chapter 7 - ionization many electrons

8/12/2019 Chapter 7 - Ionization Many Electrons

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C h a p t e r 7

I o n i z a t i o nE l e c t r o n s M a n y

7 . 1 I n t r o d u c t i o nT h e p r e v i o u s c h a p t e r d i s c u ss e d i o n i z a t i o n o r e x c i t a t i o n o f a o n e - e le c t r o nion by a f u l ly ion ized p r o jec t i l e . He r e , we concen t r a t e on "many- e l ec t r on"s i t u a t i o n s , w h e r e e i t h e r t h e t a r g e t c a r r i e s m o r e t h a n o n e e l e c t r o n ( s t r u c -t u r e d t a r g e t ) o r t h e r e a r e e l e c t r o n s o n t h e p r o j e c t i l e ( s t r u c t u r e d p r o j e c t i l e ) .T h e n e e d f o r t r e a t i n g t h e s e m o r e c o m p l i c a t e d s i t u a t i o n s b e c o m e s c l e a r i fw e c o n s i d e r b r i e f l y t h e e x p e r i m e n t a l s i t u a t i o n .

I n n e r - s h e l l i o n i z a t i o n c r o s s s e c t i o n s a r e t y p i c a l l y d e t e r m i n e d f r o m t a r -g e t x - r a y y i e ld s ( C h a p . 1 3) . U n f o r t u n a t e l y , t a r g e t v a c a n c i e s ( a n d x - r a y s )a r e a ls o p r o d u c e d i f a ta r g e t i n n e r -s h e l l e le c t r o n i s c a p t u r e d b y a s t r i p p e dp r o j e c ti l e , a p r o c e s s t h a t i n c re a s e s s t r o n g l y w i t h Z p ( C h a p . 8 ). T h e t a r g e ti o n i z a t i o n a n d t h e c a p t u r e p r o c e s s e s c a n n o t b e d i s t i n g u i s h e d e x p e r i m e n -t a l l y , e x c e p t b y a c o i n c i d e n c e m e a s u r e m e n t w i t h t h e p r o j e c t i l e c h a r g e s t a t e :t a r g e t i o n i z a t i o n g e n e r a l l y l e a v e s t h e p r o j e c t i l e c h a r g e s t a t e u n c h a n g e d ,w h e r e a s c a p t u r e d e c r e a s e s t h e c h a r g e s t a t e b y u n i t y ( u n l e s s s i m u l t a n e -o u s p r o j e c t i l e e l e c t r o n l o s s o c c u r s ) . S u c h e x p e r i m e n t s r e q u i r e t a r g e t s t h i nenough to a s sur e s ing le - co l l i s ion cond i t ions .

A c l e a n w a y t o t e s t t h e t h e o r y is t o s t u d y t h e i o n i z a t io n o f a r e la t i v i s ti co n e - e l e c t r o n p r o j ec t i l e b y a t h i n s o l id or g a s e o u s ta r g e t . C a p t u r e o f t h ep r o j e c t i l e e l e c t r o n b y a n e u t r a l t a r g e t t h e n i s f o r b i d d e n b y t h e P a u l i e x c l u -s ion p r inc ip l e ( nega t ive - ion f o r ma t ion i s comple t e ly neg l ig ib l e a t r e l a t iv i s t i cv e l o c i t i e s ) . A c o m p l i c a t i o n a r i s e s , t h o u g h , b e c a u s e t h e t a r g e t e l e c t r o n s c a na l so a ff ect t he e l ec t r o n loss o f t he p r o j ec t i l e . ( Hen ce f or th , i on iza t ion o f

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188 C H A P T E R 7. I O N I Z A T I O N - M A N Y E L E C T R O N S

ZT ~,

e vt

, X

F i g u re 7 . 1 . No n re l a t i v is t ic c o o rd i n a t e sy s t e m for t h e t wo -e l e ct ro n c a se . Th eor ig in o f the coord ina te s ys tem is chosen to be a t the t a rge t nuc leus (Z T) . Th epro jec t i l e nuc leus (Zp) moves wi th cons tan t ve loc i ty v para l l e l to the z -ax i s , a ta n i m p a c t p a ra m e t e r b. O n e e l e c t ro n h a s t h e c o o rd i n a t e r w wi t h r e s p e c t t o ZT ,t h e o t h e r e l e c t ro n h a s t h e c o o rd i n a t e ~p with respec t to Zp .p o t e n t i a l i n E q . ( 6 .1 )

i s r e p l a c e d b y

Xp e 2 Zp e 2r p I R - r T I

Z p e 2 ZT e 2 e 2V ( R ' r T ' ( P ) - - ] R - r T] -- ] R + ( P I + ] R - r T + ( p ] ( 7.4 )

a n d t h e w a v e f u n c t i o n s ~ i a n d qpf a r e n o w f u n c t i o n s o f r T a n d ~p. C o r r e -s p o n d i n g l y , t h e i n t e g r a l o v e r d 3 r T h a s t o b e r e p l a c e d b y a n i n t e g r a l o v e rd 3 r T d 3~ p . I f t h e c o l li s io n i s f a s t w i t h r e s p e c t t o t h e B o h r v e l o c i t i e s o f t h ep r o j e c t i l e a n d t a r g e t e l e c t r o n s , e l e c t r o n e x c h a n g e c a n b e n e g l e c t e d a n d t h ei n i ti a l a n d f in a l w a v e f u n c t io n s c a n b e w r i t t e n a s p r o d u c t s o f t a r g e t a n d p r o -j e c ti le w a v e f u n c ti o n s ~ i ( r T ) ~ 0 ( ( p ) a n d ~ f ( r T ) P n ( ( p ) , r e sp e c ti v el y . H e re ,t h e q u a n t u m n u m b e r s o f t h e i n it ia l a n d f i na l t a r g e t s t a t e s a r e d e n o t e d b yi a n d f, r e s p e c t i v e l y , a n d t h o s e o f t h e p r o j e c t i l e s t a t e s b y 0 a n d n .

I n a p r o c e s s d e s c r i b i n g t a r g e t e x c i t a t i o n o r i o n i z a t i o n , o n e a l w a y s h a sfT~i, bu t th er e i s no a p r i o r i r e q u i r e m e n t t h a t t h e p r o j e c t i l e s t a t e h a s t oc h a n g e . I n fa c t , o n e f i n ds t h a t t h e c r o s s s e c t i o n s f or n = 0 a n d f o r n r 0 c a nb e e q u a l l y i m p o r t a n t . I n m o s t e x p e r i m e n t a l s i t u a t i o n s , t h e f i n al s t a t e o f t h ep r o j e c t i l e e l e c t r o n i s n o t o b s e r v e d , s o t h a t a s u m o v e r c r o s s s e c t i o n s w i t h


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7.2. N O N R E L A T I V I S T I C O N E - E L E C T R O N P R O J E C T I L E 189n -~ 0 has to be exe cu ted . I t is con ven ien t to t r e a t th e n - 0 an d }--~n#0c r o ss s e c t io n s s e p a r a t e l y . F o r t h i s re a s o n , t h e y h a v e b e e n g i v e n d i f fe r e n tn a m e s , su c h a s " ( t a r g e t ) i n e l a s t i c - ( p r o je c t il e ) e l as t ic " an d " ( t a r g e t ) in -e l a s t i c - ( p r o j e c t i le ) i n e l a s t i c " , r e s p e c t iv e l y , o r " sc r e e n i n g " a n d " a n t i s c r e e n -ing . " H e r e , t he y a r e c a l l e d " s ing le i ne l a s t i c ( s ine l )" a nd " dou b le i ne l a s t i c(d ine l ) . " 1

O n e f u r t h e r a l t e r a t i o n n e e d s t o b e m a d e in E q . ( 6 .5 ) . T h e e n e r g yc h a n g e E f - E i w h i ch o cc u r s i n t h e m i n i m u m m o m e n t u m t r a n s f e r ( 5 .1 0 5)h a s t o b e r e p l a c e d b y t h e t o t a l e n e r g y c h a n g e o f t h e s y s t e m :

E - - + - 7 . 5 )

w h e r e t h e s u p e r s c r i p t s r e f e r t o t h e t a r g e t a n d t h e p r o j e c t i l e , r e s p e c t i v e l y .C o r r e s p on d i n g l y , t h e m i n i m u m m o m e n t u m t r a ns f e r is r e p la c e d b y

h q o ( n ) = A E / v , (7.6)w h e r e , f o r l a t e r u se , t he d e p e nd e n c e o f q0 on n i s no t e d .

