chapter 5 - ion-atom collisions

8/12/2019 Chapter 5 - Ion-Atom Collisions

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C h a p t e r 5I o n - A t o m C o l l i s i o n sI n t h e p r e c e d i n g t w o c h a p t e r s , w e d i s c u s s e d t h e e l e c t r o m a g n e t i c f i e l d p r o -d u c e d b y a p o i n t c h a r g e m o v i n g w i t h r e l a t i v i s t i c v e l o c i t y a n d , s u b s e q u e n t l y ,t h e r e l a t iv i s t ic m o t i o n o f a D i r a c e l e c t r o n a s a f re e p a r t ic l e o r i n a C o u l o m bp o t e n t i a l . I n t h e p r e s e n t c h a p t e r , w e m e r g e t h e r e s u l t s of C h a p s . 3 a n d 4b y a n a l y z i n g t h e e l e c tr o n m o t i o n i n t h e c o m b i n e d p o t e n t i a l o f t h e s p a c e -f ix e d t a r g e t a n d t h e m o v i n g p r o j e c t i l e n u c l e u s. B e f o r e d is c u s s i n g sp e c if icp r o c e s s e s in C h a p s . 6 - 1 0, w e c o ll e ct v a r io u s r e s u l t s w i t h g e n e r a l a p p l i c a -bi l i ty .

W e s t a r t o u t i n S ec . 5 .1 b y w r i t i n g d o w n t h e c o u p l e d e q u a t i o n s o f m o -t i o n w h i ch g o v e r n t h e i n t e r a c t i o n a m o n g t h e n u c l e a r , e le c t ro n , a n d p h o t o nf ie ld s. B y a d o p t i n g a f i e l d -t h e o r e t i c a l f o r m u l a t i o n , t h e a p p r o x i m a t i o n s l e a d -i n g t o t h e t i m e - d e p e n d e n t q u a n t u m d e s c r i p t i o n a r e c la r if ie d . I n S ee . 5 .2 w ei n v e s ti g a t e u n d e r w h i c h a s s u m p t i o n s a r e p r e s e n t a t i o n o f t h e n u c l e a r m o t i o nb y c l a s s i c a l t r a j e c t o r i e s c a n b e d e r i v e d f r o m a q u a n t u m m e c h a n i c a l w a v ef o r m u l a t io n . O n e a r r iv e s a t a t i m e - d e p e n d e n t D i r a c e q u a t i o n w h o s e p r o p e r -t i e s a r e s t u d i e d i n S e e . 5 . 3 . T h i s e s t a b l i s h e s t h e b a s i s f o r t h e d e v e l o p m e n t so f S e e. 5 . 4, w h e r e w e d is c u s s s o m e g e n e r a l f e a t u r e s o f p e r t u r b a t i v e t r a n s i -t i o n a m p l i t u d e s f o r v a r i o u s p r o c e s s e s , a g a i n s t a r t i n g f r o m a f i e l d - t h e o r e t i c a lf o r m u l a t i o n . O n e m o r e r e m a i n i n g p r o b l e m o f s o m e g e n e r a l i t y is t h e e v a l-u a t i o n o f t r a n s i t i o n m a t r i x e l e m e n t s . T o t h is e n d , i n S ee . 5.5 , it is s h o w nh o w g a u g e t r a n s f o r m a t i o n s c a n b e u s e d t o r e f o r m u l a t e m a t r i x e l e m e n t s o fr e l a t i v is t i c o p e r a t o r s s o t h a t t h e i r e v a l u a t i o n b e c o m e s l es s s e n s i ti v e to a p -p r o x i m a t i o n s . F i n a ll y , th e g e n e r a l f o r m u l a t i o n o f m a n y - e l e c t r o n s y s t e m s i sd i scussed in See . 5 .6 .

97


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98 C H A P T E R 5 . I O N - A T O M C O L L I S I O N S

5 .1 F i e l d - t h e o r e t i c b a c k g r o u n dI t i s u se ful , a s a s t a r t i n g po i n t , t o g i ve a b r i e f f i e l d - t heo re t ic desc r i p t i on o ft h e c o u p l e d e q u a t i o n s g o v e r n i n g t h e e l e c t r o n f i e l d a n d t h e p h o t o n f i e l d a n dt o d is cus s som e o f t he consequen ces . Th i s p roce du r e a l l ows u s t o add res sm o r e c l e a r ly t h e a p p r o x i m a t i o n s w h i c h a r e a d o p t e d i n t h e r e s t o f t h e b o o k .T h e f o r m a l i s m d e s c r i b e d h e r e is e s s e n ti a l a s a b a c k g r o u n d f or so m e o f t h em o r e a d v a n c e d t r e a t m e n t s i n t h e l i t e r a t u r e , b u t i n t h i s b o ok , i t s f ul l p o w e ri s no t needed .

5 .1 .1 C o u p l e d e q u a t i o n s o f m o t i o nFor s i m p l i c i t y , we as sum e t ha t t he p ro j ec t i l e nuc l eus m oves a l ong a c l a s s i ca lr e c t i l in e a r t r a j e c t o r y . T h i s a s s u m p t i o n is j u s ti f ie d i n C h a p . 3 a n d i s t a k e nup i n m ore de t a i l i n Sec . 5 .2 . Wi t h i n a c l a s s i ca l p i c t u re , t he e l ec t rom agne t i cc u r r e n t p r o d u c e d b y t h e m o v i n g n u c l e i c a n b e r e g a r d e d a s a n e x t e r n a l c u r -r e n t . T h e s y s t e m t h e n is r e p r e s e n t e d : ( i) b y t h e n u c le i w h i c h c a n e m i t a n da b s o r b p h o t o n s v i a t h e c o u p l i n g o f t h e i r c l a s si c al c u r r e n t s t o t h e e l e c t r o -m agne t i c f i e l d , ( i i ) by t he e l ec t ron f i e l d wh i ch i s a l so coup l ed t o t he pho t onf ie ld and w h i ch a l lows fo r t he c rea t i o n o r ann i h i l a t i on o f e l ec t ron - pos i t ro npa i r s , and ( ii i) by t he ph o t on s w h i ch a re coup l ed bo t h t o t he nuc l e i and t ot h e e l e c t r o n s a n d w h i c h m e d i a t e t h e e l e c t r o m a g n e t i c i n t e r a c t i o n b e t w e e nt h e m .

Th e coup l ed equ a t i on s [ see Eqs . (4 .1 )a n d (3 .9 ) ] fo r t he nuc l ea r , e l ec -t ron , and pho t on f i e l d s a re

0 e -- ) Ai h % - ~ x " + c%A I l t e C ~ 2 - - 0 (5 .1)a n d 0 ~- 47r~-

Ox----;F~'" = --J"c (5 .2)iw i t h t h e r e l a t i o n b e t w e e n t h e f i e l d - s tr e n g t h t e n s o r F " ~ a n d t h e v e c t o r p o -

t e n t i a l A " b e i n g g i v en b y E q . ( 3 . 8 ). W e u se a h a t s y m b o l t o c h a r a c t e r i z et he f i e ld ope ra t o r s ~ , A , F , and j . S i nce t he nuc l e i a re t r ea t e d c l a ss i cal ly ,t h e o p e r a t o r s f o r t h e e x t e r n a l n u c l e a r c u r r e n t s a r e r e p l a c e d b y t h e i r ex -p e c t a t i o n v a l u e s

3ext > ( 3 e x t} - 3ext , (5 .3)SO th at the fu ll cu rre n t en ter i ng in Eq . (5 .2 ) is g iven by

J"P - - J P e l -t- ~'tL3ext > - - ec ~ t ') 'O 3 'P ~ -}- Je~xt (5 .4 )


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5 . 1 . F I E L D - T H E O R E T I C B A C K G R O U N D 99Here , the de f in i tion (4 .8 ) has been in s e r t ed . L ik ewise , the to ta l r a d ia t io nf ie ld i s decomposed as

A A A AA " - Ar + A ePxt + A free. 5 . 5 )A AOwing to E q . (5 .2 ) , A~ i s p roduced by the e l ec t ron cu r r en t j~ l , t he f i e ld

#Aex is pro du ce d by the nuc lear cu r ren t 3~xt,~ wh ile A~frr re pr es en ts th e freerad ia t ion f ie ld .

As the gene ra l coup led equa t ions canno t be s o lved , one i s fo rced to t ak erecour s e to a num ber o f app rox ima t ion s : ( a ) Th e e l ec t ron cu r r en t i s d i sr e -ga rd ed as a s ou rce o f pho to ns s ince i ts e ff ect s a r e s ma l l com pare d to thos e o fthe nuc lea r cu r r en t fo r h igh -Z nuc le i and , by a s s umpt ion , the e l ec t ron cu r -ren t does not ac t back on the nuclear mot ion , which is descr ibed c lass ica l ly .

i(b ) Cor res pond ing ly , the f i e ld A~ i s neg lec ted a s compared to Ae t . Th esetwo ap pro xim at io ns are suf fic ien t to decou ple Eqs . (5 .1) an d (5 .2). (c ) Sincethe e l e c t rom agne t i c f ie ld o f the nuc lea r m o t ion p roduce s a l a rge num ber o f

# # A photo ns , f luc tuat io ns can be d is re gard ed an d Aex --~ (A~xt) - , wh ich isin acco rd wi th E q . (5 .3 ) and imp l i e s the co r r es pond ing r e l a t ions h ip fo r theF ~" . (d ) R ad ia t iv e Q E D co r rec t ions , a s for exam ple vacuu m po la r i za t ionand s e l f- ene rgy co r r ec t io ns a r i s ing f rom the f ree pho ton f ie ld , a r e no t t a k eni n to a c c o u n t h er e , h e n c e A f r ~ - 0 .H a v i n g s p e l l e d o u t t h e v a r i o u s a p p r o x i m a t i o n s , w e r e t u r n t o t h e n o wd e c o u p l e d e q u a t i o n s

0 e ) I .i h " / , - ~ z " + -%A'c - m e C ~ ) - 0 (5.6)a n d 0 47rcOx---~F ~ ' " = --Jext ,c (5.7)wh ere E q . (5 .6) i s the op era tor vers ion of Eq . (4 .1) for an e lec t ron f ie lds ub jec t to an ex te rna l e l ec t roma gne t i c po ten t i a l , wh i l e in E q . (5 .7 ) theex te rn a l nuc lea r cu r r en t d e te rm ines the f ie ld F "~ and the po te n t i a l A , .5 . 1 .2 E x p a n s i o n o f t h e f ie ld o p e r a t o r sI f the p e r tu rba t io n by a t ime -va ry in g ex te rna l po ten t i a l is weak , i t is pos -s ib le , s imi lar ly as in Eq . (4 .64) , to expand the f ie ld opera tors

~ (r , t) - - ~ by+ ~j+ (r , t ) + ~ d%* ~j _ (r , t ) ,j+ j - ( 5 . 8 )

i n t e r m s o f t i m e - i n d e p e n d e n t ( S c h rS d i n g er ) c r e a t i o n a n d a n n i h i l a t i o n o p -e r a t o rs . T h e p o s i t i v e - e n e rg y w a v e f u n c ti o n s ~ j+ a n d t h e n e g a t i v e - e n e r g y


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100 C H A P T E R 5. I O N - A T O M C O L L I S I O N Sw a v e f u n c t i o n s r c o m p r i s e a c o m p l e t e s e t o f t i m e - d e p e n d e n t s t a t i o n a r yso lu t ion s o f t he D i r ac equa t ion ( 4 .75) , s ay in t he t a r g e t sy s t em r = r T , sot h a t 0i h N C j ( r , t ) - H r t ) ( 5 . 9 )w i t h

a n dC j ( r , t ) - - ~ j ( r ) e - i E S t / h (5.10)

H T - - i h c o ~ . V Z w e 2-- ~- m ~c2 ",/~ (5.11)rA c o r r e s p o n d i n g s e t of e q u a t i o n s c a n b e w r i t t e n i n t h e p r o j e c t i le s y s t e m .

T h e b j+ a r e d e s t r u c t i o n o p e r a t o r s f o r e l e ct ro n s , a n d t h e d ' ; a r e c r e a t i o no p e r a t o r s f or p o s i t r o n s , e a c h c h a r a c t e ri z e d b y t h e s e t s o f q u a n t u m n u m b e r sj + a n d j _ , r e sp e c ti v e ly . T h e c o m m u t a t i o n r e l a t io n s a n d t h e d e fi n i ti o n o fthe va cu um 10} a re the sa m e as in Eqs . (4 .65) to (4 .68) for f ree e lec t rons .

5 . 1 . 3 T i m e e v o l u t i o n o f t h e v a c u u mI n m o s t p a r t s o f t h e b o o k , i t is a s s u m e d t h a t p e r t u r b a t i o n t h e o r y i s a p p l i c a -b l e, a n d h e n c e t h e p r o b a b i l i t y o f p r o d u c i n g e l e c t r o n - p o s i t r o n p a i r s re m a i n sneg l ig ib l e a t a l l t imes . Then i t i s na tu r a l t o cons ide r t he vacuum 10) t o beu n a l t e r e d d u r i n g t h e c o ll is io n so t h a t t h e d i s cu s s i o n o f t h i s a n d t h e f o ll o w in gs u b s e c t i o n i s n o t n e e d e d .

