Chapter 3 - Fields of Moving Charges

8/12/2019 Chapter 3 - Fields of Moving Charges

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C h a p t e r

F i e ld s o f M o v i n g h a r g e s

I n a r e l a t i v i s t i c a t omic co l l i s i on , bo th t he p ro j ec t i l e mo t ion w i th r e spec t t ot h e t a rg e t a n d t h e o r b i t a l m o t i o n o f i n n er - sh e ll e l e c tr o n s a r o u n d a h i g h - Zta rge t o r p ro j ec t i l e nuc l eus p roceed w i th ve loc i t i e s t ha t a r e compa rab l e t o

t h e s p e e d o f l ig h t. To a g o o d a p p r o x i m a t i o n , t h e n u c l e a r m o t i o n c a n b edesc r i bed by c l a ss ic a l mec han i c s wh i l e t he r e l a t i v i s ti c m o t ion o f i nne r- she l le l e c t rons i n h igh -Z a toms o r i ons i s de sc r i bed by t he D i r ac t heo ry.

In t h i s chap t e r, we conf ine ou r se lve s t o t he t r e a tm en t o f r e l a t i v is t i c a l lymo v ing c l as s ic a l cha rge s . W hi l e i n Ch ap t e r 2 we di s cuss t he consequenceso f L o r e n t z t r a n s f o r m a t i o n s o n e ne rg ie s a n d m o m e n t a , w e he r e c o n s id e r t h eco l l i s i on i n space and t ime and s t udy t he e l e c t romagne t i c f i e l d s p roducedby t he co l l i s i on pa r t ne r s .

A f t e r b r ie f ly s u m m a r i z i n g t h e b a si c e q u a t i o n s o f e l e c t r o d y n a m i c s , w ed i s cuss t he c l as s ic a l mo t ion o f nuc le i und e r t h e m u tu a l i n fl uence o f t he i re l ec t rom agne t i c f ie ld s. W e show tha t i t is app rop r i a t e f o r r e l a t i v is t i c a t om icp roces se s t o a s sume t ha t t he p ro j ec t i l e moves w i th a cons t an t ve loc i t y a l onga s t r a ig h t - li n e t ra j e c t o r y. C o o r d i n a t e s y s t e m s a t t a c h e d t o t h e ta rg e t a n dto t he p ro j ec t i l e nuc l eus t hen a r e i ne r t i a l f r ames , and t he Lo ren t z t r an s -fo rma t ion a s we l l a s t he Lo ren t z - t r an s fo rmed Cou lomb f i e l d s a r e exp re s sedin a s im ple form . For large value s of the Lo ren tz fac tor ~/ [Eq. 2 .14)],t he f i e l d gene ra t ed i n t he l abo ra to ry sy s t em by a mov ing cha rge i s qu i t e

s im i l a r t o a pu l s e o f e l e c t rom agne t i c r ad i a t i on . Th e r ep l ace me n t o f t he set ime -dependen t f i e l d s by e l ec t romagne t i c waves cons t i t u t e s t he ba s i s o ft he equ iva l en t -pho ton me thod , a l so known a s We iz sS~cke r-Wi l l i ams me thod ,wh ich i s d i s cus sed f i r s t i n a space - t ime and subsequen t l y i n a momen tum-space de sc r i p t i on .

35


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36 C H A P T E R 3. F I E L D S O F M O V I N G C H A R G E S

3 1 B a s i c e q u a t i o n s o f e l e c t r o d y n a m i c s

F o r l a te r r e f er e n c e a n d f o r e s t a b l i s h i n g t h e n o t a t i o n , w e s u m m a r i z e h e r es o m e o f t h e b a s ic e q u a t i o n s o f e l e c t r o d y n a m i c s a d o p t i n g G a u s s i a n u n i t s[ Ja c7 5 ]. I n t h e L o r e n t z f a m i l y o f g a u g e s , t h e w a v e e q u a t i o n s f o r t h e v e c t o rp o t e n t i a l A a n d t h e s c a l a r p o t e n t i a l (I) a r e

1 0 2 A 4 ~V 2 A = ~ j

c 2 t 2 c

1 02(I)c 2 t 2

V2(I) - 47rp, (3.1 )

w h e r e t h e c u r r e n t d e n s i t y j ( r , t ) a n d t h e c h a rg e d e n s i t y p ( r , t ) s a t is f y t h ec o n t i n u i t y e q u a t i o n

Op0--t + V . j - 0 (3.2 )

a n d t h e p o t e n t i a l s a r e s u b j e c t t o t h e L o r e n t z c o n d i t i o n

1 0 I )c Ot + V - A - O . ( 3 .3 )

T h e e l e c t r i c f ie ld E a n d t h e m a g n e t i c f i el d B a r e e x p r e s s e d i n t e r m s o f t h epo ten t i a l s A an d (I) a s

1 0 AE = V(I)

c OtB - V 3 . 4 )

I f w e n o w i n t r o d u c e t h e f o u r- p o t e n t i a l A U - ( A 0 , A ) = ((I), A ) , t h e f o u r-c u r r e n t j u - ( cp , j ) , a n d t h e f o u r- d e r i v a t i v e 0 u -O/Oxu = O/O c t ) , -V ) ,s e e E q . ( 2 .9 ) , w e c a n w r i t e E q s . ( 3 .1 ) i n t h e c o v a r i a n t f o r m

I-9 A ~ - 4_~ j , , (3.5 )c

w h e r e t h e d A l e m b e r t i a n I--1 - 0 ~ 0 ~ - ~ -] ~ 0 ~ 0 ~ . F u r t h e r m o r e , t h e c o n t i-n u i t y e q u a t i o n , E q . ( 3 . 2 ) , is r e w r i t t e n a s

O~j ~ - 0 , (3 .6 )

w h e r e w e h a v e a g a i n a d o p t e d t h e c o n v e n t i o n t o s u m o v e r s u b s c r i p t s o c c u r -r i n g t w i c e . S i m i l a rl y, t h e L o r e n t z c o n d i t i o n , E q . ( 3 .3 ) , b e c o m e s

O~A ~ -0 . (3 .7)


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3 .2 . M O T I O N O F I N T E R A C T I N G P O I N T C H A R G E S 37

Th e e l ec tr i c and ma gne t i c f ie ld s (3 .4 ) a re ob t a in ed a s t he co m po nen t so f t h e s e c o n d - r a n k , a n t i s y m m e t r i c f i e l d - s tr e n g t h t e n s o r

F ~ ' = 0 A ~' _ 0 ~'A ~ =

0 - E x - E y - E zE x 0 - B z B yE y B z 0 - B xE z - B y B x 0

a . 8 )

I n t e r m s o f t h e f i e l d - s tr e n g t h t e n s o r, t h e M a x w e l l e q u a t i o n s c a n b e w r i t t e nin covar ian t fo rm as

0 , F = 4Z j ( 3. 9)

a n d3 . 1 0 )) 'F ~ + O ' F ~ ')' + O~'F ) '~ - O,

wh ere A, # , u a re any th re e o f the in tegers O, 1 , 2 , 3.

3 2 M o t i o n o f i n t e r a c t i n g p o i n t c h a r g es

Th e p ro to ty pe sy s t e m fo r an a to mic co ll is ion is com pose d o f t h r ee bod i e s:

two nuc l e i and one e l ec t ron . S ince t he e l e c t ron i s so much l i gh t e r t han t henuc le i, i t s in f luence on t he nuc l ea r m o t ion ca n u su a l l y be i gno red . As af i r s t s t ep , we t he r e fo r e cons ide r two po in t cha rge s , t he t a rge t nuc l eus w i thc h a rg e n u m b e r Z T a n d m a s s M T a n d t h e p r o j e c t i l e n u c l e u s w i t h c h a rg en u m b e r Z p a n d m a s s M p . I n Se c. 3.2 .1 w e a s s u m e t h a t t h e m o t i o n o f t h enuc le i c an be de sc r i bed by c la s s ic a l mec han i c s an d e l ec t rod ynam ics , andin Sec. 3 .2 .2 we d i scuss the l im i ta t ion s imp osed by qua n ta l e ffec ts . Th econn ec t i on w i th a f ul l qu an tu m t r e a tm en t is p r e se n t ed i n Sec. 5. 2.

3 2 1 P a r t i c l e s c a t t e r i n g i n c l a s s i ca l m e c h a n i c sTh e s c a t t e r i ng o f c l a ss ic a l pa r t ic l e s by one an o the r i s de sc r i bed i n t e rm so f c l as s ic a l t r a j e c to r i e s . W e have i n m ind two pa r t i c l e s i n t he i r c en t e r-o f -m a s s c o o r d i n a t e s y s t e m w i t h a n i n t e r a c t i o n f or ce b e tw e e n t h e m d e fi n ed b ya p o t e n t i a l V. I t s h o u l d b e r e m a r k e d t h a t r e l a t i v i s t i c p o t e n t i a l s c a t t e r i n gin gene ra l i s a r a t he r a r t i f i c i a l cons t ruc t i on , because s t a t i c po t en t i a l s a r eno t L o ren t z i nva r i an t. How eve r, s ince e l ec t ro dyn am ics is a r e l a t i v i st i c a l lyinvar ian t theory, i t i s poss ib le to t rea t e lec t romagne t ic fo rces in a phys ica l ly

cons i s t en t ma nne r. Th i s ho ld s p r ec i s e ly fo r i on -a tom co ll is ions in wh ich wea re i n t e r e s t ed .

