SWIFT ION ENERGY LOSSES IN DENSE PLASMAS

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SWIFT ION ENERGY LOSSES IN DENSE PLASMAS

Yu. Sayasov

To cite this version:

Yu. Sayasov. SWIFT ION ENERGY LOSSES IN DENSE PLASMAS. Journal de Physique Colloques,

1983, 44 (C8), pp.C8-1-C8-15. �10.1051/jphyscol:1983801�. �jpa-00223308�

JOURNAL DE PHYSIQUE

Colloque C8, supplement au n O 1 l , Tome 44, novembre 1983 page C8-1

SWIFT I O N ENERGY LOSSES I N DENSE PLASMAS

I n s t i t u t f u r Kernphysik II, Kernforschitngszentmun KarZsruhe Gmbii, F.R.G.

Resume

On e t u d i e l e r a l e n t i s s e m e n t d ' i o n s r a p i d e s dans l e s plasmas non-ideaux e t degeneres

a l ' a i d e de t h G o r i e s du champ l o c a l . On montre que l e s e f f e t s de c o r r e l a t i o n d i f f e - r e n c i e n t l e s champs moyens des champs l o c a u x . Lorsque l a v i t e s s e de l ' i o n p r o j e c t i l e e s t t r e s i n f e r i e u r e a l a v i t e s s e t h e r m i q u e des e l e c t r o n s du plasma, l e p o u v o i r d ' a r r e t e s t m u l t i p l i e p a r 1+(2g/9), g e t a n t l e paramPtre plasma c l a s s i q u e . Oans l e s plasmas degeneres, l e p o u v o i r d l a r r @ t augmente de quelques d i z a i n e s de pourcents,

a basse v i t e s s e ( v < < vF), e t meme d ' u n f a c t e u r 2-3 dans c e r t a i n s metaux.

A b s t r a c t

S w i f t i o n energy l o s s e s a r e c a l c u l a t e d f o r n o n i d e a l c l a s s i c a l and degenerate quantum plasmas on t h e b a s i s o f l o c a l f i e l d t h e o r i e s . I t i s shown t h a t c o r r e l a t i o n a l e f f e c t s l e a d i n g t o a d i f f e r e n c e between average f i e l d s and l o c a l f i e l d s may i n f l u e n c e essen- t i a l l y energy l o s s e s i n such plasmas. I n p a r t i c u l a r , f o r i o n s whose v e l o c i t y v i s much s m a l l e r t h a n t h e plasma e l e c t r o n thermal v e l o c i t y t h e s t o p p i n g power i n c r e a s e s ( f o r c l a s s i c a l plasmas) by a f a c t o r 1 + ( 2 g / 9 ) , g i s t h e c l a s s i c a l plasma parameter, as con]- pared w i t h Vlasov a p p r o x i m a t i o n . f o r d e ~ e ~ i e r a t e quantum plasmas t h e s t o p p i n 9 power i n - creases (under a s i m i l a r assumption v << v F, vF i s t h e Fermi v e l o c i t y ) i n r e s u l t o f t h e l o c a l f i e l d e f f e c t s by a f a c t o r w h i c h can r e a c h a few t e n s o f p e r c e n t (as com- p a r e d w i t h L i n d h a r d s t o p p i n g power) f o r some compressed ICF plasmas. Corresponding f a c t o r s f o r n o n i d e a l d e g e n e r a t e plasmas i n some s i m p l e m e t a l s a r e as much as 2-3 and a r e i n agreement w i t h e x p e r i m e n t .

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

C a l c u l a t i o n s of t h e s w i f t i o n energy l o s s e s i n dense gas-plasmas have r e c e n t l y g a i n e d importance i n c o n n e c t i o n w i t h t h e problems o f i n e r t i a l c o n f i n e m e n t f u s i o n 1 ^1,2,3].

These c a l c u l a t i o n s as we1 l as e x i s t i n g r e v i e w s ( s e e e.g. A k h i e s e r , 4 1 ) a r e r e s t r i c t e d by t h e assumption t h a t t h e plasmas a r e i d e a l , i . e . t h a t t h e y a r e c h a r a c t e r i z e d .by a

4 3

s m a l l plasma parameter g = l/- nr o, where r d = ( ~ / 4 n n e ~ ) l " i s t h e Debye-length, n i s

3 d

t h e e l e c t r o n d e n s i t y . T i s t h e plasma t e m p e r a t u r e , ( f o r c l a s s i c a l plasmas). F o r quantum

*on l e a v e from t h e U n i v e r s i t y of F r i b o u r g , S w i t z e r l a n d

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983801

C8-2 JOURNAL DE PHYSIQUE

plasmas t h e energy l o s s e s were c o o r i d e r e d under a s i m i l a r assumption r s = ( 3 / 4 7 n a 3 ) 1 / 3 < < l i13,14]. (Here a i s t h e e o h r r a d i u s ) . The assumption o f t h e i d e a l p l a s c a can be v i g l z t e a f o r some cases o f I C F plasmas. O f p a r t i c u l a r i n t e r e s t i n t h i s r e s p e c t a r e t h e I C F plasma!

produced a t t h e i n i t i a l s t a g e s o f t h e i m p l o s i o n processes. As an example we can c i t e deu.

t e r i u m - t r i t i u m plasmas formed a t two d i f f e r e n t moments i n s i d e o f t h e v o i d - c o n t a i n i n g p e l l e t a c c o r d i n g t o c a l c u l a t i o n s d e s c r i b e d i n [ 51.

