Correlated ion stopping in dense classical plasma

6 March 1995 PHYSICS

LETTERS

A

Physics Letters A 198 (1995) 347-351

Correlated ion stopping in dense classical plasma Claude Deutsch, Patrice Fromy ~~ruf~ire

de Physique des Gw, ef Plays

I, B&t. 212, Universite’Paris XI. 91405 Orsay, France

Received 11 October 1994; revised m~usc~pt received 21 November 1994; accepted for publi~tion 20 L&ember 1994 ~mmuni~t~

by M. PorkoIab

Abstract

Correlated ion stopping within N-clusters of ion debris resulting from a swift fragmentation of a weakly ionized cluster impacting a dense and fully ionized classical plasma is considered in the high velocity limit. Elaborating upon dicluster polarized stopping with a Fried-Conte plasma dielectric function, N-cluster stopping is taken as a linear superposition of every available dicluster pair. The main focus is on the conspicuous enhancedcorrelation stopping and its dependenceon N, N-cluster arrangement as well as target plasma coupling. Results are of potential significance in enlarging substantially the parameter range requested for achieving particle-driven thermonuclear fusion.

The linear acceleration of ionized cluster structures at energies of the order of a few tens of keV1a.m.u. is becoming presently a rather routine operation [ l-51. Amongst the many available cluster configurations, it is worthwhile to mention the fullerene, carbon-like and also the metallic ones such as Au:+ worked at high energy on tandem van de Graaf accelerators. It should be recalled that low velocity acceleration is already pretty well documented [ 61. It offers many novel practical applications such as metal alloying, surface treatment and preparation and a lot more. Here we intend to focus on the interaction of more energetic particles with energy/nucleon 2 a few keV1a.m.u. Then their interaction with cold or hot matter may be mostly considered as a stopping process through Coulomb interaction with target electrons. It is now a quite well accepted view that energetic clusters fragmentate very rapidly, on a femtosecond time scale [ 71, which yields a resulting Coulombic N-cluster of ion debris flowing within a few atomic units of relative distance. ’ Asso& au C.N.R.S. 037%9601/95/$~.50

0 1995 Elsevier Science B.V. All rights reserved

SS~IO375-9601(94)01022-6

The resulting correlated stopping in dense and cold target has already been given a lot of attention with a dielectric formulation for the electron fluid responsible of the projectile energy loss. Initially, investigations were mostly confined to dicluster stopping [ 81. More recently, they have also been extended to arbitrarily large N-clusters [g-15]. Then the conspicuous fact is the obvious departure from the usually considered single ion energy loss, through a simultaneous flow of closely connected pointlike ion charges. All the above considered works are exclusively dedicated to cold matter stopping with a fully degenerate electron jellium at T=O. However, very recently systematic attention has also been paid to dicluster stopping and vicinage effect in a hot and dence classical electron fluid with a Maxwellian velocity distribution [ 16,171. The main goal of the present work is to investigate N-cluster ions, interacting with such a classical electron stopping medium by using a linear superposition of every available dicluster. Such a procedure has already recently heen the subject of much attention at T= 0 I 13-I 5 I .

C. Deutsch P. Fromy /Physics

348

The basic physical situation we intend to model is the compression of thermonuclear and spherical pellets through intense cluster ion beams [9,13,14,17-191. A huge hammer-like source of efficient shock waves has thus already been identified [ 13,l S] , which arises directly from the enhanced correlation stopping (ECS ) alluded previously. Those rather encouraging results are by now essentially documented for a T=O target. So, their potential extension to an arbitrarily hot plasma target is of crucial significance in asserting the feasability of particle-driven inertial confinement fusion through cluster ion beams. Such a statement stems from the obvious observation that the enormous compression expected through cluster beams will very rapidly heat cold target matter in a dense and hot ionized fluid. Here, we restrict ourselves to a nondegenerate electron stopping medium, which is also of relevance for further applications in magnetically confined the~onucle~ fusion [ 16,171. Recently, we have also investigated dicluster ion energy loss in a partially degenerate electron fluid [ 201. Dicluster stopping in a classical electron plasma, according to the recent treatment by d’Avanzo et al. [ 16,171 is first outlined, as a preliminary step toward N-cluster stopping explained as a linear superposition of dicluster cont~butions. The enhanced co~elation stopping (ECS) is then thoroughly documented. Its specific dependence on N topological ion debris configurations, velocity and target electron coupling are thus stressed. The cubic box arrangement with eight unit charges often appears as a convenient working example. Here we briefly review a basic formulation for dicluster stopping in a dense and hot classical plasma [ 16,171, which closely parallels a similar derivation for a T= 0 electron jellium [ 81. The correlation con-

i/l:

