Chapter 9 - Radiative Electron Capture

8/12/2019 Chapter 9 - Radiative Electron Capture

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C h a p t e r 9R a d i a t i v e E l e c t r o nC a p t u r eIn Chap. 8, we discussed Coulomb or nonradi ative electron capture. Inthe nonrelativistic energy regime, the cross section for this process falls offasy mptoti cal ly w ith increasing projecti le veloc ity v as v -12 (or as v -11 insecond order). This rapid decrease is caused mainly by the requirementthat a given momentum component in the initial electronic wave functionhas to find its counterpart in the final momentum wave function displacedby the momentum mev of an electron traveling with the speed of the pro-jectile. If, however, the electron tr ansfer is acco mpan ied by the emission ofelectromagnetic radiation, the emitted photon acts as a third body carryingaway energy and momentum released by the formation of the final boundstate. Hence, the condition of mo mentu m matc hing will be relaxed so tha tthe cross section for radia tive electron captu re (REC) falls off as v -5 forhigh nonrelativistic projectile velocities.

For free electrons, capture cannot take place at all without the emissionof photons owing to energy and mo me ntu m conservation. This means,qualitatively, that electrons loosely bound in low-Z target atoms or in outershells are more likely to be captured with photon emission than without.Hence for low-Z target atoms at high projectile energies (e.g., for ZT -- 1with Ep _> 10 MeV/ u, [SpiT9]), th e REC cross section exceeds the crosssection for Coulomb capture. From this point of view, the R EC mechanismdeserves particular attention.

Starting in Sec. 9.1 with a preliminary pictorial description of RECwithin the impulse approximation, we turn in Sec. 9.2 to the direct per-turb ativ e formulation. An insight into qualitative features, in particular

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248 C H A P T E R 9. R A D I A T I V E E L E C T R O N C A P T U R Et h e a n g u l a r d i s t r i b u t i o n , is o b t a i n e d f r o m a n o n r e l a t i v i s t i c t r e a t m e n t i nS e e. 9 .3 . A s a c o n s e q u e n c e o f t h e s e c o n s i d e r a t io n s , w e a r g u e t h a t p r e s e n t l yt h e b e s t d e s c r i p t i o n o f R E C is p r o v i d e d b y t r e a t i n g i t e s s e n ti a l ly a s t h ei n v e r s e o f t h e p h o t o e l e c t r i c e f fec t. Th e re fo re , i n See . 9 .4 , w e d i s cu s s s ev e ra lw ay s t o o b t a i n p h o t o e l ec t r i c c ro s s s ec t i o n s s o t h a t i n See . 9 .5 w e can u s et h e s e r e s u l t s t o d e r i v e t h e d e s i r ed REC c ro s s s ec t i o n s . F i n a l l y , i n See . 9 .6 ,w e p r e s e n t f u r t h e r 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 r e s u l t s .

W e u s e f ul l u n i t s i n th 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 o r t h e m o r et e c h n i c a l s e c t i o n s 9. 4 .3 a n d 9 .5 .3 , w h e r e w e a d o p t r e l a t i v i s ti c ( n a t u r a l ) u n i t sh = me = c = 1 , s ee A p p e n d i x .

9 . 1 G e n e r a l f e a t u r e sA l o o s e l y b o u n d t a r g e t e l e c t r o n m a y b e c o n s i d e r e d a s a p p r o x i m a t e l y f r e e i na h i g h - e n e r g y c o l li si on . I n t h i s l im i t , R E C is i d e n t i c a l w i t h r a d i a t i v e r e c o m -b i n a t i o n ( R R ) , i n w h i c h a n e l e c t r o n i n i t ia l l y m o v i n g w i t h t h e v e l o c i ty - vi n t h e p r o j e c t il e f r a m e i s c a p t u r e d i n to a b o u n d s t a t e w i t h t h e s i m u l t a n e o u se m i s s i o n o f a p h o t o n o f e n e r g y h a / a n d w a v e n u m b e r k ~. E x p e r i m e n t a l l y ,t h e p r o c e s s o f R E C h a s b e e n f ir st i d e n ti f ie d b y R a i s b e c k a n d Y i o u [ R aY 7 1 ],b y Sc h n o p p e r e t a l . [SCB7 2 ], an d b y K i en l e e t al . [K iK 7 3 ] , an d s i n ce t h a tt i m e h a s b e e n e x t e n s i v e l y s t u d i e d b o t h e x p e r i m e n t a l l y a n d t h e o r e t i c a l l y .

I n t h e f o ll ow i n g, w e s t a r t w i t h a b r i e f o u t l i n e o f t h e t r e a t m e n t . W ea s s u m e t h a t t h e c r o s s s e c t i o n a R R f o r r a d i a t i v e r e c o m b i n a t i o n i s k n o w ns i n ce it is e s s en t i a l l y t h e i n v e r s e o f t h e a t o m i c p h o t o e l ec t r i c e f fec t. I t is t h e nn a t u r a l t o r e fe r al l m o m e n t a t o t h e p r o j e c t i le f r a m e . I f t h e t a r g e t e l e c t r o nh a s t h e m o m e n t u m q w i t h r e s p e c t t o t h e t a r g e t n u c l e u s a n d t h e p r o j e c t i l em o m e n t u m i s c h a r a c t e r i z e d b y t h e L o r e n t z f a c t o r ~ , , i t s m o m e n t u m i n t h ep r o j e c t i l e f r a m e q~ is o b t a i n e d f r o m E q . ( 2. 29 ) . I f t h e m o m e n t u m 3 'm e vo f a n e l e c t r o n t r a v e l i n g w i t h t h e r e l a t i v e s p e e d o f t h e t a r g e t t o w a r d s t h ep r o j e c t i l e i s 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 o m e n t u m q i n a l o w - Z t a r g e ta t o m , o n e m a y u s e t h e i m p u l s e a p p r o x i m a t i o n [ K 1J 75 ] t o w r i t e t h e d o u b l e -d i f f e r e n t i a l R E C c r o s s s e c t i o n i n t h e p r o j e c t i l e f r a m e ( p r i m e d c o o r d i n a t e s )a s

d2 C rR EC - - / d 3 dC rR a q') [@i (q)]2 6 (?tW ' + E [ - E [) (9 1)df~ ' d (h a / ) - q df~ ' ' "w h e r e @ i ( q ) is t h e F o u r i e r t r a n s f o r m o f t h e i n i t i a l e le c t r o n i c t a r g e t w a v ef u n c t i o n . T h e d e l t a f u n c t i o n e x p r e s s e s t h e e n e r g y c o n s e r v a t i o n b e t w e e nt h e f i n a l e l e c t r o n i c e n e r g y E [ a n d t h e p h o t o n e n e r g y h c J ' i n t h e p r o j e c t i l ef r a m e o n t h e o n e h a n d a n d t h e i n i t i a l e l e c t r o n i c e n e r g y E { ( a l s o i n t h ep r o j e c t i l e f r a m e ) o n t h e o t h e r h a n d . A c c o r d i n g t o E q . ( 2 .2 9 ), t h e s e e n e rg i e s


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9.1 . G E N E R A L F E A T U R E S 249a r e w r i t t e n a s

E ( - " ) / ( E i - V q z )- ~ m e c2 - ~ [ ~ i l - ~ yvq z ( 9 . 2 )

a n dE~ - me c 2 - l e} l . (9 .3 )

H e r e I ~ il ~ n d I ~ 1 ~ r e t h e b i n d i n g e n e r g i e s o f i n i ti a l t ~ r g e t ~ n d f in ~ l p r o j e c t i l es t a t e s. I n s e r t i n g t h e e x p r e s s i o n s ( 9. 2) a n d ( 9 . 3 ) i n t o E q . ( 9 . 1) , w e h a v e

d 2 C r a E Cd ~' d ( ~ ' )

d a R a ( q , )d 3 q df2~

J dc raa (q , )d a q d f ~

I@i(q)125(hw + E~ - ")/Ei -1--")'VqzI@i(q)125( h w ' - 141 + ~ l ~ [ - T e + ~ V q z ) ,

(9.4)w h e re Te = mec2 ( ') - 1 ) is t h e k i n e t i c en e rg y o f an e l ec t ro n t r a v e l i n g w i t ht h e s a m e s p e e d a s t h e p r o j e c ti l e. S i nc e t h e in i ti a l m o m e n t u m d i s t r i b u t i o no f t h e t a r g e t e l e c t r o n is p e a k e d a r o u n d q ~ 0 a n d s in c e we u s u a l l y c o n s i d erZ T < Z p , t h e r e c o m b i n a t i o n c r o s s s e c t i o n v a r i e s s l o w l y a n d c a n b e t a k e no u t s i d e t h e i n t e g r a l o v e r t h e tr a n s v e r s e m o m e n t u m . I n t h i s a p p r o x i m a t i o n ,we ge t

dCrRE C _ 1 ( d f f R R ) , f f i q z ) , (9.5)d f t ~d (h w ') " /v d f t ~ qlz=3,(_rneV+qz)w h e r e f, .7i(qz) - J d2 qb l@ i(q)l 2 (9.6)is t h e C o m p t o n p r of il e. T h e i n t e g r a t i o n e x t e n d s o v e r t h e t r a n s v e r s e m o -m en t u m q b- In Eq . ( 9 .5 ) , t h e l o n g i t u d i n a l m o m en t u m q~ i s f ix ed b y w ~, s ot h a t t h e s h a p e o f t h e p h o t o n s p e c t r u m

h ~ - 1 4 t - ~ t~ i l + T o - Z V q z (9 .7)is d e t e r m i n e d b y t h e s p r e a d o f t h e l o n g i tu d i n a l m o m e n t u m qz a r o u n d i t sm e a n v a l u e q z - O , w h i c h y i e l d s t h e p e a k e n e r g y

~ ; - ~ E ~ - E ~ - I ~ f l - ~ /[ (~ i[- n - re- (9.8)T h e t e r m ~/vqz i n E q . ( 9. 7) d e s c r ib e s t h e D o p p l e r b r o a d e n i n g t h r o u g h t h eC o m p t o n p r o f i l e .

H o w e v e r , s i n c e t h e c r o s s s e c t i o n a a R d e c r e a s e s w i t h i n c r e a s i n g x - r a yen e rg y , s ee See . 9 .5 , t h e ac t u a l l y meas u red p eak i s u s u a l l y s h i f t ed t o l o w erenergy .


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250 C H A P T E R 9. R A D I A T I V E E L E C T R O N C A P T U R E

I n t h e l a b o r a t o r y s y s t em , t h e p h o t o n e n e r g y h w d e p e n d s o n t h e e m i s s i o na n g l e 0 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 v ia

h ( . U - - I C } I - - 3 ` l ~ i I + r e - - 3 ` V q z3'(1 - - 3 COS 0) ' (9. 9)

w h ic h f o l low s f r om E q . ( 2 . 51 ) . S imi l a r ly , if E qs . ( 9 .4 ) o r (9 . 5 ) l e a d to a na n g u l a r d i s t r i b u t i o n i n t h e p r o j e c t i le s y s t e m a s a f u n c t i o n o f t h e e m i s s io na n g le 0 ' w i t h r e s p e c t t o t h e b e a m a x is , w e o b t a i n t h e a n g u l a r d i s t r i b u t i o ni n t h e l a b o r a t o r y s y s t e m b y t h e t r a n s f o r m a t i o n s

cos 0 ~ = cos 0 - 3 an d df t ~ = 1 (9 .10)1 - 3 co s 0 df t 3 2 (1 - 3 co s 0) 2,g ive n by E qs . ( 2 . 53 ) a nd ( 2 . 54 ) , r e spe c t ive ly .

9 . 2 T h e t r a n s i t i o n a m p l i t u d e f o r R E CW h i l e S e c . 9 . 1 d e s c r i b e s a n i n t u i t i v e l y a p p e a l i n g a p p r o a c h b a s e d u p o n t h ei m p u l s e a p p r o x i m a t i o n , a c o m p l e t e t h e o r e t i c a l d e s c r i p t i o n s h o u l d s t a r t f r o mt h e g e n e r a l t r e a t m e n t o f S ee . 5.4 .2 . W e s e t o u t f r o m t h e f o r m u l a t i o n o f t h ep 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 . I n c o n t r a s t t o t h e ca s es t r e a t e d t h e r e ,t h e p e r t u r b a t i o n g i v i n g r is e t o R E C is 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 o ft h e a c t i v e e l e c t ro n w i t h t h e p h o t o n f ie ld . A f t e r a n a l y z i n g t h i s m e t h o d i nS e e . 9 . 3 , w e tu r n t o a d i f f e r e n t de sc r ip t ion in S e e . 9 . 4 .