T h e d e r i v a t i o n o f t h e c r os s s e c t i o n n o w f ol lo w s t h e d e v e l o p m e n t i n S ec .6 .1 .1 w i t h t h e f o l l ow i n g d i ff e re n c es . T h e m a t r i x e l e m e n t o f t h e p e r t u r b i n gp o t e n t i a l ( 7 . 4 )

(n, f lVli, O> - - e 2 . I [ d3rTd3~P~ ( ~ P ) IR-rTI IR pI- R--rT P[ 99 i(rT)r (7.7)

h a s c o n t r i b u t i o n s o n l y fr o m t h e f i rs t a n d t h i r d t e r m s b e c a u s e o f t h e o r t h o g -ona l i t y o f t he i n i t i a l a nd f ina l t a r ge t w a ve f unc t ions . I f n = 0 , t he f i r s t a ndt h i r d t e r m s b o t h c o n t r i b u t e , b u t i f n -r 0, o n l y t h e t h i r d t e r m c o n t r i b u t e sb e c a u s e o f t h e o r t h o g o n a l i t y o f t h e p r o j e c t i l e w a v e f u n c t i o n s qPn(~p) a n d~P0(~p). H e nc e , i n t he p r e se n t a pp r o a c h , t h e dou b le - in e l a s t i c p r oc e s s iss o l e l y d u e t o t h e p r o j e c t i l e - e l e c t r o n - t a r g e t - e l e c t r o n i n t e r a c t i o n .

I n de t a i l , one ha s

(0, f lV li , 0) - - e 2 d 3 r T ~ f ( r w ) I R - - r T J1physically, one must distinguish in the double-inelastic process the ionization (orexcitation) of both electrons by the electron-(other) electron interaction and that by theelectron-(other) nucleus interaction. Only the former is nonzero in the PWBA or SCA,but the latter has been observed and has been called "two-center double ionization"[MoM92b].


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190 C H A P T E R 7 . I O N I Z A T I O N - M A N Y E L E C T R O N S

(n # 0, f[VIi, 0)

, 1 ~ 0 ~ p ) ]~ 0 ~ P ) I R - r T + ~p[ 9 9 i ( r T ) ,e2 / / d3rTd3~p~*(~p)qp~(rT)

1 ~ g i ( r w ) ~ 0 ( ~ p ) 9I R - r~ + ~ 1Applicat ion of the nonrelat ivist ic a ppro xima tion to Eq. (5.103), ob-

tain ed by the rep lacem ents rp ~ rp, 7 ~ 1, and /3 --~ 0, converts thes eexpressions to(o, flv li, o) = e a / d a q " R ( f [ e i q . r T27r2 - ~ e -zq" li)

x (Zp -(01e -~qt~P I0)) , (7.8)

(n r 0 , fIVl i ,0) = e 2 / d a q e _ i q . R { f l e i q . r ~ l i )27r2 - ~ -iqt~p [0). (7.9)

The remaining development follows Eqs. (6.4) and (6.5), leading to thedifferential cross sections for ionization, with d2qb replaced by d a q - 2 7 r qd q ,dafi(sinel) _ 8 ~ a ~ ~ ~ dqlFTi e 2

d E f - - rI--TKK o(O) ~ IZp - F~ , (7.10)

daft (dinel)d E f 8 7ra ~ f q ~ d q? ]2 n ~ r o ( n ) ~ I F T I 2 IF~I 2 , (7.11)

wher e ?]K is defined in Eq. (6.27) an d the form factor s F are given byFT- (f]e--~q'~wli),

F ~ - (ol~ -~ qrq2a~ -2- - ( 1 + - ~ p ) , (7.12)F# - (nle-~q~P[0).


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7.2 . N O N R E L A T I V I S T I C O N E - E L E C T R O N P R O J E C T I L E 191T h e e x p r e s s i o n fo r F T is i d e n t i c a l t o t h a t f o r F i n E q . ( 6 .2 1 ). E q u a t i o n(7 .12 ) is va l i d fo r t he g rou nd s t a t e o f a hyd roge n i c i on. C losed exp res -s io n s f o r F ~, fo r e x c it e d a n d c o n t i n u u m s t a t e s o f s u c h i o n s a re g i v e n b y[ B a G 5 3 , C h K 6 6 ] .

W i t h o u t g o i n g f u r t h e r , o n e c a n se e f r o m E q . ( 7 .1 0 ) t h a t i n t h e s i n gl e -i ne l as t i c c ros s s ec t i on t he p ro j ec t i l e e l ec t ron f rozen i n i t s g round s t a t e e f f ec -t i v e l y s h i e l d s t h e p r o j e c t i l e n u c l e a r c h a r g e a n d m a k e s t h e t a r g e t i o n i z a t i o nl es s e f fec ti ve. O n t he o t he r ha nd , Eq . (7 .11 ) shows t h a t t h e doub l e - i n e l as t i cc r o s s s e c t i o n , w h i c h i n t h e p r e s e n t t r e a t m e n t i s s o l e l y d u e t o t h e i n t e r a c t i o nb e t w e e n t h e t a r g e t e l e c t r o n a n d a n e x c i t e d ( o r i o n i z e d ) p r o j e c t i l e e l e c t r o n ,e n h a n c e s t h e t a r g e t i o n i z a t i o n .

Al though t he re i s no d i f f i cu l t y i n eva lua t i ng t he s i ng l e - i ne l as t i c c ros ssec t i on (7 .10 ) , t he doub l e - i ne l as t i c c ros s s ec t i on i n genera l canno t be eva l -u a t e d . T h e d i f f i c u l t y l i e s i n t h e s u m m a t i o n o v e r e x c i t e d p r o j e c t i l e s t a t e s nin Eq . (7 .11) , i n wh ich t he l ower l im i t q o ( n ) o f t h e i n t e g r a l d e p e n d s o n t h ee n e r g y E p o f e a c h e x c i t e d s t a t e . O n l y f or h y d r o g e n i c i o n s , c a n t h e s u m b ecom pu ted exac t l y , u s ing c lo sed exp res s ions fo r F ~ , . F o r a l l o the r i ons , som ea p p r o x i m a t i o n h a s t o b e m a d e .

