im at ion i s u sed , is th a t t h e frequ ency o f osc i lla t ion s i s invers e ly

p ro p o r t i on a l t o t h e p a r t i c le s en e rg y. Th u s , p a r t i c le orb i t s u n d e r b o t h

so len o ida l an d qu adr u po le focu s ing can be des cr ibed b y th e sam e

equat ion, s imply wi th d i fferent f requencies of osci l la t ion.

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zx

y

F O D O F O D

F O D O FOD

X-axis

Y-axis

2.1.e Monitors.

As with an y o th er soph i s t ica ted dev ice , it i s imp or ta n t to ha ve

d iag n o s t ic s an d s en s o r s in accele r a t o r s . Sen s o r s a r e r eq u i r ed t o

d e t e r m i n e w h e r e t h e b e a m is i n t h e b e a m p ip e a s w ell a s t o d e t e r m in e

t h e con d i t io n o f t h e b eam . Acce le r a t o r o p e ra t o r s a s w e ll a s p h y s ic is t s

n eed t o k n o w va r iou s b eam p a ram e t e r s s u ch as cu r r en t , t r a n s ve r s e

velocity, frequ en cy of bet a t ron osci l la t ions , e tc . We wil l br iefly d iscu ss

two types o f sensors ( somet ime ca l l ed beam-bugs ), a s t h e y a r e u s e d t o

m e a s u r e b e a m b r e a k - u p . Th e fir s t s e n s o r e m p l oy s c a p a c it ive e le c t r ic

f i e l d p i ck u p p l a t e s ; t h e s eco n d i s t h e b - do t l o o p . Th es e co lo r fu l

n am es a r e r a t h e r d e s c r ip t i ve .

Figure 2 .7 . A l a t t i c e o f eq u a l ly s p aced q u ad ru p o l e l en s e s .Each quadrupo le focuses (F) one t ransverse d i rec t ionwhi le defocu s in g (D) th e o ther . Th e d r i ft sp aces (O) a r e a l las s u med equa l . Then la t t i ce is m ad e up of equ a l FODOlenses .

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Beam Pipe

Beam Monitor

Cut away

Th e fir s t t y p e can b e t h o u g h t o f a s s e t s o f p a r a lle l p l a t e s w h ich d e t ec t

th e beam pos i t ion (see F igu re 2 .8 ) .

Th e p o t en t ia l d i ffe r en ce in d u ced b e t w een t h e p l a t e s i s p r o p o r t i on a l t o

t h e p o s it i on o f t h e t r an s v e r s e lo ca t i on o f t h e b eam . By u s i n g t w o s e t s

o f p la t es , one fo r the x d i rec t ion and one fo r the y d i rec t ion , the

loca t ion o f th e beam s cen te r o f ch ar ge a t tha t pos i t ion in th e

acce le r a t o r c an b e k n o wn . In p r a c t ic e , t h e b eam -b u g is o ft en m ad e of

cylind rical p lates wh ich are s ect ioned off in to four arcs . By

p e r fo rmin g t h e p ro p e r s i gn a l an a ly s is , b o t h t r an s v er s e p o s it io n s can b e

d e t e r m in e d . As t h e b e a m g oe s b y t h e m o n i t o r , a r e s p o n s e vo lt a g e is

d i s p lay ed ve r s u s t i me o n a n o s c illo s cop e . Th is en ab l e s t h e o p e ra t o r t o

k n o w t h e p o s it io n t h e b eam w it h in t h e b eam p ip e .

Th e o t h e r t yp e of s e n s o r w e c o n s i d e r is t h e b - d ot lo op (s e e F igu r e

2 . 8 ). Th e n a m e r e fe r s t o a d e v ic e w h i c h m e a s u r e s t h e m a g n e t ic flu x

Figure 2 .9 . A d i agram o f a b eam -b u g s h o win g t h e t wo s e t s

o f p a r a l le l p l a t e s . Each s e t mo n i t o r s on e of t h e t r an s v e r s ed i rec t ions .

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th rou gh a loop o f wire . Th e ma gnet i c flux in t h e az im u th a l d irec t ion i s

p ro p o r t i o n a l t o t h e b eam cu r r en t p a s s i n g t h ro u g h a t a g i v en p o i n t

a long th e accele ra to r. Nam ely, in cgs un it s ,

= 1c d d t = 1c dd t B d Aloop (2 .8 )w h e r e is t h e E MF in d u c e d in t h e lo op : t h e c h a n g e in t h e m a g n e t i c

fie ld in du ces an elect r ic fie ld ac cord ing to Farada ys law. Th e

m ag n e t i c fie ld o f t h e b eam i s p ro p o r t io n a l t o t h e b eam cu r r en t . Th u s ,

the vo l t age induced in the wi re loop i s p ropor t iona l to the ra t e o f

c h a n g e o f c u r r e n t , or d B / d t . Th e n a m e of t h is d e vic e a r i s e s fr o m t h e

co mm o n u s a g e o f B (B d o t ) t o i n d i ca t e t h e d i ffe r en t i a t i o n o f B w i t h

r e s p e c t t o t i m e .