I f pa i r p r o du c t io n o ccur s w i th l a r ge p r oba b i l i t i e s ( Secs. 10 . 3 .2 and 10 .4 ) ,p e r t u r b a t i o n t h e o r y b r e a k s d o w n , a n d w e h a v e t o e x a m i n e t h e v a c u u mmor e ca r e f u l ly . I n t he D i r ac ho l e t heor y , t he vacuum r epr e sen t s t he occu-p i e d D i r a c s e a c o m p o s e d o f n o n i n t e r a c t i n g n e g a t i v e - e n e r g y s i n g l e - p a r ti c l es t a t e s . The se s ing l e - pa r t i c l e s t a t e s a r e a f f ec t ed by an ex t e r n a l f ie ld and soi s t he vacu um . Con sequen t ly , f o r t he ca se o f s t r ong e r e l ec t r om agn e t i c i n -t e r ac t ions , i t i s conven ien t t o cons ide r a modi f i ed vacuum which t ake s t h i sp r o c e s s in t o a c c o u n t . R e t a i n i n g t h e e x p a n s i o n i n t e r m s o f s t a t i o n a r y s o lu -t i ons ( 5 .10 ) o f t he t a r g e t ( a l t e r na t ive ly , t he p r o j ec t i l e ) D i r ac equa t ion , t h i sm e a n s t h a t w e h a v e t o r e p l a c e t h e t i m e - i n d e p e n d e n t c r e a t i o n a n d a n n i h i l a -t i o n o p e r a t o r s b y t h e c o r re s p o n d i n g t i m e - d e p e n d e n t ( H e is e nb e rg ) o p e r a t o r ss o t h a t

~ ( r , t ) - E b i + (t)~ b i+ ( r , t ) + E d ; _ ( t) ~2 i _ ( r , t ) (5.12)j+ j-

i n s t ead o f ( 5 . 8 ) . S ince f o r a su f f i c i en t ly s t r ong in t e r ac t ion dur ing the co l l i -s ion , t he v acu um 10} de f ined a t t ~ - o c w i l l a l so evo lve , we can no t r equ i r e

h hth a t a t a ny given ins tan t o f t ime , the def in i t ions b j+ 10} - 0 and d j _ 10} - 0


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5 .1 . F I E L D - T H E O R E T I C B A C K G R O U N D 1 01s t il l ho ld . R ath er , th e "dressed" ins t an tan eou s vac uu m 10} evolved f romthe in i t i a l vacuum 10} m ay con ta in pa i r and m u l t ip le pa i r am p l i tudes .I n t r o d u c i n g a d r e s s e d v a c u u mI t is conven ien t , t he re fo re , to de fine a new s e t of t im e- ind epe nde n t c r ea t ion~t ~top era tor s bk+ , dk_ , so tha t in the exp ans io n

A ~ ~t(r, t) - E bk+ Xk+ (r, t) + E dk - Xk_ (r, t)k+ k_

(5.13)

the t ime evo lu t ion i s abs o rbed in to the wave func t ions Xk + and Xk _ , wherek + a n d k _ d e n o t e s e t s of q u a n t u m n u m b e r s f or p o s it iv e - a n d n e g a t i v e -ene rgy s t a t e s , r e s pec tive ly . The new wave func t ions a r e s o lu t ions o f the fu llt ime-dependen t Di rac equa t ion (5 .6 ) wh ich have evo lved f rom the o r ig ina ls t a t iona ry s o lu t ions (5 .10 ) . There fo re , the in i t i a l cond i t ions a r e

l ira 1 0 ( t) } - 10) (5.14)t---~-- oca n d

l im x k ( r , t ) - ~ k ( r t ) . ( 5 1 5)t ----* - - cx:~ 'Th e t im e evo lu t ion o f the d res s ed in s t an tane ous vac uum 10} en ta i ls a

c a n o n i ca l t r a n s f o r m a t i o n o f t h e a s s o c i a te d a n n i h i l a t io n a n d c r e a t io n o p e r -a tors , which are def ined as

bk+10(t)} - 0, b k +] O ( t ) ) -- I~k+(t)}Atdk_ [6(t)) - 0, dk_ 10(t)) -- f~k_ (t)) (5.16 )

and p res e rve the commuta t ion r e l a t ions (4 .65 ) to (4 .67 ) .S ince the r ep res en ta t ions (5 .12 ) and (5 .13) a re equ iva len t , we have

j+ j -= ( t ) + ( t ) ,

k + k _(5.17)

whe re the s pace dep ende nce o f the w ave func t ions is s upp res s ed for b rev i ty .U s i n g t h e o r t h o g o n a l i t y o f t h e s t a t i o n a r y w a v e fu n c t io n s ~ j + , ~ j _ , o n e de -r ives the t r ans fo rm at ion s [WeO92]

by+ ( t ) - E bk+ (~#+ IXk+ } + E d k- (~J+ IXk_ }k+ k_


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5 .1 . F I E L D - T H E O R E T I C B A C K G R O U N D 103T h e f i rs t t e r m is a p r o d u c t o f i n d e p e n d e n t p r o b a b i l i t ie s w h i l e th e s e c o n dte r m desc r ibe s cohe r en t co r r e l a t i ons .

U s ing the com ple t ene ss r e l a t i ons f o r t he se t s o f IC j) and Xk} ( com pr i s ingp o s i t i v e a s w e l l a s n e g a t i v e - e n e r g y s t a t e s ) , o n e c a n s h o w t h a t t h e t o t a ln u m b e r o f e l e c t ro n s p r o d u c e d e q u a l s t h e n u m b e r o f p o s i t r o n s p r o d u c e d , a sr equ i r ed by cha r ge conse r va t ion , i . e . ,

j+ j -S imi l a r ly , by summing Eq . ( 5 . 21 ) ove r a l l s t a t e s j + , j _ , one f i nds t ha t onep a r t o f t h e c o r r e l a t i o n t e r m s c a n ce l s t h e p r o d u c t t e r m , s o t h a t t h e t o t a ln u m b e r o f p a i r s c re a t e d e q u a l s t h e t o t a l n u m b e r N + o r N _ o f e l e c tr o n s o rp o s i t r o n s p r o d u c e d .5 . 1 . 4 T i m e e v o l u t i o n o f s i n g l e - e l e c t r o n s t a t e sAccor d ing to Eq . ( 5 . 16 ) , a s ing l e - e l ec t r on s t a t e i s g iven by

~ tIXk+ (t)} - bk+ 10(t)}. (5.23 )I f, in i t ia l l y , t he r e is on ly one e l ec t r on p r e sen t , ad m ix tu r e s o f two- e l ec t r on-o n e - p o s i t r o n s t a t e s a n d h i g h e r - o r d e r t e r m s w i l l d e v e l o p d u r i n g t h e c o ll is io n .T h e e x p e c t a t i o n v a l u e f or t h e n u m b e r o f o b s e r v a b l e e l e c tr o n s w i t h q u a n t u mn u m b e r j + c o n t a i n e d i n t h e s i n g l e - p a rt i c le s t a t e ( 5. 23 ) is o b t a i n e d a s

( X j + ( t ) } - - (0( t ) bk+ bJ+ ( t )b j+( t )bk+10 ( t ) )= l ( r x ~ + ( t ) ) l ~ + Y ~ I ( r I x , _ ( t ) ) I ~ . ( 5 . 2 4 )

l_H e r e , { r X k + ( t) } is t h e o c c u p a t i o n a m p l i t u d e o c c u r r i n g i n c o u p l e d -channe l c a l cu l a t i ons d i scussed in Secs . 6 . 5 , 8 . 7 , and 10 . 3 . 2 . The to t a l num-b e r o f e l e c t ro n s i s o b t a i n e d b y s u m m i n g o v e r j + a s

N + ( t ) - 1 + E I { r I X l _ ( t ) } l 2 . ( 5 . 2 5 )j + , l _

I n a d d i t i o n t o t h e o n e e l e c t r o n o r i g i n a ll y p r e s e n t, t h e r e a re e l e c t ro n s c r e a t e dd u r i n g t h e c o ll is i on f r o m t h e v a c u u m . C o r r e s p o n d i n g l y , t h e t o t a l n u m b e rof po s i t r on s c r ea t e d i s

N _ ( t ) - ~ I ( r I x ~ + t ) } l 2. (5.26)j_ , l+r


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104 C H A P T E R 5 . I O N - A T O M C O L L I S I O N SS i n ce t h e s t a t e k + is o c c u p i e d i n t h e e n t r a n c e c h a n n e l, i t c a n n o t d e v e lo pi n t o a h o le i n t h e D i r a c s e a a n d h e n c e i s e x c l u d e d f r o m t h e s u m m a t i o n .

I n m o s t c a s e s c o n s i d e r e d in t h i s b o o k ( C h a p s . 6 t o 9 ), t h e p e r t u r b a t i o nb y t h e p r o j e c t i l e is s u ff ic i en t ly w e a k s o t h a t p a i r p r o d u c t i o n d o e s n o t o c c u rw i t h a p p r e c i a b l e p r o b a b i l i t y . I n t h e s e c a s e s , t h e v a c u u m r e m a i n s t h e s a m et h r o u g h o u t t h e c o l li si on , i.e ., 1 0 ( t) } - 1 0 ) , a n d t h e n u m b e r o f e l e c tr o n s isc o n s e r v e d . A f ie l d - t h e o r e t i c d e s c r i p t i o n i s n o t n e e d e d i n t h i s ca s e. W h i l et r e a t i n g t h e c o l li si o n w i t h s t a n d a r d q u a n t u m m e c h a n i c s in C h a p s . 6 t o 9,w e r e t u r n t o t h e f i e l d - t h e o r e t i c m e t h o d s i n C h a p . 1 0. S t il l i n S e e. 5. 4, w eu s e t h e s e m e t h o d s t o f o r m u l a t e t h e p e r t u r b a t i v e t r a n s i t i o n a m p l i t u d e s , a n di n Se e. 5 .6 , w e a d o p t s i m i l a r te c h n i q u e s t o d e s c r i b e m a n y - e l e c t r o n s y s t e m s .

5 .2 R e l a t i v i s t i c i m p a c t p a r a m e t e r t r e a t -m e n t

I n C h a p . 3 , w e e x p l a i n o n a n in t u i t i v e b a s is t h a t a s e m i c l as s i ca l t r e a t m e n t o ft h e n u c l e a r m o t i o n s h o u l d p r o v id e a g o o d a p p r o x i m a t i o n . B y a s e m i c l a s si c a l( or 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 n e u su a l ly m e a n s a d es c r ip t i o n i n w h i c ht h e m o t i o n o f t h e n u c l e i i s a s s u m e d t o f ol lo w a c l a s s ic a l t r a j e c t o r y w h i l et h e e l e c t ro n i c m o t i o n o b e y s q u a n t u m m e c h a n i c s . I n t h i s s e c ti o n , w e s h o wh o w u n d e r s u i t a b l e a s s u m p t i o n s a se m i c l a ss i c a l t r e a t m e n t o f t h e c o l li si o nc a n b e d e r i v e d f r o m a w a v e t r e a t m e n t . T h i s d e r i v a t i o n i s o f a m o r e f o r m a ln a t u r e a n d i s n o t e s s e n ti a l f or u n d e r s t a n d i n g t h e s u b s e q u e n t d e v e l o p m e n t s .