A s i n t h e p r e c e d i n g c h a p t e r s , w e u s e a s t r a i g h t - l i n e v e r s i o n o f t h e i m p a c tp a r a m e t e r m e t h o d t o d es c r ib e t h e p r o j e c ti l e m o t i o n , s ee F i g. 3 .2 . I t isc o n v e n i e n t t o d e s c r i b e t h e r a d i a t i n g s y s t e m i n i ts o w n r e s t s y s t em ; t h e r e f o r ew e r e fe r a ll c o o r d i n a t e s t o t h e p r o j e c t i l e fr a m e . I n o r d e r t o e n s u r e t h et r a n s v e r s e p o l a r i z a t i o n o f t h e p h o t o n s , w e w o r k i n t h e r a d i a t i o n g a u g e ( al soc a ll e d C o u l o m b g a u g e ) f or t h e e l e c t r o m a g n e t i c v e c t o r p o t e n t i a l A ~ w i t hV ~ . A ~ - 0 . W e t h e n c a n w r i t e t h e i n t e r a c t i o n o p e r a t o r a s - e 3 ` " A ~ =e " /. A ~ = e @ a . A ~, w h e r e t h e v e c t o r p o t e n t i a l a p p r o p r i a t e f or t h e e m i s s io no f a p h o t o n w i t h a w a v e v e c t o r k ~, e n e r g y h w ~ = h c U a n d u n i t p o l a r i z a t io nv e c t o r f l~ w i t h ( ~ - 1 , 2 ) i s g i v e n b y [ Sa k6 7]

1

V w ' ua e . ( 9 .11 )H e r e , V is t h e n o r m a l i z a t i o n v o l u m e , f or w h i c h p e r i o d i c b o u n d a r y c o n d i-t i o n s a r e r e q u i r e d . I f a ll i n t e g r a t i o n s a r e c a r r i e d o u t i n t h e p r o j e c t i l e s y s t e m ,


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9.2 . T H E T R A N S I T I O N A M P L I T U D E F O R R E C 251t 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 a m p l i t u d e b e t w e e n s pe c if ic in i-t i a l a n d f i n a l a t o m i c s t a t e s i s w r i t t e n a s

A (~ ) - i f d r ' / ' d 3 ' - - ' ' t ' ' .r p ~ f ( r p , ) ( e U ~ . A x ) S ~ i ( r T , t ) ( 9. 12 )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 s a r e u s e d f o r t h e m o t i o n o f a s in g l e e le c t r o ni n t h e f ie ld o f t h e r e s i d u a l t a r g e t a t o m a n d t h e p r o j e c t il e , t h i s e x p r e s s i o n ise x a c t w i t h i n t h e o n e - p h o t o n a p p r o x i m a t i o n , i . e . , w i t h i 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 i n t h e p h o t o n f i e l d .

S ince , i n p rac t i ce , i t i s no t pos s ib l e t o eva l ua t e Eq . (9 .12 ) wi th ex ac ts o l u t i o n s o f a n e ff e ct iv e t h r e e - b o d y p r o b l e m , a p p r o x i m a t e w a v e f u n c t i o n sh a v e to b e i n t r o d u c e d . F o r t h e t i m e b e i n g , i t is i n s t r u c t i v e t o a d o p t t h eB o r n a p p r o x i m a t i o n , t h a t is , t o u s e u n d i s t o r t e d e i g e n s t a t e s ~ i a n d ~ } o ft a r g e t a n d p r o j e c t i le w i t h e i g e n e n e r g ie s ( i n t h e i r r e s t s y s t e m ) E i a n d E l .T h e i m p l i c a t i o n s o f t h i s d r a s t i c s i m p l i f ic a t i o n a r e d i s c u s s e d f o l lo w i n g E q .( 9 .1 9 ) . I n s e r t i n g e x p l i c i tl y t h e t i m e d e p e n d e n c e o f t h e w a v e f u n c t i o n s ( 4. 74 )a n d w r i t i n g E i - - h02i a n d E[ - hw~ , we ge t

A ( ) ) ( b ) :1

h Va; ' d r ' 3 , ' ? (r p )e uvft^t --i(k' ' -uv't ')X ~ ' U A e .r p ~ i ( r T )e - -z w i t . (9.13)

A f t e r i n t r o d u c i n g t h e F o u r i e r t r a n s f o r m s q ~ i (q ) - ~ i ( q b , q z ) an d qS~(p) -~ ( P D , P z ) o f t h e i n i ti a l a n d f in a l s t a t e s , r e s p e c t iv e l y , a n d t r a n s f o r m i n g t h et a r g e t - f r a m e c o o r d i n a t e s r T , t to t h e p r o j e c t il e s y s t e m b y t h e L o r e n t z t r a n s -f o r m a t i o n ( 3 . 3 1 ) , w e a r e a b l e t o p e r f o r m t h e s p a c e a n d t i m e i n t e g r a t i o n s t oo b t a i n

1

- 7 V w ' q p f ( q b - hk~b , q - - hk ' z )^ lX a - U A S @ i(q) 6( hoot + Ef -- ~/E i -t- " ) ' V q z ) e i q b ' b / h ,

(9.14)w h e r e q _ = 7 q z - ~ / ( v / c 2 ) E i 9 T h e d e l t a f u n c t i o n e s t a b l i s h e s t h e r e l a t i o n( 9 . 8 ) f o r t h e p h o t o n e n e r g y a n d a l l o w s o n e t o e l i m i n a t e qz so t ha t one i sl e f t w i t h t h e t w o - d i m e n s i o n a l i n t e g r a l

, , t ) e i q b . b / h (9.15) a . u ~ S ~ i ( q b , q +


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^ !F ig u r e 9 . 1 . O r i e n t a t i on o f t he w a ve ve cto r k p a nd the po la r i z a t ion ve c to r s u 1^ tand u2 .

w i t hq+ = [e i - ( E ; + h ~ ' ) / . y ] / vq _ = [ E i / ~ - ( E l + h w l ) ] /v. (9.16)

F o r C o u l o m b c a p t u r e , t h e d e f i n i t i o n s ( 9 . 1 6 ) r e d u c e t o t h e d e f i n i t i o n s ( 8 . 6 ) .W i t h p h o t o n e m i s s io n , th e r e s o n a n c e c o n d i t i o n is q + = 0 . F r o m t h e t r a n -s i t io n a m p l i t u d e ( 9 .1 5 ), t h e s t a t e - t o - s t a t e c ro s s s e c t io n i s o b t a i n e d a s

d2o -d a ' d ( h ~ ' )

v(~')~ j(27rhc )3 ~~1 deb ]A(~) (b)]2"~2

(9 .17)

W h e n i n s e r t i n g E q . ( 9. 15 ) i n t o (9 .1 7 ), th e i n t e g r a t i o n o v e r t h e i m p a c tp a r a m e t e r p l a n e c a n b e p e r f o r m e d t o y ie l d t h e c r o s s s e c t io n i n t h e p r o j e c t i l ef r a m e a s

d 2 ad a ' d ( h ~ ' )

e 2 1 f I2 - -

--1 12. c ~ . u ~ S ~ i ( qb , q+ ) ( 9 .18 )I f t h e i n i t i a l a n d f in a l e l e c t r o n p o l a r i z a t i o n s a r e n o t d e t e c t e d , o n e h a s t o

a v e r a g e o v e r i n i t i a l a n d t o s u m o v e r f i n a l e l e c t r o n s u b s t a t e s i n o r d e r t o g e tt h e c r o s s s e c t i o n p e r e l e c t r o n . L e t 0 / b e t h e e m i s s i o n a n g l e o f t h e p h o t o n


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9.2 . T H E T R A N S I T I O N A M P L I T U D E F O R R E C 253

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 in t h e p r o j e c t i le s y s t e m . A f t e r t h e b -i n t e g r a t i o n h a s b e e n c a r r i e d o u t , i t is p o s s i b l e a n d c o n v e n i e n t t o c h o o s e ac o o r d i n a t e s y s t e m i n w h i c h t h e w a v e v e c t o r k ~ o f t h e p h o t o n a n d t h e l i n ea rp o l a r i z a t i o n v e c t o r fl~ lie i n t h e x - z p l a n e , a n d t h e o t h e r p o l a r i z a t i o n v e c t o r^ tu 2 p e r p e n d i c u l a r t o i t , t h a t is i n t h e y d i r e c t io n , s ee F ig . 9 . 1. E x p l i c i t l yw r i t t e n in c o m p o n e n t s , w e t h e n h a v e

k ' - ( k ' s in 0 ' , O, k ' cos 0 ' )fi~l - (c os 0 ' , 0 , - s in 0 ' )^ ' - ( 0 1 0) .2 , , (9 .19)

T h e a n g u l a r d i s t r ib 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 is o b t a i n e d f ro m E q .( 9 .18 ) w i t h t he a id o f E qs . ( 9 .9 ) a nd ( 9 . 10 ) .

W e n o w t u r n t o a d is c u s s io n o f t h e l i m i t a t i o n s i n t r o d u c e d b y t h e u s eo f a p p r o x i m a t e w a v e fu n c t i o n s . T h e f ir st p o i n t i s t h e r o le p la y e d b y t h ec u r r e n t o p e r a t o r - e c c ~ e n t e r i n g i n t h e f o r m u l a t i o n g i v e n a b o v e . I n g e n -e r a l , i n a c h a r g e t r a n s f e r c o l l i s i o n , o n e w i l l t r y t o u s e a p p r o x i m a t i o n s i nw h i c h t h e i n i ti a l s t a t e i s a n e i g e n s t a t e o f t h e c h a n n e l H a m i l t o n i a n H i , s eeE q . (5 . 53 ) , w h i le t h e f i na l s t a t e is a n e i g e n s t a t e o f a n o t h e r H a m i l t o n i a nH f r H i , s e e E q . ( 5 .5 4 ) . I f t h i s is i n d e e d t h e c a s e, t h e o p e r a t o r - e c ~w i ll n o l o n g e r s a t is f y a c o n t i n u i t y e q u a t i o n li ke E q . ( 4 .1 7 ) t o g e t h e r w i t ht h e c h a r g e d e n s i t y a n d h e n c e w i ll n o t q u a l if y a s a c u r r e n t o p e r a t o r j t o b eu s e d w i t h t h e e l e c t ro m a g n e t i c i n t e r a c t io n j . A / c . I n f a c t , o n e m a y c o n -v i n c e o n e s el f t h a t a s o u r c e t e r m r e s u l t in g f r o m ( H i - H f ) a p p e a r s i n th ec o n t i n u i t y e q u a t i o n . S i n c e i n s u g g e s t i v e a p p r o x i m a t i o n s , e . g. , t h o s e s u c -c e s s f u l l y u s e d i n C o u l o m b c a p t u r e , s e e C h a p . 8 , t h e H a m i l t o n i a n s H i a n dH f b e l o n g t o d i f f e r e n t c e n t e r s a n d h e n c e a r e d r a s t i c a l l y d i f f e r e n t , t h e r e i sn o c u r r e n t o p e r a t o r c o n s i s t e n t w i t h s u c h an a p p r o a c h . A s a c o n s e q u e n c e , iti s n o t p o s s i b l e t o d e f i n e a B o r n a p p r o x i m a t i o n o r 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 f or R E C , s ee , e .g ., [ H i W 8 7 , H i W 8 9 ] , c o n s i s t e n t w i t h t h e u s u a lde f in i t i o n j - - e c c ~ f o r t h e c u r r e n t o p e r a t o r .

I n t h e a b s e n c e o f a n e x a c t t h r e e - b o d y w a v e fu n c t i o n , a c o n s i s t e n t d e -s c r i p t i o n i s o b t a i n e d i f b o t h i n i t i a l a n d f i n al s t a t e s a r e c o n s t r u c t e d a s e i g en -s t a t e s o f e i t h e r H i o r H r . S i n c e u s u a l l y i n R E C o n e c o n s i d e r s s y s t e m s w i t hZ T < < Z p , t h e o n l y p r a c t i c a l w a y t o t r e a t t h e p r o c e s s i s t o c o n s i d e r i n i -t i a l a n d f in a l s t a t e s a s c o n t i n u u m a n d b o u n d s t a t e s o f t h e pro j e c t i l e . T h et a r g e t a t o m i s t a k e n i n t o a c c o u n t o n l y i n s o f a r a s i t g e n e r a t e s a n i n i t i a l e l e c -t r o n ic m o m e n t u m d i s t r ib u t i o n . T h i s is t h e im p u l s e a p p r o x i m a t i o n in t h ep r o j e c t i l e f r a m e d i s c u s s e d i n S e c t i o n 9 . 1 .


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254 C H A P T E R 9. R A D I A T I V E E L E C T R O N C A P T U R E9 . 3 A n o n r e l a t i v i s t i c t r e a t m e n tF o r q u a l i t a t i v e p u r p o s e s , i t i s i n s t r u c t i v e t o s p e c i a l i z e t h e t r a n s i t i o n a m p l i -t ud e (9 .15 ) t o a non re l a t i v i s t i c co l l is ion . In v i ew o f t he fo rego ing d is cus s ionw e ch o o s e t h e p r o j e c t i l e s y s t e m a s a r e f er e n c e f r a m e . S in c e , f u r t h e r m o r e ,i t i s a l s o k n o w n [ K 1 J 7 5 , B r D 7 4 ] t h a t t h e p l a n e - w a v e i m p u l s e a p p r o x i m a -t i o n o r , e q u i v a l e n t l y , t h e B o r n a p p r o x i m a t i o n l e a d s t o a c c e p t a b l e r e s u l t s i nt h e p r o j e c t il e f ra m e , w e u s e u n p e r t u r b e d n o n r e l a t iv i s ti c h y d r o g e n i c - b o u n ds t a t e s a s i n i t ia l and f i na l s t a t e s . A ssu m ing sp in les s pa r t i c l e s fo r t he m o-m en t , we r ep l ace c~ by v d / c = P e l / (m eC ) .

