EFFECTIVE CHARGE OF ENERGETIC HEAVY IONS IN GASES, SOLIDS AND PLASMAS

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EFFECTIVE CHARGE OF ENERGETIC HEAVY IONS IN GASES, SOLIDS AND PLASMAS

N. Cowern

To cite this version:

N. Cowern. EFFECTIVE CHARGE OF ENERGETIC HEAVY IONS IN GASES, SOLIDS AND PLASMAS. Journal de Physique Colloques, 1983, 44 (C8), pp.C8-107-C8-121.

�10.1051/jphyscol:1983807�. �jpa-00223314�

JOURNAL DE PHYSIQUE

Colloque C8, supplement au n O 1 l , Tome 44, novembre 1 9 8 3 page C8-107

EFFECTIVE CHARGE OF ENERGETIC HEAVY IONS I N GASES,SOLIDS AND PLASMAS

N .E. B. Cowern

AERE HameZZ, Oxon OX11 ORA, U . K .

La c h a r g e e f f e c t i v e q d ' i o n s p a r t i e l l e m e n t 6 p l u c h 6 s e s t d g t e r - e f f

minge p a r l ' 6 q u i l i b r ' e e n t r e l a c a p t u r e Q l e c t r o n i q u e e t l e s t r i p p i n g du m i l i e u r a l e n t i s s e u r . Les cons6quences d e s v a r i a t i o n s d e s s e c t i o n s e f f i - c a c e s c o r r e s p o n d a n t e s s u r q e f f , s o n t C t u d i 6 e s p o u r d e s i o n s quasi-compls- Cement Bpluchds, e t d ' a u t r e s , d e c h a r g e a r b i t r a i r e . Les deux p a r a m s t r e s s i g n i f i a n t s s o n t l e s e f f e t s d e d e n s i t 6 d a n s l a c i b l e , e t l e s e f f e t s d u s P l a t e m p e r a t u r e du plasma. I c i , o n s e l i m i t e

B

l a d e n s i t s . On d i s c u t e l e s o b s e r v a t i o n s e x p 6 r i m e n t a l e s e n c i b l e s o l i d e e t g a z e u s e , r e s p e c t i v e - ment. On p r 6 s e n t e un nouveau modsle d e s e f f e t s d e d e n s i t 6 s u r d e s i o n s quasi-complstement s t r i p p 6 s . On d i s c u t e l e s dE/dx c o r r e s p o n d a n t s e n m i - l i e u d e n s e , q u i peuvent v a r i e r s u r 20 %. Le t a u x d e s t r i p p i n g d ' u n i o n l o u r d d e f a i b l e c h a r g e e s t beaucoup p l u s & l e v 6 d a n s un s o l i d e que d a n s un g a z .

A b s t r a c t

The e f f e c t i v e stopping-power c h a r g e q e f f f o r p a r t i a l l y s t r i p p e d i o n s i s d e t e r m i n e d by a b a l a n c e of t h e e l e c t r o n c a p t u r e and l o s s c r o s s s e c t i o n s w i t h i n t h e s t o p p i n g medium, The c o n s e q u e n c e s f o r q

e f f r e l a t i v e changes i n t h e s e c r o s s s e c t i o n s a r e c o n s i d e r e d f o r n e a r l y f u l l y s t r i p p e d i o n s a n d f o r p a r t i a l l y s t r i p p e d i o n s .

Two p h y s i c a l f a c t o r s l e a d i n g t o s u c h changes a r e d e n s i t y e f f e c t s i n t h e s t o p p i n g medium, and plasma t e m p e r a t u r e e f f e c t s . T h i s p a p e r c o n c e n t r a t e s on t h e r o l e o f d e n s i t y e f f e c t s . E x p e r i m e n t a l o b s e r v a t i o n s o f a g e n u i n e g a s - s o l i d d e n s i t y e f f e c t a r e d i s c u s s e d , and a new r i g o r o u s model o f t h i s e f f e c t f o r n e a r l y f u l l y s t r i p p e d i o n s i s p r e s e n t e d .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983807

I m p l i c a t i o n s f o r dE/& i n dense media a r e d i s c u s s e d , and a r e shown t o b e s i g n i f i c a n t a t t h e 20% l e v e l . F u r t h e r m o r e t h e s t r i p p i n g r a t e f o r a heavy i o n of low c h a r g e s t a t e e n t e r i n g a medium w i l l b e c o n s i d e r a b l y h i g h e r i n a s o l i d t h a n i n g a s .

1. I n t r o d u c t i o n

The s t o p p i n g power of a medium f o r a p o i n t p r o j e c t i l e of c h a r g e Z e may be e x p r e s s e d a s 1

1

where v i s t h e p r o j e c t i l e v e l o c i t y , m t h e e l e c t r o n mass, and N t h e d e n s i t y of atoms of a t o m i c number Z2. The q u a n t i t y L depends s l i g h t l y on t h e d e n s i t y o f t h e medium ( s o l i d t a r g e t w a v e f u n c t i o n s a r e d i f f e r e n t from t h o s e i n g a s e s 2 ) , and depends s t r o n g l y on w h e t h e r t h e t a r g e t i s

i n a t o m i c o r plasma form. F u r t h e r m o r e i n plasmas L i s s t r o n g l y t e m p e r a t u r e dependent 3

.