W i t h i n t h e r e a l m o f a t o m i c c o l li si on s , t h e b a s i c f e a t u r e o f t h e n u c l e a rm o t i o n i s t h e f a ct t h a t t h e m a s s e s M w a n d M p o f t h e t a r g e t a n d t h e p r o j e c -t i l e n u c l e u s , r e s p e c t i v e l y , a r e v e r y l a r g e c o m p a r e d t o t h e e l e c t r o n m a s s m ~ .A s s u m i n g t h a t t h e n u c l e i c a n b e d e s c r i b e d a s D i r a c p a r t i c l e s ( o t h e r q u a n -t u m d e s c r i p t i o n s w o u l d n o t l e a d t o d i f f e r e n t r e s u l t s ) , w e m a y s a f e l y i g n o r et h e n e g a t i v e - e n e r g y s t a t e s o f t h e n u c l e i. M o r e o v e r , t h e c l a ss i c a l c o n s i d e r a -t i o n s o f C h a p . 3 s h o w t h a t i n r e l a t i v i s t i c c o ll is i o n s ( a) t h e t a r g e t r e c o i l i sn e g l i g i b l e u n d e r m o s t c o n d i t i o n s a n d ( b ) t h e p r o j e c t i l e d e f l e c t i o n d o e s n o tp l a y a n y r o l e . W e t h e n c a n t a k e t h e t a r g e t n u c l e u s a s f i x e d i n t h e l a b o r a -t o r y f r a m e ( M T ~ cx~) a n d u s e t h i s c o o r d i n a t e s y s t e m f or d e s c ri b i n g t h ep r o c e s s . W e f o ll o w t h e a r g u m e n t s g i v e n i n [ D eE 9 4 ] fo r t h e n o n r e l a t i v i s t i cs i t u a t i o n .F o r a p r o j e c t i l e m o v i n g i n t h e p o s i t i v e z - d i r e c t i o n , l e t t h e c o o r d i n a t e sb e R - ( X , Y , Z ) a n d t h e c o r r e s p o n d i n g d e r i v a t iv e s V R - - ( ox~ oY~ oz~ ) .S i m i l a r l y , t h e e l e c t r o n c o o r d i n a t e s a n d d e r i v a t i v e s a r e r = ( x , y , z ) a n do o o r e spe c t ive ly .v - ( -a x , O y , O z ) ,


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5 . 2 . R E L A T I V I S T I C I M P A C T P A R A M E T E R T R E A T M E N T 105

I n a t im e - i n d e p e n d e n t w a v e d e s c r i p t io n o f t h e p r o j e c t i l e a s w e ll a s o ft h e e l e c t r o n m o t i o n a t a t o t a l e n e r g y E , t h e c o m p l e t e w a v e f u n c t i o n f o r a ni n c o m i n g i n i ti a l s t a t e ~ + ( E ) s a ti sf ie s t h e t i m e - i n d e p e n d e n t D i r a c e q u a t i o n

( H - ( 5 . 2 7 )w h e r e t h e D i r a c H a m i l t o n i a n is g i v en b y

H Z T Z P e 2 0- - i h c ~ R " V R + R ~ - M P c 2 ~ R

- - i h c ~ . V Z T e 2 ~ r n e c 2 7 ~ + Vp,e (R , r ) .r (5.2s)H ere , th e D i rac m at r ice s c~R, 7 ~ ac t in the nuc lear sp ino r spa ce w hi le c~ , ,~0

ac t i n t he e l ec t ron sp i no r space , and t he e l ec t ron -p ro j ec t i l e i n t e rac t i on Vp ,ei s y e t t o b e d e t e r m i n e d .

I n w r i t i n g d o w n t h e H a m i l t o n i a n , i t i s a s s u m e d t h a t , o w i n g t o t h e h e a v ym a s s o f t h e p r o j e c t i le , t h e p r o j e c t il e m o t i o n is i n d e p e n d e n t o f t h e e l e c t r o nm o t i o n . T h e n t h e s o l u ti o n s ~ + ( E ) o f ( 5 .2 7 ) c a n b e w r i t t e n a s d i r e c t s p in o r1p ro du c t s fo r t he ( a s su m e d sp i n ~ ) nuc l eus and fo r t he e l ec t ron . S i nce i tis show n i n Sec. 3 .2 t h a t t he de f l ec t ion o f t he hea vy nuc l eus i s neg l ig i b l e ,t h e n u c l e a r w a v e f u n c t i o n s o b e y t h e s t a t i o n a r y f r e e - p a r t i c l e D i r a c e q u a t i o n ,viz. (4 .46),

( - i h c ~ R 9 V R + M p c 2 ~ /~ ( I ) E p, K - - E p ( I ) E p , K , (5 .29)w h e r e

a n dE 2 __ M 2 c 4 + ~ 2 K 2 c 2

( I )E p , K - - lt(p ) e i K ' R (5 .30)w i t h a p l a n e - w a v e s o l u t i o n f o r t h e w a v e v e c t o r K = K ~ z . T h e f r e e - p a r t i c l esp i no r w i t h p = 1 , 2 fo r t he sp i n pa ra l l e l a nd an t i pa ra l l e l t o t he z -ax i s isdef ined in Eqs . (4 .54) and (4 .55) , respect ive ly .

B e c a u s e o f t h e l a r g e r a t i o b e t w e e n t h e i r m a s s e s , e l e c t r o n a n d p r o j e c t il ea r e n o t t r e a t e d o n e q u a l f o o ti n g I i n t h e H a m i l t o n i a n ( 5 .2 8 ). W e s t a r t f r o mt h e i n t e r a c t i o n

j ' A , / c - - e ( ~ / t ~ / ~ - - e ( r t ~ A 0 _ ~ tc~ ~ . A ) , (5 .31 )1 A s y m m e t r i c t r e a t m e n t , s u c h a s f or t h e t w o e l e c t r o n s i n h e l i um , w o u l d l e a d t o t h e

a p p r o x i m a t e B r e i t i n t e r a c t i o n [ B eS 57 ].


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106 C H A P T E R 5 . I O N - A T O M C O L L I S I O N Ss ee S ec . 4 .4 . 1, b e t w e e n t h e e l e c t r o n i c c u r r e n t d e n s i t y j n a n d t h e r e t a r d e dv e c t o r p o t e n t i a l A n g e n e r a t e d b y t h e t i m e - i n d e p e n d e n t r e la t iv i s ti c p r o -j e c ti le c u r r e n t . F o r t h e h e a v y p r o j e c ti le w h o s e s ta t e r e m a i n s u n c h a n g e ddu r i ng t he co l li s ion , i t i s j u s t i f i ed t o r ep l ace t he fou r -cu r r en t by i t s expec -t a t i o n v a l u e

0 #( O E p , K [ Z p e C T R ' T R I O E p , K } - - Z p e v n - - Z p e ( '7 c , 'TV)b e t w e e n t h e p l a n e w a v e s t a t e s ( 5 . 3 0 ) , w h e r e v = h K / M p and ~/ = ( 1 -/32) - 1 / 2 is t h e L o r e n t z f a c t o r. T h e L i ( ~ n a r d -W i e c h e r t p o t e n t i a l c a l c u l a t e df rom t h i s con t i nuous cha rge f l ow i s c l o se l y r e l a t ed t o t he po t en t i a l s (3 .34 )p r o d u c e d b y a s in g l e m o v i n g c h ar g e : a t e a c h p o s i t i o n R o f t h e c u r r e n t , t h ec o n t i n u o u s c h a r g e f l o w p r o v i d e s c h a r g e s Z p e whi ch , a t t he t i m e t = Z / v ,a r r i v e a t t h e z - c o o r d i n a t e Z . A s a re s u l t, w e o b t a i n t h e e f fe c ti v e e l e c t ro n -p r o j e c t i l e i n t e r a c t i o n

V p ,e (R , r ) -- - ~ ( 1 - / 3 a z ) Zp (?2v / ( x - x ) : + ( y - + z : ( z -I n o r d e r t o s o l v e E q . ( 5 . 2 7 ) , w e i n s e r t t h e p r o d u c t a n s a t z

9 ( 5 . 3 2 )

9 + ( E ) - f ( r , R ) ( I) E p, g, ( 5 .3 3 )w h e r e f ( r , R ) is y e t t o b e d e t e r m i n e d . W h e n m u l t ip l y i n g t h e r e s u l ti n ge q u a t i o n w i t h t h e a d j o i n t p r o j e c t i l e s p i n o r U(p ') t f rom the l ef t , we get an o n z e r o c o n t r i b u t i o n o n l y i f p = p ' b e c a u s e K x / K z a n d K y / K z a re ze ro i nt h e i n i ti a l s t a t e ( a n d n e g l ig i b le i n th e f in a l s t a t e e v e n if a s m a l l C o u l o m bdef l ec t i on occu r s ) . The r e su l t i s

U ( p ) t ( H - E ) ~ i + ( E ) - -0 Z T Z p e 2 _ i h c o ~ . V- i 9-5-2 + R

- - ~ -~- Y p , e ( R , r ) + m e c 2 7 0 - ~ i f ( r , R )( 5 . 3 4 )

c iK RZ T e 2

rO,

w h e r e Z = R . e z a n d 5 i - - E - E p is t h e d i f fe r e nc e b e t w e e n t h e t o t a l e n e r g yo f t h e s y s t e m a n d t h e t o t a l e n e r g y o f t h e p r o j e c t i l e a l o ne . T h e v e c t o r s Ra n d r a r e t r e a t e d a s i n d e p e n d e n t v a r i a b l e s, so t h a t O / O Z is p e r f o r m e d w i t hr k ep t f ixed . I f we dec om pos e

R - b + Z ~ z


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108 C H A P T E R 5 . I O N - A T O M C O L L I S I O N Sp ic tu r e on the o the r hand . I nsof a r , t he r e su l t ( 5 .36 ) is no t su r p r i s ing . U pt o t e r m s o f t h e o r d e r o f t h e r a t i o b e t w e e n t h e e l e c t r o n m a s s t o t h e n u c l e a rm a s s , t h e t w o d e s c r i p t i o n s a r e fu l ly eq u i v a le n t . I t s h o u l d b e m e n t i o n e d t h a tin t he co r r e spond ing nonr e l a t i v i s t i c de r iva t ion [ DeE94] , a l i nea r i za t i on o fthe k ine t i c - ene r gy t e r m i s nece ssa r y ( which i s no t needed in our r e l a t i v i s t i ct r e a t m e n t ) .

5 . 3 M o v i n g C o u l o m b p o t e n t i a l sW i t h E q . ( 5 . 3 6 ) w e h a v e e s t a b l i s h e d w h a t m i g h t h a v e b e e n w r i t t e n d o w nf r o m t h e o u t s e t , n a m e l y a t i m e - d e p e n d e n t D i r a c e q u a t i o n f o r a n e l e c t r o nm o v i n g i n e l e c t r o m a g n e t i c p o t e n t i a l s g e n e r a t e d b y t w o C o u l o m b s o u r c e swhich pa ss e ach o the r a t a r e l a t i v i s t i c speed . Th i s equa t ion i s t he ba s i s f o rt h e w h o l e s u b s e q u e n t t r e a t m e n t .

5 . 3 . 1 E x p l i c i t f o r m o f t h e e q u a t i o n sAdopt ing the coor d ina t e s spec i f i ed i n Sec . 3 . 3 and in F ig . 3 . 2 , we wr i t edow n the equ a t ion ( 5 .36) f o r an e l ec t r on in t he f ie ld o f two m ovin g po in tc h a r g e s , b o t h i n t h e t a r g e t f r a m e a n d i n t h e p r o j e c t i l e f r a m e . I n t h e t a r g e tf r ame , we have

0 t ~ ( r w , t ) - - H ~ ( r w , t ) ( 5. 39 )w i t h

H - i h c O L" V w Z T e 2 S 2 Z p e 2- + m e c 2 7 ~ (5.40)r T r ~T h e H a m i l t o n i a n ( 5 . 4 0 ) d e s c r i b e s a n e l e c t r o n s u b j e c t t o t h e C o u l o m b p o -t e n t i a l - - Z T e 2 / r w o f t h e t a r g e t n u c l e u s a n d t o t h e t i m e - d e p e n d e n t p r o j e c t il ep o t e n t i a l

$ 2 Z p e 2 Z p e 2- = - 7 ( 1 - / ~ a z ) ~ ( 5.4 1)!r p ( t ) r p ( t ) 'w h i c h i s o b t a i n e d b y L o r e n t z t r a n s f o r m i n g t h e p o t e n t i a l - Z p e 2 / r ~ ( t ) f r o mthe p r o j ec t i l e t o t he t a r ge t f r ame , see Eqs . ( 4 . 34 ) and ( 4 . 32) .

S imi l a r ly , i n t he p r o j ec t i l e f r ame , one ha s

0 ~ ' ( r ~ , t' ) H ' ~ ' , t'i h ~ 7 - ( r ~ ) (5.42)w i t h

H ~ - i h c ~ . ~ Tp _ S - 2 Z T e 2 Z P e 2- ~ + m~ c27 ~ (5 .43)t7"T ~'p


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5 .3 . M O V I N G C O U L O M B P O T E N T I A L S 109T h e H a m i l t o n i a n ( 5 . 4 3 ) d e s c r i b e s a n e l e c t r o n s u b j e c t t o t h e C o u l o m b p o -t e n t i a l - Z p e 2 / r p o f t h e p r o j e c t il e n u c l e us a n d t o t h e t i m e - d e p e n d e n t t a r g e tp o t e n t i a l

_ S - 2 Z T e 2 Z T e 2rT ( t , ) ---- --~/ (1 +/~(~ z) rT ( t ' ) ' (5 .44)w h i c h i s o b t a i n e d b y L o r e n t z t r a n s f o r m i n g t h e C o u l o m b p o t e n t i a l- - Z T e 2 / r T ( t ' ) f r o m t h e t a r g e t t o t h e p r o j e c t i l e f r a m e , s e e E q s . ( 4 . 3 4 ) a n d(4 .32) .