Letters A I98 (1995) 347-35I

tribution to stopping is conveniently highlighted by taking the ion pair in a cylindrical geometry (Fig. 1) defined by the common and overall drift velocity Vand interparticle distance R. In this work, we neglect any velocity mismatch between the two charges .Zi and z;?. Preliminary considerations [ 2 I] show that if one writes the stopping power as S= - AElh, the magnitude of the mean energy-loss per unit path length, one can define @ as the square of the standard deviation of the energy-loss distribution per unit path length. For a plasma target with kgT > EF (Fermi energy), one has within a good approximation @- 2k,TS. So, computing the stopping power already yields a reasonable estimate for the straggling. Moreover, with a view toward an experimental program developed on the Orsay tandem linear accelerator, we think it useful to restrict to classical plasma targets of fully ionized deuterium at a rather moderate temperature T= a few eV, while the corresponding free electron density ranges from 2X 10” cmP3 to 102’ cmA3. At such modest temperatures, the straggling may thus be neglected in a first approximation. Dicluster stopping calculation starts with the electrostatic potential expression in a reference system where the two ions are at rest [ 8,16,17], and the leading ion is located at the origin. It reads (see Fig. 1) eiK.r 1 ‘i@) = (2~)’

X (Z, +Z2 epikeR) ,

E(k, w) = 1 -t-

t V Fig. 1. Cylindrical coordinates along projectile velocity V for dicluster stopping with intercharge distance R. B and D denote R projections ~~ndicul~ and parallel to Vrespectively.

0 1 - 7,

0 k k

wt&l =xt83

+iY(@

(2)

denotes the longitudinal dielectric function for a Maxwellian and classical electron plasma [ 221. According to Fig. 1, the dicluster stopping is defined by (N,, = number of electrons in a Debye sphere) dE

D

(1)

where

1

d3k k2@k, k-V)

a hind

ah,,

+-we-g-z =zlNDar r==r1(1)

1 r-c?(t)

(3) where r, (t) = Vt and rz( t) = Vr + R respectively. c& denotes the electrostatic potential of one unit charge acting on the other. So, from expression ( 1) and for Zr =Z,=Zonegets

349

C. Deutsch, P. Fromy / Physics Letters A 198 (1995) 347-351

dE -iii

quantify the expected ECS through ratios. The first one reads

Z2ND

?T2 =47r-

a few obvious

in terms of the total stopping Stop. An obviously related quantity of great interest is

x [ 1 +~os~~~~)~~(~~~ = 2Z2S( 1) + 2Z2S,(B,

B)] D) = Point + Corr

in terms of the ordinary Bessel function Jo(x) and k ~~=4~(V2+2)N~/Z. Also, we have split the co~elation (Corr) contribution to stopping from the unco~elated pointlike (Point) one. B and D are, respectively, the transverse and longitudinal projections of R on V (Fig. 1) . At this juncture, it is worthwhile to notice that the presently considered frozen geometry (Fig. 1) is well justified because we restrict ourselves to a high velocity range for the projectile. For instance, if one considers adicluster with a 10 keV/A.m.u. kinetic energy, and if we admit that the whole Coulomb repulsion potential within ion pair contributions to a velocity transverse to V, one has approximately V, IV= 0.79 ZA - lf2, where A is the cluster ion atomic mass. So, the present polarized approximation appears well justified for heavy ion fragments with A >> 1, and not for hydrogen ones [ 51. Now, we consider the simultaneous stopping of N ion debris flowing with the same projectile velocity V in close vicinity of each other, i.e. within a few atomic units of relative distance. A first and obvious extension of the above and polarized two-body treatment is to superimpose the $N( N- 1) dicluster energy losses available in a Coulomb N-cluster. Such a simple generalization has already produced a huge variety of charge configurations leading to the ECS, in cold target stopping [S-15] provided N>2. Rewriting the second right hand side of Eq. (4) for two distinct charges, the corresponding N-cluster expression is then obtained as

-f=

5z;s(l)+2 i=,

= Point + Corr = Stop ,

c ZiZjSt,(B,, I =Si<;
(7)

(4)

denoting the ratio of correlated to unco~elated stopping. The ratios (6), (7) are inde~ndent of target coupling. They also apply to a T= 0 jellium [ 13 ] and to a partially degenerate electron fluid as well. They produce a priori guidelines of great interest for the ECS numerical estimation. For instance, symmetrical charge configurations with Z1=Z2=... =ZN=Z, fulfill R,,
D,) 100

(5)

As shown by previous studies of the ECS in a degenerate jellium target [13], it will prove also useful to

Fig. 2. Ratio Rap, (Eq. (IO)) for a diciuster with Z, =Z,=Z and R=Ao in terms of B/l00 in AD and V in V,,. T=2 eV and ne= 2X 10”cm-“.