H i g h l y i o n i z e d a b l a t o r plasrnas o r t h e plasmas formed by i r r a d i a t i o n c f m e t a l l i c f o i l s w i t h i n t e n s i v e i o n beams, can posses ;In even h i g h e r degre.3 o f n o n i d e a l i t y . Another examples o f i n t e r e s t a r e n o n - i d e a l plasmas produced i n s t r o n g snock waves[8].

I t seems t h a t a t h e o r e t i c a l i n v e s t i g a t i o n o f t h e charged p a r t i c l e energy l o s s e s i n such n o n - i d e a l plasmas has n e v e r been performed. (J.n e x c e p t i o n a r e papers o f N a r d i e t a l . , 6 1 and of ~ e h l h o r n , [ 7 l w h i c h , however, g i v e r i s e t o some o b j e c t i o n s ; see b . 1 0 ~ ) . A complete i n v e s t i g a t i o n o f t h i s p r o b l e m s e m s t o be i n p o s s i b l e a t t h e mo- ment, s i n c e t h e t h e o r y o f s t r o n g l y n o n - i d e a l plasnias lia; n n t y e t Seen developed -

s u f f i c i e n t l y . However, f o r n o t t o o l a r g e p l a s ~ i i a p a r a r e t 2 r ; ( g < 9 o r r < 6 ) one s can s t i l l o b t a i n some g e n e r a l r e s u l t s w h i c h a r e de:criSed i n t h i s a r t i c l e .

I n t h e framework o f t h e d i e l e c t r i c f o r m a l i s n i we w i l l use t h r o u g h o u t (see 2.9.

~ k h i e s e r t 4 1 p. 231) t h e energy l o s s o f an i o n dU/du h a v i r g t h e energy W v e l o c i t y

-P

v, charge Ze, w i t h i n a n i n t e r v a l dx o f i t s s t r a i g h t o a t h , i s g i v e n by

where E & ( D I , ~ ) i s t h e l o n g i t u d i n a l d i e l e c t r i c f u n c t i o n ~f t h e p l ? s n a . The e x p r e s s i o n (1) d e s c r i b e s c o l l e c t i v e e x c i t a t i o n (Gun t o i n t e r a c t i o n o f t h e p r o j e c t i l e w i t h t h e plasma e l e c t r o n s ) o f t h e l o n g i t u d i n a l plasma waves h;.,; ?g t k e wave v e c t o r 2 and t h e frequency U. I n p r i n c i p l e , c o l l e c t i v e e x c i t a t i o n o f t h e t r d n s v e r s j l e l e c t r o m a g n e t i c waves (Cherenkov e f f e c t ) as a r e s u l t o f t h e p r o j e c t i l e ~-\(;ticl; < n t l i e plasma can o c c u r as we1 l . R e s t r i c t i n g o u r s e l v e s t o n o n - r e l a t i v i s t ~ c >rl?,jccci l e v e l o c i t i e s we can d i s r e g a r d t h i s e f f e c t . I t w i l l be assumed, however, t h d t t h e p r o j e c t i l e i o n s a r e s w i f t enough, i . e . v i , (m / ~ n . ) l / ~ u ~ , U = ( z T / ~ T I ~ ) ~ / ~ b ~ i a g t h e thermal v e l o c i t y

e i e

of t h e plasma e l e c t r o n , mi t h e mass o f t h e plasma i o r ~ s . T h i s assumption a l l o w s t o

n e g l e c t t h e i n t e r a c t i o n between t h e p r o j e c t i l e and t h e plasma i o n s . O t h e r w i s e t h e

formula ( 1 ) i s v e r y g e n e r a l and w i t h a p r o p e r d e f i n i L i o n o f t h ~ ? d i e l e c t r i c f u n c t i o n

cL i s v a l i d a l s o f o r t h e n o n - i d e a l plasmas.

I f one d e f i n e s cl i n ( 1 ) via t h e Vlasov equation v a l i d in zero approximation in the plasma parameter, one g e t s , n a t u r a l l y , the expre5sion f o r (dW/dx) Valid i n the

0

same approximation. May [ 221 c a l c u a l ted (dW/dx), i n a divergence-free manner while taking i n t o accoutn t h e quantummechanical e f f e c t s a r i s i n q i f t h e wavelength l / k of t h e longitudinal wave becomes comparable to the plasma e l e c t r o n de Broglie wave- length f ~ / n i ~ ~ ; ~ According to t h i s paper.

where

and ,20(v/u,) i s a Csu:cri~S 1ogari:iim tabulated by May 22 which can be defined a s follows

The f u n c t i o n f ( v / u e ) ? r s il'e asymptotocis f

l n ( l / ~ y e ) ' / ~ (y - i s t h e E u l e r l i c o n s t a n t ) f o r v -. L . ~ 311d f - 1rl(v/ue) f o r 2 v > ue. This d e f i n i t i o n of t h e Coulomb logs!-ithni i s 1'6; iti i f the p a r t i c l e v e l o c i t y v s a t i s f i e s t h e condition v > Ze / h . The f b n c t i c n s G(v,'u 2 ) and f ( v / u , j a r e displayed i n Fig. 1.

e

To take i n t o eicc!lnt the i ! ? f l u e n c ~ of the c o l l i s i o n a l e f f e c t s on dW/dx Nardi e t a l . [ ⁶¹ and llehlhorn! 7 1 introduced t h e d i e l e c t r i c function cl defined by

the Boltzmann e q a a t i c n wlth t n e c o l l i s i o n i n t e g r a l I defined i n t h e tau-approxi- mation

Here ; i s t h e ~ i e c t r o n v e l o c i t y , f i s the e l e c t r o n d i s t r i b u t i o n f u n c t i o n , f o i s t h e Yaxwell d i s t r i b u t i , ) n f u n z t i o n , v = const i s t h e c o l l i s i o n frequency.

e

As follows from ( S ) , the corresponding d i e l e c t r i c function cL i s given by

(1231, P . 1 9 ) :

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F i g . 1: F u n c t i o n s G(v/u ) and f ( v / u e ) e n t e r i n g t h e f o r m u l a ( 2 ) f o r t h e energy l o s s e s i n a n i d g a l plasnla.