Beam Monitor

Cut away B-dot loop

Beam B

Figure 2 .9 . A cu t awa y view of a b-d ot loop. Th e wire loopi s pos i t ioned wi th i t s p lane para l l e l to the d i rec t ion o f b eam p ro p ag a t i on s o t h a t t h e az imu t h a l ma g n e t ic fie ld w i lllink t o th e loop .

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2.2 Lon git ud inal Mot ion.

We have a l ready d i scussed the t ransverse mot ion a s ing le par t i c l e

ex ecu t e s w h en i t i s t r a ve r s i n g a m ag n e t i c fie ld . An a cce le r a t ed b eam

will a lso ha ve a longitu dina l oscil la t ion. This is ass ociated wi th

s y n c h r o t r o n m o t io n a n d is c a u s e d b y a p h e n o m e n o n k n o w a s p h a s e

stability .

Pha se s tab ility is a s imp le con cep t to gras p . Each cavity ha s a RF

p h as e a s s o cia t ed w it h it . 1 6 Th e p h a s e i s m e a s u r e d w it h r e s p e c t t o t h e

t ro u g h o f t h e acce le r a t in g fi eld . In a n RF lin ac , t h e cavit y p h a s e s a r e

t u n e d s o t h a t a n id e a l b e a m w ill a r r ive s o m e w h e r e a h e a d o f t h e c r e s t ,

a t t h e t h r e s h o l d e n e r g y (s ee Figu re 2 .10 ). The ph as e-energy well s

form region s of s ta bi l ity. 1 7 ,1 8 Th es e r eg io n s a l l ow fo r a r an g e of s t ab l e

p h a s e s .

Energy

RF Phases

Threshold Late

Early

A par t i c l e may a r r ive too ear ly a t a cav i ty i f i t s ve loc i ty i s too g rea t .

However, s ince i t a r r ived a h ead o f th e th res h o ld en ergy, i t w il l rece ive

F igu r e 2 .10 . A m odel o f ph as e s t ab i li ty. Par t i c les a r r ivingah ead (b eh in d ) o f s r e c e ive l e s s (m o r e ) a c c e l e r a t i o n t h a nth ose a r r ivin g a t s .

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less acce le ra t ion tha n th ose a r r iving a t th e co r rec t t ime . Th e oppos i t e

i s t r u e fo r a pa r t i c le a r r iving too la t e . Thu s , pa r t i c les wi ll o sc i lla t e

about a com m on velocity. The s e oscil la t ion s in velocity gen era te

o s c illa t i o n s i n t h e p a r t ic le p o s it io n s : s y n ch ro t ro n m o t io n o f t h e b eam .P h a s e s t a b ilit y e n a b le s a b e a m t o b e a c c e le r a t e d i n a R F lin a c w it h a

s t a t i c re la t ion be tween th e cavity pha ses .

Longi tudinal beam mot ion is negl ig ible in h ighly relat iv is t ic beams.

Sy n ch ro t ro n m o t io n i s m i n i ma l b ecau s e t h e b eam s v elo c it y is a lm o s t

a t t h e s p eed o f l ig h t . In o t h e r w ord s , fo r a r ea l is t i c s p r ea d in en e rg y,

th e veloc ity sp read , vz, of th e beam is m in im al .

Co n s i d e r a b eam w it h an av e rage en e rg y o f mc 2 , a n d w it h a s p r e a d o f

2 . Th en ,

vz = c 1 - 1( ) + 21/ 2

- c 1 - 1( ) - 21/ 2

. (2 . 9 )

For

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2.3 Phot ographs of Devices.

F igu r e 2 .11 . A top v iew of an in jec to r u sed a t th eacce l e r a t o r r e s ea r ch cen t e r (A RC) a t t h e Law ren ceLiverm ore Nat iona l La b (LLNL). The in jec to r is u sed fo rthe sub-harmonic dr ive relat iv is t ic k lys t ron (SHARK). 21

Th e c a b l e s fe e d i n g i n t h e e x t e r n a l RF p o w e r t o t h e fir s ttwo cel ls are v is ib le on the r ight .

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F igu r e 2 .12 . An ar t i s t ' s con cep t ion of a re lat ivis t ick l ys t ro n . cav it y It is d e s i gn ed t o ext r a c t RF en e rg y fro m arelat iv is t ic beam pass ing through i t s axis .

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F igu r e 2 .13 . A co n v en t i on a l k ly s t ro n u s ed a t SLAC on t h eSta n ford l in ear col l id er (SLC). Th e device is he ldver t i ca l ly with th e e lec t r on gun a t th e bot tom. It canp r o d u c e a p e a k p o w e r o f 6 7 M W a n d a n a v e r a g e p o we r o f 42 KW.

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Figure 2 .14 . A typ ica l qu ad ru po le magnet . Th e beam wouldt r av e l t h ro u g h t h e cen t e r, g oin g i n t o t h e p ag e .

