chapter 8 - charge exchange

8/12/2019 Chapter 8 - Charge Exchange

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C h a p t e r 8

C h a r g e E x c h a n g e8 . 1 I n t r o d u c t i o nThe transfer of an electron between target and projectile is one of thebasic processes in atomic collision physics. In this chapter, we considernonradiative or Coulomb capture, while the discussion of another process,namely electron transfer with the simultaneous emission of a photon or"radiative electron capture," is deferred to Chap. 9.

In general, charge changing (nonradiative and radiative) reactions canbe studied experimentally by passing an ion beam of known charge-statecomposition through a target and measuring the fraction of ions that haveeither captur ed or lost an electron. Such a measurement yields total crosssections. For testing details of the theory, it would be desirable to specifyexper imenta lly the states between which captur e occurs. This has beendone in a few cases, see Sec. 8.7.2.

From nonrelativistic capture theory it is known (see, e.g., [BrM92]) thatfor large projectile velocities v the nonradiative charge changing cross sec-tion decreases asy mpt otic ally as v -12 or E -6 in first-order and as v -11 orE-11/2 in second-order perturbation theory. This dramatic velocity depen-dence occurs because the overlap in momentum space between the initialand final electron wave functions quickly diminishes as the relative veloc-ity increases. The question then arises whet her there is any measurablecross section left at relativistic velocities. There is a simple quali tativereason why the relativistic cross section decreases less rapidly with the col-lision energy tha n th e nonrelativistic e stimat e would indicate: owing tothe Lorentz contraction of the electron space wave function in the beamdirection, the momentum wave functions acquire (Lorentz-) extended tails

207


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208 C H A P T E R 8 . C H A R G E E X C H A N G E

1 . 21 . 00 . 8

9 0.6

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0 . 20.0

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5.0 /i " \iiiiiiiiiiiij.1

. . . . . . . . . 9

' l 0 2 a ' ' ' J 0 ' 6 j '0 . 0 1 . . 0 3 1 0 4 1 0 5 . . 0 7 . 0E l e c t r o n m o m e n t u m p

F ig u r e 8 . 1 . O ve r l a p o f t a r ge t a nd p r o j ec t i le l s mo m e n tu m w a ve f unc tions a longthe be am ax is. The a tom ic wave func t ions for ZT = Zp = 40 a re taken to be non-re la t iv is t ic , the momenta a re g iven in un i t s o f m c c . Sol id l ine : momentum wavefunc t ion of the ta rge t ; do t ted , sho r t -dashed , and long-dashed l ines r epresen t thep r o j e c t i l e mome n tum w a ve f unc t ion L o r e n tz t r a ns f o r me d to t he l a bo r a to r y sy s -tem for p ro jec t i le ene rg ie s o f 0 .2 , 1 .0 , and 5 .0 GeV /u , r e spec t ively . The m axi m umampl i tudes a re normal ized to un i ty .

i n t h i s d i r e c t i o n a n d t h u s g i v e r i s e t o a m o r e s l o w l y d e c r e a s i n g o v e r l a p i nm o m e n t u m s p a c e .

T h i s i s i l l u s t r a t e d i n F ig . 8 . 1, i n w h i c h w e h a v e u s e d E q . ( 2 .2 8 ) t o c a l c u -l a t e t h e p r o j e c t i l e m o m e n t u m d i s t r i b u t i o n a s s e e n f r o m t h e t a r g e t s y s t e mf or v a r i o u s p ro j e c t i l e e n er g ie s . W i t h i n c r e a s in g e n er g y , t h e m o m e n t u m w a v ef u n c t i o n o f t h e p r o j e c t i l e a l o n g t h e b e a m a x i s i s s t r e t c h e d w i t h t h e L o r e n t zf a c t o r - 7 . C o n s e q u e n t l y , t h e o v e r l a p b e t w e e n t a r g e t a n d p r o j e c t i l e m o m e n -t u m w a v e f u n c t i o n s d e c r e a s e s l e s s r a p i d l y t h a n a c c o r d i n g t o n o n r e l a t i v i s t i ck i n e m a t i c s . I n t h e l a t t e r c as e, th e s h a p e o f t h e m o m e n t u m w a v e f u n c t i o nw o u l d b e r e t a i n e d a n d s i m p l y s h i f te d b y t h e p r o j e c t i l e v e lo c it y . A c t u a l c a l-c u l a ti o n s s h o w , a l m o s t i n d e p e n d e n t l y o f t h e a p p r o x i m a t i o n s i n vo l ve d , t h a tt h e r e l a t i v i s t i c c a p t u r e c ro s s s e c t i o n d ec r e as e s o n l y w i t h a b o u t E - 1 r a t h e rt h a n w i t h E - a .W e s t a r t t h e p r e s e n t a t i o n i n Se e. 8 .2 w i t h a d i sc u s s i o n o f t h e B o r na p p r o x i m a t i o n (o r t h e O p p e n h e i m e r - B r i n k m a n - K r a m e r s a p p r o x i m a t i o n )w h i c h i s t h e s i m p l e s t a n d h i s t o r i c a l ly f ir st a p p r o a c h t o r e l a t iv i s t i c e l e c t r o nc a p t u r e . I t a l r e a d y y ie l d s t h e c o r r e c t a s y m p t o t i c E - 1 e n e r g y d e p e n d e n c e ,


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8 .2 . THE B O RN APPRO XIM ATIO N 209b u t c r o ss s e c t i o n s t h a t a r e b y f a r t o o l a rg e . T h i s a p p r o x i m a t i o n i g n o r e s t h ed i s t o r t i o n o f t h e a t o m i c s t a t e s b y t h e l o n g - ra n g e C o u l o m b i n t e r a c t i o n . B yi m p o s i n g p r o p e r C o u l o m b b o u n d a r y c o n d i t i o n s i n S e c . 8 . 3 , o n e c a n o b t a i na s h o r t - r a n g e e f f ec t iv e i n t e r ac t i o n a n d r ea l i s t i c c ro s s s ec t i o n s . I n Sec . 8 .4w e t r e a t t h e e i k o n a l a p p r o x i m a t i o n w h i c h i s c o m p a r a t i v e l y e a s y t o i m p l e -m e n t a n d a l s o y i e l d s r e a s o n a b l e c r o s s s e c t i o n s . I n S e c . 8 . 5 , w e c o m p a r e t h er e s u l t s o f t h e t h e o r i e s d e r i v e d s o f a r w i t h o n e a n o t h e r a n d w i t h e x i s t i n gm e a s u r e m e n t s . I n S ec . 8 .6 w e d i sc u ss a t t e m p t s t o i m p r o v e th e t h e o r e t i c a lc r o ss s e c ti o n s b y u s i n g v a r i o u s d i s t o r t e d - w a v e m e t h o d s . I n S ec . 8 .7 , tw o -c e n t e r c o u p l e d - c h a n n e l c a l c u l a t i o n s a r e p r e s e n t e d a n d c o m p a r e d w i t h o t h e rt h e o r e t i c a l r e s u l t s a n d w i t h m e a s u r e d c r o s s s e c t i o n s .

W e u s e f ul l u n i t s i n t h e m a j o r p a r t o f t h i s c h a p t e r , e x c e p t f or t h e m o r et ech n i ca l s ec t i o n s 8 .2 .2 , 8 .4 .1 , an d 8 .4 .3 , w h e re w e ad o p t a t o mi c u n i t s h =me = e = 1, s ee A p p en d i x .

8 . 2 T h e B o r n a p p r o x i m a t i o n8 . 2 . 1 T h e c r o s s s e c t i o n i n f ir s t o r d e rA s s h o w n i n C h a p s . 6 a n d 7 , 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 , f i r s t - o r d e r p e r -t u r b a t i o n t h e o r y is a va l id a p p r o x i m a t i o n , e x c e p t f or v e r y sm a l l i m p a c tp a r a m e t e r s w h e r e t h e i n t e r a c t i o n m a y b e c o m e e x t r e m e l y st r o ng . F o r r e-a r r a n g e m e n t c o ll is io n s , t h e s i t u a t i o n is m o r e c o m p l i c a t e d . W e s t a r t o u rd i s c u s s i o n w i t h t h e s i m p l e s t a n d h i s t o r i c a l l y e a r l i e s t a p p r o a c h t o r e l a t i v i s -t i c e l e c t r o n c a p t u r e , n a m e l y t h e r e l a t i v i s t i c v e r s i o n [ M i t 6 4 , S h a 7 9 , M o S 8 0 ]o f t h e O B K a p p r o x i m a t i o n [ O p p 28 , B r K 3 0 ] . I t a ls o s e rv e s t o i n t r o d u c es o m e o f t h e c o n c e p t s a n d t h e n o t a t i o n n e e d e d f or a n y o t h e r t r e a t m e n t . I nt h i s m e t h o d , t h e i n i t i a l a n d f i n a l e l e c t r o n i c s t a t e s a r e a s s u m e d t o b e u n d i s -t o t t e d a t o m i c e i g en s t a t e s ~ai an d ~ af o f t a r g e t an d p ro j ec t i l e , r e s p ec t i v e l y .S t r i c t l y s p e a k i n g , t h i s a s s u m p t i o n i s n o t v a l i d s i n c e a C o u l o m b f i e l d l e a d st o d i s t o r t i o n s e v e n a t i n f i n it e d i s t a n c e s . T h e p r o b l e m o f a s y m p t o t i c d i s t o r -t i o n s i s d i s cu s s ed i n Secs . 6 .1 .2 an d 6 .5 .4 fo r ex c i t a t i o n an d i o n i za t i o n , an di t s r e l ev an ce fo r e l ec t ro n cap t u r e i s s t u d i ed i n Sec . 8 .3 .

F o r t h e t i m e b e i n g , w e m a y ju s t a d o p t t h e a m p l i t u d e ( 5.7 8) g i v e n b yp e r t u r b a t i o n t h e o r y a n d r e w r i t e i t i n t h e p r i o r f o r m ( 5 . 8 3 ) , w i t h t h e p e r -t u r b a t i o n ( 6 . 5 6 ) , a s

e2/ /~ - i - ~ d t d 3 rT lff (r~p) e iEf t, /hS Zp e_ ig i t / hr--~p qPi ( rT ) , (8 .1 )w h e r e E i a n d E f a r e t h e a t o m i c e i g e n e n e r g i e s i n c l u d i n g t h e e l e c t r o n r e s tm a s s , a n d 99i a n d p } a r e i n i ti a l a n d f in a l b o u n d - s t a t e s p i n o r w a v e f u n c t i o n s


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2 1 0 C H A P T E R 8. C H A R G E E X C H A N G Ei n t h e t a r g e t a n d p r o j e c t i l e f ra m e , r e s p e c ti v e ly . A t t h is p o i n t, w e m a yconv ince ou r se lves t ha t i n t he non re l a t i v i s t i c l im i t ( ene rg i es ci, e f) the phased e s c r i b i n g t h e t i m e o s c i l l a t i o n i n t h e p r o j e c t i l e f r a m e

( V Z T ) e 2 1 (v rw) + (8 .2)f t ~ - 7 E f t - ~ - me t + c f t + ~ rn ev 2 t - me . . . .c o n s i s ts o f a n i m m a t e r i a l c o n t r i b u t i o n f r o m t h e e l e c t r o n m a s s ( w h i c h is c a n -c e ll e d b y t h e c o r r e s p o n d i n g t e r m f r o m E l ) , t h e n o n r e l a t i v i s t i c e i g e n o sc i ll a -t i on , and t he t r an s l a t i on f ac to r , see Eqs . (1 .6 ) and (1 .9 ) . T hu s t he non re l -a t i v i s t ic l i m i t o f t h e t r a n s i t i o n a m p l i t u d e ( 8. 1) c a n b e o b t a i n e d a s a sp e c i a lcase. !B y i n t r o d u c i n g t h e F o u r i er t r a n s f o r m s ~ a n d ~ ' t h r o u g h

~ ( r T ) - - ( 2 7 r h )- ~ / @ ~ ( q ) e i q 'r T / h d 3 q (8.3)a n d

, 1 / ,qp~' (r~)2-7- -- (27 rh)- ~ O~ (P) ei pr ~ /h d 3 pPp ( 8 . 4 )w h e r e # , #~ d e n o t e t h e a n g u l a r m o m e n t u m p r o j e c ti o n s ( s u p p re s s i ng t h e! !l abe l j ) o f t he fou r - sp ino r s ~ [ ' , @i , ~ , and ~ whose sp in -an gu l a r pa r t isc o n s t r u c t e d a c c o r d in g to E q . ( 4. 87 ) w i t h a p p r o p r i a t e s p a ce o r m o m e n t u ma n g le s . F u r t h e r m o r e , b y d e c o m p o s i n g q = ( qb , q z ) as wel l as p = (PD ,Pz ) ,w e a r e a b l e t o m o v e a l l s p a c e - t i m e v a r i a b l e s t o t h e e x p o n e n t a n d t o p e r f o r mt h e i n t e g r a t i o n s o v e r r T a n d t , w h i c h y i e l d D i r a c d e l t a f u n c t i o n s i n t h em o m e n t u m v e c t o r a n d i n t h e e n e rg y . A s a r e s u l t, w e o b t a i n t h e t r a n s i t i o na m p l i t u d e

A~ ' - i27 rv7 --1 2Z e f ~ , t ( q b , q_ ) S~.U l ( q b , q+ )e i q b . b / h d 2qb , ( 8 . 5 )w h e r e t h e z - c o m p o n e n t s o f t h e m o m e n t a q a n d p , r e s p ec t i ve l y , a r e

q+ = (El - E f / 7 ) / vq - = ( E i / 7 - E f ) / v (8.6)

w i t h t h e n o n r e l a t i v i s t i c l i m i t sq - + 1 -~ m e V . ( 8 . 7 )

H e r e , th e + o r - s ig n i n d i c a t e s t h a t t h e r e l e v a n t l o n g i t u d i n a l m o m e n t u mp o i n t s f o r w a r d o r b a c k w a r d , w i t h r e s p e c t t o t h e b e a m d i r e c t i o n .