W e d i s c u s s b r ie f ly f iv e a p p r o x i m a t i o n s w h i c h h a v e b e e n p r o p o s e d . ( 1)I g n o r e t h e n d e p e n d e n c e o f q0 b y r e p l a c i n g t h e p r o j e c t i l e e x c i t a t i o n e n e r g i e sb y a s u i t a b l y c h o se n m e a n e n e r g y / ~ P . T h e s u m m a t i o n s i g n i n E q . ( 7.1 1)c a n t h e n b e m o v e d u n d e r t h e i n t e g r a l s ig n a n d ~ lE V I 2 c a n b e e v a l u a t e d b yc l o su r e . T h e r e s u l t a n t t o t a l c r o s s s e c t io n f it s v e r y p o o r l y w i t h e x p e r i m e n tn e a r t h e f r e e - e l e c tr o n t h r e s h o l d . ( 2) A c o r r e c t i o n f a c t o r fo r t h e d o u b l e -i n e l a s t i c c r o s s s e c t i o n i s s u g g e s t e d b y t h e f r e e - e l e c t r o n m o d e l t o r e m o v et h i s d e f e ct . ( 3) T h e a c t u a l p r o j e c t i l e s t a t e s a r e r e p r e s e n t e d b y a p p r o x i m a t eh y d r o g e n i c s t a t e s , s o t h a t t h e s u m o v e r p r o j e c t i l e s t a t e s c a n b e e v a l u a t e d .( 4 ) T h e s u m o v e r p r o j e c t i l e e x c i t e d s t a t e s i s p a r t l y e x e c u t e d b y u s i n g ah i g h e r - o r d e r s u m r u le d u e t o B e t h e a n d J a k i w [ B eJ 6 8] . ( 5) T h e i m p u l s ea p p r o x i m a t i o n i s u s e d t o e v a l u a t e t h e d o u b l e - i n e l a s t i c c r o s s s e c t i o n , b ya s s u m i n g t h a t t h e p r o j e c t i l e e l e c t r o n s a c t a s f re e e l e c t ro n s , b u t h a v e a ni n i ti a l m o m e n t u m d i s t r i b u t i o n c o r r e s p o n d i n g to t h e i r in i t ia l s t a t e . [In t h ef r e e - e l e c t r o n m o d e l e x p r e s s i o n ( 7 . 1 ) , t h e p r o j e c t i l e e l e c t r o n s a r e a s s u m e dto be i n i t i a l l y a t r e s t w i th r e spec t t o t he p ro j ec t i l e . ]

C l o s u r e a p p r o x i m a t i o nI f E p an d q o ( n ) [Eq. (7 .6) ] are rep lac ed b y /~ P an d q-0 , respe ct ively , in Eq .( 7 . 1 1 ) t h e s u m m a t i o n s i g n c a n b e m o v e d i n s i d e t h e i n t e g r a l o v e r dq a n d


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192 C H A P T E R 7. I O N I Z A T I O N - M A N Y E L E C T R O N Sclosure can be invoked [Sch68]:

E I F ~ [ 2 - 1 o r E I F ~ 1 2 - 1 - ] F ~ 2, (7.18)n n : f i 0

l ead ing todafi(dine l) _ 8rra~ fo ~ dq 12d E f - r /7 o q glFTI2(1 - IF~ )" (7.14)

In the in t egr a t io n over fina l e lec t ron ene rgies Er in Eqs . (7 .10) and (7 .14)i t i s conve nient to make sub s t i tu t ion s equivalen t to Eqs . (6 .24) and (6 .25) ,such as W - 2 E T - E T 4- ~ P - E~ (7.15)Zgd/aoOn e then a r r ives a t a r e su l t ana logous to the nonre la t iv i s t i c pa r t o f Eq .(6 .26) , wi th FK replaced by FT:

a~ = a~(sin el) 4- cra(dinel)

---- ' ~:4ca 2 r i n d W / ~ m in --~dQ IF T I2 S p ( Q ) , (7.16)w h e r e

Sp(O) = IZp - - F~ 2 + (1 - I F ~ (7 .17)One s ees tha t the e s sen t i a l d i f f e r ence be tween the c ros s s ec t ion induced bya ba re p ro jec t i l e and a one-e lec t ron p ro jec t i l e i s the r ep lacemen t in thefo rmer o f the p ro jec t i l e square d cha rge Z 2 by an e f f ec t ive square d cha rge

(Q).In gq . (7 .15) , one usua l ly se ts / )P - E0 - 0 , especia l ly if Zp


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7 .2 . N O N R E L A T I V I S T I C O N E - E L E C T R O N P R O J E C T I L E 193w h e r e c r y ( p ) i s t h e t a r g e t i o n i z a t i o n c r o s s s e c t i o n b y p r o t o n s m o v i n g w i t hthe p ro j ec t i l e ve loc it y . B u t , t h e f r ee -e l ec t ron m ode l (7 .1 ) , wh ich i s va l i d ifb ind ing i s neg l i g ib l e , p red i c t s

i _ _ Z2paiT p)+ i (7.20)Here , a iT (e ) i s t he e l ec t ron - induced i on i za t i on c ros s s ec t i on wh ich has at h r e s h o l d a t t h e p r o j e c t i l e v e l o c i t y ( 7 . 3 ) , w h e r e a s t h e r e i s n o t h r e s h o l di n e x p r e s s i o n ( 7 . 1 9 ). H e n c e t h e l a t t e r e x p r e s s i o n m u s t b e i n c o r r e c t n e a rt h r e s h o l d .

C o r r e c t i o n fa c t o r a p p r o x i m a t i o nA r e m e d y f o r t h e p r e c e d i n g d e f e c t i s s u g g e s t e d b y m o d i f y i n g e x p r e s s i o n(7.17) [Anh86]

s~nod(Q) _ I Z P - f ~ o e r ( P ) q - (1 - - I F ~ ii ( p )cr T (7.21)

T he c ros s s ec t i ons c rlr( e ) and C r~r(p ) a re t o be eva l ua t e d a t t he p ro j ec t i l ev e l o c i t y a n d c a n b e c a l c u l a t e d w i t h E q . ( 6 .2 6 ) o r t h e e q u i v a l e n t n o n r e l-a t i v i s ti c e x p r e s s i o n s . T h i s p r o c e d u r e i m p r o v e s t h e f it o f t h e c r o ss s e c ti o nw i t h e x p e r i m e n t a b o v e t h e f r e e - e l e c t r o n t h r e s h o l d , b u t i n t r o d u c e s a n a r t i -f i ci a ll y sha r p k ink i n t he t o t a l i on i za t i on c ros s s ec t i on s i nce a~r (e ) has ana b r u p t t h r e s h o l d a t ~)thr.A p p r o x i m a t e h y d r o g e n i c s p e c t r u mI t h a s b e e n s u g g e s t e d t h a t i n e v a l u a t i n g t h e s u m i n E q . ( 7 .1 1 ), i t is n o tn e c e s s a r y t o u se th e e x a c t e x c i t a t i o n s p e c t r u m , b u t t h a t i t is s u ff ic ie n t tom i m i c t h e s p e c t r u m b y a h y d r o g e n i c o n e . A l s o , i n c a l c u l a t i n g t h e f o r m f a c -t o r s , o n e m a y b e a b l e t o u s e l s h y d r o g e n i c w a v e f u n c t i o n s f o r t h e d o m i n a n tt e r m s . T h e e f f ec ti v e v a l u e o f t h e n u c l e a r c h a r g e is o b t a i n e d f r o m t h e a c t u a li o n i z a t i o n e n e r g y o f e a c h s t a t e u n d e r c o n s i d e r a t i o n . W e r e fe r t o [ H a W 8 7 ]f o r d e t a i ls . T h e r e s u l t s a r e i n r e a s o n a b l y g o o d a g r e e m e n t w i t h a n a c c u -r a t e P W B A c a l c u l a t i o n f o r t h e c a s e o f e l e c t r o n l os s o f H ~ i n c i d e n t o n a H et a r g e t .U se o f a n e x t e n d e d s u m r u leO n e c a n e v a l u a t e t h e s u m o v e r n in E q . ( 7 .1 1 ) m o r e a c c u r a t e l y t h a n p r o -v i d e d b y t h e c l o s u re r e l a t i o n i f o n e b r e a k s u p

q0 (n) : E~r hv- E ~ + E P by- E~


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= q T + 7 . 2 2 )o c q TT h e i n t e g r a l i n E q . ( 7 . 1 1 ) i s n o w w r i t t e n a s f % + fq p(n )" S u m m i n g o v erp r o j e c t i l e s t a t e s , t h e s u m o v e r t h e f i r s t i n t e g r a l c a n b e e v a l u a t e d b y c l o s u r e

a n d t h e s u m o v e r t h e s e c o n d i n t e g r a l c a n b e e v a l u a t e d a p p r o x i m a t e l y b yu s i n g a h i g h e r - o r d e r s u m r u le . T h e r e s u l t s a r e i n r e a s o n a b l y g o o d a g r e e m e n tw i t h a n a c c u r a t e P W B A c a l c u l a t i o n fo r H + H io n i z a t i o n [ M o M 9 1 a ] .I m p u l s e a p p r o x i m a t i o nT h e i m p u l s e a p p r o x i m a t i o n is e s s e n t i a ll y a n i m p r o v e m e n t o f t h e f r ee -e l e c t r o n m o d e l , E q . ( 7. 1) .2 I t t a k e s in t o a c c o u n t t h e i n i t i a l m o t i o n o f t h ee l e c t r o n ( s ) i n t h e p r o j e c t i l e b y a v e r a g i n g t h e t a r g e t i o n i z a t i o n c r o s s s e c t i o no v er t h e m o m e n t u m d i s t r i b u t i o n o f t h e s e e le c tr o n s. T h e d o u b l e - in e l a s ti cc r o s s s e c t i o n i s c o m p u t e d b y 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 f o l l o w s .