F o r h i g h p r o j e c t i l e v e l o c i t i e s , R E C i n t o p r o j e c t i l e l s s t a t e s w i l l b e d o m -i n a n t , a n d , a l t h o u g h t h i s l i m i t a t i o n is e a s i ly re m o v e d , w e a ls o a s s u m e l st a r g e t s t a t e s . W i t h t h e l s h y d r o g e n i c m o m e n t u m w a v e f u n c t io n b e in g g i ve nby [BeS57] 2V/-2F)5/29~z (9 .20)l s (q) - +w h e r e Q z - m e e 2 Z / h , t h e l s - l s R E C c r o ss s e c t i o n p e r e l e c t r o n ( 9 .1 8 ) int h e p r o j e c t i l e f r a m e c a n b e w r i t t e n a s

d ~ ' d ( h w ' ) h e (meV)2 ~ d 2 q b[@f(p)(p. uA)@i(q)l , (9.21)w h e r e

q = (qb, q+)p = (qb , q _ ) - h k ' (9.22)

and , i n ana log y t o Eq . (8 .7 ) ,q+ - [ e i - (e~ + hw ' ) ] / v + l ~m eV . (9.23)

T h e e n e r g y c o n s e r v a t i o n e x p r e s s e d b y E q . ( 9. 23 ) is a re s u l t o f r e t a i n i n gt h e f ull s p a c e - t im e d e p e n d e n c e e x p ( - i k ~. r ~+ iw~t~), s o m e t i m e s d e n o t e d a sr e t a r d a t i o n , o f t h e p h o t o n w a v e f un c t io n .

K l e b e r a n d J a k u b a s s a [ K 1 J7 5 ] s h o w t h a t t h e B o r n a p p r o x i m a t i o nE q . ( 9 . 2 1 ) i n t h e i m p a c t p a r a m e t e r p i c t u r e i s i d e n t i c a l t o t h e p l a n e - w a v ei m p u l s e a p p r o x i m a t i o n E q . ( 9 . 1 ) .

Ang u l a r d i s t r i bu t i onL e t u s n o w c o n c e n t r a t e o n t h e a n g u l a r d i s t r i b u t i o n o f t h e e m i t t e d p h o t o n s ,a s s u m i n g t h a t t h e p h o t o n p o l a r i z a t io n is n o t d e t e c te d . W h i le t h e m o m e n -t u m w a v e f u n c t i o n ~ i o f t h e i n i ti a l s t a t e e n t e r s o n l y i n s o f a r a s it c u t s o f f t h e


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9 .3 . A N O N R E L A T I V I S T I C T R E A T M E N T 255

1.2 I i I i i I i_

d a l e\ d a l g O o l . s in Z e"~

- \1 . 1 - \ /\ /\

- " . . . . / /. . . . . .

/- tf- I

tt

0 . 9 0 5 ; a , I I , I I , I3 6 0 9 0 1 2 0 1 5 0 8 00

F ig u r e 9 . 2 . Ca lc u l a t e d angu la r d is t r i bu t ions of K RE C pho tons in 197 -M e V /uXe + Be co ll is ions d iv ided by s in 2 0 and norm al ized a t 90 ~ Dash ed curve : Sau te rformula (9 .32) , see Sec . 9 .4 .2 ; so l id curve : nonre la t iv is t ic Bo rn ap prox ima t ionca lcu la t ion , Eq . (9 .24); dash-dot curve : r e la t iv is t ic approx im a t ion ca lcu la t ion fornon-spin-flip transitions only, Eq. (9.18). See also Fig. 9.8.

i n t e g r a t i o n o v e r q b a t l a r g e v a l u e s o f % , t h e e m i s s i o n a n g l e 0 ~ e n t e r s d i r e c t l yi n t o gSf w h e n p is i n s e r t e d f r o m E q . ( 9 .2 2 ) . I f X d e n o t e s t h e a z i m u t h a l a n g l eo f q b , i .e ., t h e a n g l e b e t w e e n q b a n d k ~ i n t h e i m p a c t p a r a m e t e r p l a n e , w em a y w r i t e

E q3 f ( p ) ( p - f i~ , )@ ~ ( q ) 1 2 oc),( q b COS X COS 0' - - q_ s in 0 ' )9 4- q~ sin 2 X

( q~ - 2 h q b k ' c o s X s i n 0 ' + h 2 k '2 + q 2 _ 2 h q _ k ' c o s 0 ' + Q 2 p ) 41

(q2 4- q_2_ 4- Q 2 T)4 " (9 .2 4)I n o r d e r t o i l l u s t r a t e t h e m a i n f e a t u r e s o f t h i s e x p r e s s i o n , i t is u s e f u l

t o in t r o d u c e s o m e a p p r o x i m a t i o n s . T h e m a i n c o n t r i b u t i o n t o t h e i n te g r ali n E q . ( 9 . 2 1 ) a r i s e s f o r q + = 0 , i n a g r e e m e n t w i t h t h e r e s o n a n c e c o n d i t i o n( 9 .8 ) . T h e w i d t h A ( h c ~' ) ~ v Q z w o f t h e r e s o n a n c e is d e t e r m i n e d b y t h eD o p p l e r s p r e a d i n q + p e r m i t t e d b y t h e t a r g e t c h a r g e Z T . I n t h e l im i t o fa v a n i s h i n g t a r g e t c h a r g e Z T o r o f ci ~ 0 , t h e c o n t r i b u t i o n s o f t r a n s v e r s em o m e n t a qb a r e s t r o n g l y p e a k e d a r o u n d qb = 0 . A t t h e r e s o n a n c e , w e t h e nt 1h a v e c f + h a l - - ~ r n e v 2 a n d q _ ~ - m e v . S i nc e f u r t h e r m o r e h a / < < 2 rn e c 2


10/34

256 C H A P T E R 9. R A D I A T I V E E L E C T R O N C A P T U R Ew e c a n w r i t e t h e p h o t o n a n g u l a r d i s t r i b u t i o n i n t h e p r o j e c t i l e s y s t e m a s

do" si n 2 0 tdf t ' e< (1 + /3 cos Or) 4" (9.25)

U s i n g E q . ( 9 .1 0 ) t o t r a n s f o r m E q . ( 9. 25 ) i n t o t h e l a b o r a t o r y s y s t e m , o n ef i n d s a t t h e r e s o n a n c e f r e q u e n c y

dodf~ e( sin 2 0. (9.26 )

H e n c e t h e c r o s s s e c t i o n v a n i s h e s i n t h e f o r w a r d a n d b a c k w a r d d i r e c t i o n s .I n d e e d , s in c e a t r a n s v e r s e p h o t o n c a r r ie s t h e a n g u l a r m o m e n t u m -1-1 i ni ts d i r e c t i o n o f p r o p a g a t i o n , t r a n s i t i o n s i n to a t o m i c s - s t a t e s o f a s p in l e sse l e c t r o n m o v i n g a l o n g th e z - a x i s a r e f o r b i d d e n i f t h e p h o t o n is e m i t t e da long t he pos i t i ve o r nega t i ve z -ax i s [F aM59] . In a r e l a t i v i s t i c desc r i p t i on( Se e. 9 . 5. 3 ) w h e r e t h e e l e c t r o n s p i n is t a k e n i n t o a c c o u n t , f o r w a r d a n db a c k w a r d e m i s s io n is c o n s i s te n t w i t h a n g u l a r m o m e n t u m c o n s e r v a ti o n .

I n a n e x a c t n o n r e l a t i v i s t i c c a l c u l a ti o n , i f t h e i n t e g r a t i o n o v e r qb is ac-t u a l l y c a r r i e d o u t , o n e o b t a i n s a n a n g u l a r d i s t r i b u t i o n v e r y c l o s e t o E q .(9 .26) , see F ig . 9 .2 . In orde r to der ive the re su l t (9 .25) , i t i s essen t ia l tot a k e i n to a c c o u n t t h e r e t a r d a t i o n o f t h e p h o t o n m o m e n t u m , i.e ., n o t t o u set h e d i p o l e a p p r o x i m a t i o n r e p l a c i n g e x p ( - i k ~- i f ) ~ 1. T h i s w o u l d le a d to ap h o t o n a n g u l a r d i s t r i b u t i o n i n t h e p r o j e c t il e f r a m e pro po r t i o na l t o s i n 2 0 ~.A s E q . ( 9.2 6) sh o w s, t h e f o r w a r d - b a c k w a r d a s y m m e t r y ( 9 . 2 5 ) i n t h e p ro -j e c ti le f r a m e c a u s e d b y t h e r e t a r d a t i o n is f u ll y c o m p e n s a t e d b y t h e L o r e n t zt r a n s f o r m a t i o n t o th e l a b o r a t o r y s y s t e m . T h i s w a s f ir st p o i n t e d o u t b ySpindler [Spi79 , SpB79] .

T h e a n g u l a r d i s t r i b u t io n ( 9.2 6 ) is i n g o o d a g r e e m e n t w i t h e x p e r i m e n t a lda t a fo r i n t e rm ed ia t e p ro j ec t i l e cha rges , s ee , e .g . , F ig . 9 .8 , even a t r e l a t i v i s -t i c v e l o c i t i e s w h e 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 i s r e s u l t a r e n o l o n g e rju s t i f i ed .T h e r o l e o f e l e c t r o n s p i nS o fa r , i n Eq . (9 .21 ), we have d i s reg a rd ed t h e e l ec t ron sp in . I f one t akesi n to a c c o u n t t h e i n t e r a c t i o n o f t h e e l e c tr o n m a g n e t i c m o m e n t w i t h t h em a g n e t i c f ie ld o f t h e p h o t o n , t h e s i t u a t i o n c h a n g e s . T h e c o n s e r v a t i o n o f t h ea n g u l a r m o m e n t u m a l o n g t h e p o l a r a x is n o lo n g e r p r o h i b i t s p h o t o n e m i s si o ni n t h e f o r w a r d o r b a c k w a r d d i r e c t i o n , b e c a u s e a s p i n - f l i p t r a n s i t i o n c a nc o m p e n s a t e f or t h e a n g u l a r m o m e n t u m c a r ri e d aw a y b y t h e p h o t o n . T h i sm e a n s t h a t i n a r e l a t i v i s t i c d e s c r i p t i o n R E C i n t o 1 = 0 s ta t e s a t f o r w a r d o rb a c k w a r d a n g l e s c a n o c c u r o n l y b y s p i n - f l i p p r o c e s s e s m e d i a t e d b y m a g n e t i c


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9.4 . T H E P H O T O E L E C T R I C C R O S S S E C T I O N 257i n t e r a c t i o n s . T h e r e f o r e , a n g u l a r d i s t r i b u t i o n s m a y c h a n g e s i g n i f i c a n t l y , s e e ,e .g . , F ig . 9 .10 .

I n l e a d i n g o r d e r , s p i n - f l i p t r a n s i t i o n s a r e m e d i a t e d b y t h e m a g n e t i cd i p o le t e r m i n t h e m u l t i p o l e e x p a n s i o n [M o s6 6] o f 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 , w h i c h d o e s n o t a c t o n t h e s p a c e p a r t o f t h e w a v e f u n c t io n s .T h e r e f o r e , i f t h e m a g n e t i c d i p o l e i n t e r a c t i o n is e v a l u a t e d b e t w e e n e x a c tb o u n d a n d c o n t i n u u m r a d i a l w a v e f u n c t i o n s , i t s m a t r i x e l e m e n t v a n i s h e so w i n g t o o r t h o g o n a l i t y s o t h a t o n l y th e n e x t o r d e r i n t h e e x p a n s i o n o f t h ep h o t o n p l a n e w a v e c a n g iv e a f in i te c o n t r i b u t i o n . O n t h e o t h e r h a n d , i f t h em a g n e t i c d i p ol e i n t e r a c t i o n i s e v a l u a t e d w i t h a p p r o x i m a t e w av e fu n c t i o n s ,a s p r o v i d e d b y t h e B o r n o r d i s to 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 , t h e r e w i llb e , in g en e ra l , a f i n it e o v e r l ap g i v i n g r i se t o s p u r i o u s co n t r i b u t i o n s . Th ea r g u m e n t s h o w s t h a t s p i n - f l i p t r a n s i t i o n s a r e e x t r e m e l y s e n s i t i v e t o t h e u s eo f a c c u r a t e w a v e f u n c t i o n s .