F r o m t h e e x p l i c i t f o r m o f t h e o p e r a t o r s S 2 a n d S - 2 a n d f r o m E q . ( 4 .3 4 ) ,o n e s e e s t h a t t h e m o v i n g p o i n t c h a r g e s Z p e a n d Z T e , r e s p e c t i v e l y , p r o d u c ea n e l e c t r o m a g n e t i c i n t e r a c t i o n o f t h e f o r m p( I) - j - A / c w h e r e (I) - A ~ a n dA a r e t h e L i S n a r d - W i e c h e r t p o t e n t i a l s g i v e n i n E q s . ( 3 .3 4 ) . T h e L o r e n t zi n v a r i a n c e o f t h e s e e q u a t i o n s is d e m o n s t r a t e d i n S ec . 4. 1. 3.5 . 3 . 2 A b r i e f d i s c u s s i o n o f t i m eA t t h i s p o i n t , a b r ie f d is c u s s i o n o f t i m e m a y b e i n o rd e r , b e c a u s e t h e r o le o ft h e t i m e v a r i a b l e m a y e a s il y g iv e r is e t o m i s u n d e r s t a n d i n g s . I n t h e t a r g e ts y s t e m ( o r l a b o r a t o r y s y s t e m ) , t h e c o o r d i n a t e s r T a n d t a r e i n d e p e n d e n tv a r i a b le s , t h a t is , w e m a y a s s ig n t o e a c h s p a c e p o i n t r T o f t h e e l e c t r o nd e n s i t y d i s tr i b u t i o n t h e s a m e t i m e c o o r d i n a t e t . T h e c lo c k t o m e a s u r e t h et i m e t m a y b e p r o v i d e d , f o r e x a m p l e , b y t h e m o t i o n o f t h e p r o j e c t il e . I np a r t i c u l a r , t h e z e r o - p o i n t o f t h e t i m e a x i s c a n b e c h o s e n t o c o i n c id e w i t ht h e m o m e n t w h e n t h e p r o j e c t i l e is c l os e s t t o t h e t a r g e t n u c l e u s . T h e n t h ep r o j e c t i l e c o o r d i n a t e i n t h e t a r g e t f r a m e is g i v e n b y R = b + v t , s ee F i g . 3 .2 .N e v e r t h e l e s s , t is t h e t i m e c o o r d i n a t e o f t h e e l e c t ro n , n o t o f t h e p r o j e c t il e .

T h e d i s t i n c t i o n b e t w e e n t h e t i m e v a r i a b l e s a s s o c i a t e d w i t h t h e e l e c t r o nm o t i o n a n d a s s o c i a t e d w i t h t h e p r o j e c t i l e m o t i o n b e c o m e s c l e a r w h e n w ep e r f o r m a L o r e n t z tr a n s f o r m a t i o n t o th e p r o j e c ti le i n e rt ia l fr a m e . F r o mE q . ( 3 . 3 1 ) , t h e r e l e v a n t c o o r d i n a t e s t r a n s f o r m a s

- v t )p - ~ (ZTt = ( 5 . 4 5 )

O n e s e e s t h a t t h e t i m e c o o r d i n a t e t ' o f t h e e l e c t r o n i n t h e p r o j e c t i l e D a m ed e p e n d s o n t h e p o s i t i o n o f t h e e l e c t r o n in t h e t a r g e t f r a m e . T h i s s h o w st h a t , e v e n t h o u g h t h e v a r i a b l e t c a n b e u s e d t o d e s c r i b e t h e t r a j e c t o r y , i ti s the e l e c t r o n i c t i m e a s s o c i a t e d w i t h t h e s p i n o r ~ p (r , t ) .

I f o n e w i s h e s t o a s s i g n a t i m e c o o r d i n a t e to t o t h e p r o j e c t i l e i n t h et a r g e t f r a m e , c h o o s i n g a g a i n t h e c l o s e s t a p p r o a c h t o t h e t a r g e t n u c l e u s f o rt h e d e f i n i t i o n of t h e z e r o - p o i n t , t h e n t h e z - c o o r d i n a t e Z o f t h e p r o j e c t i l e


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1 1 0 C H A P T E R 5 . I O N - A T O M C O L L I S I O N S

i n t h e t a r g e t f r a m e is a l so ze r o a t t h i s m o m e n t , a n d R = b + v t 0 . I n it so w n r e s t s y s t e m , i n s t e a d o f E q . ( 5 . 4 5) , t h e s p a c e - t i m e c o o r d i n a t e s o f t h ep r o j e c t i l e a r e

Z I = 0t~o - t o / ? . (5 .46)

T h e l a st e q u a t i o n e x p r es s es t h e f a c t t h a t t h e p r o p e r t i m e i s s h o r t e r t h a n t h et i m e m e a s u r e d i n a c o o r d i n a t e f r a m e m o v i n g w i t h r e s p e c t t o t h e r e s t f ra m e .I t is i m p o r t a n t t o d i s t in g u i s h t h e t w o a s s i g n m e n t s o f t h e t i m e c o o r d i n a t ed i s c u s s e d h e r e . W h e r e a s i n t h e f o r m e r c a se , E q . ( 5 .4 5 ) , fo r a q u a n t u ms y s t e m , s p a c e a n d t im e c o o r d i n a t e s a r e i n d e p e n d e n t , i n t h e l a t t e r c a s e,Eq . (5 .46 ) , fo r a c l a s s i ca l t r a j ec t o ry , t he space coo rd i na t e s a re a func t i on o ft i m e . F o r o u r c o n s i d e r a t i o n s , w e w il l a l w a y s u s e t h e i n d e p e n d e n t e l e c t ro n i ct im es t a nd t ~.5 . 3 . 3 A s y m p t o t i c s o l u t i o n sT h e p o t e n t i a l s a p p e a r i n g i n t h e t w o - c e n t e r t i m e - d e p e n d e n t D i r a c e q u a t io n s( 5 .3 9 ) a n d ( 5 .4 2 ) a r e o f l o n g r a n g e s o t h a t c o n v e n t i o n a l s c a t t e r i n g t h e o r yi s no l onger app l i cab l e w i t hou t spec i a l p recau t i ons . In gene ra l , i t i s conve-n i e n t i f n o t n e c e s s a r y t o e l i m i n a t e f r o m t h e i n it ia l - a n d f i n a l - s ta t e e l e c t r o n i cw a v e f u n c t i o n s t h e a s y m p t o t i c t i m e d e p e n d e n c e c a u s e d b y t h e l o n g - r a n g eC o u l o m b i n t e r a c ti o n . T h i s c a n b e d o n e b y i m p o s i n g a p p r o p r i a t e C o u l o m bb o u n d a r y c o n d i t i o n s, w h i c h is t h e s a m e a s a p p l y i n g a n a p p r o p r i a t e g a u g et r a n s f o r m a t i o n , s o t h a t t h e r e s u l ti n g tr a n s i t i o n a m p l i t u d e c o n t a i n s a s h o r tr a n g e i n t e ra c t io n . H e r e " s h o r t r a n g e " m e a n s t h a t t h e p e r t u r b a t i o n f allso ff m o r e r a p i d l y t h a n 1 / r a s a f u n c t i o n o f t h e e l e c t r o n - n u c l e u s s e p a r a t i o n .T h e i m p o r t a n c e o f s u c h a p r o c e d u r e f or t h e t r a n s i t i o n a m p l i t u d e s h a s b e e nde m on s t r a t e d fo r non re l a t i v i s t i c [DEE85 , DEE8 6] and fo r r e l a t i v i s t i c co ll i-s i ons [E i c8 7a , ToE90b] .

I n o r d e r t o e s t a b l i s h w e l l - d e f i n e d c h a n n e l w a v e f u n c t i o n s t h a t a r e d e -v o i d of t h e a s y m p t o t i c t i m e d e p e n d e n c e c a u s e d b y l o n g - ra n g e i n t e r a c ti o n s ,w e h a v e t o d e r iv e t h e a s y m p t o t i c s o l u t i o n s of E q s . ( 5 .3 9 ) a n d ( 5 .4 2 ) fo rt ~ W e n o t i c e f ir st t h a t w h e n t h e i n t e r n u c l e a r s e p a r a t i o n is l a r g ec o m p a r e d t o t h e s e p a r a t i o n o f t h e e l e c t ro n f r o m t h e t a r g e t o r t h e p r o j e c t il enuc l eus , r e spec t i ve l y , we m ay rep l ace

r w ~ R = v / b 2 + v 2 t 2' R ' ~ / b 2 + v 2 t '2 (5 .47)p -- , =

H e r e , t a n d t ~ a r e t h e e l e c t ro n i c t i m e c o o r d i n a t e s , r e f e r r e d t o t h e t a r g e tan d p ro j ec t i l e f r am e , r e spec t i ve l y , s ee Secs. 5 .2 and 5 .3 .2 . T hu s R and


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5 .3 . M O V I N G C O U L O M B P O T E N T I A L S 111R ' meas u re the in te rnuc lea r s epa ra t ions a t t imes a s s oc ia ted wi th the e l ec -t ron ic po s i t ions r T o r r~ , r e s pec t ively . Thes e t imes a r e r e l a t ed acco rd ing toEq. (5 .45) in contra s t to (5 .46), va l id for the re la t ive m ot io n of the nucle i .A s y m p t o t i c e q u a t i o n sIn the l imi t s exp res s ed by E qs . (5 .47 ) , t he fo l lowing as ym pto t i c equa t ion shave to be sat isf ied [Eic87a]:in the t a r g e t f r ame , fo r an e l ec t ron a s ympto t i ca l ly bound to the t a r g e tnucleus :

r T R ' -~- meC2"y0 (I)~C FT ' t);( 5 . 4 s )

in the t a r g e t f r a m e , f o r a n e l e c t r o n a s y m p t o t i c a l l y b o u n d t o t h e p r o j e c t i l enucleus :

, + meC 2~ 0 (I)~ (rT , t);h~-t~ I)~(rT, t ) - - i h c ( ~ . V T(5.49)

in the p r o j e c t i l e f r a m e , f o r a n e l e c t r o n a s y m p t o t i c a l l y b o u n d t o t h e t a r g e tnucleus :

i h ~ O ~T ( r~ , t ' ) - i h c o z . V ~ p_S _2 Zwe 2

r TZp e 2

R ' + rnec27 ~ O~ r~(r~, , t ') ; (5 .50)

in the p r o j e c t i l e f r am e , f o r a n e l e ct r o n a s y m p t o t i c a l l y b o u n d t o t h e p r o j e c t i l enucleus :i h 0____~o~p ( r~ t ' )O t I - i h c o z . V ~

_ S _ 2 Z T e 2R

Z p e 2 \} q ~ p ~ , t .~- r n e c 2 7 ~ ( r } , ) ( 5 . 5 1 )rp /Since Eqs . (5 .48) an d (5 .49) are com ple te ly analo gous to Eqs . (5 .50)

and (5 .51) , we conf ine our d iscuss ion to the former express ions in the ta r -ge t s y s tem . Fo r conven ience , we in t roduc e the d imens ion less Som mer fe ldp a r a m e t e r s

ZT e2 Zp e 2a nd Up = ~ . (5 .52)t/T- -- V Vh


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112 C H A P T E R 5 . I O N - A T O M C O L L I S I O N STh e sam e Eqs. (5 .48) to (5 .51) a l so apply i f the e lec t ron i s not b o u n d to

the t a rge t o r p ro j ec t i l e nuc leus , bu t i n s t ead i s rep resen ted by a con t inuumsta te referred to the ta rget or to the pro jec t i le . This i s so , because in a fas ta tomic co l l i s ion , t he a sympto t i c i n t e rnuc lea r sepa ra t ion i s l a rge comparedto the co r respond ing d i s t ance o f t he wave packe t desc r ib ing the em i t t ede l ec t ron (o r pos i t ron ) f rom the emi t t i ng a tom.D e r i v a t i o n o f t h e a s y m p t o t ic s o l u t i o n sWe next show that the as ym pto t ic so lu t ions of Eqs . (5 .48) and (5 .49) fac-to r i ze i n to a phase fac to r t imes an unper tu rbed e igen func t ion sa t i s fy ing thesingle-center Dirac equat ion in the ta rget or pro jec t i le sys tem, respect ive ly ,which are g iven by

ih~-~ CT ( rT , t ) - - - - i h c ~ . V Ta n d

Z T e 2 c2~/o)~- m e ~pT (r T , t) (5.5 3)r T

c9 ' t / (i h ~ r ) = - i h c c ~ . V ~ p r ~ ~ - ? T t e 2 ~ 0 ~ p~ , r ~ , t ' ) . ( 5 . 5 4 )The asympto t i c so lu t ions to be cons t ruc t ed a re sa id to sa t i s fy Cou lomb

boundary cond i t i ons because they t ake in to accoun t t he a sympto t i c l ong-range po ten t i a l s caused by the d i s t an t cha rges . Indeed , t he pa ren thesesoccur r ing in Eqs . ( 5 .48)and (5 .53 ) d i f fe r on ly in t he t e rm - S 2 Z p e 2 / R ' ,while those o f Eqs. (5.51) an d (5.54) in the p rojec t i le fra m e differ only int h e t e r m - S - 2 Z T e 2 / R . The remova l o f an e l ec t romagne t i c fou r -po ten t i a lmay genera l ly be ach ieved by a gauge t rans fo rmat ion (4 .7 ) . We on ly haveto rewr i t e t he long- range po ten t i a l t e rm as t he fou r -de r iva t ive o f a gaugefunct ion X(x) in order to be ab le to speci fy the corresponding phase fac torof the w ave funct ion ac cord in g to Eq. (4 .7) .