R e f e r r i n g t o F i g . 7.1 fo r t h e c o o r d i n a t e s y s t e m , w e d e n o t e t h e m o m e n -t u m o f t h e p r o j e c t i l e e l e c t r o n 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 n u c l e u s b y q p .T h e n , n o n r e l a t i v i s ti c a l l y , t h e m o m e n t u m o f t h e p r o j e c t i l e e le c t r o n w i t h re -s p e c t t o t h e t a r g e t i s g i v e n b y r n e v + q p , a n d i ts k i n e t ic e n e r g y b y

[ m e V + q p [ 2Tep = 2 m e 2 @ y 2__ 1 q P x q P z- - ~rne v 2 + q e . v + ~ + ~ + - -'2me (7 .23)

T h e t e r m s ( @ x + @ y ) / ( 2 m e ) can b e n eg l ec t ed i n mo s t co l l i s i o n s , b u t r e -t e n t i o n o f t h e t e r m @ ~ / ( 2 r n e ) a n d s u b t r a c t i o n o f t h e p r o j e c t i l e e l e c t r o nb i n d i n g e n e rg y I p f r o m T e e i m p r o v e s a g r e e m e n t w i t h e x p e r i m e n t . T h i sg ives 2

_ _ 1 ~ - - I p . 7 . 2 4 )e p -~ r n e v 2 + q p z v + q p z2 m eE a c h p r o j e c t i l e e l e c t r o n h a s a m o m e n t u m d i s t r i b u t i o n w i t h r e s p e c t t o t h ep r o j e c t i l e n u c l e u s g iv e n b y t h e s q u a r e o f i ts m o m e n t u m w a v e f u n c t io n@ ( q p ). 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 , t h e c o n t r i b u t i o n o f e a c h p r o j e c t i l ee l e c t r o n t o t h e t a r g e t d o u b l e - i n e l a s t i c i o n i z a t i o n c r o s s s e c t i o n i s t h e n g i v e nb y a n e x p r e s s i o n o f t h e f o r m

i (e, Zep)l@(qp)12(din el) - d aqP crT7T2 T h e a p p r o x i m a t i o n u s e d h e r e is a n i m p u l s e -l i k e a p p r o x i m a t i o n b u t c a n n o t b e j u s -t i f ied by the derivations of the usual imp ulse approximation [GoW 64,McC70]. Thesederivations typically assume a rigid projectile imping ing on a target with loosely boundelectron s. In the present case , though, the projectile is struc ture d and i t is assumed tohave loosely bound electrons [MoM94].


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7.3 . M O R E E L E C T R O N S . R E L A T I V I S T I C P R O J E C T I L E S 195

~ 0 ~ i ( e , T e p ) , f f ( p z ) d q p z , ( 7 . 2 5 )c r Tw h e r e t h e s e c o n d e x p r e s s i o n m a k e s u s e of t h e d e f in i ti o n o f t h e C o m p t o npro f il e func t i on o f t he p ro j e c t i l e e l ec t ron :

f ff ( q P z ) - d q p x d q P y l @ ( q p ) l 2. (7.26)The cross sec t ion o- I (e , Zep) is t h e s a m e f r e e - e le c t r o n t a r g e t i o n i z a t i o n c r o s ssec t i on as occu r s in Eq . (7 .1 ), bu t he re is eva lua t e d a t t he e l ec t ron k ine t i c1e n e r g y T e p , E q . ( 7 .2 4 ) , r a t h e r t h a n a t t h e e n e r g y ~ m e v 2 a s in t h e f re e -e l e c t r o n m o d e l . F o r t h e e v a l u a t i o n o f E q . ( 7 .2 5 ), s e m i e m p i r i c a l f o rm u l a s f ore l e c t r o n - i n d u c e d i o n i z a t i o n c r o s s se c t i o n s o f n e u t r a l a t o m s [ JA G 8 7, L eB 8 8 ]o r of h y d r o g e n i c i o n s [ M o G 8 0 ], o r t a b u l a t i o n s [ S co 7 8] a r e u se f ul . C o m p t o npro f il e func t i ons can be looked up in [B iM75 , Lee77] . T he r e su l t s , w hena d d e d t o t h e s i n g l e - i n e l a s t i c c r o s s s e c t i o n , a r e i n g o o d a g r e e m e n t w i t h e x -pe r im en t [LeZ9 2] .

I n s u r v e y i n g t h e v a r i o u s a p p r o x i m a t i o n s , ( 4) a n d ( 5 ) a p p e a r t o b e b e s tf o u n d e d t h e o r e t i c a l l y , a l t h o u g h ( 2) is c o n v e n i e n t a n d is o f t e n u s e d f o r t h eeva lua t i o n o f r e l a t i v i s t i c c ros s s ec t ions , a s desc r i be d i n t he nex t s ec t i ons .7 . 3 M o r e e l e c t r o n s . R e l a t i v i s t i c p r o j e c t i l e sI f t h e p a r t n e r t o b e io n i ze d , i n t h e p r e s e n t d i s c u s s i o n t h e t a r g e t , h a s m a n ye l e c t r o n s , e a c h s h e l l o r s u b s h e l l w h i c h i s i o n i z e d c a n b e t r e a t e d s e p a r a t e l y .O n e i n t r o d u c e s a n i n n e r s c r e e n i n g c o n s t a n t w h i c h r e d u c e s t h e n u c l e a rc h a r g e [ S l a 6 0 ] , a n d a n o u t e r s c r e e n i n g c o n s t a n t 0 d e f i n e d s i m i l a r t o t h eK - s h e l l e x p r e s s i o n i n E q . ( 6 .2 8 ) , a s t h e r a t i o o f t h e e x p e r i m e n t a l e l e c t ro n -b i n d i n g e n e r g y d i v i d e d b y t h e h y d r o g e n i c - b i n d i n g e n e r g y o f t h e s h e ll o rsubshel l [MeL58] .

T h e ( n o n r e la t i v is t ic ) m e t h o d s k e t c h e d in t h e p r e c e d i n g s e c t io n c a n b ee x t e n d e d t o a p r o j e c t i l e w i t h m o r e t h a n o n e e l e c tr o n . T o m a k e u s e o f t h eo r t h o g o n a l i t y r e l a t i o n s s im i l a r to t h o s e u s e d i n o b t a i n i n g E q s . ( 7. 10 ) a n d( 7. 11 ), o n e h a s t o a s s u m e v e r y si m p l e in d e p e n d e n t - e l e c t r o n p r o d u c t w a v ef u n c t i o n s f o r t h e t a r g e t a n d t h e p r o j e c t i le . M a k i n g a g a i n u s e o f c l o su r e toe v a l u a t e t h e d o u b l e - i n e l a s t i c c r o s s s e c t i o n , t h e m o d i f i e d e f f e c t i v e s q u a r e dp ro j e c t i l e cha rge equ i va l en t t o Eq . (7 .21 ), for a p ro j e c t i l e w i th Np e l ec t rons ,i s [ B r T 7 8 , M c S 8 1 , A n h 8 5 a ]

S ~ ~ - N P 1 2 ( )p - ~ F ~ + N p - ~ IY~ ~ ~ e )i - - 1 i - - 1 ~ T T ( P - - - ) "

(7.27)


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Here , the co r r ec t ion f ac to r expres s ion (7 .21 ) has been used . Tabu la t ions by[HuV75] and ana ly t i c expres s ions by [SAM 87] a re use fu l in eva lua t ing thefo rm fac to r sums .