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Referen ces an d Notes(C h a p t e r 2 )

1More accura tely, most electron gu ns in televisions w ork in a man ner similar to those in electronaccelerators, since cathod ray tu bes have been in existence longer th an televisions.2The range of frequencies defined as m icrow aves varies in the literatu re.3The term excited is used to indicate that electromagnetic fields ar e prod uced in a cavity, andsustain for a finite length of time.4The term RF (from rad io frequen cy) is used in this pap er to connote pow er sources operating inthe microwave region.5Copp er not only has the excellent condu ctivity (second on ly to silver amon g the elements atroo m tem pe ratu re), it is also economical and h as excellent th erm al pr operties, is easy tomachine, electrop late, etc.6Field strength in a cavity is usually limited by th e amou nt of RF pow er that can be generatedand delivered to i t . More fund am ental ly, the field strength is l imited by multipactoring(mu ltiple electron collisions). In other words, a resonant growth of electron current (from

second ary emission) can occur. This current pu ts an upp er limit on the fields in the cavitybecause all the pow er is absorbed by the curren t. Anoth er limit on the field strength is fieldemission . If the local fields in a cavity w all are sufficiently high, electrons can be liberat ed .Eventually, the electrons sp ark across to an other su rface.7In cases wh ere the BBU m ode cannot prop agate throu gh th e irises in th e disks, the cross-talk can be ignored .8Bun ching or ph ase comp ression can be indu ced in a suitable structure wh ose phase velocity isless than th e speed o f light. See R. H . Helm an d R. Millers article, Particle Dyn am ics, inLinear Acceler ato rs , op . cit ., p .119.9This mod el for an ind uction cell, along with th e figur e, wher e derived from ones given b y R. J.Briggs, et al. , Beam Dyn amics in the ATA 10 kA Linear Ind uction Accelerators: Observationsand Issues, IEEE Transactions on Nuclear Science, NS-28 3 (1981).10R. J. Brigg s, Deflection of Beam Electro ns From Sym m etric Excitation s of an Accelerating

Gap, Lawrence Livermore National Lab ATA Note 204 (1983).11The use of ma gnets in linacs may become less pr evalent if ion-chann el focusing and othernext-generation schemes prevail.12In the injector, it is often u nd esirable to h ave magnetic fields because the beam w ill bed efocused (see S. H um ph ries, Pr inciples of Charg ed Par ticle Acceleration , pp .156-157 for abrief discussion). Thus, electric fields are used to focus and steer. In circular machines, wh eresmall changes in a b eams orbit can be m ore useful, electric fields ar e also used. Parallel plates,called kickers , are employed to m ake minor orb it corrections, and to stochast ical ly cool beams.13Betatron oscillations der ive their nam e from the betatron accelerator wh ere they were f i rstobserved . How ever, the actual frequen cy differs from that in a typical betatron.14See S. Hu mp hries, Principles of Charged Particle Acceleration (Ne w York : John W iley &Sons, 1986), Ch. 6 an d 8.15Fortun ately, this is often th e case in linacs.16In non -relativistic LIAs ph ase stability is ensur ed by pr oper tim ing of the ind uction cells, andramp ing the app lied voltage rather than using a squ are pu lse. At relat ivist ic velocit ies, thisbecomes unn ecessary: longitud inal spr ead of the beam is negligible.17A bucket refers to the region of ph ase stability. A bunch is an area in p hase spa ce fi l led w ithpart icles.18It is often helpful to th ink of relat ivist ic pha se stabil ity as ana logous to a su rfer riding aw ave. If the sur fer s ini t ial speed is close to that of the w ave, the surfer w ill be able to ridethe w ave to the beach. This, of course, is only an analogy.19See S. H u mp hr ies, op. cit , p p.430-436.

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20For a considera tion of long itud inal wake effects in a lin ac see R. B. Pa lmer , Th eInterdep end ence of Parameters for TeV Linear Colliders, Stanford Linear A ccelerator CenterPU B-4295 (1987).21A brief review art icle on relat ivist ic klystron research at LLNL can be foun d in Energy &Technology Review, LLNL (July, 1988), p p.58-59.

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C h a p t e r 3T h e B BU P r o b l e m

Th e p r o b le m o f b e a m b r e a k - u p is fo rm u la t e d .

Th e c on c e p t o f a w a k e is i n t r o d u c e d , a n d t h eBBU equ at ion is d er ived.