These e f f e c t s i n t h e s t o p p i n g p r o c e s s

r e l a t i n g t o L have r e c e i v e d c o n s i d e r a b l e a t t e n t i o n i n r e c e n t y e a r s . F o r p a r t i a l l y s t r i p p e d , high-energy heavy i o n s , t h e i o n s t i l l b e h a v e s v e r y n e a r l y a s a ~ o i n t - l i k e c h a r g e a s r e g a r d s t h e s t o p p i n g p r o c e s s 4

.

F o r t h e s e i o n s , i t i s t h e r e f o r e s u f f i c i e n t t o r e p l a c e Z

l i n t h e dE/dx f o r m u l a by an e f f e c t i v e c h a r g e q e f f , which i s s i m p l y g i v e n by t h e r o o t mean s q u a r e c h a r g e q of t h e i o n s a v e r a g e d

r.m.s.

o v e r t h e c h a r g e - s t a t e d i s t r i b u t i o n w i t h i n t h e t a r g e t medium.

I n d i l u t e g a s t a r g e t s , one can measure q d i r e c t l y by r . m . s .

o b s e r v i n g t h e c h a r g e - s t a t e d i s t r i b u t i o n a f t e r t h e i o n s emerge from t h e t a r g e t . However i n s o l i d s , c h a r g e - s t a t e measurements w i t h many- e l e c t r o n i o n s a r e a f f e c t e d by p o s t - t a r g e t a u t o i o n i s a t i o n , which l e a d s t o a s p u r i o u s i n c r e a s e i n t h e a p p a r e n t v a l u e of q

.

Only i n v e r y

r.m.s.

r e c e n t measurements w i t h n e a r l y f u l l y s t r i p p e d i o n s 1 5 ( h y d r o g e n - l i k e and f u l l y s t r i p p e d ) h a s i t been e s t a b l i s h e d t h a t a d i s t i n c t g a s - s o l i d d e n s i t y e f f e c t i n q does i n d e e d e x i s t .

r .m. S .

Since both d e n s i t y e f f e c t s and plasma temperature e f f e c t s i n qr.m.s. a r i s e through changes i n "charge-changing c r o s s s e c t i o n s " , we begin i n S e c t i o n 2 by looking a t t h e p r o j e c t i l e charge s t a t e d i s t r i b u t i o n i n terms of t h e s e c r o s s s e c t i o n s . Transformation formulae a r e g i v e n , r e l a t i n g changes i n t h e e l e c t r o n c a p t u r e and l o s s c r o s s s e c t i o n s , t o changes i n q e f f . I n S e c t i o n 3 we c o n s i d e r t h e e f f e c t of t a r g e t d e n s i t y on t h e charge-changing c r o s s s e c t i o n s , and make a q u a n t i t a t i v e e s t i m a t e of t h e d e n s i t y e f f e c t i n t h 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 o r h i g h l y s t r i p p e d i o n s . F i n a l l y i n S e c t i o n 4 t h e s e r e s u l t s a r e combined t o e s t i m a t e t h e expected changes i n q e f f r e s u l t i n g from d e n s i t y e f f e c t s , and some b r i e f remarks on plasma temperature e f f e c t s a r e p r e s e n t e d .

2 . Charge-changing c r o s s s e c t i o n s and t h e e f f e c t i v e charge V i s u a l i s e an i o n of v e l o c i t y v and atomic number Z1, i n c i d e n t on two t a r g e t s A and B having t h e same atomic number Z2. T a r g e t s A and B may have d i f f e r e n t d e n s i t i e s o r t e m p e r a t u r e s , and t h e

charge-changing c r o s s s e c t i o n s w i t h i n t h e s e t a r g e t s may t h e r e f o r e be d i f f e r e n t . We c o n s i d e r t h e e f f e c t o f such d i f f e r e n c e s i n c r o s s s e c t i o n s on t h e charge q

,

f o r two r e g i o n s of p r o j e c t i l e

r.m.s.

v e l o c i t y where s t r a i g h t f o r w a r d approximations a r e p o s s i b l e . (a) Nearly f u l l y - s t r i p p e d ions

For v e l o c i t i e s i n t h e r e g i o n v

2

Z1 vo, p r o j e c t i l e i o n s a r e n e a r l y f u l l y - s t r i p p e d of t h e i r e l e c t r o n s 6

.

I n t h i s case we may assume t h a t t h e c h a r g e - s t a t e d i s t r i b u t i o n i s composed of two charge s t a t e s ; t h e f u l l y s t r i p p e d i o n ( l a b e l l e d 0) and t h e one-electron i o n ( l a b e l l e d 1 ) . The r o o t mean square charge i s then given by

where $ . d e n o t e s a c h a r g e - s t a t e f r a c t i o n .