In p rinc ip le , Eq . (7 .27 ) can be su bs t i tu ted on ly in to the non re la t iv i s t i cexpres s ion (7 .16 ) . I f one wan ts to con s ide r t a rg e t ion iza t ion by r e la t iv i s t i cp ro jec t i l e s , one i s on s ecu re theo re t i ca l g rou nds for ba re p ro jec t i l e s , i f oneuses one o f the " r e la t iv i s t i ca l ly exac t " expres s ions de r ived in Chap . 6 .S i n c e t h e s e e q u a t i o n s c a n a l s o b e u s e d t o c o m p u t e t h e i o n i z a t i o n c r o s ssec t ion fo r r e la t iv i s t i c e l ec t rons , un der the neg lec t o f exchange , one i s a l soon good theo re t i ca l g rounds in app ly ing the f r ee -e lec t ron expres s ion (7 .1 )to a r e la t iv i s t i c p ro jec t i l e w i th Np e lec t rons . Of cour se , the k ine t i c ene rgyo f t h e p r o j e c t i l e e l e c tr o n is o n l y ~ m e / M p t i m e s t h e k i n e t i c e n e rg y o fthe p ro jec t i l e , e .g . , fo r a 1 -G eV /u p ro jec t i l e , the k ine t i c ene rgy o f a f reep ro jec t i l e e l ec t ron i s on ly 0 .5 M eV. In tha t case , an expres s ion l ike Eq .(6 .26 ) migh t no t be su f f i c ien t ly accu ra te , and i t migh t be be t t e r to u se oneof the r e f e rences c i t ed a f t e r Eq . (7 .26 ) to eva lua te O~r(e .

A more comple te many-e lec t ron expres s ion fo r r e la t iv i s t i c p ro jec t i l e s i ssugges ted by the subs t i tu t ions , in Eq . (7 .16 ) ,

Sp (Q ) ~ s~n~ (7 .28 )

(~ZT ct)2( 1 - - Q m i n / Q ) 2 1 2IfT I2 ~ IFTI2 + (1 --/32Qmin/Q)2 (IGTI + ~Q[FTI ), (7.29)f rom Eqs . (7 .27) an d (6 .26) , respe ct ively [Anh85a]. In the secon d expres -s ion , which is va l id only for target K-shel l ioniza t ion , [FT[ 2 a n d [GT[2 aregiven by Eqs . (6 .29) and th e o th er qu an t i t ie s are a lso def ined in See . 6 .2 .2 .Be low , we conf ron t the r e su l t ing c ros s s ec t ion w i th exper imen t .

7 . 4 E l e c t r o n l o s s .m e n t

C o m p a r i s o n w i t h e x p e r i -

To fac i l i t a t e the compar i son w i th exper imen t , c ros s s ec t ion expres s ions fo rK - e l e c t r o n l os s o f a p r o j e c ti l e i n d u c e d b y a n e u t r a l t a r g e t a t o m a r e d e r iv e df rom the above expres s ions fo r t a rge t ion iza t ion , by in te r chang ing the ro leso f the t a rge t a nd the p ro jec t i l e . I t is no t d i f fi cu lt to ex ten d the r e su l t sto L- she l l lo s s [AnM 84b] , bu t the expres s ions g iven be low app ly on ly toK-she l l lo s s , excep t fo r the f r ee -e lec t ron mode l expres s ion wh ich i s genera l .We s ta r t by r ewr i t ing the l a t t e r , Eq . (7 .1 ) :

4 ~ 1 z ~ ,- Z w a p ( p ) + (e) (7.30)


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7. 4 . E L E C T R O N L O S S . C O M P A R I S O N W I T H E X P E R I M E N T 197wher e a lp ( p) and a~ ( e ) deno te the p r o j ec t i l e l os s c r oss sec t ions induced byp r o t o n s a n d f re e e l e c tr o n s , r e s p e c ti v e ly , b o t h m o v i n g w i t h t h e p r o j e c t i leve loc i ty v . T h e t a r g e t i s a s su m ed to be neu t r a l , w i th Z T " fr ee" e l ec t r ons .

F o r t h e m o r e c o m p l e t e , m a n y - e l e c t r o n e x p r e s s i o n , w e u s e t h e o n e p r o -posed in the p r eced ing sec t ion , which g ives , pe r t a r ge t K e l ec t r on ,

4 1 r a ~ / i mo l ( ~ P ) 2 Z ~ :) in Q d Q [Fl~el]2 s ~ o dd W min Q-~ (Q ), (7.31)w h e r e

s ~ ~ ( Q ) = ZTi=1 2 ( )Z T - - ~ I F ~ 2 s ( 7 . 3 2 )i :1 O ' I (P ) '

i F h e l l 2 _ _ i F P i 2 + ( /~ZPa)2(1 - Qmin/Q) [ 2 1( 1 ~2Qmin/Q)2 ( I G P + ~ Q I F P I ~ ) ,W - 2 E P - E P + ( E T - E T )

e 2 Z ~ / a oW 2 2 E ~

Qmin - - 4 ~ 7 p , W m in -- 0 P = e 2 Z 2 / a o ,~P - - ( ~ )2 - - Z bareOlZp , Z p - 0.3,

wh e r e ( E T - E T ) i s a su i t ab ly chosen t a r ge t exc i t a t ion ene r gy . T h e f o r mfac tors IFp[ 2 an d IGpI 2 a re given by EQ. (6 .29) ap pl ied to t he pro jec t i lee l ec t r on . I n the ex pr e ss ion f o r W , the d e s ign a t ions i and f f o r t he in i t i a land f ina l s t a t e s o f t he ion ized pa r tne r , and 0 and n f o r t he ion iz ing pa r tn e r ,have bee n ke p t . I n the c a l cu la t ions show n be low, ( E w - E T ) ha s b een se tequa l to zero .

I n exam in in g ex pe r im enta l ly th e va l id i ty o f a t heo r e t i c a l l oss c r oss sec -t ion exp r e ss ion l ike E q . ( 7 .31) , one can a sk the f o l lowing ques t ions o fphys i ca l i n t e r e s t . ( 1) I s t he r e ex pe r im enta l ev idence f o r t he r o l e o f t hedo ub le- in e las t i c c ross sec t ion, i . e ., for the in crease of the pro jec t i le lossc r oss sec t ion due to the t a r g e t e l ec t r ons? ( 2) I s t he r e ev idence f or t hesc r een ing e ff ec t o f t he t a r g e t e l ec t r ons in the s ing le - e l a s t ic c r oss sec t ion ,i .e ., fo r t he r edu c t ion o f t he e f fec tive squ a r ed t a r g e t cha r ge du e to the f o r mf a c to r s F ~ C o n s i d e r i n g t h e r e la t i v e s iz es o f t h e e l e c tr o n d i s t r i b u t i o n sin the p r o j ec t i l e and the t a r ge t , one would expec t t a r ge t sc r een ing e f f ec t sto be la rge in projec t i le loss i f Zp > ZT. (3) I f