W e m a y i m m e d i a t e l y d r a w a c o n c l u s i o n r e g a r d i n g r e l a t i v i s t i c c o l l i s i o n s .S i n c e t h e s t a r t i n g e q u a t i o n ( 9 . 1 2 ) f o r r e l a t i v i s t i c R E C i s b a s e d o n t h e D i r a cc t o p e r a t o r w h i c h e m b r a c e s b o t h n o n - s p i n - f l i p a s w e l l a s s p i n - f l i p t r a n s i -t i o n s, a n y B o r n t y p e a p p r o x i m a t i o n w i ll l e a d t o i n c o r r e c t s p in - fl ip c o n t r i-b u t i o n s t o t h e a n g u l a r d i s t r i b u t i o n .

O n t h e o t h e r h a n d , i f o n e c o n f in e s o n e s e lf t o n o n - s p i n - fl ip t r a n s i t i o n s ,a r e l a t i v i s t i c B o r n t r e a t m e n t l e a d s t o r e s u l t s t h a t a r e s im i l a r t o th o s e o f an o n r e l a t i v i s t i c t h e o r y , d i s r e g a r d i n g t h e s p i n d e g r e e o f f r e e d o m . F i g u r e 9 . 2s h o w s th e d e v i a t i o n o f v a r i o u s a n g u l a r d i s t r i b u t i o n s f r o m a p u r e s i n 2 0 d e -p e n d e n c e i n t h e l a b o r a t o r y s y s t e m . B e s i d e s r e l a t i v i s ti c a n d n o n r e l a t i v i s t i cc r o s s s e c t i o n s i n t e g r a t e d o v e r t h e R E C r e s o n a n c e , t h e f i g u r e a l s o i n c l u d e st h e d i f f e r en t i a l c ro ss s ec t i o n d e r i v ed f ro m t h e Sa u t e r t h e o ry [Sau 31 ] of t h er e l a t i v i s t i c p h o t o e f f ec t , s ee Sec . 9 .4 .2 . I n al l c a s e s , t h e d ep a r t u r e f ro m as i n 2 0 b eh av i o r a ro u n d 9 0 o i s v e ry s m a l l a t t h i s en e rg y . Th e c ro s s s ec t i o n sv a n i sh a t f o r w a r d a n d b a c k w a r d a n g l e s a s is r e q u i r e d b y a n g u l a r m o m e n t u mc o n s e r v a t i o n .

Mu ch mo re t h an fo r n o n - s p i n - f l i p t r an s i t i o n s , i s i t c ru c i a l f o r s p i n - f l i pt r a n s i t i o n s a n d h e n c e f o r s p i n - a v e r a g e d r e l a t i v i s t i c a n g u l a r d i s t r i b u t i o n s t od e s c r i b e t h e i n i t i a l e l e c t r o n i c c o n t i n u u m s t a t e ( w i t h r e s p e c t t o t h e p r o j e c -t i l e ) b y an e x a c t C o u l o m b w a v e f u n c t i o n o r a t l e a s t b y a w a v e f u n c t i o n t h a ti s o r t h o g o n a l t o t h e f i n a l b o u n d s t a t e .

9 . 4 T h e p h o t o e l e c t r i c c r o s s s e c t i o nIn o rd e r t o av o i d t h e p ro b l ems d i s cu s s ed i n Sec . 9 .3 , i t i s r e a s o n ab l e t od e s c r i b e r a d i a t i v e e l e c t r o n c a p t u r e a s t h e m o d i f i e d in v e r se o f t h e a t o m i cp h o t o e l e c t r i c e f fe ct : a c o n t i n u u m e l e c t r o n is c a p t u r e d b y t h e p r o j e c t i le


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w i t h t h e s i m u l t a n e o u s e m i s s i o n of a p h o t o n . A s is d i s c u ss e d p r e l i m i n a r i l yi n Se c. 9 .1 , t h e o r i g i n al t h r e e - b o d y p r o b l e m is t h e n r e d u c e d t o a t w o - b o d yp r o b l e m , a n d t h e t a r g e t m e r e l y e n t e r s b y p r o v i d i n g a n i n i t i a l m o m e n t u md i s t r i b u t i o n o f t h e c o n t i n u u m e l e c tr o n s a s s ee n f r o m t h e p r o j e c t i l e n u c le u s .T h e i n it ia l m o m e n t u m s p r e a d o f t h e e l ec t ro n s b o u n d i n t h e t a r g e t a t o ml e ad s t o a b r o a d e n i n g o f t h e p h o t o n r e s o n a n c e.

A s a f i r s t s t e p i n d e r i v i n g R E C c r o s s s e c t i o n s , w e d i s c u s s t h e p h o t o -e l e c t ri c c r o s s s e c t io n . F o r t h e p h o t o e l e c t r i c e ff ec t, a p p r o x i m a t e f o r m u l a sf o r t h e d i f f er e n t i al c ro s s s e c t i o n a r e a v a i la b l e [ P r L 6 4 a , b , P r R 7 3 ] , m o s t l yf or K - e l e c t r o n e m i s s io n . W h e n a p h o t o n w i t h e n e r g y hw i n t e r a c ts w i t h ah y d r o g e n i c t a r g e t a t o m o f c h a r g e Z , a n e l e c t ro n w i t h t o t a l e n e r g y E ( in -c l u d i n g t h e r e s t m a s s ) m a y b e e m i t t e d . I f I%1 is t h e b i n d i n g e n e r g y o f t h ei n i ti a l ly b o u n d e l e c tr o n , t h e c o n s e r v a t i o n o f e n e r g y r e q u i re s t h a t

E - hc~ + m ~ c 2 - I % 1 , (9 .27)s o t h a t t h e r e e x i s t s a o n e - t o - o n e r e l a t i o n b e t w e e n E a n d h w . B e f o r e d i s-c u s s i n g t h e g e n e r a l p r o c e d u r e f o r c a l c u l a t i n g t h e e x a c t p h o t o e l e c t r i c c r o s ss e c t i o n i n S e c . 9 . 4 . 3 , w e f i r s t c o n s i d e r a n o n r e l a t i v i s t i c a p p r o x i m a t i o n i nS e c . 9 . 4 . 1 a nd in S e c . 9 . 4 . 2 a r e l a t i v i s t i c a pp r oa c h va l id f o r l ow - Z a toms .

9 . 4 . 1 T h e S t o b b e f o r m u l aA s s u m i n g h w < < m e c2 a n d a Z


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9.4 . T H E P H O T O E L E C T R I C C R O S S S E C T I O N 259The cor responding d i f ferent ia l c ross sec t ion is

d(~ph : 3df~p h O'ph ~ sin 20 . (9.31)The S tobbe c ros s s ec t ion p roves to be qu i t e u se fu l in REC ca lcu la t ions upto p ro jec t i le ene rg ies o f a f ew hun dred M eV /u , co r r espon d ing to e lec t ronk i n e t i c e n e r g i e s ( 7 - 1)m e c2 , well be low the e lec t ron res t energy .9 . 4 . 2 T h e S a u t e r f o r m u l aSauter [Sau31] has der ived an analy t ic express ion for the cross sec t ion forthe r e la t iv i s ti c pho toe f f ec t a s s um ing /3 ~ 1 and d i s r ega rd ing t e rms o f theo rde r a2 Z2 in an c~Z expans ion o f the m a t r ix e lemen t . Fo r a d i s cus s ionof the ap pro xim at io ns , see Fano e t a l . [FaM59] an d Pr a t t e t al . [PrR73] .T h e S a u t e r f o r m u l a h a s b e e n u s e d t o d e r i v e t h e c o r r e s p o n d i n g R E C c r o s ssec t ions and to compare them wi th exper imen ta l da ta [RAY71 , AnM 84] .Accord ing to the Sau te r fo rmula [Sau31 , Nag60 , P rR73] , the d i f f e r en t i a lpho to ion iza t ion c ros s s ec t ion pe r K-she l l e l ec t ron fo r unpo la r i zed pho tonsand e lec t rons is

dCrphdf~ph

) 5 ; ~ 3-- O~6 Z 5 m e c2ha~ )~ 7

sin 20 [x (1 - ~ e o s 0 ) 4 1 + 7( 7 - 1)(72 - 2)(1 - ~ co s0 )] . (9.32)

By in teg ra t ing over the so l id ang le , one ob ta ins the to ta l c ros s s ec t ion pe rK-e lec t ron as

Crphx / 3 3 7 a { 4 7 ( 7 - 2 ) [~ + 17 + 1

1 1 + ; ] }- 2/3,)--------- in 1 - . (9.33)The Sau te r fo rmulas (9 .32 ) and (9 .33 ) have been improved by Gavr i l a

[Gav59, Gav60] , by Nagel [Nag60 , HUN67] , and by Pra t t [PrR73] , whoi n c l u d e d h i g h e r - o r d e r t e r m s i n a Z .

9 . 4 . 3 E x a c t r e l a t i v i s t i c c a l c u l a t i o n sTh e ex ac t ev a lua t ion o f the r e la t iv i s t i c pho toe lec t r i c c ros s s ec t ion r equ i r esa p a r t i a l- w a v e e x p a n s i o n o f t h e C o u l o m b c o n t i n u u m f u n ct io n . T h i s m e a n s


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260 C H A P T E R 9. R A D I A T I V E E L E C T R O N C A P T U R Et h a t c l o s e d - f o r m e x p r e s s i o n s c a n n o l o n g e r b e d e r i v e d , a n d o n e h a s t o r e -s o r t to n u m e r i c a l m e t h o d s . D e t a i l e d f o r m u l a t i o n s a n d t a b u l a t i o n s o f c ro s ss ec t i o n s can b e fo u n d i n [N ag 6 0 , A 1 J 6 5 , H UN 6 7 , P rR7 3 , Sco 7 3 ] . H o w ev e r ,t h e e x i s t i n g t a b u l a t i o n s a r e n o t a l w a y s s u f f i c i e n t f o r u s e i n c o n n e c t i o n w i t hREC, w h i ch r eq u i r e s d i f f e r en t i a l c ro s s s ec t i o n o n a s u f f i c i en t l y d en s e mes h .S i n c e f o r t h i s p u r p o s e i n d e p e n d e n t c o m p u t e r c o d e s h a v e t o b e d e v e l o p e d[IC S9 4], w e h e r e p r e s e n t a n o u t l i n e o f t h e f o r m u l a t i o n , a s s u m i n g u n p o l a r i z e dp h o t o n s a n d a p o l a r i z a t i o n - i n s e n s i t i v e d e t e c t o r f o r t h e e m i t t e d e l e c t r o n s .

Fo r a g i v en p h o t o n en e rg y h c~ an d b i n d i n g e n e rg y l eb l o f t h e e l ec t ro ni n t h e b o u n d i n i ti a l s t a t e b , t h e f i n a l e l e c t r o n e n e r g y E ( i n c l u d in g t h e r e s tm a s s ) i s d e t e r m i n e d b y E q . ( 9 .2 7 ). I n t r e a t i n g t h e e l e c t r o n s p in , o n e m a ye i t h e r a d o p t a q u a n t i z a t i o n i n a f i x e d s p a c e d i r e c t i o n o r , a l t e r n a t i v e l y , i nt h e d i r e c t io n o f t h e e l e c tr o n m o m e n t u m . E a c h o f t h e s e a p p r o a c h e s h a sc e r t a i n a d v a n t a g e s .Q u a n t i z a t i o n o f t h e el ec t r o n s p i n i n t h e z - d i r e c t i o nW e f i r s t c o n s i d e r t h e f o r m a l l y s i m p l e r r e p r e s e n t a t i o n , i n w h i c h t h e e l e c t r o ns p i n is q u a n t i z e d w i t h r e s p e c t t o a n a r b i t r a r y z - d i r e c t i o n . T h i s i s p o s s ib l ei f t h e e l e c t r o n s p i n is n o t m e a s u r e d a n d s o m e t i m e s a l lo w s fo r a n u m e r i c a l l yf a s t e r e v a l u a t i o n .T h e d i f f e r en t i a l c ro s s s ec t i o n fo r a s in g l e e l ec t ro n i s [P rR7 3 ] 1

d a p h me c2_= o~ E E IM p ,b (m s )~+ ' #b)12 (9 .34)d ~ 4 hw 2 j b + 1 ' 'ptb msw h e r e w e h a v e a v e r a g e d o v e r t h e ( 2 j b + 1 ) 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 s1# b i n t h e b o u n d s t a t e a n d h a v e s u m m e d o v er t h e sp i n c o m p o n e n t s m s - + ~o f t h e e m i t t e d e l e c t r o n . F u r t h e r m o r e , w e h a v e a v e r a g e d o ve r t h e c i r c u l a rp o l a r i z a t i o n s ) % = 1 a n d A _ = - 1 o f t h e i n c o m i n g p h o t o n . 2 B e c a u s e o ft h e s u m m a t i o n o v e r a l l o t h e r 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 , # b , m s , t h i si s eq u i v a l en t t o t ak i n g o n e p h o t o n p o l a r i z a t i o n , e .g ., A+ = 1 o n ly . T h eq u a n t i t y c~ d e n o t e 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 .