As an expl ic i t i l lus t ra t ion , we der ive the asymptot ic so lu t ion ofEq . ( 5 .48) . The s imp les t way to remove the long- range pe r tu rb ing L iSnard -Wiecher t i n t e rac t ion

$2 Zp e 2 Zp e 2R ' = - 7 ( 1 - 3 ( ~ z ) R 1 (5.55)i s by t rans fo rming Eq . ( 5 .48) i n to the p ro j ec t i l e sys t em thus ob ta in ingEq . (5 .50 ). Here the pe r tu rba t ion has the ope ra to r - f ree Cou lom b fo rm- Z p e 2 / R I. In order to cancel th i s te rm, we observe the ident i ty

0 - - i ~ p l n ( v / b 2 + v 2 t ' 2 - v t ' ) 1 - i v p l n ( x / b 2 + v 2 t ' 2 - v t ' ) ( 5 . 5 6 )e - - i b , p V eO t ! v / b 2 + v 2 t I2


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5 .3 . M O V I N G C O U L O M B P O T E N T I A L S 113w i t h t he d e f i n i t i on (5 .52 ) o f Up. Us i ng Eq . (5 .47 ) , t h i s can be wr i t t en a s

0 e _ i u p l n ( R ' - v t ' ) _ Z P e 2i h - ~ _ _ t~----- - -~ e- iup ln(R'-vt ' ) . (5 .57)

I f w e i n s e r t t h e a n s a t z 2

m !O~ r~(r~ , t ' ) e - i ~ p l n ( R ' - v t ' ) ~ ( r p , t ' ) (5.5s)i n t o E q . ( 5 .5 0 ) , w e f in d t h a t t h e t i m e d i f f e r e n t i a ti o n p r o d u c e s a t e r m o n th el e f t - h a n d s i d e , w h i c h i s i d e n t i c a l t o t h e l o n g - r a n g e p o t e n t i a l o n t h e r i g h t -h a n d s id e , s o t h a t t h e s e t e r m s c a n c e l e a c h o t h e r . T h e e x p o n e n t i a l s t h e na p p e a r a s c o m m o n f a c t o rs a n d c a n b e d i s c a rd e d , s o t h a t

ih ff--~ ~ r ( r ~ , t ' ) - ( - i h c a . V I p - S - 2 ZTe2rT t l ) ./ ( 5 . 5 9 )

B y L o r e n t z t r a n s f o r m i n g th e d i ff e r e n ti a l o p e r a t o r s a c c o r d i n g t o E q . ( 4 . 38 )we ge t

0i h s - l - ~ s - l ~ ( r ~ p , t ' ) - i h c S - l o z . XTTS-1_ S -1 ZT e2 ~ t~ S -1 -4- ?TteC2")/0 ~ r ( r p , t ' ) . (5 .60)P T /

U p o n m u l t i p l y i n g E q . ( 5 .6 0 ) w i t h S f r o m t h e l ef t, i d e n t i f y i n g t h e i n v e r s eL o r e n t z t r a n s f o r m a t i o n S - l ~ r ( r ~ , t ' ) - ~ T ( r T , t ) , a n d r ea li zi ng t h a t a c-c o r d i n g t o E q . (4 .3 1 ) $ 7 ~ 7 ~ - 1 , w e f in d t h a t ~ T ( r T , t ) i n d e e d s at is fi est h e s in g l e - c e n t e r D i r a c e q u a t i o n ( 5 . 5 3 ). T h i s m e a n s t h a t E q . ( 5 .5 8 ) e x -p r e s s e s t h e e x a c t s o l u t i o n o f t h e a s y m p t o t i c e q u a t i o n ( 5 .5 0 ) a s a p r o d u c to f a p h a s e f a c t o r a n d a s i n g le - c e n t e r e i g e n f u n c t i o n .

2A constant phase factor depending on the un it of length is dropped her e and in thefollowing phase factors.


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T h e p h a s e - d i s t o r t e d a s y m p t o t i c s o l u t i o n sF i n a l ly , t h e s o l u t i o n o f E q . ( 5 .4 8 ) i s o b t a i n e d b y t r a n s f o r m i n g b a c k t o t h el a b o r a t o r y s y s t e m , s o t h a t

(I)~ (rw , t ) - - e - - i u p l n ( R ' - v t ' ) @ T ( r T , t ) . (5 .61)T h e s a m e r e s u l t m a y a l s o b e d e r i v e d b y w o r k i n g i n t h e t a r g e t f r a m e f r o mt h e o u t s e t . I n t h is c a s e, b e c a u s e o f t h e t r a n s f o r m a t i o n t ~ = 7 ( t - V Z w / C 2 ) ,t h e t i m e d i f f e r e n t i a t i o n O / O t l e a d s to t h e c a n c e l l a t i o n o f t h e s c a l a r p o t e n t i a l- T Z p e 2 / R ' c o n t a i n e d in E q . ( 5 .4 8 ) t h r o u g h t h e id e n t i t y ( 5 . 5 5) , a n d t h ek i n e t i c e n e r g y t e r m c ~. V w l e a d s t o t h e c a n c e l l a t i o n o f t h e v e c t o r p o t e n t i a lp a r t 7/3 c~ z Z p e 2 / R ' .

W e st il l h a v e t o o b t a i n t h e e x a c t s o l u t i o n o f t h e a s y m p t o t i c e q u a t i o n(5 .49 ). S i nce i n t h i s case , t he l ong - ran ge i n t e ra c t i o n i s a s i m p l e C ou l om bp o t e n t i a l , t h e r e is n o n e e d fo r a t r a n s f o r m a t i o n t o t h e p r o j e c t il e s y s t e m . I nana l ogy t o Eq . (5 .61 ) we f i nd

(I)~ ( rT , t ) - - e i ' T l n ( R + v t ) ~ ) P ( r T , ) . (5 .62)T h e p h a s e f a c t o r o f E q . ( 5 .6 1 ) , in c o n t r a s t t o it s c o u n t e r p a r t i n E q .

( 5 .6 2 ) , c a r r ie s a d e p e n d e n c e o n t h e s p a c e c o o r d i n a t e s i n t h e l a b o r a t o r ys y s t e m t h r o u g h t h e r e l a ti o n t t = 7 ( t - - v z T / c 2 ) . W h i le t h i s s pa c e d e p e n d e n c ei s expected to g ive r i se to smal l e f fec t s in mos t cases , i t i s essen t ia l fo rr i g o r o u s l y c a n ce l li n g t h e m a g n e t i c p a r t o f t h e a s y m p t o t i c i n t e r a c t i o n ( 5 .5 5 ) .

A s h a s b e e n s a i d a b o v e , t h e e l i m i n a t i o n o f t h e a s y m p t o t i c s c a l a r a n dv e c t o r p o t e n t ia l s a c c o m p a n i e d b y a n a p p r o p r i a t e p h a s e c h a n g e of t h e w a v ef u n c t io n m a y b e i n t e r p r e t e d a s a g a u g e t r a n s f o r m a t i o n . O f c o u rs e , g a u g et r a n s f o r m a t i o n s m a y n o t o n l y b e a p p li e d t o c o n s t r u c t s o l u ti o n s sa t is f y i n ga s y m p t o t i c b o u n d a r y c o n d i t i o n s , b u t q u i t e g e n e r a l l y m a y b e e m p l o y e d t oa d d o r to r e m o v e t e r m s i n t h e H a m i l t o n i a n in o r d e r to f a c i li ta t e t h e p r o b l e ma t h a n d . F u r t h e r a p p l i c a t i o n s a r e g i v e n i n S ee . 5 .5 . S i n ce t h e s e t e r m sr e a p p e a r i n t h e c o n c o m i t a n t p h a s e f a c t o r s , t h e p h y s i c s r e m a i n s u n c h a n g e d .I n f l u e n c e o f t h e i n t e r n u c l e a r i n t e r a c t i o nI n t h e H a m i l t o n i a n ( 5 .4 0 ) , w e h a v e n o t i n c o r p o r a t e d t h e i n t e r n u c l e a r i n t e r-a c t i o n , w h i c h i s o b t a i n e d f r o m E q . ( 3 . 3 4 ) a s V pw - Z p Z T e 2 / (b 2 + ~ 2 v 2 7)89S i n c e t h i s t e r m d o e s n o t d e p e n d o n t h e e l e c t r o n i c s p a c e c o o r d i n a t e s , i t d o e sn o t c o n t r i b u t e t o d ir e c t t r a n s i t i o n s b e t w e e n o r t h o g o n a l s t a t e s . H o w e v e r , i nan e i k ona l t r ea t m en t , s ee See . 5 .2 , i t m ay be shown [B rM 92] t ha t t he i n -t e r n u c l e a r i n t e r a c t i o n e n t e r s i n t o t h e d i f f e r e n t i a l c r o s s s e c t i o n . F o r c h a r g et r a n s f e r , V p T c a n b e r e m o v e d b y a s u i t a b l e g a u g e t r a n s f o r m a t i o n , n a m e l yb y in c l u d i n g i n E q s . ( 5 .6 1 ) a n d ( 5 .6 2 ) a n o t h e r p h a s e f a c t o r w i t h a p h a s e


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5 .4 . T R A N S I T I O N A M P L I T U D E S 1 1 5p r o p o r t i o n a l t o + l n [( b 2 + ~ 2 v 2 t 2 ) - } ~ : ~ / v t] , r e s p e ct iv e l y . I n t h e t r a n s i t i o na m p l i t u d e , t h e s e a d d i t i o n a l p h a s e f a c t o r s a r e c o m b i n e d t o a t e r m ( b / a o ) 2 i~w i t h u = Z p Z T e 2 / v h , w h i c h d e p e n d s o n l y o n t h e i m p a c t p a r a m e t e r b . T h ei n t e r n u c l e a r i n t e r a c t i o n , t h e r e f o r e , d o e s n o t a f f e c t t h e t o t a l c r o s s s e c t i o nbu t on ly the d i f f e r e n t i a l c ross se c t ion (5 .84 ) , s e e Se e . 5 .4 .2 .

5 . 4 T r a n s i t i o n a m p l i t u d e sI n S e c. 5 .3 w e d e a l w i t h t h e s o l u t i o n s o f t h e t w o - c e n t e r D i r a c e q u a t i o n f ora n e l e c t r o n . O n c e w e k n o w t h e e x a c t w a v e f u n c t i o n , w e a r e a b l e t o d e r i v eq u a n t i t i e s o f p h y s i c a l i n t e re s t , f o r e x a m p l e , t h e p r o b a b i l i t y a m p l i t u d e s A ftf o r a t r a n s i t i o n f r o m a n i n i t i a l s t a t e i t o a f in a l s t a t e f . B e f o r e w e e s t a b l i s ha p e r t u r b a t i o n d e s c r i p t io n , w e w a n t t o d e fi ne t h e e n e r g y r a n g e i n w h i c hs u c h a n a p p r o a c h i s a p p l i c a b l e .

5 . 4 . 1 S l o w a n d f a s t c o l l i s i o n sF o r n o n r e l a t i v i s t i c c o l l i s i o n s , i t i s c o n v e n i e n t t o d i s t i n g u i s h l o w - e n e r g ya n d h i g h - e n e r g y c o l l i s i o n s , d e p e n d i n g o n w h e t h e r t h e p r o j e c t i l e v e l o c i t yi s s m a l l e r o r l a r g e r t h a n t h e v e l o c i t y o f t h e a c t i v e e l e c t ro n . I n t h e f o r m e rc a s e a n d i n t h e t r a n s i t i o n r e g i o n , t h e e l e c t r o n i s a b l e t o f o l l o w t h e n u c l e a rm o t i o n . T h i s r e q u i r e s a t h e o r e t i c a l d e s c r i p t i o n t h a t p u r s u e s i n d e t a i l t h et i m e e v o l u t i o n o f t h e s y s t e m d u r i n g t h e c o l li si on . T h e m o s t c o m m o n l y u s e dm e t h o d , t h e c o u p l e d - c h a n n e l ( o r c l o se - co u p l in g ) a p p r o a c h , e m p l o y s a n ex -p a n s i o n o f t h e t o t a l t i m e - d e p e n d e n t w a v e fu n c t io n i n t e r m s o f a p p r o p r i a t es t a t i o n a r y s t a t e s a n d t h e n c o n s id e rs t h e t i m e e v o l u ti o n o f t h e e x p a n s i o ncoeff ic ients, see See. 1.2.

I f t h e p r o j e c t i l e v e l o c i t y is m u c h l a r g e r t h a n t h e e l e c t r o n v e l o c i ty , t h ee l e c t ro n c a n n o t f ol lo w t h e p r o j e c t i le m o t i o n . C o n s e q u e n t l y , t h e p a s s ag eo f t h e p r o j e c t i l e a c t s as a s m a l l p e r t u r b a t i o n . T h i s i s t h e r e g i m e w h e r ep e r t u r b a t i v e m e t h o d s a r e a d e q u a t e .