2

where k d = 2"' u p / u e r and Z ( b ) = 7: -'I2 ^-m r e-Z d z / ( z - 5 ) i s t h e plasma d i s p e r s i o n f u n c t i o n . We must n o t e f i r s t o f a l l t h a t f o r 1.51 >> 1, i . e . f o r v >> U, , t h e d i e l e c t r i c f u n c t i o n ( 6 ) possesses wrong a s y m p t o t i c s .

i n s t e a d o f t h e a s y m p t o t i c s

-.

w h i c h would f o l l o w f r o m ( 5 ) w i t h a c o r r e c t c o l l i s i o n i n t e g r a l I. ( T h i s de- f i n c i e n c y o f ( 6 ) i s connected w i t h t h e f a c t t h a t t h e c o l l i s i o n i n t e g r a l i n

( 5 ) does n o t s a t i s f y t h e r e q u i r e m e n t 1.; I d; = O ) . * Moreover, t h e u s e of t h e Boltzmann e q u a t i o n ( 5 ) r i s e s i n i t s e l f f o l l o w i n g o b j e c t i o n s . T h i s e q u a t i o n

* A b e t t e r d e s c r i p t i o n o f c o l l i s i o n a l e f f e c t s can be o b t a i n e d w i t h a h e l p of

c o l l i s i o n a l i n t e g r a l o f Bhatnagar-Gross-Krook ( s e e Appendix).

a l l o w s , a t t h e b e s t , t o t a k e i n t o a c c o u n t t h e b i n a r y c o l l i s i o n a l e f f e c t s o f t h e o r d e r o f (v,/w ) g'. However, i n plasmas c o r r e l a t i o n a l e f f e c t s o f t h e same o r d e r i n g e x i s t i n a d d i t i o n . These e f f e c t s l e a d t o a d i f f e r e n c e between P t h e average f i e l d f e n t e r i n g ( 5 ) and an e f f e c t i v e f i e l d Cff a c t i n g on t h e e l e c t r o n . A c c o r d i n g t o Kadomtsev [ 91, K l i m o n t o v i c h [ l 0 1 t h i s l o c a l e f f e c t i v e f i e l d zeff d i f f e r s from i! b y t h e f a c t o r 1-6, where 6 = g / ~ ' / ~ 9 ( f o r ^U -. 0 ) . Re- p l a c i n g i n ( 5 ) b y feff we g e t ( K l i m o n t o v i c h , ( 1 0 1 , p. 189) f o r v >> ue

(Here o i s t h e plasma c o n d u c t i v i t y ) . Thus, t o t a k e i n t o a c c o u n t c o r r e c t l y t h e i n f l u e n c e o f plasma n o n i d e a l i t y on t h e energy l o s s e s dN/dx, one must d e f i n e t h e d i e l e c t r i c f u n c t i o n E ~ by , t h e Boltzmann e q u a t i o n ( 5 ) w i t h an e f f e c t i v e f i e l d teff i n t r o d u c e d i n s t e a d o f ?.

A g e n e r a l d e s c r i p t i o n o f d i e l e c t r i c p r o p e r t i e s of t h e n o n i d e a l plasmas v a l i d f o r any f r e q u e n c i e s ( i . e . f o r any p r o j e c t i l e v e l o c i t i e s ) seems t o be i m p o s s i b l e a t t h e moment. T h e r e f o r e we r e s t r i c t o u r s e l v e s h e r e i n s e c t . 2, w i t h c a l c u l a t i o n of t h e s t o p p i n g power o f t h e c l a s s i c a l n o n i d e a l plasmas f o r p r o j e c t i c l e v e l o c i - t i e s v u u s i n g t h e d i e l e c t r i c t h e o r y o f S i n g w i [ 111 w h i c h i s w e l l s u i t e d

e

i n t h i s case. I n t h i s way we a r r i v e a t t h e c o n c l u s i o n t h a t plasma n o n i d e a l i t y l e a d s t o some enhancement o f t h e s t o p p i n g power. I n a s i m i l a r way u s i n g t h e t h e o r y o f S i n g w i Ill] we a r r i v e a t c o n c l u s i o n i n s e c t . 3 t h a t i n d e g e n e r a t e quantum plasmas t h e l o c a l f i e l d e f f e c t s l e a d t o a n i n c r e a s e o f s t o p p i n g power as compared w i t h L i n d h a r d a p p r o x i m a t i o n [ 13,141.