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8.2 . T H E B O R N A P P R O X I M A T I O N 2 1 1In an exper im en t , i t w i l l no rm al ly no t be pos s ib l e t o spec i fy t he i n i -

t i a l a n d f i n al m a g n e t i c s u b s t a t e s . W e a r e t h e r e f o r e i n t e r e s t e d i n t h e c r o sssec t i on

1 f IA ~,. ]2rfi = 2 j + 1 ~ d2b (8.8)ave raged over t he i n i t i a l and sum m ed over t he f i na l p ro j ec t i ons p , p~ , r e -s p e c ti v e ly , o f t h e e l e c t r o n a n g u l a r m o m e n t a j , j~ . I n e v a l u a t i n g E q . ( 8. 8)w i t h t h e a m p l i t u d e ( 8 .5 ) , it is c o n v e n i e n t t o c o n s t r u c t d e n s i t y m a t r i c e s f r o mthe i n i t i a l an d f i na l fou r - sp ino r s 93i and ~ f and t o pe r f o rm the sum over t he( c u r r e n t ly u n o b se r v a b le ) a n g u l a r m o m e n t u m p r o j e c ti o n s w i t h i n each o f t hedens i t y m a t r i ces [E i c85 ] . Hence we de f i ne t he m a t r i ces

Pi -- E P'~ - E @~ (qb, q+ ) ~5~ (qb, q+ ) (8.9 )p #

a n dr f - E Pf~' - E 0~' (qb, q -) 0~ t (qb, q- ) . (8 .10)

t d t t ~T h e s e e x p r e s s i o n s a r e d i a g o n a l i n t h e m o m e n t a ( qb , q + ) b e c a u s e t h e b -i n t e g ra t i on i n Eq . (8 .8 ) p rod uce s a D i ra c de l t a fu nc t i on i n qb , q~ owing t ot h e e x p o n e n t i a l e x p ( i q b , b / h ) i n Eq . (8 .5 ) . Th e t o t a l c ros s s ec t i on i s noww r i t t e n i n t h e c o m p a c t f o r m

crfi = 2j + 1 fl2,,/2 hcc Z~ T r { S Pf S Pi} d 2 q b , (8.11)w h e r e t h e i n t e g r a n d is e x p r e s s e d a s t h e t r a c e o f a p r o d u c t o f 4 x 4 m a t r i c e s .T h e t a s k i s r e d u c e d t o c o n s t r u c t i n g t h e m a t r i c e s P i a n d P f fo r e a c h i n d iv i d -u a l c as e . O w i n g t o a x i a l s y m m e t r y o f t h e c o l li si o n, t h e r i g h t - h a n d s id e o fEq . (8 .11 ) is a one -d im ens io na l i n t eg ra l . I t t u r ns o u t t ha t i n t h i s case , t hepos t and p r i o r fo rm s o f t he c ros s sec t i on as de fi ned i n Eqs . (5 .82 ) and (5 .83 )a r e i d e n t ic a l . F r o m E q . ( 8. 11 ) a n d t h e "y d e p e n d e n c e o f t h e t r a n s f o r m a t i o nm a t r i x S i t f o l l o w s t h a t t h e c r o s s s e c t i o n d e c r e a s e s a s y m p t o t i c a l l y a s ? - 1o r E -1 a s t he co l li s ion ene rgy E i nc reases . Th e genera l dep end enc e ont h e p a r a m e t e r s c h a r a c t e r i z i n g t h e c o l l i s i o n c a n b e b e s t s e e n i n t h e s i m p l eana ly t i ca l exp res s ion (8 .52 ) .

8 . 2 .2 C a p t u r e f r o m i n it ia l 1 s - s t a t e s i n to f in a l 1 s - s t a t e sT h e p r e s e n c e o f h i g h - m o m e n t u m c o m p o n e n t s i n i n i ti a l a n d f i n al s t a t e s r e n -d e r s l s - l s c a p t u r e t h e m o s t i m p o r t a n t c a s e a t c o l l i s i o n e n e r g i e s h i g h a b o v ethe m a tch ing ene rgy , s ee F ig . 5 .1 .


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212 C H A P T E R 8. C H A R G E E X C H A N G EFro m the spinor wave func t io ns Eq. (4 .86) for the hydrog enic 1s-s ta te

1 (we use a tomic un i t s i n t h i s subsec t i on)

~ s ( r ) - ( 4 7 r ) - 8 9 ( i f ( r ) ( a . ~ ) Xu ~ (8.12)!wi th ~ = r / r , one obta ins the Four ie r t ransforms ~5~s and ~ by i nve r t i ng

the def ining equa t ions (8 .3) and (8 .4) , respec t ive ly , and by inser t ing thepar t ia l -w ave expa ns io n (6 .42) of the plane wave . On ly the L = 0 andL = 1 t e rms i n t he expans io n co n t r i bu t e t o t he i n t egra l s ove r g and f ,respec t ive ly , and leave us wi th radia l in tegra l s whose eva lua t ion yie lds therad i a l momentum func t i ons [E i c85] [ q ) ]N g F ( s T + 1) sin ( 8 -~- 1)arctan(~--TTai(q) = q ( Z 2 + q2)(ST+l)/2

1 N fF(ST) [ qbi(q) - - q-~ ( Z 2 + q 2 ) s T / 2 s i n 8 T a r c t a n ( ~ T )1 N fF (s T + 1) [ qq ( Z ~ + q2)(ST+1)/2 COS (8 T q- 1)arctan(~--TT ) , ( 8 . 1 3 )

wh ere q = (qb, q+ ), 8T i s the p ara m eter s def ined in Eq. (4 .94) for Z = ZT,and t he norma l i za t i on fac tor s N g and N f a re def ined in Eq. (4 .102) . As are su l t , we ge t t he Four i e r - t r ans formed sp inor

l a i q ) ~ s ) 8 .1 4 )~5~ (q) - - 7rv~ - b i (q)( t r . ( t )X ~

in a form tha t c lose ly resembles Eq. (8 .12) . W he n ins er t ing into Eq. (8 .9)a n d p e r f o r m i n g t h e s p i n s u m m a t i o n w e o b t a i n t h e d e n s i t y m a t r i x1 a 2 a b , 0 , )

Pi - 2~ 2 - a b ( a . ( t ) b 2 (8.15)

wh ich i s bui l t f rom blocks of 2 2 mat r ices . Here an d in the fol lowing,we do no t exp l i c i t l y d i sp l ay t he un i t ma t r i x I mul t i p ly ing t he d i agona lqua nt i t i es . In der iving Eq. (8 .15) , use i s m ade of the re la t ion (~r .u)(e r-v) =

11We use the notation X~s for the Pauli spinor with ~s --


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8.2 . T H E B O R N A P P R O X I M A T I O N 2 13

u . v + i c r - ( u x v ) . T h e s p i n s u m m a t i o n w i t h i n t h e d e n s i t y m a t r i x l e a ds t o as u b s t a n t i a l s i m p l i f i c a ti o n of t h e s u b s e q u e n t d e v e l o p m e n t . I n a n a n a l o g o u s

tf a sh i o n , w e m a y c o n s t r u c t ~ a n d P f. O n e o b t a i n s t h e q u a n t i t i e s a f a n dbf by repla c ing ZT --~ Zp , s T ~ S p - 1 , an d q = (q b , q + ) --+ P = (q b , q - )i n E q . ( 8 .1 3 ). F r o m P f w e t h e n a s s e m b l e t h e m a t r i x p r o d u c tS P f S = 3 ` + 187r4

~ t l l- - m 1 2 ( o " " ~1) + m i 2 ( O ' " e z )

- -? Y t12 O 1 )? Tt22n- ? T t i2 t T { ~ z ) )( 8 . 1 6 )

wher e ~z i s a un i t vec to r i n t he beam d i r ec t ion , and the coe f f i c i en t s a r er o l lf n l 2

tr r t l 2m 2 2

- - a ~ + 2 5 a f b f q _ / p + 52 b~a f b f ( 1 - 5 2 )

= - ~ 4 + 2 e a f b f q _ / p + b~ )- e ~ 4 + 2 e a f b f q _ / p + h i. (8.17)

Her e , 5 - [ ( 3 ` - 1 ) / ( 3 ' + 1)1 ~/2 a n d p - ( q~ + q2_ )1 /2 . I n t h e f i n a l m a t r i xm u l t i p l i c a t i o n a c c o r d i n g t o E q . ( 8 .1 1 ), w e o n l y n e ed c o n s i d e r t h o s e t e r m st h a t c o n t r i b u t e t o t h e t r a c e .T o t a l l s - l s c r o s s s e c t i o n sT h e O B K c ro s s s e c t io n f o r c a p t u r e o f a n ( u n p o l a r i z e d ) 1 s - e le c t ro n i n t o b o t hposs ib l e 1s - s t a t e s o f t he p r o j ec t i l e i s t he n ob ta in ed ( in a tom ic un i t s ) a s

.OBK __ 271.Z2 3` nt- 1 fo ~l s l s V2 ~ 2 Q ( q ~ ) d ( q ~ ) (8.18)w h e r e t h e i n t e g r a n d i s

2 m 1 2Q - r n ll a ~ + a i b i ( q ~ + q + q - ) 2 rn i 2 a i b i q+ + rn22b2 i (8.19)P q qa n d q - ( q~ + q 2 _ )1 / 2 . W e have u sed t he re la t i on Zp + q2_ _ Z~ + q~_ wh ichi s r i gor ous ly va l id f o r l s - l s t r ans i t i ons .I t i s poss ib l e t o de r ive a s imple ana ly t i c a l expr e ss ion ( 8 .52) f r om E q .(8 .18) which i s va l id for a Z


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214 CH APT ER 8. CHARGE EXC HAN GE

@in-flip cross sectionsI t i s i n s t ruc t i ve t o u se t he exp l i c i t fo rm ulas g iven i n [MoS 80] t o com parethe c ros s s ec t i ons a m p fo r sp in - f l ip and a non fo r non - sp in - f l i p t r an s i t i o ns i nl s - l s c a p t u r e . F o r a Z ~ 1 t h e r a t i o i s g i v e n b y

0 "flip 1 5 4= 1 (~2 Z p ) ( ~ ] " ( 8 .2 0 )O n~ 16 ( 1 - 8 9 [ + ~87 roz(ZT -l-

In t he l im i t o f l ow ve loc i t ie s , t h i s r a t i o q u i ck ly app roa che s ze ro a s onew o u l d e x p e c t s i n c e t h e m a g n e t i c f i el d g e n e r a t e d b y th e p r o j e c t i l e m o t i o n i nt h e t a r g e t f r a m e d e c r e a s e s w i t h d e c r e a s i n g v e l o c i t y . W h i l e s o m e c o r r e c t i o nt e r m s e n t e r i n g i n E q . ( 8 .2 0 ) a r e t r e a t e d i n t h e h i g h - v e l o c i t y l i m i t, t h eeq ua t i o n sho u ld never th e l es s g ive a u se fu l e s t im a t e fo r l ow ve loc i ti e s . F o rexa m p le , i n p + H co l li s i ons a t t he ene rg y o f 100 keV, Eq . (8 .20 ) y i e ld sa r a t i o o f 1.8 x 10 - l ~ w h e r e a s a n e x a c t n u m e r i c a l e v a l u a t i o n g i ve s t h er a t i o 3 . 6 x 1 0 - l ~ [ T oE 9 0 a] . O n t h e o t h e r h a n d , i n t h e e x t r e m e - r e l a t i v i s t i creg im e , 7 ~ oo o r 5 ~ 1 , t he r a t i o t en ds t o 1 /4 i n t he l im i t o f sm a l lc h a r g e s a n d d e c r e a s e s fo r h i g h e r c h a r g e s . T h e a p p r o x i m a t e v a l i d i t y o f E q .(8 .20 ) has be en co n f i rm ed [Dec90] for U + U co l li s ions by exac t n um er i ca le v a l u a t i o n o f t h e O B K c r o s s s e c ti o n s . F o r U + U c o ll is io n s a t 5 0 0 M e V / u ,t h e r a t i o s f o l l o w i n g f r o m v a r i o u s o t h e r a p p r o a c h e s a r e c o l l e c t e d i n T a b l e8.2, see See. 8.7.