Now l e t t h e r a t i o s o f t h 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 ue and of

t h e e l e c t r o n c a p t u r e c r o s s s e c t i o n U i n t a r g e t s A and B be given by

and

Then i t can e a s i l y be shown t h a t

The combination of eqns. (2) and (5) shows how changes RR and RC i n t h e charge-changing c r o s s s e c t i o n s i n f l u e n c e t h e r o o t mean square charge of t h e p r o j e c t i l e .

(b) P a r t i a l l y - s t r i p p e d i o n s

For v e l o c i t i e s i n t h e r e g i o n v << V << Z1vO, p r o j e c t i l e i o n s i n atomic t a r g e t s possess many bound electrons! A reasonable approximation t o t h e c h a r g e - s t a t e d i s t r i b u t i o n i n t h i s case i s given b y 1

2 -L

+i = ( 2 n ~ ) exp C- (q - < ) 2 / ( 2 ~ 2 ) 1 ( 6 )

where i denotes t h e number of e l e c t r o n s bound t o t h e i o n (q= Z1 - i ) . I n o r d e r t o e s t i m a t e t h e s h i f t i n

<

due t o changes i n t h e e l e c t r o n l o s s and c a p t u r e c r o s s s e c t i o n s , i t i s convenient t o make two supplementary assumptions. F i r s t l y , we assume t h a t t h e s i n g l e - e l e c t r o n l o s s and c a p t u r e c r o s s s e c t i o n s f o r a l l i n i t i a l charge s t a t e s i ( w r i t t e n U ( i ) and o c ( i ) ) ,

R

a r e s t i l l given by eqns. (3) and ( 4 ) i r r e s p e c t i v e of t h e v a l u e of i. I n o t h e r words t h e q u a n t i t i e s RR and R a r e assumed independent of t h e i n i t i a l charge s t a t e . Secondly, we assume t h a t t h e c h a r g e - s t a t e d i s t r i b u t i o n i s dominated by s i n g l e e l e c t r o n l o s s and c a p t u r e p r o c e s s e s , r a t h e r t h a n by double charge-exchange. This assumption i s good f o r high-energy i o n s , where charge-exchange c r o s s s e c t i o n s a r e reasonably s m a l l . It may then be shown t h a t t h e s h i f t i n t h e mean charge s t a t e between t a r g e t s B and 7

A i s given by

For p r e s e n t purposes, i t i s convenient t o use t h e v a l u e of S given by Betz and Schmelzer8, namely

We s h a l l a l s o n e g l e c t t h e s m a l l d i f f e r e n c e between

q

and q r e m e s

,

and hence f e e l f r e e t o apply eqn.

( 6 ) t o q r .m. s

.

3 . E f f e c t of t a r g e t d e n s i t y on charge-changing c r o s s s e c t i o n s I t i s customary t o d e s c r i b e the charge-exchange p r o c e s s i n terms of "charge-changing c r o s s s e c t i o n s " , a s d i s c u s s e d above, t r e a t i n g t h e i o n i c charge s t a t e s a s i f they were w e l l d e f i n e d i n i t i a l and f i n a l s t a t e s Bohr and Lindhard argued i n an e a r l y paper t h a t t h i s procedure l e a d s t o 9 c r o s s s e c t i o n s which a r e dependent on t a r g e t d e n s i t y . I n t h e i r p i c t u r e , a low-density t a r g e t allowed p r o j e c t i l e e x c i t e d s t a t e s (formed e i t h e r by ground-state e x c i t a t i o n o r by e l e c t r o n c a p t u r e ) t o decay r a d i a t i v e l y t o t h e ground s t a t e b e f o r e f u r t h e r c o l l i s i o n s occurred. Conversely i n a dense t a r g e t , they saw t h e s e e x c i t e d s t a t e s a s being r a p i d l y i o n i z e d . This i o n i s a t i o n could occur f o r many-electron ions i n a dense gas t a r g e t as a r e s u l t of r a p i d a u t o i o n i s a t i o n , and i n even denser t a r g e t s such a s s o l i d s i t would occur due t o t h e high frequency of i o n i s i n g c o l l i s i o n s experienced by e x c i t e d p r o j e c t i l e s t a t e s .

I n 1 9 7 0 , Betz and Grodzins s t r e s s e d t h e importance of p o s t - f o i l a u t o i o n i s a t i o n on observed c h a r g e - s t a t e d i s t r i b u t i o n s of i o n s emerging from s o l i d s ? Although t h i s p r o c e s s occurs o n l y i n many-electron i o n s , i t s e f f e c t s on t h e observed c h a r g e - s t a t e s a r e then s o dramatic t h a t d e n s i t y e f f e c t s o c c u r r i n g w i t h i n t h e t a r g e t have tended t o be n e g l e c t e d i n r e c e n t y e a r s . U n c e r t a i n t y t h e r e f o r e remains as t o t h e magnitude of any d e n s i t y e f f e c t s i n t h e charge-changing c r o s s s e c t i o n s .