2. C l a s s i c a l Plasmas

A c c o r d i n g t o S i n g w i 11 t h e l o n g i t u d i n a l d i e l e c t r i c f u n c t i o n f o r a system o f e l e c t r o n s embedded i n a s t a t i c u n i f o r m b a c k g r o u n d formed b y i o n s can b e f o r m u l a t e d as f o l l o w s . I n t h e e x a c t e q u a t i o n f o r t h e o n e - p a r t i c l e d i s t r i b u t i o n f u n c t i o n fl(;,j;.t) ( f i r s t e q u a t i o n o f t h e BBGKY h i e r a r c h y )

2 + +

( @ = e /Ir-r'l a n d Fe, i s an e x t e r n a l f i e l d ) , t h e two p a r t i c l e d i s t r i b u t i o n + + + +

f u n c t i s n f 2 ( r , p , r ' . p a , t ) i s assumed t o have t h e f o r m f 2 = fl(?,;,t).

f

, t ) Y(?-?I ^). H e r e '+'(;-F1) i s a s t a t i c p a i r c o r r e l a t i o n f u n c t i o n accoun- 1

t i n g f o r t h e c o r r e l a t i o n a l e f f e c t s . (The V l a s o v a p p r o x i m a t i o n c o r r e s p o n d s t o

CB-6 JOURNAL DE PHYSIQUE

t h e a s s u n p t i o n ?'(;-;')=l). As follows from equation ( 1 0 ) s i m p l i f i e d with t h i s

"Ansatz" and l l n e a r i z e d i n t h e usual f a s h i o n ( f l = fo-:f, f i i f o )

t h e l o n g i t u d i n a l d i e l e c t r i c f u n c t i o n t a k e s t h e form

The f u n c t i o n G(Z) i n ( 1 2 ) i s defined by t h e s t r u c t u r a l f a c t o r S ( ; ) :

tF

S(;) = 1 + n i d r (I(;) - ^{1) (13)}

which i s connected w i t h cL(0,X) through a r e l a t i o n

P u t t i n g i n (12) G = 0 we recover the Vlasov approxinlation f o r e t = 0 = 1 + Q .

0

The appearanie i n ( 1 2 ) of t h e denoninator l-G(k)Qo takes i n t o account t h e loca!

f i e l d e f f e c t s . Note t h a t ( 1 2 ) i s s i m i l a r t o t h e well-known Lorentz-Lorenz f o r - mula d e s c r i b i n g t h e s e e f f e c t s in t h e theory of d i e l e c t r i c a

Here a ( w ) i s t h e p o l a r i s a b i l i t y of d i e l e c t r i c ( S l a t e r . ~ r a n k 1 1 5 1 p. 1 1 5 ) .

R e l a t i o n ( 1 4 ) allows to c z l c u l a t e t h e p a i r - c o r r e l a t i o n f u n c t i o n '?(F - F ' ) and thus the d i e l e c t r i c f u n c t i o n E a. ( a , i f ) . This program was executed n u n i e r i c a l l j t.j Berggren (1970), f o r a broad i n t e r v a l of g-values. On t h e o t h e r hand, the func- t i o n ~ ( i ) can be r e p r e s e n t e d a n a l y t i c a l l y w i t h t h e help of an i t e r a t i o n pro- cedure (Kalman 1173 ) .

We w i l l r e s t r i c t o u r s e l v e s t o not too l a r g e plasma parameters g , allowing t h i s

p e r t u r b a t i o n a l t r e a t m e n t t o be v a l i d . According t o Kalman 1 7 1 , we havc. i n

t h e f i r s t approximation in g :

In t h i s a p p r o x i ~ a t i o n io m u l a ( 1 2 ) c d n be r e p r e s e n t e d a s f o l l m s

The energy l o s s e s (1) can be ncvr be w r i t t e n

(We a r e using here 2 xetnod of i n t e g r a t i c n exp a i n e d i n Akhieser, 4 2. 240).

d Z -,2 S

dZ -s2'

Using t h e r e l a t i o n s Re - = - Z(1-2e s j d t e t ) , Im -- = - ~ n ' / ~ s e we f i n d

ds o d s

from (18)

where

The Coulonib logarithm In ,2 in ( 1 9 ) i s defined 4s

being a niaxircal wave nui11bcv intrccuced i s s t t h e d;vcl.,: i n t h e i n t e g r a l i n ( 2 1 ) a r i s e s . Neglectirig thi: temis of t h e o r d c r of I/inA, g/9 l n h one f i n d s In.! = l ~ i ( k ~ , ; ~ ~ / k ~ ~ ) . CI-13~sing k

a s knlax=kd h.

,)lax

with h, defined by (4) we g e t ? o r dUldx t h f c . x p : - e s s h which d i f f e r s from t h e Vlasov approximation ( 2 ) o:!lj by the :dct;r 1

(2gH/?). The f u n c t i o n H(>) d i s p l a y e d i n F i g . 2 has the l i m i t H = ? f c : v

U. i . e . f o r v

e U,.

The p e r t u r b a t i o n a l t r e a t m e n t lead i n 9 t o tt;: f v r r ~ l a s (l!?), ( 2 2 ) i s e v i d e n t l y

W

a ~ o l i c a b l e only i f ( g / 9 ) < 1. A c c o r d i ~ ~ q 1::. ",l? 1 i i n e r i c 3 l c a i c u l a t i o n s by Berggren 16 , scme b a s i c a l l y new ;~r.il.ncr,ie.;il a r i s e f o r $ r e a t e r values of g.

One can a s a m a t t e r of f a c t connect chc 1 : ~ ; i i n g ( f o r 3- +o) expression f o r

k u e

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w i t h t h e c o m p r e s s i b i l i t y K o f t h e i n t s a c t i n g e l e c t r o n gas. (Here i s t h e c o m p r e s s i b i l i t y o f t h e non- i n t e r a c t i n g g z s ) . As fc:lows f r o m ( 1 7 ) , i n t h e

f i r s t a p p r o x i m a t i o n i n g t h e c o n p r e s s i b i l i t y i!!creases b y a f a c t o r ~ / ~ ~ = l + ( g / 9 ) . F o r a g r e a t e r v a l u e o f g t h e c o m p r e s s i b i i i ty can e,den become i n f i n i t e and

t h e n n e g a t i v e ( a c c o r d i n g t o ~ e r g g r ? n [ ~ ~ ' ; j [ h i s happens f o r g 2 30). F o r such h i g h l y n o n i d e a l plasmas t h e e n e r g y 1o;ses may d i f f e r g r e a t l y f r o m t h a t g i v e n b y t h e p e r t u r b a t i o n a l f o r n l u l a s ( 1 9 ) , ( 2 2 ) .