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198 C H A P T E R 7. I O N I Z A T I O N - M A N Y E L E C T R O N Ss c r e e n i n g e ff ec ts a r e s m a l l, i s t h e r e a n y e v i d e n c e t h a t p e r t u r b a t i o n t h e o r yi sv a l i d i n r e l a t i v i s t i c co l li si ons, wh ich im p l i e s t ha t a l ( s i ne l ) c< Z~? S o f a r,o n l y p o i n t s ( 1 ) a n d ( 3) h a v e b e e n e x a m i n e d e x p e r i m e n t a l l y , b u t w e g iv eca l cu l a t ed r e su l t s fo r (2 ) .R o l e o f d o u b l e - in e l a st i c c r o s s se c t i o nF ig u re 7 .2 shows m e asu rem en t s o f t he on e -e l ec t ron l os s c ros s s ec t i on o f100- an d 3 80 -M eV / u Au 52+ i ons i n ( a) H2 and (b ) He t a rge t s . Th e da shedcu rves a re t he s i ng l e - i ne l as ti c a nd t he so l i d cu rves a re t he su m s o f s i ng l e -a n d d o u b l e - i n e l a s t ic c r o s s s e c ti o n s. B e l o w t h e " t h r e s h o l d " a t 1 0 M e V / u ,on ly t he fo rm er i s e f f ec t i ve . The c ros s s ec t i on ca l cu l a t i on fo r t he H2 t a rge ti nc l udes a sm a l l co r rec t i on fo r t he m o lecu l a r n a tu re o f t he t a r ge t [MeH9 1] .T h e t h e o r e t i c a l r e s u l ts a g r e e w i t h e x p e r i m e n t o n l y i f t h e d o u b l e - i n e l a s t icc ros s s ec t ion i s i nc l uded . Th i s i s s een m ore s t r i k ing ly i n F ig . 7 .3 wh ichg ives t he r a t i o o f t he H2 t a rg e t t o t he H e t a rge t c ros s s ec t i on ( a ) fo r t hea b o v e i o ns a n d ( b) f o r 4 0 5 - M e V / u U s 6+ i on s. W i t h o u t t h e d o u b l e - i n e l a s t icp roces s , t he r a t i o wou ld fo l l ow the dashed l i ne .

E f f e c t o f t a r g e t -e l e c t r o n s c r e e n i n gTh e ro l e of s c reen ing i n t he t o t a l l o ss c ros s s ec t i on o f va r i ous o ne -e l ec t ronions co l l id ing wi th a A1, Ag, and U target s i s shown in F ig . 7 .4 . For pro jec-t i l e s w i t h Zp < ZT , t he t a rge t s c reen ing e f f ec t i n t he s i ng l e - i ne l as t i c c ros ssec t i on is qu i t e d ra m at i c ( for t he t a rg e t s show n , t he doub l e - i ne l as t i c c ros ssec t i on p l ays on ly a m ino r ro l e ) . I t e f f ec t i ve ly r em oves t he t r ansve r se andsp in - fl i p con t r i bu t i ons t o t he c ros s s ec t ion , so t h a t i n s t ea d o f i nc reas ing asIn 7 , t he l os s c ros s s ec t ion beco m es i nd epe nd en t o f 7 .V a l id i ty o f p e r t u r b a t i o n t h e o r yT h e v a l id i ty o f p e r t u r b a t i o n t h e o r y h a s b e e n e x a m i n e d i n m e a s u r e m e n t s o fi s - e l e c t r o n l o s s f o r 8 2 - t o 2 0 0 - M e V / u X e a n d 1 0 5 - t o 9 5 5 - M e V / u U i o n si n c i d e n t o n v a r i o u s s o l i d t a r g e t s b e t w e e n B e a n d A u . F i g u r e 7 . 5 s h o w s t h eq u a n t i t y

~ c o r r / , - ~ 2 - [ 4 - 7 .3 3 )which acco rd ing t o t he f r ee -e l ec t ron m ode l exp res s ion (7 .30 ) shou ld be equa lt o a l p ( p ) . T h e d a t a p o i n t s h a v e b e e n o b t a i n e d f r o m si ng l e K - e l e c t r o n lo ssm e a s u r e m e n t s u s i n g X e 53+ a n d X e 52+ i o ns ( t h e l a t t e r d a t a h a v e b e e n c o r-r ec t e d fo r t he i n f l uence of doub l e K -e l ec t ron l o ss ) . O n e s ees t h a t i n t hee n e r g y r e g i o n e x a m i n e d , p e r t u r b a t i o n t h e o r y b r e a k s d o w n f o r t h e h i g h e r -ZT t a rge t s , bu t i nc reas ing ly l e s s so , a s t he p ro j ec t i l e ve loc i t y i nc reases .


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7.4. ELEC TRO N LOSS. COMPARI SON WITH EXPE RIME NT 1 9 9I I I ~ I I I I I I I I I I I I I I I

/ ",N\ ~ (a)2

_oI--.0 0 I , ~ J , , , l i i , l l , l l l I t J , l , ,U J( D

1 . 0 ( b ) _o

0

0.5

10 100 1000PROJECTILE ENERGY ( MeV /nuc leon )

F i g u r e 7 . 2 . P r o j e c t i l e o n e - e l e c t r o n l o ss c r o s s s e c t i o n o f A u 52+ in ( a) H e a n d ( b )H e 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 e e n e r g y . T h e s o l i d c u r v e s a r e t h e s u m s o f t h es i n g l e - a n d d o u b l e - i n e l a s t i c c r o s s s e c t i o n s , t h e d a s h e d c u r v e s a r e o n l y t h e f o r m e r .F r o m [ H u F 9 1 ] .

0.80i - 0.7r rz 0.6OI - -o 0 .5u Jmoo 0.709onO 0.6(1)' - r 0 .5"1-

0.4

I I I J l i t I I I I 1 i i t l I i I i i l l i l

- - a )

(b)

, I , i J , , I i , J , J l , , 1 i J i t , , , ,10 100 1000P R O J E C T IL E E N E R G Y ( M e V / n u c l e o n )

F i g u r e 7 . 3 . R a t i o o f H 2 t o H e o n e - e l e c t r o n l o s s c r o s s s e c t i o n f or ( a) A u 52+ a n d( b ) U s 6+ i o n 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 e e n e r g y . W i t h o u t a d o u b l e - i n e l a s t i cp r o c e s s , t h e r a t i o w o u l d f o ll o w t h e d a s h e d l i n e. F r o m [ H u F 9 1 ].


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. 0q .r

t :>4A I -

No . .Nv

1 0987654

so

=

. . . . 1, , - - F : ~ : 4 , : ~ 4 ~ I ~ ~ I I , ~ ' - ~ - : ,

( b ) Z 1 - =4 7 Z P = 9 2

5 0

~ . . . . . - _ - . . . . . . . . . . . . .

3 02 01 0

. . . . . . . . . | , , ~ I

i ( c ) Z ' r = 9 2 _ Z P =9 2

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3 2. . . . . . . . . .

3 4 101 2 a 4 102 2 3 4 103 2

Figure 7 .4 . Loss c ross sec t ions mul t ip l ied by (Zp/ZT) 2 for var ious one-e lec tronprojec t i les col l iding wit h (a) hl , (b) Ag, and (c) W targ ets as a funct ion of theprojec t i le energy. Adapted f rom [AnB87] .


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7.5. M U L T I P L E I O N I Z A T I O N 201Q u a l i t a t iv e l y , o n e e x p e c t s a s m a l l e r p e r t u r b a t i o n o f t h e p r o j e c t i l e e l e c t ro nw a v e f u n c t i o n b y t h e t a r g e t ( n u c le u s ) w i t h i n c r e a s i n g p r o j e c t i le v e l o ci ty .T h e d a t a a r e c o m p a r e d w i t h t h e G l a u b e r a p p r o x i m a t i o n [M c g8 2], i n w h i c ht h e d i s t o r t i o n o f t h e p r o j e c t i le w a v e f u n c t i o n b y t h e t a r g e t i s p a r t i a l l y t a k e ni n t o a c c o u n t b y m e a n s o f a n e i k o n a l p h a s e ( Se c . 8 . 4. 1 ).