For de scr ib ing a c lass ica l co l l is ion of a par t i c le w i th a f ixed ta rg e t , i t isu se fu l t o i n t roduc e t he concep t o f t he im pac t p a r a m e te r and o f t he de fle c -t ion ang le . D ur in g the co l li s ion , the p ro jec t i l e ge nera l ly fo llows a curv ed


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38 C H A P T E R 3 F I E L D S O F M O V I N G C H A R G E S

J /

R { R m i

F ig u re 3 .1 . Class ical t ra jec tory of a par t ic le in the labora tory system.

t r a j e c t o r y w i t h a n in c o m i ng a n d a n o u tg o i n g b r an c h . T h e a s y m p t o t e tot h e i n c o m i n g b r a n c h is p a r a ll e l to t h e b e a m d i r e c ti o n w h il e th e a s y m p t o t eto t he ou tgo ing b r anch de f i ne s t he de f l e c t i on ang l e O w i th r e spec t t o t heb e a m d i re c t io n . T h e d i s t a n c e o f c lo s es t a p p r o a c h o f t h e i n c o m i n g a s y m p -t o t e f r o m t h e s c a t t e r i n g c e n t e r is d e n o t e d a s t h e i m p a c t p a r a m e t e r b. F i g u r e3 .1 i l l u s t r a t e s a c l a s s i c a l t r a j e c to ry fo r a r epu l s i ve Cou lomb po t en t i a l .

Fo r a gene ra l po t en t i a l , w i t h a t t r a c t i ve and r epu l s i ve pa r t s , t h e de f l e c -t i on ang l e O ( a l so den o t ed a s de f le c t ion func t i on ) m ay have any va lueb e t w e e n - o c a n d 7 r, w h e r e t h e n e g a t i v e v a lu e s o c c u r w h e n t h e t r a j e c t o r y isb e n t a r o u n d t h e s c a tt e r i n g ce n te r. O n t h e o t h e r h a n d , th e o b s e rv e d s c a t-t e r in g ang le 0 , by def in i t ion , l ie s a lways be twe en zero and 7r. Th e twoangles a re connec ted b j~

0 = + (O + 27rm), (3 .11)

wh ere m is a pos i t ive in teger o r ze ro . For a g iven def lec t ion ang le O , theva lues a re so chosen tha t the sca t te r ing ang le 0 l i e s be tween zero and 7r.

Fo r a g iven co l l i s i on ene rgy E and a g iven impac t pa r ame te r b , t hes ing le -va lued func t i on O(b ) is c a l cu l a t ed f rom the c l as s ic a l equa t i on o f mo-t ion , see , e.g ., [McC70, New66] . I f one has a bea m of par t i c les (o f equa le n e rg y ) u n i f o r m l y d i s t r i b u t e d o v e r a l l i m p a c t p a r a m e t e r s a n d a l l a z i m u t h a lang les , the d i ffe ren t ia l c ross sec t ion may be ca lcu la ted as

d~ bd a s in 0I d O / d b 3.12)

where b (O ) i s ob t a ined by i nve r t i ng t he o r i g ina l l y c a l cu l a t ed func t i on O(b ) .


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3.2. M O T I O N O F I N T E R A C T I N G P O I N T C H A R G E S 39

Coll is ions between two bare nucle i

T h e r e s u l t s o b t a i n e d f or p o t e n t i a l s c a t t e r i n g c a n b e a p p l i e d t o t h e r e l a t i v em o t i o n o f t w o p a r t i c le s i n t e r a c t i n g v i a a p o t e n t i a l . F o r th e p u r e l y r e p u l si v ei n t e r a c t i o n b e t w e e n t w o p o s i t i v e p o i n t c h a rg e s i n w h i c h w e a r e i n t e r e s t e d ,de f l e c t i on ang l e 0 and s ca t t e r i ng ang l e 0 a r e iden t i c a l and , w i th in c l a ss i -c a l m e c h a n i c s , t h e r e is a o n e - t o - o n e r e l a t i o n s h i p b e t w e e n t h e i m p a c t p a -r a m e t e r a n d t h e s c a t te r i n g a n gl e. S m a l l i m p a c t p a r a m e t e r s c o r r e s p o n d tol a rg e s c a t t e r i n g a n g l e s a n d v i c e v e r s a. H o w e v e r, a s is k n o w n f r o m R u t h e r -fo rd ' s c l a s s i c s ca t t e r i ng expe r imen t f o r p ro j ec t i l e ene rg i e s a s l ow a s abou t1 M eV /u , l a rge de f l e c ti on ang l e s o f t he o rde r o f 7 r/ 2 co r r e spo nd t o im-

p a c t p a r a m e t e r s o f t h e o r d e r a f ew f in ( = 1 0 - 1 3 c m ) . O n t h e o t h e r h a n d ,t he l eng th s ca l e f o r a t omic p roce s se s i s g iven by t he a tomic K- she l l r ad iu sa K Z l a o Z 1 0 . 5 3 1 0 - S c m w h e r e Z is t h e n u c l e a r c h a rg e n u m b e ri nv o lv e d . S i n ce t h e c o n t r i b u t i o n t o th e c r o s s s e c t io n o f a n i m p a c t p a r a m -e t e r r a n g e s b e t w e e n b a n db db i s w e i g h t e d w i t h27r b db , w e e x p e c t t h a ti m p a c t p a r a m e t e r s o f t h e o r d e r o f a fe w fm w h i c h a r e a s s o c i a t e d w i t h l a rgede f l ec t ion ang l e s g ive a neg l i g ib l e con t r i b u t i on . Mo reove r, f o r a t om ic co ll i-s i o n s t u d i e s , t h e s e s m a l l i m p a c t p a r a m e t e r s h a v e t o b e a v o i d e d b e c a u s e t h eb a c k g r o u n d f r o m n u c l e a r r e a c t i o n s w o u l d t e n d t o m a s k a t o m i c p r o c e s s e s .

Th e re fo r e , a s i de f rom d ev i a t i on s d i s cus sed i n Sec.3 .4 .2, i t is pe r f e c t l y ju s t i -f i ed fo r m os t pu rpose s t o d i s r ega rd t he f i n i t e nuc l ea r s i z e and t o subs t i t u t ea pos i t i ve po in t cha rge fo r t he nuc l eus .

F o r t h e i m p a c t p a r a m e t e r s r e l e v a n t to m o s t a t o m i c p ro c e s se s , t h is m e a n st h a t t h e t a rg e t n u c l e u s e ff e c ti v el y s t a y s a t r e s t i n t h e l a b o r a t o r y s y s t e mt h r o u g h o u t t h e c ol li si on , t h e r e b y p r o d u c i n g a s t a t i c C o u l o m b p o t e n t i a l . I tis t h e n c o n v e n i e n t t o e x p r e s s a ll q u a n t i t i e s i n t h e l a b o r a t o r y ( t a rg e t ) s y s t e m .F o l lo w i n g S o m m e r f e l d ' s c la s si c t r e a t m e n t [S om 3 1] o f t h e r e l a t i v i s t ic K e p l e ro rb i t , one m ay de r i ve [MaS87 , New66] t he de f l e c t ion o f a p ro j ec t i l e o f ma s sMp and k i ne t i c ene rgy Tp by a space - f i xed cha rge Zpe ( i . e . , w i t h an i n f i n i t et a rg e t m a s s ) a s a f u n c t i o n o f t h e i m p a c t p a r a m e t e r b . A l t e r n a t iv e l y, o n em a y c o n s i d e r t w o p a r ti c l e s in t h e i r c e n t e r - o f -m a s s s y s t e m . T h e s y m b o l M pt h e n h a s t h e m e a n i n g o f t h e r e d u c e d m a s s o f t h e t w o p a r t ic l e s , a n d t h ede f l ec t i on ang l e r e f e r s t o t he cm f r ame .

I t is o f t en conve n i en t t o ex p re s s t he Lo ren t z f a c to r y d i r ec t l y by t hek i n e t ic e n e rg y o f t h e p r o j e c t i le a s

1 = 1 + Tp [M eV /u] (3 .13)

V / I _ ~ 2 9 3 1 . 4 9 4

a n d t o w r i t e t h e d e f l e c t i o n a n g l e a s a f u n c t i o n o f t h e r a t i o

Z p Z Te 2 / b e 2X Mpc 2 = h c Z p Z T b (3 .14)


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40 C H A P T E R 3. F IE L D S O F M O V I N G C H A R G E S

be twee n t he Co u lom b r epu l s i on ene rgy o f t he nuc l e i a t t he s epa ra t i on b andthe r e s t ene rgy o f t he p ro j ec ti l e . I n t he s econd equa t i on , ) =h / M p c i s theC om pto n w ave l eng th o f t he p ro j ec t i le ( it is mo re conven i en t t h i s exp re s s ion ,r a t h e r t h a n h / M p c ) . U s i n g t h e a b b r e v i a t i o n

~/ xe (3 .15)r / - 1 3 ` 2 1 ,

one ob t a in s t h e de f le c t ion ang l e O a s a f unc t i on o f t he p a r a m e te r s 3` and xa s

2O - - 0 = T r

or for x 1

- - a r c t a n ( 3 ` 2 _ 1 ) r/ , ( 3. 16 )3`x

X0 ~ 2 - . ( 3 . 1 7 )

3'Le t u s cons ide r U + U col li si ons a s an example . Th e C om pto n wave -

leng th ( in the cm f ram e) fo r th i s sys te m is ) = 1 .78 x 10 -3 fm. Fro mEqs . (3 .14) and (3 .17) we the n ob ta in th e cm def lec t ion ang le

0.220 3.83 x 10 -40 - = (3.18)

3` b [ f m ] 3`(b/ag)wh ere i n t he f i r st equa t i on , b is m easu red i n fm , wh i l e in t he s econd equ a t i oni t is m easu red i n un i t s o f t he u r an ium K-she l l r ad iu s aK . I t is s een t ha td o w n t o i m p a c t p a r a m e t e r s b v e r y sm a l l o n th e a t o m i c s c ale , th e C o u l o m bdefle ct ion is negl igible for re la t iv is t ic col l is ions . Ev en for b = 15 fm, wh ent h e u r a n i u m n u cl ei b e g i n to t o u c h , a n d f or a k i n e ti c e n e rg y o f 1 G e V / uin t he cm sys t em, t he more accu ra t e e s t ima t e u s ing Eq . ( 3 . 16 ) y i e ld s ade f l e c t i on o f on ly 9 mrad .