F i g . 2: F u n c t i o n H ( v / u ) e n t e r i n g the f o r n ~ u l a ( 1 9 ) f o r t h e ener-gy l o s s e s i n an n o n i d e a l pl%sma.

The t h e o r y o f S i n g w i ill] and c o n s e q u e n t l y the f o r m u l a s (19), ( 2 2 ) do n o t t a k e i n t o a c c o u n t c o l l i s i o n a l e f f e c t s . I n d s i m p l i f i e d way t h e s e e f f e c t s can be e v a l u a t e d b y i n t r o d u c i n g t h e c o l l i s i o n a l i n t e g r a l o f t h e BGK t y p e

I = - v e ( f - f o ( i fd; /i fad;)) i n t o t h e e q u a t i o n ( 5 ) . We g e t i n t h i s way (Be- k e f i 1 2 3 1 ) t h e d i e l e c t r i c f u n c t i o n

where 5 = (w+ive)/kue, y = ve/kue. I n s e r t i n g ( 2 4 ) i n t o (1) we o b t a i n a f t e r some

i n t e g r a t i o n s (see Appendix):

where

A c o r r e c t c o m b i n a t i o n o f t h e f o r m u l a ( 2 5 ) , a c c o u n t i n g f o r c o l l i s i o n a l e f f e c t s o n l y , and o f t h e f o r m u l a (22), a c c o u n t i n g f o r t h e l o c a l f i e l d e f f e c t s , seems t o be

The Coulomb l o g a r i t h m l n ( 1 . 2 T/hw ) i n ( 2 7 ) m u s t b e c o n s i d e r e d as a b i g quan- t i t y . On t h e o t h e r hand t h e l o g a r i t h m e n t e r i n g t h e r a t i o ve/w P i s r a t h e r s m a l l f o r n o t t o o s m a l l plasma p a r a m e t e r s g 7 3. Thus, t h e i n f l u e n c e o f t h e c o l l i s i o n a l P e f f e c t s on t h e s t o p p i n g power f o r n o t so s m a l l plasma p a r a m e t e r s g p r o v e s t o be s l i g h t and d e c i s i v e r o l e , as f o r m u l a ( 2 7 ) shows, p l a y i n t h i s r e s p e c t t h e l o c a l f i e l d e f f e c t s .

The eq. ( 2 7 ) can be j u s t i f i e d t o some d e g r e e as f o l l o w s . The i n c r e a s e o f s t o p p i n g power dW/dx b y t h e f a c t o r 1+(2g/9) as g i v e n b y t h e eq. ( 2 2 ) a l l o w s t o be i n t e r - p r e t e d as a r e s u l t o f i n c r e a s e of t h e plasma e l e c t r o n mass due t o i t s i n t e r - a c t i o n w i t h t h e plasma p a r t i c l e s ( s e e i n t h i s c o n n e c t i o n 1 1 8 1 , p. 6 4 ) . Thus, i n t h e l i m i t v << ue we c a n u s e a model o f e l e c t r o n s w i t h a n e f f e c t i v e mass

meff = ( 1 + 2g/9) me. U s i n g t h e B o l t z m a n n eq. 2 ( 5 ) w i t h me r e p l a c e d b y t h e e f f e c t i v e mass meff We a r r i v e a g a i n i n t h e l i m i t v << ue t o t h e eq. ( 2 7 ) .

The eq. ( 2 2 ) a l l o w s a l s o a d i f f e r e n t i n t e r p r e t a t i o n . We can, e v i d e n t l y , r e w r i t e ( 2 2 ) i n t h e form

where P = m v i s t h e p r o j e c t i l e i o n i m p u l s e and m i s i t s mass. The q u a n t i t y

P P

T = T / ( l + 2g/9) i s a n e f f e c t i v e t i m e o f t h e p r o j e c t i l e i o n i m p u l s e t r a n s f e r

0

t o t h e plasma e l e c t r o n s . As we see f r o m ( 2 8 ) , t h e i n t e r a c t i o n between e l e c t r o n s l e a d s t o a d e c r e a s e o f t h i s r e l a x a t i o n t i m e b y t h e f a c t o r 1 + ( 2 g / 9 ) as con~pared w i t h t h e r e l a x a t i o n t i m e T~ i n t h e i d e a l plasmas.

The d i e l e c t r i c f u n c t i o n ^ca ( 1 7 ) based on t h e t h e o r y o f S i n g w i 11 appear t o be

w e l l a p p l i c a b l e f o r n o t t o o f a s t p r o j e c t i l e whose v e l o c i t i e s s a t i s f y t h e con-

03-10 JOURNAL D€ PHYSIQUE

d i t i o n (w/kue) a ( v / u e ) 1. The c a s e o f h i g h p r o j e c t i l e v e l o c i t i e s v >U e i s more d i f f i c u l t t o t r e a t a n a l y t i c a l l y and we w i l l r e s t r i c t o u r s e l v e s h e r e i n t h i s r e s p e c t w i t h some q u a l i t a t i v e remarks. T h e a s s u m p t i o n (w/kue) = ( v / u e ) > 1 corresponds t o h i g h f r e q u e n c e s w and s m a l l wave - numbers k. Thus t h e space - d i s p e r s i o n m u s t b e n e g l i g i b l e i n t h i s case and we can use t h e d i e l e c t r i c f u n c t i o n ( 9 ) t a k i n g i n t o a c c o u n t e f f e c t s o f f i r s t o r d e r i n g. F o r f r e q u e n c e s u >> v t h e