I n a n a l o g y t o E q . ( 9 . 1 2 ) , t h e t r a n s i t i o n m a t r i x e l e m e n t ( a d o p t i n g r e l a -t i v i s t i c u n i t s h = c = me = 1 i n t h i s s u b s ec t i o n f ro m h e re o n ) i s

M p,b (ms, )~+, #b) -- / ~ptp,ms (r ) a . fi+ @ik.r~]jb,lab( r ) d3r , ( 9 . 3 5 )1 E q u a t i o n ( 4 .2 . 2 2) o f [ P r R7 3 ] f o r t h e n o r m a l i z a t i o n o f t h e r a d i a l w a v e f u n c t i o n s

c o n t a i n s a m i s p r in t . T h e e x p r e s si o n s [ ( E i 1)1/2/2E] s h o u l d b e r e p l a c e d w i t h [ ( E 1)/2E] 1/2. I n t h i s c as e , a g r e e m e n t w i t h [H U N 67 , A1J65] is ob t a i ne d . T h e e xp r e s s i on( 9 . 3 4 ) i s c o n s i s t e n t w i t h t h e c o r r e c t e d n o r m a l i z a t i o n ( 9 . 3 7 ) .

2 I n c o n j u n c t i o n w i t h a n a n g u l a r m o m e n t u m r e p r e s e n ta t i o n , i t is c o n v e n ie n t t o s u mo v e r t h e c i r c u l a r r a t h e r t h a n o v e r t h e l i n e a r p o l a r i z a t i o n s A 1 , 2 .


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9.4 . T H E P H O T O E L E C T R I C C R O S S S E C T I O N 261w h e r e , a c c o r d i n g t o E q . ( 4 . 1 2 3 ) , b u t w i t h a d i f f e r e n t n o r m a l i z a t i o n ,

11Vp , m ~ ( r ) - - 4 7 ~ E i l e - i A "~,tt # -- m s ms

i f ~ ( r ) X ' -~

J )p

(9 .36)

is t h e p a r t i a l - w a v e e x p a n s i o n o f t h e w a v e f u n c t i o n d e s c r i b i n g t h e r e l a t i v is t i ce l e c t r o n e m i t t e d w i t h a s y m p t o t i c m o m e n t u m p (p2 = E 2 - 1, Pz = p c o s 0 )a n d s p in p r o je c t i o n m s w i t h r e s p e c t t o t h e z -a x is . T h e s u m m a t i o n in1E q . ( 9 .3 6 ) e x t e n d s o v e r t h e r e l a ti v i s t ic q u a n t u m n u m b e r s ~ - : i: (j + ~ ) a n dt h e a n g u l a r m o m e n t u m p r o j e c t io n s p . A s s o c i a t e d w i th g a r e t h e o rb i t ala n g u l a r m o m e n t a 1 = 1A a n d 1~ = 1B o f t h e u p p e r a n d l ow e r c o m p o n e n t ,r e s p e c ti v e ly , g i ve n in E q . ( 4 .8 5 ). T h e X ~ a r e t h e n o r m a l i z e d t w o - c o m p o n e n ts p i n - a n g u l a r f u n c t i o n s ( 4 . 8 7 ) , a n d t h e g ~ ( r ) a n d f ~ ( r ) a r e t h e ( r e a l ) r a d i a lc o n t i n u u m w a v e f u n c t io n s ( 4 .1 1 4) .

T h e C o u l o m b w a v e f u n c t i o n s a r e n o r m a l i z e d o n t h e e n e r g y s c a l e a n dt h e i r a s y m p t o t i c b e h a v i o r is g i v en b y E q . ( 4 .1 1 8 ), t h a t i s/1 E + 1t / c o s (pr + a~ )r V 7rp

/t / E - 1 s i n (p r + a ~ ) . ( 9 .3 7 )f~ -~ r V ~P

I n t h e l i m i t o f a p o i n t n u c l e u s , t h e C o u l o m b p h a s e s h i ft ( 4 . 1 2 0 ) isa ~ - - A ~ + r / l n ( 2 p r ) , ( 9 .3 8 )

w h e r eA t ~ z 1 1a r g - a r g F ( s + i T ) - ~ ss + i ~

a Z E= ~ , s - V /~ ~ - ( ~ Z ) ~. (9 .3 9)PS i n c e a n g u l a r d i s t r i b u t i o n s a r e s e n s i t i v e t o p h a s e s , o n e h a s t o m a k e s u r et h a t t h e p h a s e c o n v e n t i o n s E q s . ( 9 .3 7 ) a n d ( 9 .3 8 ) ( o r a n o t h e r s e t o f c o n s is -t e n t p h a s e c o n v e n t i o n s ) a r e s i m u l t a n e o u s l y s a ti s fi e d . 3

3In [ICS94] the gen eraliza tion for an exte nde d nucleus g ive n by M iiller et al. [MuR73]has been used.


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262 C H A P T E R 9 . R A D I A T I V E E L E C T R O N C A P T U R EReturn ing to Eq . (9 .35) , t he exac t hydrogen ic bound-s t a t e wave func-

t ion i s ident if ied wi th E q. (4 .86) . I f we now inser t the e xpan sion (6 .42) ofthe ph o ton p lane wave and pe r fo rm the sp inor a lgebra , we a rr ive a t i n t eg ra lsover th ree spher i ca l ha rmonics which can be expressed by Clebsch-Gordancoeff ic ients (4 .88) . Col lec t ing the resul t s , we obta in

OOM p ,b ( m s , ~ + , # b ) - - 4 7 r V~ E E i L + l - 1 eiA'~

L=01 )1 2 J F + ( L , n ) Y1 + 1x (0, 0) (9.40)Pb n - 1 - - m s m s Pb + 1 ~b , ,b - - '~s ,w h e re

F + ( L n )~t b ' [ / oL ( j b P b j ; l'b l) g ~ ( r ) j L ( k r ) f b ( r ) r 2 d r- - A L ( j b P b j ; l b l ' ) f ~ ( r ) j L ( k r ) g b ( r ) r 2 d r , (9.41)

a n dA L ( j b # b j ; / 1 1 2 ) - (2L + 1) V /2/ 1+ 1 ( /1 12/2 + 1 ~b -t-

12x 1#b -t- ~ ( l l L

0 0

1 ) (~ j l l1 1

1 )-~ jb1[2 #b

1# b + ~(9.42)

I t i s not d i ff icul t to carry the analy t ica l reduct ion fur ther [PrR73,HuN67] bysquar ing the mat r ix e l emen t (9 .40 ) and by expand ing the resu l t ing p roduc t so f spher i ca l ha rmonics . In th is way , add i t iona l su m ma t ions a re in t roduced .However , for numerica l evaluat ion , i t i s of ten convenient to take d i rec t lythe m odu lu s square o f the m at r ix e l em en t (9 .40 ) and to in t roduce i t i n toEq. (9.34).H e l i c i t y r e p r e s e n t a t i o nAn al ternat ive descr ip t ion , convenient for ca lcula t ing to ta l cross sec t ions(9.46) and necessary for angular correlat ions, see, e .g. , [Eic94], employs thehe l ic i ty rep resen ta t ion (4.124) o f the e l ec t ron con t inu um wave func tion ,


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9 .4 . T H E P H O T O E L E C T R I C C R O S S S E C T I O N 263i n w h i c h t h e e l e c t r o n s p i n i s q u a n t i z e d a l o n g t h e a s y m p t o t i c m o m e n t u m .A d o p t i n g a d i f f e r e n t n o r m a l i z a t i o n t h a n i n E q . ( 4 . 1 2 4 ) ,

1C p , ~ (r ) - ~ i ' e -izx~ V/4 7r(2 /+ 1) 1

~,# 0 O"

i f ~ ( r ) x~_~

J )O"

(9.43)

i s t h e p a r t i a l - w a v e e x p a n s i o n w i t h t h e s p i n p r o j e c t i o n a w i t h r e s p e c t t o p .T h e r o t a t i o n m a t r i x D ~ ( 2 ~ 15) c a r ri e s e a c h p a r t i a l w a v e f r o m a n o ri g i n a lq u a n t i z a t i o n a l o n g th e z - a x is ( e x p re s s ed b y t h e C l e b s c h - G o r d a n c o ef fi ci en t)i n t o a q u a n t i z a t i o n a x i s a l o n g p .

The d i f f e r en t i a l c ros s s ec t ion i s then ob ta ined as an expans ion in t e rmso f L e g e n d r e p o l y n o m i a l s P ~ ( c o s 0 ) a s

dO 'ph __ OL 7/~eC2 ~ E B~ P~ (cos 0) . (9.44)d f~ 4h w 2 jb + 1 12with the expans ion coe f f i c ien t s

S/] - - 87r ~ '~ ( - 1 ) ~b -1 /2 ~ E ( - 1 ) ( L - 1 - L + l ) / 2 cos (A~ - A~)ttb ~ ~

x ( 2 / + 1 ) (2 j + 1 ) ( 2 1 + 1 ) ( 2 ~ + 1) 0 0) { 1 }2 v 1 jX p b + l - - p b - - 1 0 j 1 V rL (L , U (Z,

0

( 9 . 4 5 )w h e r e F + ( L , n ) i s g iven by Eq . (9 .41 ) . Th e sum m at io ns occu r r ing inEq . (9 .45 ) a r e l im i ted by the s e lec t ion ru les em bod ied in the C lebsch -G ord an coe f fi c ien t s and 6 j - sym bo ls { . - . } . In pa r t i cu la r , i f the e lec t ron isi n i ti a l ly b o u n d i n a l s 1/2 s ta t e , a r a t h e r s m a l l n u m b e r o f t e r m s c o n t r i b u t e s .Never the les s , the doub le sums occur r ing in Eq . (9 .45 ) u sua l ly r equ i r e longerc o m p u t i n g t i m e s t h a n n e e d e d f or s q u a r i n g t h e u n d e v e l o p e d m a t r i x e l e m e n t(9 .40 ) . Fo r numer ica l ly ca lcu la t ing the d i f f e r en t i a l c ros s s ec t ion , the l a t t e ri s the re fo re a more p rac t i ca l s t a r t ing po in t .

M o s t o f t h e p u b l i s h e d c a l c u l a t io n s h a v e b e e n p e r f o r m e d f or t h e K - s h e llpho toe f f ec t , t ak ing in to accoun t s c reen ing co r r ec t ions wh ich a re needed fo rn e u t r a l a t o m s . H o w e v e r , i f o n e w a n t s t o u s e e x is t i n g t a b u l a t i o n s t o d e ri v e


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264 C H A P T E R 9. R A D I A T I V E E L E C T R O N C A P T U R ER E C c r o s s s e c t i o n s f o r bare p r o j e c t i l e s , o n e h a s t o c o m p e n s a t e f o r t h e s eu n d e s i r e d s c r e e n i n g c o r r e c t i o n s [ M e A 8 5 ] .T h e t o t a l c r o s s s e c t i o nT h e t o t a l p h o t o e l e c t r i c c r o s s s e c t i o n i s o b t a i n e d b y i n t e g r a t i n g E q . ( 9 . 4 4 )o v e r t h e p h o t o n a n g le s . T h e n o n l y t h e t e r m w i t h u = 0 c o n t r i b u t e s i n t h es u m , a n d t h e t o t a l c r o s s s e c t i o n p e r e l e c t r o n i s

O~Tr m e c2 )~ B0 (9.46)~p h : h~d 2 j b + 1w i t h t h e p o s i t i v e - d e f i n i t e e x p r e s s i o n

ttb t~E i L F~ L , ~ )

L(9.47)

R a d i a t i v e c o r r e c t i o n sR a d i a t i v e c o r r e c t i o n s t o t h e a t o m i c p h o t o e f f e c t [ B O G 8 2 , M c G 7 7 ] c a n b eexp res sed b y a co r rec t i ve f ac t o r [1 (~ /~ )5 ] m u l t i p ly ing t he bas i c d i ff e r-en t i a l c ro s s s ec t i on (9 .34 ) o r (9 .44 ) . Th e qu an t i t y 5 , ca l cu l a t ed t o l owes to r d e r i n a a n d a Z , i n c r e a s e s w i t h t h e p h o t o n e n e r g y a n d t h e e l e c t r o n e j e c -t i o n a ng le . T h e c o r r e c t io n t e r m ( c ~ / ~ ) 5 is a lw a y s n e g a t i v e a n d o f t h e o r d e r0 . 2 % f o r 1 0 0 - k e V , 1 . 5 % f o r I - M e V , a n d 5 % f o r 5 - M e V p h o t o n e n e r g y . Fort h e R E C e x p e r i m e n t s d i s c u s s e d i n Se c. 9 .6 , t h e r a d i a t i v e c o r r e c t i o n i s w e llw i t h i n t h e e x p e r i m e n t a l u n c e r t a i n t y . S i n ce f or l a r g e p h o t o n e n e rg i e s t h ec o r r e c t i o n b e c o m e s q u i t e a p p r e c i a b l e , i t w o u l d b e i n t e r e s t i n g t o i n c r e a s et h e e x p e r i m e n t a l a c c u r a c y t o a p o i n t t h a t a ll o w s f o r a te s t o f t h e s e r e s u l ts .