Effects of retardation and of magnetic fields

T h e e s t i m a t e s g i v e n a b o v e a r e b a s e d o n p o t e n t i a l s c a t t e r i n g w i t h a p u r eCo u lom b r epu l s i on , t h a t is , e f fec ts o f r e t a rda t i on and o f ma gne t i c f ie ld sa r e d i s r ega rded . Le t us ske t ch he r e a more comp le t e t r e a tm en t [MaS87 ,M aF90] . Desc r i b ing t he co ll is ion i n t he l ab o ra to ry sy s t em , the t a rg e t nu -c le u s e x pe r ie n c e s t h e L i 6 n a r d - Wi e c h e r t p o t e n t i a l p r o d u c e d b y t h e p r o j e c ti lem o v i n g a l o n g it s t r a j e c t o r y X p = x p ( t ) . A t t h e p o s i t io n x T o f t h e t a rg e tnuc leus , th i s po ten t ia l has the fo rm [Jac75]

Zpe )I ) p X T , t ) - - (1 - ~ p - n ) R re t

)p X T , t ) - - 1 - - ~ p : n ) / ~ re t (3.19)


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3.2. M O T I O N O F I N T E R A C T I N G P O I N T C H A R G E S 41

Here R = x T t ) - - X p t r e t )is t he s e pa ra t i on o f t he t a r ge t nuc l eus f romt h e p r o j e c t i l e a t t h e r e t a r d e d t i m et re t = t R//C n i s the un i t vec tor inthe d i r ec t i on o f R , and ~p =v p / c i s t he r educed p ro j ec t i l e ve loc i t y a t t her e t a rde d t ime . Th e e l ec t ri c and mag ne t i c f ie ld s r e su l t i ng f rom the po t en t i a l s(3 .19 ) de t e rm ine t h e t a rg e t m o t ion . Th e t a rg e t n uc l eus is i n i ti a l ly ( a t t =- o o ) a t r e s t a n d s t a r t s m o v i n g as t h e r e p u l s io n f r o m th e p r o j e c t il e b e c o m e seffec t ive . Conv erse ly, the reco i l ing ta rg e t nuc leu s a t th e p os i t ion x T is thes o u rc e o f L i 6 n a r d - W i e c h e r t p o t e n t i a l sI ) T a n d A T a c t i n g o n t h e m o v i n gp ro j ec t i le . I n t h i s way, t he m u tu a l p ro j ec t i l e - t a rge t i n t e r ac t i on l e ads t o ase t o f e igh t coup led l inear d i ffe ren tia l equa t ion s [MaS87, MaF90] fo r then u c l e a r m o t i o n w h i c h m a y b e s o lv e d n u m e r ic a ll y. A r i g o ro u s t r e a t m e n t o ft h i s k ind t ake s i n to a ccou n t t he e f fec ts of r e t a rda t i on and ma gne t i c f ie ld scaused by t he r eco i l i ng t a rge t .

As one migh t expec t , i t t u rn s ou t t ha t magne t i c f i e l d co r r ec t i ons t o t hecross sec t ion fo r e las t i c sca t t e r i ng be twe en nuc le i a re sma l l (


8/25


t e r m s o f t h e u n c e r t a i n t y A p o f t h e tr a n s v e r s e m o m e n t u m w i t h r e sp e c t tot h e l o n g i t u d i n a l m o m e n t u mp z , the un cer ta in ty re la t ion A p A b _> h y ields

A 0 - A p > 1 h _- h 3 .21)p z - p z A b 7 M p c A b

w h e n c eAb 1 h>

b - A b 3 , M p c O

Inse r t ing the angle 0 f rom Eq. 3 .20) we have

k 2 Z p Z Te2

In order to ob ta in suffic ien t p rec is ion in b, the r ig h t -h an d s ide of Eq . 3 .22)m us t be suffic ien t ly smal l com pa red to un i ty. Th is requ i rem ent i s conve-n i e n t ly e x p r e s s ed w i t h t h e a id o f t h e S o m m e r fe l d p a r a m e t e r

Zp ZT e 2 Ze ZTu ~ = h c = 137 3.23)

wi th t he p ro j ec t il e ve loc i ty r ep laced by t he speed o f l igh t . I n add i t i onto t ak ing i n to accoun t t he l im i t a t i ons impo sed by Eq . 3 .22 ), we have t or equ i r e t ha t t he ene rgy l o ss A E o f t he pa r t i c l e s du r ing t he i on -a tom co ll is ionis neg l ig ib le . In sum m ary, th e con di t ions for the va l id i ty o f the co ncep t o fc lass ica l t ra jec tor ies a re

2Uc >> 1A EE


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3.3. L I I ~ N A R D - W I E C H E RT P O T E N T I A L S 43

ion -a to m co l li s ions [McC70 , Han90 ] t ha t a s emic l a s s i c a l app rox im a t io n , i n

w h i c h t h e p r o j e c t i le m o t i o n i s t r e a t e d c l a ss ic a l ly b u t t h e e l e c t r o n m o t i o nq u a n t u m m e c h a n i c a l l y ( se e S e es . 1.1 a n d 5 .2 ) , is a n e x t r e m e l y p o w e r f u lm e t h o d e v en w h e nu = Z p Z Te 2 / h v < 1. T he u t i l i t y of t h i s concep t r e s t so n th e B o r n - O p p e n h e i m e r s e p a r a t i o n w h i c h a ll ow s o ne t o s e p a r a t e c o n si d-e r a t i o n o f e l e c t ro n i c s t a t e s f r o m c o n s i d e r a t i o n o f t h e n u c l e i of t h e a t o m i cs y s t e m s t a k i n g p a r t i n t h e a t o m i c c o ll is io n . S i nc e t h e n u c l e a r m a s s e s M pa n d M T a r e l a rg e c o m p a r e d t o t h e e l e c t r o n i c m a s s m e , t h e e n e rg y t r a n s -f e r r ed t o t h e i n t e r n a l d e g r e e s o f f r e e d o m o f t h e a t o m s is m u c h s m a l l e r t h a nt h e i n i ti a l k i n e ti c e n e rg y o f t h e r e l a t iv e m o t i o n o f t a rg e t a n d p r o j e c ti l e .

Consequen t l y, t he r e l a t i ve ve loc i t y p r ac t i c a l l y does no t change i n t he co l l i -s i on and s ca t t e r i ng i s e s s en t i a l l y c l a s s i c a l .

T h e s e p a r a t i o n o f n u c l e ar a n d e l ec t ro n i c m o t i o n m e a n s t h a t t h e e l e c tr o nm a y b e r e g a r d e d t o m o v e i n a t i m e - d e p e n d e n t p o t e n t i a l , t h e t i m e d e p e n -dence a r i s i ng f rom the m o t ion o f t he nuc l e i ( see See . 5 . 1) . Th e m e th od isg e n e r a ll y c a ll ed t h e i m p a c t p a r a m e t e r t r e a t m e n t o r t h e i m p a c t p a r a m e -t e r p i c t u r e b e c a u s e t h e i m p a c t p a r a m e t e r b s e rv e s t o s p e c if y t h e t r a j e c t o r ye v e n t h o u g h t h e l a t t e r m a y n o t b e d i r e c tl y m e a s u r a b l e . I m p a c t p a r a m e t e rm e t h o d s h a v e a s t r o n g a p p e a l i n a i d i n g t h e p h y s i c a l i n t u it i o n . U s u a l ly,

c a l c u la t io n s b a s e d o n t h is m e t h o d a r e m a t h e m a t i c a l l y m o r e tr a c t a b l e t h a nf ul ly q u a n t u m m e c h a n i c a l t r e a t m e n t s . B o t h m e t h o d s y i el d i d e n ti c a l r e s ul tsfo r qu i t e g ene ra l c a se s o f i on -a tom co ll is ions .

I n t h e i m p a c t p a r a m e t e r p i c t u r e , t h e c l a s s i c a l o n e - t o - o n e r e l a t i o n b e -t w e e n s c a t t e r i n g a n g l e 0 a n d i m p a c t p a r a m e t e r b is r e p l a c e d w i t h a n in -t eg r a l ( s ee See. 5 .4 . 2) ove r t he r e l evan t am p l i t u des i n an a tom ic p roce s s .T h a t i s , t h e s c a t t e r i n g a m p l i t u d efn O), Eq. (5 .85) , be tween in i t i a l and f i -na l s t a t e s i , f f o r a g iven ang l e 0 r e ce ive s con t r i bu t i o ns f rom the t r an s i t i ona m p l i t u d e s An b) f or a ll i m p a c t p a r a m e t e r s . C o n v e r se l y, t h e i m p a c t p a r a m -e t e r d e p e n d e n c e o f A n (b) c a n b e r e c o n s t r u c t e d , if t h e s c a t t e r i n g a m p l i t u d e sfn O), i n c lud ing t he phase s , a r e known fo r a l l ang l e s .

S in c e t h e i m p a c t p a r a m e t e r m e t h o d is i n t i m a t e l y c o n n e c t e d w i t h t h ee lec t ron ic mot ion , we defe r a d i scuss ion to Sec . 5 .1 . Here , i t may suff ice tos a y t h a t i n c o n j u n c t i o n w i t h th e i m p a c t p a r a m e t e r p i c t u r e , a c l a ss ic a l t r e a t -m e n t o f t h e n u c l e a r m o t i o n is a p p r o p r i a t e f or re l a t i v is t i c a t o m i c c o l li si on sup t o a h igh l eve l o f a ccu racy.