I t s h o u l d b e r e m a r k e d t h a t s p i n - f l i p c a p t u r e o c c u r s f o r v e r y h i g h v e l o c -i ti e s e v e n w h e n t a r g e t a n d p r o j e c t i l e c h a r g e s t e n d t o z er o . T h i s i s d u e t ot h e D i r a c m a g n e t i c m o m e n t o f t h e f r ee e l e c tr o n . H e n c e , i f t h e w a v e f un c -t i o n s ( 8 . 1 2 ) a r e r e p l a c e d b y s o m e d i s t o r t e d w a v e f u n c t i o n s , t h i s w i l l o n l ya f fe c t th e t e r m s o f t h e o r d e r aZT a n d aZp i gno red in Eq . (8 .20 ). In o the rw o r d s , w e w il l e x p e c t E q . ( 8 .2 0 ) t o b e a p p r o x i m a t e l y v a l i d fo r m o r e g e n e r a ld i s t o r t e d - w a v e a p p r o a c h e s p r o v i d e d OLZT< < 1 a n d a Z p


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8.2. THE BO RN APPROXIM ATION 215T h e t r a n s i t i o n a m p l i t u d e i n t h e p o s t f o rm w i t h i n t h e s e co n d - o r d e r O B K

can be wr i t t en [ DeE 91a ] a sA (+) i f d t fd 3r T[~ (r~ p t ' ) ] t s - l (Z Te 2) @i(rT, t )OBK2 -- ~ ~ rT

i / d t / d 3 r T / d t l f darT [~b}(r~ t t ) ] t s - l ( Z T ( 3 2 )h 1 , r Tx G + ( r T , r T1 ; t - - t l ) S 2 D - - ~ i ( r r l , t l ) . ( 8 .21)fP1

Whi le the f i r s t t e r m desc r ibe s a f i r s t - o r de r o r s ing le - s t ep t r ans i t i on , t he sec -o n d t e r m c o m e s a b o u t b y a t w o - s t e p m e c h a n i s m . I n t h e f ir st s t e p , m e d i a t e db y t h 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 , t h e b o u n d t a r g e t s t a t e r i s e x c i te d t oa n i n t e r m e d i a t e f r e e - p a r t i c l e s t a t e w i t h m o m e n t u m k . I n t h e s e c o n d s t e p ,m e d i a t e d b y th e e l e c t r o n - t a r g e t i n t e r a c t i o n , t h e fr ee e l e c t ro n i s c a p t u r e di n t o t h e b o u n d p r o j e c t il e s t a t e ~ f. T h e e l e c t r o n m o t i o n in t h e i n t e r m e d i a t es t a t e i s de sc r ibed by the f r ee - f i e ld Feynman pr opaga to r [ Sak67 , Sch61]

1 / G+ (rT, rT1;S) e - i e ( t - t l ) / h d c (8.22)Go ( r T , rT 1; t - - t l ) - - 27rhw h e r e

(+) ( + ) t ( - ) ' r ' ( - ) t , ]9 ~k,s (rT) ~k, s (rT1) qPk,s~, T)99k,s ~,rT1)G0+ (rT, rT l , e) - - ~ + ~(8.23)

Her e , one ha s to t ake the l imi t r/ + 0 . T h e sp ino r s ~ ( +)-+ k ,s ( rT) an d ~( . 2 ( rT)f o r pos i t i ve and nega t ive ene r gy , r e spec t ive ly , sa t i s f y the f r ee - f i e ld D i r ace q u a t i o n

( - ich~. VT q- meC2"y0) ~(+) E(k=k) ~(,s ( rT) - - + l I k,s ( rT)" (8.24)T h e in i t i a l an d f inal wave f unc t ions ~ i an d ~ f i n E q . ( 8 .21) a r e t ake n a s hy-d r o g e n i c l s l / 2 D i r a c w a v e f u n c t i o n s . T h e s u m m a t i o n i n ( 8 . 2 3 ) e x t e n d s o v e ra l l f r e e - p a r t i c l e s p i n o r s w i t h i n t e r m e d i a t e m o m e n t u m k a n d b o t h p o s i t i v eand nega t ive ene r gy .

F r o m t h e a n a l y t i c a l f o r m o f E q . ( 8 .2 1 ), o n e m a y o b t a i n t h e a s y m p t o t i cbeh av io r o f t he O BK 2 c r oss sec t ion f or ve r y la r ge ene r g ies . E xp r e ssed int e r m s o f t h e L o r e n t z f a c t o r 7 , t h e a s y m p t o t i c a l l y d o m i n a n t d e p e n d e n c e i s[ JaA85 , DeE 91a ]

( lnT) 2(YOBK2 O( - - . (8.25)


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2 16 C H A P T E R 8. C H A R G E E X C H A N G E

E x cep t f o r t he s lowly va r y ing f ac to r ( ln 7 ) 2 , which i s c aused by th e in fi -n i t e r a n g e o f t h e C o u l o m b i n t e r a c t i o n a n d w o u l d d i s a p p e a r f o r a s c r e e n e dp o t e n t i a l , t h e e n e r g y d e p e n d e n c e i s t h e s a m e a s i n f i r s t - o r d e r p e r t u r b a t i o ntheor y .

T h e i n t e g r a l s o c c u r r i n g i n ( 8 . 2 1 ) h a v e b e e n e v a l u a t e d b y u s i n g a " p e a k -i n g a p p r o x i m a t i o n " [H U M 8 4, H U M 8 5], in w h i c h t h e i n t e g r a l is e v a l u a t e da t t h e m a x i m u m o f t h e i n t e g r a n d , a n d a re l a t ed " a v e ra g i n g a p p r o x i m a -t i o n " [M o i8 8, M o i 89 ]. W h i l e t h e f o rm e r a p p r o x i m a t i o n g a v e a m o d e r a t er e d u c t i o n o f t h e c r o s s s e c ti o n s w i t h r e s p e c t t o t h e O B K 1 v a lu e s , th e l a t-t e r a p p r o x i m a t i o n b r o u g h t a b o u t a l a r g e e n o u g h r e d u c t i o n t o a c h i e v e g o o da g r e e m e n t w i t h t h e e x p e r i m e n t a l d a t a [ C ra 7 9 , A n h 8 5 a ] i n m o s t ca s es . H o w -eve r , i t ha s been shown [ DeE 91a ] tha t an e x a c t n u m e r i c a l e v a l u a t i o n o f t h e7- d imens iona l i n t egr a l s r equ i r ed f o r t he second- or de r c r oss sec t ion us ing aM o n t e - C a r l o t e c h n i q u e [K aw 86 ] p o i n t s in t h e o p p o s i t e d i r e c ti o n : t h e e x a c tO B K 2 c r o s s s e c t i o n s a r e n o t s m a l l e r b u t l a r g e r t h a n t h e O B K 1 c r o s s s e c -t i o n s b y a b o u t o n e o r d e r o f m a g n i t u d e . T h i s i s, i n f a ct , a n a lo g o u s t o t h en o n r e l a t i v i s t i c c a s e w h e r e e x a c t O B K 2 c r o s s s e c t i o n s a r e l a r g e r t h a n t h ec o r r e s p o n d i n g O B K 1 c r o s s s e c t i o n s f o r h i g h b u t n o t y e t a s y m p t o t i c c o l l i -s ion ene r g ie s . For a mor e de t a i l ed d i scuss ion , see [ E ic90 , DeE 91a ,b , Moi92 ,DEE93] .W e c o n c l u d e t h a t p e a k i n g - t y p e a p p r o x i m a t i o n s , w h i c h a r e w i d e l y u s e df o r m a t h e m a t i c a l c o n v e n i e n c e , a r e o f t e n n o t a d e q u a t e e x c e p t f o r a s y m p t o t -i c a ll y h ig h e n e r g ie s . M o r e i m p o r t a n t l y , h o w e v e r, w e l e a r n f r o m t h e s t r i k i n gd i s c r e p a n c y b e t w e e n t h e f i r s t - o r d e r a n d s e c o n d - o r d e r O B K c r o s s s e c t i o n sfo r h i g h - Z t a r g e t a t o m s t h a t t h e O B K e x p a n s i o n is n o t a s u i ta b l e p e r t u r -ba t ion se r i e s a t t he ene r g ie s cons ide r ed .

8 . 3 T h e b o u n d a r y - c o r r e c t e d B o r n a p p r o x i -m a t i o nS i m i l a r t o i t s n o n r e l a t i v i s t i c c o u n t e r p a r t , t h e r e l a t i v i s t i c O B K a p p r o x i -m a t i o n d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n h a s a s e r i o u s d e f e c t : t h e w a v ef u n c t i o n s i n e n t r a n c e a n d e x i t c h a n n e l s d o n o t s a t i s f y t h e c o r r e c t a s y m p -t o t i c w a v e e q u a t i o n s , E q s . ( 5.4 8 ) a n d ( 5. 49 ). I n d e e d t h e l o n g - r a n g e n a t u r eo f t h e C o u l o m b i n t e r a c ti o n d o e s n o t a ll ow o n e t o u s e u n p e r t u r b e d a t o m i cw a v e fu n c t i o n s e v e n a t in f i n it e s e p a r a t i o n s . T h e v i t a l i m p o r t a n c e o f p r o p e rb o u n d a r y c o n d i t i o n s f o r n o n r e l a t i v i s t i c c h a r g e t r a n s f e r h a s f i r s t b e e n r e c -o g n i z e d b y D e w a n g a n a n d E i c h l e r [ D E E 8 5 , D E E 8 6 ] w h e n t h e y s h o w e d t h a td i s r e g a r d o f b o u n d a r y c o n d i t i o n s m a y l e a d t o s i n g u l a r it i e s in t h e t r a n s i t i o na m p l i t u d e s o f s e c o n d - o r d e r t h e o r ie s . S u b s e q u e n t l y , a b o u n d a r y - c o r r e c t e d


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2 18 C H A P T E R 8. C H A R G E E X C H A N G Ewhere ~ + i s t he exac t so lu t ion o f the fu ll D i rac equa t ion (5.40) . W i ththe a id o f the co r respond ing exac t so lu t ion ~ - and by app ly ing the exac ta n d a sy m p t o t i c D i r a c e q u a t i o n s w e m a y d e r i v e t h e p r i o r form (5.81) in as imi l a r way . In sum mary , t he exac t am pl i tudes may be cas t i n to one o f thea l t e rna t ive fo rms

F { ZW ZTr T R ~ + > (post form) (8 .29)o r

F{( - ) = i e 2 d t ~ fh s ~ ( z , z , )r ~ R ' 9 F > (prior form ). (8.30)The pe r tu rb ing po ten t i a l i n the m at r ix e l emen t s i s no longer of the long-range Cou lom b fo rm b u t o f a su tt i ci en tly shor t range to rend er conven t iona lsca t t e r in g theory app l i cab le . Th i s i s i ndeed the ve ry reason to choosea pa r t i cu la r ly conven ien t gauge by in t roduc ing the phase -d i s to r t ed wavefunctions (I)~ In o rde r to ob ta in a f i r s t-o rde r appro x im at ion wi th re la -f , i "t i v i s t i ca l ly mod i f i ed Cou lomb boundary cond i t ions , we have to rep lace ~+wi th (I)~ and ~ f w i th (I)~. Inse r t ing Eqs. (8 .26) and (8 .27) , we ob ta in theexpl ic i t ampl i tudes [Eic87a]/ ? /( + ) - - i e 2 dt d 3 r T [r t ')] te -iuT n R+vt) (post form)~B 1B h cx~

x , 1 r T Ra n d / ? /(-) - i e2 d t d 3 r r [ ~ ( r ' p t ' ) ] t e - i~ Tln (R+ ~ t ) (pr ior form)B I B ~ o o

( Z P Z P ) iupln(R'-v t') S r~, R ' e - r t ) . (8.32)T h e se t r a n s i t i o n a m p l i t u d e s a r e m a t r i x e l e m e n t s t a k e n b e t w e e n e x a c t so -lu t ions ( I )~ and ( I )~ o f the asym pto t i c D i rac eq ua t ions (5.48) an d (5 .49),re spec t ive ly. In th i s po in t, t he B1B appr ox im at ion i s d i s t ingu i shed f romother d i s to r t ed -wave approx imat ions fo r wh ich the d i s to r t ed waves a re n o te x a c t e i g e n s t a te s o f t h e a sy m p t o t i c H a m i l t o n i a n .

S imi l a r ly as fo r the OB K approx imat ion , one may conv ince onese lf t ha tthe pos t and p r io r am pl i tudes a re iden ti ca l, p rov ided r and r obey Eqs .(5.53) and (5.54), respect ively.