I t i s s o m e w h a t d i s t u r b i n g t h a t f or Z T _< 1 3, w h e r e p e r t u r b a t i o n t h e o r yis e x p e c t e d t o b e v a li d , t h e e x p e r i m e n t a l r e s u l t s d o n o t a g r e e b e t t e r w i t hthe t h eo re t i ca l ones . Th i s is no t on ly t he case fo r X e p ro j ec t i l e s , a s iss h o w n b y F i g . 7 .6 , w h i c h g i ve s t h e r a t i o o f e x p e r i m e n t a l t o t h e o r e t i c a lc ros s s ec t i ons fo r r e l a t i v i s t i c X e , B i, and U p ro j ec t i l e s . A t p rese n t , t he rei s n o e x p l a n a t i o n f o r t h e s e e f f e c t s , e v e n t h o u g h s o m e t i m e s t h e y h a v e b e e nb l a m e d o n e x p e r i m e n t a l u n c e r t a i n t ie s [ M oG 9 0] .

7 . 5 M u l t i p l e i o n i z a t i o nW i t h r e l a t iv i s t ic p r o j e c t i l e s o f h i g h a t o m i c n u m b e r , t h e r e is a l w a y s th e p o s -s i b i l it y o f m u l t i p l e - e l ec t ron t a rg e t i on i za t i on an d p ro j ec t i l e l o s s i n a s i ng leco l li s ion , i n ad d i t i o n t o t he s i ng l e -e l ec t ron p roces ses . (M ul t i p l e p ro j ec t i l ee l ec t ron l o s s i s o f t en ca l l ed s t r i p p i n g . ) I n t h e p r e s e n t s e c t i o n , t h e s e p r o -cesses are d i scussed br ief ly .7 .5 . 1 M u l t i p l e t a r g e t i o n i z a t io nM u l t i p l e i o n i z a t i o n o f g a s e o u s t a r g e t s ( N e , A r , K r , I 2 ) b y 4 2 0 - M e V / uU 9 1+ i ons un de r s i ng l e -co ll i si on con d i t i ons has bee n i nve s t i ga t e d t o f i ndo u t w h e t h e r h i g h l y c h a r g e d , s l o w ( e V ) t a r g e t r e c o i l i o n s c a n b e p r o d u c e dc o p i o u s l y e n o u g h s o t h a t t h e s e r e c o il io n s c o u l d b e u s e d a s s e c o n d a r y b e a m sf o r o t h e r s t u d i e s . F i g u r e 7 .7 g iv e s a n o v e r v i ew o f t h e p r o d u c t i o n c r o ss se c -t i o n s o f t h e h e a v i e r i o n s a s a f u n c t i o n o f t h e i r c h a r g e q . C h a r g e s t a t e s u pt o q ~ 0 . 5 Z T a r e i n d e e d p r o d u c e d w i t h r e a s o n a b l e c r o s s s e c t io n s .

I n t h e s e c o ll is io n s , Z p > Z T , so t h a t p e r t u r b a t i o n a p p r o a c h e s o f t h ek i n d p r e s e n t e d i n C h a p . 6 a n d i n t h i s c h a p t e r a r e n o t a p p l ic a b l e . H o r -b a t s c h [ H o r 8 6 , H O D 8 6 ] h a s m a d e a c a l c u l a t i o n i n w h i c h t h e t i m e e v o l u t i o no f t h e t a r g e t m a n y - e l e c t r o n d e n s i t y is o b t a i n e d f r o m a c l a ss ic a l e q u a t i o n f o rt h e p h a s e - s p a c e d i s t r i b u t i o n f u n c t i o n . T h e c a l c u l a t i o n is s t a t i s t i c a l in n a -t u r e a n d u s e s a n a v e r a g e o v e r s h e l l s t r u c t u r e , s i m i l a r t o t h e T h o m a s - F e r m id i s t r i b u t i o n [Sch68]. Th e p rob ab i l i t y o f i on i za t i on pe r e l ec t ron , P 1 (b) , fo r ag i ve n n u c le a r t r a j e c t o r y w i t h i m p a c t p a r a m e t e r b is o b t a i n e d b y a M o n t e -C a r l o m e t h o d w h i c h u s e s a t i m e - d e p e n d e n t c l a s s i c a l H a m i l t o n i a n

p2 Z~ f ( t ) e 2 Zp e 2H = - (7.34)2me r IR ( t ) - r ]


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2 0 2 C H A P T E R 7. I O N I Z A T I O N - M A N Y E L E C T R O N S

405020

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- - - - - L ~ - - ~ - -

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0 2 0 4 0 6 0 8 0Z TF i g u r e 7 . 5 . S i ng le K - e l e c t r o n lo ss c r o ss s e c t io n , c o r r e c t e d f or t a r g e t - e l e c t r o nc o n t r i b u t i o n s a c c o r d i n g t o E q . ( 7 .3 3 ) , f o r o n e - a n d t w o - e l e c t r o n X e i o n s c o l l id i n gw i t h v a r i o u s t a r g e t s a t t h e e n e r g ie s i n d ic a t e d , a s a f u n c t i o n o f t h e t a r g e t a t o m i cn u m b e r . T h e c r o s s s e c t i o n i s d i v i d e d b y Z ~ . I n a d d i t i o n t o t h e s t a t i s t i c a l e r r o r ss h o w n , t h e r e s u l t s h a v e p o s s i b l e s y s t e m a t i c e r r o r s u p t o 2 0 % . T h e s o l id c u r v e sh a v e b e e n c a l c u l a t e d w i t h t h e G l a u b e r m o d e l . F r o m [ A n M 8 7 ] .

g 2 -l o _

c " q i . . _N 1-

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0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6R e d u c e d V etocityj vp/Z p

9 B i - > A t9 B i - > CO X e - > A t

X e - > C0 X e - > B eA U - > A tCV U > B e

F i g u r e 7 . 6 . R a t i o b e t w e e n e x p e r i m e n t a l p r o j e c t i le l os s c r os s s e c t i o n a n d t h e o -r e t i c a l s e m i c l a s s i c a l c a l c u l a t i o n s f o r p r o t o n s m u l t i p l i e d b y Z ~ . [ Tr R S0 ]. A b o v e t h er e d u c e d v e l o c i t y v p / Z p - 1 ( in a to m ic un i t s ) , see EQ . (7 .3 ) fo r t he case o f t a rg e ti o n i z a t i o n , t h e c r o s s s e c t i o n s h a v e b e e n c o r r e c t e d f o r t h e d o u b l e - i n e l a s t i c p r o c e s sa c c o r d i n g t o E q . ( 7 .3 3 ) . 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 [ M o G9 0 ] l ie w i t h i n 1 0 % o ft h e s e m i c l a s s i c a l c a l c u l a t i o n s u s e d h e r e . A d a p t e d f r o m [ R y S 9 3 ] .


19/21

7 .5 . M U L T I P L E I O N I Z A T I O N 203

1 0 - - 1 ~ I I I

( a l

1 0 1 6E

1 0 wD

1 04 s

1 0 -1 s

i

h i

|

io10"1 9 .. , i , I i I . i i I , i0 1 0 2 0 3 0 1 0 2 0 3 0

F ig u re 7 .7 . M easured c ross sec t ions for the prod uc t ion of s low reco i l ions by420 -M eV /u U 91+ pro jec t i le s im pinging on va r ious gas ta rge ts a s a func t ion of thecharge s ta te q of the recoil ions. D at a po ints (solid c irc les) for (a) K r and (b)I ta rge ts a re comp ared to ca lcu la t ions by [Hot86 , HOD 86] (open c ir c le s ). F rom[KeU86].

t o d e t e r m i n e t h e p h a s e - s p a c e d i s t r i b u t i o n f u n c t i o n a t p o s i t i o n r a n d m o -m e n t u m p . T h i s e q u a t i o n i m p l i e s t h a t e a c h t a r g e t e l e c t r o n s ee s t h e p r o j e c -t i le fi el d a n d a n e f fe c ti v e s c r e e n e d t a r g e t f i el d w h i c h is t i m e d e p e n d e n t . T h ev a l u e o f t h e t a r g e t c h a r g e p a r a m e t e r Z ~ ~ ( t) a t t = 0 is a d j u s t e d t o g i v e t h eb e s t f it t o t h e e x p e r i m e n t a l c r o s s s e c ti o n s . U n f o r t u n a t e l y , t h e d e p e n d e n c eo f t h e c a l c u l a t e d c r o s s s e c t io n s o n Z ~ f ( 0) i s v e r y s e n s i t iv e . T h e c a l c u l a t i o n sshow n in F ig . 7 . 7 u se Z ~ ,~ (0) = 3.5.

