DIELECTRIC RESPONSE AND ENERGY LOSS FOR AN INTERMEDIATE QUANTUM PLASMA

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DIELECTRIC RESPONSE AND ENERGY LOSS FOR AN INTERMEDIATE QUANTUM PLASMA

N. Frankel, K. Hines, R. Speirs

To cite this version:

N. Frankel, K. Hines, R. Speirs. DIELECTRIC RESPONSE AND ENERGY LOSS FOR AN INTER- MEDIATE QUANTUM PLASMA. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-513-C7-514.

�10.1051/jphyscol:19797248�. �jpa-00219232�

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JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au n07, Tome 40, ~ u i l Z e t 1979, page C7- 513

DELECTFUC RESPONSE AND ENERGY LOSS FOR AN INTERMEDIATE QUANTUM PLASMA

N.E. Frankel, K.C. Hines and R.D.B. Speirs.

University of MeZboumze, SehooZ of Physics, ParkuiZZe, 3052 MeZbome, AustvaZia.

1. INTRODUCTION : With t h e advent of t h e use of laser-driven p e l l e t s t o obtain therinonuclear fusion, we have i n the lab- oratory a plasma i n which the electrons have a fugacit y, Z x 1 , the intermediate quantum regime. When a highly compressed deuterium plasma i s obtained by l a s e s com- pression, t h e f i n a l s t a t e of t h e system

corresponds t o p a r t i c l e number d e n s i t i e s and tempeGatures

T-lo7

-

^lo8 ^/1/,/2/.

It i s perhaps i n t e r e s t i n g t o note t h a t t h e ions i n the deep i n t e r i o r of J u p i t e r corr- espondin l y reach a fugacity regime around unity /3$

.

Using t h e above data, we f i n d t h a t t h e Fermi temperature, TF, i s v i r t - u a l l y equal t o t h e system's temperature, T. The temperature at which t h e fugacity of an electron gas reaches unity i s

To(Z = 1 )- = 0.99fF /47;

thus these lases-driven fusion conditi- correspond t o a plasma of intermediate ( p a r t i a l l y degenerate ⁾electrons and cla- s s i c a l ions. What is more, the plasma parameter

r ⁼ e 2 ( 4 / 3 n p ) l / 3 ~ ~ i s , f o r t h e above values, such t h a t

los2 ,< r;S.

+.

Thus t h e p a r t i c l e s t o a f i r s t approximation a r e weakly coupled and a l s o t o a f i r s t approximation we can t r e a t them using standard Linear response theory.

Hore and Frankel /4/ have shown t h a t a l l q u a n t i t i e s which a r e thermodynamically averaged over the Femi-Dirac distribution function can r e a d i l y be expanded about t h e intermediate quantum region, ⁼1 , using standard Mellin i n t e g r a l t r a n s f o m techniques. Hore and Frankel / 5 / have a l s o

studied t h e d i e l e c t r i c response of t h e charged Bose gas about t h e condensation region, Z = 1. I n t h i s paper we report on s i m i l a r calculations using t h e techniques of reference /5/ along with t h e exp- ansions appropriate f o r a gas of fermions about Z = 1 given i n reference /4/.

Work up t o now on t h i s region of compell- ing i n t e r e s t i n fusion research has ess- e n % i a l l y only been accessible by n u m e r i d techniques / 6 / , /7/.

2.8 RESULTS: We give here a b r i e f suium- ary o m t s obtained by t h e above anal- y t i c a l technfques f o r : ^(a)the Ion itud- i n a l d i e l e c t r i c response function Tb) plasma dispersion r e l a t i o n s h i p s , [ c > ^the

ion-accoustic sound, (dl t h e energy l o s s t o c o l l e c t i v e modes, t e ) t h e energy l o s s t o binary collisi'ons and ( f ) the electron- ion contribution t o the thermal conduct- i v i t ^y

.

&

Given the standard longitudi3al l e e c t r i c response function G (2, w ) from l i n e a r RPA theory f o r an electron

where 1 i s t h e spin,

n

the volume of t h e system and P0(2) the Permi d i s t r i b u t i o n function

Using t h e methods and techniques of r e f - erences /4/ and /5/ we now expand € ^(CJ, ^&) about t h e region Z = 1 t o obtain small 9,

T " ^Tr,asymptotic expressions

where T ( s ) = ( I - 2'")j ( s ) , { s ) being the Zeta function and

0 ⁼

-

^1.