9 . 5 F r o m t h e a t o m i c p h o t o e f f e c t t o R E C9 . 5 . 1 R a d i a t i v e r e c o m b i n a t i o nO n c e t h e c r o ss s e c t i o n O p h ( ~ ' , 0 ' ) of t he pho toe l e c t r i c e f fec t in t he p ro -j e c t il e f r a m e ( w e r e i n t r o d u c e p r i m e d q u a n t i t i e s ) h a s b e e n c o m p u t e d , i t isa s i m p l e m a t t e r t o w r i t e d o w n t h e c r o s s s e c t i o n a R R ( E I , 0 t) f o r r a d i a t i v er e c o m b i n a t i o n . T h i s i s t h e c r o s s s e c t io n f o r c a p t u r i n g a fr e e e l e c t r o n o fe n e r g y E ' i n t o a b o u n d a t o m i c s t a t e ( b i n d i n g e n e r g y I% 1 ) w i t h t h e s im u l -t a n e o u s e m i s s i o n o f a p h o t o n o f e n e r g y J . S i nc e a c c o r d i n g t o E q . ( 9 .2 7 ) ,h u t = E t - m e c2 ]s we m a y use E ~ as we l l a s J - k t c t o cha rac t e r i ze


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9.5. F R O M T H E A T O M I C P H O T O E F F E C T T O R E C 265t h e p r o c e s s . W e n o w c h o o s e t h e d i r e c t i o n o f t h e i n c o m i n g e l e c t r o n a s t h e zd i r e c t i o n , s o t h a t 0 ' is t h e e m i s s i o n a n g l e o f t h e p h o t o n .

B y t h e p r i n c i p l e o f d e t a i l e d b a l a n c e , t h e c r o s s s e c t i o n f or r a d i a t i v e r e -c o m b i n a t i o n i n t o a v a c a n t s h e ll is w r i t t e n a s

d 2 O'a a ( E , )d E ' d ~ ; h

( w ' / c ) 2 d2(Yph(E , 0 ' )= (2 jb + 1) p ,2 d E' drY'el

(2jb + l) [" hW ' "~2 1 d2 0 ph (E' ,O ' )~, 2 / 3 2 7 2 d E ' d ~ t e l . (9 .48)

Here, O-ph i s t h e p h o t o e l e c t r i c c r o s s s e c t i o n p e r e l e c t r o n a n d t h e f a c t o r(2 jb + 1 ) c o n v e r t s t h e a v e r a g i n g o f E q . (9.34) o v e r b o u n d - s t a t e a n g u l a rm o m e n t u m p r o j e c t i o n s i n t o a s u m m a t i o n o v e r a f i l l e d s h e l l w h i l e t h e f a c t o r( w ' / c ) 2 / p '2 r e p l ac e s t h e p h a s e s p a c e f a c t o r o f t h e o u t g o i n g e l e c t r o n b y t h a to f t h e e m i t t e d p h o t o n .

I n a p p l i c a t i o n s t o r e l a t i v i s t i c i o n - a t o m c o l l i s i o n s , t h e z - d i r e c t i o n i s u s u -a l ly d e f i n e d as t h e d i r e c t i o n o f t h e p r o j e c t i l e m o t i o n . T h i s i s o p p o s i t e t ot h e d i r e c t i o n o f t h e e l e c t r o n m o m e n t u m a s s e en f ro m t h e p r o j e c t il e . H e n c ef o r R E C , t h e a n g l e 0 ' o f t h e p h o t o e l e c t r i c e ff ec t o r o f t h e r a d i a t i v e r e c o m -b i n a t i o n h a s t o r e p l a c e d b y 7 r - 0 ', o r co s 0 ' is r e p l a c e d b y - c o s 0 '.T h e t o t a l c r o s s s e c t i o n f o r r a d i a t i v e r e c o m b i n a t i o n i n t o a n e m p t y p r o -j e c t i l e s u b s h e l l J b i s o b t a i n e d f r o m E q s . ( 9 . 4 8 ) a n d ( 9 . 4 6 ) i n t h e f o r m

e x a c t CtTl" ~tC0 g UoO 'RR -- /~2~2 ?TteC2 (9 .49)w he r e B0 i s g ive n by E q . ( 9 . 47 ) .The S tobbe cross sec t ionI n t h e n o n r e l a t i v i s t i c l i m i t , t h e f a c t o r i n E q . ( 9 . 4 8 ) i s ( ha / / r nec 2 ) 2 (1 / /3272)= ( hw / 2 r nec 2 ) ( 1 + u 2 ) , s o t h a t t h e S t o b b e c ro s s s e c t io n ( 9. 30 ) f or t h e p h o t o -e l e c t r i c e f f e c t c a n b e r e w r i t t e n f o r r a d i a t i v e e l e c t r o n c a p t u r e i n t o a n e m p t y1K - s h e l l ( J b - 5 ) as

( /23 ) 2 -4uarctan (1/u)Stobbe __ 287r20~ g e (9 .50)a RR 3 l + u 2 1 - - e - 2 ~ 'w he r e ~ : - - h / m e c is t h e C o m p t o n w a v e l e n g t h o f t h e e l e c t r o n a n d u -Z e 2 / h v is t h e S o m m e r f e l d p a r a m e t e r . T h e c o n s t a n t s i n f r o n t o f t h e u -d e p e n d e n t t e r m s m a k e u p a f a c t o r of 9 1 6 4 .7 b a r n .

A s m e n t i o n e d a b o v e , t h e S t o b b e c r o s s s e c t i o n p r o v e s t o b e q u i t e u s e f u lt o e s t i m a t e R E C i n t o t h e K - s h e l l u p t o p r o j e c t i l e e n e rg i es o f a fe w h u n d r e d


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266 C H A P T E R 9. R A D I A T I V E E L E C T R O N C A P T U R EM e V / u , c o r r e s p o n d i n g t o e l e c t r o n k i n e t i c e n e r g i e s ( ~ / - 1 ) m e c 2 wel l be lowthe e l ec t ron r e s t ene rgy . T he an gu l a r d i s t r i bu t i on i s d i s cus sed i n S ec. 9 .4 .1 .T h e S a u t e r c r o s s s e c t i o nF o r r e l a t i v i s t i c e l e c t r o n e n e r g i e s , o n e m a y a p p l y t h e a p p r o x i m a t e S a u t e rfo rm ula (9 .33 ) . In se r t i ng i n to Eq . (9 .48 ) , one ob t a in s t he c ros s s ec t i on fo rr a d i a t i v e r e c o m b i n a t i o n i n t o a n e m p t y K - s h e l l i n t h e f o r m

o . S a u t e r 4 7r c~ 6 Z5 ( r n e C 2 ) 3RR -- ha2 gx / 7 7 g + 1 - ~ l n . (9 .5 1)7 + 1 2 /372 1 - / 3

T h e a n g u l a r d i s t r i b u t i o n is g i v e n b y E q . ( 9 .3 2 ). W e n o t i c e t h a t f o rm od er a t e ly r e l a t i v i s t i c co l l is i ons (7 -< 2 ) , t he l ead i ng t e rm in Eq . (9 .32 )h a s t h e s a m e s i n 2 0 a n g u l a r d e p e n d e n c e i n t h e l a b o r a t o r y s y s t e m a s t h en o n r e l a t i v i s t ic R E C e x p r e s s i o n o f E q . ( 9 .2 5 ) . H o w e v e r , w i t h in c r e a s i n gv a l u e s o f 7 , t h e s e c o n d t e r m i n (9 .3 2 ) b e c o m e s m o r e i m p o r t a n t a n d l e a d s t oa s h i ft o f t h e m a x i m u m i n t h e a n g u l a r d i s t r i b u t i o n t o f o r w a r d a n g le s . I n t h ee 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 e g i m e , t h e S a u t e r a n g u l a r d i s t r i b u t i o n ( 9 . 3 2 )b e h a v e s a s c o s 2 ~ e x c e p t i n th e v e r y f o r w a r d d i r e c t i o n . T h i s d e p e n d e n c e ,i n i ts e f f ec t on R EC , is i l l u s t r a t ed i n F igu r e 9 .3 fo r va r i ou s va lues o f 7 .I n t h e e x a c t f o r w a r d d i r e c t i o n 0 = 0 a s w e l l a s i n t h e b a c k w a r d d i r e c t i o n0 = ~ , t h e S a u t e r c r o s s s e c t i o n v a n is h e s , a l t h o u g h , w i t h t h e i n c l u si o n o f t h ee l e c tr o n s p i n, t h e c o n s e r v a t io n o f a n g u l a r m o m e n t u m d o e s n o t e n f or c e su c ha b e h a v i o r . I t h a s b e e n c o n j e c t u r e d [N a g6 0] t h a t t h i s c u r io u s f e a t u r e o f t h ed i f fe r e n t ia l c r o s s s e c t i o n c a l c u l a t e d i n fi rs t o r d e r o f a Z , o b s e r v e d a l r e a d yb y F a n o e t a l. [ Fa M 5 9] , m a y b e a c c i d e n t a l fo r t h e C o u l o m b p o t e n t i a l .H i g h e r - o r d e r t e r m s i n a Z y i e ld a f in i te c o n t r i b u t i o n a t 0 = 0 a n d a t 0 = 7r[S AW55, Nag60 ] . Ex ac t l y ca l c u l a t ed d i f f e ren t ia l c ro s s s ec t i ons a re d i s cus sedin Sec. 9 .6.3.

F o r t he exac t r e l a t i v i s t i c c ros s s ec t i on , S ec . 9 .4 .3 , no c lo sed fo rm ulac a n b e g iv e n . N u m e r i c a l l y e v a l u a t e d t o t a l c r os s s e c t io n s a r e p r e s e n t e d i nS e c . 9 . 6 . 2 a n d a r e c o m p a r e d w i t h t h e S t o b b e c r o s s s e c t i o n s .9 . 5 . 2 T h e m o m e n t u m d i s t r i b u t i o n i n t h e t a r g e tI n r e l a t i v is t i c e l e c t r o n c a p t u r e , t h e v e l o c i ty o f t h e c a p t u r e d e l e c t r o n i s e s-s e n t ia l ly d e t e r m i n e d b y t h e v e l o c it y - v o f t h e t a r g e t a t o m w i t h r e s p e c t t ot h e p r o j e c t i le . H o w e v e r , a n e l e c t ro n b o u n d i n t h e t a r g e t ( w h i c h is a s s u m e dt o h a v e a lo w c h a r g e Z T < < Z p ) h a s a m o m e n t u m d i s t r i b u t i o n p ( q ) , s o t h a tt h er e is a m o m e n t u m s p r ea d a ro u n d th e m o m e n t u m - P 0 = - T m e v .


21/34

9.5 . F R O M T H E A T O M I C P H O T O E F F E C T T O R E C 267I I I I I I I I I I I I j

J.00 . 8

k 0 . 6N~ 0 . 4~ _

O.2

3 0 ~ 6 0 ~ 9 0 ~ 1 2 0 ~ 1 5 0 ~ 1 8 0 ~p h o f o n a n g l e

F i g u r e 9 . 3 . No rm a l i z e d S a u t e r a n g u l a r d i s t ri b u t i o n (9.3 2) for t h e em i s si o n o fR EC p h o t o n s i n t e g ra t e d o v e r t h e r e s o n a n c e a n d t r a n s fo rm e d t o t h e l a b o ra t o rys y s t e m . Th e c urv e s a r e l a b e le d by t h e p roj e c t il e e n e rg y Tp in G e V / u . Th eangu la r d i s t r ibu t ions fo r Tp < 1 G eV /u a re a lmos t ind i s t ingu ishab le f rom thec u rv e fo r 1 Ge V/ u .

H y d r o g e n ic w a v e f u n c t i o n sI f w e a s s u m e a s p h e r i c a l l y a v e r a g e d d i s t r i b u t i o n f o r a s i n g le e l e c t r o n ( a v -e r a g i n g o v e r a c o m p l e t e p r i n c i p a l s h e l l n ) , t h e n o n r e l a t i v i s t i c h y d r o g e n i cm o m e n t u m d i s t r i b u t io n is g iv e n b y

1 8Q5nPn(q2) -- --~ E E 1~Sntm(q)12 = (9 .52 )71-2 (q2 -t- Q 2 ) 4 ,l rnw h e r e ~ Sn l,~ (q) i s t h e F o u r i e r t r a n s f o r m o f t h e e l e c t r o n i c w a v e f u n c t i o n i nt h e s t a t e { n l r n } a n d

m e (?2 ZTQ n - ~ . (9 .53 )n hI n o r d e r to a p p r o x i m a t e t h e f ull m o m e n t u m d i s t r i b u t io n p ( q 2 ) i n a m u l -t i e l e c t r o n t a r g e t a t o m , o n e h a s t o s u m ( 9. 52 ) o v e r a ll t a r g e t e l e c t r o n s ,i n s e r t i n g t h e a p p r o p r i a t e e f f e c t i ve c h a r g e s in E q . ( 9 .5 3 )

T h e m o m e n t u m d i s t r i b u t io n ( 9.5 2) h a s a m a x i m u m a t q = 0 a n d aw i d t h d e t e r m i n e d b y Q n : t h e d i s t r i b u t i o n b e c o m e s m o r e a n d m o r e p e a k e da r o u n d q = 0 , t h e l o w e r Z T , a n d t h e h i g h e r n .