3.1 A Qualitat ive Explana t ion of BBU.

Perh ap s t h e m o s t fu n d am en t a l lim i t p laced o n b eam p ro p ag a t io n is t h e

Al fvn cur ren t l imi t ( the space-charge l imi t ing cu r ren t ) : 1 t h e

h ig h es t cu r r en t p o s s i b l e fo r a b eam p ro p ag a t i n g in fr ee s p ace . At t h is

cu r r en t , t h e b ea m s s e l f fi e ld w i ll d e f lec t t h e b ea m i n t h e d i r ec t i o n

o p p o s i t e t o p ro p a g a t i on . 2 Fo r r e fe r en ce , t h e Alfvn cu r r en t is , I A =

(m c 3 / e ) 5 .1 x 1 0 1 3 s t a t a m p s ( 1 7 . 1 k A), w h e r e m a n d e a r e t h e m a s sa n d c h a r g e of t h e e le c t r o n a n d c is t h e s p e e d o f l ig h t i n c gs (m k s )

uni ts .3

Pro p ag a t in g a b eam in fr ee s p a ce d o es n o t a l lo w fo r mu ch co n t ro l ove r

t h e b eam . Th e re fo re , b eam s a r e u s u a l ly s en t t h ro u gh evacu a t ed p ip e s

(s ee ch a pt er 2 for m ore d eta i ls ) . Maxwel l s equa t ions a fford us a

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so lu t ion to th e fi elds o f a p ar t i c le in a cy l in dr i ca l ly sym m et r i c p ipe . 4 If

t h e w a l l s w e re mad e f ro m p e r f ec t co n d u c t o r s , t h en t h i s g eo me t ry

w ou ld n o t b e m u c h m o r e in t e r e s t i n g t h a n fr e e s p a c e (a s l o n g a s t h e

p a r t i c le d o e s n o t s t r i k e t h e p ip e w a l l). S i n c e , t y p ic a l ly, b e a m p i p e sare no t made f rom superconducto rs , t he wal l s have a f in i t e

condu ct ivity an d p r ob lems in p ropa ga t ion ex is t . 5 If t h e p a r t i c le c o m e s

n ear th e wal l, even fu r th er d i fficu lt i es a r i se ; an d a s s oon as th e pa r t i cle

t r a v e r s e s o t h e r b e a m lin e c o m p o n e n t s s t i ll m o r e co m p l ic a t i on s e n s u e .

When a b e a m o f p a r t i c le s i s con s i d e red t h e s e s a m e p ro b l em s ex is t a n d

n ew ones a re c rea ted . An y o f th ese d i fficu lt i es can lead to beam brea k-

u p.

Th is t h e s i s o n l y con ce rn s it s e l f w it h b eam b reak -u p d u e t o w ak e fie ld s :

t h o s e p ro b l ems cau s ed b y t r av e r s in g b eam lin e co mp o n en t s . 6

Beam Line

Cavity

Cut-awayParticle

RF FeedPort

A p a r t i c l e t r a v er s i n g a cav it y can ex c it e fi el d s in t h e cav it y. In t u rn ,

th ese fi elds can a ffec t th e pa r t i c le . However, fo r a re l a t iv is t i c pa r t i c le ,

Figure 3 .1 . An off-axis pa rt icle tr a vers ing a cavity canexc i t e fie lds . In a dd i t ion , exte rn a l RF can be u sed toexcite t h e ca vity.

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th e t ra n s i t t im e th r ough th e cavi ty is s u ffi cien t ly sh or t th a t a n y e ffec t s

a r e n e g li gi b le . In o t h e r w o r d s , t h e fi e ld s r e q u i r e a fi n i t e t im e t o b e

es ta b l ish ed . An d , by cau sa lity, th e e ffec t o f th e f ie lds cann o t t r ave l

fa s t e r t h a n t h e s p e e d o f li gh t , s o t h e s i gn a l c a n n o t c a t c h - u p t o t h ep a r t i c le . A cav i t y can a l s o b e d r i ven b y a n ex t e r n a l fi e ld (s ee F i gu re

3 .1 ). Th is fi el d ce r t a i n l y can , an d d o es , a ffec t t h e p a r t i c le . H o w eve r,

w e s h a l l a s s u m e t h a t t h e e n gin e e r s h a v e d o n e t h e ir jo b a n d t h a t t h i s

e ffe c t h a s b e e n m i n i m i ze d . No w, h a d a s e c o n d p a r t i c le fo ll ow e d t h e

fir s t o n e w i t h i n t h e t i m e t h a t t h e fi eld s w e r e s t i ll e xc it e d , t h e s e c o n d

p a r t i c l e co u l d h av e b een d e fl ec t ed b y t h e fie ld s ex c it ed b y t h e fir s t

pa r t i c le (see F igur e 3 .2 ). In a dd i t ion , th e secon d pa r t i c le wou ld exc it e

th e cavity fu r th er. Th i s is th e s im ple two-pa r t i c l e m odel of

cumulat ive 7 b eam b rea k -u p d u e t o w ak e fie ld s . It is s o lved an a l yt i ca l ly

in s ec t io n 4 o f t h i s ch ap t e r.