2 2 ^e

formula ( 9 ) c a n b e w r i t t e n as c = 1 - ( W /U ) ( 1 - 6 - i v / W ) and hence i t c o i n c i d e s

P e

w i t h t h a t d e r i v e d b y Oberman e t a l . [ 191, Oawson, Oberrnan [ 2 0 ] ( s e e a l s o ~ c k e r [ 2 l ] p. 283) on t h e b a s i s o f t h e BBGKY t h e o r y . As f o l l o w s f r o m c a l c u l a t i o n s i n Dawson, Oherman C20 ] t h e c o e f f i c i e n t & ( U ) r e m a i n s a l m o s t ' c o n s t a n t i n t h e f r e q u e n c y i n t e r - v a l 0 ( w ? u Thus, t h e p r o b a b i l i t y o f a l o n g i t u d i n a l plasmon e m i s s i o n on t h e

P' 2

p a i i l element dx o f a p r o j e c t i l e , p r o p o r t i o n a l t o t h e f a c t o r I m ( l / ~ ) = - I ~ E / ] E ~ ^%

2 2 2 2 3

l / ( ( l - o Q - 6 ) / w ) + 4 3 p ~ e / ~ ) ) has a maximum f o r t h e f r e q u e n c y w=w ( 1 - & ( u p ) ) .

P P

I t means t h a t t h e plasmons w i t h r e d u c e d e n e r g y hw (1-6(w ) ) w i l l be m a i n l y

P P

e m i t t e d by t h e c h a r g e d p a r t i c l e on i t s passage t h r o u g h plasma i . e . t h a t t h e p a r t i c l e energy l o s s e s w i l l b e t h e r e b y r e d u c e d b y a f a c t o r 1 - & ( U ) as compared

P

w i t h t h o s e f o l l o w i n g f r o m t h e plasma i d e a l i t y a s s u m p t i o n ( 6 = 0 ) . A c c o r d i n g t o i i i i ~ n i n t o v i c h 10 6 ( 0 ) = g / 9 J 2 and, hence, f o r an example o f t h e ICF d e u t e r i u m - t r i t i u m plasma m e n t i o n e d i n t h e i n t r o d u c t i o n (g=3,1) we have 6 ( w ) = 6 ( 0 ) = 0.25.

P

However, one can c o n j e c t u r e t h a t i n f a c t t h e e f f e c t can b e more pronounced. As t h e e x p e r i m e n t s f ⁸ 'l show, s t a t i c a l n o n i d e a l plasma c o n d u c t i v i t i e s appear t o by a few t i m e s l o v e r t h a n t h o s e f o l l o w i n g f r o m t h e S p i t z e r f o r m u l a f o r a f u l l y i o n i z e d -

plasma, i f g S 1. T h i s d i s c r e p a n c y may be p a r t l y due ( a t l e a s t f o r many - e l e c t r o n atoms l i k e Xe) t o i n c r e a s e o f t h e c o l l i s i o n f r e q u e n c y o c c u r i n g i n a r e s u l t of f r e e e l e c t r o n s c a t t e r i n g b y t h e i o n - c o r e o f a complex atom. ( T h i s e f f e c t can be impor- t a n t f o r s u f f i c i e n t l y h i g h t e m p e r a t u r e s ) . However, a p a r t o f t h i s descrepancy i s p r o b a b l y due [ 2 4 1 t o l o c a l i s a t i o n o f f r e e e l e c t r o n s w h i c h c a n o c c u r i n n o n i d e a l plasmas. As a r e s u l t , d e n s i t y o f c o n d u c t i n g e l e c t r o n s w i t h p o s i t i v e e n e r g i e s i s reduced by a f a c t o r exp ( - g/3.2'/') a c c o r d i n g t o 1 2 4 3 .

Assuming t h a t o n l y e l e c t r o n w i t h p o s i t i v e e n e r g i e s can t a k e p a r t i n c o l l e c t i v e processes l e a d i n g t o t h e p r o j e c t i l e i o n e n e r g y l o s s e s we c a n c o n j e c t u r e t h a t i n n o n i d e a l plasmas t h e e n e r g y l o s s e s w i l l b e r e d u c e d b y t h e same f a c t o r . T h i s e f f e c t appear t o be more e s s e n t i a l t h a n c o l l i s i o n a l e f f e c t s l e a d i n g t o some r e - d u c t i o n o f t h e Coiilonb l o c j a r i thm e n t e r i r g t h e s t o p p i n g power dU/dx ( s e e Appendix). *

r These c o n s i d e r a t i o n s concern, e v j d e n t l y , o n l y c o l l e c t i v e energy l o s s e s . The q u e s t i o n a b o d t i n f l u ~ n c e o f t h e c o t - - e l ; l t i z ~ n e f f a c t s on t h e t o t a l energy l o s s e s f u r V U r e r ~ a i n s , i n f a c t , o p c n .