Since t h e charge-changing process f o r many-electron ions i s

h i g h l y complex: a r i g o r o u s d i s c u s s i o n of t h e d e n s i t y e f f e c t i s r e s t r i c t e d i n t h i s paper t o the case of n e a r l y f u l l y - s t r i p p e d i o n s . However t h e r a t h e r c o n c r e t e r e s u l t s o b t a i n e d i n t h i s way w i l l provide a u s e f u l p o i n t e r towards t h e p o s s i b l e behaviour f o r p a r t i a l l y s t r i p p e d ,

e n e r g e t i c heavy i o n s . To d e r i v e t h e charge-changing c r o s s s e c t i o n s i n d i l u t e and dense media i n terms of fundamental c r o s s s e c t i o n s f o r p r o j e c t i l e e x c i t a t i o n , i o n i s a t i o n and e l e c t r o n c a p t u r e , we c o n s i d e r only t h e f o l l o w i n g p r o j e c t i l e s t a t e s . These a r e t h e f u l l y - s t r i p p e d i o n ( l a b e l l e d 0 ) and a number of s t a t e s of t h e one-electron i o n ( l a b e l l e d l i , where i = 0 r e f e r s t o t h e ground s t a t e and i

>

1 r e f e r s t o e x c i t e d s t a t e s . S t a t e f r a c t i o n s a r e r e p r e s e n t e d by $ 0 o r O l i , and c r o s s s e c t i o n s a r e r e p r e s e n t e d f o r example by ,:U where t h e s u b s c r i p t 0 r e f e r s t o t h e i n i t i a l s t a t e (here t h e f u l l y - s t r i p p e d i o n ) and t h e s u p e r s c r i p t 10 r e f e r s t o t h e f i n a l s t a t e (here t h e ground s t a t e of t h e one-electron i o n )

.

The e v o l u t i o n of t h e s t a t e p o p u l a t i o n s w i t h depth s i n t h e t a r g e t i s governed by t h e t h r e e e q u a t i o n s

where we have assumed f o r t h e sake of formal s i m p l i c i t y t h a t only one e x c i t e d s t a t e , b l l , need be considered. By comparison, t h e more f a m i l i a r r a t e e q u a t i o n s based on charge-changing c r o s s s e c t i o n s a r e 10

given by only two e q u a t i o n s ,

where t h e nomenclature i s t h e same as i n S e c t i o n 2 .

Since t h e e x c i t e d s t a t e

oll

i s more r a p i d l y i o n i s e d than t h e ground s t a t e $10, t h e p o p u l a t i o n $11 may be expected t o a d j u s t r a t h e r promptly t o changes i n 010 and $O. I t i s t h e r e f o r e r e a s o n a b l e t o assume t h a t , a f t e r a s h o r t d i s t a n c e i n t o a t a r g e t , $11 i s i n q u a s i - e q u i l i b r i u m with r e s p e c t t o $10 and $0, and one can then make t h e approximation d$ /dx = 0 i n e q u a t i o n ( 9 ) . I t i s f o r t h i s reason t h a t t h e

11

phenomenological d e s c r i p t i o n given by eqns. (11) and (12) can produce an a c c e p t a b l e approximation t o t h e c h a r g e - s t a t e e q u i l i b r a t i o n p r o c e s s . I t i s now p o s s i b l e t o d e r i v e o and oR i n terms of t h e more fundamental c r o s s s e c t i o n s of e q u a t i o n s (8) and ( 9 ) . S e t t i n g d$ /dx = 0 i n

11 e q u a t i o n (9) y i e l d s

Combining e q u a t i o n s (8) and (13) l e a d s t o the r e s u l t s

These formulae i n c l u d e t h e e f f e c t of r a d i a t i v e decay of t h e 11 s t a t e t o t h e 10 ground s t a t e , i f we d e f i n e u1° t o be an e f f e c t i v e c r o s s s e c t i o n

11

where A l l i s t h e r a d i a t i v e decay r a t e of s t a t e 11, N i s t h e number

d .

d e n s i t y of t a r g e t atoms, v i s t h e i o n v e l o c i t y and oll 1 s t h e c o l l i s i o n a l d e - e x c i t a t i o n c r o s s s e c t i o n from s t a t e 11 t o t h e ground s t a t e . (Since ull i s i n f a c t too small t o b e s i g n i f i c a n t f o r our purposes, i t w i l l be d

dropped i n t h e remaining d i s c u s s i o n ) . The r e l a t i v e importance f o r t h e charge-changing c r o s s s e c t i o n s of e x c i t e d - s t a t e i o n i s a t i o n and r a d i a t i v e d e - e x c i t a t i o n , depends on t h e v a l u e of N , t h e t a r g e t d e n s i t y . I n t h e l i m i t i n g c a s e of a low-density t a r g e t (N + O), we have o1° + m and hence