3 .3 L i nard W iecher t p o t e n t i a l s

As a conseque nce o f t he e s t im a t e s g iven in Sec . 3 .2 , we a s sum e in t hef o l l o w i n g t h a t t h e p r o j e c t i l e m o v e s a l o n g a c l a s s i c a l r e c t i l i n e a r t r a j e c t o r y


10/25


w

Z T T x

L . J

~v

t

L

F ig u re 3 .2 . Coord ina te sys tems for a re l a t iv i s ti c co ll is ion be tween two a toms .Th e po s i t ion of the ta rge t nucleus (charge ZT) i s chosen as the or ig in of thecoord inates x , y, z . Th e projec t i le nucleus (charge Zp ) moves wi th th e co nstan tveloci ty v para l le l to the z-axis a long a t ra jec tory tha t i s d isplaced f rom the targ et

by the im pac t par am ete r b a long the x-axis . Th e projec t i le is located a t the or ig inof the moving iner t ia l f ram e w i th the coordina tes x , y~, z ~. Th e e lec t ron e- hasthe coord ina te r T wi th r e spec t to the t a rg e t f r ame and rp wi th r e spec t to theprojec t i le f rame. Owing to the i r def in i t ion in d i fferent iner t ia l f rames associa tedwith d i fferent t imes , the vectors R (def ined in the target f rame) and r~ (def inedin the projec t i le f rame)do no t add vector ia l ly to rT as might be sugges ted by thefigure.

w h i c h i n t h e l a b o r a t o r y s y s t e m i s g i v e n b y

R - b + v t , (3 .25)

w h e r e b i s t h e i m p a c t p a r a m e t e r v e c t o r a n d v i s a c o n s t a n t v e l o ci ty. T h i st r e a t m e n t o f t h e p r o j e c t i le m o t i o n i m p l i es a n i m p o r t a n t s i m p l i f ic a t i o n fo rh e a v y - i o n c o l li si on s w i t h a n e n e rg y e x c e e d i n g a f ew M e V / u . W h e n d e f i n i n gt h e c o o r d i n a t e s y s t e m s , i t is c o n v e n i e n t t o p l ac e t h e t a rg e t n u c l e u s a t t h eo r i gi n o f t h e l a b o r a t o r y s y s t e m w i t h t h e x a n d z a x e s t a k e n i n t h e d i r e c t i o n s

o f b a n d v, r e sp e c t i v e l y. F i g u r e 3 .2 i l l u s t r a t e s t h e g e o m e t r y. T h e i n e r t i a lf r a m e , i n w h i c h t h e p r o j e c t i l e n u c l e u s i s a t r e s t , m o v e s w i t h t h e s p e e d vw i t h r e s p e c t t o t h e l a b o r a t o r y f r a m e , a n d i t s x ~, y~, z ~ a x e s a r e p a r a l l e l t o t h ex , y, z l a b o r a t o r y a x e s , r e s p e c t i v e l y. F o r t h e t i m e c o o r d i n a t e s , w e c h o o set h e c o n v e n t i o n t h a t t h e t i m e s t o, t~ a s s o c i a t e d w i t h t h e n u c l e a r p o s i t i o n s


11/25

3 .3 . L I E N A R D - W I E C H E R T P O T E N T I A L S 45

a r e b o t h z e r o w h e n t h e p r o j e c t i l e r e a c h e s i t s c l o s e s t a p p r o a c h t o t h e t a rg e t

n u c l e u s d u r i n g t h e c o l l i s i o n .L e t

a n d

X - - X = ( x O , x l , x 2 , x 3 ) :(C t, x , y , z ) o r ( c t , r )

X = X r - - ( x tO , x t l , x t 2 , X ' 3 ) - -(c t t , x t , y t , z t ) or (c t ' , r ' )

(3 .26)

(3 .27)

d e n o t e t h e s p a c e - t im e c o o r d i n a t e s o f a p o i n t w i t h r e s p e c t t o t h e t w o L o r e n t zf r a m e s . I n a n a l o g y t o E q . ( 2 .1 0 ), t h e s e c o o r d i n a t e s a r e r e l a t e d b y t h e i n -

h o m o g e n e o u s L o r e n t z t r a n s f o r m a t i o n

(3.28)i P = A P t ~ z V - 5 l i tb ,

w h e r e w e a d o p t t h e c o n v e n t i o n t o s u m o v e r s u b s c r i p t s o c c u r r i n g t w i c e a n dw h e r e t h e A S s a t is f y t h e o r t h o g o n a l i t y r e la t i o n ( 2 .11 ) . D e f i n in g t h e u n i tv e c t o r f i = v / v , E q . ( 3 .2 8 ) c a n a l so b e w r i t t e n i n v e c t o r f o r m a s

r ' = r + ( 7 - 1 ) ( r . 9 ) ~ r - T v t - b

t ' = 7 ( t - r ' v / c 2 ) ,

o r , f o r t h e i n v e r s e t r a n s f o r m a t i o n ,

r = r ' + ( 7 - 1 ) ( r ' . ~ r)~ r + 7 v t ' + b

t = 7 ( t ' + r v / c 2 ) .

(3 .29)

(3 .30)

C h o o s i n g t h e z - a x i s in t h e d i r e c t i o n o f v, w e h a v e e x p l i c i tl y

c t 7 0 0 - 3 7 c t 0x ' _ 0 1 0 0 x _ b (3 .31)

y ' 0 0 1 0 y 0 'z ' - 3 7 0 0 7 z 0

a n d t h e d i s t a n c e o f a p o i n t ( x , y, z , t ) f r o m t h e o r i g in o f t h e m o v i n g f r a m e ,m e a s u r e d i n t h e m o v i n g f r a m e , i s

r ' - V/(x - b) 2 + y2 + 72 (Z_ v t i 2 . (3 .32)

T h e s a m e t r a n s f o r m a t i o n s a ll ow u s t o c o n s t r u c t t h e f o u r - p o t e n t i a l g e n er -a t e d b y a u n i f o r m l y m o v i n g c h a rg e . 1 In t h e p r o j e c t i l e f r a m e , t h e p r o j e c t i l e

c h a rg e Z p e g i v e s r i s e t o t h e e l e c t r o s t a t i c p o t e n t i a l s(I) ' (r ' t ) - Z p e A ,' r ' ' ( r ' t ' ) - 0 . (3 .33)

1 We a lw a y s ta k e t h e c h a rg e e > 0 , i n a g r e e m e n t w i t h [ J ac 7 5 , R o s 6 1] a n d o p p o s i t e t o[BjD64] and [Sak67] .


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46 C H A P T E R 3 . F I E L D S O F M O V I N G C H A R G E S

By the de f in ing r e l a t i on (2 .7 ) o f a f ou r-vec to r, t h e co r r e spo nd in g fou r-p o t e n t i a l s (I) a n d A i n t h e t a rg e t f r a m e a r e o b t a i n e d b y th e h o m o g e n e o u sp a r t o f t h e s a m e L o r e n t z t r a n s f o r m a t i o n ( 3.3 0 ) w h i ch c a r ri e s t h e f o u r-c o o r d i n a t e s t ' a n d r ' i n t o t a n d r. B y s im p l e s u b s t i t u t i o n , w e d e r iv e th eL i 6 n a r d - W i e c h e r t p o t e n t i a l s p r o d u c e d b y t h e p r o j e c ti l e in t h e t a rg e t f r a m ein t he fo rms

9 ZpeI ) r , t ) =

v / x - b ) : + y : + - v t)v

A (r , t) = - (I)(r, t) (3.34)

wh ich s a t i s fy t he Lo ren t z gauge con d i t i on (3 .3 ). I n pa r t i cu l a r, a t t he o r ig ino f t h e t a rg e t f ra m e , t h e e l e c tr ic a n d m a g n e t i c f ie ld s d u e t o t h e m o v i n gp ro j ec t i l e a r e g iven by

Z p e y bEX --

(b 2 + ~ / 2 v 2 t 2 ) 3 / 2

E v - 0

Z p e 7 v t

E z = - ( b ~ + ~ 2 v 2 t 2 ) 3 / ~ (3 .35)

a n d

gx ~ 0

B y F z

B z - 0, (3.36)

r e spec t ive ly. W e see t h a t t he pea k t r ans ve r se e l ec tr i c fi eld Ex is i nc r ea sed

by a fac to r o f 7 w i th r e spec t t o i t s non re l a t i v i s t i c va lue wh i l e t he d u ra t i o n

3 . 3 7 )

of app rec i ab l e f ie ld s t r eng th a t t he t a rg e t nuc l eus i s dec rea sed by t he s amea m o u n t . We n o t e t h a t t h e r a t ioE x / E z = - b / ( v t ) is j u s t t h e t a n g e n t o ft h e an g l e 0 - a r c c o s ( - R - 9 ) f o rm e d b e t w e e n t h e v e c to r s - R a n d t h ez -ax i s (F ig . 3 .1 ) . Hence t he e l ec t r ic fi eld p ro du ced by t he cha rgeZ p e a tt h e p o s i t i o n o f t h e t a rg e t n u c l e u s i s d i r e c t e d r a d i a l l y f r o m t h e p r o j e c t i le ' sp r e s e n t ( i. e., n o t r e t a r d e d ) p o s i t i o n t o t h e o b s e r v a t i o n p o i n t a t t h e t a rg e tnuc l eus . W r i t i ng b = R s in 0 andv t = R cos 0 , we ob ta in the e lec t r ic f i e ld

- Z p e RE - .y2R3 ( 1 _ /32 sin2 0)3/2 . (3.38)


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3 .4 . E Q U I VA L E N T P H O T O N M E T H O D 47

u 3 ~

F ig u re 3 .3 . Polar d iagrams for the angular dependence of the radia l e lect r icfield strength produced by a point charge moving with the velocity v to the r ight.The numbers give the Lorentz factors -y.

T h e a n g u l a r v a r i a t i o n o f t h e r a d i a l e l e c t ri c fi el d s t r e n g t h is i l l u s t r a t e din F ig . 3 .3 fo r va r ious p r o jec t i l e ve loc i t i e s in t e r m s o f the Lo ren tz fac to r

~ . Along th e d i re c t ion o f m ot ion , 0 = 0 o r 0 = 7r, the f ie ld s t r en g th i sd e c r e a s e d b y a f a c t o r of ~ /- 2 as c o m p a r e d t o a c h a rg e a t r e s t . O n t h e o t h e rh a n d , p e r p e n d i c u l a r t o t h e t r a j e c t o r y , 0 = 7 r/ 2, t h e f ie ld is i n c r e a s e d b y af a c t o r o f ? . T h e f l a t t e n i n g o f t h e s u r f a c e s i n t o d is k s h a p e s is e s s e n t i a l ly a ne f fe c t o f t h e L o r e n t z c o n t r a c t i o n o f t h e e l e c t r o m a g n e t i c f ie ld s.