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HeadParticle

Cavity

Cut-away

TestParticle Wake Field

Beam Line

I f w e n o w co n s i d e r a b ea m o f p a r t i c le s t r av e r s i n g s eve ra l c avit ie s , a

m o re co m p l e t e p i c t u r e o f w a k e f ie l d BBU in in d u c t io n lin ac s can b e

as ce r t a i n ed . Th e b eam m ay b e t h o u g h t o f a s a s t r eam o f s l ic e s . Each

s l ic e wi ll e x c it e fi el d s in t h e s t r u c t u r e , c o u p l in g t h r o u gh t h e d i p o le

m o m e n t o f t h e a x i a l c u r r e n t d e n s i t y t o t h e a x ia l e le c t r i c fi e ld o f t h e

m o d e s ( J E ). Th e t r a n s v e r s e B- fi e ld ex c it ed b y t h e fi r s t n s l i c e s w il l

d e f l ec t t h e (n + 1 ) t h s l i c e v i a t h e Lo ren t z ( vx B) fo r ce . Th e fu r t h e r o ff-

ax is a s l i ce is , t h e g rea t e r i t s d ipo le -mom en t - ind u ced f ie lds in

down s t rea m cav it i es wi ll be . Th ese la r ger fi elds cau se la r ger impu lses

on fol low-on part ic les and an ins tabi l i ty ensues (see Figure 3 .3) .

Figure 3 .2 . Th e two-pa r t i c le model descr ip t ion ; an o ff-ax i spa r t i c l e t ra verses a cavity an d exc i t es wakefie lds . As eco n d p a r t i c le fo ll ow in g t h e fi r s t o n e r ece ive s a n im p u ls efr o m t h e vx B fo rce o f t h e cav it y fi el d s ex c i t ed t h ro u g h t h eJ E cou pl in g of th e fi rs t .

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Beam Line

Cavity

Cut-away

HeadSlice

Cavity

Cut-away

3.2 Notat ion

M os t o f t h e n o t a t i on u s e d i n t h is t r e a t is e a d h e r e s t o c on v e n t io n a n d

t r a d i t io n . U n fo r t u n a t e l y, d i ffe r en t au t h o r s u s e d i ffe r en t n o t a t io n . Byn eces s it y, t h e u n it s a r e a m ix t u re o f cgs a n d mk s (an d s o me t i mes even

inch es ). Th i s n ecess i ty a r ises becaus e th eor is t s p r e fe r cgs whi l e

ex p e r im en t a lis t s p r e fe r mk s . Th es e p r e f er en ces a r e n o t a rb it r a ry

cho ices . Cgs (Gau ss ian ) u n i t s a re ap pea l ing in th eore t i ca l work

beca u se o f th e equ a l foo t in g of th e e lec t r i c an d m agn et i c fie lds (am ong

oth er reason s ). However, u s ing cgs u n i t s fo r exper imen ta l

m e a s u r e m e n t s w ou ld n o t b e p r a c t ic a l: t e s t eq u i p m e n t , m a n u fa c t u r i n g

too ls , et c . a re a ll in m ks o r Eng lish u n i ts . Th es e a re fu nc t iona l

t r ad i t io n s . In k eep i n g w it h t h is , t h is w o rk u s e s mk s u n i ts fo r

p h y s ic a lly s ign i fic a n t q u a n t i t ie s s u c h a s c u r r e n t a n d l e n gt h , b u t m o s t

F i gu re 3 .3 . Wakefield BBU for a beam . A beam can bec o n s id e r e d a s a s t r e a m o f s l ic e s . Th e fir s t s l ic e e xc it e s afi e ld which def lec t s th e n -1 s l ices wh ich fo llow. Th es eco n d s l ic e ex c it e s a f ie l d w h i ch d e fl ec t s t h e n -2 s l ic e t oit s r ear, e t c . .

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an a lyt i ca l ca lcu la t ions a re in cgs . Prob lems as soc ia ted wi th conven t ion

ar e overcom e wi th exp l ic i t defin i t ions .

Th e g eom e t ry o f an acce l e r a t o r c a n b e co n s i d e ra b l y s im p l ified for o u r

p u rp o s es . Th e b eam s t r an s v er s e d is p l acemen t fro m t h e b eam lin e ax is

w il l b e m o d e l ed b y o n e d i men s i o n , . Th i s r eq u i r es t wo a s s u m p t i on s .

Th e f ir s t is t h a t t h e d i s p lacemen t o f a c ro s s s ec t io n o f t h e b eam can b e

m o d e l ed b y t h e d is p lacem en t o f t h e cro s s s ec t i on cen t e r o f mas s .

This a ss u m pt ion is wel l ju s t i fied for BBU becau se we a re only

co n ce r n ed w it h t h e mean m o t io n o f t h e b eam. Th e s eco n d

as s u mp t i on is t h a t t w o t r an s v er s e d imen s i on s o f t h e b eam , i. e . (x ,y) o r

(r , ) , can be approx imated by a s ing le one , i . e . . Th i s s eco n d

assumpt ion i s va l id i f t he t ransverse coord ina tes a re no t coup led .