11

We s h a l l c a l l t h i s t h e ' g a s t a r g e t ' case although i n r e a l i t y i t only a p p l i e s t o reasonably d i l u t e gases. I n t h e l i m i t i n g c a s e N + a p p r o p r i a t e t o a s o l i d ) 5:: 'O and hence

Thus e q u a t i o n s (14)

-

(16) a r e a p p l i c a b l e t o both gas and s o l i d t a r g e t s , w i t h q = 0 and Q as given by equation (16b). S u b s t i t u t i o n

g

of TI = I-

"

i n e q u a t i o n s ( 1 4 ) and (15) y i e l d s t h e d i f f e r e n c e between g'

t h e s o l i d and gas t a r g e t charge-changing c r o s s s e c t i o n s , Ao = (5)solid-(5)gas, i n t h e very simple form

It may be shown t h a t t h e r e i s good j u s t i f i c a t i o n f o r i n c o r p o r a t i n g many e x c i t e d s t a t e s i n t o t h e p r e s e n t d e s c r i p t i o n , simply by adding t h e i r c o n t r i b u t i o n s t o Aoc and AuQ l i n e a r l y . For n e x c i t e d s t a t e s we have, dropping t h e s u b s c r i p t s from 0

S '

The corresponding g a s - t a r g e t c r o s s s e c t i o n s simply g e n e r a l i s e t o

The r e s u l t s p r e s e n t e d i n eqns. (21 - 25) give d e f i n i t e meaning t o t h e charge-changing c r o s s s e c t i o n s i n both gases and s o l i d s . The d e n s i t y e f f e c t , eqns. (21

-

23) c l e a r l y r e f l e c t s t h e q u a l i t a t i v e

l i

arguments of Bohr and Lindhard. The a. term i n eqn. (21) corresponds t o t h e p h y s i c a l f a c t t h a t c a p t u r e t o an e x c i t e d s t a t e i n a s o l i d i s followed r a p i d l y by c o l l i s i o n a l i o n i s a t i o n , a channel which does n o t

l i

e x i s t i n a d i l u t e gas. The o10 term i n eqn. (22) corresponds t o t h e a d d i t i o n a l channel i n t h e s o l i d - t a r g e t l o s s c r o s s s e c t i o n due t o e x c i t a t i o n followed by r a p i d i o n i s a t i o n . The numerical q u a n t i t y ?l i

(17;

<

1) accounts p h y s i c a l l y f o r t h e time d e l a y between t h e formation of t h e e x c i t e d s t a t e i and i t s subsequent i o n i s a t i o n . ?l. i s u n i t y i n

0 11

t h e l i m i t i n g case of oli +

-,

a s has been d e s c r i b e d p r e v i o u s l y . Some u s e f u l g e n e r a l o b s e r v a t i o n s can be made about e x c i t a t i o n and i o n i s a t i o n c r o s s s e c t i o n s , from which a q u a n t i t a t i v e e s t i m a t e . of t h e d e n s i t y e f f e c t r a t i o i n t h e l o s s c r o s s s e c t i o n o R can be made. The t h e o r y of e l e c t r o n l o s s from f a s t one-electron i o n s i n d i c a t e s t h a t t h e

0 0

i o n i s a t i o n c r o s s s e c t i o n s oI0, uli a r e i n v e r s e l y p r o p o r t i o n a l t o t h e b i n d i n g energy of t h e i n i t i a l s t a t e 6 Evidence from r e s o n a n t coherent e x c i t a t i o n of channelled i o n s has i n d i c a t e d t h a t a f i n i t e number of 12

i o n i c bound s t a t e s can e x i s t w i t h i n a s o l i d , and t h a t t h e b i n d i n g e n e r g i e s f o r t h e low-lying e x c i t e d s t a t e s a r e only moderately p e r t u r b e d

from t h e i r vacuum e q u i v a l e n t s . F u r t h e r evidence f o r t h i s view comes from t h e r e c e n t o b s e r v a t i o n s of non-equilibrium e f f e c t s i n t h e proton n e u t r a l f r a c t i o n emerging from s o l i d s bombarded w i t h MeV H beams, which s u g g e s t t h a t even f o r protons a s t r o n g l y c o r r e l a t e d proton-

1 3

e l e c t r o n c o n f i g u r a t i o n can e x i s t i n a s o l i d . Consequently i f one i d e n t i f i e s t h e s t a t e l a b e l i with (n - l ) , where n i s t h e p r i n c i p a l quantum number, one has