: From t h e a n a l y t i c a l r e s u l t given i n we have obtained t h e following dispersion r e l a t i o n s h i p f o r electron o s c i l l - a t i o n s i n t h e small region:

~ ( g ) = + i y ⁽⁹⁾

where

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797248

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( c ) : Defining x =

-

_c?-

t h e expansion given i n ( a ) above corresponds t o x >) I and 8 << ^{I .} ^NOW i n consid- e r i n g i o n so-and modes we r e q u i r e x <<

?.

Using t h e a p p r o p r i a t e asymptotic expanslo- n# f o r 6 (9, ^{W )}f o r t h e e l e c t r o n s , x and

s m a l l , and t h e s t a n d a r d form of (5 (g, W ) f o r t h e c l a s s i c a l i o n s /8/, we t h e n o b t a i n f o r t h e i o n sound modes

w ($1 = U t ( * ) + ⁱ^d%($)^;

where

0:

-

^4/[1⁺e # ( e ) J ,

u:

The d i e l e c t r i c response f u n c t i o n i n ( a ) can be used t o c a l c u l a t e t h e energy l o s s from Mev i o n s i n a Kev e l e c t r o n plasma t o plasmons / 9 / . Using t h e r e s u l t f o r (dE/dt )pl from r e f e r e n c e /9/, we have obtained t h e a p p r o p r i a t e asymptotic expansion about Z = 1 ( 6 = 0 ) which we g i v e below

where t h e a n a l i c a l expansion f o r t h e p r e f a c t o r

(hf),l

^{i s}

and C i s E u l e r l s c o n s t a n t .

( e ) : We have a l s o s t u d i e d t h e c o n t r i b -

=on t o t h e energy l o s s of an i o n i n an i n t e r m e d i a t e quantum e l e c t r o n g a s due t o b i n a r y c o l l i s i o n s , (dE/dt lbc. We have g e n e r a l i z e d t h e r e s u l t s f o r a c l a s s i c a l plasma /lo/ a l o n g t h e l i n e s of r e f e r e n c e /I I/. We have obtained a d e t a i l e d a n a l y t i c a l expression f o r (dE/d$), a l o n g w i t h t h e a p p r o p r i a t e Coulomb

f a c t o r ( 1 n A )bc f o r t h e l a s e r - d r i v e n plasma.

asymptotic expansion f o r t h i s expression . i n t h e regime of Z r 1 ( 8 % 0 ) :

where a l l q u a t i t i e s a r e a s i n r e f e r e n c e /12/.

3. DISCUSSION: D e t a i l e d comparison w i l l be given of t h e b i n a r y c o l l i s i o n and co1lecti;e energy l o s s r a t 6 s i n t h e f i n a l s t a t e of a l a s e r - d r i v e n f u s i o n e l e c t r o n - i o n plasma. We w i l l a l s o make s p e c i f i c corn a r i s o n s f o r t h e q u a n t i t i e s presented i n ?a) - ( f ) above with t h e i r

corresponding form i n t h e cases Z = 0 ( c l a s s i c a l ) and Z = 00 ( t o t a l l y degener- a t e ) .

REFEQENCES:

/I/ BRUECXNER,K. A. and JORNA, S. ,Rev.

Mod. Phys. 46 (1974) 325.

/2/ MINO0 ,H , ^DEUTSCH, ^C.and HANSEN, 3. P. ,

Phys.Rev.A 74 (1976) 840,

/3/ GEHRELS, T. , e d i t o r , JUPITER (Univ, of Arizona P r e s s , 1976)--

/4/ HORE,S.R. and FRANKEL,N.E., Phys.

Rev. A 2 (1975) 1617.

/5/ HORE,S.R. and FRANKEL,N.E., Phys.

Rev. B 2 (1975) 2619.

/ 6 / GOUEDARD, C. and DEUTSCH, C. , ^J,Math.

Phys. 2 (1978) 32.

/7/ BRYSK,H. , CAMPBELL, P.M. and HAMMERLING, P., Plasma Phys. 17

(7975) 473.

/8/ ICHIMARU, S , , BASIC PRINCIPLES OF PLASMA PHYSICS ^(W.A. Benjamin, Reading, Mass. , ^I⁹⁷³1.

/9/ SKUPSKY,S., Phys. Rev. A 16

(1977) 727.

/lo/ ^FRANKEL, ^N^.E., Plasma Phys, 1

(1965) 225.

/I I / BRYSK R. , Plasma Phys. 16

(1974) 927.

/12/ GOUERABD, C., J. Phys. A (1977) L143.

: A s p e c i f i c e x p r e s s i o n f o r t h e c o n t r i b u t i o n t o t h e thermal c o n d u c t i v i t y f o r a plasma of a r b i t r a r y degeneracy ( a l l Z ) h a s been given i n r e f - erence /12/. We p r e s e n t t h e following