T h e c a l c u l a t i o n a s s u m e s t h a t t h e t a r g e t e l e c t r o n s c a n b e t r e a t e d a si n d e p e n d e n t p a r t i c l e s . I n t h a t c a se , t h e o v e r a ll p r o b a b i l i t y o f i o n i z i n g qe l e c t r o n s o f a t a r g e t h a v i n g i n i t i a l ly N ( = Z T ) e l e c t r o n s is g i v en b y t h eb i n o m i a l d i s t r i b u t i o n

N~P ( q , N , b ) - [Pl(b)] q [1 - P l( b)] N - q . (7 .35)q ( N q )T h i s e q u a t i o n h a s t o b e i n t e g r a t e d o v e r 2 7 r b db t o o b t a i n t h e q - f o l d i o n i z a -t i o n c r o s s s e ct i o n . A t r e l a t i v i s t i c v e l o c i t ie s , t h e e f fe c t o f c a p t u r e c a n b en e g l e c t e d , b u t a t l o w e r v e l o c i t ie s i t m u s t b e i n c l u d e d . F i g u r e 7 .7 s h o w st h a t t h e o v e r al l m a g n i t u d e a n d t r e n d o f t h e e x p e r i m e n t a l c r o s s s e c t io n a r er e p r o d u c e d . O n e c a n a l s o i n t r o d u c e r e f i n e m e n t s i n t h e t h e o r y , s u c h as s t r u c -


20/21


A

103

102

1 1

I

- - m :

- - X ~ ~ - . - -

I I I

/m=5 I . - -/ i / ~ , , ~ ~ o

/ ,; Az

0 2 0 4 0 6 0 8 0ZTF ig u r e 7 .8 . Single- and m ul t ip le-elect ron loss cross sect ions for 95 5-M eV /u U 6s+ions passing th rough var ious t a rge t s as a func tion o f the t a rg e t a tomic num ber .On eac h curve, m indicates the m ul t ip l ic i ty of the loss. Th e sol id curves havebeen ca lcu la ted on the bas i s o f the independe n t -par t i c le m odel. The d ashedcurves include correct ions for the LMM Auger effect . From [AnM87].

t u r e d p r o je c t il e s , sh e ll s t r u c t u r e o f t h e t a r g e t , a n d q u a n t u m - t h e o r e t i c a l a p -p r o a c h e s b e y o n d t h e i n d e p e n d e n t - p a r t i c l e m o d e l f o r t h e e l e c t r o n s [ H o r 9 1 ] .

7 . 5 . 2 M u l t i p l e p r o j e c t i l e l o ssM u l t i p l e e l ec t ro n l o s s i n a si n g l e co l li s io n o f r e l a t i v i s t i c X e an d U p ro j ec t i l e sb y a v a r i e t y o f t a r g e t s h a s b e e n o b s e r v e d u p t o s ix f ol d. F i g u r e 7 .8 sh o w se 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 l ts f or 9 5 5 - M e V / u U 6 s+ .

T h e c a l c u l a t i o n s a r e a g a i n b a s e d o n t h e i n d e p e n d e n t - p a r t i c l e m o d e l .B u t h e r e , c o n t r a r y t o t h e c a s e o f t a r g e t i o n i z a t i o n b y h i g h - Z p p r o j e c -t i l e s , p e r t u r b a t i o n t h e o r y c a n b e a p p l i e d t o c o m p u t e t h e o n e - e l e c t r o n i o n -i za t i o n p ro b ab i l i t y P (S ) (b ) i n e ach s h e l l o r s u b s h e l l S . Th i s p ro ced u re i sa p p r o x i m a t e f o r t h e U p r o j e c t i l e s u s e d , e v e n t h o u g h t h e r a t i o Z T / Z ( s )


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7 .5 . M U L T I P L E I O N I Z A T I O N 205(Z (p ) = sc r een ed p r o jec t i l e cha r ge ) i n a l l t he subshe l l s S cons ide r ed neve rexceeds un i ty and the p r o j ec t i l e ve loc i ty i s h igh enough so tha t P~S) ( b)nev er exceeds uni ty , ev en a t sm al l va lues of b .

For h igh ly sc r eened p r o jec t i l e s , r e l a t iv i s t i c e f f ec t s on the impac t pa r am-e te r de pen den ce r em a in sma l l f o r /~ _< 0 .9 , so th a t SC A tab le s o f [HaJ75]can be used to e s t ima te P~S) ( b) f o r e ach r e l evan t K , L , and M subshe l l .E q u a t i o n ( 7 . 3 5 ) c a n t h e n b e a p p l i e d t o e a c h s u b s h e l l , w i t h N e q u a l t o t h eo r i g in a l n u m b e r o f e l e c tr o n s N ( s) i n t h e s u b s h e l l a n d q e q u a l t o t h e n u m -be r n ( s ) o f e l ec t r ons s t r ip pe d f r om the subshe l l S . If e l ec t r ons a r e s t r i pp edf r om mor e than one subshe l l , e . g . , t h r ee , t he c r oss sec t ion f o r s t r i pp ing me lec t r ons f r om the p r o jec t i l e i s g iven by :

O" m z oi - - I - n l i - } - n l i i - - m p ( I ) ( n i , N I , b) p ( H ) ( n i l , N I I , b )x p ( II I ) (n i l1 , NI II , b) 27rbd b , (7.36)

whe r e the subsc r ip t s and supe r sc r ip t s I , I I , and I I I r e f e r t o t he th r ee sub-she l l s considered.

F i g u r e 7 .8 s h o w s t h a t t h e m a g n i t u d e a n d t h e t r e n d o f t h e c r o ss s ec t io n sa r e qu i t e w e ll r epr od uce d by the theor y , bu t on ly i f a side ef fec t, t he e j ec t ionof e l ec t r ons by the L M M Au ge r e f fec t, i s i nc luded . S ince p [ S ) ( b ) sca le sa p p r o x i m a t e l y a s Z ~ , o n e e x p e c t s t h a t f o r o n e - e l e c t r o n s t r i p p i n g t h e r a t i oa l / Z ~ , as show n in F ig . 7 .8 , shou ld be ind epe nd en t o f Z T . A t t h i s h ighp r o j e c t il e v el o ci ty , t h e p e r t u r b a t i o n t h e o r y s h o u l d b e v a li d . I n c o n t r a s tto the d a t a i n F ig . 7 .5 , t he d r op- of f w i th inc r ea s ing Z T i s due he r e tothe in f luence o f m ul t ip l e - io n iza t io n e f fec ts , i n pa r t i cu l a r , due to the f ac to r[ 1 - in xpr s sion 73 5 ) As increases ,i nc r ea se s , t h i s f ac to r becomes appr ec i ab ly l e s s t han un i ty .On e o the r e ff ec t c an in f luence am. A t sma l l va lue s o f Z T , t he t a r g e te l ec t r ons can enh ance the loss p r o bab i l i t y P1 ( Secs . 7 .2 and 7 .3) . T h i s e f fec th a d n o t b e e n t a k e n i n t o a c c o u n t i n t h e c a l c u l a t i o n o f [ A n M 8 7 ] b e c a u s e t h er e l evan t expr e ss ions we r e no t ava i l ab le a t t he t ime [ MoM91b, MoM92a] .

chapter 7 - ionization many electrons

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