I n o r d e r t o e s t i m a t e d e n s i t y e f f e c t s , a d d i t i o n a l i n f o r m a t i o n i s needed on t h e f r a c t i o n of t h e ground-state d e s t r u c t i o n c r o s s s e c t i o n which l e a d s t o e x c i t a t i o n of t h e one-electron i o n , a s opposed t o

i o n i s a t i o n . P r e d i c t i o n s f o r t h i s i o n i c e x c i t a t i o n f r a c t i o n f o r c o l l i s i o n s i n which t h e t a r g e t atom undergoes e x c i t a t i o n s i m u l t a n e o u s l y , have been made by G i l l e s p i e using t h e Born approximationt4 The e x c i t a t i o n f r a c t i o n

v a r i e s from about 0.2 f o r H i o n s t o n e a r l y 0.6 f o r i o n s with Z = 30, and i s only a very weak f u n c t i o n of t a r g e t atomic number. For i l l u s t r a t i v e p u r p o s e s , t h e r e f o r e , we t a k e t h e v a l u e s of f e x from r e f . 14 f o r t h e c a s e of an N t a r g e t atom, and use e q u a t i o n s (26) and ( 2 7 ) t o e s t i m a t e l i m i t s t o t h e d e n s i t y - e f f e c t r a t i o DL - - ('L)solidf (aR)gas-

The r e s u l t s a r e shown i n F i g . 1 a s a f u n c t i o n of i o n atomic number Z1. The upper s o l i d curve corresponds t o t h e assumption t h a t a l l e x c i t a t i o n l e a d s t o h i g h l y e x c i t e d s t a t e s ( n + m ) , while t h e lower s o l i d curve corresponds t o t h e assumption t h a t a l l e x c i t a t i o n l e a d s t o t h e n = 2 s t a t e . Since t h e t r u e s i t u a t i o n l i e s somewhere between t h e s e two extremes, we a r b i t r a r i l y choose an "adopted value" curve f o r Fig. 1, t h e dashed l i n e , by drawing a s t r a i g h t l i n e through t h e average of t h e extreme curves a t Z1 = 3 and Z1 = 30.

2 5 10 2 0 Ion atomic number, Z,

F i g . 1 The d e n s i t y e f f e c t i n t h e l o s s c r o s s s e c t i o n f o r o n e - e l e c t r o n i o n s , a s a f u n c t i o n of i o n a t o m i c number Z1. The u p p e r and lower s o l i d c u r v e s r e p r e s e n t e x t r e m e t h e o r e t i c a l e s t i m a t e s a s d e s c r i b e d i n t h e t e x t of S e c t i o n 3 . The dashed l i n e r e p r e s e n t s a n "adopted v a l u e " c u r v e a s d e s c r i b e d i n t h e t e x t . The e x p e r i m e n t a l datum comes from r e f . 1 5 , c o n c e r n i n g d e n s i t y e f f e c t s f o r 3 MeV/u C i o n s i n g a s e o u s and s o l i d carbon t a r g e t s .

F i g u r e 1 a l s o shows e x p e r i m e n t a l d a t a o b t a i n e d u s i n g 3 MeV/u 1 5

C i o n s i n c i d e n t on gaseous and s o l i d C t a r g e t s . (The gas t a r g e t r e s u l t s were o b t a i n e d by e x t r a p o l a t i n g d a t a o b t a i n e d w i t h a number of hydrocarbon g a s e s o f d i f f e r e n t s t o i c h i o m e t r y ) . The t h e o r e t i c a l p r e d i c t i o n s a r e s e e n t o b e i n e x c e l l e n t agreement w i t h t h i s e x p e r i m e n t a l r e s u l t .

As r e g a r d s t h e d e n s i t y e f f e c t i n t h e e l e c t r o n c a p t u r e c r o s s 1 6

s e c t i o n , t h e c o m p l e x i t y i n t r o d u c e d by v e l o c i t y m a t c h i n g r u l e s o u t t h e p r o s p e c t o f a s t r a i g h t f o r w a r d g e n e r a l r e s u l t f o r t h e d e n s i t y - e f f e c t r a t i o D

.

S u f f i c e i t t o s a y t h a t s i n c e D 1 ( e q u a t i o n s ( 2 1 ) , ( 2 4 ) ) , t h i s e f f e c t can o n l y i n c r e a s e t h e d e n s i t y - e f f e c t i n t h e r.m.s. c h a r g e ( e q u a t i o n s ( 5 ) , ( 6 ) ) a l r e a d y a r i s i n g from t h e enhanced l o s s c r o s s s e c t i o n

4. I m p l i c a t i o n s f o r t h e e f f e c t i v e c h a r g e ( a ) D e n s i t y e f f e c t s

For n e a r l y f u l l y - s t r i p p e d i o n s , a d e n s i t y e f f e c t i n t h e c h a r g e - c h a n g i n g c r o s s s e c t i o n s a s e x e m p l i f i e d by F i g . 1 f o r t h e l o s s c r o s s s e c t i o n , w i l l h a v e o n l y a modest e f f e c t o n q

r.m. S .

.