22/34

268 C H A P T E R 9. R A D I A T I V E E L E C T R O N C A P T U R EU s i ng Eq . (9 .52 ) , t he C om pt o n p ro f i le (9 .6 ) can be w r i t t e n i n t he fo rm

f f n ( q z ) - 2 7 r p n ( q 2 ) q d q 37r (q2 + Q 2n)3. (9. 54)z

Roo thaan -Har t ree -Fock wave func t i on sF o r m u l t i e l e c t r o n a t o m s , t h e h y d r o g e n i c d e n s i t y d i s t r i b u t i o n m a y n o t b ea c c u r a t e e n o u g h . M o r e r ea l is ti c m o m e n t u m d i s t ri b u t io n s c a n be o b ta i n e df ro m R o o t h a a n - H a r t r e e - F o c k ( R H F ) w a v e f u n c ti o n s t a b u l a t e d b y C le m e n t iand R o e t t i [C 1R 74]. In t hese ca l cu l a t i ons , s i ng l e -e l ec t ron o rb i t a l s i o f sym -m e t r y l, m a r e r e p r e s e n t e d a s l i n ea r c o m b i n a t i o n s o f S l a t e r - t y p e o r b i t a l s( S T O ) in t h e f o rm

7~i,lm(r) -- E Ci,lnRln(?~)Ylm(r) ' (9.55)n

w h e r e n - { n , p } r e fe r s t o t h e i n te g e r n o d a l q u a n t u m n u m b e r n a s w e lla s t o t he l abe l p d i s t i ngu i sh ing bas i s func t i ons w i th t he s am e n , lm , whi leRln (r ) i s t h e S T O r a d i a l w a v e fu n c t i o n

R l n ( ?~ ) - - N l n r n - l e - r (9.56)w i t h t h e n o r m a l i z a t i o n f a c t o r N l n , the expans ion coeff ic ien t s Ci ,zn , and theop t im ized va l ues ~n be i ng t a bu l a t ed i n [C 1R 74] fo r a l l a t o m s and spec i fi edp r i n c i p a l q u a n t u m n u m b e r s i .

F r o m E q s . ( 9 . 5 5 ) a n d ( 6 .4 2 ), w e o b t a i n t h e F o u r i e r tr a n s f o r m o f t h ee l e c t r o n w a v e f u n c t i o n a s

~bi,zm(q) - (27rh) -3 /2 47 ri 1Yzm (0) ~ CilnXln(q) ,n

w h e r e I ln ( q ) - - N l n r n - l e - r d r

(9.57)

(9.58)

l l n - -

N o n n ( q )q (~2 _jr_q2)(n+1) /2 s in ( n + 1 ) a r c t a nN l n ( n - 1 ) ' ( q )q2 ( ~ + q2 )n /2 s i n n a r c t a n

W i n n ' ( q )q (r _~_ q2)(n-[-1)/2 COS (Tt -Jr- 1 ) a r c t a n ~nn " (9.59)

0 n - -

a n d j l ( q r ) i s t he spher i ca l B es se l func t i on .F o r conven i ence , we g ive t he i n t eg ra l s (9 .58 ) fo r s - and p - s t a t e s , wh ich

a re de r i ved i n t he s am e way as Eq . (8 .13 ) .


23/34

9.5 9 F R O M T H E A T O M I C P H O T O E F F E C T T O R E C 269I n t e g r a l s fo r h i g h e r a n g u l a r m o m e n t a a r e si m i la r . T h e s p h e r i c a ll y a v e r a g e dm om en tu m dens i ty for a s ing le e lec t ron in an i , 1 she ll i s

1 , 1 2p(q2) = 2 1 + l E l ~ i ~ ' c ( q )m

2 C i I n I l n 9 (9.60)71" n

U s i n g t h e t a b u l a t e d v a l u e s , i t i s n o t d i f f i c u l t t o c a l c u l a t e t h e m o m e n t u md i s t r i b u t i o n . I n a n a l o g y t o E q . ( 9 . 5 4 ) , o n e m a y a l s o c o m p u t e t h e C o m p t o npro f i l e by numer ica l in teg ra t ion .9 . 5 . 3 C o n v o l u t i o n w i t h t h e t a r g e t m o m e n t u m d e n s i t yS i n c e , w i t h a p r o b a b i l i t y g i v e n b y p ( q ) , t h e e l e c t r o n m o m e n t u m w i t h i n t h et a r g e t h a s a c o m p o n e n t qb t r a n s v e r s e t o t h e b e a m d i r e c t i o n , t h e m o m e n -t u m s e e n f r o m t h e p r o j e c t i l e d e v i a t e s s l i g h t l y f r o m t h e n e g a t i v e z d i r e c t i o n( w h i c h i s d e f i n e d b y t h e b e a m ) a n d t h e c o r r e s p o n d i n g a n g u l a r d i s t r i b u t i o nwi l l be t i l t ed by th i s amoun t . Fo r example , fo r the r a the r low-energy co l l i -s ion o f 197 M eV /u X e 54+ on Be , s ee F ig . 9 .5 , one es t im a tes a dev ia t ion f romt h e n eg a ti v e z - di re c ti o n b y a r c t a n ( Q 1 / / ~ ) ~ 4.3 ~ while for h igher energ iesthe ang le w i l l be l e s s and fo r h igher t a rge t cha rges g rea te r . I n genera l , thee f f ec t on the angu la r d i s t r ibu t ion w i l l be smal l , bu t fo r comple tenes s , weg ive a fo rmula t ion wh ich inc ludes th i s e f f ec t .

F o r a g i v e n e l e c t r o n m o m e n t u m q i n t h e t a r g e t , t h e L o r e n t z t ra n s f o r m e dm o m e n t u m q ~ i n t h e p r o j e c t i l e f o r m s t h e a x i s o f a n a x i a l l y s y m m e t r i c p h o -t o n a n g u l a r d i s t r i b u t i o n . I f w e d e n o t e t h e p o l a r a n d a z i m u t h a l a n g l e s o ft h e p h o t o n w i t h r e s p e c t t o q~ a s a n d r a g i v e n p h o t o n d i r e c t io n 0 ~ ( wec h o o se t h e x - a x i s s u c h t h a t t h e p h o t o n a z i m u t h a l a n g l e r = 0 ) in t h e p r o -jec t i le f r ame w il l r eceive con t r ibu t ion s f rom var ious axes r ( 0q , Cq) and ,a s s o c i a t e d w i t h t h e m , f ro m c o r r e s p o n d i n g a n g le s 0~ a n d r T h e a n g l e s a r ei l lu s t r a ted in F ig . 9 .4 . Le t u s cons ide r the spher ica l t r i ang le composed o fthe pole P0 = (0 = 0) , of the po in t P~ = ( 0 ' , 0 ' = 0 ) g i v in g t h e d i re c t i o n o ft h e e m i t t e d p h o t o n , a n d o f t h e p o i n t Pq - (Oq , Cq) g iv ing the d i r ec t ion o f

9 m 0! !t h e e l e c t ro n m o m e n t u m q~ T h e s id e s i n t h i s t r i a n g l e a re So , s .~ - Oq,!a n d Sq - 0 ~. Knowing the ang le Cq a t the po le , we can expres s the oppos i t e

_ ~s ide So by the law of cos ines asO ' ' 0 ' ' 0 ' ' . (9.61)o s - - c o sO q c o s + s i n O q s i n cO SO u

I n t h i s w a y , w e m a y e l i m i n a t e t h e a n g l e O ' b y w h i c h t h e p h o t o n i s e m i t t e dwi th r e spec t to the d i r ec t ion C l ' .


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ZP0 1

e

o

F ig u re 9 .4 . The angles cha rac te r iz ing the photo n d i rec t ion and the e lec t ronmome n tum in t he p r o j e c t i l e sy s t e m.

W e t h u s g e t a g e n e r a l i z a t i o n o f E q . ( 9. 1) w h i c h i n d o R R / d f t ~ t a k e s i n t oa c c o u n t t h e d e v i a t i o n s o f t h e e l e c t ro n m o m e n t a q ~ f r om t h e n e g a t i v e b e a ma x i s a n d i n t e g r a t e s o v er t h e i r c o n t r i b u t i o n s . A d o p t i n g r e l a t i v i s t i c u n i t s ,h = c = me = 1 , i n t h i s subse c t ion , t he d i f f e r e n t i a l RE C c r os s s e c t ion int h e p r o j e c t i l e f r a m e i s

d2 aR EC (a / , 0 ' ) 1 d2aaEC (W ' , 0 t ) 1d a /d f F 27r dw ~d co s 0 ~ 27r f d 2qb f d q z / d c o sO

d 2 a R a ( q ' , )d ~ ' d c o s 0 '

0 COS ~t0 cos 0 ~ p(q 2) 5 (w ' + E[ - ")'Ei -+- 7 vq cos Oq)

x 5(c os 0 ' ' 0 ' ' 0 ' ' .c O S 0 q C O S - s i n 0 q s i n c O S C q ) (9.62)H e r e , r i s m e a s u r e d w i th r e spe c t t o r c hose n to de f ine t he x a x i s .

I n o r de r t o e va lua t e E q . ( 9 . 62 ), w e a pp l y E q . ( 2 .29 ) t o de r ive t h eL o r e n t z - t r a n s f o r m e d v e c t o r q~ f r o m q . F o r t h e a n g l es , w e g e t

, _qs in Oq - qt s in Oq, 7__ V/1 +q2OS0q = q, (q co s Oq - v ), (9.63)


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9.6 . 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 R E S U L T S 271w h e r e

t _ ( . U 'q c o s O q - - qz - - w~ = qo , (9.64)7 vwit h the reson anc e ph oto n ene rgy c~{) - 7 E i - E } given by Eq. (9 .8) . Cor-r espond ing ly , the k ine t i c ene rgy o f the t a rg e t e l ec tron , wh ich , w i th in th eimpu lse approx imat ion , i s a s sumed to be f r ee , i s fu l ly de te rmined by thet r a n s v e r s e m o m e n t u m qb, SO t h a t

E - v/q~ + q02 + 1. (9.6 5)F r o m t h e L o r e n t z t r a n s f o r m a t i o n ( 2. 29 ), w e o b t a i n

q , 2 _ E , 2 - 1 _ 72 ( v / q ~ + qg + 1 w ~ 7 - ~ , ) 2 _ 1. ( 9 . 6 6 )With these r e la t ions and (9 .61 ) , we a re now ab le to eva lua te Eq . (9 .62 ) .The REC c ros s s ec t ion i s ob ta ined as a two-d imens iona l in teg ra l ove r thec ros s s ec t ion fo r r ad ia t ive r ecombina t ion

d2aREC 0')d w ' d f F 1 d 2 o a l ~ ( q ' 0 ' )2 7 r 3 ' v d 2 qb d w 'd cos 0 COS 0'0 cos 0' p (q~ + q~ ) . ( 9 . 6 7 )In the l imi t ing c a s e qb


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. J 3c -O00

0I , _o

n , "n , "

1 03

1 02

101

1 0 0

10-I

1 0 - 2

1 0 - 3

' ' . . . . I ' . . . . . . . I '

Z = l O 0 2 ~ . - ~ : . . .0 ~ . . . .60 ~ . , , , ~ . ~ _ " " ' " " ' " ' " " " " . ,, , . , . . ., ., . . . . .4 o . . . . . . . . . .

\I I I I ' ' 1 , i i , , i i i I10-1 100

Ene rgy (GeV /u )F ig u re 9 .5 . Energy an d charge dependence o f the c ross sec tion for rad ia t ive re -combinat ion ( in barns) wi th capture in to the project i le K-shel l . Sol id l ines : exact(relat ivist ic) calculated cross sections; short-dashed l ines: Stobbe cross sections,Eq. (9 .50) , assuming the correct re lat iv is t ic re lat ion between project i le energyand veloci ty ; long-dashed l ines : Eq. (9 .50) , however assuming the nonrelat iv is t ic1re la t ion E - -~ M v 2 between project i le energy and veloci ty .

t h e e l ec t ro n i s a s s u m ed t o b e a t r e s t in t h e i n i t i a l s t a t e . I n t h i s c a s e , n ot a r g e t p a r a m e t e r s e n t e r . T h e e x a c t t o t a l c r os s s e c t io n s a r e g i v e n f or t h eK-she l l by the so l id l ines in F ig . 9 .5 and fo r the L-she l l s in F ig . 9 .6 .