Reca l l tha t in the case o f so leno ida l focus ing , bend ing magnet s ,

a s y m m e t r i c (fla t ) b e a m s , e t c . , t h e t r a n s v e r s e m o t i on i s c ou p l e d . F o r

cy l indr i ca l ly symmet r i c s t ruc tu res and beams wi th no coup l ing

b e t w een co ord i n a t e s , t h e o n e d i men s i on a l ap p ro xima t io n i s q u it esuitable.

The nex t f igure i l lu s t ra t es the re l evan t coord ina tes and l eng ths :

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Z BeamlineStart

BeamHead

Zmax

BeamTailSmax

BeamlineEnd

S

Magnification

of Beam

L

R

Th e n o t a t i o n u s e d i n t h e a b o v e fi gu r e a n d fo r t h e r e m a i n d e r of t h is

t h e s is i s p r e s en t ed i n Tab l e 3 .1 .

Figure 3 .4 . A s a m p l e s ec t io n o f t h e b eam p i p e s h o win g t h eu s e o f t h e s an d z co ord i n a t e s . Th e z co ord i n a t e meas u re st h e d i s t an ce d o w n t h e acce le r a t o r. Th e s co ord i n a t em e a s u r e s t h e d i s t a n c e a l on g t h e b ea m s t a r t in g f r om t h eb eam h ead . Th e in s e r t s h o ws t h e coord ina te .

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Th e b eam co ord i n a t e , s , i s a co n ven i en t v a r i ab l e u s ed i n p lace o f t i me .

Th e r e la t i on b e t w ee n t im e , t h e b e a m a n d t h e b e a m lin e c o o r d in a t e i s

given by,

t = zvz- s zc - s ( 3 . 1 )

w h e r e v z is t a k e n t o b e c on s t a n t a n d c . Th i s d e f in i t i on w i ll b e r e -

in t ro d u ced d u r in g t h e d e r iva t i on s in t h e n ex t s ec t io n . For

c o m p l e t e n e s s , t h e n e x t t a b l e g ive s s o m e a d d i t io n a l n o t a t ion u s e d in

t h i s p ap e r.

Al thou gh d efin i t ions o f th e t ra n sverse wa kefield var y, 8 t h i s p a p e r u s e s

a wa kefie ld wi th u n it s cm -2 s ec -1 in cgs (Vo lt s / Cou lo mb / me t e r i n m k s ).

Table 3 .1 . A l is t o f n o ta t ion us ed in th is pap er. MKS un it sa r e l is t ed i n t h e t h i rd colu mn .

s b ea m (p u ls e) coord in a te in u n it s of t im e [s ec]

s m a x p u ls e len gth in u n it s of t im e [s ec]

Z b ea m lin e coor d in a t e [m ]Z m a x t ot a l b ea m lin e (a ccelera t or ) len gt h [m ]

t r a n s v er s e b ea m d i s p la c e m en t fr om p i p e a x is . [m ]r b b ea m ra d iu s [m ]

L ca vit y len gt h [m ]

D ca vity s pa cin g in u n it s of t im e (D=s pa cin g/ c) [s ec]

R ca vit y r a d iu s [m ]

Table 3 .2 . Fu r t h e r n o t a t i on u s ed in t h i s p ap e r. MKS u n i t sa r e l is t ed i n t h e t h i rd colu mn .

c s p eed of ligh t in va cu u m [m / s ec]

e th e ch a rge of a n elect ron (a p os it ive va lu e) [C]

m th e elect ron m a s s [k g]

beam energy (E= m c 2 ) [-] ch a rge d en s ity (ch a rge/ u n it len gth ) [C/ m ]W t r a n s ver s e wa k efie ld [V/ C/ m ]

k b e t a t ro n waven u mb er, k = / c [m -1 ] beta t ron wavelength , =2 / k [m ]

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3.3 Derivation of t he BBU Equat ion.

Th e d e r iva t i on o f t h e b eam b rea k -u p eq u a t io n r eq u i r e s t h r ee s t ep s . In

s t e p o n e w e c a l cu la t e t h e im p u ls e im p a r t e d t o a p a r t ic le c r os s in g a

cav ity du e to th e fie lds in th e cavity. S tep two requ i res ca lcu la t ing th e

fi elds in a ca v ity du e to th e mom en t o f an o ff-ax i s par t i c le . S tep th ree

i n v ol ves co mb in i n g t h e r e s u lt s o f s t ep s o n e an d t w o b y u s e o f a f or ce

equat ion.

3.3.a Step One: The Im pulse

Le t u s b egin b y a s s u mi n g t h a t w e k n o w t h e fie ld s in an ex cit ed cav it y.