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8.4 . T H E E I K O N A L A P P R O X I M A T I O N 219I n g e n e r a l , t h e r e a r e i n f i n i t e l y m a n y w a y s t o g e n e r a t e a s h o r t - r a n g e p e r -

t u r b i n g p o t e n t i a l . F o r e x a m p l e , i n E q . ( 8 .3 2 ), o ne m a y r e p l a c e t h e r e s i d u a li n t e ra c t i o n - Z p / R ~ b y

U ( R' ) = Z p [ ( W R ' + 1 ) e x p ( - 2 W R ' ) - 1] /R ' , (8 .33)w h e r e W > 0 is a s u i t a b l e p a r a m e t e r . T h e m o d i f i e d r e s id u a l in t e r a c t i o nU ( R ' ) t e n d s t o - Z p / R ' as R ' ~ o c , b u t b eco m es a co n s t a n t f o r R ' ~ 0 . I fU ( R ' ) i s u s e d i n E q . ( 8 .3 2 ), t h e p h a s e f a c t o r d i s t o r t i n g t h e i n i t i a l s t a t e h a st o b e mo d i f i ed acco rd i n g l y ; i n g en e ra l , i t w i l l b e d e f i n ed b y a t i me i n t eg ra l[ To i8 7] . T h e c h o i c e a m o n g v a r i o u s d i s t o r t e d - w a v e a p p r o a c h e s i s g u i d e d b yp h y s i c a l a r g u m e n t s a n d b y c a l c u l a t i o n a l c o n v e n i e n c e , s e e a l s o S e c . 8 . 6 .F r o m E q s . ( 8. 31 ) a n d ( 8. 32 ) o n e m a y c a l c u l a t e c a p t u r e c r o s s s e c t io n s .O w i n g t o t h e o s c i l l a t o r y f a c t o r s i n t h e i n t e g r a n d , t h e e x a c t n u m e r i c a l e v a l -u a t i o n i s d i f f icu l t. C ro s s s ec t i o n s h av e b een o b t a i n e d i n t h e l i m i t c~Z


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220 C H A P T E R 8. C H A R G E E X C H A N G Ef a c t o r d e p e n d i n g o n t h e e l e c t r o n i c c o o r d i n a t e s . I n a m o r e g e n e r a l c o n t e x t ,t h i s a p p r o a c h i s c a ll e d th e e i k o n a l m e t h o d w h i c h h a s a ti m e - h o n o r e d h i s t o r yin op t i c s . I t was in t rod uce d in po ten t i a l s c a t t e r i ng by M ol i~re [M o147], inh igh -energy s ca t t e r ing by G lauber [G la58] and in f a s t a tomic co l l i s ions byW i l e t s a n d W a l l a c e [ W i W 6 8 ]. T h e e i k o n a l a p p r o a c h , i n i t s a s y m m e t r i c f o rmin wh ich the d i s to r t ion i s inc luded in on ly one o f the channe l s , was app l i edto nonre la t iv i s t i c cha rge t r ans fe r in [Dew75 , ChE79a , ChE79b , E ic81] andw a s s h o w n t o b e a r a t h e r a c c es si b le d i s t o r t e d - w a v e t h e o r y w h i c h y i e ld s g o o do v e r a l l a g r e e m e n t w i t h e x p e r i m e n t a l d a t a a t h i g h b u t n o t y e t a s y m p t o t i cve loc i t i e s . I t i nc ludes h igher -o rde r Born t e rms desc r ib ing mul t ip le so f t co l -l i s ions in an approx imate f a sh ion [E iN80] wh i le a t the s ame t ime p rese rv ingt h e s i m p l i c i t y o f t h e O B K a p p r o x i m a t i o n , w h i c h d r a s t i c a l l y o v e r e s t i m a t e sthe cross sec t ions .

T h e r e l a t i v i s t i c e i k o n a l a p p r o x i m a t i o n h a s b e e n i n t r o d u c e d a n d w o r k e dou t fo r a rb i t r a ry in i t i a l and f ina l s t a t e s in [E ic85]. S ince a r e la t iv i s t i cs y m m e t r i c a l t r e a t m e n t g i v e s r i s e t o s e r i o u s p r o b l e m s a n d m i s r e p r e s e n t s t h ee x p e r i m e n t a l d a t a , s ee Se c. 8.6 , it is r e a s o n a b l e t o a d o p t t h e u n s y m m e t r i c a lt r e a t m e n t , i n w h i c h o n l y t h e i n i t ia l o r t h e f i na l w a v e f u n c t i o n i s d i s t o r t e d .Th i s co r r esponds to r ep lac ing the a sympto t i c s t a t e s ( I )~ in Eq . (5 .82 ) fo rthe pos t fo rm and ( I )~ in Eq . (5 .83 ) fo r the p r io r fo rm, r e spec t ive ly , byu n p e r t u r b e d b o u n d f i n a l a n d i n i t i a l s t a t e s , w h i l e f o r t h e e x a c t s t a t e s ~ +a n d ~ f p h a s e - d i s t o r t e d e i k o n a l w a v e f u n c t i o n s a r e s u b s t i t u t e d .

To be spec if ic , le t u s choose the p r io r fo rm. Fo l lowing the p re sen ta t io no f t h e O B K a p p r o x i m a t i o n i n Se c. 8.1 , t h e e i k o n a l t r a n s i t i o n a m p l i t u d e i nt h e p r i o r f o r m i s w r i t t e n a s+//f t - i - ~ d t d 3 r T [r t ) l t S Z__~p t)" (8 .34)' r ~In the p r io r fo rm, the in i t i a l an d f ina l wave func t ions 4

r t ) - - ~ i ( rT ) C i E i t / h (8.35)a n d r t ' ) - 7~( r~) e - i E ~ t ' / h e i:+ ln(rT+ZT) (8.36)a r e e x p r e s s e d b y t h e u n p e r t u r b e d t a r g e t a n d p r o j e c t i l e w a v e f u n c t i o n s~ i an d ~ in the i r re spec t ive f r ames . Th e Somm er fe ld pa ram ete r u~: =Z ~ e 2 / h v and the cha rge number Z~: in the e ikona l wave func t ion (8 .36 ) a r el a b e l e d b y a p r i m e i n o r d e r t o p r o v i d e a u n i q u e s i g n a t u r e f o r t h e e l e c t r o n -t a r g e t i n t e r a c t i o n i n t h e e i k o n a l p h a s e . T h i s a ll ow s o n e t o r e c o ve r t h e O B Ka p p r o x i m a t i o n b y s im p l y s e t t i n g Z ~: - 0 o r u~: - 0 .

4A constant p hase factor depending on the unit of leng th is dropp ed in the eikonalphase factor.


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8.4. THE EIKON AL APP ROX IMA TIO N 221In the pr ior vers ion of the e ikonal app roac h, the f ina l wave funct ion

i s phase -d i s to r t ed by the e l ec t ron- t a rge t i n t e rac t ion . The d i s to r t ing phasefactor exp[iu~: ln(r T + ZT) ] is the asy m pto t ic form of the Co ulom b wavefunc t ion (8 .57) . I t may a l so be rep resen ted by an e ikona l i n t eg ra l t h roughthe re l a t ion

l n ( r T + z T )] c( e x p - i ~ r Txp[iuT (s.37)where the p ropor t iona l i ty fac to r i s an i r re l evan t cons tan t phase fac to r and

i i s kept f ixed wh en evalu at ing the in tegra l . Th is i s in l ine wi th thehere rpcond i t ion for the v al id i ty of the e ikonal ap proa ch [EiNS0] th at the coll is ionis "fast" in the se nse of Sec. 5.4.1.

Equa t ion (8 .37) shows tha t t he e l ec t ron- t a rge t i n t e rac t ion (occur r ing inthe expo nen t ) ac t ing f rom the t ime o f cap tu re to in f in i ty is t rea t ed nonper -turbat ively , whi le the e lec t ron-project i le in teract ion in Eq. (8 .34) expl ic i t lyenters in f i rs t order . Al th oug h the wave funct ion (8 .36) has the correctasympto t i c behav io r , t he t rea tmen t a s a who le does no t sa t i s fy Cou lombbo un dar y cond i t ions and the in t e rac t ion in Eq . (8.34) is no t o f shor t range ,because the in i t i a l wave func t ion (8 .35) remains und i s to r t ed .

The e ikona l approx imat ion may a l so be fo rmula ted in the post vers ion ,which i s ob ta in ed by invert ing the ro le of targ et and p roject i le in Eqs. (8 .34)to (8 .36) , tha t i s by replac ing Zp by ZT and vice versa and by in t e rchang ingin i ti a l and f ina l s t a t e s . As the theory i s no t pos t -p r io r sy mm et r i c , bo thvers ions y ie ld d i fferent resul t s .

S ince in an asymmet r i c theory l i ke the e ikona l approx imat ion , t hes t ronger one o f the e l ec t ron - t a rge t and e l ec t ron-p ro jec t i le i n t e rac t ion shou ldbe t rea t ed nonper tu rba t ive ly , t he usua l p resc r ip t ion , wh ich we deno te a sZ-cr i ter ion , i s

i f Zp < ZT then use the prior fo rmi f Zp > ZT then use the post form. (8.38)

An a l te rna t ive p rescr ip t ion , Eq. (8 .49), which devia tes f rom (8 .38) forh igher pr inc ipal shel l s has a lso been proposed [MeA85]; however , in th issec t ion , we adop t the s t andard fo rm (8 .38) .

8 . 4 . 1 C a p t u r e f r o m i n i t i a l i n t o f i n a l l S l / 2 - s t a t e sSimi lar ly as for the OBK approximat ion , Sec . 8 .2 .1 , we expl ic i t ly considerthe s imples t case wh ich i s a l so the mos t impor t an t one , namely l s1 /2 -1s1 /2t rans i t i ons . We adop t a tomic un i t s t h rou gho u t th i s subsec tion .


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222 C H A P T E R 8. C H A R G E E X C H A N G ET h e a n a l y t i c a l t r e a t m e n t o f t h e e i ko n a l a p p r o x i m a t i o n s t a r t s f r o m t h e

re la t ion [GaM74] L X ~-ivy, ln(rT+ZT) __ [ r ( i v ~ ) ] - i ,~iu@--i--A(rT+ZT)d/~ (8.39)and the cor r esp ond ing Four i er t ra ns f orm ~ti (q , )~) def ined by

y ) ~ s ( r T ) e - x ( r T+ z T ) = (2~)-~r(iu~) ~t~ (q, A)e iq'rT d 3 q (8.40)whi ch replace s qSi(q) in Eqs . (8 .3) and (8 .5) . Th e pr io r form of the e ikonalt r ans i t ion am pl i tu de i s the re fo re g iven by

A # : t ~ - - - - Z p d 2 ' t ~ . ) ~ ) / ~ i v ~ - l e i q b . bf i v '7 qb dA O~s (qb, q - ) S (qb, q+ , ,(8.41)

where q+ and q_ are def ined in Eq. (8 .6) .Th e exp res s ion (8 .41 ) i s va l id for a rb i t r a ry in i t i a l and f ina l s t a t e s . The

cross s ec tion i s aga in conven ien t ly wr i t t en in t e rm s o f dens i ty m a t r i ces inthe fo rm o f Eq . (8 .11 ). Th e den s i ty m a t r i x for the f inal s t a t e i s the s am e asfor the O BK appro x im at ion , Eq . (8 .10 ), and on ly the in i t ia l s t a t e dens i tym at r ix has to be mod i f i ed com pare d to Eq . (8 .9 ). Th e cons t ru c t ion o fPr i s com pl ic a ted by the presenc e of the fac tor e -~(~T+zT) in Eq. (8 .40)and by the in teg ra t ion over A. Whi le the r ad ia l pa r t o f the expo nen t canb e c o m b i n e d w i t h t h a t o f t h e w a ve f u n c ti o n , t h e t e r m e - ~ T is h a n d l e dconven ien t ly by abso rb ing i t i n to the p lane wave th rough the de f in i t ion o fa complex wave vec to r

k~ , = q - i A 6z = (qb, q+ - iA) (8.42)w it h k~ - q2 + q~_ _ A2 _ 2iAq+ . As a res ult , th e q ua nt it ie s ai, bi, an d ftin Eqs . (8.14 ) and (8 .15 ) a r e r ep laced w i th complex in teg ra l s . W i th th ea b b r e v i a t i o n A = A + Z T + i k ~ , and w i th s = 8 w [see Eq . (4.94) for Z = ZT]and u = u~ , the in teg ra l s fo r 1 s - l s t r ans i t ions t ake the fo rm

Ao(q) = N g ~( s + 1) L ~ d A A _ ( s + l )r ( i u ) 2 - ~ ( - _ _ ) ~ ; s + 1 ) ) ) ~ i u - - 1

B i n ( q ) p~~;b)) L cx) d)~ ~iu ~l+mN f 2 - ~ ( s - A +~ )_Nfr(s+ ) L ~ d A ( )k __(s+ l ) a i u _ l + mr ( i . - - - - - ~ ~ + a + ( ~ + ~ ) ) . ( 8 . 4 3 )


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8.4. T H E E I K O N A L A P P R O X I M A T I O N 223I f t he in t egr a l s B m w i th m = 0 , 1 a r e comb ined to bu i ld a vec to r

K = (B0qb, B oq + - i B 1 ) , (8.44)t h e A - i n te g ra l o v er t h e s p i n o r f u n c t i o n / t ~ s a s s u m e s a s t r u c t u r e a n a l o g o u sto E q . ( 8 .14) , nam e ly

f 0 a ( , s ( q , a ) d a - 1 )r r v ~ - (cr 9 K ) X , s " (8.45)T h e d e n s i t y m a t r i x f o r t h e i n i t i a l s t a t e i s t h e n o b t a i n e d a s

1 ( [A o l - A o ( c r .K * ) )Pi = 27r2 - A ; ( e r . K ) ( e r . K ) ( t r . K* ) (8.46)

i n p l a ce o f E q . ( 8 .1 5 ). T h e m a t r i x m u l t i p l i c a t i o n i s n o w c a rr i e d o u t u s i n gE qs . ( 8 .16) and ( 8 .1 7) in th e expr e ss ion ( 8 .11) f o r t he c ross sec t ion . T h ee i k o n a l cr o ss s e c t io n f or th e c a p t u r e o f a n u n p o l a r i z e d 1 s - e le c tr o n i n t o b o t hposs ib l e p r o j ec t i l e i s - s t a t e s t hen r eads [ E ic85]

j~0 x )_elk -- 2rr Z~, ? + 1 Qeik (q2O - l s _ l s - - V2.y 2 ) d ( q ~ ) (8.47)w i t h t h e i n t e g r a n d

Qeik m 11 1 Ao l 2 + ( 2 m 1 2 / q ) [ R e ( A o B ~ ) ) ( q ~ + q + q _ ) - I m ( A o B ' ~ ) q _ ]+ 2 m ' 1 2 [ - R e ( Ao B ~ ) ) q + + I m ( d o B ~ ) ]+ m 22[IBo12(q~ + q~_) - 2 I m ( B o B ~ ) q + + IB112]. (8.48)

T h e m o m e n t a q + a r e d e f in e d in E q . ( 8. 6) , t h e p a r a m e t e r s rn~k in Eq. (8 .17) ,a n d t h e m a t r i x e l e m e n t s A 0 , B k i n E q . ( 8 . 4 3 ) .