A c u r i o u s f e a t u r e w a s n o t e d i n [ S t K 9 2 ] b y c o m p a r i n g t a b u l a t e d e x a c tp h o t o i o n i z a t i o n c r os s s e c ti o n s w i t h t h e s i m p l e r e s u l t ( 9. 50 ) o f t h e n o n r e l a -t i v i s ti c d i p o le a p p r o x i m a t i o n . A l t h o u g h t h e s e t r e a t m e n t s a r e w i d e l y d if fe r-e n t , t h e t o t a l c r os s s e c ti o n s f or r a d i a t i v e r e c o m b i n a t i o n a g r e e r a t h e r w e ll f ore n e r g i e s b e l o w 1 G e V / u , a r e s u l t t h a t w a s c o n f ir m e d i n s y s t e m a t i c r e l a t iv i s -t i c c a l cu l a t i o n s [ ICS9 4] . F i g u re 9 .5 p r e s en t s a co m p ar i s o n o f t h e e x ac t c ro s ss ec t i o n ca l c u l a t e d f ro m E q . ( 9 .4 9 ) b y i n c l u d i n g a l l p a r t i a l w av es I~1 __ 2 0( s ol id l in e ) w i t h t h e c r os s s e c ti o n c a l c u l a t e d a c c o r d i n g t o t h e n o n r e l a t i v i s t i cS t o b b e f o r m u l a ( 9 . 5 0 ) f o r r a d i a t i v e e l e c t r o n c a p t u r e i n t o a n e m p t y K - s h e l l .T h e s h o r t - d a s h e d l i n e s a r e c o m p u t e d b y a s s u m i n g t h e c o r r e c t r e l a t i o n b e -


27/34

9.6 . 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 R E S U L T S 273

1021 0 1

~ - 1 0 0t -O= ,= 1 0 - 1030303o 1 0 - 2orrn 1 0 . 3

1 0 - 4

~ - . . . . . I . . . . . . . . IZ= lO0" 8 0r 6 0 ~ ,

=

4 0 ' ' ,

, .. " ,~, . - , ~

\ -

1 0 . 5 1 0 - ~ 1 0 0Energy (GeWu)

Fig ur e 9.6. Energy and charge dependence of the cross section for radiativerecombination (in barns) with capture into the projectile L-shell. All results areobtained from exact relativistic calculations. Solid lines: capture into the L 1-shell;long-dashed lines: capture into the L2-shell; and short-dashed lines: capture intothe L3-shell.

tween the velocity v and the projectile energy, while the long-dashed line1 2uses the relation Ekin -- - ~ M p v , as if the projectile speed were nonrela-

tivistic. For the la tte r case, the results of the nonr elativ istic dipole approx-imation do not deviate much from those of the exact calculations, even forrelativistic energies (up to 1 GeV/u), where the energy-velocity relationis far from the nonrelativistic limit. This astonishing agreement must beregarded as fortuitous.

A similar equivalence cannot hold for the differential cross section, whichnonrel ativis tical ly would be represented by a strict sin 2 0 dis trib ution (9.31),while in the exact cross sections, considerable deviations from this formoccur. One reason is that spin-flip con tributions to the cross section yieldnonzero values at forward and/or backward angles, see Fig. 9.10.

In analogy to Fig. 9.5, Fig. 9.6 shows the exact relativistic RR crosssections for capture into the L_l, L2-, and L3-shells corresponding to final


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274 C H A P T E R 9. R A D I A T I V E E L E C T R O N C A P T U R E2 s l /2 , 2 p l / 2 , a n d 2 p 3/2 s t a t e s . H e r e , c a p t u r e i n t o th e L l - s h e l l d o m i n a t e sfo r a l l ch a rg es an d en e rg i e s .

9 . 6 . 2 T o t a l R E C c r o s s s e c t i o n sI n o r d e r t o p r e s e n t t o t a l R E C c r o s s s e c t i o n s i n a s y s t e m a t i c w a y , i t i s u s e -f u l t o r e n d e r d i f f e r e n t c o l l i s i o n s y s t e m s c o m p a r a b l e ( a ) b y n o r m a l i z i n g t h ec ro s s s ec t i o n s per t a rge t e l ec t ron , ( b) i n t h e c a s e o f i n c o m p l e t e l y s t r i p p e dp ro j ec t i l e s , b y co n s i d e r i n g c ro s s s ec t i o n s p e r p r o j e c t i l e v a c a n c y i n the she l lco n s i d e r ed , a n d ( c) b y d i s p l ay i n g c ro s s s ec t i o n s a s a f u n c t i o n o f t h e " ad i a -b a t i c i t y p a r a m e t e r " ~], w h i ch d ec i d e s w h e t h e r t h e co l l is i o n i s f a s t (~ > 1 ) o rs low (~ < 1 ) in the sense o f Sec . 5 .4 .1 . A cco rd in g to Eq . (5 .63) , we def ine

- - E k in ( M e V / u )m~c2(~, 1) ~ 40.3 1 . (9.70 )~/ - - Eb in d Z~T h e l a s t a p p r o x i m a t i o n a s s u m e s a n o n r e l a t i v i s t i c t o t a l b i n d i n g e n e r g y a n da n o n r e l a t i v i s t i c e n e r g y - v e l o c i t y r e l a t i o n .

E x p e r i m e n t a l c ro ss s e c ti o ns n o r m a l i z e d p e r t a r g e t e l e c tr o n a c c o r d i n g t o( a ) m a y b e c o m p a r e d w i t h t h e o r e t i c a l c r o s s s e c t i o n s f o r r a d i a t i v e r e c o m -b i n a t i o n , s ee Eq . ( 9 .4 9) . I n t h i s w ay , t a r g e t p ro p e r t i e s d o n o t en t e r s in cet h e e l e c t r o n s a r e a s s u m e d t o b e a t r e s t w i t h r e s p e c t t o t h e t a r g e t n u c l e u s ,w h i c h is a g o o d a p p r o x i m a t i o n f or l o w - Z t a r g e t s . F i g u r e 9 . 7 s h o w s t h a tt h e a v a i l a b l e e x p e r i m e n t a l t o t a l R E C c r o s s s e c t i o n s p e r t a r g e t e l e c t r o n a sa f u n c t i o n o f t h e a d i a b a t i c i t y p a r a m e t e r r / f al l a p p r o x i m a t e l y o n a s i ng l eu n i v e r s a l c u r v e [ S tK 9 2 ]. T h i s b e h a v i o r f in d s i ts c o u n t e r p a r t i n t h e e x a c tt h e o r e t i c a l t o t a l c ro s s s e c t i o n s f or r a d i a t i v e r e c o m b i n a t i o n . I t t u r n s o u tt h a t f o r c a l cu l a t i n g t h e t o t a l c ro s s s ec t i o n , i t i s s u f f i c i en t t o ad d u p t h ec o n t r i b u t i o n s f r o m t h e K - , L - , a n d M - s h e l l s [ IC S 9 3, S t K 9 5 ] . W h i l e f o r t h eK - a n d L - s h e l l s , t h e c u r v e s f o r t h e d i f f e r e n t p r o j e c t i l e c h a r g e n u m b e r s a r ew e l l s ep a ra t ed a s a f u n c t i o n o f t h e co l li s io n en e rg y , a s s h o w n i n F i g s . 9 .5an d 9 .6 , an d s i mi l a r l y fo r t h e M-s h e l l s , t h ey fo rm a c l o s e - l y i n g b u n d l e a s af u n c t i o n o f t h e p a r a m e t e r ~ i n t h e r e p r e s e n t a t i o n o f F i g . 9. 7. T h e t h e o r e t -i c a l c u r v e s , w h i c h a l l r a n g e f r o m 1 0 0 M e V / u t o 1 . 0 G e V / u , a r e i n r a t h e rg o o d a g r e e m e n t w i t h th e e x p e r i m e n t a l d a t a p o i n ts . W i t h o n e e x c e p ti o n ,o n l y e x p e r i m e n t a l d a t a f o r p r o j e c t i l e e n e r g i e s l a r g e r t h a n 1 4 0 M e V / u a r ei n c l u d e d . H o w e v e r, t h e r e e x i st m a n y m o r e d a t a p o i n t s a t lo w e r e n e r g ie sw h i ch fo l lo w t h e b u n d l e o f t h eo re t i c a l cu rv es t o h i g h e r v a l u e s o f r/.W e c o n c l u d e t h a t t o t a l R E C c r o s s s e c t i o n s c a n b e r e l i a b l y e s t i m a t e d b yt h e o r e t i c a l c r o s s s e c t i o n s f o r r a d i a t i v e r e c o m b i n a t i o n , w h i c h a r e d o m i n a t e dw i t h i n e a c h p r i n c i p a l s h e ll b y c a p t u r e i n t o s -s t a t e s . N o w i t is a g e n e r a lo b s e r v a t i o n [ R o G 9 4 ] t h a t n o n r e l a t i v i s t i c d i p o l e t o t a l c r o s s s e c t i o n s ( b u t


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9.6 . 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 R E S U L T S 275

1 0 3 . . . . . . ~ . . . . . . . . ~ ,

. . o12:0o(D

ooOk _( D

10 2

101

1 0 o

%

8 O

"~"'~" 6 00 X e " ' -9 A u , , ' , .a Pb "99 Bi@ U

10-1 , ,I O 0

I I i I I I I I I101A di( ]bet ic i ty p(]rum eter

40x

x x x x

F i g u r e 9 . 7 . E x p e r i m e n t a l t o t a l R E C c ro s s s e c ti o n s f or c a p t u r e i n t o b a r e p r o-j e c t i l e s p e r t a r g e t e l e c t r o n i n b a r n a s a f u n c t i o n o f t h e a d i a b a t i c i t y p a r a m e te r r/d e f in ed in E q . (9 .7 0 ). T h e p r o j e c t i l e s a r e i n d i c a t e d e x p l i c i tl y ; t h e t a r g e t a to m srang e f rom Be to A1. In der iv ing to ta l c ross sec t ions , r ea l i s t ic theo re t ica l an -g u l a r d i s t r i b u t io n s a r e t a k e n i n to a c c o u n t . F r o m [ StK 95 ]. T h e c u r v e s r e p r e s e n te x a c t t h e o r e t i c a l c r o s s s e c t i o n s f o r r a d i a t i v e r e c o m b in a t io n i n to t h e K + L + Mshe l l s ca lc u la t ed ac co rd in g to Eq . (9 .49 ) [ ICS94]. Th e nu m bers g ive the p ro jec t i lecharges .

n o t a n g u l a r d i s t r i b u t i o n s ) f or t h e p h o t o e l e c t r i c e f f ec t o f s - s t a t e s ( a n d o t h e rp r o c e s s e s i n v ol v i n g s - s t a t e s ) a r e v a l id f ar b e y o n d t h e i r a n t i c i p a t e d e n e r g yl i m i t s , se e a l s o F i g . 9. 5. T h e r e f o r e i t i s n o t s u r p r i s i n g t h a t e x a c t t h e o r e t i c a lR E C c r os s s e c t i o n s a r e w e ll a p p r o x i m a t e d b y c o n s i s t e n t l y n o n r e l a t i v i s t i cS t o b b e c r o s s s e c t i o n s . I f o n e i n t r o d u c e s t h e l a t t e r i n F ig . 9 .7 , t h e v e l o c i t i e sa n d c h a r g e n u m b e r s e n t e r o n l y i n t h e c o m b i n a t i o n u o( 1 /x / ~ , s e e E q . ( 9. 50 ) ,s o t h a t t h e b u n d l e o f e x a c t t h e o r e t i c a l c u r v e s c o n t r a c t s t o a s in g l e u n i v e r s a lc u r v e .


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276 CHAPTER 9 . RADIATIVE ELECTRON CAPTURE

10

,.s 8r.g 6

0 0

4o0

2

0 0 3 0 6 0 9 0 1 2 0 1 5 0 1 8 0A n g le ( d e g )

F ig u re 9 .8 . Di f fe ren t ia l RE C c ross sec t ion in ba rn /s r fo r 197 MeV X e 54+ on Beas a funct io n of the lab ora tor y angle . M easured values are f rom [AnA84] . Th esolid curve represen ts ca lcu la ted resul ts [ ICS94] mu lt ipl ied w ith a norm aliza t ionfac tor o f 0 .77 in orde r to f it the d a ta . Ha r t r ee -Fock wave func t ions for the ta rge ta to m [C1R74] are used.

9 . 6 . 3 D i f f e r e n t i a l R E C c r o s s s e c t i o n sA s d i s c u s s e d i n Se c. 9 .3 , i n t h e t h e o r e t i c a l d e s c r i p t i o n o f n o n r e l a t i v i s t i cR E C , t w o i n g r e d ie n t s c o m b i n e in s u c h a w a y t h a t t h e r e s u l t i n g p h o t o n a n -g u l a r d i s t r i b u t i o n is s im p l y p r o p o r t i o n a l t o s i n 2 0 in t h e l a b o r a t o r y s y s t e m .T h e t w o i n g r e d i e n t s a r e ( a ) t h e i n c l u si o n o f t h e f u ll r e t a r d a t i o n , t h e r e b yk e e p i n g t h e c o m p l e t e e x p o n e n t i a l o f t h e p h o t o n w a v e f u n c t i o n i n E q . ( 9 .1 1 ),a n d ( b ) t h e t r a n s f o r m a t i o n ( 9. 10 ) o f t h e a n g l e s f r o m t h e p r o j e c ti l e s y

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Chapter 9 - Radiative Electron Capture