We n o w w is h t o ca lcu l a t e t h e t r an s v er s e i mp u ls e i mp a r t ed t o a p a r t i c le

t ra vers in g su ch a ca vi ty. Th e Loren t z fo rce in teg ra t ed a long th e cav ity

g ives t h e ch an g e in m o men t u m,9

p = e E + 1c (v B) d t t0t

, (3 . 2 )

w h e r e t 0 an d t a re the t im e of en t ry an d ex i t th rough th e cav ity,

r e s p ec t i ve l y. It i s a s i m p l e m a t t e r t o co n v e r t t h is in t o an eq u a t ion

w r it t en in t e rm s o f t h e vec t o r p o t en t i a l , 1 0

p = ec vz - t A + ( )v ( ) A d z0L

, (3 . 3 )

w h e re d z= v z d t , E =- A / ct , an d B= x A. Th i s equa t ion can be

s im pl ified by u se of th e vector re la t ion

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v ( A) = (v A) - A ( v) - ( v )A - (A )v, (3 . 4 )

an d b y t h e a s s u mp t io n t h a t t h e b eam v e lo c it y in t h e t r an s ve r s e

d i r e c t io n , v , is co n s t a n t . Th is am ou n t s t o an a s s u mp t ion t h a t t h e

t ransverse veloci ty i s not a l tered in the cavi ty (an accelerat ion, of

co u r s e , c an s t i ll b e im p a r t ed ). In o t h e r w ord s , w e a r e w ork i n g t o fi r s t

o rd e r in A. Th en ,

v ( A) = (v A) - ( v )A (3 .5 )

a n d ,

p = - ec vz t + v A - ( )v A d z0L

(3 .6 )

Th e le ft t e rm in th e in teg ra n d i s th e convec t ive der iva t ive ,

t + v A =

d Ad t . (3 . 7 )

H e n c e ,

p = - ec vz

d Ad t d z0

L

- ( )v A d z0L

(3 .8 )

Assu ming tha t v

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d u r i n g t h e p a r t i c le s fi n i t e t r a n s i t t i m e . If t h e ve c t or p o t en t i a l h a d

e t t i me d ep en d en ce , t h en A ( t )= A (0 )e L/ c . Th es e fir s t t e rm s

rep re s en t t h e fr i n g e fie ld s o f a c av it y d u e t o p o r t s o r o t h e r n o n s mo o t h

fea tu res . Un der su it ab le geomet r y, th i s t e r m may be neg lec ted .Sp ec i f i c a l l y, i f t h e cav i t y en d s a r e p e rp en d i cu l a r t o t h e w a l l s an d t h e

p o r t s a r e s m a ll co m p a r e d t o t h e c a v it y r a d i u s , t h e n t h e fr i n g e fie ld s

are negligible.

Th e s eco n d t e r m can b e eva lu a t ed b y co n s i d e r in g t w o or t h o n o rma l

m o d e s o f e x c it a t i on : t h e t r a n s v e r s e e le c t r i c (TE ) a n d t h e t r a n s v e r s e

m ag n e t i c (TM) m o d es . Fo r TE mo d es , t h e lo n g it u d i n a l e lec t r i c fi e ld s

vanish , E z=0, so A z=0 an d t h e in t eg ran d van is h es . H en ce ,

p =0 for TE m od es . (3 .1 0 )

For TM m odes we h ave 1 1

* p = ec

Az d z

0

L

(3 .11 )

w h i ch can b e w r it t en a s

p = - ec E z d z0L

( 3 . 1 2 )

i f w e co n s i d e r t h e vec t o r p o t en t i a l (an d h en ce t h e fi el d s ) t o h av e e -t

t i m e d e p e n d e n c e . E q u a t i on (3 . 1 2 ) c o n s t i t u t e s a s o lu t io n t o s t e p on e :

t h e i mp u ls e d u e t o t h e fie ld s in a c a vi t y i s n o w k n o w n (t o fir s t o rd e r ).

Th es e r e s u lt s w e re o b t a in ed b y n eg lec t i n g t h e t r an s v e r s e v e lo c it y, v .

H a d t h is n o t b e e n d o n e , t h e r e w o u l d h a v e b e en a c on t r ib u t ion fr o m

t h e t r an s ve r s e co mp o n en t s of t h e vec t o r p ot en t ia l . Th u s , t h e TE

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m odes would , in fac t , im pa r t som e imp u lse to th e par t i c le . Th i s

con t r ibu t ion i s n eg ligib le fo r re la t ivis t ic beam s . As s u ch , we sh a l l on ly

co n s i d e r TM mo d es fo r b eam b reak -u p .

N ow w e a r e r e q u ir e d t o p e r f or m s t e p 2 . A l in e a r i ze d s o l u t io n fo r t h e

fields du e to an off-axis pa r t ic le can be obt ained u s ing any on e of

severa l m eth ods . The der iva t ion p r es en ted her e fo llows from a

dom ina n t m ode ca lcu la t ion o f th e fi elds in a n idea l cavi ty. 1 2 A n o t h e r

d e r iva t i on is p r e s en t ed in Ap p en d i x 2 . Th e m e t h o d u s ed in t h e

App end ix is d u e to W. Pan ofsk y 1 3 a n d i s o n e o f t h e e a r lie r t r e a t m e n t s

o f th e th eory o f beam break-up . The ap proach re l i es on ener gy

co n s e rva t io n an d p h y s ica l in s i gh t w h i ch o n l y p h y s i c is t s o f Pan fo s k y s

cal iber s eem to posses s . Th e Pan ofsky der ivat ion , a l thou gh very

in t u it i ve , is n o t in c lu d e d i n t h e m a in t e x t b e c a u s e i t a r r i ve s a t a le s s

useful model of BBU.