T h e c a l c u l a t i o n o f t h e c r o s s s e c t i o n r e q u ir e s t h e n u m e r i c a l e v a l u a t i o n o ft h e A - in t eg r a ls i n A 0 , B 0 , B 1 a n d t h e a d d i t i o n a l o n e - d i m e n s i o n a l i n t e g r a t i o nove r q2. E xp l i c i t ex pr e ss ions inc lud in g in i t i a l and f ina l L - she l ls have a lsobeen g iven [ E ic85] . Numer i ca l r e su l t s have been pub l i shed in [ AnE 85] and[MeA85] and a re presented in Sec . 8 .5 .1 .

8 . 4 . 2 T r a n s f e r b e t w e e n h i g h e r i n i t i a l a n d f i n a l s h e l l sT h e m e t h o d o u t l i n e d i n S e c t i o n 8 . 4 . 1 c a n b e g e n e r a l i z e d t o a r b i t r a r y i n i t i a land f ina l s t a t e s . I n f ac t, t he e ikona l app r ox im a t io n , owing to it s i n t r ins i c


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224 C H A P T E R 8. C H A R G E E X C H A N G Es i m p l i c i t y , i s p a r t i c u l a r l y s u i t e d f o r t h i s e x t e n s i o n , a n d n o o t h e r p e r t u r b a -t i v e a p p r o a c h h a s b e e n c a r r i e d s o f a r. F o r d e t a il s o f t h e f o r m u l a t i o n a n d o fthe resu l t ing express ion , we refer to [Eic85] and [Eic90] .

W e j u s t m e n t i o n h e r e a n a l t e r n a t i v e p r e s c r i p ti o n f o r c h o o s in g b e t w e e nt h e p r i o r a n d t h e p o s t f o r m o f t h e e i k o n a l a p p r o x i m a t i o n . I f t h e i n i t ia l a n df ina l she l ls a re d if f e ren t, one m ay t h ink o f r ep l ac ing t he p resc r i p t i o n (8 .38 )b y a n o t h e r o n e in w h i c h t h e w e i g h ti n g o f t h e i n t e r a c t i o n w i t h t h e C o u l o m bw a v e f u n c t i o n s is t a k e n i n t o a c c o u n t . S i n c e t h e e x p e c t a t i o n v a l u e o f t h eC o u l o m b p o t e n t i a l is e q u a l t o < Z / r> . = ( Z / n ) 2 i r r e spec t i ve o f t he subshe l l ,t h e e f f ec t iv e s t r e n g t h o f t h e p o t e n t i a l s h o u l d b e m e a s u r e d b y Z / n r a t h e rt h a n b y Z . T h i s l e a d s t o a n a l t e r n a t i v e p r e s c r i p t i o n [ M eA 8 5 ], w h i c h w ed e n o t e a s Z / n -c r i t e r i on :

i f Z p / n p < Z T / n T t h e n u s e t h e p r i o r fo rm ,if Z e / n p > Z T / n T t h e n u s e t h e pos t form. (8 .49)

Th i s ru l e m ay en t a i l d i f f e ren t cho i ces fo r d i f f e ren t com bin a t i o ns o f i n i t i a la n d f i na l s t a t e s w i t h i n t h e s a m e p r o j e c t i l e - t a r g e t c o m b i n a t i o n . I n F i g s .8 . 9 - 8 . 1 1, w e c o m p a r e t h e p r e d i c t i o n o f t h e u s u a l Z - c r i t e r i o n ( 8. 38 ) w i t ht h a t o f t h e Z / n - c r i t e r i o n (8.49).T h e e i k o n a l a p p r o a c h h a s b e e n c o m p a r e d w i t h o t h e r t h e o r e t i c a l c a l c u -l a t i o n s a s w e ll a s w i t h e x p e r i m e n t a l d a t a . F o r e l e c t r o n c a p t u r e b y 1 . 05G e V / u N e 1 ~ i o n s f r o m t a r g e t a t o m s w i t h c h a r g e s Z T = 1 3 , 3 0 , 4 7, 73 , a n d9 2 , t h e e i k o n a l c a l c u l a t i o n s a r e i n r a t h e r c l o s e a g r e e m e n t w i t h r e l a t i v i s t i cb o u n d a r y - c o r r e c t e d B o r n c a l c u l a ti o n s , se e F i g . 8 .8 , w h i c h a r e p o s t - p r i o rs y m m e t r i c , i n c o n t r a s t t o th e e i k on a l t h e o ry . F u r t h e r m o r e , t h e a p p r o a c his s u p p o r t e d b y t h e r e a s o n a b l e a g r e e m e n t w i t h a v a r i et y o f e x p e r i m e n t a lda t a [Eic90 ]. In som e cases, de t a i l ed tw o-ce n t e r cou p l ed -cha nne l ca l cu l a -t i ons [ToE88a ,b ] , s ee S ee . 8 .7 .2, y i e ld c ros s s ec t i ons o f a s im i l a r , b u t no ts u p e r i o r q u a l i t y .

W e n o w p r e s e n t a s u r v e y o f c a l c u l a t e d c r o ss s e c ti o n s i n t h e i r d e p e n d e n c eo n t h e v a r i o u s p a r a m e t e r s e n t e r i n g i n t o t h e t h e o r y . S i n ce c a p t u r e c r o s ss e c ti o n s v a r y o v e r m a n y o r d e r s o f m a g n i t u d e , e v e n a p p r o x i m a t e r e s u lt s w i llh e l p t o a s s e s s t h e p a r a m e t e r d e p e n d e n c e a n d t h e q u a l i t a t i v e b e h a v i o r o ft he c ros s s ec t i on .

I n F i g . 8. 2, w e s h o w t h e d e p e n d e n c e o n t a r g e t a n d p r o j e c t i l e c h a r g e s fo rt h e l e a d i n g t r a n s i t i o n ( 1 s1 /2 - 1s 1 /2 ) a t a f ix e d p r o j e c ti l e e n e r g y o f 1 0 G e V / u .A s c a n b e s e e n f r o m t h e s u b s e q u e n t f i g u r e s , t h e o t h e r c r o s s s e c t i o n s f o l l o wa s i m i l a r p a t t e r n . T h e s e t o f c u r v e s r e fl e ct s t h e a p p r o x i m a t e d e p e n d e n c eas Z~rZ~ o f t h e c a p t u r e c r o ss s e c t io n , w h i c h i s c h a r a c t e r i s t i c f o r a ll c a p t u r etheor ies , see , e .g . , [DeE94] .

I n F i g s. 8 . 3 - 8 .7 , w e d i s p l a y t h e c a p t u r e c r o ss s e c t i o n s fo r c o m p l e t e l y


19/40

8.4 . T H E E I K O N A L A P P R O X I M A T I O N 225

.J3EOo ~o(/3

1 0 2

1 0 1

1 0 o

1 0 - 1

1 0 - 21 0 - 3

1 0 - 4q~L . 1 0 - 5EL 10 sO(J

1 0 - 71 0 - 8

1 0 - 9

Z p = 1 O0

f/

f L t/ /

///

f l If l

/ ~ Z p = 1 0

0 2 0 4 0 6 0 8 0 1 O 0T o r g e t c h o r ge n u m b e r ZT

F i g u r e 8 .2 . Cross sec t ions ( in barn ) fo r l s l /2 -1Sl /2 e lec t ron cap tu re a t a p ro -jec t i l e energy o f 10 G eV /u as a func t ion o f the t a r ge t and p ro jec t i l e charges ZTand Zp in s teps o f 10 un i t s . Th e c ross sec t ions a re ca lcu la ted acc ord ing to thep r e s c r i p t i o n ( 8 .3 8 ) a n d r e f e r t o a s i n g l e t a r g e t e l e c t r o n c a p t u r e d i n t o a v a c a n tp r o j e c t i l e K - sh e l l. T h e y a r e a v e r a g e d o v e r t h e i n i t i al a n d s u m m e d o v e r t h e f in a la n g u l a r m o m e n t u m p r o j e c t i o n s . F r o m [IC S93].

s t r i p p e d A u i o n s i m p i n g i n g o n h y d r o g e n - l i k e C , A 1 , C u , A g , a n d A u t a r g e t sa s a f u n c t i o n o f e n e rg y . T h e d a t a a r e a v e r a g e d o v er t h e i n i t i al a n d s u m m e do v e r t h e fi n al a n g u l a r m o m e n t u m p r o j e c t i o n s . F o r o t h e r h e a v y p ro j e c t il e s ,o n e c a n s c al e t h e s e c u r v e s a p p r o x i m a t e l y a s Z p . I n t h e s t a t e - t o - s t a t e c r os ss e c t i o n s p r e s e n t e d , e i t h e r t h e i n i t i a l o r t h e f i n a l s t a t e i s a l S l / 2 s t a t e ( i no r d e r t o r e d u c e t h e n u m b e r o f c u r v e s ). A s a f u n c t i o n o f t h e p r o j e c t i l ee n e r g y , t h e c u r v e s c l e a r l y e x h i b i t t h e t r a n s i t i o n f r o m a n e s s e n t i a l l y n o n r e l -a t i v i s t i c t o a r e l a t i v i s t i c b e h a v i o r : a t r a t h e r l o w c o l li s io n e n e r g i e s a r o u n d1 G e V / u , t h e c r o s s s e c t i o n c u r v e s s t i ll re f l e c t t h e r a p i d d e c r e a s e a s E - 6k n o w n f r o m 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 , w h i l e i n t h e e x t r e m e r e l a t i v i s t i c e n -e r g y r a n g e a b o v e 1 0 G e V / u , t h e c u r v e s b e n d o v e r t o t h e E - 1 d e p e n d e n c e .


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2 2 6 C H A P T E R 8. C H A R G E E X C H A N G E

1 0 - 1 - 1 8 1 / 2 A u 79+ + C 5+1 0 - 2

1=~ 1 ~ / , - 1 S l / 2

\ \ 1 s l / 2 - 2 p 3 ,

~ ~ 1 S l / 2 - - 2 p l / 2i 2 p l / 2 - 1 s l / 21 0 -1 1 ~ ~ , ~ 12 p 3 /z - 1 S l / 2 " ~ ~1 0 - 1 2 ~ ~ . 11 0 _ 1 3 , i , , l l i , l ' I l l L l , i l , , * * i H , l i i , , i H , l = i i , l l i , l1 0 - 1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4

Energy (CeV/u )

1 0 - 3J::3r - 1 0 - 40

; ~ 1 0 - 5uq~u ~ 1 0 - s0 1 0 - 7L _o 1 0 - 6q)

- ~ 1 0 - 9C L0C . ) 1 0 - 1 0

r'~v ro~( 9O,)83( / )030IL( 9( 9I,,,_::3I : L0r,,3

1 0 11 0 0

1 0 - 11 0 - 21 0 - 31 0 - 41 0 - -~1 O - S1 0 - 71 0 - e

I 0 - 9 I1 0 - l o1 0 - 1 11 0 - 1

\ ~ x . . . . . . t . . . . . . . . I . . . . . . . . I . . . . . . . . i . . . . . . . . t "9 - 1 , , / 2 A u 79 + + A I 12+

% '% \'~ x~ X , 1,,- 1,,,,\ \ \ ~ 1 , I / 2 - 2 s z / 2

x / 2 - P / ~ ~

2 p l / 2 - 1 S l / 2 ~ ~

~ ~ l l ~ l t ~ = 1 11 1 ~ ~ I ~ 1 J = ~ = = ~ .1 ~ ~ J = ~ = H L1 0 0

]

1 0 1 1 0 2 1 0 3 1 0 4Energy (CeV/u )

F i g u r e s 8 . 3 ( t o p ) a n d 8 . 4 ( b o t t o m ) . C r os s s ec ti on s fo r e le c tr o n c a p t u re b yAu79+ ions as a funct io n of the col l is ion energy. See cap t ion of Fig. 8 .2 .