e

3. D e w e r a t e Quantum --- Plasmas

The w e l l known L i n d h a r d t h e o r y o f charged p a r t i c l e energy l o s s e s i n a degenerate n o n i n t e r a c t i n g e l e c t r o n gas [ls]. i s v a l i d under t h e a s s u ~ n p t i o n th a t t h e i n t e r - e l e c t r o n d i s t a n c e s r s a r e s m a l l , rs = (,/a:-r ^{a3)1'3 <<} 1, n b e i n g t h e e l e c t r o n d e n s i t y and a t h e Bohr r a d i u s . T h i s a s s u n p t i o n i s n o t f u l f i l l e d f o r m e t a l s ( t h e parameter rS r a n g e s from 2,! f o r A1 t o 5,5 f o r 2 s ) and a l s o f o r some i n e r t i a l con- finement f u s i o n ( I C F ) plasmas [ 2 5 1 . C o r r e l a t i o n e f f e c t s due t o t h e i n t e r a c t i o n between t h e e l e c t r o n s ( C o u l c x b r e p u l s i o n , P a u l i p r i n c i p l e ) , n e g l e c t e d i n t h e t h e o r y o f L i n d h a r d [ 1 3 1 l e a d t o a r e d u c t i o n o f t h e e l e c t r o n d e n s i t y i n v i c i n i t y o f a p a r t i c u l a r e l e c t r o n . A: a r e s u l t r e a l f i e l d e x p e r i e n c e d by t h e e l e c t r o n d i f f e r s f r o m t h e average f i e f c i , c o n s i d e r e d i n t h e L i n d h a r d t h e o r y , by some l o c a l f i e l d c o c t r i b u t i o n . T!ie I c c a l f i e i d e f f e c t s e v i d e n t l y a l s o i n f l u e n c e t h e i n t e r a c t i o n betiveen t h e p r o j e c t i l e i o n and t h e plasma e l e c t r o n s and, c o n s e q u e n t l y , t h e i o n energy l o s s e s . i n t h e f o l l o w i n g t h e energy l o s s e s o f i o n s w i t h v e l o c i t i e s

V << vF a r e c a l c u l a t e d on t h e b d s i s o f t h e d i e l e c t r i c t h e o r y of Singwi 111 w h i c h t a k e s i n t o a c c o u n t t h e l o c a l f i e l d e f f e c t s i n a r e l a t i v e l y s i m p l e way. A c c o r d i n g t o 1111 t h e l o n g i t u d i n a l d i e i e c t r i c f u n c t i o n c L o f an i n t e r a c t i n g e l e c t r o n gas embedded i n a s t a t i c u n i f o r r i ~ 3.ckgr-ound o f p o s i t i v e charges formed by i o n s ( j e l l i u m m o d e l j can be dei-incd 3s f o l l o w s

where Q, i s t h e 1-indhard p o l a r i s a b i l i t y ( r e e eq. ( 6 ) i n [ l 4 1 ) , and G(k) i s t h e l o c a l f i e 1 d c o r r e c t i o n .

Assuming v <: vF ( w 3 i c h means t l i a t t h e v a r i a b l e u = w/kvF e n t e r i n g Q. i s s m a l l ) we c a n s i m p l i f y ti;o Lin"1at.d p , , i a r i s a b i l i t y Q as f o l l o w s

0

1 l 1 2 z + l ,

where f ( z ) = -i(lt ? ( l - z ) I n ! - - . ) , f

G U f o r z < 1. f = 0 f o r Z ) 1.

1 2 - 1 ' 2 L - 2

I n s e r t i n g now ( 2 9 ) i n t o ( 1 ) and p e r f o r m i n g i n t ~ g r a t i o n s o v e r t h e v a r i a b l e cos9 ( - 1 ( cos0 ( 1 ) . and o v e r t h e v a r i a b l e z ( 0 - z 2 l ) , we o b t a i n

where C ( r S ) i s t h e f u n c t i o n

JOURNAL DE PHYSIQUE

The f u n c t i o n C ( r s ) d i f f e r s from t h e corresponding e x ~ r z s s i o n (1 3 ) i n [ 1 4 1 by t h e term containing t h e l o c a l f i e l d c o r r e c t i o n G ( z ) . One of the b e s t d e r i v a t i o n s of the l o c a l f i e l d c o r r e c t i o n was given in 1 1 2 1 and shows t h a t G(k) i s an universal f u n c t i o n of t h e v a r i a b l e z = k/2kF behaving a s G z2 f o r z + and reaching a maximum Gmax 2 f o r z = 1.

Using t a b l e 1 in 1121 f o r t h e l o c a l f i e l d c o r r e c t i o n L ( k ) we obtained by nu- merical i n t e g r a t i o n t h e c o e f f i c i e n t C a s a f u n c t i o n o f r S , represented by t h e curve 2 in Fig. 3. As i s seen from Fig. 3 t h e energy l o s s e s following from t h e theory of Singwi appear t o be appreciably higher than those given by t h e Lind- hard approximation f o r a n o n i n t e r a c t i n g e l e c t r o n gas (cl~i-ve 1 in Fig. 3 ) , i f the parameter rS i s n o t too small. O f i n t e r e s t in t h i s r e s p e c t a r e f i r s t of a l l t h e energy l o s s e s of a l p h a - p a r t i c l e s a r i s i n g i n f u s i o n r e a c t i o n s i n compressed ICF plasmas. (The v e l o c i t i e s v of these a - p a r t i c l e s s a t i s f y t h e condition v << v F ) . For i n s t a n c e , t h e degenerate DT plasmas a r i s i n g in p e l l e t s conlpressed by heavy ions beams a r e c h a r a c t e r i s e d 1 5 1 by t h e parameters r s

U,3 ( r s = 0,35 f o r a = l l l g / c m 3

) . According t o curve 2 i n Fig. 3 the s t a p ? i n g po.,~er f o r t h e a - p a r t i c l e s should i n c r e a s e by about 20% i n this case a s compared with the Lindhard approximation.