The l a r g e s t e f f e c t f o r n e a r l y f u l l y - s t r i p p e d i o n s o c c u r s f o r l i g h t i o n s , s i n c e t h e c h a r g e d i f f e r e n c e between t h e f u l l y - s t r i p p e d i o n and t h e o n e - e l e c t r o n i o n i s t h e n a n a p p r e c i a b l e f r a c t i o n o f q

.

F o r example, u s i n g R -

r.m.s. L - D~

a s g i v e n by F i g . 1, and making t h e c o n s e r v a t i v e a s s u m p t i o n R = DC = 1, e q u a t i o n s ( 2 ) and (5) y i e l d t h e f o l l o w i n g p r e d i c t i o n s f o r t h e d e n s i t y e f f e c t i n t h e r . m . s . c h a r g e , D =

4 (qr.m. s . ) s o l i d ' ( q r . m . s . ⁾ g a s .

.

^When

t h e c h a r g e f r a c t i o n s @O and a r e e q u a l , one o b t a i n s f o r C i o n s , D = 1 . 0 2 ; and f o r Li i o n s , D = 1 . 0 4 . F o r H and He i o n s , t h e d e n s i t y

q 4

e f f e c t i s u n l i k e l y t o o p e r a t e a s d e s c r i b e d by o u r model, s i n c e t h e r e a r e p r o b a b l y no bound e x c i t e d s t a t e s i n a s o l i d . F o r t h e s e i o n s , any d e n s i t y e f f e c t i s more l i k e l y t o r e s u l t from t h e m o d i f i c a t i o n o f t h e p r o j e c t i l e g r o u n d - s t a t e b i n d i n g e n e r g y a s a r e s u l t of s c r e e n i n g by

1 7 t a r g e t e l e c t r o n s

.

For p a r t i a l l y s t r i p p e d , m a n y - e l e c t r o n i o n s a t MeV/u e n e r g i e s , e q u a t i o n s (6) and (7) a r e v a l i d . Combining t h e s e e q u a t i o n s , one o b t a i n s

( a g a i n assuming D = l ) ,

A p p l i c a t i o n of t h e d e n s i t y e f f e c t D _.E p r e s e n t e d i n F i g . 1 t o such i o n s i s n o t s t r i c t l y v a l i d . However i n t h e a b s e n c e of more r e l i a b l e p r e d i c t i o n s i t forms a w o r t h w h i l e b a s i s f o r e s t i m a t i n g h i g h energy heavy-ion e f f e c t i v e c h a r g e s , e s p e c i a l l y s i n c e v e r y l i t t l e e x p e r i m e n t a l

dE/dx d a t a i s a s y e t a v a i l a b l e f o r t h e s e i o n s .

As a n i l l u s t r a t i o n , we a p p l y t h i s e s t i m a t e t o t h e c a s e of 3.6 and 7 . 9 MeV/u U i o n s , t h e o n l y example t o d a t e of a high-energy heavy i o n

1 8 f o r which b o t h g a s and s o l i d - t a r g e t s t o p p i n g powers have b e e n measured

.

The p r e d i c t i o n i s

AT

⁼ 4.6 a t 3.6 MeV/u and A< = 4.9 a t 7 . 9 MeV/u f o r U i o n s , i n e x c e l l e n t agreement w i t h t h e e x p e r i m e n t a l d a t a . These v a l u e s c o r r e s p o n d t o d i f f e r e n c e s o f about 20% between t h e s t o p p i n g powers i n s o l i d and g a s e o u s t a r g e t s .

So f a r we h a v e c o n s i d e r e d o n l y t h e d e n s i t y e f f e c t on t h e i o n e f f e c t i v e c h a r g e a f t e r t h e i o n h a s r e a c h e d c h a r g e - s t a t e e q u i l i b r i u m . However, a s Fig. 1 shows, t h e s t r i p p i n g r a t e f o r an i o n i n c i d e n t i n a low c h a r g e s t a t e may b e e x p e c t e d t o b e s u b s t a n t i a l l y h i g h e r i n a dense medium t h a n i n a d i l u t e g a s . F o r example, on t h e b a s i s of measured charge-changing c r o s s s e c t i o n s f o r 3 ~ e V / u C i o n s i n gaseous and s o l i d C t a r g e t s , one e x p e c t s e q u i l i b r i u m t o b e approached % 1.6 t i m e s f a s t e r i n t h e s o l i d t a r g e t s . C o n s i d e r a t i o n s of t h i s k i n d may b e

of i m p o r t a n c e i n t r e a t i n g t h e a b l a t i o n b e h a v i o u r of ion-beam h e a t e d ICF t a r g e t s .

( b ) Plasma t e m p e r a t u r e e f f e c t s

D e t a i l e d d i s c u s s i o n o f plasma t e m p e r a t u r e e f f e c t s on t h e e f f e c t i v e c h a r g e h a s b e e n g i v e n r e c e n t l y by N a r d i and ZiLlamon 19

.