3 . 4 E q u i v a l e n t p h o t o n m e t h o d

T h e e l e c t r i c f i e l d p r o d u c e d b y a m o v i n g c h a rg e b e c o m e s a l m o s t t r a n s v e r s ei f ~ >> 1 , see Eqs . 3 .35) and 3 .36) , and i s acc om pa nie d by a t r an sve rsem a g n e t i c f ie ld p e r p e n d i c u l a r t o i t a n d o f a l m o s t e q u a l s t r e n g t h . T h e s e e le c -t r i c a n d m a g n e t i c f i el ds c a n b e r e p l a c e d a p p r o x i m a t e l y w i t h t h e f ie ld s o f ap u l s e of p l a n e l i n e a r l y p o l a r iz e d r a d i a t i o n p r o p a g a t i n g i n t h e z - d i r e c t i o n .


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Thi s r ep l acem en t fo rms the bas is o f t he equ iv a l en t -pho ton m e th od w h ichis o r ig ina l ly due to Ferm i [Fer24] and was dev e loped by von W eizs /icker[Wei34] and W il l iams [Wi134, Wi135]. Th e m eth od can be appl ied to a va-r ie ty of e lec t ro m agn et ic processes indu ced b y charged pro jec t i l es in nuc le iand a tom s , such a s exc i t a t i on o f a toms and nucl ei , i on i za t ion o f a tom s ,emis s ion o f b r em ss t r ah lung , and the c r ea t i on o f l ep ton pa ir s . These ando the r a pp l i ca t i ons have been r ev iewed by Be r tu l an i and Bau r [BeB88]. TheWeizs / i cke r-Wi l l i ams o r equ iva l en t -pho ton me thod i s va luab le a t ex t r eme lyhigh energ ies in s i tua t ions where r igorous methods a re d i ff icu l t to imple-m ent . I t is the n poss ib le to use the k now n cross sec t ions for ph oto n- in du ced

reac t ions t o e s t ima te t he co r r e spond ing c ros s s ec t i ons i nduced by mov ingcharges, see Sec.10.2.3.In Sec . 3.4.1 we fol low the in tu i t ive ly a ppe a l ing space- t im e d escr ip t ion of

von Weizs / i cke r and Wi l l i ams in o rde r t o exp l i c i t l y exh ib i t t he dependenceo f t he f ie ld s and o f t he f r equency spec t r a on the imp ac t pa r am e te r b. I nSec . 3.4.2 we tu rn i n s t ead to t he sho r t e r b u t more fo rma l de r iva t ion o f t hewave-number descr ip t ion which leads to essen t ia l ly the same f ina l resu l t fo rthe ph o to n spec t rum . The l a t t e r ha s t he advan tag e t ha t f in i te - si ze ef fec ts o fthe cha rge d i s t r i bu t ion can be accou n ted for by i n t roduc ing a fo rm fac to r i n

a n a t u r a l w ay. T h e i m p a c t - p a r a m e t e r d e p e n d e n c e o f t h e f r e qu e n c y s p e c t racan then be der ived for any g iven form fac tor.

3 . 4 . 1 F o r m u l a t i o n i n t h e s p a c e r e p r e s e n t a t i o n

B a s i c a s s u m p t i o n s

The ma in i dea beh ind the equ iva l en t -pho ton me thod i s t o r ep l ace t he t r an -

s ien t f ie lds 3 .35) and 3 .36) by pulses of l inear ly po la r ized rad ia t io n . Th ef ie ld prod uce d a t a po in t X by the m oving par t ic le X ~ is ind ic a ted inFig . 3.4. As has been m ent io ned above , the t im e in te rva l for which thef ie lds are ap prec iable are of the orde r A t ~_b i l l Y. Dur ing th i s t ime ,I E zl : I E x l v t / b I E x l /

Ima gine now a pu lse P1 of p lane ra d ia t i on t rav e l ing para l le l to v wi than e lec t r ic field Ex t ) and a ma gne t ic f ie ld By t ) w i th IByl =l e v i xand an o the r pu lse P2 t r ave l ing pe rpend icu l a r t o v and Ex t ) t ha t is i nthe pos i t ive or nega t ive y -d i rec t ion) wi th an e lec t r ic f ie ld equa l to Ez t )

and a magn e t i c f ie ld Bx t ) w i thI B x l= IEz l . These rad ia t ion pu lses a recon s t ruc t ed t o accu ra t e ly r ep rodu ce the e l ec t ri c fie lda . a a )or ig ina t ing f romX ~ a t t he po in t X , bu t t hey do no t r ep rodu ce the m agne t i c f ie ld3 . 3 6 ) .

The ma gne t i c f ie lds o f t he pu l se s dev ia t e f rom those o f t he mo v ing cha rgeb y a c o m p o n e n t ] pulse _ _ y R c h a r g e II E x l 1 - ~ )in the y-d i rec t ion and by a


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3.4. E Q U I VA L E N T- P H O T O N M E T H O D 49

l

v

E X

- ~ E z

f - Ou

F ig u re 3 .4 . The electromagnet ic fie lds (3 .35) and (3 .36) produced at the pointX by the charge at X mov ing with velocity v.

com po nen t IB pulse- B c h a r g e I ~ ~I z l ~ IExl / yin the x-d i rec t ion . S ince ther a t i o o f ma gne t i c t o e l ec tr ic f o rces on a pe r tu r be d sys t em ( e.g., an a tom )loca ted a t po in t X is o f the ord eru / c , where u i s a represen ta t ive ve loc i tywi th in t he pe r tu rbed sys t em, i t f o l l ows t ha t t he e r ro r s i n t he fo r ce s a r i s i ngf rom the m i s in t e rp re t a t i on o f t he m agne t i c f ie ld o f t he m ov ing cha rge bythe pu l s e s P I and P2 a r e o f t he o rde ru / cT. These e f f ec t s may be neg l ec t edif u / c 7 > 1.

Ano the r e r ro r i n t he r ep l acemen t i s c aused by t he f ac t t ha t t he f i e ldp roduced by t he mov ing cha rge va r i e s i n t he t r ansve r se d i r ec t i on w i th inthe space reg ion occup ied by the pe r tu rb ed sys t em. Thu s , i f Ab is i tst rans vers e ex tens ion , we have to requ i re tha t Ab


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A I 1

B )

E

b

t = 0

v

v ~

tE

P P P

F ig u re 3 .5 . A projecti le wi th veloci ty v follows a t ra jectory shifted by the imp actpara me ter b wi th respect to the or igin of the target system indicated by a dot . ) .The arrows indicate the electric f ields at posit ions before and after the projecti lereaches the dis tance of c losest approach. A) Fie ld s t rength acco mpan ying theprojecti le, see Fig.3.3. B) Eq uivale nt-ph oton pulses P1 and P2 pro pag ating inthe z and y directions perp endicu lar to the plane of the figure), respectively.

f lux in the f ie ld o f v i r tu a l p ho ton s i s g iven by

S = c E x B )47r

c

= 4--4 + E z ) 3.39)

wh e re ~z is t he u n i t ve c to r i n t he z -d i r ec t i on an d i s a s soc i a t ed w i th apacke t o f p l ane waves po l a r iz ed i n t he x -d i r ec t i on an d p r op aga t i ng i n t hez -d i r ec t ion . I t r ep r e sen t s t he pu l s e P1 . Th e second t e rm in Eq . 3 .39 )w i t h t h e u n i t v e c t o r + ~ y c o r r e s p o n d s t o t h e p r o p a g a t i o n i n t h e p o s i t i v e o r

nega t i ve y -d i r ec t i on and t o a po l a r i z a t i on i n t he z -d i r ec t i on . I t r ep r e sen t sthe pu lse P2 . I t tu rn s o u t th a t fo r h igh pro jec t i l e ve loc i ti es wi th ~, >> 1,t he pu l s e P2 is o f m ino r imp or t a nce and can be d is ca rded , s ee Eq . 3 . 46 ) ,bu t f o r t he t ime be ing , we r e t a in t he pu l s e P2. Bo th pu l s e s o f r ad i a t i on a r eschemat ica l ly i l lus t ra ted in F ig . 3 .5 .


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3 .4 . E Q U I VA L E N T- P H O T O N M E T H O D 51

In order to ana lyze th e t ran s ien t f ie lds in te rm s of ph oto n spe c t ra , i t i snecessary to in t ro duc e the Four ie r t ran sfor m of the e lec t ric f ie ld as

faJ) - 1 E ( t ) e i ~ t d t 3.40)and i ts inverse

? E a~)e_i~tdaj 3 .41)E t ) - x / ~ oo

and c o r r e spond ing r e l a t ions fo r t he m agne t i c f ie ld B . W e d is t i ngu i sh thefunc t ion f rom i t s t r ans fo rm by exp l ic i t ly d i sp l ay ing i t s a rgum en t s ) i n ques-t ion, in this case, co an d t . Since E r, t ) is a real qu ant i ty, we ma y avoid ref-e r ence to nega t ive f r equenc ie s by no t ing tha t In -w ) l = In* w) l = In w) l .By Four ie r t r ans fo rm ing the Poyn t in g vec to r 3 .39 ), we ob ta in t he am oun to f ene rgy pe r un i t a r ea and un i t f r equency in te rva l i nc iden t on t he t a rg e tnucleu s in the c ourse of the col l ision as

CS l ~ , b ) = 2 ) I E x ~ , b ) l ~ ,

s ~ ~ b ) = 27r IEz(aJ b)[2, 3.42)

where I1 and/2 re fe r to the pu lses P1 and P2 , respec t ive ly, and a fac tor o ftwo accoun t s fo r t ak ing the f r equenc i e s t o be pos i t i ve . The dependence onthe impa c t pa ra m e te r b and the charge num ber Z we d rop the l abe l P ) isexpl ic i t ly d i sp layed in Eq . 3 .35) . Th e Four ie r t ran sfo rm

_ Ze~ ,b f ~ e i~ t d tE x (aJ,b) 3 . 4 3 )J - ~ b + ~ / 2v 2t 2) 3/ 2

can be pe r fo rme d conven ien t ly by chang ing the i n t eg ra t i on va r i ab le t o 7- =~ /v t / b and us ing the i n t eg ra l r ep re sen t a t i on o f t he mod i f i ed Besse l func t ionsK 0 a n d / 1 o f o r d e r 0 a n d1 [Erd54, AbS65]. The resul t is [Jac75]

Z eEx(a J b ) - - ~ / :

v r

cob~--~ K1 ~ v b ) l . 3 .44 )

In a s imilar fashion, one der ives

e vz ~ , b ) = - i ~ - g wb cob

The f i e ld s t r eng th E z i n Eq . 3 .45 ) has an add i t i ona l f ac to r 1 /7 co m paredto Eq . 3 .44) . Indeed , f rom the d i scuss ion fo llowing Eq. 3 .37) and f rom


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E3L

, = r

EL

E0

0c

O .