Deuterium plasmas with higher parameters rS : 1 can ? r i s e according t o [ 25,263 in hollow spheres f i l l e d with deuterium gas a s a r e s u l t of i r r a d i a t i o n by l a s e r p u l s e s . I t i s a l s o of i n t e r e s t t o apply e q u a t i o n s (31), ( 3 2 ) f o r i n t e r p r e t a t i o n of slow ion energy l o s s e s in metal f o i l s . As experiments (27,231 show, energy l o s s e s of ions with e n e r g i e s W = 1 keV appear t o be s y s t e m a t i c a l l y higher than those following from t h e I-indhard theory by a f a c t o r 2-3. In Fig. 3 ere i n d i c a t e d a v a i l a b l e ex-

d

W 4 2 3

perimental values of t h e c o e f f i c i e n t C ( r S ) = - T ~ F ; / ( 4 2 mev/3rrh ) f o r some simple

metals belonging t o t h e f i r s t t h r e e groups of t h e p e r i o d i c system. (The parameter

rs was c a l c u l a t e d under the assumption t h a t t h e f r e e e l e c t r o n d e n s i t y i s equal to

t h e atomic d e n s i t y m u l t i p l i e d by t h e number of valence e l e c t r o n s per atom). As

Figure 3 shows, accounting f o r t h e local f i e l d e f f e c t s removes the discrepancy

with experiment almost completely. The remeining s l i g h t discrepancy may be due t o

l a t t i c e e f f e c t s which a r e neglected i n t h e j e l l i u m model. Another reason i s ,

probably, t h a t we used a s t a t i c f o r n ~ u l a t i o n of t h e local f i e l d c o r r e c t i o n G ( k ) ,

n e g l e c t i n g i t s dependence upon t h e frequency W. ( I t must be noted t h a t a v a i l a b l e

dynamic formulations of t h e l o c a l f i e l d c o r r e c t i o n G(k,w) a r e c o t s a t i s f a c t o r y

because they d i s r e g a r d c o l l i s i o n a l e f f e c t s ) . Further improvement of t h e local

f i e l d t h e o r i e s , i n which t h e l a t t i c e and c o l l i s i o n a l e f f e c t s must be taken i n t o

c o n s i d e r a t i o n , would allow t o develop a systematical theory of t h e charged par-

t i c l e energy l o s s e s i n t h e nonideal plasma formed by conducting e l e c t r o n s i n

metals.

F i g . 3: Curve 1 - L i n d h a r d a p p r o x i m a t i o n ( f o r m u l a ( 6 ) i n 14 ) c o r r e s p o n d i n g t o assumption G=O i n o u r f o r m u l a ( 3 2 ) ; c u r v e 2 - f o r m u l a (32

c i r c l e s - e x p e r i m e n t a l v a l u e s t a k e n f r o m t h e t a b l e 3 . 1 i n [ 281;

f e r r e d f r o m e x p e r i m e n t d e s c r i b e d i n . Energy l o s s e s dkl/dx a r e

i[

expressed i n u n i t s eV/ . ^(Z=l).

APPENDIX

I n f l u e n c e o f t h e c o l l i s i o n a l e f f e c t s on t h e stoppjng-power

F o r s m a l l c o l l i s i o n f r e q u e n c e s v e t h e BGK d i e l e c t r i c f u n c t i o n ( 2 4 ) c a n be s i m p l i f i e d as f o l l o w s

where q~ = R e ( l + s Z ( s ) ) , y = ve/kue, s = w/ku I n s e r t i n g t h i s f u n c t i o n i n t o (1) e '

we can t r a n s f o r m t h i s e x p r e s s i o n f o r s = ( v / u e )

o as f o l l o w s

0

C8-14 JOURNAL DE PHYSIQUE

( N o t e t h a t t h e second i n t e ~ r a l c v e r k i n ( k 2 ) p r o v e s t o be c o n v e r g e n t f o r k m ) . Choosing i n (A2) t h e maximal wave - number klnax as kmax =o,85m u /h we a r r i v e

e e a t t h e f o r m u l a ( 2 5 ) .

T h i s method o f i n t e g r a t i o n i s n o t a p p l i c a b l e f o r v > ue when t h e space d i s p e r s i o n i s n o t e s s e n t i a l and t h e d i e l e c t r i c f u n c t i o n reduces t o ( 8 ) . i t i s p r e f e r a b l e t o c a l c u l a t e t h e e n e r g y l o s s e s d e f i n e d by ( l ) , ( 8 ) u s i n g i n s t e a d o f t h e v a r i a b l e s k,s t h e v a r i a b l e s ^W, q = ( k 2 - ( W ~ / V ~ ) ~ / ~ . I n s e r t i n g t h e s e v a r i a b l e s i n t o (1) and u s i n g t h e formula dz = 2aq dq $ we can b r i n g ( l ) . t o t h e f a r m

qmax

dW - - qdq - - ^e2z2

aT -

2 ^W 2- - ( Y 1

Y 2 ) q

2 kmax'V

F o r Ve << W t h e i n t e g r a l Y 1 i s e q u a l t o 2 1 : ~ In(--

P ) a c c o r d i n y t o t h e r e s i d u e

P u p

theorem. The i n t e g r a l Y 2 can be c a l c u l a t e d by employing a v a r i a n t o f t h e

, a c c o r d i n g t o which we must r e p l a c e c o n s i d e r t h e l i m i t p o . I n t h i s way we

these e x p r e s s i o n s f o r Y 1 and Y 2 i n t o (A3) P e '

and i n t r o d u c i n g kmax = 2mev/% we o b t a i n

The f o r m u l a (A4) c o i n c i d e s w i t h t h e f o r m u l a o b t a i n e d by Kramers [30] i n

a n o t h e r way

i f we t a k e i n (A5) kmax = (2mev/"n) and assume ve ^W i . e . In A =

P '

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