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8.4. THE EIKONA L APPR OXI MATI ON 227

1 0 3

1 0 21 0 1

.D 1 O 0t -O: . ~ 1 0 -1( 9u 3 1 0 . 2oOoo0 1 0 . 3t,_( 9@ 0 - 4Q _0 1 0 -5

C.)1 0 - S

1 0 - 7

1 0 - 81 0 - 1

~ ' ' ' I ' ' ' ' ' ' I ' ' ' ' ' ' I ' ' ' ' ' ' I ' ' ' ' ' ' I\ ~ / z - , , , , / , A u 79 + + C u 2 8 +X X 1 S l / 2 / - 1 S l / 2

2p1/2 - I s+ /2 ~2ps/z - IsI/2 ~ ~

-

1 0 o 1 0 + 1 0 2 1 03 1 0 +E n e rgy ( C e V /u )

1 0 +

1 0 3

1 0 2. Q 1 O+c -0~ .~ 1 0 ~(,.)q,)e 3 1 0 - 1o ooo0 1 0 . 2t,_( 9

0 - 3IC1 1 0 - 40( D

1 0 - 5

1 0 - s

1 0 . 71 0 - 1

, , , , , , , , I , , , , , , , , I , w t w i r l | v i I l l v v l | I I w v v v v v |

2 / - s / A u 79 + + A g 4 6+

\ \ %

2 p l / 2 - / ~ ~2 p 3 / 2 - / 1 s l / 2

i I i i * | l | l i i I I l l l l l I I I I I l | l l I I I I I I l l l I I I l l l l l l1 0 o 1 03 1 0 +0 1 1 0 2

E n e r gy (C e V /u )Fi gu re s 8.5 (top ) a nd 8.6 (b ot to m) . Cross sections for electron capture byAu79+ ions as a function of the collision energy. See caption of Fig. 8.2.


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228 C H A P T E R 8. C H A R G E E X C H A N G E

1 0 .5

1 0 41 0 3"z"1 0 2g

~ . ~ 101m lOO

0 1 0 - ~0Q. } 0- 21r~O 1 0 - 3

r j1 0 - 41 0 - 5

, , ;w l l . I w I ' ; ' ' " l 9 w I l , , , n | ! , , , , s , , | I i , , , , , , |

A u 7 9 + 4 - A u 7 8 +2 s I / 2 - 1 $ I / 2

1 s l / ? - I s l / 2

~ ' , ~ 1 s i / 2 - 2 S l / 2

1 s l / 2

2 p 3 / 2 - 1 s l / 2 ~

1 0 - 6 i i , , , l l i l l I i l , , , * I , , , , , , , , I , , , , , , , , I , , , , , , , , i1 0 - 1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4

E n e rgy ( C e V /u )F ig u re 8 .7 . Cross sec tions for e lec tron cap ture by Au 79+ ions as a funct ion ofthe coll ision energy. See ca ptio n of Fig. 8.2.

F o r th e m o r e a s y m m e t r i c s y s t e m s , t h e s p r e a d o f c u r v e s i s m u c h l a rg e r t h a nfo r sy m m et r i c co l l is i ons. On e a l so obse rve s t he e f f ec t o f i nc rea sed sp i n -o rb i ts p l i t t i n g w i t h i n c r e a s i n g c h a r g e o f t h e c o ll is i on p a r t n e r .

T h e e i k o n a l c r o s s s e c t i o n s p r e s e n t e d h e r e a r e a v a i l a b l e i n t a b u l a r f o r mfo r m ore i n i t i a l and f i na l s t a t e s and a l so fo r u ran ium p ro j ec t i l e s [ IC S 93] .W e b e li e v e t h a t t h e s e c r o s s se c t i o n s a r e u s e fu l g lo b a l e s t i m a t e s . W h i l e i t isd i ff ic u lt t o q u a n t i t a t i v e l y a s s es s th e a c c u r a c y o f t h e r e s u l ts , i t i s e x p e c t e dto i nc rease wi th t he co l l i s i on ene rgy .

8 . 4 . 3 A n a p p r o x i m a t e f o r m u l a fo r l s - l s t r a n s i t i o n sA l t h o u g h t h e e i k o n a l c r o s s s e c t i o n E q . ( 8. 47 ) c a n b e e v a l u a t e d n u m e r i c a l ly ,i t i s u s e f u l t o g iv e a n a p p r o x i m a t e a n a l y t i c a l e x p r e s s io n . A n y r e l a ti v i s t ict r e a t m e n t i n c l u d e s t w o k i n d s o f r e l a t iv i s t i c e f fe c ts , n a m e l y t h e r e l a t i v i st i cp r o j e c t i l e m o t i o n , c h a r a c t e r i z e d b y th e p a r a m e t e r 5 - [ ( ~ - 1 ) /( 'T + 1)] 1 /2 ,a n d t h e r e l a t i v i s t i c m o t i o n o f t h e e l e c t r o n , c h a r a c t e r i z e d b y a Z , i n i ts in i-t i a l a n d f i n a l a t o m i c o r b i t s . C o m p a r e d t o t h e n o n r e l a t i v i s t i c c a s e , t h e r e l a -t i v i s t ic e l e c tr o n m o t i o n h a s t w o i m p o r t a n t c o n s e q u e n c e s . ( 1) T h e e l e c tr o n


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8.4. T H E E I K O N A L A P P R O X I M A T I O N 229o r b i t a l s a n d t h e i r b i n d i n g e n e r g i e s a r e m o d i f i e d . ( 2 ) T h e e l e c t r o n a c q u i r e sa D i r a c m a g n e t i c m o m e n t w h i c h , in t u r n , i n t e r a c t s w i t h th e i n d u c e d m a g -n e t i c fi el d p r o d u c e d b y t h e p r o j e c t i l e m o t i o n . T h e o c c u r r e n c e o f a D i r a cm a g n e t i c m o m e n t is , o f c o u rs e , i n d e p e n d e n t o f t h e c h a r g e Z .

I n o r d e r t o d e r i v e a n a p p r o x i m a t e c r o s s s e c t i o n f o r m u l a w e e x p a n d t h ee l e c t r o n i c w a v e f u n c t i o n s i n p o w e r s o f a Z p , T > Zp , T a n d a r c t a n ( q / Z T ) ~ 7 v / 2 . C o r r e s p o n d i n g l y ,i n ( 8 . 4 3 ) w e a s s u m e t h a t c o n t r i b u t i o n s t o t h e i n t e g r a l s a r i s e m a i n l y f r o mv a l u e s A


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230 C H A P T E R 8. C H A R G E E X C H A N G ET h e r e f o r e , i t is i n t e r p r e t e d a s a m a g n e t i c c o n t r i b u t i o n t o c a p t u r e m e d i a t e db y t h e i n t e r a c t i o n b e t w e e n t h e r e l a t i v i s t i c a ll y i n d u c e d m a g n e t i c f ie ld o f t h ep r o j e c t i l e a n d t h e D i r a c m a g n e t i c m o m e n t o f t h e e le c t r o n . ( d ) T h e q u a n t i t yS orb is com pose d o f t e rm s t h a t exp l i c i t l y i nc lude C ~Zp o r aZ T and hencea r e i n t e r p r e t e d a s c o r r e c t i o n t e r m s a r i s i n g f r o m a r e l a t i v i s t i c m o d i f i c a t i o no f t he e l ec t ron i c o rb i t a l s .

S e t t i ng Z ~ = 0 , t he ap p ro x im ate O B K c ros s s ec t i on [MoS 80] i s s im u l -t a n e o u s l y d e r i v e d a s

~OBK _ 2 s~ z~ 5 7 + 1 OBK r (8 . 5 2 )l s - I s 5 ~ o 2( Z 2 + q~_)5 272 (1 + Sm ag -Jr- ~o rb )w i t h

s O B K __ 5 2 5 4- - + i - 6a gSo - ~ ~ ( Z p + z ~ ) - ~ ~ ( Z p + z ~ ) ( 8 .5 a)rb

B oth i n t he r e l a t i v i s t i c and i n t he non re l a t i v i s t i c case [DEE94] , t he OB Kc r os s s e c t io n s c o n s i d e r a b l y o v e r e s t i m a t e t h e e x p e r i m e n t a l r e s u l ts . F o r aroug h es t im a t e o f t o t a l c ros s s ec t i ons i nvo lv ing h igher p r i nc ipa l she l l s, onem ay use Eq . (8 .50 ) by app ly ing a s ca l i ng ru l e . One s t a r t s f rom the obse r -v a t i o n t h a t i n t h e n o n r e l a ti v i s ti c O B K a p p r o x i m a t i o n a n d i n t h e n o n r e la -t i v i s ti c e i k o n a l a p p r o x i m a t i o n u p t o s e c o n d o r d e r i n t h e e x p a n s i o n o f t h ee ikona l phase f ac to r , t he c ros s s ec t i on s ca l es w i th Z / n where n i s t he p r i n -c i p a l q u a n t u m n u m b e r o f t h e s he ll i n q u e s t io n . T h u s o n e m a y f o r m u l a t e a na p p r o x i m a t e s c a li n g r u le [ M eA 8 5] .

A p p r o x i m a t e r e l a t i v i s t i c c a p t u r e c r o s s s e c t i o n s a v e r a g e d o v e r s u b s h e l l swi th in a rb i t r a ry i n i t i a l and f i na l p r i nc ipa l she l l s spec i f i ed by t he i r p r i nc ipa lq u a n t u m n u m b e r s n T a n d n p , r e s p ec t iv e l y , c a n b e o b t a i n e d f r o m E q . ( 8. 50 )b y r e p l a c i n g

Z TZT --~ n TZ pZp --~ (8.54 )n p

i n t h e p r i o r f o rm . T o o b t a i n t h e p o s t f o r m , t h e t a r g e t a n d t h e i n i t i a l a n df ina l s t a t e s a re i n t e rchanged .I f t h i s s c a li n g r u le i s a d o p t e d , t h e d e c i s i on b e t w e e n t h e p r i o r a n d t h e p o s tf o r m w i l l b e a u t o m a t i c a l l y b a s e d o n t h e Z / n - c r i t e r i o n (8 .49 ) . A num er i ca lc o m p a r i s o n b e t w e e n t h e e x a c t a n d t h e a p p r o x i m a t e f o rm o f t h e e i k on a lc ros s s ec t i on i s g iven i n Tab l e 8 .1 and , m ore sys t em at i ca l l y , i n [E i c90 ] .


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8.5. THE ORE TIC AL AND EXP ERI MEN TAL CROSS SECTIONS 231T a b l e 8.1 . Theo re t ica l c ross sec t ions pe r e lec t ron ( in ba rn) for l s - l s e lec t roncap ture by 1 050 M eV /u Ne 1~ ions . The n um ber in squa re bracke ts gives thepow e r o f 10 mu l t i p ly ing the p r ec e d ing numbe r . T he O BK , e xa c t, a nd a pp r ox im a tee ikonal resul ts a re ca lcula ted f rom Eqs. (8 .18) , (8 .47), and (8.50), respect ive ly,the BIB resul ts a re taken f rom [Eic87a] .

Z T O B K e x a c t e i k o n al a p p r o x , e i k o n a l B I B13 2.00[-4] 3.93[-5] 3.85[-5 ] 5.53[-5]30 1.19[-2] 1.81[-3] 1.76[-3] 2.31[-3]47 8.47[-2] 1.10[-2] 1.10[-2] 1.31[-2]73 3.35[-1] 4.10[- 2] 4.81[-2] 4.86[-2]92 4.15[-~] 5.6 012] S.3412] 7.24[-2]

8 . 5 T h e o r e t i c a l a n d e x p e r i m e n t a l c r o s s s e c -t i o n s

8 . 5 . 1 T h e o r e t i c a l t o t a l c r o s s s e c t i o n sW e n o w h a v e a t o u r d i s p o s a l s e v e r a l t h e o r i e s w h i c h a r e a l l b a s e d o n ap e r t u r b a t i v e t r e a t m e n t . I t is w o r t h w h i l e to c o m p a r e t h e i r p r e d ic t i o n s fo rc a p t u r e c r o s s s e c t i o n s a m o n g e a c h o t h e r a n d w i t h e x i s t i n g e x p e r i m e n t a ld a t a .

I n o r d e r t o c o m p a r e v a r i o u s t h e o r e t i c a l a p p r o x i m a t i o n s , w e p r e s e n t i nT a b l e 8 . 1 c r o s s s e c t i o n s f o r l s l / 2 - 1 s l / 2 e l e c t r o n c a p t u r e b y 1 0 5 0 M e V / ub a r e N e i o n s f r o m v a r i o u s s i n g l e - e l e c t r o n ( i d e a l iz e d ) t a r g e t s . A c o m p a r i s o nw i t h e x p e r i m e n t a l d a t a is g i ve n b e lo w . B e s id e s t h e O B K a p p r o x i m a t i o n ,w h i c h y i e l d s c o n s i d e r a b l y t o o l a r g e c r o s s s e c t i o n s , w e s h o w r e s u l t s f o r t h ee i k o n al a p p r o x i m a t i o n i n a n e x a c t [ Eq . ( 8. 47 )] a n d a n a p p r o x i m a t e [E q.( 8. 50 )] e v a l u a t i o n , a n d t h e B 1 B a p p r o x i m a t i o n , s ee S ec . 8 .2 . I t i s s e e n t h a tt h e e i k o n a l a n d t h e B I B a p p r o x i m a t i o n s y i e ld s i m i l a r r e s ul t s. O f c o u r s e t h en u m b e r s f or t h e h i g h e s t c h a r g e s Z T h a v e t o b e ta k e n w i t h s o m e c a u t i o n ,s i n c e o n e g e t s c l o s e t o t h e e n e r g y m a t c h i n g , s e e S e c . 5 . 5 . 1 , b e t w e e n e l e c t r o na n d p r o j e c ti l e , a n d a l so t h e a Z e x p a n s i o n u s e d fo r t h e a p p r o x i m a t e e i k o n a la n d f or t h e B I B a p p r o x i m a t i o n w i l l l ea d t o i n a c cu r a c ie s . N e v e r th e l e s s , ac o m p a r i s o n b e t w e e n t h e e x a c t a n d t h e a p p r o x i m a t e e i k o n a l r e s u l t s s h o w st h a t t h e d e f i c ie n c ie s o f t h e c t Z e x p a n s i o n a r e n o t t o o d r a m a t i c i n t h i s c a se .