T h e s u b d i v i s i o n i n t o t w o e n e r g y r e g i m e s i m p l i e s t h a t t h e t e r m " s l o w "o r " f a s t " h a s a m e a n i n g o n l y f o r a s p e c i f i e d c h a r g e n u m b e r Z , s i n c e t h en o n r e l a t i v i s t i c e l e c t r o n v e l o c i t y i n a B o h r o r b i t a l i s V n = ( ~ Z / n ) c w h e r ec~ = 1 / 1 3 7 . 0 3 6 i s t h e f i n e - s t r u c t u r e c o n s t a n t a n d n t h e p r i n c i p a l q u a n t u mn u m b e r . I n o t h e r w o r d s , if K - s h e ll e le c t r o n s o f h i g h - Z a t o m s a r e i n v ol v ed ,m u c h h i g h e r p r o j e c t i l e e n e r g i e s a r e n e e d e d t o r e n d e r p e r t u r b a t i o n t h e o r ya p p l i c a b l e t h a n f o r l o w - Z a t o m s . F o r e x c i t a t i o n a n d i o n i z a t i o n , t h e a t o m i cc h a r g e n u m b e r Z m a y b e i d e n t i f i e d w i t h t h e t a r g e t a t o m i c n u m b e r , w h i l ef o r e l e c t r o n t r a n s f e r , t h e l a r g e r v a l u e Z / n o f t a r g e t o r p r o j e c t i l e s e r v e s toa s se s s t h e a p p l i c a b i l i t y o f p e r t u r b a t i o n t h e o r y .


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1 I I I I I 1 I I I I I I I

Retativistic matching

_ls

2 ~ 6 flO 1 12 I~At mic charge number ZF ig u re 5 .1 . Col li sion energy in M eV /u for which energy ma tch ing accord ingto E q. (5 .63) is obta ined as a funct ion of the a tomic ch arge num ber for variouselectronic eigenstates .

T h e f o r e g o in g d is c u s s i o n re f e rs t o a n o n r e l a t i v i s ti c s i t u a t i o n . F o r r e la -t i v i s t i c co l li s ions , a ll ve l oc i t ie s ap p ro ac h t h e spe ed o f l i gh t so t h a t a com par -i son o f ve l oc i t ie s is no l onger m ea n i ng fu l . In t h i s case , i t is m o re ap p r op r i a t et o d e f i n e a m a t c h i n g e n e r g y T m p e r a t o m i c m a s s u n i t o r , c o r r e s p o n d i n g l y ,"~'m = 1 + T m [ M e V / u ] / 9 3 1 . 5 , s ee E q . ( 3 .1 3 ) , b y d e m a n d i n g t h a t t h e k i n e t i ce n e r g y o f a f re e e l e c tr o n t r a v e l i n g w i t h t h e s p e e d o f t h e p r o j e c t i le , e q u a l st h e b i n d i n g e n e r g y E b i n d of t he e l ec t ron i c s t a t e , o r

TrteC2("/m -- 1) - E b in d - - m e c 2 [ 1 - V /1 - ( a Z ) 2 ] , ( 5 . 6 3 )w h e r e t h e s e c o n d e q u a t i o n r e f er s t o a t o m i c l s 1 /2 s ta t e s , s ee T a b l e 4 .3 , w h i c ha l so c o n t a i n s t h e b i n d i n g e n e r g ie s o f o t h e r l o w - l y i n g h y d r o g e n i c s t a t e s . T h ed e f i n i t io n ( 5 .6 3 ) h a s t h e c o r r e c t n o n r e l a t i v i s t i c li m i t /3 = a Z . F i g u r e5 .1 shows t he m a t ch i ng ene rgy Tm ca l cu l a t ed f rom Eq . (5 .63 ) fo r d i f f e r -e n t a t o m i c s t a t e s a s a f u n c t io n o f t h e c h a r g e n u m b e r Z . T h e r e a r e o t h e rp o s s i b l e p r e s c r i p t i o n s l e a d i n g t o s i m i l a r e s t i m a t e s .F o r e x a m p l e , f o r a c o ll is io n i n v o lv i n g l s 1/2 s t a t e s o f u r a n i u m , t h e m a t c h -i n g e n e r g y o f t h e p r o j e c t i l e is T m = 2 4 0 M e V / u , s o t h a t " f a st " c o l l is io n s s t a r tf ro m , s ay , 1 G e V / u . A b o v e t h i s e n e rg y , w e e x p e c t p e r t u r b a t i v e m e t h o d s t ob e a p p l i c a b l e .


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5 .4 . T R A N S I T I O N A M P L I T U D E S 1 1 75 . 4 . 2 P e r t u r b a t i v e t r a n s i t i o n a m p l i t u d e s a n d c r o s s s e c -

t i o n sIn t h i s s ec t ion , we as sum e t h a t t he co l l i si on is su f f ic i en t ly ene rg e t i c an d t hei n t e r a c t i o n s u ff ic i e nt ly w e a k t o r e n d e r p e r t u r b a t i v e m e t h o d s v a li d. S i nc e w ewi sh to t r ea t a l l pos s i b l e t r ans i t i ons , i nc l ud i ng t he p rod uc t i on o f e l ec t ron -pos i t ron pa i r s , on t he s am e foo t i ng , i t i s conven i en t i n t h i s subsec t i on t o u set h e l a n g u a g e o f f ie ld t h e o r y i n t r o d u c e d i n S ec . 4. 2. 2 a n d a p p l i e d t o i o n - a t o mcol l i s ions in Sec . 5 .1 . In cont ras t to Sec . 5 .1 .3 , however , we assume, wi th inp e r t u r b a t i o n t h e or y , t h a t t h e p r o b a b i l i ty f o r p a i r p r o d u c t i o n r e m a i n s s m a l la t a l l t i m e s a n d h e n c e t h e v a c u u m s t a t e i s u n a l t e r e d d u r i n g t h e c o l l i s i o n .A s b e f o r e , w e a d o p t t h e i m p a c t p a r a m e t e r r e p r e s e n t a t i o n .I n o r d e r t o e x p r e s s m e a s u r a b l e q u a n t i t i e s , w e f i r s t h a v e t o f o r m u l a t e t h et r a n s i t io n a m p l i tu d e s A n t h a t p e r m i t u s to c a lc u l a te 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 t t r a n s i t i o n p r o b a b i l i t y

P ~ ( b ) - IA ~ ( b) t ~ ( 5. 64 )a n d t h e t o t a l c r o s s s e c t i o n

- / P (b )d2b ( 5 . 6 5 )' fibe t w een any pa i r o f spec i fi ed i n i t ia l and f ina l s t a t e s , i and f , r e spec t i ve ly .T h e S - m a t r i x , f r o m w h i c h th e t r a n s i t i o n a m p l i t u d e s m a y b e d e r i v ed , i sg e n e r a l l y g i v e n b y

S - T e x p - ~ d t H i ( t )o o

- 1 - - ~ d t l J ~ I t l )~ - - - ~ d r1 d t 2 U i ( t l ) H i ( t 2 ) + ' - - ( 5.6 6)

AH e r e T is t h e D y s o n t i m e - o r d e r i n g o p e r a t o r , a n d t h e i n t e r a c t i o n H a m i l t o -n i a n

H i ( t ) - / ~ ~ i n t d 3 r (5 .67)i s e x p r e s s e d a s a s p a c e i n t e g r a l o v e r t h e b a s i c H a m i l t o n i a n d e n s i t y

A A ~ Ai n t - - j A . I c - ( 5 . 6 8 )d e s c r i b i n g t h e i n t e r a c t i o n o f a n e l e c t r o n o r p o s i t r o n c u r r e n t ( 4. 8 ) w i t h a ne l ec t rom agn e t i c f ie ld . Th e f i e ld m a y be ex t e rn a l , t ha t i s c l a s si ca l, o r m a y


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1 1 8 C H A P T E R 5 . I O N - A T O M C O L L I S IO N Shb e q u a n t i z e d if p h o t o n s a r e e m i t t e d o r a b s o r b e d . T h e q u a n t i t i e s ~ a n d

A

_ ~ t ? 0 a r e t i m e - d e p e n d e n t f ie ld o p e r a t o r s . T h e s e o p e r a t o r s a r e u s u -a l ly e x p a n d e d a c c o r d i n g t o E q . ( 5 .8 ) , a n d b y c o n s t r u c t i n g s t a t e s o f o n e o rmor e i ndependen t pa r t i c l e s i n ana logy to Eqs . ( 4 . 69 ) and ( 4 . 70) , we mayc a l c u l a t e S - m a t r i x e l e m e n t s f r o m E q s . ( 5 . 6 6 ) .F i r s t - o r d e r t r a n s i t i o n a m p li tu d e i n a c o v a r i a n t f o r mI n p a r t i c u l a r , t h e f i r s t - o r d e r t r a n s i t i o n a m p l i t u d e a p p e a r s i n a m a n i f e s t l yc o v a r i a n t f o r m

Afi - S(fi ) i / A-- ~t ((I)fl -- e d 4 x - ~ 7 " ~ A , I(I)i), (5.69)w h e r e I O i) a n d IO f) d e n o t e s t a t e s i n t h e o c c u p a t i o n n u m b e r r e p r e s e n t a t i o nin t r o du ced in Sec. 4 . 2 .2 . For e l ec t r on exc i t a t i on , i on i za t ion , o r t r ans f e r ,we use i n i t i a l and f i nal s t a t e s o f t he t yp e ( 4 .69) and ob ta in

Ae,tr i /- ~ d a x E < 0 1 b f b 1 + b l + b i I 0> 5 . 7 0 )k+,/+w h e r e k + , l + a re q u a n t u m n u m b e r s c h a r a c t e r i z i n g p o s i t i v e - e n e r g y s t a te s( w h i le k _ , l _ a r e u s e d t o d e n o t e n e g a t i v e - e n e r g y s t a t e s ) . F o r e x c i t a ti o nand ion i za t ion , we r e fe r al l s t a t e s t o an o r th og on a l ba s i s o f t a r g e t s t a t e s .Af t e r c a r r y ing ou t a l l f i e ld - theor e t i c man ipu la t i ons , Eq . ( 5 . 70 ) s impl i f i e s t o

A e,tr i J ' d 4 e ~ f ~ P ~ i A , ( 5.7 1)fi ~ xF o r e l e c t r o n t r a n s f e r , w e i m a g i n e t h a t t h e w a v e f u n c t i o n s c e n t e r e d a r o u n dt h e p r o j e c t il e a r e e x p a n d e d i n t e r m s o f t a r g e t f u n c t io n s , s o t h a t o r t h o g o -na l i t y i s p r e se r ved and the vacuum i s de f ined w i th r e spec t t o t h i s ba s i s s e t .A t t h e e n d , o n e c a n r e t u r n t o p r o j e c t i l e - c e n t e r e d s t a t e s . W e a s s u m e f r o mt h e b e g i n n i n g t h a t t h i s h a s b e e n d o n e .

For pa i r c r ea t ion we use 10) a s t he i n i t i a l s t a t e and dAtp_b tp+ 0> as thef i n a l e l e c t r o n - p o s i t r o n s t a t e t o o b t a i n

APair i f d4 A= x < 0 1 bp+3p_ I 0 > 5 . 7 2 )k+,l_or Apair i / ~= ~ d a x e -~ p+ C p _ d , . (5.73)

Th e h igh e r - o r de r t e r m s in Eq . ( 5 .66 ) al low f o r exc i t a t i on , i on i za t ion ,o r c a p t u r e o f m o r e t h a n o n e e l e ct r o n o r f or t h e c r e a t io n o f m o r e t h a n


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5 .4 . T R A N S I T I O N A M P L I T U D E S 119o n e e l e c t r o n - p o s i t r o n p a i r. I n a d d i t i o n , w i t h i n t h e s p a c e o f s i n g l e -e l e c tr o ns t a t e s , t he se t e r ms y i e ld co r r ec t ions t o t he unpe r tu r bed wave f unc t ions ~P ior ~pf ap pe ar i ng in Eq s . (5 .71) an d ( 5 . 7 3 ) . A c o m p l e t e s u m m a t i o n o f t h eco r res po nd ing m at r ix e lem ents of Eq. (5 .66) replaces e i th er ~Pi or ~pf w i ththe exac t wave f unc t ions .

B e f o re l e a v i n g t h e g e n e r a l d i s cu s s i o n o f t h e i n t e r a c t i o n H a m i l t o n i a n( 5 . 67) , l e t us r e ca l l t ha t t he gauge t r ans f o r ma t ion ( 4 . 7 ) l e aves t he ba s i cDi r ac equ a t io n ( 5 .6 ) t o be sa t is f i ed by the f ie ld op e r a to r s ~p inva r i an t . Th ei n v a r i a n c e a l s o h o l d s f o r t h e i n t e r a c t i o n H a m i l t o n i a n

O X ) d 3 x _ HI (5.74)s i n c e i n t e g r a t i o n b y p a r t s t u r n s t h e t e r m O ) i/ O x ~ i n to a f our - d ive r gence

AO j ' / O x ~ o f t h e f o u r - c u r r e n t r 1 6 2 B y c o n s t r u c t i o n , t h i s f o u r -d i v e r g en c evan i shes f o r any e x a c t s o l u t i o n of t he D i r ac equa t ion , a nd h ence the i n t e r-a c t i o n H a m i l t o n i a n i s g a u g e i n v a r i a n t .