1 0

0 8

0 6

0 . 4

0 2

0 01 0 2

9 i , i , i , i i i 1 i i i I , i , i , 1 , 1 1 i i i i i I , ! , i , i , i i i i i I i

P 1 _

, I 9 J . , . ' ' ' ! : ' . ,, , I , i , , , . , I I I I I

1 0 1 1 0 o 1 0 1

R e d u ce d f r e q u e n c y

F ig u re 3 .6 . The shape o f the equ iva len t-pho ton spec t ru m fo r the pu lses P1 and

P2 as a function of ~ - w b / T vand assum ing a Lorentz fac tor 7 - 2 . The cut -offfrequ ency 3.47) corre spon ds to ~ ~ 1.

F i g . 3 . 3 , o n e s e e s t h a t t h e r a t i oI E z l / I E x l d e c r e as e s w i t h 1 / 7 . W i t h t h ed i m e n s i o n l e s s q u a n t i t y ~ - ~ b / 7 v , E q . 3 .4 2 ) y i e l d s t h e f r e q u e n c y s p e c t r a

1 Z 2 e 2-I1 W b) -- 7r2 cb 2

1 Z 2 e 2 / 3 _ 2 1b = . 3.46)

W e s e e t h a t t h e i n t e n s i t y o f t h e p u l s e P 2 is r e d u c e d b y a f a c t o r o f 7 - 2c o m p a r e d t o t h a t o f P 1 , s o t h a t f o r e x t r e m e r e l a t i v i s t i c c o l li s io n s , th i s p u l s ec a n b e i g n o r ed . F i g u r e 3 .6 s ho w s t h e f r e q u e n c y s p e c t r a o f t h e p u l s e s P1 a n dP 2 . T h e s h a p e o f t h e p u l s e P1 r e f l e c ts t h e f a c t t h a t a c c o r d i n g t o E q . 3 . 3 5 ),t h e d u r a t i o n o f t h e p a s s a g e is g i v e n b y A t _~b / T v s o t h a t f r e q u e n c ie sa p p r e c i a b l y h i g h e r t h a n

_ 7c 3 .47)2m a x m b

c a n n o t o c c u r. I n c o n t r a s t , s i n c eE z p a s s e s t h r o u g h z e r o a t t - 0 , t h e p u l s eP 2 b e h a v e s a p p r o x i m a t e l y a ss i n 2 T v t / b ) n e a r t - 0 , a n d t h e r e f o r e t h es p e c t r u m w i ll b e l o ca l iz e d a r o u n d ~ -7 v / b or ~ ~ 1. F igu re 3 .6 conf i rms


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3 .4 . E Q U IVA L E N T- P H O TO N M E T H O D 53

th a t t he i n t ens i t y o f t he pu l s e P2 is cons ide rab ly r educe d w i th r e spec t t oth a t o f P1 , even for a Loren tz fac tor as low as 7 - 2 . For l a rger va lues o f7 , the co nt r ib u t io n of P2 beco m es increas ing ly neg l ig ib le .

The cut-off frequency and the tota l radiated energy

Th e cu t -o ff f r equency g iven by Eq . 3 .47 ) depen ds on t he impac t pa r am -e t e r b. Th e ma x im um f r equency in t he pu l s e o f r ad i a t i o n i s de t e rm inedb y th e m i n i m u m i m p a c t p a r a m e t e r b m in . T h i s q u a n t i t y i s n o t a l w a y s v e r ywel l de te rm ined . For exam ple , in the co l l is ion be tw een two nuc lei , one m aytake bra in - R T - t - R Pwhere R T and R0 a r e the nuc l ea r r ad i i o f t a rge t andpro jec t i l e . S ince the ch arge d i s t r ib u t io n of a nuc leus i s no t p rec i se ly th a tof a un i fo rm ly charge d sphere , nu c lear fo rm fac tors see Sec. 3 .4.2) m ayen te r i n to t he f r equency cu t-o ff . Moreove r, the u nde r ly in g a s s um pt ion o f ar ec t i li nea r p ro j ec t i le t r a j e c to ry and o f a cons t an t f ie ld ac ros s t he t r ansve r seex t ens ion o f t he s y s t em b reaks dow n fo r such close encoun te r s . Th e p rob -lem of the f reque ncy cu t -off o f ten l im i t s the usefu lness o f the W eizs / icker-W i l li a m s m e t h o d . W i t h t h e s e r e s e rv a t io n s i n m i n d , w e m a y i n t e g r a t e t h ein t ens i ti e s 3 .42 ) ove r t he imp ac t pa r am e te r p l ane t o ge t t he t o t a l r ad i a t edenergy wi th in a g iven f requency range dc~ as

I ( w ) d w - 2 7 : f ib ]~n

[11 a;, b) + h w, b)]b db da:.

By us ing r ecu r s ion fo rmu la s [GrR80] fo r t he func t i ons K0 , K1 , and t he i rde r iva t i ve s , one can show tha t

2z2e2- [ K 1 ~ m i n ) - K 2 ~ m i n ) ] ,m i n K O ~ m i n ) K l ~ m i n ) - - - - ~ 2 i n 2

3.48)w h e r e t h e c o n t r i b u t io n s o f I1 a n d / 2 a r e n o lo n g e r s e p a r a t e d a n d w h e r e

~ m i n - -w b m i n

7 v

In the l imi t ing cases o f~min ~ 1 and ~min ~>~1 , app rox ima te ana ly t i c a l ex -

press ions for the modi f ied Besse l func t ions Ko and K1 a re ava i lab le lAbS65,GrR80] wh ich l ead t o t he e s t ima te s

I a~ )_ 2 Z2e2 [ 1 .1 2 3 7 v) ~~1 ~/v 3.49)7r cfl 2 In ~mm ~n -- for c O < b m i n


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a n d

Z2c2 ~2)-2wbmin/~V~VI (w) ~ c/32 1 - -~- e for co bmin (3.50)T h e e n e rg y s p e c t r u m 2 is d o m i n a t e d b y t h e l o w - f r e q ue n c y p a r t w i t h a t a ile x t e n d i n g t o a b o u t co ~ 2 )'v/bmin.

T h e n u m b e r o f v i r t u a l q u a n t a p e r e n e rg y i n te r v a l is o b t a i n e d f r o m t h ef r e q u e n c y s p e c t r u m b y t h e r e l a t i o n I (a ~) dc o = h a J N ( h a J) d ( h co ) , s o t h a t

N ( h ~ ) - ~ I ( ~ ) , (3.51)f t -w

a n d i n t h e l o w - f r e q u e n c y l i m i t

T h e m i n i m u m i m p a c t p a r a m e t e r b m in p l a y s t h e r o le of a p a r a m e t e r w h i c hh a s t o b e c ho s en to s u it th e p r o b l e m a t h a n d . D e p e n d i n g o n t h e s y s t e mcons idered , bminm a y b e t h e s u m o f t w o n u c l e a r r a d ii , a n a t o m i c r a d i u s , o rt h e C o m p t o n w a v e l e n g t h o f a n e l e m e n t a r y p a r t i c l e c r e a t e d i n t h e c o ll is io n .I n s o m e c a s e s , o n e m a y t r y t o u s e a d i f f e r e n t a p p r o x i m a t i o n f o r i m p a c t p a -r a m e t e r s a r o u n d o r b e l o wbmin. A s w a s s t a t e d a b o v e , t h i s i s a n u n a v o i d a b l ea m b i g u i t y i n t h e a p p l i c a t i o n o f t h e e q u i v a l e n t - p h o t o n m e t h o d .

3 . 4 . 2 F o r m u l a t i o n i n t h e w a v e - n u m b e r r e p r e s e n t a t i o n

I n S e c . 3 . 4 . 1 , w e s t a r t f r o m t h e L i ~ n a r d - Wi e c h e r t p o t e n t i a l w i t h t h e f i e l d s(3 .35) an d (3 .36) , o f a po in t ch argeZ e m o v i n g a l o n g a s t r a i g h t - l i n e t r a j e c -to ry. I t i s ins t ruc t ive to recons ider the p rocess in a fo rmula t ion [BeL82] inw h i c h ( a) t h e L o r e n t z t r a n s f o r m a t i o n o f t h e e l e c t r ic f ie ld s p r o d u c e d b y ag e n e r a l s p h e r i c a l c h a rg e d i s t r i b u t i o n is e x p l i c i t ly c a r r i e d o u t , a n d ( b ) f ro mt h e v e r y b e g i n n i n g , o n l y t h e w a v e n u m b e r s a n d f r e q u e n ci e s of t h e e m i t t e dv i r t u a l p h o t o n s p l a y a r ol e. T h e i m p a c t - p a r a m e t e r d e p e n d e n c e is r e c ov e r e dw h e n f o r m f a c t o r s a r e i n t r o d u c e d .