F o r a c o m p a r i s o n o f v a r i o u s t h e o r i e s f o r t h e ( b e c a u s e o f t h e s m a l l n e s so f t h e c r o s s s e ct i o n s , s o m e w h a t a c a d e m i c ) c a s e o f p + H c o l li s i o n s u p t o 1


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232 CHAPTER 8. CHARGE EXCHANGE1 I I

1 0 0 /- 1 0 5 0 M e V l a m u I[ N e O B K / /

/- / .,.,r~.,

10 -1 _-- / ,~T~':- / # - ] /Z I /# 'K+'L

7 - / / . / 'lO - Z _ _ / / / ; K . / 'k _- , / , ; ' . / '

_ i / I / 1 . 1 _1 0 - 3 E - / / 2 ,' -'=-L / 1 # / _~~ / / / . /

I O - L . V / / I / I I1 0 2 0 3 0 5 0 1 0 0Z TF ig u re 8 .8 . Expe r imenta l and theore t ica l cap ture cross sec tions for 1050 M eV /uNe 1~ ions versus targe t charge num ber. D ata from [Cra79, Anh85a]; dashedcurve: O BK ap prox ima t ion for K-K capture ; dash-do t curve: eikonal calculat ions(with shielding) for targe t K , L, and the sum of K an d L cap ture [AnE85, Eic90];solid curve: K-K cap ture from the B IB appro xim at ion [Eic87a], see Sect ion 8.2.

GeV , we re fe r t o [E ic90 ]. In t h a t r ev i ew, a sys t em at i c co m p ar i so n ( i nc lud ingh ighe r p r i nc ipa l she ll s ) i s m ad e be twe en t he exac t [Eq. (8 .47 )] and t heap pro x im ate [Eq . (8.50 ) ] eva lua t i on o f t he e ikona l c ros s s ec t i on .

8 . 5 . 2 C o m p a r i s o n w i t h e x p e r i m e n t a l d a t aT h e r e i s a q u a l i t a t i v e d i f f e r e n c e b e t w e e n c h a r g e e x c h a n g e i n v o l v i n g l o w - Zand h ig h -Z i ons In t he fo rm er case , l s 1 2 -1S l/ 2 ca p tu re i s c l ea r l y do m ina n t9 /w h e r e a s i n t h e l a t t e r c a se c h a rg e e x c h a n g e b e t w e e n e x c i t e d t a r g e t a n d p r o -j e c t il e s t a t e s i s o f e q u a l i m p o r t a n c e i n t h e e n e r g y r a n g e b e l o w 2 G e V / u . F o rb o t h c a s e s , w e p r e s e n t a c o m p a r i s o n b e t w e e n e x p e r i m e n t a l a n d t h e o r e t i c a lt o t a l c ros s s ec t i ons .

F i g u r e 8 . 8 s h o w s a c o m p a r i s o n b e t w e e n e x p e r i m e n t a l c a p t u r e c r o s s s e c -t i o n s a n d t h e o r e t i c a l r e s u l t s f o r 1 0 5 0 M e V / u b a r e N e i o n s i m p i n g i n g o nv a r i o u s t a r g e t s . T h e d a s h e d l in e r e p r e s e n t s t h e O B K a p p r o x i m a t i o n f or K -


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8.5. THEORE TICAL AND EXPERIM ENTAL CROSS SECTIONS 233Xe54. Xe52*, o t : : , 4

x \X 0:3 "1051 , " 1

, o 4 , , - F/eb i 0 5 , \ % ~ l I ' - - I05 t, C1 0 4 - \ ' \ ' \ % 1 I 2 _ 1041 0 3 i \ I i I I I I 1 0 3 I t \ \5 0 I 0 0 2 0 0 5 0 I 0 0 2 0 0 5 0 I 0 0 2 0 0 5 0 I 0 0 2 0 0Ep (M eV /amu) Ep (MeV /omu)

F i g u r e s 8 . 9 ( l e f t ) a n d 8 . 1 0 ( r i g h t ) . T o t al ( n o n ra d i a ti v e p lu s r a d i a t i v e ) c a p -tur e cross sec t ion [MeA85] for Xe 54+ and X e 52+, respect ive ly, im ping ing on v ar i-ous target s (My deno tes m ylar foi ls). Sol id l ines den ote e ikonal ca lcula t ion s usingthe Z/n cr i ter ion Eq. (8 .49) , dashed l ines the conventional Z cr i te r ion (8.38) forse lect ing be tween the pos t a nd pr ior fo rm. For low-Z ta rge ts such as Be andmyla r , the r ad ia t ive e lec t ron cap ture (dash-dot l ines ) i s dominant .

K c a p t u r e w h i c h o v e r e s ti m a t e s t h e d a t a b y a b o u t o n e o r d e r of m a g n i t u d e .T h e d a s h - d o t l i n e s i n d i c a t e e i k o n a l c a l c u l a t i o n s u s i n g S l a t e r - s c r e e n e d e f f e c -t i v e c h a r g es f o r K - K a n d L - K c a p t u r e a n d a l s o s h o w t h e s u m o f b o t h . T h e s er e s u l ts a g r e e r e a s o n a b l y w e l l w i t h e x p e r i m e n t a l d a t a . T h e s o li d c u r v e re p r e -s e n t s K - K c a p t u r e c r o s s s e c t i o n s c a l c u l a t e d f r o m t h e r e l a t i v i s t i c b o u n d a r y -c o r r e c t e d B o r n a p p r o x i m a t i o n B 1 B , s e e S e c t i o n 8 . 3 . T h i s c u r v e ( b a s e d o nS l a t e r - s c r e e n e d e f f e c t i v e c h a r g e s ) l i e s s l i g h t l y a b o v e t h e e i k o n a l K - K c u r v ea n d r e p r e s e n t s t h e d a t a q u i t e w el l c o n s i d e r i n g t h a t a c o n t r i b u t i o n f r o m L - Kc a p t u r e h a s t o b e a d d e d .

F i g u r e s 8 . 9 a n d 8 . 1 0 s h o w a c o m p a r i s o n b e t w e e n m e a s u r e d a n d t h e o r e t -i c a l e ikon a l c r o s s s e c t i ons f o r X e 54+ a nd X e 52+ p r o j e c t i l e s [ Me A 85] . T h ed i f fe r e n ce b e t w e e n t h e t w o s e t s o f c u r v e s c a n b e a s c r i b e d t o c a p t u r e i n t olS l /2 s t a t e s a s d i s c us se d in S e c . 8 . 7 . 2 . I t i s s e e n tha t i n mos t c a se s t he she l l -s ize Z/n c r i t e r i o n , E q . ( 8 .4 9 ) , ( s o l id l in e ) fo r c h o o s i n g t h e p o s t o r p r i o r


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234 C H A P T E R 8. C H A R G E E X C H A N G E

f o r m i s i n b e t t e r a c c o r d w i t h t h e d a t a t h a n t h e c o n v e n t i o n a l Z - c r i t e r i o n( d a s h e d l i n e ) . I n s o m e c a s e s , t h o u g h , t h e Z - c r i t e r i o n s e e m s t o y i e l d a b e t -t e r a g r e e m e n t w i t h t h e d a t a . I n a ll c a l c u l a t i o n s , t h e fu ll n u c l e a r c h a r g ew a s u s e d f o r t h e p r o j e c t i l e . F o r t h e t a r g e t a t o m s , e f f e c t i v e n u c l e a r c h a r g e sZ ~ = Z T - A Z b a s e d o n t h e S l a t e r ru l e s w e r e a d o p t e d , w i t h A Z = 0 . 3 f orthe K-she l l , 4 .15 fo r the L-she l l , 11 .0 fo r the 3s and 3p she l l s , and 21 .0 fo rt h e 3 d s h e l l . F o r t h e L - a n d M - s h e l l s , n o n r e l a t i v i s t i c t a r g e t w a v e f u n c t i o n sw ere u s ed , w h i ch w as fo u n d t o h av e a v e ry s ma l l e f f ec t o n t h e t o t a l c ro s ss e c ti o n s u m m e d o v er s u b sh e ll s. E x p e r i m e n t a l a n d t h e o r e t i c a l r e s u lt s in F ig -u r e s 8. 9 a n d 8 . 1 0 i n c l u d e r a d i a t i v e e l e c t ro n c a p t u r e ( R E C ) , s e e C h a p . 9 ,i n d i c a t e d b y a d a s h - d o t l in e. F o r B e a n d m y l a r ta r g e t s , R E C is d o m i n a n t ,w h e r e a s f o r h i g h e r t a r g e t c h a r g e s , R E C i s n e g l i g i b l e .

8 . 5 . 3 T h e o r e t i c a l a n g l 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 sF o r m o s t t h e o r i e s , a n g l 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 s h a v e n o t b e e n c a l c u l a t e db e c a u s e , a t p r e s e n t , m e a s u r e m e n t s w i t h s u f f i c i e n t a n g u l a r r e s o l u t i o n a r en o t f e a s i b l e . H o w e v e r , w h e n e x p e r i m e n t s h a v e d e v e l o p e d t o t h i s p o i n t , i t i sn o ma j o r p ro b l em t o ca l c u l a t e d i f f e r en t i a l c ro s s s ec t i o n s u s i n g Eq s . ( 5 .8 4 )o r ( 5 .8 7 ) w i t h i n t h e t h eo r i e s d e s c r i b ed h e r e .Fo r p u re l y t h eo re t i c a l r e a s o n s , d i f f e r en t i a l c ro s s s ec t i o n s a r e o f i n t e r e s t i no r d e r t o s t u d y t h e T h o m a s p e a k [ T ho 27 ], w h i c h c a n b e i n t e r p r e t e d a s a r is i n gf r o m a d o u b l e - s c a t t e r in g m e c h a n i s m a t a s y m p t o t i c a l l y h ig h v el oc it ie s . I nt h e f ir s t s t e p o f t h e c l a ss i c a l T h o m a s m o d e l , t h e e l e c t r o n s c a t t e r s o ff t h ep r o j e c t i l e b y 6 0 ~ w i t h a s p e e d e q u a l t o t h e s p e e d o f t h e p r o j e c t i le . I n t h es e c o n d s t e p , t h e e l e c t r o n i s e l a s t i c a l ly r e s c a t t e r e d o ff t h e t a r g e t n u c l e u s b ya g a i n 6 0 ~ s o t h a t i t t r a v e l s a lo n g w i t h t h e p r o j e c t i le . T h e c o n n e c t i o n o f t h i sc l a s s i c a l m o d e l w i t h t h e n o n r e l a t i v i s t i c s e c o n d - o r d e r O B K a p p r o x i m a t i o nh a s b e e n d i s c u s s e d in g r e a t d e t a i l i n th e l i t e r a t u r e , s e e e .g . [ S H S 79 , D r i 5 5 ,B r M 9 2 ] .

S i n c e t h e s e d i s c u s s i o n s i n v a r i a b l y a s s u m e a s y m p t o t i c a l l y h i g h n o n r e l a -t i v i s t i c v e l o c i t i e s , i t i s n e c e s s a r y t o m a k e a p r o p e r r e l a t i v i s t i c t r e a t m e n t .A c co rd i n g t o c l a s s i ca l mec h an i c s [Sh a7 9 , MO S79 ], a p ea k o f t h e d i f f e r en t i a lc r o s s s e c t i o n i n t h e l a b o r a t o r y f r a m e s h o u l d o c c u r a t t h e T h o m a s a n g l e

O Th ~ V/2~ -~- 1 me (8 .55)7 + 1 M p 'w h e r e m e i s t h e e l e c t r o n m a s s a n d M p t h e p r o j e c t i l e m a s s . I t is i n t e r e s t i n gt o n o t e t h a t t h e c l a s s i c a l t o t a l c r o s s s e c t i o n f o r t h e d o u b l e - s c a t t e r i n g m e c h -an i s m d ec rea s e s fo r l a rg e v a lu e s o f ~ / a s 3 . 3 [Sha7 9] . H e n ce t h i s m ec h a n i s mi s n o t a s y m p t o t i c a l l y d o m i n a n t a s i t i s i n n o n r e l a t i v i s t i c c h a r g e t r a n s f e r .