I n t h i s su bsec t io n so f ar , we have used the l angu age o f f ie ld t heo r y inor de r t o de r ive , f r om a s ing l e f o r mula , t he f i r s t - o r de r ampl i t udes f o r a l lp r ocesse s cons ide r ed in t h i s book . I n t he f o l l owing , howeve r , we w i l l r e tu r nt o t h e l a n g u a g e o f w a v e f u n c t io n s .E x p l i c i t t r a n s i t i o n a m p l i t u d e s f o r v a r i o u s p r o c e s s e sI t i s use f u l he r e t o r ewr i t e t he expr e ss ions f o r t he t r ans i t i on ampl i t udes o fthe m os t r e l evan t p r ocesse s i n som e mo r e de t a il . S ince , fo r conven ience ,w e w o r k i n t h e l a b o r a t o r y s y s t e m , w e h a v e t o e x p r e s s a l l q u a n t i t i e s i n E q s .( 5 .71) and ( 5 .73) i n t he t a r ge t sys t em . On the o th e r hand , t he p r o j ec t i l ep o t e n t i a l i n i t i a l l y i s b e s t f o r m u l a t e d i n t h e p r o j e c t i l e s y s t e m , s i m i l a r l y t h ef ina l - s t a t e wave f unc t ion f o r cha r ge t r ans f e r .(a) E x c i t a t i o n a n d i o n i z a t i o n

A ~ - - - - ~ d 4 x - ~ f ( r T , t ) e S - I - , / " A ; ( F ~ , t ' ) S r ). ( 5.7 5)S i n c e i n t h e p r o j e c t i l e s y s t e m t h e i n t e r a c t i o n i s s i m p l y C o u l o m b i c , w e g e tf r om Eq . ( 4 . 34 ) t he r e su l t

A ~ - i - ~ d t d 3 r T~ p / ( r T t ) S 2 Zp, -_ -7 - ~bi(rT, t) . (5.76)rp

(b) C h a r g e t r a n s fe r.L F = ( )d 4 x ( S - l C f ( r p , t ' ) ) e S - 1 7 " A , ( r ~ p , t ' ) S ~i ( rT , t ) (5 .77)


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120 C H A P T E R 5 . I O N - A T O M C O L L I S I O N Sw h e r e t h e f i n a l - st a t e e l e c t ro n i c w a v e f u n c t io n is t r a n s f o r m e d b a c k t o t h et a r g e t s y s t e m ( t h e b a r i n d i c a t e s t h e D i r a c a d j o i n t ~ - ~ p t ~ 0 ) , h e n c e

A t~ - - i V d t d 3 rw r ( r ; , t ' ) S ~ i ( r T , t ) . ( 5 .7 8 )rp(C) E l e c t r o n - p o s i t r o n p a i r c r e a t i o nApair - i - - ~ d t d 3 r T ~ t ( r T t ) S 2 Zp..~ = p+ , - r-Tp- ~p _ (r T , t) , (5 .79 )

s i m i l a r l y a s fo r exc i t a t i on , however w i t h t he i n i t i a l wave func t i on r ep l acedb y t h e p o s i t r o n w a v e f u n c t i o n ~ p _ a n d t h e f i n a l w a v e f u n c t i o n r e p l a c e d b yt h e e l e c t r o n w a v e f u n c t i o n ~ p+ . T h e t r a n s f o r m a t i o n m a t r i x S a n d i ts s q u a r ea re g i ven i n Eqs . (4 .29 ) and (4 .32 ) . T he i nde pen den t i n t e g ra t i on va r i ab l esi n al l c a s es a r e r w a n d t , so t h a t t h e L o r e n t z - t r a n s f o r m e d c o o r d i n a t e s h a v et o b e i n s e r t e d f o r r ~ a n d t t w h e n t h e i n t e g r a ls a r e e v a l u a t e d . F u r t h e ra p p r o x i m a t i o n s d e p e n d o n t h e k i n d o f w a v e fu n c t i o n a c t u a l l y i n s e r t e d f o rt he fou r - sp i no r s ~ i and ~ f .D i s t o r te d - w a v e B o r n a p p ro x i m a t i o n . T h e po s t a n d t h e p r i o r f o r mFor t he d i s cus s ion o f s ca t t e r i n g p roces ses , i t is o f t en u se fu l t o s t a r t f romt h e e x a c t e x p r e s s io n s f or t h e t r a n s i t i o n a m p l i t u d e . L e t ~ + ( t) b e t h e e x a c to u t g o i n g - w a v e s o l u t i o n i n t h e l a b o r a t o r y s y s t e m e v o l v i n g f r o m t h e i n i t i a ls t a t e i a n d ( I ) ~ ( t) b e th e e x a c t a s y m p t o t i c f in a l s t a t e . T h i s f in a l s t a t em a y b e a n a s y m p t o t i c t a r g e t eig en s ta te [e.g . (5 .61)] , occ urr i ng in a "d i rec tr eac t i on " l ik e exc i t a t i on , i on i za t i on , o r pa i r p rodu c t i on , o r i t m a y be ana s y m p t o t i c p r o j e c t i l e e i ge ns t a t e [ e.g . (5 .62 ) ], occu r r i ng fo r cha rge t r ans fe r .T h e n , b y d e fi n it io n , t h e e x a c t t r a n s i t i o n a m p l i t u d e is g i v en b y

A (+) ~- l im ( O ~ 1 ~ + ) .t---*o~ (5.s0)Thi s ve r s i on i s ca l l ed t he p o s t f o r m o f t h e t r a n s i t i o n a m p l i t u d e . A l t e r n a -t ive ly , the p r i o r f o r m o f t h e t r a n s i t i o n a m p l i t u d e is o b t a i n e d a t t ~ - o c a st he p ro j ec t i on o f t he ex ac t i ncom i ng -wa ve so l u t i on evo l v i ng i n t o the f i na ls t a t e f o n t o t h e e x a c t a s y m p t o t i c i n i t i a l s t a t e ( 5 . 6 1 ) , n a m e l y

A ( - ) = l im ( ~ f l ( I ) ~ ) . ( 5.8 1 )t--+ --OoFrom t he de f i n i ng i den t i ca l exp res s i ons , (5 .8 0 ) and (5 .8 1 ) , one m ay de r i vet h e b a s i c e q u a t i o n f o r t h e d i s t o r t e d - w a v e B o r n a p p r o x i m a t i o n .


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5 .4 . T R A N S I T I O N A M P L I T U D E S 1 2 1P r o v i d e d th e o v e r la p m a t r i x e l e m e n t s w i t h i n E q s . ( 5 .8 0 ) a n d ( 5 . 8 1)

van i sh fo r t ---, - o c and t ---, + oc , r e spec t i ve l y , one m ay w r i t e t he p o s t f o r ma s

i F)(+ - d th o~= d th

a n d t h e p r i o r f o r m a s_ _i f fh

h

i h ~oteP) (5.s2); I o }t qy -f- H - i h - ~ O ~ . o t 7 } ) (5.sa)

B e i n g e x a c t , b o t h e x p r e s s i o n s a r e i d e n ti c a l. I f a p p r o x i m a t e w a v e f u n c t i o n sa re subs t i t u t ed fo r t he exac t ones , t h i s i s no l onger t rue i n a l l cases . Th i st hen i s r e fe r r ed t o a s a p o s t - p r i o r d i s c r e p a n c y .

T h e p h y s i c a l m e a n i n g o f t h e e q u a t i o n s i s t h e f o ll ow i n g . I n t h e p o s tf o r m , t h e o p e r a t o r ( H - i h O / O t ) a c t i n g o n ( I ) ~ y i e l d s t h e p o t e n t i a l t e r m ,t h e r e s i d u a l i n t e r a c t i o n , n o t i n c l u d e d i n t h e d e f i n i t io n o f t h e a s y m p t o t i cf ina l s t a t e ( I)~ a t t ~ + oc ( fo r cha rg e t r ans fe r , t h i s is t he e l ec t ron - t a rg e ti n t e r a c t i o n ) . I n t h e p r i o r f o r m , t h e o p e r a t o r ( H - i S O / O t ) ac t i ng on ( I )~y i e ld s t h e p o t e n t i a l t e r m n o t i n c l u d e d i n t h e d e f i n it i o n o f t h e a s y m p t o t i c i n i-t i a l s t a t e ( I) ~ a t t ~ - o c ( fo r c h a r g e tr a n s f e r , t h i s is t h e e l e c t r o n - p r o j e c t i l ei n t e r a c t i o n ) .

T h e B o r n a p p r o x i m a t i o n is o b t a i n e d b y s u b s t i t u t i n g u n p e r t u r b e d in i-t ia l a n d f in a l s t a te s . E x a m p l e s o f t h e c o r r e s p o n d i n g o p e r a t o r ( H - i h O / O t )a re g i ven i n Eqs . (6 .56 ) an d (6 .57 ). A va r i e t y o f d i s t o r t e d - w a v e B o r n a p -p r o x i m a t i o n s c a n b e c o n s t r u c t e d b y i n s e r t i n g d i s t o r t e d s t a t e s , i n w h i c h as u i t a b l e p a r t o f t h e t o t a l H a m i l t o n i a n H is i n c l u d e d i n t h e d e f i n i ti o n o ft h e i n i t i a l a n d f i n a l s t a t e s . T h e o p e r a t o r ( H - i h O / O t ) t h e n p r o d u c e s t h ep e r t u r b a t i o n a s s o c i a t e d w i t h t h e a s s u m e d d i s t o r t i o n .D i f f e r e n t i a l c r o s s s e c t i o n sI n p r i n c i p l e , i t i s p o s s i b l e t o p e r f o r m m e a s u r e m e n t s i n w h i c h n o t o n l y t h ea t o m i c p r o c e s s, c h a r a c t e r i z e d b y t h e t r a n s i t i o n a m p l i t u d e A ~ ( b ) , i s iden-t i fi ed bu t a l so t he de f l ec t i on o f a heav y p ro j ec t i l e i s de t e c t ed d i r ec t l y , o rt h r o u g h t a r g e t r e co il (s ee S ec . 1 3. 2. 3) . W e f u r n i s h h e re th e a p p r o p r i a t efo rm u l as . Fo r t he h i gh co l l is i on ene rg i es cons i de red an d fo r sm a l l de fl ec -t i on ang l es , when t he e i k ona l app rox i m a t i on d i s cus sed i n See . 5 .2 i s va l i d ,


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122 C H A P T E R 5. I O N - A T O M C O L L I S IO N St h e d e p e n d e n c e o f t h e s c a t te r i n g a m p l i t u d e o n t h e t ra n s v e r s e m o m e n t u mt r a n s f e r r l a n d h e n c e o n t h e s c a t t e r i n g a n g l e 0 m a y b e o b t a i n e d b y F o u r i e rt r a n s fo r m i n g 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 t t r an s i t io n a m p l i t u d e Aft(b) .U s i n g t h e f a ct t h a t t h e i n i t i a l a n d f in a l p r o j e c t il e m o m e n t a h a v e v e r y n e a r l yt h e s a m e m a g n i t u d e Mp') 'v, o n e m a y w r i t e t h e d i f f e r e n t i a l c r o s s s e c t i o n i nt h e c e n t e r - o f - m a s s s y s t e m a s [ B r M 9 2 , D E E 9 4 ]

do"ftdf t = I f ( O ) l ( 5 . 8 4 )

w i t h 3- - i M p T " / v / Aft(b) b2i~'e i~7b d2b, (5.85)A (O ) - 2 ~

w h e r e M p T - - M p M T / ( M p + M T ) is t h e r e d u c e d m a s s o f t a r g e t a n d p r o -j e c t il e a n d t h e t r a n s v e r s e m o m e n t u m r l w i t h r l . v - 0 is r e l a t e d t o t h es c a t t e r i n g a n g l e 0 i n t h e c e n t e r - o f - m a s s s y s t e m b y

2MpT~/V 0s in - . ( 5 .86 )r / - h 2

T h e p h a s e f a c t o r b 2 i " w i t h t h e S o m m e r f e l d p a r a m e t e r u = Z T Z p e 2 / h vr e s u l t s f r o m t h e C o u l o m b d e f le c t io n o f t h e p r o j e c t i le b y t h e t a r g e t n u c l eu s .S i n ce t h e C o u l o m b r e p u l s io n t e r m c o n t r i b u t e s m o s t l y t o la r g e r s c a t t e r i n ga n g l e s , w h e r e t h e c r o s s s e c t i o n b e c o m e s v e r y s m a l l , i t i s o f t e n n e g l e c t e d[M oi85 , De E 91a ] .

F o r t r a n s i t i o n s o f t h e e l e c t r o n b e t w e e n s t a t e s w i t h t h e m a g n e t i c q u a n -t u m n u m b e r s p i a n d p f , t h e i n t e g r a t i o n 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 nr / a n d v i n E q . ( 5 .8 5 ) c a n b e c a r r ie d o u t , so t h a t

d odr2 iM pT,.,/v f A ft(b ) ~.,2i,T ]2U J ] . f - - , i [ (~]b) b db ( 5 . 8 7 )hw h e r e J l t t f - t z i [ d e n o t e s t h e B e s s e l f u n c t i o n o f t h e f ir s t k i n d .