W e p r o c e e d a s f ol lo w s. W e w r i t e t h e p r o j e c t i l e c h a rg e d i s t r i b u t i o n i n i tsr e s t s y s t e m i n a c o v a r i a n t f o r m , b o t h i n t h e s p a c e r e p r e s e n t a t i o n a n d i n t h eF o u r i e r - t r a n s f o r m e d w a v e - n u m b e r re p r e s e n t a t io n . T h e c o v a r ia n t f o rm a l-l ow s u s t o e v a l u a t e t h e c o r r e s p o n d i n g f o u r- c u r r e n t i n a c o o r d i n a t e f r a m e i nw h i c h t h e p r o j e c t i l e m o v e s a n d t h e t a rg e t i s a t r e s t. F r o m t h e f o u r- c u r r e n t

2The numerical factor 1.123 in Eq. (3.49) arises from the occurrenc e of Euler's con-stan t in the se ries expansion of K0 andK1.


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3 .4 . E Q U I VA L E N T - P H O T O N M E T H O D 55

thus ob t a ined , one can in f e r d i r ec t l y t he e l ec t romagne t i c fou r-po ten t i a l i nthe wave -number r ep re sen t a t i on wh ich , i n t u rn , y i e ld s t he e l ec t r i c and mag-ne t ic f ie lds and hence the Po yn t ing v ec tor and , f ina lly, the nu m be r of equiv-a l en t pho tons .

T h e f o u r - c u r r e n t f o r a n e x te n d e d c h a r ge d i s t r ib u t i o n

As be fo re , quan t i t i e s i n t he mov ing p ro j ec t i l e sy s t em a re marked wi th ap r ime wh i le quan t i t ie s i n t he l abo ra to ry sys t em a re unp r imed . To beg inwi th , t he dens i ty o f t he fou r- cu r r en t o f t he cha rgeZ e in i t s res t sys tem i s

j ' . ( x ' ) - p ( I r ' l ) u ' , ( 3 . 5 3 )

wh ere x I - x ' deno tes the four-vec tor of the space - t ime co ord ina tes , p ( [ r '[ )= p( r ~) is a t im e- in de pe nd en t spher ica l charg e dens i ty, an d u ~ - u / =(c, 0 , 0 , 0 ) repre sen ts the four-ve loc ity, h ere in the res t s ys te m of the chargedens i ty. In order to der ive the dec om pos i t ion of j / (x ~) wi th respe c t towave num be rs k an d f requenc ies a~, which a re com bined as a four-vec tork - ( a ~ / c ,k) , we t ake t he fou r-d imens iona l F our i e r t r ans fo rm

1 /j t p k t ) - - (2 71 _) d 3 x d t / ) ( r t ) u t e - i ( k ' ' r ' - c J ' t ' )

= v / ~ 6(co') p (k ' ) u ' . (3 .54)

Here , p (k ' ) - p ( k ) i s t h e t h r e e - d i m e n s i o n a l F o u r i e r t r a n s f o r m o fp ( r ) . Forex tend ed spher ica l cha rge d i s t r ibu t ion s , i t is conve nien t to define a formfac to r f (U2) , wh ich i s aga in sphe r i ca l , by wr i t i ng

p ( k ) - - (2 7 r ) - 3 / 2 Z e f ( k 2 ) , (3.55)

or, expl ic i t ly,1 / r

f ( k 2 ) - ~ p ( r ) e - i k d 3 r . (3.56)

In pa r t i cu l a r, f o r a po in t cha rgep ( r ) = Z e 6 ( r ' ) w e h a vef o ( k 2 ) = 1 . Aba re nuc l eus, howeve r, ha s an ex t en ded cha rge d i s t r i bu t ion and , fo r t hep r e s e n t p u r p o s e , m a y b e r e p r e s e n t e d b y a h o m o g e n e o u s l y c h a rg e d s p h e r e(hcs ) w i th r ad ius R0 and to t a l cha rgeZ e . By d i rec t eva lua t ion of Eq . (3 .56)one ob t a in s t he nuc l ea r fo rm fac to r

fhcs(k ,2 ) _ 3 j l (k 'R 0 ) , (3 .57)k 1 R o

wh ere j l is a spher ica l B esse l func t ion . In the l imi t R0 ~ 0 , the form fac torf0 (k I2) = 1 for a point cha rge is re t r iev ed.


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56 C H A P T E R 3. F IE L D S O F M O V I N G C H A R G E S

I f o n e a s s u m e s a G a u s s i a n c h a rg e d i s t r ib u t i o n , t h e c o r r e s p o n d i n g f o r mf a c t o r i n w a v e - n u m b e r s p a c e is a l so G a u s s i a n a n d o f t h e f o r m

U : )f G a u s s( k 2 ) - - e x p - ~ - ~ , ( 3 .5 8 )

w h e r e Q 0 = 6 0 M e V ( in e n e rg y u n i t s ) f o r P b - P b [ D rE 8 9 ].T h e f r e q u e n c y J , w h i c h o c c u r s i n t h e d e l t a f u n c t i o n o f E q . ( 3 .5 4 ) ,

is t h e z e r o t h c o m p o n e n t o f a f o u r - v e c to r. B y f o r m a l l y w r i t i n g i t a s aL o r e n t z s c a l a r ~ t = k ~ u ~, t h e e v a l u a t i o n i n a n y c o o r d i n a t e s y s t e m isg r e a t l y s im p l i f ie d . S i m i l a r ly, s i n ce E q . ( 3 .5 4 ) r e q u i r e s J = 0 , w e c a n w r i t e

U 2 = - k ~ . k ~ = - k ~2 a s a L o r e n t z i n v a r i a n t q u a n t i t y. T h i s m e a n s t h a tE q . ( 3 .5 4 ) c a n b e w r i t t e n i n a c o v a r i a n t f o r m w h e n ( 3 .5 5 ) is i n s e r t e d . I nt h e l a b o r a t o r y s y s t e m , t h e f o u r - c u r r e n t r e a d s

jU(k ) - ~ Z e S k . u ) f ( - k 2) u u , 3 . 5 9 )

w h e r e t h e v e l o c i t y f o u r- v e c t o r u = u u = -y (c , v ) w i t h u - u = c 2 e x p l i c i t l yr e f e r s t o t h e p r o j e c t i l e v e l o c i t y v.

The electromagnetic four-potential

T h e c u r r e n t d e n s i t y ( 3 .5 9 ) a ll o w s u s t o c a l c u l a te t h e v e c t o r p o t e n t i a l f r o mE q . ( 3 .5 ) w h i c h in w a v e - n u m b e r s p a c e t a k e s t h e f o r m

471

k . k A U ( k ) = - ~ j U ( k ) . ( 3 .6 0 )c

B y i n s e r t i n g E q . ( 3 .5 9 ) i n t o E q . ( 3 .6 0 ) w e g e t t h e w a v e - n u m b e r d e c o m p o -s it io n o f t h e e l e c t r o m a g n e t i c f o u r - p o t e n t i a l p r o d u c e d b y a m o v i n g c h a r g ea s

A U ( k ) - _ _ 2Z e S k . u ) f ( - k 2 ) u u ( 3.6 1)c k 2

F r o m ~ r = 0 , w e k n o w t h a t t h e L o r e n t z i n v a r i an t a r g u m e n t o f t h e d e l t af u n c t i o n k . u = 7 ( ~ - k - v ) = J = 0 a n d h e nc e t h a t

k z = w / v. (3 .62)

T h e r e f o r e ,

- k - k + k z = k + , 3 . 6 3 )

w h e r e kz d e n o t e s t h e w a v e - n u m b e r c o m p o n e n t in th e d i r e c ti o n o f t h e m o v -i ng c h ar g e w h il e k is t h e c o m p o n e n t p e r p e n d i c u l a r t o v.


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3.4 . E Q U I VA L E N T - P H O T O N M E T H O D 57

W e h a v e n o w c a l c u l a t e d t h e e l e c t r o m a g n e t i c f o u r - p o t e n t i a l i n w a v e -n u m b e r r e p r e s e n ta t i o n in t h e l a b o r a t o r y s y s t e m w i t h o u t t a k i n g r e co u r set o t h e e x p l ic i t L i h n a r d - W i e c h e r t p o t e n t i a l s 3 .3 4 ) . T h e l o n g i t u d i n a l c o m -p o n e n t k z is f ix e d b e c a u s e t h e c h a rg e d i s t r i b u t i o n i n i ts r e st s y s t e m ist i m e - i n d e p e n d e n t s o t h a t i ts o s c i ll a t io n f r e q u e n c y cz~ = 0 in Eq . 3 .54) . Int h e s p a c e r e p r e s e n t a t i o n o f A U t h is m e a n s t h a t t h e f ie ld s d o n o t d e p e n do n t h e z - c o o r d i n a t e .

T h e t o t a l p h o t o n f l u x

T h e n e x t t a s k is t o c o n s t r u c t t h e P o y n t i n g v e c t o r a n d t h e p h o t o n f lu xf r o m t h e e l e c t ri c a n d m a g n e t i c f ie ld s . T h e t r a n s v e r s e e l e c t ri c f ie ld E =

E x , E y , 0 ) is o b t a i n e d f r o m E q . 3 .4 ) a s

E a J, k ) = i k A ~ a~, k)

i n t h e w a v e - n u m b e r r e p r e s e n t a t i o n , w h e r e u s e h a s b e e n m a d e o f A - 0[which i s a co nse qu en ce o f v - 0 in 3 .61) ]. In a f i r st s t ep , we pe r fo rmo n l y t h e i n v e rs e F o u r i e r t r a n s f o r m i n t h e z - d i r e c t i o n t o g e t

E a ~, k z ) 2 - 2 Z 2 e 2 k 2 f 2 _ k 2 )7l V 2 k4 , 3.64 )

w h e r e k 2 a n d k~_ a r e c o n n e c t e d b y E q . 3 .6 3 ) . S o f ar , n o a p p r o x i m a t i o nh a s b e e n i n t r o d u c e d . W h e n c a l c u l a t i n g t h e l o n g i t u d i n a l e l e ct ri c f ie ld a n dt h e m a g n e t i c f i el d f r o m E q . 3 . 5 ), w e t a k e t h e l i m i t v ~ c . T h e n u ~ =~ c , 0 , 0 , c ) in Eq . 3 .61) , so th a t A~ - A ~ 0, 0 , A ~ and , be cau se o fkz =a~/c, t h e l o n g i t u d i n a l f ie ld i n E q . 3 . 8) is

Ez - - F ~ - i ~ A 3 - k z A ~ - 0 .