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8.6. O T H E R D I S T O R T E D - W A V E T H E O R I E S 235

E x a c t q u a n t u m m e c h a n i c a l c a l c u l a t i o n s o f d i f fe r e n ti a l cr o s s s e c t i o n sw i t h i n a s e c o n d - o r d e r O B K a p p r o x i m a t i o n ( O B K 2 ) y i e l d t h e f o ll ow i n g r e-su l t s [DeE91a] : ( a ) F o r p + H co l li s ions a t 1 GeV , t he d i f f e ren t i a l O B K 2c r o ss s e c t i o n h a s a s h a r p s p i k e a t t h e T h o m a s a n g l e O T h . T h i s c o n f i r m se a r l i e r r e s u l t s [ H U M 8 4 , M o i 8 5 ] w h i c h a r e b a s e d o n p e a k i n g a p p r o x i m a t i o n s[A1888, D eE91a] . (b ) On t h e o th e r ha nd , fo r U + U co ll i si ons a t ene rg i eso f 0 .5 , 5, a n d 1 00 G e V / u , t h e d i f f e r e n ti a l O B K 2 c r o ss s e c ti o n s d o n o t s h o wa n y i n d i c a t i o n o f a s t r u c t u r e t h a t m i g h t s u g g e s t a T h o m a s p e a k [ D eE 9 1b ].

8 . 6 O t h e r d i s t o r t e d - w a v e t h e o r i e sW h i l e t h e e i k o n a l a p p r o x i m a t i o n d i s c u s s e d a b o v e i s r e a s o n a b l y s u c c e s s -fu l i n accoun t i ng fo r exper im en ta l c ros s s ec t i ons , i t i s no t qu i t e s a t i s -f a c t o r y t h e o re t ic a l ly . I t d o e s n o t s a ti s fy C o u l o m b b o u n d a r y c o n d i t i o n s( 8 .2 6 ) a n d ( 8 .2 7 ) a n d i s n o t p o s t - p r i o r s y m m e t r i c . W e d i s c u s s b r i ef l y s o m eo t h e r a p p r o a c h e s i n w h i c h m u l t i p l e e l e c t r o n - t a r g e t o r e l e c t r o n - p r o j e c t i l e i n -t e r a 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 i n a n a p p r o x i m a t e w a y . W e u s e a t o m i cun i t s i n t h i s s ec t i on .

I m p u l s e a p p r o x i m a t i o nO n e o f t h e e a r l ie s t t h e o r i e s f o r r e l a t iv i s t i c e l e c t r o n c a p t u r e , d u e t o J a k u b a s -s a - A m u n d s e n a n d A m u n d s e n [ Ja A S0 ], is b a s e d u p o n t h e i m p u l s e a p p ro x i -m a t i o n w h i c h in c l u d e s a n i n f in i t e n u m b e r o f e l e c t r o n - t a r g e t i n t e r a c t i o n s . I nt h i s a p p r o x i m a t i o n , t h e e x a c t p r o j e c t i l e e i g e n f u n c t i o n s ~ f i s r e p l a c e d w i t h as u p e r p o s i ti o n o f c o n t i n u u m t a r g e t C o u l o m b e i g e n fu n c ti o n s ~ , k ( r w , t ) w i t he n e r g y E = hco a n d m o m e n t u m h k w e i g h t e d w i t h th e m o m e n t u m d i st ri -b u t i o n o f t h e b o u n d p r o j e c t i l e s t a t e ~ b~ (r~ , t ' ) . T h i s is i m p l e m e n t e d i n t w os t eps . F i r s t , t he p ro j e c t i l e s t a t e ~b~ i s ex pa nd ed i n t e rm s o f f ree p ro j e c t i l e

' , t ' - ex p ( ik ' ' - ico ' t ' ) w h e r e u 'E , , p , is a free-ige ns t a te s v~, ,k , ( r ' ) u~, ,v , 9 r eIp a r t i c l e sp i n o r . S u b s e q u e n t l y , t h e f r e e - p a r ti c l e w a v e f u n c t i o n s v ~ , , k , a re

t r a n s f o r m e d i n t o t h e t a r g e t s y s t e m a n d e x p a n d e d i n t o f r e e t a r g e t e i g e n -s t a t e s v ~ , k = U E , p e x p ( i k , r T - i co t ) . F i n a l l y , e a c h m o m e n t u m c o m p o n e n ti n t h e t a r g e t s y s t e m is w e i g h t e d w i t h t h e t a r g e t C o u l o m b e i g e n f u n c ti o ni n m o m e n t u m s p a ce , s o t h a t o n e a r ri v e s a t a n io n i z a t io n m a t r i x e l e m e n tl e a d in g f r o m a b o u n d t a r g e t s t a t e r t h r o u g h t h e p r o j e c ti l e p e r t u r b a t i o nt o a c o n t i n u u m t a r g e t s t a te . D e n o t i n g s p a c e - ti m e i n t e g r a t i o n s b y a n g u -l a r b r a c k e t s a n d i n t r o d u c i n g t h e s p i n o r l a b e ls s , s ' o n e g e ts t h e t r a n s i t i o n


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236 C H A P T E R 8. C H A R G E E X C H A N G Ea m p l i t u d e i n t h e i m p u l s e a p p r o x i m a t i o n a s

A / A _ i / dak, , l (s ' ) l v l( S ' ) S _ l v ( S )8.8 p

T h i s e q u a t i o n i s s t u d i e d a t a s y m p t o t i c a l l y h i g h c o l l i s i o n e n e r g i e s w i t h i n ap e a k i n g a p p r o x i m a t i o n . I t i s f o u n d t h a t t h e c r o s s s e c t i o n d e c r e a s e s a s y m p -t o t i c a l l y a s ( l n ~ ) 2 / ~ / i n a g r e e m e n t w i t h t h e r e s u l t ( 8.2 5 ) o f t h e s e c o n d - o r d e rO B K a p p r o x i m a t i o n a n d i n c o n t r a s t t o t h e O B K 1 , t h e B 1 B , a n d t h e e ik o n ala p p r o x i m a t i o n s w h i c h al l g i v e a d e c r e a s e a s ~ - 1 . O w i n g to t h e c o m p l i c a t e di n t e g r a l s a p p e a r i n g i n t h e e x p r e s s i o n ( 8 . 5 6 ) , t h e i m p u l s e a p p r o x i m a t i o n h a sno t ye t been eva lua t ed a t f i n i t e ene r g ie s .T a r g e t c o n t i n u u m d i s t o r t e d - w a v e a p p r o x i m a t i o nA " t a r g e t c o n t i n u u m d i s t o r t e d w av e" ( T C D W ) a p p r o x i m a t i o n d e r i v ed f ro mE q . ( 5. 82 ) is p r o p o s e d b y M c C a n n [M c c8 5]. I n t h i s a p p r o x i m a t i o n , t h ee ikona l phase f ac to r mu l t ip ly ing the f ina l - s t a t e wave f unc t ion o f E q . ( 8 .36)i s r ep laced by

e l - - el~'~r(1 + iuT) 1FI[ - - iPT, 1 , - - i~ / (VrT + V Z T ) m e / h ] (s.57)wh ich f o r l a rge va lues o f r T ha s th e sam e be hav ior a s t he e ikona l ph asef ac to r and f o r f i n i t e va lues r e f l ec t s t he Coulomb d i s to r t i on . I n i t i a l and f ina lw a v e f u n c ti o n s a r e a p p r o x i m a t e d b y th e D a r w i n f o r m ( 4 .1 0 9). E v e r y t h i n ge lse i s the sam e as in Eqs . (8 .34) , (8 .35) , an d (8 .36) . As a conseq uen ce ,in the nonr e l a t iv i s t i c l imi t , sp in - f l i p t r ans i t i ons a r e f o r b idden , a s expec ted ,s i n c e t h e m a g n e t i c f i e l d p r o d u c e d b y t h e p r o j e c t i l e m o t i o n t e n d s t o z e r o i nth i s l imi t . For 500 MeV/u p + H co l l i s ions , t he r a t i o o - f l i p / o - n ~ - - 3 X 10 . 3w h i c h i s i n r e a s o n a b l e a g r e e m e n t w i t h t h e O B K v a l u e (8 .2 0 ). N u m e r i c a lca l cu la t io ns have bee n pe r f o r m ed f o r p + H and p + Be 4+ co l li s ions be tween1 an d 105 MeV .S y m m e t r i c e i ko n a l a p p r o x i m a t io nT h e T C D W t r e a t m e n t a s w e ll a s t h e e i k o n a l t r e a t m e n t , s ee S ec . 8 .4 , a rea s y m m e t r i c i n t a r g e t a n d p r o j e c t i l e w h i c h c e r t a i n l y i s a n u n d e s i r a b l e f e a -tu r e . Moi se iw i t sch [ Moi86 , Moi87] and Deco and R iva r o la [ DeR87b] haves t u d i e d a s y m m e t r i z e d e i k o n a l a p p r o x i m a t i o n ( S E ) a n d D e c o a n d R i v a r o l aa s y m m e t r i c " c o n t i n u u m - d i s t o r t e d w a v e " m o d e l [ D eR 8 6]. B o t h a p p r o a c h e s


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8.6 . O THER D ISTO RTED -WAVE THEORIES 237u s e d i s t o r t e d i n i t i a l a n d f i n a l w a v e f u n c t i o n s i n t h e g e n e r a l f o r m

Hi -- gi @i(r T , t)~f - - gf~)f ( r~ , t ' ) , (8 .58)

w h e r e @i a n d ~pf a r e t h e i n i t ia l a n d f i n a l D i r a c b o u n d - s t a t e w a v e f u n c t i o n s .C o n s i d e r i n g f i r st t h e S E a p p r o x i m a t i o n , t h e f u n c t i o n s 5

Ui _ e-i tJp ln(r~, +z~,)Uf = e ivT ln(rT+Z T) (8.59)

a r e ( p h a se ) d i s t o r ti o n fa c to r s. 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 na m p l i t u d e i n t h e pr ior fo rm then i s g iven by [Moi86 , Moi87 , DeR 87b]

ASE (b) = ~ d t(N3

S 1 Vf~)~ ZPe2 /, S 1?~p 1 a 'r~P-7+ C~zr~) S U i ~ i > 9(8.60)

R em em be r in g t h a t t he D i rac m a t r i ce s C~x an d C~y m ed ia t e sp in - f li p t r ans i -t i o n s , t h e e x p l i c i t f o r m ( 8 . 6 0 ) w i t h t h e f a c t o r / 3 = v / c i n t h e d e n o m i n a t o ro f t h e s e c o n d t e r m s u g g e s t s t h e e x i s t e n c e o f l a rg e s p i n -f li p c o n t r i b u t i o n s a tsm a l l ve loc i t i e s . Th i s i s un rea l i s t i c s i nce sp in - f l i p i s caused by t he m o t ion -i n d u c e d m a g n e t i c f ie ld o f t h e p r o j e c t i l e i n t e r a c t i n g w i t h t h e m a g n e t i c m o -m e n t o f t h e e l e c t r o n . T h e s e f e a t u r e s a p p e a r a s a n a r t i f a c t o f t h e S E a p p r o x -i m a t i o n , w h i c h a l s o s i g n i f i c a n t l y u n d e r e s t i m a t e s t h e e x p e r i m e n t a l d a t a .

T h e s p u r i o u s c o n t r i b u t i o n s i n t h e e x p r e s s i o n ( 8 . 6 0 ) a r e a n a l y z e d i n[ToE90a], w ho show tha t t h ey van i sh if e i t he r U i = 1 o r Uf = 1 , ac h o i c e w h i c h c o r r e s p o n d s t o t h e u s u a l ( n o n s y m m e t r i c ) e i k o n a l a p p r o x i m a -t i o n . T h e c o n c l u s i o n s r e g a r d i n g t h e n o n r e l a t i v i s t i c l i m i t o f t h e S E a p-p r o a c h a r e c o n f i r m e d b y e x a c t n u m e r i c a l e v a l u a t i o n s o f c a p t u r e c r o ss s ec -t i o n s [ T o E 90 a ]. W h i l e th e O B K , e ik o n a l , a n d B I B r e s u l t s s h o w t h a t t h esp in - f l i p con t r i bu t i on i s exceed ing ly sm al l fo r p + H co l l i s i on a t 100 keV,t h e S E a p p r o x i m a t i o n g i v e s a s p i n - f l i p c r o s s s e c t i o n t h a t i s b y m a n y o r d e r so f m a g n i t u d e t o o l a r g e a n d i s o f t h e s a m e o r d e r a s t h e n o n - s p i n - f li p c r o sssec t i on .

I n o r d e r t o a v o i d t h e s p u r i o u s s p i n - f l i p c o n t r i b u t i o n s i n t h e n o n r e l a t i v i s -t i c l i m i t , a m o d i f i e d s y m m e t r i c e i k o n a l a p p r o x i m a t i o n h a s b e e n i n t r o d u c e d

5Constant phase factors depending on the unit of length are dropped in the eikonalphase factors.


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238 C H A P T E R 8. C H A R G E E X C H A N G E[G1M92] wh ich , how ever , does no t po s ses s t he co r rec t r e l a t i v i s t i c b ehav io r .

C o n t i n u u m d i s t o r t e d - w

chapter 8 - charge exchange

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