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C. Deutsch, P. Fromy /Physics

charge distance R = A,,. One already sees an ECS of the order of 30 percent on most of the parameter range. ECS increases steadily with V up to a nearly constant plateau values. Also, for small V ( G 5) one observes a larger stopping for an oblique dicluster than for a parallel one, with respect to V. It is useful to recall that a plasma with T= 2 eV and n,=2 X 10” cme3 has V&=0.25 a.u. and A,=444 a.u. From the above it is already clear that a mere twobody vicinage effect already brings in a substantial ECS. So, now we implement the N-cluster stopping through Eq. (5). In order to emulate a typical arrangement of closely packed ion debris, we found it very instructive to look at a cubical display of eight unit point like charges at vertices, as shown on Fig. 3. The given cubic box has sides B = C = D running from 0 to A,. The projectile velocity is V= 3V,. As previously D IIV while Z3and C denote the two transverse dimensions. At complete coagulation (B = C = D = 0) the ECS ful~lls the upper funds (6) and (7) for a symme~ic charge distribution with RapI= 7 at any target density. When the intercharge distances increase, total stopping and ECS decay as well. Nonetheless, even at the largest size (B = C = D = A,), the ECS remains at work with

Le f fersA I98 (I 995) 347-351

R,, > 1. It should be kept in mind that for the given plasma targets, the ion debris relative distances are pretty large on atomic scales (of the order of several tens of a.u.), We now contrast the cubic box with regular arrays of point-like charges propagating either parallel or transverse to the overall velocity V. Fig. 4 displays a target electron density available on linear plasma column and Z-pinch devices installed on ion accelerator beam lines. A first overall and striking observation is given by the rather scale invariant increase with N of the stopping/charge although its value differs almost by an order of magnitude. Despite of the fact that the three considered topologies share a same connectivity, the cubic box is the best performer because its charge-charge distances have a more compact distribution than in linear arrays. In the last two, the transverse N-chain exhibits the second largest stopping. Also, it proves rather illuminating to explore the opposite situation (Fig. 5) with a large size (B = C= D = A,). Now, one witnesses behaviour dif-

-A 2

4

6

8

Number N Unit Charges

Fig. 4. Stopping of%clusters in terms ofN with B= C=L)=O.lho and V=3 V,. 7’=2 eV. D/IV. n,=2X lOI7 cme3. (a) Cubic Box; (b) transverse N-chain; (c) longitudinal N-chain. 8 7 6 5

a

4

rr”

3 2 1 1.2

Stoppingof eight unit charges and ratio Rap, (I?+ (8)) arranged on a cubic box with side ranging form 0 to AD. V= 3 V,, T=2 eV.I)IIVand n,2X 10’7cm-3. Fig. 3.

Number N Unii Chaqes

Fig.5.S~~inFig.4wi~~=C=~=~~~==l~cm-3.

C. Deutsch, P. Fromy /Physics

ferent from previous ones, so topology manifests itself globally. The overall N-cluster extends now over several screening lengths. The order of the relative stopping efficiency remains unchanged. Curves b and c in Fig. 5 anticipate specific N-chain trends detailed below. Those results demonstrate that the ECS remains nonnegligible at large particle interdistances. We made use of a complete random phase approximation for the linear response of a dense classical electron target, to estimate the interaction with incoming Coulomb N-clusters of ion debris. The given stopping is taken in a high velocity formalism based on a linear superposition of every correlated dicluster contribution within N-cluster. We restricted to even distributions of unit charges in order to stress out the basic features and properties of the enhanced correlation stopping (EC?) such as ion debris topology and target plasma coupling. The nearly ubiquitous ECS is thus expected to be of considerable and potential significance in providing new range-energy relationships useful in designing novel scenarios of interest for the cluster ion beamdense matter interaction in particle-driven inertial confinement fusion. The presently considered model bears direct relevance to the plasma phases of the compressed pellet containing the thermonuclear fuel.

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Letters A 198 (I 995) 347-351

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