3.3.b Step Two: The Field.

We c o n s i d e r t h e e le c t r ic fie ld a s a s u m o ve r o r t h o go n a l m o d e s o f t h e

e lec t r i c fi e lds . In o th er words , 1 4

E ( r , t ) = p (t ) E ( r ) + m (t ) F( r ) ( 3 . 1 3 )

w h e r e = (m , n , p ) is t h e m o d e n u m b e r t r i p le t , E a n d F a r e t h e t im e -

i n d ep en d en t , d i v e rg en t - f r ee an d cu r l - f r ee e l ec t r i c f i e l d s ,

respect ively. 1 5 (Do not confuse F for fie ld with F for forc e.) The vector

po ten t ia l can b e wr it t en in th e Loren tz gau ge as

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A(r , t ) = p (t ) E ( r ) . (3 .1 4 )

(N ot e , h e r e p i s mer e ly u s ed t o d i ffe r en t i a t e i t fr o m p , n o t t o i n d i ca t e

a com plex variab le .) We also requ ire th e ma gnet ic fie ld ,

B( r , t ) = h (t ) B( r ) . (3 .1 5 )

In ord er to s olve for th e fie lds in a ca vity excited by an off-axis p a r t ic le

o r bea m , we em ploy Maxwel l s equ a t ion s . Afte r ob ta in ing a su it ab le

secon d o rder d i ffe ren t i a l equ a t ion , we t ake in to accou n t th e losses in a

ca v it y d u e t o o h m i c h ea t i n g . As it w i ll b e s h o wn , it is s u ffic ien t t o

co n s i d e r = (1 , n , 0 ) m o d e s ; i n o t h e r w o r d s , d i p o le m o d e s wi t h n o z

va r i a t i on . Ne x t , w e s o l ve t h e e q u a t i on t o fir s t o r d e r i n t h e p a r t i c le s

d i s p l acemen t .

Let u s b egin by ap plyin g Fara da ys law wh ich yields

p d h d t B -1

k ( ) E

, (3 .1 6 )

w h e r e =k c as u su a l . S imi la r ly, Am peres l aw, equa ted to the cu r l o f

Far ad ays law, produ ces t h e fo llowin g relat ion :

h k

2E = 1c d p d t

E + d m

d t F + 4 c J ( 3 . 1 7 )

w h e r e J is t h e cu r r en t d en s i t y, an d t h e o r t h o go n a l it y of t h e m o d es h a s

a l lo wed t h e s u mm at i on t o b e s u p p re s s ed (i. e . t h e eq u a t i on h o l d s fo r a l l

). In fa c t , w e c a n n o w c o n s id e r a s i n g l e m o d e t o d o m in a t e . If w e

n o w ex ami n e t h e eq u a t io n b y comp o n en t s ( E a n d F), d i ffe ren t i a t e wi th

res pec t to t ime , an d s impl ify, we a r r ive a t

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p k

2E - 1c d 2p d t 2

E = 4 c d Jd t ( 3 . 1 8 )

w h e r e J is u n d e r s t o od t o b e t h e z (c u r l -fr e e ) c o m p o n e n t o f t h e t o t a l

cu r r en t d en s i t y.1 6

Next , we requ ire th e wave equa t ion ,2E + k

2 E = 0 , (3 .1 9 )

in o rder to ob ta in th e re la t ion

2 + d2

d t 2p = - 4 dd t ( )J E d V

Cavity . (3 .2 0 )

Equat ion (3 .20) ho lds fo r an a rb i t ra ry cav i ty geomet ry under the

ass u mpt ion tha t th e fi elds a re cu t off in th e beam p ipe . (Th i s

a s s u m p t i on a llo ws u s t o n e gle c t t h e b e a m - p o r t lo a d i n g a n d b e a m -

wal l load ing t e rm s . ) In t e rm s o f th e vec to r po ten t i a l coeffi c ien t ,

2 + d2

d t 2p = - 4 c ( )J E d V

Cavity . (3 .2 1 )

We h ave a r r ived a t a n equ a t ion fo r th e ra t e o f en ergy flow in to a ca vity

d u e t o a b e a m w it h c u r r e n t d e n s it y J .1 7 Unfortunately, th is re lat ion

does n ot tak e in t o accou nt t h e losses of a cavity du e to a fin i te Q

value .1 8 Th e b es t w ay t o in t ro d u ce t h e s e l os s e s i s b y ad d i n g a d a mp in g

t e r m 1 9 to give,

2

+Q

dd t +

d 2

d t 2 p = -4 c ( )J E d VCavity . ( 3 . 2 2 )

Th i s can be inver t ed to g ive ,

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p = -

exp - 2 Q

( t - t ) s in ( t - t )

4 c ( )J E d V

Cavity

d t 0

t