E q u a t i o n s ( 5.8 5 ) a n d ( 5 .8 7 ) p r o v i d e a r e l a t i o n 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 t e r i n g a n g l e w h i c h i s n o t s i m p l y b a s e d o n t h e c l a s s i c a lo n e - t o - o n e r e l a t io n ( 3. 20 ) b e t w e e n t h e t w o q u a n t i t i e s . I n s t e a d , t h e w h o l er a n g e o f i m p a c t p a r a m e t e r s c o n t r i b u t e s t o a g i v e n s c a t t e r i n g a n g le , a n d b y aF o u r i e r t r a n s f o r m a t i o n , i t is s e e n t h a t t h e o p p o s i t e i s a ls o t r u e . E v e n t h o u g hf i n i t e d e f l e c t i o n s a r e t a k e n i n t o a c c o u n t b y t h e i n t e g r a t i o n o v e r i m p a c t p a -r a m e t e r s , t h e s t r a i g h t - l i n e a p p r o x i m a t i o n f o r t h e p r o j e c t i l e t r a j e c t o r y c a nb e u s e d t o c a l c u l a t e t h e a t o m i c m a t r i x e l e m e n t s .

3An irrelevant phase factor ( l/a 0 ) 2iv is discarded he re and in the following phasefactors.


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5 . 5 . M A T R I X E L E M E N T S A N D G A U G E T R A N S F O R M A T I O N S 1235 . 5 M a t r i x e l e m e n t s a n d g a u g e t r a n s f o r m a -

t i o n sI n S e c . 5 . 4 , w e g i v e p e r t u r b a t i v e t r a n s i t i o n a m p l i t u d e s f o r v a r i o u s p o s s i b l er e a c t i o n c h a n n e l s . I n a ll t h e s e c as e s, a n d a c t u a l l y in m a n y o t h e r s , t h ep r o b l e m a r is e s o f c a l c u l a t i n g m a t r i x e l e m e n t s o f t h e D i r a c o p e r a t o r c~ o r it sc o m p o n e n t s . A s is k n o w n f ro m n o n r e l a ti v i st i c q u a n t u m t h e o ry , t h e r e a res o m e t i m e s d if f e re n t w a y s t o c a lc u l a t e a c e r t a i n m a t r i x e l e m e n t . A t y p i c a le x a m p l e i s t h e " l e n g t h f o r m " a n d t h e " v e l o c it y f o r m " o f t h e e l e c t ri c d i p o lem o m e n t . B o t h f o rm s ar e re l a t e d b y t h e o p e r a t o r i d e n t i t y

p = i ( m e / h ) [ H , r],w h i c h is v a l id if t h e H a m i l t o n i a n H d o e s n o t c o n t a i n v e l o c i t y - d e p e n d e n tp o t e n t i a l s ( a s , f or e x a m p l e , t h e s p i n - o r b i t in t e r a c t i o n ) . B y t a k i n g m a t r i xe l em en t s o f bo t h s i des o f t he i den t i t y be t we en s t a t e s li} an d If} w i t h e i genen -e r g i e s E i a n d E l , w e o b t a i n t h e r e l a t i o n

{fl P I } - i ( m e / h ) ( E f - Ei)(f l r l i )b e t w e e n t h e d i p o l e - v e l o c i t y a n d t h e d i p o l e - l e n g t h f o r m s o f t h e m a t r i x e l-e m e n t s . T h i s r e l a t i o n i s r i g o r o u s l y va l id a s l o n g a s e x a c t w a v e f u n c t i o n sa r e e m p l o y e d in th e e v a l u a t i o n ; h o w e v e r , if a p p r o x i m a t e w a v e f u n c t i o n s a r eu s e d , b o t h s i d e s m a y d if fe r. T h i s r e l a t i o n c a n a l so b e i n t e r p r e t e d a s t h er e s u l t o f a g a u g e t r a n s f o r m a t i o n , a s is s h o w n b e lo w .

I n a s i m i l a r w a y , w e d i s c u s s g e n e r a l r e c i p e s f o r t r a n s f o r m i n g m a t r i x e l -e m e n t s i n v o l v i n g t h e D i r a c c~ o p e r a t o r . T h i s i s o f p a r t i c u l a r i m p o r t a n c e i ft h e L i ~ n a r d - W i e c h e r t i n t e r a c t i o n o c c u r s i n t h e t r a n s i t i o n a m p l i t u d e s s u m -m a r i z e d i n S e c . 5 . 4 , b e c a u s e l a r g e c a n c e l l a t i o n s a p p e a r f o r h i g h p r o j e c t i l ev e l o c it ie s , so t h a t g r e a t c a r e m u s t b e e x e r t e d i n t h e e v a l u a t i o n . F o l l o w i n gt h e g e n e r a l tr a n s f o r m a t i o n , w e d i sc u s s t h e r e p r e s e n t a t i o n a n d t r a n s f o r m a -t i o n o f t h e L i S n a r d - W i e c h e r t i n t e r a c t io n in w a v e - n u m b e r s p ac e .5 . 5 . 1 R e f o r m u l a t i o n o f a - m a t r i x e l e m e n t sL e t u s c o n s i d e r e i g e n s t a t e s li} a n d If} o f t h e a t o m i c H a m i l t o n i a n f o r t h en u c l e a r c h a r g e n u m b e r Z T , t h a t i s

H o - i h c o z . V Z T e 2- - ~ m ~ c 2 " ~ ~ (5 .88)rw i t h e i g e n e n e r g i e s Ei a n d Ef. I f g ( r ) is a n a r b i t r a r y d i f fe r e n t ia b l e f u n c t i o no f r , we have

[H0, g ( r ) ] = - i h c o z . V g ( r ) ( 5 . 8 9 )


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124 C H A P T E R 5 . I O N - A T O M C O L L I S I O N Sa n d t h e c o r r e s p o n d i n g m a t r i x e l e m e n t

E f - E i{f lc~ . V g( r ) l i } - i hc {f lg ( r )l i}" (5 .90)I n p a r t i c u l a r , f or g ( r ) = x j , w h e r e Xj is one o f t he spa ce co m p one n t s o fr = (x , y , z ) , we have t h e r e l a t i on

E f - - E i(fl aj li} - - i h--------~ f Xj l i) (5 .91)b e t w e e n t h e v e l o c i t y a n d t h e l e n g t h f o r m o f t h e r e l a t i v is t i c d i p o le m a t r i xe l e m e n t .W he rea s t h e r i g h t -h an d s i des o f Eqs . (5 .91 ) and (5 .90 ) y i e l d f i n it e re -su l t s if non re l a t i v i s t i c Sch rhd i ng e r wave func t i ons a re i n se r t ed , t he l e f t-h a n d s id e s w o u l d y i e ld z e ro , b e c a u s e t h e D i r a c c~ m a t r i x c o n n e c t s l a r gea n d s m a l l c o m p o n e n t s o f t h e w a v e f u n c ti o n s . W e m a y th e r e f o r e e x p e c tt h a t t h e r i g h t - h a n d s i d e s a r e l e s s s e n s i t i v e t o a p p r o x i m a t i o n s m a d e i n t h ee v a l u a t i o n .C o n n e c t i o n w i th g a u g e t r a n s f o r m a t i o n sA m o r e g e n e r a l a n d m o r e p o w e r f u l m e t h o d f o r r e f o r m u l a t i n g m a t r i x e l e -m e n t s is p r o v i d e d b y g a u g e t r a n s f o r m a t i o n s . I n a n a l o g y to th e n o n r e l a -t i v i s t i c c a s e [ S a A 5 1 , E i c 7 4 ] , t h e c o n n e c t i o n w i t h g a u g e t r a n s f o r m a t i o n s i sa s fol lows : i f t he sy s t em desc r i bed b y Eq . (5 .8 8 ) i s sub j ec t e d t o an ex t e rn a lv e c t o r p o t e n t i a l A , s e e E q . ( 4 .1 3 ) , th e r e s u l t i n g H a m i l t o n i a n is a f u n c t i o n a l

H { A } = H 0 + H I { A } ( 5.9 2)w i t h H i { A } = e c ~ - A . T h e g a u g e i n v ar i an c e e x p r e ss e d b y E q . (4 .1 5) c a nn o w b e r e w r i t t e n a s

e eH { A } - e xp (i~ c cX ) H { A + V X } e x p ( - i ~ c c X ) , ( 5. 93 )o r , i n t he a bsence o f an ex t e rna l po t en t i a l , A = 0 ,

e . eH { V X } - e x p ( - i ~ c X ) H 0 e x p ( ~ c X ) . (5 .9 4)T h e r i g h t - h a n d s i d e c a n b e r e p l a c e d b y a n e s t e d c o m m u t a t o r e x p a n s i o nw h o s e f i r s t - o r d e r t e r m H i f o r a t i m e - i n d e p e n d e n t g a u g e f u n c t i o n ) / ( r ) i sw r i t t e n a s eH I { V X } - - e o ~ . V X - - i h cc [ H 0 , X ] , ( 5 . 9 5 )whi ch i s r ecogn i zed a s be i ng i den t i ca l t o Eq . (5 .90 ) i f g ( r ) = x ( r ) .


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5 .5 . M A T R I X E L E M E N T S A N D G A U G E T R A N S F O R M A T I O N S 1255 .5 .2 L i ~ n a r d - W i e c h e r t i n te r a c t io n in w a v e - n u m b e r

r e p r e s e n t a t i o nIn the t r ans i t ion ampl i tudes (5 .76 ) fo r exc i t a t ion and ion iza t ion and inE q . (5 .79 ) for pa i r p rodu c t ion , we have to ca lcu la te m a t r ix e l emen t s o f theL i ~ n a r d - W i e c h e r t i n t e r a c t i o n S 2 ( - Z p e / @ ) be tween the r e l evan t in i t i a l andf ina l s ta te s li) and I f} . I t is usefu l to cons ider the m atr ix e lem ents g enera l lybefore deal ing wi th the speci f ic cases .

S ince the t ime dependence r e s id ing in the e l ec t ron -p ro jec t i l e s epa ra t ion!rp ( t ) i s d i f f i cu l t t o hand le , one u s ua l ly r e s o r t s to the wave-number r ep re -

s e n t a t i o n ( o b t a i n e d b y F o u r ie r t r a n s f o r m a t i o n ) , i n w h i c h t h e c o m p l e t e t i m ed e p e n d e n c e i s m o v e d t o a n e x p o n e n t i a l. T h e t i m e i n t e g r a t i o n c a n t h e n b eca r r i ed ou t . I n s e r t ing th e op e ra to r S 2 f rom E q . (4 .32 ) and ad op t in g theb rack e t no ta t ion fo r the s pace in teg ra l in the t a rge t f r ame , we can wr i t ethe gene r i c ma t r ix e l emen t fo r exc i t a t ion , ion iza t ion , and pa i r p roduc t iona s

/ ? ~ 1 )M ~ ( b ) - c i ( E f- E i ) t/ u / f ( 1 - ~ C t z ) 2 - ; - i dr , (5.96)oc rp

s o tha t in E qs . (5 .76 ) and (5 .79 ) the t r ans i t ion ampl i tude i s exp res s ed a sA e , p a i r (b ) - i ~ v L , M ~ ( b ) . The coo rd ina tes a r e exp la ined in F ig . 3 .2 and- - + + - ( s . 9 7 )

i s the e l ec t ron -p ro jec t i l e d i s t ance me as u re d in the p ro jec t i l e f r ame .T h e r e l a t i v i s t i c B c t h e i n t e g r a lI n o rd e r t o c o n s t r u c 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 o f t h e i n t e r a c t io n , w eder ive the r e l a t iv i s t i c Be the in teg ra l lAmA8 1 , Moi8 5 ] . We s t a r t f rom thenonre la t iv i s t i c Be the in teg ra l

f ( 5 . 9 s )iqrr da r - q2i n w h i c h a c o n v e rg e n c e -e n f o rc i n g f ac t o r e x p ( - p r ) w i t h # ~ + 0 i s a s s u m e dwh en pe r fo rm ing th e in teg ra t ion . In the fo l lowing , we have to ma k e us e o fthe coo rd ina te t r ans fo rm at ion s (3 .29 ) and (3 .30 ) . W e a l so need the r e l a t ion

_ _ R- r p + R i n t h e t a r g e t s y s t e m a n d r p - r i n t h e p r o j e cti l e s y s t e m .Here , we decom pos e R = (b , v t ) and R ' = (b , v t ' ) i n to the t r ans ve r s eand the long i tud ina l co mp onen t s . S imi lar ly , we wr i t e q = ( qb , qz ) a n d

_ _ /r T (rTb, Zw), w her e r T b - - r p b - t- b .


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126 C H A P T E R 5 . I O N - A T O M C O L L I S I O N S

W e n o w c o n s i d e r t h e i n t e g r a l" 1 eiq .(rw_ R) d3 !r 7 r p f f l e i [ q b . r l p b + q z ( Z T _ V t ) ] 3 tm d r p

- - f f 7-#1 e i ( q b . r , p b + q z @ / 7 ) d 3 @r p (5 .99)!A l l c o o r d i n a t e s n o w b e i n g e x p r e s s e d b y r p , w e c a n a p p l y t h e B e t h e i n t

chapter 5 - ion-atom collisions

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