I n t h i s li m i t, t h e r e a r e o n l y t r a n s v e r s e f ie ld s w i t h I E I - ] BI , a n d t h e p u l s eP2 d i scussed in Sec . 3 .4 .1 i s d i sca rded .

W e h a v e n o w e v e r y t h i n g n e e d e d t o c a l c u l a t e t h e e n e rg y f lu x 3 .3 9 ) a n dt o i n t e r p r e t i t a s a n e q u i v a l e n t f lu x of p h o t o n s w i t h p o s i t i v e f r e q u e n c ie s w.I n o r d e r t o e s t a b l i s h t h e i d e n t i f i c a t i o n , w e i n t e g r a t e S o v e r t i m e a n d o v e rt h e c o o r d i n a t e p l a n e p e r p e n d i c u l a r t o v.

da~ d x dy [E a~ r)]edt d x dy S = 27r

- - ek x 27r

~ 0 :x:)d h ) 3 . 6 5 )


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I n t he f i r s t equ a t i on , a f a c to r o f two a r i s e s f r omIE -w)l-IE w) l , s im i l a r l ya s i n E q . 3 .4 2 ); i n t h e s e c o n d e q u a t i o n , t h e r e m a i n i n g F o u r i e r t r a n s f o r m a -t i on i s pe r f o rm ed i n t he x , y -p l ane , w i th t he a id o f t he Pa r s eva l r e l a t i on[Mes62 ], t ha t is , t h e i nva r i ance o f t he s ca l a r p r od uc t und e r F ou r i e r t r an s -f o r m a t i o n s . T h e la s t e q u a t i o n d e f in e s t h e n u m b e r o f e q u i v a l e n t p h o t o n s p e rene rgy un i t . I t l e ads t o t he i den t i f i c a t i on

N hw) - 2 e 2 Z 2 f km~xk 3 f 2 - k 2 ) dk 3.66)[ k +

wh ere we r eca l l t h a t f - k 2 ) = 1 fo r a mo v ing po in t c ha rge . Fo r a con -s t a n t f o r m f a c t o r , t h e i n t e g r a l d i v e rg e s l o g a r i t h m i c a l l y a t t h e u p p e r l i m i ta s km ~ x ~ o c. I t is t h e r e f o r e n e c e s s a r y t o i n t r o d u c e t h e c u t - o f f p a r a m e t e rkm~x = 1/row i th r0 hav ing t he d im ens ion o f a l eng th . Th i s cu t -o f f p r ec i s e lyr e fl e ct s t h e c u t - o ff b min t h a t w e h a d t o i n t r o d u c e f or t h e i m p a c t p a r a m e t e ri n Eq . 3 . 48) . Fo r a po in t cha rge , t he in t eg ra l 3 .66 ) c an be ca r r i ed ou t .D i sca rd ing t e rm s t ha t a r e neg l i g ib l e f o r l a rge va lue s o f km~x we ob t a in

N h w ) _ 2 e2 Z 2 7 c 1 )7~ hc hw in . 3.67)

wro 2Th is resu l t i s iden t ica l to Eq . 3 .52) in the l im i t v ~ c i f the c u t -o ff rad iusr0 i s iden t i f i ed wi th min and t h e num er i ca l f a c to r 1 . 123 o f Eq . 3 . 52 ) isr ep l ac ed w i th 1 . T hese m ino r d i f f e rences a r e a l l w i t h in t he l im i t s o f a cc u ra cyi n h e r e n t i n t h e m e t h o d .

I f the fo rm fac to r fhcs - -k 2 ) o f Eq . 3 .57) o r fG au ss -k 2 ) o f Eq . 3 .58) i si n se r t e d i n 3 . 66) , t he i n t eg ra l conve rges a t t he up pe r l im i t , s o t h a t we cant a k e k m a x - o c. N e v e r t h e l e s s , v e r y l a rg e w a v e n u m b e r s m a y c o r r e s p o n d

t o im p a c t p a r a m e t e r s t h a t a r e s m a l l e r t h a n t h e r a d i u s R 0 of t h e p r o j e c t il en u c l eu s . I n t h i s ca s e, t h e a s s u m p t i o n o f a r e c t i li n e a r t r a j e c t o r y b r e a k s d o w na n d , f u r t h e r m o r e , a m u t u a l p e n e t r a t i o n o f t h e c o l li d in g n u c le i w il l o c c u r,s o t h a t n u c l e a r r e a c t i o n s c a n p r o c e e d i n a d d i t i o n t o t h e e l e c t r o m a g n e t i cp roce s se s .

The impact parameter-dependent photon f lux

F o r so m e a p p l i c a t i o n s , i t is u s e fu l t o k n o w th e i m p a c t p a r a m e t e r d e p e n d e n c e

o f t he ph o to n f l ux. S im i l a r l y as i n Eq . 3 . 64 ), we f ind by an add i t i ona lF o u r i e r t r a n s f o r m a t i o n i n t h e i m p a c t p a r a m e t e r p l a n e t h a t

I E a , , b , z ] 2 _ 1- 2 ~ 3 Z e ) 2f k f 2 - k 2 ) e ik d2k k 2V

2

b3.68)


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3 .4 . E Q U I VA L E N T- P H O TO N M E T H O D 59

In t he l im i t v - -, c and w i th t he i den t i f i c a t i on o f Eq . 3 . 65 ) f o r pos i t i vef r equenc i e s w ) , we ge t f o r t he ph o to n f lux in t he i n f i n i t e s ima l a r e a o f t hei m p a c t p a r a m e t e r p l a n e b e t w e e n b a n db + dba n d b e t w e e n t h e a z i m u t hand p + d ~ t he r e su l t

a Z 2N hw, b) - 4 f d k k f k~ + w2/~2v 2)eik bk ~ ~ 2 / ~ 2 v 2

a Z 2= 4 - ~

~o ~ k2 dkf k~_ w2 /,,/2 v 2)27r k~ ~2 ~ ,2v2

Jl k , 3.69)

wh e re J1 is t he Bes se l f unc t i on o f t he f ir s t k ind and o rde r one . I n t he f i rs ti n t e g r a l of E q . 3 .6 9 ), o n l y t h e c o m p o n e n t o f k p a r a l l e l t o b c o n t r i b u t e st o t h e i n t e g r a l o v e r t h e a z i m u t h a l a n g l e b e t w e e n k a n d b . B y in t e g r a t -ing N hw, b)o v e r t h e i m p a c t p a r a m e t e r p l a n e, w e r e t r ie v e E q . 3 .6 6 ). T h ere su l t 3 . 69 ) c an , o f cou r se , a lso be ob t a in ed d i r ec t l y f r om Eq . 3 . 43 ) ; how-e v e r, f or in t r o d u c i n g t h e f o r m f a c t o r f - k 2 ) a n d t h e e x t r e m e r e l a t i v i s ti cl im i t , t h e w a v e n u m b e r r e p r e s e n t a t i o n a p p l i e d h e r e is m o r e c o n v e n i e n t .

Fo r a po in t cha rge , Eq . 3 .46 ) t e ll s u s t h a t t h e co r r e sp ond ing ph o to n

f l u x c a n b e w r i t t e n a s

N hw , b ) - ~2 hw [ l- : K ~ ~ ) + V K g ~ ) , 3 .7 0)w h e r e = wb/~/v,whi l e K1 and / 4o a r e t he mod i f i ed Bes se l f unc t i ons o ford er one and ze ro , respec t ive ly. In F ig . 10 .12 of See . 10 .2 , we co m pa ret h e i n fl u e nc e s o f v a r i o u s f o r m f a c t o rs o n t h e i m p a c t p a r a m e t e r - d e p e n d e n tp h o t o n d i s t r i b u t i o n sN hw , b)f o r a f i xed Lo ren t z f a c to r ~ /= 3500 .

I n th i s s e c ti o n , w e h a ve c a l c u l a t e d t h e f r e q u e n c y s p e c t r u m o f e q u i v a l e n tp h o t o n s r e p r e s e n t i n g t h e t r a n s i e n t e l e c t r o m a g n e t i c f ie ld o f a m o v i n g c h a rge .I n C h a p . 1 0, w e a p p l y a n d e x t e n d t h e s e r e s u l ts t o e s t i m a t e t h e c r o ss s e c ti o nf or e l e c t r o n - p o s i t r o n p a i r p r o d u c t i o n . B y r e p l a c i n g t h e e l e c t r o m a g n e t i cf ie ld s ge ne ra t e d b y bo th o f t he co l l i d ing nuc l e i by equ iva l en t p ho ton s , onec a n c a l c u l a t e p a i r p r o d u c t i o n c r o s s s e c t i o n s f r o m t h e k n o w n c o r r e s p o n d i n gc ros s s ec t i ons fo r ph o to n -p ho ton co ll is i ons .

I n a p p l y i n g t h e e q u i v a l e n t - p h o t o n m e t h o d , o n e ha s to b e a w a r e o f t h el i m i t a t i o n s a r i si n g f ro m t h e a s s u m p t i o n o f r e a l p h o t o n s a n d f r o m t h e a s -

s u m p t i o n o f a c o n s t a n t f ie ld a cr o s s t h e t r a n s v e r s e e x t e n s i o n o f t h e p e r t u r b e ds y s t e m . T h e l a t t e r r e s t r i c t i o n m a y in v a l i d a t e t h e m e t h o d f or i m p a c t p a -r a m e t e r s s m a l le r t h a n t h e C o m p t o n w a v e l e n g t h o f t h e e l ec t ro n .

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Chapter 3 - Fields of Moving Charges