However f o r c o m p l e t e n e s s we d e s c r i b e b l - i e f l y how s u c h e f f e c t s come a b o u t from t h e s i m p l i f i e d v i e w p o i n t g i v e n i n S e c t i o n 2 . E q u a t i o n ( 6 ) p r e d i c t s f o r a p a r t i a l l y - s t r i p p e d i o n t h e s h i f t i n t h e e f f e c t i v e c h a r g e a r i s i n g from changes i n t h e u n d e r l y i n g charge-changirig c r o s s s e c t i o n s . R and R now r e p r e s e n t t h e r a t i o s of t h e e l e c t r o n l o s s ( R )

R

and c a p t u r e ( c ) c r o s s s e c t i o n s i n t h e plasma and t h e c o l d t a r g e t s . I t i s c l e a r t h a t , u n l e s s some d r a m a t i c e f f e c t o c c u r s i n e i t h e r RI o r RC,

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

AT

^{w i l l}

b e modest. The i m p o r t a n t p h y s i c s b e h i n d t h e t e m p e r a t u r e e f f e c t i s t h a t , i n a h o t p l a s m a , t h e e l e c t r o n c a p t u r e c r o s s s e c t i o n may i n d e e d b e d r a m a t i c a l l y r e d u c e d 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 r e l e v a n t t o e l e c t r o n c a p t u r e may no l o n g e r b e bound. Thus i n s t e a d o f d i r e c t c a p t u r e o f bound e l e c t r o n s a s i n c o l d m a t t e r , c a p t u r e may o n l y p r o c e e d by REC o r by d i e l e c t r o n i c r e c o m b i n a t i o n , p r o c e s s e s whose c r o s s s e c t i o n s a r e o r d e r s o f magnitude s m a l l e r . As a r e s u l t , plasma t e m p e r a t u r e e f f e c t s i n t h e e f f e c t i v e c h a r g e can make a s u b s t a n t i a l c o n t r i b u t i o n t o w a r d s e n h a n c i n g t h e s t o p p i n g power i n h o t p l a s m a s .

R e f e r e n c e s

1. J . L i n d h a r d , N u c l . I n s t r . and Mechods

132,

1 ( 1 9 7 6 ) .

2. J . F . Z i e g l e r , The S t o p p i n g and Ranges o f I o n s i n M a t t e r , Vol. 5 , Pergamon P r e s s , New York ( 1 9 8 0 ) .

3 . G.Maynard and C.Deutsch, Phys.Rev. A

2,

665 (1982)

4. I n p r o c e s s e s where s m a l l i m p a c t p a r a m e t e r s a r e o f Idominant

i m p o r t a n c e , s u c h a s t a r g e t i n n e r - s h e l l i o n i s a t i o n o f heavy t a r g e t s , some c o n s i d e r a t i o n o f t h e form f a c t o r o f t h e i o n h a s t o be i n c l u d e d . T h i s , however, i s n o t t h e c a s e f o r s t o p p i n g powers o f MeV/u i o n s . S e e f o r example W.Brandt a n d M.Kitagawa, Phys-Rev. B

5,

5631 ( 1 9 8 2 ) . H.D.Betz and L - G r o d z i n s , Phys.Rev. L e t t .

25,

211 ( 1 9 7 0 ) .

N.Bohr, Kgl.Dan.Vid.Selsk,Mat.-Fys.Medd.

c,

No. 8 (1948)

H.D.Betz, Rev.Mod.Phys. 4 4 , 465 ( 1 9 7 2 ) .

H.D.Betz and C.R.Schmelzer, UNILAC R e p o r t 1-67.

N.Bohr and J . L i n d h a r d , Kgl.Dan.Vid.Selsk.Mat.-Fys-Medd.

26,

No.12 (1954) S . K . A l l i s o n , Rev.Mod.Phys.

E,

1137 ( 1 9 5 8 ) .

N.E.B.Cowern, D . P h i l t h e s i s , Oxford ( 1 9 7 9 ) ; UK Atomic Energy R e p o r t AERE R-9500 (1979).

C.D.Moak e t a l . Phys.Rev. A

19,

977 ( 1 9 7 9 ) ; N u c l . I n s t r . and Methods

194,

327 ( 1 9 8 2 ) .

13. M.J.Gaillard, J.C.Poizat, A.Ratkowski, J.Remillieux and M.Auzas, Phys.P.ev. A - 16, 2323 (1977).

14. G.H.Gillespie, Phys.Rev. A

18,

1967 (1978).

15. C.J.Woods, D.Phi1. thesis, Oxford (1983); UK Atomic Energy Report R-10795 (1983).

16. See for example G.Lapicki and W.Losonsky, Phys.Rev. A

15,

896 (1977) 17. W.Brandt, Atomic Collisions in Solids,

1,

261 (Plenum Press, New

York, 1975) 18. H.Geisse1, Y.Laichter, W.F.W.Schneider and P.Armbruster,

Nucl.Instr. and Methods

194,

21 (1982).

19. E.Nardi and Z-Zinamon, Phys.Rev.Lett.

49,

1251 (1982).